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Poincaré’s philosophy of geometry and its reception in the twentieth-century philosophy of science Soltani, Mojitaba 2020

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 Poincaré’s philosophy of geometry and its reception in the twentieth-century philosophy of science  by Mojtaba Soltani M.A., University of Wisconsin, Milwaukee, 2010 M.Sc., Sharif University, 2006 B.Sc., Iran University of Science and Technology, 2003   A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Philosophy)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  April 2020  © Mojtaba Soltani, 2020 ii  The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:  Poincaré’s philosophy of geometry and its reception in the twentieth-century philosophy of science  submitted by Mojtaba Soltani in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Philosophy  Examining Committee: Steven Savitt  Supervisor  Paul Bartha Supervisory Committee Member  Alan Richardson Supervisory Committee Member Christopher Stephens  University Examiner Alexei Kojevnikov University Examiner  Additional Supervisory Committee Members:  Supervisory Committee Member  Supervisory Committee Member  iii  Abstract Most twentieth-century philosophers of science, whose works are surveyed in this work, argue that Poincaré’s most consequential contribution to the philosophy of geometry is his discovery that one can choose one’s favourite geometric language/framework in order to formulate physical theories not based on epistemological considerations (truth) but practical ones (convenience).  According to these philosophers, Poincaré uses such freedom in choosing the geometric framework to argue that axioms of (physical) geometry – which (allegedly) describes the (spatial) structure of the scene in which the drama of physics unfolds (physical space) – are a matter of convention. The main goal of this work is, however, to establish that, properly understood in its intellectual and historical context, Poincaré’s contribution to the philosophy of geometry is not limited to offering a novel epistemological category – that is, conventional truths.  Instead, I argue that Poincaré offers a comprehensive philosophical account of geometry dealing with all major philosophical questions about or raised by geometry: (1) What kind of objects are geometric objects? (2) What can be known about geometric objects? How is this kind of knowledge possible? What is the epistemic status of this kind of knowledge? (3) How are the geometric concepts acquired? (4) How can one explain the applicability of geometry? How is it possible to use geometry to describe/explain the natural world? What is the nature of the relation between physics and geometry? We will see how Poincaré uses four elements of his account of geometry –  that is, (i) his theory of the nature of geometry and geometric objects where the notion of ‘free mobility’ plays an essential role; (ii) his epistemic stance according to which the axioms of geometry are conventions; (iii) his theory of the origin of spatial intuitions and geometric concepts in which the notion of group plays a constituting role; and (iv) his stance regarding the relation between geometry and physics, that is the freedom in choosing the geometric language/framework for describing the spatial relations in theories of physics – to answer these questions.    iv  Lay Summary While both are true statements, the truth of ‘magnets attract iron.’ and ‘cars stop at red lights.’ seem to have different weights. The truth that the former expresses seems to be, in some sense, more substantive than the truth that the latter does, which is a convention. Although the geometric truths, such as those expressed by statements like ‘given a triangle, we can construct a similar triangle of any size whatever.,’ seem to be more like the former than the latter, Henri Poincaré argues that they are conventions. The meaning of, the justification for, and the significance of Poincaré’s claim are the subject of this study. v  Preface This dissertation is original, unpublished, independent work by the author, Mojtaba Soltani.     vi  Table of Contents  Abstract ................................................................................................................................................................ iii Lay Summary ....................................................................................................................................................... iv Preface .................................................................................................................................................................. v Table of Contents ............................................................................................................................................... vi List of Figures ...................................................................................................................................................... xi Dedication .......................................................................................................................................................... xii Introduction ......................................................................................................................................................... 1 Part 1           Philosophers and Geometry: From Plato to Poincaré ................................................................... 5 Introduction ......................................................................................................................................................... 5 Chapter 1: The ancient era .................................................................................................................................. 7 Geometry at its cradle: from Egypt to Greece ................................................................................................ 7 A geometric manifesto: Euclid and his Elements .......................................................................................... 11 Philosophy of geometry at its cradle ............................................................................................................. 16 Plato’s philosophy of geometry ..................................................................................................................... 17 Aristotle’s conception of geometry ............................................................................................................... 20 Irrational allies: arithmetic or geometry ....................................................................................................... 21 Aristotle’s philosophy of geometry ................................................................................................................ 22 Chapter 2: The early modern era ...................................................................................................................... 26 The fall: earthly geometry .............................................................................................................................. 26 From the heavens to the earth ...................................................................................................................... 26 From pristine constructions to messy calculations: elegance versus power ................................................. 29 The ontological and epistemological status of earthly geometry ................................................................. 30 Reason versus senses: the rationalism and geometry ................................................................................... 31 To be or not to be: Newton versus Leibniz .................................................................................................. 32 From figures to space ..................................................................................................................................... 34 Senses versus reason: the empiricism and geometry .................................................................................... 37 Sensibility versus understanding: apriorism and geometry .......................................................................... 39 Space and objective knowledge ..................................................................................................................... 40 Chapter 3: The non-Euclidean era .................................................................................................................... 46 From one to many ......................................................................................................................................... 46 vii  The geometry of vision versus the geometry of measurement ..................................................................... 46 A universal geometry: the projective geometry ............................................................................................. 48 Certainty sought and lost .............................................................................................................................. 51 All that is solid melts into air ........................................................................................................................ 54 Models of the hyperbolic plane ..................................................................................................................... 56 A brave new world ......................................................................................................................................... 59 The question of ontology and epistemology of geometry ............................................................................ 64 The question of psychology and applicability of geometry .......................................................................... 64 Chapter 4: Poincaré’s philosophy of geometry in a nutshell ............................................................................ 67 The iceberg and its tip ................................................................................................................................... 67 The question of the epistemology of geometry ............................................................................................ 67 The question of the origin of spatial intuitions and geometric concepts .................................................... 70 Poincaré’s criticism of theories of the origin of spatial intuitions and concepts ........................................ 71 Poincaré’s theory of the origin of spatial intuitions and concepts .............................................................. 73 Geometry: the senses, the mind, and the will .............................................................................................. 73 It begins with the senses: the empirical contributions ................................................................................. 74 The mind guides it: the a priori contributions .............................................................................................. 76 The ‘will’ chooses one: the conventional contributions .............................................................................. 82 The question of the ontology of geometry ................................................................................................... 85 The nature of geometry as a (mathematical) science .................................................................................... 87 Geometry as a language ................................................................................................................................. 93 Poincaré’s conventionalism vs. conventionalism ......................................................................................... 95 Part 2           Philosophers and Poincaré: Poincaré in the twentieth century .................................................. 99 Introduction ....................................................................................................................................................... 99 Chapter 5: Logical Positivists’ reception of Poincaré’s geometric conventionalism ...................................... 102 The unravelling of nineteenth-century physics: space, geometry and physics in the new age .................. 102 Eddington’s Poincaré: a semantic issue? ..................................................................................................... 103 Einstein’s Poincaré: a confirmation issue? .................................................................................................. 105 Schlick’s Poincaré: an epistemic issue? ....................................................................................................... 108 Reichenbach’s Poincaré: a semantic issue, after all? ................................................................................... 111 The conventionality of physical geometry .................................................................................................. 119 Chapter 6: Michael Friedman’s reception of Poincaré’s geometric conventionalism ................................... 125 viii  The great chain of sciences .......................................................................................................................... 125 Geometry, physics and their egalitarian relationship ................................................................................. 125 Friedman’s Poincaré: a tale of two hierarchies ........................................................................................... 127 Hierarchy of sciences ................................................................................................................................... 128 Geometry, measurement, and the Lie group of free motions .................................................................... 129 Friedman’s Poincaré and mathematical idealizations ................................................................................ 130 Friedman’s Poincaré = Einstein + Klein? .................................................................................................... 132 Dubious textual evidence and false assumptions in Friedman’s account ................................................. 132 Friedman’s imagined hierarchy of sciences ................................................................................................ 132 The autonomy of mechanics ....................................................................................................................... 135 Metrics and meters ...................................................................................................................................... 138 Friedman’s Poincaré: a sloppy mathematician! .......................................................................................... 138 Groups and their magical powers demystified: the rejection of a myth .................................................... 140 The space problem, Helmholtz’ solution, and Lie’s contribution ............................................................. 141 Chapter 7: Yemima Ben-Menahem’s reception of Poincaré’s geometric conventionalism .......................... 148 Geometric peculiarity: Poincaré vs. Duhem and logical positivists ........................................................... 148 Necessity vs. apriority .................................................................................................................................. 149 Empirically inaccessible space ..................................................................................................................... 151 Ben-Menahem, Poincaré, and geometric empiricism ................................................................................. 153 Poincaré’s disregarded battle with Kant ..................................................................................................... 155 Finally acknowledged yet remains misunderstood ..................................................................................... 160 Revisiting the hot disk ................................................................................................................................. 161 It’s all about distance ................................................................................................................................... 169 Geometry and physics: a complicated relationship .................................................................................... 172 Space and group: a misunderstood relationship ........................................................................................ 174 Geometry and experience ............................................................................................................................ 177 Chapter 8: Alberto Coffa’s reception of Poincaré’s geometric conventionalism .......................................... 184 In search of meaning ................................................................................................................................... 184 New geometries: new challenges for the old tradition ............................................................................... 185 The reductionist attempt at accommodating new geometries: too many to handle ................................. 187 The fifth postulate: a revisionist history ..................................................................................................... 188 A brilliant proof or a blunder ..................................................................................................................... 190 ix  Desperate times and a desperate solution: semantic radicalism ................................................................ 191 Unravelling a blunder: imaginary problems and revisionist history .......................................................... 193 Semantic and Riemannian geometries: false motivations.......................................................................... 193 Unmotivated revisionist history: collapsing distinctions ........................................................................... 194 Logical, geometric, and non-logical terms: a similarity in mirage .............................................................. 197 Poincaré, Hilbert, and formalism ............................................................................................................... 199 Conclusion ....................................................................................................................................................... 205 Part 3           Poincaré on Geometry: in his own words .................................................................................. 209 Introduction ..................................................................................................................................................... 209 Chapter 9: Henri Poincaré .............................................................................................................................. 212 A mathematician in the making ................................................................................................................. 212 Differential equations: an unlikely road to geometry ................................................................................ 216 The popularizer and the philosopher ......................................................................................................... 220 Chapter 10: A critique of the empiricist and apriorist epistemology and psychology of geometry .............. 222 A philosophy of geometry in the making ................................................................................................... 222 Geometric intuitions: where they do not come from ................................................................................ 229 Chapter 11: The conventionalist psychology and ontology of geometry ....................................................... 235 A neglected contribution............................................................................................................................. 235 Geometric intuitions: where they do come from ....................................................................................... 236 From the sensible to geometric space ......................................................................................................... 241 From the sensible space to the spatial frame: the psychology of spatial intuition .................................... 241 The spatial frame and its structure: the epistemology and ontology of spatial intuitions ........................ 244 The group of displacements and its subgroups .......................................................................................... 247 Epilogue: a psychological fantasy or a conceptual foundation? ................................................................. 259 Chapter 12: The conventionalist methodology of (applied) geometry .......................................................... 263 Geometry in action: the geometric description of nature .......................................................................... 263 The empirical verification of geometry by experiments? ............................................................................ 263 The empirical verification of geometry on principle?................................................................................. 270 Physical space: an illusion ........................................................................................................................... 271 Bibliography ..................................................................................................................................................... 275 Appendices ....................................................................................................................................................... 287 Appendix A: Post non-Euclidean geometric apriorism: Russell’s philosophy of geometry ........................... 287 x  Appendix B: Excerpts ....................................................................................................................................... 291  xi  List of Figures Figure 1.1 Two pages of Euclid’s Elements .......................................................................................... 13 Figure 1.2 Constructing an equilateral triangle ................................................................................. 15 Figure 3.1 Perspective techniques ....................................................................................................... 47 Figure 3.2 Geometric transformations ............................................................................................... 49 Figure 3.3 Geodesics on the pseudosphere ........................................................................................ 56 Figure 3.4 Poincaré’s and Klein’s disks .............................................................................................. 57 Figure 3.5 The half-plane .................................................................................................................... 58 Figure 3.6 Walking in the hyperbolic plane ....................................................................................... 60 Figure 4.1 Change of state vs. change of position .............................................................................. 75 Figure 4.2 From impressions to geometric space ............................................................................... 80 Figure 4.3 From changes of position to the spatial frame ................................................................. 80 Figure 4.4 Euclidean transformations ................................................................................................ 90 Figure 5.1 The hyperbolic plane from the hyperbolic point of view  .............................................. 163 Figure 5.2 The hyperbolic plane from the Euclidean point of view  ............................................... 164 Figure 11.1 Changes and corrections ............................................................................................... 242 Figure 11.2 Rotations ........................................................................................................................ 248 Figure 11.3 Translations ................................................................................................................... 250            xii  Dedication     To    Pari and the memory of Mina with love and gratitude 1   Introduction Even though in their final analysis, several philosophers of science and philosophically inclined physicists reject it, they have found Poincaré’s view on geometry – which is often referred to as ‘geometric conventionalism’– too attractive to ignore. For instance, in the fifth chapter, we will see that despite his reluctance, Einstein sees no other option than rejecting geometric conventionalism as he finds it incompatible with his theory of general relativity – a conclusion that, as we will see in the fifth and sixth chapters, is reiterated, although sometimes for different reasons, by several others. According to most of these twentieth-century philosophers of science, whose works are surveyed in chapters five, six, and seven, Poincaré’s most consequential contribution to the philosophy of geometry is his discovery that one can choose one’s favourite geometric language/framework in order to formulate physical theories not based on epistemological considerations (truth) but practical ones (convenience). Freedom in choosing the geometric framework, according to these commentators, implies that axioms of (physical) geometry – which (allegedly) describes the (spatial) structure of the scene in which the drama of physics unfolds (physical space) – are a matter of convention. Any particular version of such an understanding of geometric conventionalism – as the proponents of every other version are quick to point out – amounts to either untenable or trivial claims.   Those who try hard to save Poincaré’s geometric conventionalism portray it as the penultimate step in developing a more coherent and comprehensive philosophical view waiting to come to existence – a position exemplified by views that will be discussed in chapters seven and, especially, eight.  2   The main goal of this work, in contrast, is to establish that, properly understood in its intellectual and historical context, Poincaré’s philosophy of geometry is neither merely an intriguing but nevertheless untenable view, nor a preliminary attempt at a theory yet to come. I argue that rather than being just concerned with the isolated question of the epistemology of geometry and contributing a novel epistemological category (conventional truths), Poincaré offers a comprehensive philosophical account of geometry dealing with all major philosophical questions – ontological, epistemological, conceptual/ psychological, and methodological – about or raised by geometry. Put differently, on my interpretation – contrary to existing ones most of which are surveyed in part II – rather than proposing an enticing but ultimately confused or mistaken isolated epistemological claim, Poincaré offers a full-blown philosophy of geometry.    I begin with a brief and selective survey of the history and philosophy of geometry, where we will see how, as the science of geometry evolves, the philosophical questions about or raised by it also change. We will see that, by the end of the nineteenth century, the philosophy of geometry consists of the following main questions: (1) What kind of objects are geometric objects? (2) What can be known about geometric objects? How is this kind of knowledge possible? What is the epistemic status of this kind of knowledge? (3) How are the geometric concepts acquired? (4) How can one explain the applicability of geometry? How is it possible to use geometry to describe/explain the natural world? What is the nature of the relation between physics and geometry?i1 The purpose of these chapters is, however, not to provide a historical and/or philosophical account of more than two millennia of intense intellectual history. The brief and  i See Shapiro (2005). For the excerpts, referred by to Hindu-Arabic numbers, see the appendix B. 3  selective survey of these chapters is just intended to furnish the proper intellectual and historical context in which Poincaré develops his philosophy of geometry.   The interpretation of Poincaré’s geometric conventionalism that I offer may seem suspicious – curious, at least – to those who know Poincaré’s views on geometry from the copious philosophical literature which has been produced since the early twentieth century. To address such worries among others, in the four chapters of part II, I consider four major readings of Poincaré’s views on geometry. We will see that, as different as they are, they all share the same flaw: they fail to consider the proper context of Poincaré’s writing.      Besides the recent discovery of non-Euclidean geometry, mathematics and its nature were the subjects of intense philosophical debates during the final decades of the nineteenth century –when Poincaré was thinking and writing about geometry, among other subjects. These discussions resulted in different schools of thought, which would dominate much of twentieth-century philosophy of mathematics – such as logicism, formalism, and intuitionism.i Poincaré’s works on geometry are his contribution to these foundational debates regarding the nature of geometry as a mathematical science. However, later commentators – mostly, twentieth-century philosophers of science – tried to read and present his contributions in a totally different context – that is, the implications of the theory of relativity for the philosophical debate regarding physical space,ii and the structure of physical space, often called physical or applied geometry. In order to do this, they  i See Demopoulos and Clark (2005), Detlefsen (2005), and McCarty (2005). ii Despite the fact that Poincaré explicitly rejects the idea of physical space as a misguided conception.  4  had to emphasize a few passages, from Poincaré’s writings, and to ignore the rest – that is, almost all.  In part II, we will see how spotlighting a few different passages, arguments, or observations makes it possible to develop, and defend, often contradictory, readings of Poincaré’s views. It is worthwhile to stress that not only are these readings of Poincaré’s geometric conventionalism the result of attempts to present Poincaré’s geometric conventionalism in the context of a philosophical debate that he was not part of and would arguably not have found interestingi but also render geometric conventionalism either untenable or else trivial.    Unlike these attempts at understanding Poincaré’s philosophy of geometry which often entirely disregard,ii or reject, Poincaré’s theory of the origin of spatial intuitions and geometric concepts, and his discussion of the nature of geometry – as a mathematical science, – my reading avails itself of Poincaré’s psychological/conceptual analysis in order to reconstruct a comprehensive philosophy of geometry. The close study of a number of Poincaré’s papers, in part III, is intended as the textual justification for the interpretation offered in chapter four.     i That is, the supposed philosophical implications of the theory of relativity; this issue, however, is beyond the scope of this work. See Damour (2005), Walter (2010), and Stein (2014), for instance. ii Which given the space and effort that Poincaré allocated to them should have raised serious interpretive questions even on its own.  5  Part 1           Philosophers and Geometry: From Plato to Poincaré Where the evolution of geometry and its philosophy during three crucial moments is considered first and then Poincaré’s philosophy of geometry is presented. Introduction From since ever, they started to seriously consider the philosophical questions regarding the nature of geometry, geometric objects, and geometrical propositions until the discovery of non-Euclidean geometry, philosophers have had avail themselves of two approaches. These are, broadly speaking, the rationalist approach – in its different forms from Plato’s theory of forms to Kant’s transcendental idealism – and the empiricist approach – in its different forms evolving from its Aristotlean conception to its late eighteenth-century manifestation. With the discovery of non-Euclidean geometry, however, proponents of the latter approach seemed vindicated while their rivals’ approach – Kant’s apriorism – seemed to be invalidated. It is in this context that Poincaré enters the picture – not as a professional philosopher but as perhaps the most influential mathematician of his time and a physicist gifted by exceptional artistry in explaining cutting-edge scientific discoveries to the public. Not convinced by the empiricist conception of geometry, while trying to explain non-Euclidean geometries to the public in a paper whose English translation is published in Nature, Poincaré takes advantage of the discovery of non-Euclidean geometry to argue that not only is the apriorists’ philosophy of geometry untenable but also that of the empiricists. Poincaré, instead, offers his own philosophy of geometry – an account of geometry that he alluded to in a few previously published papers and would continue to develop in a small number of future papers.     As I believe that the failure to correctly identify the historical and intellectual context of 6  Poincaré’s works has contributed significantly to misunderstanding Poincaré’s geometric conventionalism, the first three chapters of this part are devoted to the survey of the evolution of geometry and its philosophy. In these chapters, we will see how the questions that define the philosophy of geometry in Poincaré’s time emerge.  In the fourth chapter, we will see how Poincaré responds to these questions – that is, the nature of geometry as a mathematical science, the ontological status of geometric objects, the epistemic status of geometric propositions, and the origin of spatial intuitions and geometric concepts.   We will see how four elements constitute a comprehensive account of geometry: (i) Poincaré’s theory of the nature of geometry and geometric objects – where the notion of ‘free mobility’ plays an essential role; (ii) Poincaré’s epistemic stance according to which the axioms of geometry are conventions; (iii) his theory of the origin of spatial intuitions and geometric concepts – in which the notion of group plays a constituting role; (iv) his stance regarding the relation between geometry and physics – that is, the freedom in choosing the geometric language/framework for describing the spatial relations in theories of physics.       7  Chapter 1: The ancient era Geometry at its cradle: from Egypt to Greece  Although the documented use of geometrical structures to decorate objects goes far back to 40000 BCE, the dawn of geometry is lost to us in the mists of history. The systematic use of geometrical patterns containing equilateral triangles, squares, circles and even regular hexagons to decorate ornaments suggests that these geometrical objects were noticed very early on. The decorative use of geometry in Neolithic clay pots requires some knowledge of geometric figures and their properties. The practical needs and activities of early farming communities such as land surveying, building houses, dams and ditches arguably provided further interest in geometry since without knowing geometric properties of two and three-dimensional figures the building projects that the early civilizations undertook would have been impossible. Our knowledge of the extent of the geometric understanding in the Neolithic era is very speculative.  We do, however, have some understanding of old Egyptian and Mesopotamian geometry.i   Egyptian surveyors had to measure fertile expanses of the land annually flooded by the Nile.  In order to accomplish their task, these surveyors had to know how to calculate the area of rectangular, triangular, and circular pieces of land. The Rhind and the Moscow Mathematical Papyruses, the two most important sources of knowledge of Egyptian mathematics in general and geometry in particular, date to around 2000 BCE. They contain the top and side views of geometrical solids and problems concerning the calculation of the area of rectangles, trapeziums, triangles, and circles. They also contain formulas for calculating the volume of truncated pyramids  i Scriba & Schreiber (2015). 8  and some other geometrical objects, which arguably are their most remarkable geometrical achievements.i  Our sources for Mesopotamian mathematics are much more abundant. There are about five hundred known clay tablets containing information of interest to the historian of Mesopotamian mathematics.ii Mesopotamian interest in geometry, like Egyptian, mainly originated from the practical needs of land surveying, dividing fields into parts of equal area, and so on. They too had to figure out how to calculate the area of lands with different shapes and estimate the volume of their pots. What distinguishes their geometrical achievement, rather than its extent, however, is their use of the Pythagorean theorem and properties of similar triangles to solve their practical problems.iii Ancient Greeks believed that the subject that they turned into the systematic study of constructible shapes – that is, those shapes that can be constructed by different combinations of lines, circles, planes, and spheres – and their properties – that is, what they called geometry which literally means measuring the Earth – had its origin in the work of Egyptian surveyors imported to  i See Clagett (1999) for the original sources and detailed discussions of Egyptian mathematics. See Cooke (2013), Burton (2011), Joseph (2011), Roero (1994), and Eves (1990) for an interesting discussion of Egyptian mathematics. ii See Høyrup (2002) and Friberg (2007) for some original sources and detailed discussions. See Cooke (2013), Burton (2011), Joseph (2011), Høyrup (1994), and Eves (1990) for an interesting discussion of Mesopotamian mathematics.  iii Joseph (2011). For the history of geometry in non-western civilization, see Joseph (2011), Grattan-Guinness (1994), Cooke (2013), and Eves (1990). 9  Greece around 500 BCE.i They, however, transformed what they had learned from Egyptians in two essential respects – that is,  its subject matter and its method.  Geometry for ancient Greeks was not the study of the area of differently shaped pieces of land or volume of pots, etc.; it was instead the study of properties of some abstract shapes such as angles, triangles, and circles. Evidence is provided, for instance, by what they considered to be their earliest achievements in geometry – that is, Thales’s discoveries, among which are the fact that the base angles of an isosceles triangle are equal, that when two straight lines intersect, the opposite angles are equal, and that the sides of similar triangles are proportional.ii Of even more far-reaching consequence, ancient Greeks abandoned the empirical methods used in Egyptian geometry and replaced them with the deductive method – that is, a rigorous chain of reasoned steps, starting from some explicitly stated assumptions and concluding with the desired result. In other words, while Egyptian geometric knowledge was a collection of rule-of-thumb procedures producing sufficiently accurate but approximate solutions to practical needs and resulting from the method of trial and error and lacking any notion of proof or demonstration, geometry in the hands of the ancient Greeks became a deductive science.  Although the more primitive and practical aspect of geometry survived and even flourished in Euclid’s, Archimedes’, Apollonius’, and Ptolemy’s writings, it was not considered as part of the  i Proclus, in his Commentary on the First Book of Euclid’s Element, claims that Thales was the first who travelled to Egypt and brought geometrical knowledge back to Greece. ii Burton (2011). 10  science of geometry, but rather as that of practical sciences such as optics and Sphaerica – a practical science devoted to the study of the starry heavens.i   The process of transforming geometry, its subject matter and method, was started by Thales and continued by Pythagoras and others. The enormous impact of Euclid’s works helped to push the earlier works to obscurity, and, as a result, the history of the evolution of geometry at this stage remains, to a great extent, unknown.ii  At some point during the three centuries that separate Thales from Euclid, however, the method of deductive reasoning manifested in Euclid’s Elements was established.iii Euclid’s Elements is mostly a masterful compilation and systematic presentation of earlier mathematical achievements.iv Its importance, accordingly, lies not merely in its content but  i Eves (1990 b). ii Allman (1877). See Heath (1921) for the history of ancient Greek mathematics, and Mueller (2006) for the state of Greek mathematics at the time of Euclid, and also Knorr (1986). iii The origin of the Greeks’ attitude toward mathematics is somewhat obscure. Several speculative explanations have been put forward: their tradition of philosophical inquiries dealing with analyzing abstract concepts, the Hellenic love for beauty, and the disdain of manual work and practical pursuits among the social class to which mathematicians belong, for instance. See Netz (1999) for an extensive discussion of the emergence of the deductive method in Greek mathematics and Eves (1990 b) for a brief one.      iv Eves (1990). Despite the widespread belief, the thirteen books of Euclid’s Elements are not solely devoted to geometry; they contain number theory and rudimentary geometric algebra as well. See also Cooke (2013). 11  in its style, which in addition to the Pythagorean school seems to be significantly influenced by Platonic philosophy and Aristotelian methodology.i A geometric manifesto: Euclid and his Elements Although his masterpiece has been one of the most, if not the most, praised intellectual achievements of our species and has been studied continuously for two millennia, little is known about Euclid himself, except that, having probably learnt mathematics in the Platonic school at Athens, he was a mathematician working in Alexandria.ii   The first book of the Elements begins, in accordance with the Aristotelian conception of science, with the definition of the subject matter of (two-dimensional) geometry – that is, plane geometric objects/figures, such as points, lines, plane surfaces, angles, triangles, circles, and so on.iii While the definitions of different kinds of lines, triangles, squares, circles and so on, delineate the  i Scriba & Schreiber (2015). For Plato’s philosophy of mathematics, see Wedberg (1955) and Pritchard (1995) and see Heath (1970) for Aristotle’s philosophy of mathematics and also the role of mathematics in Aristotle’s philosophy.  ii Euclid wrote ten books of which a fairly complete text of five survived. Unfortunately, no copy of his Elements written in his time has been found; the modern edition of Euclid’s Elements is based on the work of Theon of Alexandria (ca. 400 CE) although it is not the oldest edition found.  Eves (1990). See Heath (1968) for the English translation of and extensive commentary on the Elements, Mueller (1981) for a philosophically oriented exposition of the Elements, and Hartshorne (2000) for a modern introduction to Euclidean and non-Euclidean geometry in an axiomatic fashion.  iii For instance, here are the first four definitions: “1. A point is that which has no part. 2. A line is breadthless length. 3. The extremities of a line are points. 4. A straight line is a line which lies evenly with the points on itself.” Heath (1968, p. 153).  12  domain of geometry and make it clear what objects are of interest to a geometer, they make no claim as to the existence of these objects – a task left to the postulates.  Euclid suggests that the following be postulated: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.i  Each postulate posits the possibility/existence of a specific geometric object. The first, the line- segment between any two points; the second, the line containing any two points; the third, the circle centred at any point passing through any point; the fourth, expressed awkwardlyii, the right angles at any point; and the notorious fifth, a common point on any two non-parallel lines.  Euclid’s geometrical investigations, in the rest of the Elements, are carried on in the form of either providing instructions for constructing specific geometric objects – that is, proving the  i Heath (1968, p. 154). There are good reasons, according to Heath (1921), to believe that the fourth and the fifth postulates, if not the first three, are Euclid’s original contributions.  ii The fourth postulate, at first glance, looks like an easy theorem. However, to prove the fourth postulate one needs to assume that, when displaced, geometric objects preserve their shapes. Yaglom (1988) speculates that the reason for including it as a postulate might be Euclid’s reluctance to include ‘displacements’ or ‘motions’ in geometry which due to   Zeno’s argument were controversial notions for Greeks. Furthermore, as noted by Heath (1968), unless right angles could be considered as determinate magnitudes, which is what the fourth postulate posits, the fifth postulate would be useless.  13  possibility/existence of those geometric objects – or proofs for statements expressing their properties, as illustrated by the following examples.  Figure 1.1 Two pages of Euclid’s Elements To accomplish his task, however, in addition to the postulates, Euclid needed additional assumptions that are introduced under the title of the common notions in the Elements.i  Euclid  i These are the explicit assumptions made in the Elements. In carrying out geometric constructions and proving theorems, however, other unstated assumptions – for instance, that two circles with centers at the distance of their radius will intersect in two points, in the very first construction – enter Euclid’s arguments. Thus, as presented, his system is logically incomplete; see Golos (1968). Particularly, it fails to state the conditions for intersecting a line and a circle, and two circles, which is not an easy task at that early stage of development of geometry. Heath (1921). The first thoroughly rigorous presentation of Euclidean geometry appeared more than two millennia later, at the eve of the twentieth century in Hilbert’s work. However, see Manders (2008). 14  mentions five of them.2 Unlike postulates which are theory-specific principles whose permission is debatable, common notions are basic general principles whose truths are supposed to be undebatable.i  While the existence of primary geometric objects – that is, points, lines, right angles and circles –are posited/guaranteed by Euclid’s postulates, any claims about their properties beyond what is asserted in postulates, as well as the existence of more complicated geometric objects built from the basic ones and their properties, must be proved in a chain of logical reasoning. Euclid’s postulates determine which geometric constructions are possible and thus delimit the proper domain of geometry, the constructible figures – that is, those that can be drawn using the so-called Euclidean toolsii– the straightedge and Euclidean compass. To see how Euclid’s geometrical investigations unfold, it is worthwhile to consider an actual example from the Elements.   i Scriba & Schreiber (2015). ‘Postulate’ literally means ‘to demand’; they are asked to be accepted as true for the sake of the study of the subject at hand. On the other hand, according to the widespread view, the common notions are supposed to be self-evident general truths, and Euclid’s first three common notions which define (or express the properties of) identity are certainly such. The fourth and the fifth common notion seem, however, to be particularly concerned with geometry. The fourth one, for instance, is a sufficient definition of geometrical equality, according to Heath (1921), and accordingly more properly should be called a postulate. For a different interpretation of common notions, according to which they provide the notion of equality for areas and volumes, see Stillwell (2016). ii Plato is credited with establishing this tradition; Burton (2011). Euclid himself, however, does not talk about these so-called Euclidean tools, but only of the line-segments and the circles, namely the geometrical objects themselves; Scriba & Schreiber (2015). It turned out that the straightedge 15  Here is how, in the very first proposition of the Elements, Euclid proves the existence of equilateral triangles with any given size:  To prove the claim, it is enough to construct an equilateral triangle with an arbitrary side, AB. It is possible to draw a circle with center A passing through B, so is a circle with center B passing through A, according to the third postulate. The first postulate makes it possible to join A to B and C, and B to C with straight lines. Therefore, it possible to draw/construct the desired triangle.                                                                              Figure 1.2 Constructing an equilateral triangle Let me conclude with three critical remarks about the subject matter of geometry as understood by Greeks. First, drawn figures are merely the tangible representation of geometrical figures and must not be equated with the geometrical objects themselves. Second, any figure which cannot be drawn/constructed by Euclidean tools – that is, cannot be warranted by the postulates – does not belong to Euclidean geometry. This is best manifested when their attitude toward the three classical problems is considered. That is, they viewed them as unsolved, even though solutions involving tools other than Euclidean ones were known.i And finally,  as we shall see in more detail  is dispensable; any figure that can be constructed with the straightedge and Euclidean compass can also be constructed with the compass alone; see Coxeter (1973). i Doubling the cube, trisecting angles, and squaring the circle are the three classical problems. Although Greeks knew how to solve them using so-called higher curves –that is, the quadratrix, 16  in the next section, geometric objects were considered on their own, in one way or another, autonomous, neither part of physical space – that is, the stage in which the derma of physics unfolds – nor a mathematical space – that is, a structured set of some sort of mathematical elements, whatever it means.i  The epistemological status of geometrical claims and the ontological status of geometric objects, as the rest of this chapter will show, have been debated by philosophically inclined mathematicians and philosophers throughout different stages of the evolution of geometry. Philosophy of geometry at its cradle Ancient Greeks tended to define a scientific discipline, at least partly, by its subject matter. Geometry,ii accordingly, was seen as the study of certain objects constructed in accordance with  conchoid of Nicomedes, and so on – these curves were never incorporated in their geometry nor seen as legitimate geometrical objects because their construction involves tools other than Euclidean tools and thus could not be justified by Euclid’s postulates. Plato, for example, found Hippias’s solution of squaring the circle using a curve called the quadratrix unacceptable, saying “For in this way the whole good of geometry is set aside and destroyed, since it is reduced to things of the sense and prevented from soaring among eternal images of thought.” Plutarch, Convivial Questions, quoted in Burton (2011, p. 122). We now know none of these three problems can be solved using Euclidean tools.  i In fact, it seems that the ancient Greeks did not have a clear concept of space; see Algra (1995), Cornford (1976), Jammer (1993), and Mendell (2015). ii In this section, unless otherwise stated, ‘geometry’ means Euclidean geometry. Moreover, to avoid unnecessary complications, whenever they do not affect my project, I confine my discussions to the two-dimensional case. In fact, the Elements begins with planimetry – that is, geometry in the plane – and only its last three books, starting with Book XI, deal with solid geometry by tracing back the 17  Euclid’s postulates, which, following Wedberg (1955), I call ‘Euclidean objects.’ Geometric knowledge – that is, the knowledge about Euclidean objects – was thought to be certain and infallible.  It was, indeed, considered as the paradigm of knowledge to which other branches of knowledge should look to. In addition to providing an ontological account – that is, explaining the nature of Euclidean objects – many philosophers took on the task of explaining the possibility and the nature of geometrical knowledge – that is, the task of providing a philosophical account for geometry, the philosophy of geometry. Both Plato and Aristotle – either directlyi or indirectly and as part of their more general philosophical discussions – considered and tried to address ontological and epistemological questions specific to geometry.   Plato’s philosophy of geometry  To study Plato’s philosophy of geometry often two different kinds of sources/texts are consulted: Plato’s writing and the so-called unwritten Platonic doctrines – that is, those views which are mainly reported and criticized in Aristotle’s writings. The problem with the former is that these brief remarks are scattered throughout writings spanning half a century, often painting incomplete and different pictures. The latter, on the other hand, appear only in the context of Aristotle’s  properties of non-planar geometric objects to the definitions of planar ones. For instance, the orthogonality of two planes is defined by the orthogonality of two lines. Janich (1992).  i For both Plato and Aristotle, geometry is one of the two mathematical sciences, the other being arithmetic. Their philosophy of geometry accordingly is part of their philosophy of mathematics. However, we only consider those aspects of their philosophy of mathematics which either are common to both arithmetic and geometry or pertain only to the latter. 18  criticism and probably would not offer the most accurate account.i With this point in mind, let us see how Plato deals with the Euclidean objects.ii  To put it bluntly, regarding the Euclidean objects, Plato, not surprisingly, is a platonist, in the sense that the term is used in the current philosophy of mathematics.iii  That is, firstly, according to his account, mathematics is about real objects which, although they do not exist in space and time, do genuinely exist – that is, independent of mathematicians’ knowledge or thoughts. Secondly, for Plato, knowing a Euclidean object is possible; in fact, mathematical knowledge is the paradigm of knowledge.  Plato’s account of ontology, his theory of forms or ideas, allows for two different kinds of objects: the perceptible – spatiotemporal objects – and the intelligible, eternal and unchanging forms which are the object of genuine knowledge. Plato’s theory of recollection, according to  i Annas (1976). Much of my report of Plato’s philosophy of geometry draws from Wedberg (1955) and Annas (1976). See Bostock (2009) for a different interpretation, however.  Needless to mention that which interpretations one favors has no bearing on my project since the goal of this chapter is merely to provide the background against which philosophical questions regarding geometry are formulated and are engaged with, not the particular answers to these question – of course, with one exception: that of Poincaré which will be the subject of the last part. ii Although Euclid’s Elements appeared later, Plato’s and Aristotle’s conception of geometry is close enough to the view expounded in the Elements that justifies the seemingly anachronic terminology of ‘Euclidean geometry’ and ‘Euclidean’ objects.    iii Annas (1976) and Bostock (2009). However, while in the current philosophy of mathematics the emphasis is on the semantic – that is, what makes the mathematical statement true – Plato’s concern was the nature of mathematical objects. The two questions, however, are, in a sense, equivalent, according to Wedberg (1955).  19  which souls, before birth, had access to the forms is supposed to explain how it is possible to acquire knowledge about non-spatiotemporal forms.i Plato’s philosophy of geometry, accordingly, seems to be committed to the following:  The possibility of having genuine geometric knowledge entails that Euclidean objects are intelligible – that is, they are among the Platonic forms. Furthermore, the geometric knowledge, acquired as the result of the souls’ prenatal acquaintance with the forms, is a priori and infallible.ii  Putting aside the implausibility of his explanation of the possibility of geometric knowledge, Plato’s account of the nature of Euclidean objects runs into at least one serious problem. For Plato, a form is often not just a characteristic but also the perfect token of that characteristic. The form of a circle, for instance, not only makes what participates in it a circle but also is the perfect circle; in fact, since nothing other than the form of a circle can be perfectly circular, nothing else could be a perfect circle. Plato’s theory of forms, therefore, accounts for only one circle. To make sense of geometric assertions – such as circles A and B intersect each other at two points – however, there has to be more than one circle, and more than one point, for that matter. Plato, thus, seems to be committed to a third kind of objects, the intermediate objects, which share their perfection and immunity to change with the forms, and their plurality with  i It seems, however, that Plato later abandoned his theory of recollection without ever proposing an alternative theory as, for example, this theory does not appear as an account of genuine knowledge in the Republic. Bostock (2009). ii See Annas (1976). 20  perceptible objects.i Aristotle’s criticism of Plato’s philosophy of geometry, however, is not limited to this flawii and takes issue with Plato’s theory of forms in general and offers a different account of the nature of Euclidean objects to which we turn now. Aristotle’s conception of geometry  Aristotle defines mathematics as a science whose subject matter is quantity – that is, that which is divisible into its constituent parts. Quantity thus defined is further divided into two kinds: multitude or plurality – that is, those quantities that can be divided into discrete parts – and magnitude – that is, those quantities that can be divided into continuous parts. Magnitudes come in three kinds: one-dimensional, lines, two-dimensional, surfaces, and three-dimensional, solids.3 Although at one point, Aristotle says that “geometry is about points and lines,”iii he often defines geometry as “the science of spatial magnitudes.”iv Thus, we need to see how Aristotle’s conception of geometry could be squared with the conception of geometry that emerged from the analysis of the Elements – that is, a science whose subject matter is Euclidean objects. To do so, we should step back to consider a turning point in the history of Greek mathematics – that is, a discovery that shattered the Pythagorean dream.   i According to Aristotle, Plato introduced intermediate objects to his later account of ontology; however, in his writing, neither is Plato committed to this so-called unwritten doctrine nor is he helpful in clarifying the nature of the intermediates. Annas (1976). ii For a discussion of Aristotle’s argument against Plato’s intermediate objects, see Bostock (2012). iii Metaphysics 76b3-76b11. iv Metaphysics 1142b36-1143a18. 21  Irrational allies: arithmetic or geometry Pythagoreans, apparently, believed that the building blocks of reality, at the most fundamental level, were numbers, and hence reality could be thus explained. Their number concept, however, was limited to positive integers.i Everything, therefore, was supposed to be explainable in terms of these numbers. The application of the Pythagorean theorem, ironically enough, led to the discovery of ‘things,’ like √2, which were called irrationals, however. This shattered the Pythagorean dream of understanding all that could be understood in terms of numbers and their ratios – that is, rationals. With the discovery of irrationals, then, it became clear that arithmetic no longer could serve as the foundation of knowledge and geometry assumed its foundational role, instead.ii    Euclid’s postulates allow constructing any desired multiple of a given line-segment. Furthermore, line-segments can be added to, subtracted from, multiplied, and divided by each other. Once a line-segment, a length, is chosen as a unit, then, the concept of length can replace that of the number. The geometric concept of (one-dimensional) magnitude – that is, constructible line-segment – thus could be identified with and assume the role of the arithmetical concept of multitude. This move, although it expands the concept of number to include constructible numbers, it ties numbers to line-segments.   i The number one, however, was the subject of much discussion, and often was not considered to be a number itself.  ii For more on the history of the discovery of irrational numbers and its consequences see Edwards (1979), Klein (1968), Kline (1972), and von Fritz (1945). 22   When we think about a square/cube, we consider its area/volume as its attribute which can be assigned a numerical value, a number. Given their concept of number, however, Greeks had to think about two and three-dimensional geometric objects differently. They did not think of area, for example, as a numerical attribute of a square; from their point of view, there was no such thing as the area of a square other than the square itself. In short, Greeks did not distinguish length from the one-dimensional Euclidean objects, area from two-dimensional Euclidean objects, and volume from three-dimensional Euclidean objects.i Aristotle’s conception of geometry, thus, is the same as the one has been assumed in our discussion. With this point in mind, we can now turn to Aristotle’s philosophy of geometry. Aristotle’s philosophy of geometry  Aristotle’s philosophy of mathematics, like much of his thought, is developed as a reaction to Plato’s philosophy. It has its roots in his criticism of Plato’s theory of forms, which according to Aristotle, not only suffers from serious internal problems ii but also fails to accomplish the task for which it is devised – that is, providing an intelligible account of the orderly world. Aristotle,  i Stillwell (2016). The fact that Aristotle does not mention conic sections and higher curves when discussing geometry,  that he considers only ‘straight’, ‘circular’, and ‘angle’ as genera of figure in Physics 188a19-188a26, and that despite the existence of the solution to the problem of squaring the circle with the help of non-constructible curves, he adds the proviso “supposing it to be knowable” in the following passage suggest that like Plato, Aristotle confines the domain of geometry, as a mathematical science, to the constructible – that is, what I call Euclidean objects: “Take, for example, the “squaring of the circle, supposing it to be knowable; knowledge of it does not yet exist but the knowable itself exists.” Categories 7b23-7b34.  ii The notion of participation in forms, the third man problem, etc.  23  instead, proposes his theory of hylomorphism, a theory according to which every knowable object consists of form and matter. Aristotle’s philosophy of geometry is based on his alternative account of ontology, naturally.  Aristotle, like Plato, believes that, unlike sensible objects, mathematical objects perfectly fulfill their defining conditions – a mathematical circle is perfectly circular, for instance – and are knowable by pure thought.i Unlike Plato, however, Aristotle does not believe that mathematical objects belong to the separate realm of unearthly forms. Nor does he believe that, in their existence, mathematical objects are dependent on human thought. In Metaphysics M. 3,4   Aristotle attempts to provide an account of mathematical objects compatible with these beliefs.ii Aristotle begins with his analysis of the subject matter of geometry. He argues that although the subject matter of geometry is not the study of sensible substances as sensible, it is the study of sensible substances as magnitudes, nevertheless. In other words, geometry is concerned with sensible objects, but only to the extent that their extensions are concerned – that is, geometry studies sensible substances as one, two, and three-dimensional Euclidean objects. He insists, moreover, that such ‘things’ – that is, sensible substances qua extension – do really exist. However, his account of their particular mode of existence is only by way of negation – that is, described as  i The epistemic status of a statement and the nature of entities that statement is about are connected, for Aristotle: “the truth of knowledge keeps pace with the actuality of its object” according to Zeller (1897, p. 339).  ii Much of my report of Aristotle’s philosophy of geometry draws from Mueller (1969) and Annas (1976). However, see Lear (1982), Hussey (1991), and Bostock (2009 and 2012), for different interpretations. 24  neither sensible substances nor separate from them. Euclidean objects seem to be a particular kind of abstractions, for Aristotle. However not as abstract thoughts/ideas in the geometer’s mind; they are abstract entities out in the world, ‘attached’ to/in the sensible substances. In short, Euclidean objects are particular entities distinct but inseparable from sensible substances, but, at the same time, are the result of abstraction. Rather than generating a general idea from observing a collection of certain particulars, Aristotelian abstraction involves the absence of attention to/neglecting some aspects of the thing in question.i Although the result of abstraction – that is, the nature of the resultant abstract objects – is not the same, Aristotle does not distinguish absence of attention to/ignoring the matter – for instance, ignoring the fact that a disk is made of bronze – from ignoring all but a specific property – for instance, ignoring all properties of a disk except its roundness. Accordingly, depending on which notion of abstraction one chooses, Aristotle’s account seems to render Euclidean objects either as properties – in our example, roundness – or as physical objects lacking particular properties – a disk such that only its roundness is attended to.ii  In any case, Aristotle seems to believe that the intelligible matter – that is, the same purely dimensional entity which underlies sensible properties – underlies geometric properties. For him,  i For Aristotle’s theory of abstraction see Bäck (2014).  ii The first interpretation – that is, mathematical objects as properties – is more often attributed to Aristotle. Mueller (1969) points to the difficulties of such an interpretation and opts for the second; however, he construes it in a way that allows him to avoid ending up with physical objects. Instead, on his reading the result of abstractions are substance-like individuals with the intelligible matter. 25  thus, Euclidean objects are, like any other knowable object, composed of matter, which is not sensible but intelligible, and forms – that is, geometric properties, such as being straight and being circular. Accordingly, the geometric knowledge could be acquired through successive abstractions. Therefore, there seems to be no radical difference between acquiring geometric knowledge and other kinds of scientific knowledge, for Aristotle.  Now that this brief review, I hope, to some extent, has introduced us to the philosophical challenges raised by questions regarding the epistemology and ontology of geometry and we have seen some attempts, though perhaps not entirely successful, at facing them, we should turn to the history of evolution of geometry in the early modern era, in the next chapter. Let me, then, conclude our review of Greek geometry, by noting that although during the period skipped here the study of higher curves, particularly conic sections, and trigonometry led to many significant results, Greeks considered them as practical achievements in optics, astronomy, and so on. What they considered as geometrical research was mainly focused on and limited to their unsuccessful attempts at solving the three classical problems, as well as proving Euclid’s fifth postulate.i We now leave the Greek era for the early modern period with the irony that Greeks did not value much what they achieved and what they most valued was not achievable.      i This, however, does not mean that no advance in Euclidean geometry was made. See  Ostermann and Wanner (2012). 26  Chapter 2: The early modern era The fall: earthly geometry During the same time as the earth was losing its central place, geometry was becoming more and more earthly. The subject matter of geometry, geometric objects, was being brought down to earth; its elegant axiomatic method was being replaced with messy algebraic calculations; and finally, what had been the geometry for millennia was being forced to assume a nondescript status as one among equals. What follows is merely an abridged, simplified, and incomplete version of that intricate story – that is, the fall of geometry – with many characters whose involvements and actual views are far subtler, convoluted, and, at times, confusing than could be, or is necessary to be, told here.i From the heavens to the earth     The descent of geometry had begun with Aristotle’s account of Euclidean objects, rendering them inseparable from earthly objects. Geometric objects, being made of intelligible matter, however, were autonomous, real, and, in a sense, superior to physical objects, for Aristotle. Nevertheless, having lost their place in the heavenly realm of the forms, the ontological status of geometric objects was put in danger. It was not long before that they were reduced to mere properties of physical objects.  Galileo Galilei (1564-1642) was not the first one to employ geometry in describing/explaining natural phenomena. Both ancient and medieval thinkers had been using geometrical descriptions, for instance, in their studies in optics and astronomy. The subject matter  i  The same, of course, could be said of the brief exposition of Plato’s and Aristotle’s views. 27  of these sciences – that is, the geometric objects and the physical objects – were kept separate, however. Planets were seen as moving in circular orbits, but not as a geometric point on a circle, for instance. For one thing, a planet and a geometric point had different natures; for another thing, geometry and astronomy were seen to be capable of producing different kinds of knowledge – that is, certain versus probable.i For Galileo, however,  Philosophy is written in this grand book – […] the universe – which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.ii   Accordingly, he initiated the geometrization of mechanics – a move for which Galileo had to face considerable resistance.iii He, in response, took an even more consequential step; in Joseph Pitt’s words:  [In the Two New Sciences] he argues that the justification for the role of geometry is its ability to provide the means for the generation of new knowledge, or, in more modern terms, it not only can function deductively as a justificatory method itself, it also can be used as a logic of discovery.iv   i Pitt (1992, pp. 6-7). ii Galilei (1623/1960, pp.183-184). iii See Hall (1963), Clavelin (1983), Shea (1983), Pitt (1992), Barbour (2001), Dijksterhuis (1961), and Marshall (2013) for discussion of geometrization of motion, its difficulties, and Galileo’s arguments.  iv Pitt (1992, pp. 24-5).  28  While Galileo’s works helped to close the gap between geometry and the physical sciences considerably, his contemporary the French geometer and philosopher René Descartes (1596-1650) both expanded the domain of the applicability of geometry and revolutionized its method.  Descartes believed that, God aside, there were two kinds of substance, each with one principal attribute. The principal attribute of material substances, Descartes argued, was extension – that is, geometric dimensions of length, breadth, and depth: [E]ach substance has one principal attribute, thought, for example, being that of mind, and extension that of body. And substance is indeed known by any attribute [of it]; but each substance has only one principal property which constitutes its nature and essence, and to which all the other properties are related. Thus, extension in length, breadth, and depth constitutes the nature of corporeal substance; and thought constitutes the nature of thinking substance. For everything else which can be attributed to body presupposes extension, and is only a certain mode {or dependence} of an extended thing; […]. Thus, for example, figure cannot be understood except in an extended thing, nor can motion, except in an extended space; […]. But on the contrary, extension can be understood without figure or motion; […] as is obvious to anyone who pays attention to these things.i  Descartes, furthermore, believed that space itself is “a sort of conceptual abstraction from [the] bodily spatial extension:” ii [n]or in fact does space, or internal place, differ from the corporeal substance contained in it, except in the way in which we are accustomed to conceive of them. For in fact the extension in length, breadth, and depth which constitutes the space occupied by a body, is exactly the same as that which constitutes the body. The difference consists in the fact that, in the body, we consider its extension as if it were an individual thing, and think that it is always changed whenever the body changes. However, we attribute a generic unity to the extension of the space, so that when the body which fills the space has been changed, the extension of the space itself is not considered to have been changed {or transported} but to remain one and the same; as long as it remains of the same size and shape and maintains the same situation among certain external bodies by means of which we specify that space.iii  i Descartes (1644/1982, pp. 23-4). ii Slowik (2017) “Descartes’ Physics”  iii Descartes (1644/1982, pp. 43-4). 29   The essence of both matter and space, according to Descartes, is extension, and thus both are in the province of geometry.5 Descartes, therefore, expands the realm of geometry to include everything but minds and God.i   Not only did he subject the entire material world to geometry, but also, Descartes devised a new powerful method for geometry, though at the cost of its ingeniously fashioned elegant constructions. From pristine constructions to messy calculations: elegance versus power Descartes opens his book on geometryii by declaring that  [any] problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction. Just as arithmetic consists of only four or five operations, namely, addition, subtraction, multiplication, division and the extraction of roots, which may be considered a kind of division, so in geometry, to find required lines it is merely necessary to add or subtract other lines; or else, taking one line which I shall call unity in order to relate it as closely as possible to numbers, and which can in general be chosen arbitrarily, and having given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity (which is the same as multiplication); or, again, to find a fourth line which is to one of the given lines as unity is to the other (which is equivalent to division); or, finally, to find one, two, or several mean proportionals between unity and some other line (which is the same as extracting the square root, cube root, etc., of the given line.iii  i See McGuire (1983) and Slowik (2002) for Descartes’ conception of space, and Garber (1992) for Descartes’ physics.  ii The first edition of Geometry was first published as an appendix to his Discourse on Method in 1637; see Bos (2001) for a detailed discussion of Geometry and for a brief one Mancosu (1996), Grosholz (1991) for a critical discussion of the place of the analytical method of Geometry within the broader context of Cartesian methodology, and Sasaki (2003) for the history of the evolution of Descartes’ mathematical thought.  iii Descartes (1637/1954, pp. 2-5). 30   In Geometry, to put it briefly, Descartes reduces all magnitudes to lines – that is, one-dimensional magnitudes – and treats their lengths as known or unknown algebraic quantities. As a general method for solving geometrical problems, then, he recovers the relevant algebraic equations by analyzing assumed solutions to those problems. The time-honoured axiomatic tradition of geometry, thus, began to be replaced with tedious but mostly straightforward algebraic calculation.i Descartes, however, did not part ways with Euclidean constructive tradition.ii Nor did he equate points with their coordinates. The relative ease with which the use of the new method led to the solutions of Pappus’ problem helped to convince mathematicians of the power of analytic methods which would be vastly improved at the hands of Newton, Leibniz, Euler, and many others.iii   The ontological and epistemological status of earthly geometry The subject matter of geometry – that is, the objects studied by geometry – as we have seen, for Descartes is extension – that is, space and the principal attribute of matter – which closely resembles Aristotle’s intelligible matter.iv Regarding the question of the ontology of geometry,  i Descartes’ innovation, of course, has a history and analytic geometry has its roots in ancient mathematical investigations; see Boyer (2012) and Rosenfeld (1988) for the history of analytic geometry.   ii Bos (2001). iii Pappus’ problem is a family of locus problems which requires finding all points that share a certain given property; see Bos (2001) for a detailed discussion of Descartes’ solution. iv Unlike Aristotle, however, Descartes does not see it as inseparable nevertheless autonomous, but instead, as the principal attribute of matter.  31  accordingly, Descartes seems to be Aristotelean. As to the question of the epistemology of geometry, however, he seems to be in the same boat as Plato. Reason versus senses: the rationalism and geometry  Being a rationalist, Descartes erects his epistemology on the ground of innate ideas. Genuine knowledge, for him, accordingly, stems from the (re)acquaintance with the intellect’s innate ideas, which is supposed to be the fruit of the Cartesian method.i In the Fifth Meditation, regarding extension and how it and its properties become known, Descartes writes:  Not only are all these things very well known and transparent to me […] but in addition there are countless particular features regarding shapes, number, motion and so on, which I perceive when I give them my attention. And the truth of these matters is so open and so much in harmony with my nature, that on first discovering them it seems that I am not so much learning something new as remembering what I knew before [.]ii  Elsewhere, he even explicitly mentions Plato and his theory of recollection: [W]e come to know them by the power of our own native intelligence, without any sensory experience. All geometrical truths are of this sort — not just the most obvious ones, but all the others, however abstruse they may appear. Hence, according to Plato, Socrates asks a slave boy about the elements of geometry and thereby makes the boy able to dig out certain truths from his own mind which he had not previously recognized were there, thus attempting to establish the doctrine of reminiscence.iii  Expounding such an epistemic stance excuses the lack of any conceptual/psychological account of geometric concepts/spatial intuitions in Descartes’ philosophy of geometry. Descartes, however,  i Newman (2016) “Descartes’ Epistemology”  ii Descartes (1641/1984, p. 44). Inclusion of ‘motion’ aside, these words could have been written by Plato.  iii Descartes (1643/1991, pp. 222-3). 32  was a pioneer in investigating vision and his theory is considered the first psychology of vision.i  The fact that he does not appeal to psychological considerations in his philosophy of geometry is significant, accordingly, in that it highlights the distinction that Descartes makes between geometry and what he calls natural geometry – that is, geometrical considerations employed in distance perception.   We now leave Descartes’ philosophy of geometry to briefly consider those of two thinkers who were greatly influenced by his philosophy – keeping in mind how Descartes was forced to accept an uncanny union of Plato’s and Aristotle’s accounts in order to address philosophical questions raised by geometry.   To be or not to be: Newton versus Leibniz Both Gottfried Wilhelm Leibniz (1646-1716) and Isaac Newton (1642-1727) took Descartes’ philosophy very seriously; however, influenced, partly, by their scientific and mathematical investigations, they ended up with different views. Regarding the ontological status of space, in particular, they took opposite positions. While, in Newton’s account space would be elevated over that of matter, in Leibniz’ account, where space was seen as merely a system of relations, it would be demoted.   De Gravitatione, an unfinished text whose date of composition is obscure and which remained unpublished until the twentieth century, is considered the best window to Newton’s  i Hatfield (1990). See Wolf-Devine (1993 and 2000) for Descartes’ psychology of vision and visual spatial perception, and Pastore (1971) for a history of theories of visual perception.  33  mind when it comes to the question of space. There, while he criticizes Descartes’ conception of space, Newton writes that space or extension  has its own manner of existing which is proper to it and which fits neither substances nor accidents. It is not substance: on the one hand, because it is not absolute in itself […]; on the other hand, because it is not among the proper affections that denote substance, namely actions, such as thoughts in the mind and motions in body. [….] Moreover, since we can clearly conceive extension existing without any subject, as when we may imagine spaces outside the world or places empty of any body whatsoever, and we believe [extension] to exist wherever we imagine there are no bodies, […] it follows that[extension] does not exist as an accident inhering in some subject. And hence it is not an accident. And much less may it be said to be nothing, since it is something more than an accident, and approaches more nearly to the nature of substance.i   This apophatic account of space – that is, describing space by what it is not – follows by Newton’s enumeration of properties of space: 1. In all directions, space can be distinguished into parts whose common boundaries we usually call surfaces; and these surfaces can be distinguished in all directions into parts whose common boundaries we usually call lines;and again these lines can be distinguished in all directions into parts which we call points. [….] 2. Space is extended infinitely in all directions. [….] 3. The parts of space are motionless. [….] 4. Space is an affection of a being just as a being. No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. [….] 5. The positions, distances, and local motions of bodies are to be referred to the parts of space. [….] 6. Lastly, space is eternal in duration and immutable in nature because it is the emanative effect of an eternal and immutable being.ii  Newton’s account of the ontology of space, thus, is very different from that of Descartes. Concerning the epistemology of space, however, it seems that Newton has not much to add to  i Newton (?/2004  pp. 21-2). ii Newton (?/2004  pp. 22-6). 34  what Descartes had offered. Regarding how the aforementioned properties of space could be known, Newton writes that we have an exceptionally clear idea of extension by abstracting the dispositions and properties of a body so that there remains only the uniform and unlimited stretching out of space in length, breadth and depth. And furthermore, many of its properties are associated with this idea [.]i  In Leibniz’ philosophy of geometry, too, the question of the epistemology of geometry and space remains, more or less, as Descartes left it.6 Epistemological considerations, however, play a crucial role in Leibniz’ philosophy of geometry and lead him to argue that space is nothing but a system of relations.7 Unlike his illustrious contemporary who has elevated space from being an essential attribute of matter to being an affection of all beings, minds and God included, Leibniz banishes space from the realm of ‘serious’ beings.8 From figures to space Leibniz, like Descartes, is a rationalist and naturally innate ideas play a fundamental role in his epistemology. His conception of truth, furthermore, is analytic – that is, for Leibniz, a proposition is true if and only if its subject contains its predicate. Proving the truth of a proposition, for him, therefore, requires establishing that the conceptual marks of its predicate are among the conceptual marks of its subject.9 This is done through conceptual analyses and involves a system of definitions.ii Given this epistemological stance, Leibniz conceives geometrical reasonings as  i Newton (?/2004  p. 22). ii Leibniz (1686/1998); for an exposition of Leibniz’ conception of truth and his principle of predicate-in-notion, the details of which need not concerns us here, see Broad (1975).  35  conceptual analyses of the (real) definitions of geometrical terms, using the law of identity.i Such a conception of geometry seems very much in line with that of ancient Greeks even though unlike Greeks, the ultimate object of geometry, for Leibniz, is space. Such a conception puts Leibniz in an almost impossible position and forces him to somehow pull the metaphysical rabbit from the hat while at the same time remaining conscious of his epistemological scruples. His conception of space is supposed to do the trick.ii  The notion of space seems to be foreign to the ancient Greek geometers. Geometric figures – that is, the subject matter of geometry – were conceived as independent individuals, not as part of a space. Leibniz breaks with this ancient tradition and insists that the objects of geometry are not individual figures or continuous magnitudes but the space where these objects live.iii Furthermore, he contends that the nature of space and its properties and structures determine which geometrical figures are possible and what properties they have.iv Leibniz, thus, proposes that  i See De Risi (2016) for Leibniz’ theory of definition. Real definition, in the case of geometry, just means nominal definitions of geometric objects along with a proof of their existence. ii The account of Leibniz’ philosophy of geometry given here is mostly based on De Risi (2015 and 2007). iii It is true that Descartes and Newton, and even some in the scholastic tradition, thought that space was the subject matter of geometry, but for them, space was a geometrically structureless and amorphous container. Leibniz was the first to conceive space endowed with structure. See De Risi (2015).  iv According to De Risi, Leibniz’ proposal is to give what he calls the essential definitions for all Euclidean objects– that is, what gives “the ground of the possibility of all geometrical objects and 36  one must begin with the analysis of space if one wants to prove any geometric propositions, Euclid’s postulates included. While how such a proposal is supposed to be carried out is not our concern here, we need to briefly discuss Leibniz’ account of the concept of space that he believes does the job and how it is acquired. To get an accurate idea of Leibniz’ conception of space, it is worthwhile to quote him at length: to give a kind of a definition: place is that which we say is the same to A and to B when the relation of the coexistence of B with C, E, F, G, etc. agrees perfectly with the relation of the coexistence which A had with the same C, E, F, G, etc., supposing there has been no cause of change in C, E, F, G, etc. It may be said also, without entering into any further particularity, that place is that which is the same in different moments to different existent things when their relations of coexistence with certain other existents which are supposed to continue fixed from one of those moments to the other agree entirely together. And fixed existents are those in which there has been no cause of any change of the order of their coexistence with others, or (which is the same thing) in which there has been no motion. Lastly, space is that which results from places taken together. And here it may not be amiss to consider the difference between place and the relation of situation which is in the body that fills up the place. For the place of A and B is the same, whereas the relation of A to fixed bodies is not precisely and individually the same as the relation which B (that comes into its place) will have to the same fixed bodies; but these relations agree only. For two different subjects, such as A and B, cannot have precisely the same individual affection, since it is impossible that the same individual accident should be in two subjects or pass from one subject to another. But the mind, not contented with an agreement, looks for an identity, for something that should be truly the same, and conceives it as being extrinsic to the subjects; and this is what we call place and space. But this can only be an ideal thing, containing a certain order, in which the mind conceives the application of relations. [….]in order to explain what place is, I have been content to define what is the same place [….] And it is this analogy which makes men fancy places, traces, and spaces, though those things consist only in the truth of relations, and not at all in any absolute reality.i   constructions [which] is to be found in space itself, its properties, and its structure.” De Risi (2015, p. 40). In other words, Leibniz seeks the proof of Euclid’s postulates.  i Leibniz (1716/2000, pp. 46-7).   37  Space is defined as a set of points structured according to a system of relations among them – that is, situations. A situation is defined as a relation between objects seen as extensionless – that is, points. The relation in question is not defined; instead, it is characterized by distance. As noted by De Risi (2015, pp. 43-4), although usually it is discussed in the context of his philosophy of physics, “Leibniz’ definition of space as an order of situations is first of all a purely geometrical claim about the mathematical structure of space. It is in fact the first conception ever of a geometrical space as a structure [. This is so because space thus defined] is roughly equivalent to the modern concept of a metric space, which is given by a set of points and a set of distances between them. A space is just a system of distances – a metric structure.”i   Having seen briefly how the questions regarding the ontology and epistemology of geometry are dealt with in the rationalist camp, let us now see how the other camp – that is, the empiricists – struggled with these questions.10 Senses versus reason: the empiricism and geometry   Unlike the rationalist who allows innate ideas and concepts to ground a priori knowledge, the empiricist believes that all ideas and concepts are inspired by and derived from sense data, and experience is the only legitimate ground for knowledge. Naturally, providing an account of the origin of spatial intuitions and geometric concepts and ideas is high on the empiricist’s agenda, and informs and influences their conception of the nature of space. Empiricists’ accounts of the origin of geometric ideas, however, at least in the early stages, were too general and sketchy.     One of the first early modern empiricists, Thomas Hobbes (1588-1679), for instance, writes   i See Aarts (2009). 38  If therefore we remember, or have a phantasm of any thing that was in the world before the supposed annihilation of the same;11 and consider, not that the thing was such or such, but only that it had a being without the mind, we have presently a conception of that we call space: an imaginary space indeed, because a mere phantasm, yet that very thing which all men call so. [….] SPACE is the phantasm of a thing existing without the mind simply; that is to say, that phantasm, in which we consider no other accident, but only that it appears without us.i  Other empiricists in the seventeenth and eighteenth centuries, although they disagreed on details, more or less, gave some version the same account of the epistemology of geometry.ii John Locke (1632-1704) whose conception of space is essentially Newtonian, George Berkeley (1685-1753) who as an idealist denies the existence of material substances, and David Hume (1711-1776) who rejects the conception of absolute space, for example, agree that the idea of space is somehow built from the sense data provided by sight and touch – that is, through perceiving the extension of physical objects and their distances.12 For more detailed accounts of the origin of spatial ideas, we have to wait for the nineteenth-century thinkers.iii We shall now turn to the apriorist’s philosophy of geometry – a reaction to rationalist/empiricist’s and absolutist/relationalist’s debates.13  i Hobbes (1655/1839, pp. 93-4). Hobbes believes that natural philosophy, done properly, begins by what nowadays called a thought experiment in which the thinker assumes the world has annihilated. ii For Hume’s philosophy of geometry; see also De Pierris (2015), Hatfield (1990), and Pastore (1971). iii One of these empiricist’s accounts– that is, that of Helmholtz – which highlights the importance of the possibility of motion without deformation turns out to greatly influence Poincaré’s philosophy of geometry. In the fourth chapter, we will briefly encounter Poincaré’s version of such an account of the origin of spatial ideas. But, for the sake of brevity, I will not consider Helmholtz’ account here. For theories of spatial perception in eighteenth and nineteenth centuries see 39  Sensibility versus understanding: apriorism and geometry Trained in Leibnizian tradition but also influenced by British thinkers – Newton and Hume, in particular – Immanuel Kant (1724-1804) attempted to straddle opposing sides of some of the most intriguing philosophical debates of the eighteenth-century – realism/idealism, rationalism/empiricism, and absolutism/relationalism.i His philosophy, as a result, was evolved as a reaction to these binary philosophical positions trying to synthesize dichotomies to go beyond them. Kant’s philosophy of geometry, in addition to being an exciting chapter in the history of philosophical thoughts about and originated by geometryii – in particular, an attempt at explaining the applicability of geometry in empirical science without reducing it to an empirical science – is one of the two contexts against which Poincaré’s account is offered and should be understood – the other one being that of the empiricists. Poincaré’s exposition of these views is, however, very brief. iii  The following exposition of Kant’s account of space and geometry – which although, I hope, does not distort Kant’s views, I prefer to sometimes call the Kantian or the Kantian version  Hatfield (1990). For a discussion of how Helmholtz’ physical research, in particular, his work on the conception of energy and its conservation, influenced his conception of geometry, see Hyder (2009) and Biagioli (2016). For Helmholtz’ own exposition of his theory of vision, see Helmholtz (1867/1924).  i See Cassirer (1981) and Kuehn (2001) for Kant’s life and ideas.  ii An enormous literature exists about Kant’s philosophy of space, geometry, and intuition; see, for instance, Buroker (2013), Carson (1997), Cummins (1968), Falkenstein (1991), Ferrarin (2012), Friedman (2012), Hatfield (1984), Kitcher (1978), Shabel (2004), Vinci (2014), Warren (1998), White (1978), and Wilson (1975).  iii See Schmaus (2003) for a discussion on Kant’s reception in the nineteenth century France. 40  of the apriorist account– attempts to paint a picture consistent with what seems to be Poincaré’s understanding.  Space and objective knowledge Space, Kant argues, “is nothing other than merely the form of all appearances of outer sense, i.e., the subjective condition of sensibility, under which alone outer intuition is possible for us[;]”i it “represents no property at all of any things in themselves nor any relation of them to each other[.]”ii Understanding Kant’s conception of space requires familiarity with his terminology.  In course of explaining how it is possible to represent objects and make judgments about them that can be said to be at the same time both synthetic and a priori, Kant distinguishes three cognitive faculties – that is, sensibility, understanding, and reason – and three kinds of representations – that is, sensations, intuitions and concepts. Provided by sensibility – that is, the capacity to be passively affected by objects –intuitions are singular representations. That is, intuitions refer to/represent their single object immediately. Concepts, on the other hand, refer to/represent one or more objects through intuitions, are employed when thinking about objects, and are provided by a cognitive faculty which Kant calls understanding.14 Kant, then, distinguishes pure representations from empirical ones, and the form of a representation from its matter.iii Sensations – that is, “[t]he effect of an object on the capacity for representation, insofar as we are  i A27/B43. ii A26/B42 iii Only pure intuitions or concepts can be the subject of synthetic a priori judgments, according to Kant. 41  affected by it”i – are the matter of empirical intuitions – that is, those intuitions which are related to objects through sensations. “The undetermined object of an empirical intuition is called appearance.”ii What “allows the manifold of appearance to be ordered in certain relations”iii is called the form of appearance. Finally, the outer sense is a property of our mind, which represents external objects as situated in space.15 Kant then argues that the notion of space, as a form of appearance, cannot be drawn from experience, built from, or abstracted from sense perception. It is instead “the condition of the possibility of appearances, not as a determination dependent on them, and is an a priori representation that necessarily grounds outer appearances.”iv The notion of space, thus, Kant concludes, “must be due to the nature of the knowing mind, in this case to the nature of our sensibility.”v Neither does space, therefore, represent an (external) entity, nor a property of an entity, nor a relation among entities. To establish these claims – that is, that the source of representation of space is non-empirical, that its content is singular, and that it is immediately given – Kant offers several arguments, only one of which we consider here – the one, as we will see, with which Poincaré takes issue.vi For in order for certain sensations to be related to something outside me (i.e., to something in another place in space from that in which I find myself), thus in order for me to  i B34/A20. ii B34/A20. iii B34/A20. iv B 39. v Paton (1936, p.102). vi For a brief discussion of these arguments, see Shabel (2010).  42  represent them as outside <and next to> one another, thus not merely as different but as in different places, the representation of space must already be their ground. Thus the representation of space cannot be obtained from the relations of outer appearance through experience, but this outer experience is itself first possible only through this representation.i  It seems that, in this argument, from the observation that certain sensations are related to things outside of the subject experiencing those sensations, it is concluded that the subject must be able to represent those things – and also oneself – as spatially distinct from one another. This is taken to imply that sensations received through outer sense must be already spatially structured, which requires that the representation of space precede empirical representations. Kant, thus, concludes that the former is the condition of the possibility of the latter.  Having argued that space is a form of sensibility, Kant makes it clear that he takes geometry to be “a science that determines the properties of space synthetically and yet a priori.”ii Then, he claims that his “explanation alone makes the possibility of geometry16 as a synthetic a priori cognition comprehensible.”iii  So far, if successful, Kant has shown that space is a form of sensibility. His goal, however, is to show that space is merely a form of sensibility. To this end, he argues that were space not a form of sensibility, geometric propositions would not be a priori.17 That is, Kant argues that only if space  i A24/B39. ii B 41. However, space as the subject of geometry is not exactly the same as the form of the utter sense. Their difference, however, need not concern us here. iii A26/B42.  43  is taken to be merely a form of sensibility, can the a priori status of geometrical propositions be justified/explained.i  Kant’s argument notwithstanding, in assessing the ramifications of the discovery of non-Euclidean geometries on his philosophy of geometry, it is crucial to note the following. All that Kant’s arguments (without making further assumptions about the epistemic status of geometric propositions), if successful, establish is that particular structural properties of one of the two forms of sensibility – that is, what is called the spatial structure – are innate contributions of cognitions, or in his words: “the subjective constitution of our mind.” When applied to geometry, Kant’s arguments imply that all those geometrical propositions that assert/describe such structural properties are a priori. This is not, however, how Kant, as we saw, presents his arguments. Instead, he starts with two assumptions about geometry – one explicit and the other implicit. He explicitly assumes that all propositions of Euclidean geometry are a priori, and implicitly that each and every proposition of Euclidean geometry describes some innate property or structure of the outer form of sensibility.  To recapitulate: Kant took upon himself to explain how it was possible to make synthetic a priori judgments, of which he took propositions of Euclidean geometry to be examples.ii In other words, he began by assuming that synthetic a priori judgments are possible and went on to  i See, Shabel (2004). ii Among these geometric propositions, Kant includes “space has only three dimensions” (B.41) which, of course, neither belongs to Euclid’s postulates nor theorems. However, Euclid must have some notion of dimensionality in his mind, as it is suggested in how a point, a line, and a plane are defined in the first book. See Janich (1992, p. 17). 44  determine the conditions that made them possible. The condition of the possibility of such judgments, according to his analysis, turned out to be that such judgments are contributed by/are about innate cognitive faculties. In the context of the philosophy of geometry, this means objective spatial knowledge – that is, ‘true’ geometrical propositions in the sense that they describe/are applicable to the reality – is possible only if the spatial concepts are provided and forced on representations by cognitive faculties. The goal of explaining the applicability of geometry aside, Kant did not need to assume that all propositions of Euclidean geometry were synthetic a priori, however. He could have, instead, examined these propositions and tried to distinguish those that in fact expressed/determined the properties of space as a pure form of sensibility. That is a step that, as we will see, Bertrand Russell (1872-1970) takes and like Poincaré, but for different reasons, ends up with a subset of propositions of Euclidean geometry – that is, those that describe the structures required for the free mobility.i  When the implications of the existence of non-Euclidean geometries for Kant’s philosophy of geometry are considered, thus, it is vital to keep in mind that Kant’s positions regarding the nature and the epistemic status of Euclidean geometry are not the binding consequences of his account; they are just his (arguably dispensable) assumptions. Kant does not need to commit himself to the strong (and arguably false) claim that all propositions of Euclidean geometry describe/determine the form of the outer sense. However, if one assumes, as Kant does, that space – that is, the form of the outer sense – is Euclidean, it follows that no perceptual/cognitive experience structurally incompatible with Euclidean geometry could be possible. In other words,  i See the appendix A.  45  the conceivability of Euclidean and non-Euclidean geometries does imply that either space is not a form of sensibility, or space as a form of sensibility is less-structured than three-dimensional spaces of constant curvatures. Poincaré, as we will see, argues for the first possibility, Russell for the second. But, I am getting ahead of myself, and first should conclude the survey of the history of the evolution of geometry with the story of the discovery of non-Euclidean geometry.                 46  Chapter 3: The non-Euclidean era From one to many The last, but probably the most crucial, episode in the saga of the descent of Euclidean geometry from its central place in the edifice of human knowledge was the discovery of non-Euclidean geometries. Not only because it demoted the former bastion of secure knowledge to one among equals, but also and more importantly, because it called into question the credibility of all geometries, as well as the kind of knowledge that geometry had been taken to represent and the method by which such knowledge could be obtained. Such an enormously consequential discovery, ironically enough, was the unwanted outcome of the millennia-long quest for securing the foundation of Euclidean geometry – that is, a proof for the fifth postulate.  Practical considerations, in addition to the failure in proving the fifth postulate, also, played a crucial role in undermining the position of Euclidean geometry. This latter trend led to the replacement of Euclidean geometry with the projective geometry whose origins lie in attempts to meet the practical need of visual artists, and later engineers, to accurately depict three-dimensional objects on two-dimensional surfaces. The geometry of vision versus the geometry of measurement The need, and the challenge, to represent spatial objects on two-dimensional surfaces are not new. Through the ages, different more or less efficient solutions to this challenge have been found by artists and artisans belonging to various cultures. One of these solutions – that is, using perspective 47  – found by Renaissance artists, inspired a new kind of geometry – the geometry of vision, which is of interest to our story.i  Unlike Euclidean geometry – which is concerned with the metrical properties of geometric figures, and has its origin in measuring distances among farms, their areas, and so on – the central question for the new geometry, the geometry of vision, is how (three-dimensional) objects would appear when seen from different locations, and how these images – that is, different two-dimensional renditions of an object, called perspectives – could be systematically depicted. The one-point perspective technique, at first, and later, the two-point perspective technique, illustrated below, were the Renaissance artists answers to these questions.  Figure 3.1 Perspective techniques Although these solutions were good enough for dealing with artists’ practical needs, making sense of them and building a consistent system required mathematicians’ contribution.ii As objects appear to have different shapes when seen from different vantage points, the central concepts of the old geometry – such as Euclidean shape, angle, and distance – had to be revised.   i Not all agree that perspective was invented in this period, however; see Andersen (2007) for a history of the mathematics of perspective.  ii See Stillwell (2014 and 2006). 48  A universal geometry: the projective geometry In the constructions suggested by artists, unlike in Euclidean constructions, parallel lines meet, but not in a point on the plane. They meet in a (seemingly) mysterious place on the horizon, a point at infinity.i To keep up with these developments theoretically, mathematicians had to invent/discover new geometric concepts and tools. They first added a special line – that is, the line at infinity – consisting of points at infinity to the Euclidean plane, and thus defined/created a new geometric object, the projective plane.ii The projective plane consists of points and lines, both the ordinary ones – that is, Euclidean points and lines – and those at infinity. It is a model for the following four axioms:iii 1. Any two points belong to a unique line.  2. Any two lines have a unique point in common. 3. There are four points, no three of which are in the same line.                                 [4.] If A, B, C, D, E, F are points of the plane lying alternately on two lines, then the intersections of the pairs of lines AB and DE, BC and EF, CD and FA, lie on a line.iv  i The vanishing point, in the artist’s terms.  ii I have limited my exposition to only two dimensions, here; however, projective geometry can be easily extended to higher dimensions. For the history of projective geometry and also its mathematics, see Ostermann & Wanner (2012), Blumenthal (1980), Richter-Gebert (2010), Kline (1972). iii One and two-dimensional linear subspaces of three-dimensional Euclidean space – that is, ordinary Euclidean lines and planes having a common point – can be, also, interpreted as projective points and lines, respectively. The common point, then, can be interpreted as the origin for a homogeneous coordinate system. iv Stillwell (2016, pp. 190-1). The first and the third axiom hold for Euclidean geometry; the second does not. The fourth is a theorem of Euclidean geometry, called theorem of Pappus. 49   Unlike the Euclidean plane which is structured enough to be endowed with a notion of distance between any two points, and a notion of identity based on congruence,i the more primitive structure imposed on the projective plane by the above axioms allows for a broader and more relaxed notion of identity for projective objects – that is, well-defined subsets of the projective plane. For instance, in the projective plane, all quadrilaterals are the same – that is, for example, it is not possible to distinguish squares from trapezoids. In general, any transformation which preserves collinearities and incidences of points and lines does not alter the identity of a projective object. Such transformations are called projective transformations of which Euclidean motions – that is, distance-preserving translations, rotations, and reflections – are only a special case.ii   Figure 3.2 Geometric transformations As with Euclidean motions, however, there is an invariant quantity associated with projective transformations; it is called the cross-ratio. Unlike the invariant of Euclidean motions – that is, distance – which is defined for any two points, the cross-ratio is defined for any four collinear points, a, b, c, and d. It can be defined in terms of Euclidean distance as (a,b;c,d) = ac.bd/bc.ad  Unlike in Euclidean geometry, circles and angles are not defined in projective geometry, and thus there are no axioms about them.  i ‘Two’ Euclidean objects which coincide when superimposed are considered to be identical – that is, the same object. ii Our discussion, without loss of generality, is often is limited to the orientation preserving motions – that is, translations and rotations.  50  where ac is the distance between a and c, and so on.i The concept of the cross-ratio, however, does not presuppose that of distance and can be used to define the latter.ii In fact, once Arthur Cayley (1821 – 1895) and Felix Klein (1849 – 1925) did so for Euclidean and non-Euclidean distances respectively,  projective geometry became ‘all geometry,’ a universal geometry.iii Euclidean and non-Euclidean planes (spaces) are contained in the projective plane (space), as are Euclidean and non-Euclidean transformations among projective ones. Different geometries, thus, form a hierarchical structure which Klein highlights in his Erlangen program.iv Geometric transformations are examples of an important algebraic structure – that is, the group structure.v The study of a specific  i Assuming (a,b;c,d)=λ, (i) (a,b;c,d)=(b,a;d,c)=(c,d;a,b)=(d,c;b,a), (ii) (a,b;d,c)=1 /λ, (iii) (a,c;b,d)=1−λ, and (iv) the values for the remaining permutations are a consequence of these three rules; Richter-Gebert (2010). ii This follows from the fact that the fourth axiom makes it possible to introduce a coordinate system which turns out to be an algebraic field; Stillwell (2006). The cross-ratio for any four points can be calculated with respect to coordinates assigned to them using a purely projective construction – von Staudt’s construction, for example – and then it can be turned to distance using appropriate formula for each geometry. To see how it could be done, see Richter-Gebert (2010), for instance. iii “Projective geometry is all geometry,” Cayley as cited in Klein (1925/2016). See Richter-Gebert (2010, Chap. 20) for a modern exposition. iv See Ji & Papadopoulos (2015). v A group, G, consists of an underlying set, G, and an associative function, * from G×G onto G, where there is an e∈G (identity) such that for every g∈G, e*g=g*e=g, and for every g∈G there is a g-1∈G (inverse of g) such that g*g-1= g-1*g=e. Note that this definition defines groups abstractly and is the modern definition of the abstract group. During the time that we are interested in, however, a group was defined, more concretely, as a given set of transformations satisfying the 51  geometry, thus, becomes the study of a particular group, and the study of the relationship among geometries, the study of the relationship among groups. That is, as is famously put by Poincaré, ‘geometry is nothing but the study of group.’i With powerful algebraic tools – that is, group theory – came new areas of research interests with which mathematicians were mostly busy until the end of the nineteenth century when the importance of Riemann’s work was understood, and the era of differential geometry was ushered in. Euclidean geometry had become history.   Now that we saw how the geometry of vision played its crucial role in helping along with the demise of Euclidean geometry, let us turn to the principal player in that story – that is, the discovery of non-Euclidean geometry.  Certainty sought and lost A scientific discipline, according to the Aristotelian account, is a hierarchical system built on a few fundamental presuppositions and developed in accordance with the accepted role of reasoning. For a given discipline, its postulates state the former, and the axioms, at least in the case of geometry, according to the widespread view, the latter. Postulates, for a given discipline, are posited, and thus are not the subject of further inquiry within that discipline.ii In the case of  aforementioned properties. See Bacry (2004). For an elementary exposition of the theory of transformation groups, see Duzhin & Chebotarevsky (2004). i See Wussing (1984) and Hawkins (2000) for a history of group theoretic approach.  ii It lies beyond the scope of our discussion to ponder on the qualifications of postulates, in general.  52  Euclidean geometry, however, geometers viewed the fifth postulatei somehow differently.ii Not its truth, but its status as a postulate was in question from early on.18  Several explanations for why the fifth postulate seemed suspicious to geometers – such as, not being self-evident, the explicit reference to ‘indefinitely,’ and so on – have been suggested.iii Given the history of the rejection of the suggested proofs, which will be briefly discussed in chapter eight, however, these explanations do not seem adequate.iv Regardless of what made the fifth postulate suspect in the eyes of numerous generations of geometers, one thing seems clear. The fifth postulate, even though it might seem superficial at first, is different from the first four in that it is longer and far more complicated than the others. More importantly, unlike others, the fifth postulate is a conditional proposition. To put it differently, it shares its form with theorems. No wonder that geometers insisted that, instead of just accepting it as a postulate, they should be able to prove it. The Euclidean edifice could be rebuilt based on presuppositions whose status as  i That is, the one that asserts that ‘if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.’ ii Euclid himself postponed its use as long as he could – that is, 29th propositions of the first book of the Elements. Others tried to dispense with it, by making it a theorem – that is, by proving it using either only the other four postulates, or in addition to those, an additional presupposition that they could have accepted as a postulate. See Bonola (1912) and Rosenfeld (1988).  iii See Rosenfeld (1988).  iv In the course of attempts at proving the fifth postulates, several of its equivalents were discovered. However, although among them were some not subject to these supposed weaknesses, they too were rejected. See also the next footnote.  53  postulates was beyond question, they believed, once such a proof was found. The quest for the proof of the fifth postulate, thus, can be seen as the quest for a more secure foundation. A quest for certainty – certainty in whatever sense the participants believed the first four postulates, unlike the fifth, could offer. It was, as far as they were concerned, a losing battle, alas. But it provided valuable insights on the nature of Euclidean geometry: Euclid showed the great strength of his genius by introducing Postulate V, which is not self-evident like the others. [….] Between his time and our own, hundreds of people, finding it complicated and artificial, have tried to deduce it as a proposition. But they only succeeded in replacing it by various equivalent assumptions, such as the following five: [1]. Two parallel lines are equidistant. (Posidonius, first century B.C.) [2]. If a line intersects one of two parallels, it also intersects the other. (Proclus, 410-485 A.D.) [3]. Given a triangle, we can construct a similar triangle of any size whatever. (Wallis, 1616-1703.) [4]. The sum of the angles of a triangle is equal to two right angles. (Legendre, 1752-1833.) [5]. Three non-collinear points always lie on a circle. (Bolyai Farkas, 1775-1856.).i   Geometers failed to prove the fifth postulate. It was a fortunate failure, however. It led them to a whole new world – a world much more exciting than a securely founded Euclidean world.  Geometers’ favourite tool, in their search for a proof for the fifth postulate, was reductio ad absurdum. Their plan was to show that absurdities – that is, contradictions – would invade geometry if the fifth postulate were false. Accordingly, they investigated the consequences of adding the negation of the fifth postulate to the other four, and on numerous occasions, announced a proof of the fifth postulate. Time after time, however, either the alleged absurdity  i Coxeter (1998, p. 2). None mentions ‘indefinite’ nor seems less self-evident than the first four postulates. The first, second and fourth do not, however assert the existence of a geometric construction, either. The third and fifth ones seem as fine as the first four postulates, at least to me. 54  turned out to be a merely counterintuitive result – not a real contradiction – or, some equivalent of the fifth postulate was implicitly assumed in the alleged proof.i In the course of these failed attempts, in addition to the various equivalent of the fifth postulate, several counterintuitive geometric propositions, the consequences of adding the negation of the fifth postulate to the first four postulates in the Elements – that is, what turned out to be theorems of non-Euclidean geometries – were accumulated. The quest for a more secure foundation for Euclidean geometry thus failed and with that the trust in the certainty promised by rational/a priori disciplines.           All that is solid melts into air Although non-Euclidean geometry has many forerunners – that is, those that contributed to its discovery through their attempts at proving the fifth postulate – a few are credited with being its founders among whom are Carl Friedrich Gauss (1777-1855), Ferdinand Karl Schweikart (1780-1859), Franz Adolf Taurinus (1794-1874), Nikolai Ivanovich Lobachevsky (1793-1856), János Bolyai (1802-1860).ii They called what they discovered/founded by different names – such as, anti-Euclidean, non-Euclidean, astral, logarithmic spherical, imaginary geometry, pangeometry, and the absolute science of space. All of these names, however, refer to what is later called the hyperbolic geometry. A geometry in which more than one parallel from a point not on a line to that line is possible. The presuppositions of the two-dimensional hyperbolic geometry, accordingly, can be  i See Greenberg (1993). ii See Gray (2006) for Gauss’ contribution, Kàrteszi (1987) for Bolyai’s appendix, Papadopoulos (2010) for Lobachevsky’s 1855 work, and Gray (1989 & 2007), Bonola (1912), Sommerville (1919) and Rosenfeld (1988) for the history of non-Euclidean geometry. The title of this section is stolen from the Communist Manifesto. 55  captured in an axiomatic system when Euclid’s fifth postulate is replaced with what usually called the hyperbolic postulate. A postulate which asserts that ‘given a line and a point not on it, there is more than one line going through the given point that is parallel to the given line.’i  The postulates which Euclid puts forward in the Elements determine/describe planar geometric objects – to put it colloquially, straight lines, circles, and their combinations. The natural question, then, is what do the postulates of the two-dimensional hyperbolic geometry determine/describe? Hyperbolic straight lines, circles, and their combinations, surely. But what are these? From a mathematical point of view, a perfectly reasonable answer could be ‘abstract mathematical entities determined/described by those particular postulates.’ However, the question remains, if the hyperbolic geometry is a geometry on a par with Euclidean geometry, what are these hyperbolic objects, intuitively? In what sense is hyperbolic geometry geometry? In 1868, in two influential memoirs, the Italian mathematician, Eugenio Beltrami (1835 – 1900), took up these questions.ii    i Cannon et al. (1997) and Stahl (1993). Like in the Elements, these postulates are not complete, of course. See Hilbert (1899/1971) and Hartshorne (2000) for rigorous axiomatic treatments of Euclidean and non-Euclidean geometries. Two-dimensional hyperbolic geometry, like its Euclidean counterpart, can be extended to three and higher dimensions. For the sake of simplicity, however, as before, I stick to two dimensions.  ii His work is often considered as a proof for the relative consistency of (two-dimensional) hyperbolic geometry by providing several models for it, including the pseudosphere. For Beltrami’s 1868 memoirs with commentaries see Stillwell (1996). 56  Models of the hyperbolic plane                Like the Euclidean (straight) line, which is the shortest distance between two points on the plane, the (straight) hyperbolic line can be thought of as the shortest distance between two points on the pseudosphere.i                                                  Figure 3.3 Geodesics on the pseudosphere A hyperbolic circle, like its Euclidean counterpart, is the locus of points on the pseudosphere equidistant from a given point on it. The hyperbolic geometry could be seen as the geometry of the pseudosphere; like Euclidean geometry which is the geometry of the plane. The pseudosphere is, however, not the only model of the hyperbolic postulates, and not even the best one. Quite the contrary: the pseudosphere does not instantiate all the properties of what is determined/described by the hyperbolic postulates – that is, the hyperbolic plane. It has, for instance, a boundary while there is nothing in the hyperbolic postulates that necessitates it. The pseudosphere, in fact, instantiates only part of the hyperbolic plane.ii   i That is, the surface of revolution of a tractrix. A tractrix is a curve studied since the seventeenth century whose defining property is that the distance between a point on it and the intersection of the tangent at that point with the asymptote of the curve is constant; see Stillwell (2010).               ii In fact, Hilbert proved that no regular smooth isometric immersion of hyperbolic plane in three-dimensional Euclidean space is possible.   57   Poincaré’s and Klein’s disks and Poincaré’s half-plane are, in a sense, better models of the hyperbolic plane.i  Points belonging to a particular subset of the Euclidean plane – that is, points included in but not on a Euclidean circle – constitute both Poincaré’s and Klein’s disks. However, unlike the pseudosphere whose metric is that of the Euclidean space restricted to the surface of the pseudosphere, Poincaré’s and Klein’s disks have their own metrics that determine their straight lines.ii In Poincaré’s disk, circular arcs orthogonal to the boundary of the disk and its diameters are considered as straight lines, and in Klein’s disk, parts of Euclidean straight lines contained in the disk are so considered – as illustrated.iii   Figure 3.4  Poincaré’s and Klein’s disks  i Although named after Poincaré and Klein, these models are first constructed by Beltrami; see Stillwell (2010). ii A metric is a function that assigns a distance to any pairs of points. The metric for Poincaré’s and Klein’s disks, ds2p and ds2K, are 4 (dx2+dy2)/(1-x2-y2)2 and [(dx2+dy2)/(1-x2-y2)]+[ (xdx+ydy)/(1-x2-y2)2], respectively; Cannon et al (1997). iii In addition to lines, circles on both Poincaré’s and Klein’s disks are also illustrated as well as the pseudosphere mapped to Klein’s disk and the construction for mapping between Poincaré’s and Klein’s disks.  58  Another useful model of the hyperbolic plane is the half-model plane – that is, a part of the Euclidean plane bounded by an arbitrarily removed line. Half-circles and half-lines orthogonal to the boundary, as illustrated below, are the straight lines; circles remain circles in the half-plane.i  Figure 3.5 The half-plane The importance of these models is twofold. On the one hand, they put the hyperbolic geometry on the same footing as Euclidean geometry, logically. That is, assuming that Euclid’s postulates are consistent, the existence of such models implies the consistency of the hyperbolic geometry and therefore the independence of the fifth postulate from the first four ones. On the other hand, it is crucially important not to forget that these are Euclidean models of a non-Euclidean geometry. In each model Euclidean objects – for instance, half-lines and circles in the half-plane model – are taken as the referent of the term ‘straight-line’ in the statements of the hyperbolic geometry.ii Accordingly, with respect to the models, there is no conflict between the assertions of these two  i To be accurate, there are three kinds of circle; also, circles’ centres seem misplaced on the half-plane. The half-plane’s metric, ds2h, is (dx2+dy2)/y2 when y-axis is orthogonal to the boundary. The mapping between different models is also illustrated in the picture.  ii Here ‘Euclidean objects’ is used in a broader sense than ‘constructible according to Euclid’s objects’ and is meant to refer to the geometric objects live in Euclidean plane and space – that is, some specific subsets of Euclidean plane and space.  59  geometries; they are merely about different things – that is, they have different meanings.i Let me, thus, conclude this section by emphasizing that Euclidean half-lines and circles are not hyperbolic straight lines; they are Euclidean truth-makers of the hyperbolic postulates. Straight lines of the hyperbolic geometry, whatever they are, are straight, pure and simple in whatever sense Euclidean lines are so. The hyperbolic geometry is a geometry on equal footing with the old good Euclidean one; its objects are autonomous – at least in whatever sense in which Euclidean objects are so – not weird concoctions of Euclidean objects. Built from something or nothing, the hyperbolic geometry is a new universe which we shall visit briefly now.ii A brave new world Let us try to find out what it is like to live in the hyperbolic plane by asking H, an imaginary, two-dimensional, hyperbolic friend.iii  If H is asked, for example, how H felt about a very long walk, on a path with a fixed direction, taken with a dog, H would say “it was not very exciting. Nothing remarkable happened. We simply went on for a long time before deciding to come back.” H’s experience, despite what H thinks, is indeed exciting. Since if someone, E, observed H and the dog  i After all, the existence of models proves the consistency.   ii And again, in chapter seven.  iii The title of this section is stolen from the title of a novel by Aldous Huxley, and its content from Richter-Gebert (2010, pp. 465-8). Needless to say, we speak metaphorically and the (mathematical) hyperbolic plane itself is a mathematical object and could not be visited.  60  from the Euclidean vantage point, E would have seen them getting smaller and smaller as they approached what would seem to E as a limiting point.i  Figure 3.6 Walking in the hyperbolic plane However, as together with H’s body, dog, shoes, and ruler, everything shrinks by the same amount, H would perceive nothing of this change of size. H would find the experience ordinary, not very exciting. This fact, however, raises an interesting question, the answer to which offers remarkable freedom in the geometric description of physical phenomena: In this little story there are two observers: [H] inside the [hyperbolic plane] and [E] looking from the outside. Both describe the same situation in different terms. [H] from the inside observes an infinite space. The person from the outside observes [H] constantly shrinking within a finite space. Both descriptions are perfectly legitimate. [H’s description] describes the inner geometry of the space [she lives] in. The person outside describes the scenario he sees in terms of the geometry he lives in (and perhaps as an irony of fate his world is also embedded in some strange way into a larger space he does not know about). [Does H] as an inhabitant of the [hyperbolic plane] have any chance to find out that the world [she lives] in is indeed [hyperbolic]? Does this question even make sense? In a way yes, in a way no, and a good answer to these questions becomes philosophical sooner or later. ii   To put it differently, the fact that H does not find her experience exciting is itself exciting because it implies that no perceptual experiences/qualitative considerations can help to distinguish the  i With their legs shrinking, and thus their step size getting smaller and smaller, every single step they took would take them increasingly smaller distances forward.  ii Richter-Gebert (2010, p. 466). 61  hyperbolic plane from the Euclidean one.i What about quantitative measurements. Can H, who lives inside the hyperbolic plane and uses her own measuring devices, gather enough geometric data – that is, length and angle measurements – to determine the geometry of her plane?  As it turns out, neither measuring distances nor measuring angles alone will do.ii Measuring distances and angles together, however, leads H to a geometry different from that of Euclid.iii Here is an example of an experiment that one can perform from within one’s world and with one’s own measuring devices:  iIn this example, only translations are considered. Adding rotations, however, would not make any difference. A full turn on the hyperbolic plane brings you back to your initial location.  ii See Richter-Gebert (2010, Chap. 21). iii That is when it is done using H’s measuring devices. However, what distinguishes H’s from E’s measuring devices – that is, their being in a certain relation with H’s body – is an empirical fact, accidental but not necessary. As we will see later, it is in this sense that Poincaré says the question of choosing a geometry – both for the (perceptual) representation of the external world, a choice which is decided by natural selection and also for the (scientific) representation/description of (some aspects of) the external world, a choice which is decided by physicists – is a question of convenience not truth. Convenience but not convention. We see the world by Euclidean ‘eyes’ not as a matter of convention but convenience. We use the Euclidean metric in classical mechanics not as a matter of convention – in the sense that we use the metric or imperial system –but as a matter of convenience. These issues have been conflated with Poincaré’s thesis of the conventionality of geometry – which are led to all sorts of confusion.      62  First construct a device with which you can measure a right angle (perhaps take a piece of paper, fold it and fold it again so that your first fold comes to lie on itself. After unfolding the paper you see four right angles). Put a pin at your start position, then choose any direction and walk 1000 steps straight in this direction. Make a right turn by exactly 90◦, walk 1000 steps straight, make a right turn again, walk 1000 steps, make a final right turn again, and finally walk once more 1000 steps. After this procedure, where are you? In Euclidean geometry you would exactly end up at the position you started. In the [hyperbolic geometry] you would end up at some different place.i  We have, now, seen different Euclidean models of the hyperbolic plane, and even visited imaginary inhabitants of a world which might seem weird to us, and gained some familiarity with it. But, H could do the same. She could build different hyperbolic models of the Euclidean plane and use her (hyperbolic) geometric intuitions/tools to develop some understanding of /feel for Euclidean geometry.ii  One geometry might seem unnatural/weird to someone whose spatial intuition and geometrical training are based on the other geometry. This subjective preference, however, should not count against either geometry.iii Not only logically, but also, in a sense, conceptually and perceptually, the hyperbolic geometry is on the same footing as Euclidean geometry.iv   i Richter-Gebert (2010, p. 467). ii For instance, see A disk model for Euclidean geometry.  iii Here is a more familiar case: ‘Grue’ and ‘bleen’ are definable in terms of ‘green’ and ‘blue,’ but seem unnatural to green/blue-speakers, and vice versa.  iv I have only discussed one of two planar non-Euclidean geometries, the hyperbolic geometry, here. The same, however, is true for the other possible one, the elliptic geometry, in which parallel lines do not exist.   63   Demoting Euclidean geometry to one among equals, for one thing, makes the Kantian version of the apriorist philosophy of geometry untenable while apparently promoting its rival empiricist account.  This is important because the Kantian version of apriorism is one of the two accounts of geometry that Poincaré presents his account of geometry against – the other one is, of course, empiricism. Furthermore, since the fact that there is more than one geometry plays a crucial role in Poincaré’s arguments, when evaluating Poincaré’s philosophy of geometry, it is of crucial importance to correctly identify where and how an argument essentially relies on the existence of non-Euclidean geometry.   However, although not its Kantian version, an apriorist account of geometry can survive the discovery of non-Euclidean geometries.i It cannot be assumed, thus, that the discovery of non-Euclidean geometries helped to mitigate philosophical confusion regarding the ontology and epistemology of geometry by making one of the two rival accounts – that is, the apriorist account – untenable. On the contrary, in addition to bringing to the front the old questions and concerns ever more forcefully, it incites new ones – remarkably, it forces both mathematicians and philosophers to consider the question of the applicability of geometry, which is often mistakenly disguised as the question of the truth of a given geometry. The philosophical debate about geometry, according, continued more vigorously and brought the need for a comprehensive account of geometry – a philosophy of geometry – to the fore. In the following section, taking stock of our long journey so far, we will see how a philosophy of geometry emerges as the result of struggling with four main questions.   i See the appendix A.  64  The question of ontology and epistemology of geometry The main philosophical question about geometry with which the philosophers struggled at the early stages was the question of ontology – that is, the true nature of geometric objects. Plato and Aristotle noted the peculiar status of geometric objects when formulating their account of ontology. Their account of epistemology could – or they so thought – explain how it was possible to know objects of that particular nature and also the nature of such knowledge. In other words, the ancient Greek philosophy of geometry strived to provide answers to the following questions.  1. What kind of objects are geometric objects?   2. What can be known about geometric objects? How is this kind of knowledge possible?  What is the epistemic status of this kind of knowledge?  In their account, geometric objects were seen as autonomous objects, and the subject of the science called geometry. No genuine and consequential connection between the behaviour physical and geometric objects were assumed. There had been no significant changes in the conception of geometry, as far as it concerns us, until the advent of the early modern era.  The question of psychology and applicability of geometry In concordance with the changing philosophical outlook, the conception of geometry started to evolve. On the one hand, the idea of the (mathematical) space, as the subject matter of geometry, started to be developed. The ever-increasing use of geometry in describing natural (physical) phenomena, on the other hand, helped to forge a strong connection among geometry and mechanics, and geometry and (physical) space. In addition to questions regarding the ontology and epistemology, moreover, some philosophers started to pay attention to the (psychological) questions regarding the origin of geometric concepts and how they are acquired. It was noted that 65  the answer that one dominant tradition in philosophy, rationalism, provided for these questions, among other things, would make the applicability of geometry in the natural sciences a mystery. The question of applicability of geometry – and more generally, the relation between geometry and physics – became a significant philosophical question, accordingly. Kant, who based his philosophy of geometry on the assumption of the possibility of synthetic (a priori) judgment is a notable case in point. In short, in addition to the questions that ancient philosophers faced, as the result of the development surveyed in the second chapter and the previous one, the following questions were added to the list of the questions with which philosophers of geometry had to struggle.  3. How are the geometric concepts acquired?   4. How can one explain the applicability of geometry? How is it possible to use geometry  to describe/explain the natural world? What is the nature of the relation between physics  and geometry?  Once it turned out that, in addition to Euclidean geometry, there are other geometries, the question of which geometry accurately describes the (physical) space is also added to the questions discussed in the fourth group, which I call methodological questions.   The primary goal of this work, which will be achieved in the third part, is presenting Poincaré’s philosophy of geometry – that is, detailed answers to these questions. We will see that a close reading of Poincaré’s works on geometry reveals that Poincaré’s philosophy of geometry is concerned with geometry as a mathematical science. Questions regarding the nature of geometry – that is, its defining characteristics /what distinguishes geometry from other branches of mathematics – and the nature of its axioms – that is the epistemic status of those geometric propositions from which all other geometric propositions follow logically motivate Poincaré's investigation. Poincaré, however, takes questions regarding the psychology and application of 66  geometry in physics, too, very seriously. Poincaré’s account of the origin of spatial intuitions and geometric concepts and the role and the (epistemic) status of geometry, when used as the language/framework for theories of physics, becomes, in fact, a crucial constituent of his philosophy of geometry, at least partly. For one thing, it is so because Poincaré’s answers to the ontological and epistemological questions put him, prima facie, at a disadvantage relative to his apriorist and empiricist rivals. Whatever problems their accounts had, they had plausible theories to explain the origin of geometric concepts and the applicability of geometry. Thus, only if Poincaré could offer a plausible explanation of the origin of geometric concepts and the application of geometry compatible with his ontological and epistemological account, his philosophy of geometry would have any chance to be taken seriously.   Put it differently, on my interpretation, rather than merely proposing an enticing but ultimately confused or mistaken epistemological category, Poincaré has a full-blown philosophy of geometry by a preview of which I shall now conclude this selective survey of the history and philosophy of geometry. I leave most of the arguments/justifications for my claims for part III.         67  Chapter 4: Poincaré’s philosophy of geometry in a nutshell The iceberg and its tip Poincaré’s philosophical investigation of geometry begins with two simple observations: the conceivability of non-Euclidean geometries and the exactness of geometric statements. From these observations, Poincaré derived his (in)famous epistemological conclusion which brought him to philosophers’ attention more than anything else that he has done in his remarkable career.  The question of the epistemology of geometry Poincaré argues that geometry, as a mathematical science, neither is a priori nor a posteriori science. Geometric axioms cannot be empirical truth because unlike the latter, which is approximate and subject to revision, they are exact. They cannot be a priori either because were they so, non-Euclidean geometries would not be conceivable. Poincaré, therefore, claims that ‘the axioms of geometry are conventions’ – a claim which I call (CG). For Poincaré, (CG) is –on its own and considered in isolation – rather an uninteresting and non-controversial claim about which he writes that “I do not insist further; for the aim of this work is not the development of these truths which begin to become commonplace.”i To the twentieth-century philosophers of science’s ear, (CG) is a bold and captivating claim. As we shall see in this chapter, however, had they paid attention to the following three points, they would have been saved from reading too much into (CG).   i Poincaré (1887, pp. 214-6). 68   First, by deeming the axioms of geometry, or any propositions for that matter, conventions, Poincaré means that neither are they empirical nor a priori.  That is, neither is the fact of the matteri nor the presupposed conditionsii enough to determine the axioms’ truth value.    Second, the axioms referred to in (CG) in the texts often quoted in the literature on the so-called Poincaré’s geometric conventionalism are left unspecified. Most interpreters, nevertheless, seem to think (CG) is about all geometric axioms. Poincaré’s discussions and arguments pertinent to (CG), however, are only about the axioms that distinguish Euclidean from non-Euclidean geometries – let us call them, parallel axioms – in particular, Euclid’s fifth postulate. There are clear passages and arguments excluding other postulates – that is, the axioms of the neutral geometry/constant curvature spaces which guarantee the possibility of free motion of geometric figures/defining the concept of geometric shapes.iii (CG) thus must be understood as claiming that parallel axioms are neither empirical nor a priori. (CG), thus understood, only claims that the notion of parallelism is neither entirely determined by factual nor a priori considerations. Or, put differently, it acknowledges mathematicians’ highly-constrained freedom in choosing their notion of parallelism. Why would any mathematician, let alone Poincaré who is (in)famous for emphasizing the role of creativity in science – especially in the light of the fact that (CG) does not imply that ‘theories of physics could be expressed in whatever geometric framework deemed  i That is, the information provided by experience/the external objects. ii That is, conditions required by mind/innate brain structures/conceptual or structural relations. iii There is a subtle point to add here, which is irrelevant to my argument here and will be addressed in the eleventh chapter.   69  practically convenient,’ a claim which I call (CPG) – find (CG) on its own an extraordinary claim? The close reading of Poincaré’s texts undertaken in the third part of this study reveals that although Poincaré, indeed, agrees with (CPG), he does not equate these claims. He does not derive (CPG) from (CG), either. And, even more importantly, neither (CPG) nor (CG) expresses the essence of Poincaré’s philosophy of geometry, which cannot and should not be reduced to either.   Third, the significance of (CG) is not in its bold claim. Reading too much into it – for instance, by equating/confusing it with (CPG) – does not elevate its status either. It only leads to, as I will argue in the next part of this study, untenable views, enticing but ultimately incorrect which must be rejected. (CG), however, rejects the epistemic claims of both philosophical accounts of geometry which were taken seriously at the time when Poincaré was formulating his own account of geometry. It is only in the context of Poincaré’s philosophy of geometry in its totality that the significance of (CG) could be understood. It is Poincaré’s attack point. Poincaré begins with the epistemic claims of the extant philosophies of geometry –geometric empiricism and geometric apriorism – and argues why they are not viable. It is his first, not the only bone to pick with them, however. He goes on to expose the issues with the accounts that empiricists and apriorists have offered for the ontology, psychology, and methodology of geometry and offers his own. Once he is done, he has refuted the empiricist and the apriorist accounts of geometry in their totality – not just their epistemology of geometry – and proposed his novel and comprehensive account of geometry – not just the so-called thesis of the conventionality of geometry. Let me now go on with presenting my interpretation of Poincaré’s geometric conventionality, which will be further argued for in the final part of this study.  70  The question of the origin of spatial intuitions and geometric concepts  On the one hand, given his epistemic stance – according to which some geometric axioms are neither empirical nor a priori – Poincaré cannot avail himself of the common empiricist theories of abstraction and the apriorist innate mechanisms. On the other hand, Poincaré believes that the (world-supplied) impressions, the (mind/brain-provided/innate) enabling conditions, and (specifically-constrained) choices all play their roles in making these conventions true.  In other words, in the sense used by Poincaré, there are more than free choices to truth-makers of conventional statements – that is, geometric objects and concepts in the case of (CG). (CG), however, might mislead some – as it did mislead Le Royi – to assume that he claims that the truth of the axioms of geometry depends solely on the will/decree of those who adopt those conventions. It is not, thus, hard to see why Poincaré feels obliged to offer a theory of origin compatible with his epistemic stance – a theory of origin in which the role of conventional choices – in addition to empirical impressions, and a priori/innate principles – is revealed.   Poincaré, however, begins his discussion of the origin of geometric concepts with the criticism of the extant theories independent of the above considerations. Poincaré’s criticism of geometric empiricism’s and apriorism’s account of the origin of spatial intuitions and concepts reveals the need for a different account of geometry regardless of his epistemic account. The further other accounts of geometry are undermined, the further Poincaré’s account is motivated. Poincaré’s account of the origin of geometric concepts, in its turn, by highlighting the role of free  i Édouard Le Roy (1870-1954), the French philosopher and mathematician. See part III in Poincaré (1905/1913).  71  choices in the construction of geometric intuitions, concepts, and objects, further supports his philosophy of geometry.  It is essential to keep in mind, then, that Poincaré’s psychological/conceptual account of geometry is an integral constituent of his philosophy of geometry, which both motivates and is supported by it. It has two purposes, destructive and constructive.  Poincaré’s criticism of theories of the origin of spatial intuitions and concepts  There are those who argue that external objects are – and can be only – represented in a spatial frame which exists prior to and independent of those representations. Furthermore, the three-dimensional Euclidean space – that is, the geometric space – is the spatial frame in which the external world is, and must necessarily be, represented. And, there are those who argue that the (idea/concept of) geometric space is constructed, by some sort of construction – abstraction, classification, etc. – from the impressions.   To refute the former theory, Poincaré considers what he calls the pure visual, tactile, and motor spaces – that is, the spaces in which the visual, the tactile, and muscular sensations are supposedly represented, according to the apriorists – and shows that none of them has any of the properties of the geometric space – that is, continuity, infinitude, three-dimensionality, homogeneity, and isotropy. Poincaré, then, shows how the claim that ‘an external object is localized/represented as localized in such a point of space,’ when correctly analyzed, is reduced to the claim that ‘we represent to ourselves the movements it would be necessary to make to reach that object.’ Such representations do not depend on the existence of a frame prior to and independent of impressions. Having argued that (a) the existence of a pre-existing spatial frame is not a necessary condition of the possibility of representation and that (b) external objects are not 72  in fact represented in the geometric space, Poincaré further argues that (c) it would have been impossible to conceive/imagine non-Euclidean spaces had it been the case that impressions are necessarily given – or, external objects are represented – in Euclidean frame. Only in the last stage of the argument, (c), I would like to highlight, does the conceivability of non-Euclidean spaces – accordingly, the discovery of non-Euclidean geometries – play a role. Poincaré’s criticism of the apriorist’s account of geometry does not solely stem from the conceivability of more than one geometry. Nor does it stem from his epistemic stance.   Having dealt with one rival theory, Poincaré takes the other one to task. Empiricists believe that, like other ideas and concepts, spatial and geometric concepts are built from the sensory inputs by operations such as classification and abstraction. Poincaré, however, shows that the principles of classification – that is, the criteria according to which impressions must be organized so that the resultant constructions give rise to spatial and geometric concepts – themselves cannot be derived from the impressions.  Sensations by themselves have no spatial character. [….] Our sensations differ from one another qualitatively, and they can therefore have no common measure [….Thus in order]  to classify sensations according to their character, and then to arrange those of the same kind in a sort of scale, according to their intensity […] the active intervention of the mind  […. the] object of [which is] to refer our sensations to a sort of rubric or category [must]  pre-exist […] in us.i  Poincaré, thus, concludes that in order to form spatial ideas, in addition to empirical ones, innate/a priori principles are also required. Poincaré’s criticism of the empiricists’ account of the  i Poincaré (1898, pp. 1-3). 73  origin of spatial and geometric ideas, so far, has nothing to do with either his epistemic stance or non-Euclidean geometries. Having rejected both accounts, Poincaré offers a new one.  Poincaré’s theory of the origin of spatial intuitions and concepts        Poincaré’s discussion regarding his theory of the origin of spatial intuitions and geometric concepts and objects is extremely obscure – in particular, ontological and psychological considerations are often conflated. Most commentators, as we shall see in part II, simply skip this part of his discussions entirely – which, given the space and effort that Poincaré allocated to it, raises serious questions about their interpretation of Poincaré’s view. A few who did not entirely ignore this part of Poincaré’s discussion mostly misconstrued it.i No one, as far as I know, has fully realized the crucial role that the so-called psychological discussion plays in Poincaré’s account of geometry. Geometry: the senses, the mind, and the will  Given that the notion of convention used in (CG) is a peculiar one – that is, the freedom in adopting conventions is highly restricted – Poincaré has to offer a theory of the origin of geometric concepts and objects – that are the truth-makers of geometric propositions – and an ontological account of them which allows freedom within a highly-constrained domain in their construction.  i Ben-Menahem (2006), for example. Vuillemin (1972) is a notable exception .On his interpretation, it is in these discussions where Poincaré suggests a new kind of association in addition to two kinds known to psychologists and philosophers – that is, association by resemblance and by contiguity – and “was the first to clearly answer the question, “How can we construct space from spaceless impressions?” (Ibid. p. 162).  74  Furthermore, Poincaré’s criticism of the extant theories of the origin of geometric concepts makes it clear that neither are these concepts exclusively empirically abstracted nor are they exclusively mind-imposed. In other words, Poincaré has committed himself to a view according to which, in addition to the constituents supplied by (i) the external world and (ii) mind/the built-in brain mechanisms, the truth-makers of geometric propositions have another kind of constituents, let us call them the conventional constituents. Poincaré’s criticism of the extant philosophies of geometry – the available epistemic, psychological, and ontological accounts of geometry – thus, commits him to a theory of the origin of geometric concepts and an ontological account of geometric objects which can accommodate (i) the empirical, (ii) the mental, and (iii) the conventional elements as their constituents. The cogency and viability of Poincaré’s epistemic account of geometry – and more generally, his account of geometry in its totality – crucially depends on his success in providing the required (so-called) psychological and conceptual/ontological accounts to which I now turn.  It begins with the senses: the empirical contributions The most primitive empirical fact to which Poincaré traces back the genesis of geometry is the fact that the impressions are subject to change. Poincaré begins his investigation with the set/space of all these changing sensations, which he calls the sensible space and has already shown that it has nothing in common with the geometric space. Then, step by step, by introducing more and more classifications, he constructs sets/spaces which have increasingly rich structures. The geometric space is the last structure in this series. As it is sketched in the following exposition of Poincaré’s discussion of the origin of geometry, these constructions, on many occasions, take advantage of a priori and conventional elements. 75   In the first/the most primitive step, the sensible space is divided into two sets. (CS) whose members are what Poincaré calls changes of states or alternations of characters and (CP) whose members are what he calls changes of positions or displacements. The changes in impressions caused when a plant grows belong to the first kind and those caused by an apple dropping from a tree belong to the second kind.  Figure 4.1 Change of state vs. change of position What distinguishes (CP) from (CS) is that the former could be reversed – in Poincaré’s own terms corrected or compensated – by the observer’s voluntary movement which is accompanied by muscular sensations. Taking advantage of these empirical facts, Poincaré thus introduces the first structure into the sensible space: Among the changes which our impressions undergo, we distinguish two classes:(1) The first are independent of our will and not accompanied by muscular sensations. These are external changes so called. (2) The others are voluntary and accompanied by muscular sensations. We may call these internal changes. […. Some] external change, accordingly, can be corrected by an internal change. External changes may consequently be subdivided into the two following classes: 1. Changes which are susceptible of being corrected by an internal change. These are displacements. 2. Changes which are not so susceptible. These are alterations.i  That the sensible space can be thus structured is not an a priori fact. Poincaré shows that the possibility of such classification depends on several contingent facts. Among them – in addition to  i Poincaré (1898, p. 7). 76  the mental power of the observer – are the following:  that there are objects that, to a good approximation, behave like solids in the environment, that humans can freely move, and that the relations among their sensory apparatuses behave approximately like the relations among solid bodies. When Poincaré claims that there could be no geometry were there no solids – or, no possibility of moving for us, for that matter – what he means is that in these cases it would not be possible to reverse one’s aggregates of impressions and thus identify/define the set/space (CP). Accordingly, no creature could evolve to develop spatial intuitions and to invent/discover a science like geometry. He does not mean that geometry as the science of deriving logical consequences would be impossible. Geometry in its logicist’s/formalist’s guise still would have been possible had an oracle gifted humanity with its axioms.     What is discussed above is just one example of the crucial role of the empirical conditions in the genesis of geometry, but it should be enough to illustrate the point. Considered more carefully, the construction of (CP) helps to illustrate another crucial point. It shows that without the intervention of a priori elements, even constructing a structure as primitive as (CP) is not possible. The mind guides it: the a priori contributions The construction of (CP) requires identifying two changes – one internal and one external – with the property that one undoes the changes in impressions caused by the other one. But impressions – like any other physical thing – lack, for want of a better term, the exactness required of the ideal/mathematical objects; so, do the required compensations. Poincaré says that they are 77  approximative. The principles of classification, however, must be exact.i Mind, thus, has to intervene, and hence, the first a priori element enters the scene.  The classification is not a crude datum of experience, because the aforementioned compensation of the two changes, the one internal and the other external, is never exactly realised. It is, therefore, an active operation of the mind, which endeavors to insert the crude results of experience into a pre-existing form, into a category. This operation consists in identifying two changes because they possess a common character, and in spite of their not possessing it exactly. Nevertheless, the very fact of the mind’s having occasion to perform this operation is due to experience, for experience alone can teach it that the compensation has approximately been effected.ii   (CP) is still a vast and unruly set. Its members are too closely tied to the individual details of the movements of individual observers. Furthermore, they are identified by the internal changes correcting external changes. The internal changes correcting external changes are, in turn, identified with muscular sensations. The sameness of muscular sensations, hence, is crucial to cps’ identity. However, neither does it provide a sufficient condition for cps, nor a necessary one. Once again, mind intervenes. Another a priori element – this time the group concept – and, for the first time, a conventional-like element, to be explained shortly, come to rescue and help to tame (CP), according to Poincaré.  As it turns out, those changes in impressions that we are interested in – that is, those we want to recognize as members of (CP) – behave, to a good approximation, like members of a group  i This is one of Poincaré’s assumption but by no means only his. It has many illustrious advocates through history. ii Poincaré (1898, p. 9) 78  – more accurately, the group of the motions of three-dimensional Euclidean geometry.i That this is the case is a contingent fact. It depends on both the way that the world actually is and on the details of our perceptual and conceptual faculties. Poincaré’s parable of imaginary habitats of the temperature world illustrates the former point – their (CP) approximates the group of the motions of three-dimensional hyperbolic geometry. Poincaré’s discussion of the possibility of developing intuitions for a four-dimensional Euclidean geometry – as the result of the lack of suitable harmony between what he calls ‘the effort of accommodation and by the convergence of the eyes’ – illustrates the latter one. That (CP) – as well as (D)ii and the geometric space – has this or that group structure, therefore, it is not a simple/pure empirical fact. Although specific empirical conditions have to be met in order that a subspace of the sensible space turns out to be a group, the satisfaction of these conditions is not enough for the impressions to be thus organized and classified.iii For that to be the case, the following conditions must also be met.   First, those cps which behave approximately like elements of a group must have been treated as if they behaved exactly in that manner. The concept of group employed here is an a priori  i We shall see the technical definition of motions in due course. For our purpose, it suffices to consider rotations and translations, which are the (direct) motions of Euclidean geometry.  ii Where (D), which I call the spatial frame and its members displacements and as we see in what follows, is constructed by identifying certain members of (CP) – that is, (D) =  (CP)/~, where ~ is an equivalence relation that identifies distinct elements of (CP). iii This should not seem like a too controversial claim. It is just a special case of rejecting the Tabula rasa theory.  79  concept and must be supplied by the mind. It cannot be abstracted from impressions. This is so because nothing – neither as a matter of fact or as a matter of necessity – requires a classification of impressions in such a way that makes such an abstraction possible. On the other hand, cps could not have begotten the intuition for spatial notions – or, concepts, such as position, changes of position (motions), which geometry is all about – had not they had such a structure. How could a concept/idea be abstracted from a set which fails to illustrate it? Thus, the group concept is necessary, in the sense that it is a condition of the possibility of geometry – that is, in the sense used in Kant’s transcendental arguments.      Second, those cps which do not behave like elements of a group must have been treated –  and accordingly, have been classified – as if they were composed of one part – as large as possible – that does so behave and another one– as small as possible – that does not. Nothing necessitates this way of classification. It is chosen by what Poincaré calls ‘an artificial convention,’ which we can alternatively understand as a result of some innate neurophysiological mechanism. What justifies such an artificial convention or the existence of such an innate mechanism is, in a sense, its usefulness. But for adopting such a principle for the classification, there would be no reason/cause for dividing the sensible space into (CS) and (CP) in the first place. It is so because there is no natural object that is only susceptible to the change of position and no other change. Take the example of the falling apple, for instance. When it hits the earth, the apple is a bit deformed. The aggregate of impressions the apple on the earth causes cannot be restored to the one which was caused when it was on the tree. Without the aforementioned optional decomposition, even a clear example of the change of position – such as our example of the falling apple – cannot be considered as such. This notion of ‘artificial conventions,’ however, should not 80  be confused with other uses of ‘convention’ – the one we saw before in our discussion of (CG) and those that we shall see later. Let us now continue with the further classification of the elements of (CP). Having the below sketch in mind should help with what follows.    Figure 4.2 From impressions to geometric space (CP) is, first, divided into subclasses (Di) such that the set consists of Dis, D, turns to be a (continuous) group where any cpij in Di could be a representative for Di.i I call (D) the spatial frame and its members displacements.ii Roughly speaking, things that we want to ultimately recognize as motionsiii are represented by several members of (CP). Constructing (D) remedies this redundancy by appropriate identification of multiple elements, which are then represented by a single element in (D).   Figure 4.3 From changes of position to the spatial frame  i Its continuity – the move from the paradoxical concept of physical to mathematical continuity – is another example of mind’s interventions, as we shall see in the last part. ii Poincaré sometimes uses ‘displacement’ and ‘change of position’ interchangeably. I try to use the former for representatives of equivalence classes.    iii We shall see the technical definition of motions in due course. For now, the example of  rotations and translations – the (direct) motions of Euclidean geometry – will do.  81  The fact that the geometric space is homogeneous, isotropic, and unbounded follows from the fact that its mental counterpart, (D), is a continuous group. The crucial point that must be emphasized is that, however, the fact that (D) forms a group – and other pertinent facts –   is “not imposed by nature upon us but are imposed by us upon nature. But if we impose them upon nature, it is because she suffers us to do so. If she offered too much resistance, we should seek in our arsenal for another form which would be more acceptable to her.”i It has, so far, been demonstrated that there are facts – and naturally, there are propositions – which are neither empirical nor a priori.   Now that (D) has been constructed/reconstructed thanks to contributions from the world and the mind, it can motivate the conception of a very crucial geometric concept, the point. As for how it is done, I cannot go into too much detail right now but shall briefly consider it in due course. For the present purpose, it suffices to know that several algebraic concepts and techniques are available that can be used to study the structure of a group. The one that plays a crucial role in our discussion takes advantage of subgroups of a group. (D) has infinitely many subgroups, but they can be classified into several different categories.  One of these categories is the rotative subgroup. A subgroup whose mathematical counterpart is O(3) – that is the group of rotation around the origin of the three-dimensional space. When the spatial frame – or, its mathematical counterpart, geometric space – is structured with respect to O(3) and its conjugates – that is, what that turn to be/will be led to rotations around any points – it appears as the familiar three- i Poincaré (1898, pp. 11-12) 82  dimensional (Euclidean) space consisting of (geometric) points. Poincaré, thus, says that the rotative subgroup is the origin of the notion of the point.  The point which is of the utmost importance to our discussion is that had the spatial frame been structured with respect to other subgroups – or metaphorically speaking, seen through the lens of other subgroups – it would have appeared differently. For instance, seen through the lens of what Poincaré calls the helicoidal subgroup – that is, rotation around a fixed axis – the spatial frame would appear as a four-dimensional space made of lines.   Hence, considerations regarding the origin and the nature of geometric intuitions and concepts, once again, reveal that certain geometric claims – such as ‘space has three dimensions.’ and ‘space consists of points.’ – are neither a priori nor empirical. They are conventional. That is, in addition to empirical and a priori constituents, a third optional element is also involved in the construction of the truth-makers of such claims. I shall, accordingly, turn to this optional element, that plays an essential role in understanding Poincaré’s peculiar version of conventionalism, which is essentially an ontological claim rather than an epistemic or semantic one. Ignoring the so-called psychological fantasies in Poincaré’s work has led to the failure in appreciating Poincaré’s unique and novel brand of conventionalism. Efforts in fitting Poincaré’s conventionalism into the so-called canonical understanding of conventionalism’s mould, as we will see in the second part of this study, has led many authors to attribute untenable views to Poincaré which he did not hold. The ‘will’ chooses one: the conventional contributions  The construction from which the idea of the geometric space arises – that is, the specific manner of organizing sense data – is possible only due to two fortunate ‘choices’ or ‘decisions,’ metaphorically speaking. Furthermore, as we have just seen, the compositionality and 83  dimensionality of the spatial frame and geometric space also depend on making certain ‘choices.’ Some essential aspects of the geometry of the spatial frame and geometric space – that is, those aspects that are expressed by what I called parallel axioms, in terms of which Poincaré choose to formulate his thesis of geometric conventionalism– also depend on certain ‘decisions.’i Poincaré does not spend any time in spelling the difference among them. I think, however, that it is worthwhile to briefly consider this issue.  The first kind of choiceii differs from the second oneiii in that adopting the alternative option in each case has vastly different consequences. In the former case, adopting alternative options, as we have seen, defeat the purpose of constructing (CP) – or (D) – in the first place, which, in a sense, means the alternative is not much of an option. They are optional not in the sense that there is more than one viable choice available, but in the sense that neither the way that  i I do not intend to imply that these are conscious choices or voluntary decisions. Accordingly, the terms such as ‘will,’ ‘choice,’ and ‘decision’ should not be understood literally. What is intended, for example in the first sentence, is simply that the sense data could not have begotten the idea of geometric space had certain innate cognitive/neurophysiological mechanisms not existed. ii That is (a) using the rule of group as the principle of classification for certain type of amalgamations of impressions, and (b) using a certain rule of decomposition – that is, what is called ‘artificial convention’ in the above discussion. iii That is (a’) using this or that subgroup to impose a given structure on the spatial frame or its mathematical counterpart, and (b’) using the invariance of this or that group to define the notion of geometric identity. 84  the world actually is nor a priori conceptual considerations require them.i In the latter case, however, there is more than one viable choice. Given the way that the world actually is, the choices that have been made are – in some relevant sense such as simplicity, practicality, convenience, and so on – preferable to the alternatives.ii   The optional/conventional constituent of geometric objects is, therefore, not a mysterious thing – a creature of psychological fantasy. It is different from the empirical and a priori constituents in that it is one among other possibilities. If one wants to put in terms of the form and matter, the a priori element is the form, and the empirical element is the matter. The conventional element is the possibility of putting the form and matter in more than one way together to build an object. As a result, we can have different things, say spatial frames, that can serve our purpose. One might be preferable to the other, however.    Given the way that the world and we are, a three-dimensional space with the Euclidean metric is preferable to us. Were we to live in the sphere of Poincaré’s parable, we would prefer a three-dimensional space with the hyperbolic metric. In a possible world exactly like the actual one  i Compare it with impressions of red and blue which can be naturally classified thus because they are originated as such – that is, by different cone cells. Metaphorically speaking, colours wear their kind on their sleeves. Things are different for changes of position and state; they do not wear their respective kind on their sleeves.  ii For instance, a three-dimensional space which is made of zero-dimensional elements, points, is – I assume – preferable to/simpler than a four-dimensional space whose most basic element is a line. That Euclidean geometry is preferable/easier than the hyperbolic one not only would probably be attested by those who studied both but also by the fact it took humanity two more millennia to come up with the latter. 85  where, for some reason, say an evolutionary accident, the helicoidal subgroup plays the role that the rotative subgroup plays in ours, a four-dimensional space is preferable. For its habitants, lines are intuitively the basic elements; and, that you need to intersect two lines to get a point is as intuitive to them as it is to us that to get a line you need two points.i   In each example, the empirical element,ii the a priori element,iii or the conventional elementiv  which constitute the truth-makers of geometric propositions – that is, geometric objects, concepts, and relations – are different.  Geometric terms, thus, refer to ontologically different entities in each case. For Poincaré, the referents of geometric terms are not some primitive appearing in some axioms that are chosen on whim or wisdom. Nor are they physical entities. What they are is the subject of the next brief section.     The question of the ontology of geometry  Throughout his numerous and on occasions lengthy discussions, Poincaré never clearly distinguishes mental constructions – aggregates of impressions constituting the class called changes of position, for example – from its mathematical counterparts – for instance, translations and rotations. Worse, he moves from discussing one to the other with such an ease that his subtle moves are hardly noticeable. It might even seem that Poincaré considers them the same. This is,  i Incidentally, the principle of duality, which was known but seemed mysterious to geometers, in a sense, states the equivalency of two cases. ii That is, the specific condition of the actual impressions. iii That is, the innate cognitive structure and/or the conceptual presuppositions. iv That is, the actual choice for the optional part which could belong to either category – the empirical or the a priori.  86  however, not the case. Mental constructions are subjective and belong to individuals. Mathematical objects are not subjective; mathematical propositions are not about mathematicians’ mental constructions. The spatial frame – even though, thanks to generalization and idealization, is, more or less, like the geometric space – is not identical to the geometric space. The problem of filling the gap between mental constructions and what they represent is a general philosophical problem and is neither peculiar to the ontology of geometry, nor is peculiar to Poincaré’s philosophy of geometry, in particular. In any case, Poincaré does not offer any clear answer to the question of the ontology of mathematics in this regard.   For what it is worth, here is how, I think, one can speculate as to how Poincaré might deal with the problem.i Pointmo – that is, the most basic element from which the spatial frame consists of – is a mental sign/symbol/place holder for a rather large class of impressions which are put together through several stages of classification. It signifies the current place of an individual. Any other constituent of the spatial frame, pointmt/pointmr, signifies the place of that individual after he or she changes his or her place via a Euclidean translations/rotations. Pointmt/Pointmr is a mental sign/symbol/place holder for a rather large class of impressions – more accurately, changes in impressions that are qualified as a change of position as the result of a movement initiated at Pointmo and eventuated at Pointmt/Pointmr – which are put together through several stages of classification. The spatial frame consisted of all (logically) possible such points. Now, Pointg is the  i The subscripts m, g, o, t, and r are for ‘mental,’ ‘geometric,’ ‘origin,’ ‘translation,’ and ‘rotation,’ respectively. ‘Pointmo’ refers to the mental point at the origin and ‘Pointmt’ to its translation by t.   87  mathematical counterpart of Pointm.  Pointg is a given, but not a special, point of the geometric space. The geometric space, using mathematical jargon, is the orbit of the group of Euclidean motions. Geometrical objects are certain well-defined parts of the geometric space.   What is important here is not such speculations, however. The crucial point is that none of the points of contention – either with regard to Poincaré’s philosophy of geometry or the interpretations of his version of conventionalism – depends on such details. What is more relevant to our concerns is Poincaré’s view of the nature of geometry. The nature of geometry as a (mathematical) science Geometry, for Poincaré, is a mathematical science. As a science, geometry is in the business of finding general patterns among particulars. It is not about this triangle or that circle; triangles, circles, and their properties are what geometers are interested in, for example. As a mathematical science, geometry deals in exact and certain generalities. What makes it possible to jump from particular statements to general ones, in empirical sciences, is (empirical) inductive reasoning, which can only produce approximate and probable generalities. Geometers, like other mathematicians, must have a way ‘to proceed from the particular to the general.’ In addition to ‘passing from the finite to the infinite,’ their method must also produce not merely approximate and probable, but exact and certain generalities. Mathematical induction – that is, the possibility 88  of repeating indefinitely the same operation – fills the bill. Mathematics, Poincaré claims, owes its certainty and generality to mathematical induction.i  The possibility of indefinite repetition of its operations, thus, must be considered among defining characteristics of mathematics. Mathematics is also an exact science. Exclusive use of constructive methods – that is, constructing the objects of the science within that science– is a necessary condition for being an exact science, according to Poincaré.19  Free mobility, thus, plays yet another essential role. It makes geometry a mathematical science. However, without the existence of good enough solid objects and advanced enough sentient being which retain their shapes there would be no cause/reason to develop geometric intuitions. This makes free mobility an empirical condition for these creatures to develop geometric intuitions. Some of them, much later, invented/discovered geometry, indeed.   Free mobility is, also, an a priori/conceptual condition for the possibility of geometry. What gives a geometric object its identity is its shape – that is, that which remains unchanged in motion. It would not be possible to provide geometric objects with their criterion of identity unless the possibility of free mobility is presupposed. In this sense, then, free mobility is a priori/conceptual condition of geometry.  i Note that this notion of mathematical induction is broader than its standard definition of logic textbooks, in which it is taken to be equivalent to the principle of well-ordering of natural numbers, which states that every non-empty set of positive integers contains a least element.   89    Lastly, motions – that is, those operations that leave shape and size unchanged – must be applicable indefinitely otherwise geometric claims lack the required generality. In other words, free mobility is a global condition – that is, it must be satisfied everywhere. The global free mobility requires geometric space to be unbounded, homogenous, and isotropic – or equivalently constantly curved.i    Geometry is the science whose subject matter is spaces of constant curvature. Roughly speaking, the defining property of an n-dimensional space of constant curvature, for our current purpose, is the existence and behaviour of specific maps, M and d, where d is certain real binary functions, called the distance function, such that d(x,y) = d(M(x),M(y)), for any two points chosen from a set of n+1 points of the space.ii In plain English, M sends any point of the space to exactly one point of that space while leaves unchanged the value of a certain real binary function applied to any two points among n+1 points of the space. M is called a motion. Motions form a group, called the isometry group of that space.   For example, the good old plane of Euclidean geometry is a space with constant (zero) curvature. The distance function for the plane is determined by the intuitive notion of distance – some measure for the separation of two points on the straight line – and the Pythagorean theorem.  i That is, to have constant curvature. That having constant curvature everywhere is the necessary and sufficient condition of free mobility is a mathematical fact due to Riemann, Helmholtz, and Lie.  ii For any three points, d(x,y)≥ 0, d(x,y)=0 iff x=y, d(x,y)=d(y,x), and d(x,z) ≤ d(x,y)+d(y,z). M must be defined on the entire space and be injective and surjective.   90  Translations, reflections, rotations, and their combinations are those transformations of the plane that leave distance unchanged. Line-segments, circles, and what that could be built using them are geometric shapes.  Figure 4.4 Euclidean transformations Another familiar example is three-dimensional Euclidean space which models physical space. And these two were the only examples known to humanity until the discovery/invention of non-Euclidean geometries. The proof of the relative constancy of these new geometries revealed that Euclidean spaces, distance, and motions are not only extensions of these concepts. The hyperbolic plane, distance, and motions are, for example, as good as them, conceptually and mathematically. These geometries, nevertheless, seemed counterintuitive. The fact that they do not lend themselves to imagination, also, counted against them. Helmholtz’ mirror world and Poincaré’s temperature world parables undermined the prejudiced status of Euclidean geometry. All in all, it became, thus, evident that our naive understanding of the notion of shape and size and the change of positions – that is, transformations that leave shape and size unchanged – can be expanded to include non-Euclidean shapes and transformations which leave them unaffected.     Doing so requires us to figure out what were the underlying intuitions that we once thought were captured and formalized by Euclidean geometry. That is, we need to find out what intuition the concept of a geometric shape is supposed to capture, which brings to the fore the importance of Poincaré’s discussion of the inception of spatial intuitions and geometric concepts 91  even more.  Free mobility, as the result of such conceptual analysis, emerges as a fundamental requirement. It turns out to be the necessary condition for the possibility of defining/constructing geometric concepts/objects. This implies that geometric axioms such that their negations contradict the possibility of free mobility are, in this sense, a priori and those that their negations are compatible with the possibility of free mobility are not a priori. The axioms of the neutral geometry – Euclid’s first four postulates, roughly speaking – are among the former and the fifth postulate belongs to the latter. The question of the epistemic status of Euclid’s parallel axiom – more generally, parallel axioms – thus reappears, for which we now have a plausible answer.  The conception of the nature of geometry thus gained, also, sheds light on two crucial issues. First, the foundational question of the relation between ‘the notion of space and the first principles of constructions in space,’20 which according to Riemann remained elusive ‘from Euclid to Legendre,’i is, finally, answered in Poincaré’s philosophy of geometry.ii    Second, it explains why, when discussing geometry, Poincaré – and even Riemann himself – does not consider those so-called ‘geometries’ of variable curvature. Having deprived of using mathematical induction, mathematicians can only derive the consequences of defining the length of a curve with this or that (admissible) formula. Put it differently, taking advantage of the genuine mathematical method of proof, which Poincaré calls demonstration, based on mathematical  i Adrien-Marie Legendre (1752-1833) was a great French mathematician whose Éléments de géométrie (1794) was taught for about a century.  ii Given our discussion of the ontology and psychology of geometry, anyone interested in this question, I think, can see how it is answered. In any case, here is not the place to discuss this issue further.   92  induction, it is possible to produce new generalities when they are doing geometry – that is, investigating spaces of constant curvature. While when they are studying spaces of variable curvature, what they have in their disposal is only the method of verification, which “is an instrument purely analytic, and incapable of teaching us anything new.”i  That is why Poincaré says these geometries are analytical.ii   To recapitulate: Poincaré’s investigation started with noticing the peculiar epistemic status of some axioms of geometry – that is, the fact that whatever they are grounded in allows for some freedom. This realization along with the shortcomings of the extant theories of the psychology of geometry, among others, led Poincaré to his conceptual analysis of fundamental notions of geometry highlighting the role of the concept of group and free mobility. Poincaré’s theory of the origin of geometry, the nature of geometric objects, and geometry as a mathematical science, in turn, reveal the multiplicity of available constituents of geometric objects, which is not only compatible with the epistemic peculiarity but also explains it.    i Poincaré (1902/1913, p. 37). ii Note that although the study of the geometry of spaces of variable curvature, differential geometry, has now advanced far from its primitive state at the time when Poincaré wrote about it, Poincaré’s point still stands. Those techniques and tools that can be called geometric, in the sense that Poincaré would have approved, could be introduced only artificially, in the following sense. To any point of a differentiable manifold, linear spaces – such as tangent or cotangent spaces – can be attached. However, there is no natural way – that is, determined by the manifold itself – to identify objects in these different spaces. It has to be done extrinsically, for example, by choosing a connection on a fiber bundle.  93   We have now come full circle, and along the way, developed an almost comprehensive philosophy of geometry. Geometry, in its applied role, however, is the language in which physical theories are articulated. To make his account genuinely comprehensive, the relation between geometry and physics is the last issue that Poincaré needs to address. It is this last piece of his work on the philosophy of geometry that captures the attention of twentieth-century philosophers of science at the expense the rest of his philosophy of geometry to the extent that almost none of the subjects I have covered so far can be found in the literature on Poincaré’s philosophy of geometry.  The irony is that it is his least original contribution.  Geometry as a language The scene on which the drama of physics unfolds, so-called physical space, had seemed very much like three-dimensional Euclidean space, at least locally. And since Galileo declared that the book of nature is written in the language of geometry, physicists have formulated their theories in geometrical terms. Objects move and interact with each other and fields. Motions and laws determining them are expressed in geometric terms. For instance, the rate of the change of position of a charged particle depends on its distances from other charged particles. Fields fill the space and change as time passes. Their magnitudes are expressed in geometric terms – such as vectors and tensors. So are the laws to which these fields are subjected. In short, “[f]undamental laws of nature are expressed as relations between geometric fields describing physical quantities.”i     i Novikov and Taimanov (2006, p. xiii). Let us not worry about non-classical theories.  94   Now that there is more than one geometry available, it is natural to wonder which one should be chosen as the language of physics. Such a natural and innocent question is, however, often presented in terms of the truth of a geometry, which can easily send us down rabbit holes if care is not taken. These two questions – that is, the question of formulating/presenting physical theories and the question of the truth of a geometry – are the same only in the context of some version of naïve empiricism, akin to the view we will encounter briefly in the next chapter when discussing Eddington’s interpretation of Poincaré’s view. Such a view, however, is precisely the view that Poincaré rejects in his discussion of the relation between experience and geometry. So do Helmholtz and Reichenbach, for instance. There is, thus, no disagreement between Poincaré and some of his critics, for instance, Reichenbach, who finds calling the view that he advocates ‘conventionalism’ misleading. And he is right. But, what Poincaré means by claiming that some ‘axioms of geometry are conventions’ is entirely different. Twentieth-century philosophers of science are to blame, as I argue in part II, for placing such a view – which Poincaré sometimes calls ‘the relativity of space’ – under the banner of conventionalism and trying to understand it as such.     Not only does Poincaré contend that geometric empiricism is in the wrong to assume that geometric ideas/concepts can be formed merely by organizing the impressions/sensory data based on their own characteristics, but also to assume that geometric claims can be verified or refuted experimentally. The chasm that separates geometrical concepts from impressions makes them inaccessible to experience, and thus impossible to coordinate/identify concrete objects with geometrical concepts in a way that is empirically verifiable. Such a coordination/identification, therefore, must be done by decrees guided by considerations of convenience, as result of which, as a special but significant case, the ‘geometrical’ structure of world, the ‘geometry’ of the ‘physical 95  space,’ becomes a matter of convention; a choice made based on practical considerations, convenience, rather than epistemic reasons. This, however, does not mean or make the axioms of geometry conventional. All it means is that any of geometric systems can be used to describe the world, of course, if it is empirically justified. If it turns out that – as it did in the presence of the strong gravitational field – actual bodies do not behave like rigid bodies of any of the geometries, one has to abandon the geometric – in the sense that Poincaré understands ‘geometry’ – description.   Poincaré’s most important or exciting contribution is, however, not the ‘discovery’ of the fact that one can choose one’s favourite geometric language/framework in order to formulate physical theories not based on epistemological considerations (truth) but practical ones (convenience). Riemann and Helmholtz, for example, had already defended such a view. To the eyes of twentieth-century philosophers of science, however, it was the jewel of Poincaré’s philosophy buried under confused and misguided psychologism of his time, which they threw away. Skimming what was left, then, they found Poincaré’s supposed arguments for his claim according to which ‘the axioms of geometry are conventions,’ by which he supposedly meant ‘physicists should feel free to choose whatever geometry they like,’ and referred to as ‘geometric conventionalism.’ Thus, not only did they misidentify Poincaré’s geometric conventionalism, but also his arguments in its support. But this will be the subject to which we turn in part two, after concluding this part with the following brief note. Poincaré’s conventionalism vs. conventionalism  As Goodman aptly observes, in its everyday usage, ‘convention’ and ‘conventional’ are ambiguous. “On the one hand, the conventional is the ordinary, the usual, the traditional, the orthodox as 96  against the novel, the deviant, the unexpected, the heterodox. On the other hand, the conventional is the artificial, the invented, the optional, as against the natural, the fundamental, the mandatory.”i In its technical usage in philosophy, ‘conventionalism’ appears in almost all areas of philosophy.ii Linguist conventionalism can be traced back to Plato and with regard to money to Aristotle.iii  Money and language, which are two favourite examples commonly cited by philosophers, are not natural in the sense that, in Aristotle’s words, they exist “not by nature but by law (nomos) and it is in our power to change it and make it useless.” They are not subjects of the natural sciences. Neither are property and justice,iv morality,v personal identity,vi ontology,vii and necessity,viii which are among things that some philosophers have argued are conventional.   Another point to note, in differentiating various notions of conventionalism, is to consider what ‘conventional’ is often contrasted with – that is, the natural, the mind-independent, the objective, the universal, the factual, and the truth-evaluable.ix It is often the case that it is not news that the concepts which are the subjects of conventionalist theories are not subjected to truth considerations. It is not the case regarding geometry. Unlike claims regarding language, money, or  i Goodman (1989, p. 80). ii See Rescorla (2019) “Convention” iii For example, see Cratylus (384c-d) and Nicomachean Ethics (V.5.II33a). iv See Hume (1740/2007), for instance. v See Harman (1996), for instance. vi See Parfit (1984), for instance.  vii See Carnap (1937), Goodman (1978), or Putnam (1987). viii See Sidelle (1989), for instance.  ix Rescorla (2019). 97  morality, geometric claims were considered truth-evaluable. The science in which the structure of physical space was studied was, moreover, considered a natural science. It is against this background that Poincaré’s geometric conventionalism – as it is usually construed, or rather as I argue, misunderstood and misconstrued – has been seen as a radical move in philosophy.   Twentieth-century philosophers of science eagerly welcomed the inclusion of geometry – seen as the science of physical space – in the expanding domain of conventional sciences – as opposed to real/natural sciences. The way that dealt with such an expansion is, however, paradoxical. On the one hand, they consider Poincaré’s geometric conventionalism as an important moment in the history of conventionalism, which entails that they consider geometry and geometric claims as radically different from those that had previously have been deemed conventional. Otherwise, Poincaré’s geometric conventionalism must be regarded just as yet another example in a long list of conventionalist theories in philosophy. Given the many supposed issues which Poincaré’s interpreters were busy to find and deal with throughout the past century, moreover, it is then hard to explain why they bother so much with just an ordinary expansion of conventionalism.  On the other hand, as we will see shortly, in the interpretations offered by these twentieth-century philosophers of science, that which distinguishes Poincaré’s geometric conventionalism from other conventionalist theories is left unexplained. Worse, even; such a question is hardly raised.i In any case, nothing in the arguments that these commentators reconstructed to support  i To be fair, it is alluded to, in a limited and misunderstood form, as the question of the difference between Poincaré’s view and Duhem’s. 98  their versions of Poincaré’s conventionalism explains the significance of Poincaré’s work in the conventionalist tradition. A few who noticed this issue – Zahar (1997&2001) and Reichenbach (1928/1958) – did not doubt their understanding of Poincaré. They concluded that Poincaré’s geometric conventionalism did not deserve the attention it had gotten.   On the reading presented above, however, the peculiarity of Poincaré’s geometric conventionalism is not a mystery. Poincaré’s version of geometric conventionalism is grounded in and built on his account of the ontology and the nature of geometry – that is, what most of Poincaré’s commentators deemed a confused and even misguided psychological fantasy.i    Having situated Poincaré’s views about geometry in the context of the long history of the evolution of geometry and its philosophy, I now turn to the reception of Poincaré’s view of geometry in the twentieth-century philosophy of science.          i See Torretti (1978), for example.   99  Part 2           Philosophers and Poincaré: Poincaré in the twentieth century Where major readings of Poincaré’s philosophy of geometry are critically considered. Introduction In philosophical circles, Poincaré is known, more than anything else, for what is called geometric conventionalism, claiming that the axioms of geometry are conventions. A more accurate version of this claim – that is, the claim according to which those axioms of geometry that distinguish different geometries of spaces of constant curvatures are conventions – I called (CG), in part one. There is, however, little agreement as to what (CG) means among the philosophers. Nor do they agree on how Poincaré justifies such a claim.   Some commentators – Schlick (1915), Reichenbach (1928), Carnap (1966), Giedymin (1982), Sklar (1977), Friedman (1995), DiSalle (2006), and Folina (2014), to name a few – believe that by geometry, Poincaré means applied geometry – that is, the theory that describes physical space. Others – Black (1942), Torretti (1978), and Ben-Menahem (2006), for instance – believe that both applied and pure geometry fall in the scope of (CG). While some find (CG) confused or even trivial – Eddington (1920), Zahar (1977), and Nagel (1961) – others find it intriguing and insightful.  The question about the grounds on which Poincaré defends (CG) is also hotly debated. Some – Black (1942), for example – believe that it is a consequence of Poincaré’s argument for the relative consistency of the hyperbolic geometry – that is, the so-called ‘dictionary’ argument.’ Others believe that (CG) follows from Poincaré’s vivid depiction of imaginary people subjected to some ‘weird’ conditions – that is, Poincaré’s parable of ‘the temperature world.’  Some – Giedymin (1982) and Torretti (1978), for example – think (CG) is a consequence of the so-called Kleinian conception of geometry while others – Grünbaum (1973), for instance – trace (CG) to the result 100  that they attribute to Riemann according to which the topology of a manifold does not determine its geometry. Some – Eddington (1920) – believe that (CG) is just an instance of the more general thesis of confirmation holism or underdetermination and follows from Poincaré’s discussion of stellar parallax; others – Carnap (1966) and Sklar (1977)– believe that (CG) is the strongest possible version of underdetermination: any two geometries are equivalent given all possible observations.  Some – Ben-Menahem (2006) – believe that Poincaré’s theory of the origin of spatial intuitions and geometric concepts is an essential part of his argument for (CG); others either ignore it or think it is nothing but misguided psychological fantasy – Torretti (1978) is a point in case. While Friedman (1995) claims that (CG) is a consequence of geometry’s peculiar status in the so-called hierarchy of sciences, Zahar (1997) believes that (CG) follows from Poincaré’s description of a mechanical device that behaves like a hyperbolic solid. Different combinations of the above arguments are also considered as the grounds on which (CG) is established – Ben-Menahem (2006) is a case in point.   The four chapters composing the second part of this study are devoted to four different interpretations of Poincaré’s so-called geometric conventionalism.  In the first three chapters, we will see that the major commentators have emphasized one interesting and provocative strand of Poincaré’s thinking – that is, the freedom in choosing the geometric language/framework for describing the spatial relations in theories of physics – and equated it with (CG) or a consequence of (CG). Such a conflated claim, then, seemed to them to be Poincaré’s geometric conventionalism. In addition to highlighting the questionable attribution of such a claim to Poincaré, I shall point to some of the internal problems of these supposed readings of Poincaré’s view. 101   The last chapter is devoted to a reading of Poincaré’s geometric conventionalism that – unlike the interpretations of the previous chapters which see it as a contribution to the philosophy of physics – construes it as a contribution to the philosophy of mathematics. It, nevertheless, I shall argue, it misidentifies what Poincaré’s contribution is.   Our survey, also, suggests that one reason why these commentators failed to understand Poincaré’s view might be the fact that they failed to consider the context in which Poincaré developed his philosophy of geometry; they, instead, used his insights within a different context in a way that was not consistent with his general philosophical outlook. Unlike our approach in this study – that is, treating Poincaré as a nineteenth-century mathematician, physicist, and science populariser dealing with the philosophical consequences of a mathematical discovery, among others – most twentieth-century commentators approached him as a philosopher of physics dealing with the philosophical consequences of Einstein’s theory of relativity. As a result of disregarding the historical and intellectual context, they ended up attributing skewed views to Poincaré. The irony is that some of these authors have been preaching otherwise their whole careers.         102  Chapter 5: Logical Positivists’ reception of Poincaré’s geometric conventionalism The unravelling of nineteenth-century physics: space, geometry and physics in the new age By the end of the nineteenth century, Newtonian mechanics had dominated physics for two centuries. However, one of its central assumptions – that is, the relativity of uniform motion – seemed to stand in tension with the other pillar of physics, the theory of electromagnetism – a merely forty-year-old theory then, proposed by James Clerk Maxwell (1831- 1879). In his attempt to resolve this conflict, Albert Einstein (1879-1955) proposed his theory of special relativity in 1905 – a theory which unveiled an unexpected connection between space and time. Its profound metaphysical implication was announced three years later by one of his former teachers, Hermann Minkowski (1864-1909), at the 80th Assembly of German Natural Scientists and Physicians: “space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”i Less than a decade later, Einstein’s theory of general relativity forged yet another connection between spacetime – the independent reality resulting from the union of space and time – and the matter it contains.    One ramification of the theory of general relativity was the resurrection of the question of the ‘true’ geometry of spacetime, just a few years after Poincaré had argued that asking such a question was meaningless. The geometry of spacetime, Einstein’s theory of general relativity implies, is non-Euclidean – in fact, of the kind that Poincaré rejected as merely analytical.ii  iMinkowski (1907/1923, p. 75). ii Since according to the theory of general relativity what is contained in spacetime determines its geometry, except for very special energy-momentum tensors, the curvature of spacetime is not 103  Accordingly, some – Arthur Eddington (1882-1944) and Einstein, for example – concluded that Poincaré’s argument must be wrong while others – like Moritz Schlick (1882-1936) and other positivistsi – found the conventionalist thesis, which of course they attributed to Poincaré, not only compatible with the recent discoveries but more importantly as a solution to the philosophical problems that they raised.21 In this chapter, we shall look at these critics and their interpretations of Poincaré’s view. Eddington’s Poincaré: a semantic issue? In the prologue to his treatise on the theory of relativity – Space, Time and Gravitation –Eddington presents his views on geometry in the form of a conversation between an experimental physicist defending a naïve empiricist view of geometry, a relativist who presumably presents Eddington’s own views, and a mathematician holding the ‘hypothetical’ viewii of mathematics as characterized by Russell (1917/1949).22 According to the relativist, the naïve empiricist’s view assumes that there is something corresponding to the notion of length,  that it can be studied by practical measurements, and that it obeys the laws of Euclidean geometry. The relativist, however, believes that the recent development in physics requires that space is viewed as a mere abstraction of the extensional relations of matter and geometry as the description of the behaviour of material scales and thus rejects the empiricist’s assumptions.  constant.   i When I use the phrase ‘logical positivists’ I mean to, in particular, refer to Carnap (1966), Reichenbach (1928/1958), and Schlick (1915&1917/1979). ii This view is sometimes called if-thenism. See Putnam (1975). 104   Moreover, the relativist reminds the empiricists of what Poincaré has supposedly demonstrated in the following argument– that is “the interdependence between geometrical laws and physical laws, which we have to bear in mind continually”i – citing the following text from Poincaré: If Lobatchewsky’s geometry is true, the parallaxii of a very distant star will be finite. If Riemann’s is true, it will be negative. These are the results which seem within the reach of experiment, and it is hoped that astronomical observations may enable us to decide between the two geometries. But what we call a straight line in astronomy is simply the path of a ray of light. If, therefore, we were to discover negative parallaxes, or to prove that all parallaxes are higher than a certain limit, we should have a choice between two conclusions: we could give up Euclidean geometry, or modify the laws of optics, and suppose that light is not rigorously propagated in a straight line. It is needless to add that everyone would look upon this solution as the more advantageous. Euclidean geometry, therefore, has nothing to fear from fresh experiments.iii  Eddington argues that Poincaré’s thesis of the conventionality of geometry is true only in a trivial sense – that is, in the sense that the meaning of every word in the language is conventional.23 The semantic conventionalism, however, as we shall see in chapter ten, is the first option that Poincaré considers and rejects. Eddington, then, rejects Poincaré’s claim according to which experiment has no bearing on the question of geometry, for the following reasons.  Assume that the parallax of a very distant star turns out to have a negative value, which is  i Eddington (1920/1953, p. 9). ii Parallax is an apparent displacement of the position of an object viewed from two different locations; it is usually measured by a function of the angle between the two different lines of sight and its value depends, among other factors, on whether light rays are taken to be straight lines of Euclidean, elliptical, or hyperbolic geometry. iii Eddington (1920/1953, p. 9); a passage quoted from Science and Hypothesis. 105  what one expects if space is positively curved. To save Euclidean geometry, Poincaré has suggested that one can assume that light rays do not traverse (Euclidean) straight-lines and thus explain the negative value of the parallax by modifying physics. According to Eddington, however, this maneuver does not save Euclidean geometry at all since what has been done amounts to a change in the very meaning of ‘straight-line.’i Once the meaning of terms used in the description of the experiment is thus changed, the theorems of Euclidean geometry remain true, but at the same time they lose their previous significance – that is, what they assert will also change. No longer do they assert the propositions which they used to assert; instead, they assert different propositions which are true in the new interpretation.ii Eddington concludes that Poincaré was mistaken in thinking that the question of the true metrical structure of the world is not an empirical issue. Einstein too arrives at the same conclusion, though via a different route to which I shall now turn.   Einstein’s Poincaré: a confirmation issue? In a lecture that Einstein delivered in 1921,iii he characterized Poincaré’s view in the following way:  Geometry (G) predicates nothing about the behavior of real things, but only geometry together with the totality (P) of physical laws can do so. Using symbols, we may say that only the sum of (G)+(P) is subject to experimental verification. Thus (G) may be chosen arbitrarily, and also parts of (P); all these laws are conventions.iv   i My purpose is not to consider whether Eddington’s argument is convincing or not. The issue he is discussing is, however, not as simple as he presents it. After all, a dog would not have four tails if the word “tail” were used to refer to legs. See Hirsch (2002) or Brueckner (1986), for instance.24 ii This interpretation of Eddington’s critique of Poincaré is due to Sklar (1977). For a different account of Eddington’s objection to Poincaré’s view see Grünbaum (1973, pp. 24-7). iii “Geometry and Experience” is the revised and expanded version of this lecture to which I refer by Einstein (1921/2007).  iv Einstein (1921/2007, p. 149).  106   After endorsing Poincaréi, however, Einstein finds it wrong to consider the question of the geometry of spacetime non-factual:24 The question of whether the structure of [the four-dimensional space-time continuum] is Euclidean, or in accordance with Riemann’s general scheme, or otherwise, is, according to the view which is here being advocated, properly speaking a physical question that must be answered by experience and not a question of mere convention to be selected purely on grounds of expediency.ii  To reject Poincaré’s conclusion, Einstein presents the argument for such a claim as follows: Why is the equivalence between the body that is practically rigid in experience and the body of geometry — which suggests itself so readily— denied by Poincaré and other researchers? Simply because under closer inspection the real solid bodies in nature are not rigid, because their geometrical behavior, that is, their potential relative positions (Lagerungsmöglichkeiten), depend upon temperature, external forces, etc. Thus the original, immediate relation between geometry and physical reality appears destroyed, and one feels impelled toward the following more general view, which characterizes Poincaré's standpoint.iii  Einstein, however, argues that it is possible “to determine the physical state of a measuring body so accurately that its behavior in relation to the relative position of other measuring bodies will be sufficiently free from ambiguity to allow it to be substituted for the “rigid” body.”iv However, while here Einstein (mistakenly) assumes that Poincaré’s rejection of the possibility of empirically determining the geometrical structure of space is based on the fact that there are no perfect rigid bodies in nature, in a later work, he rejects such an interpretation of Poincaré’s argument, as  i “Sub specie aeterni Poincaré, in my opinion, is right.” Einstein (1921/2007, p. 149). ii Einstein (1921/2007, p. 150). iii Einstein (1921/2007, p. 149). iv Einstein (1921/2007, p. 150). 107  evidenced by his criticism of Reichenbach (1949/1970).   In an imaginary conversation, Einstein makes Poincaré repeat the argument based on the nonexistence of truly rigid bodies,25 which Reichenbach rejects based on an argument very similar to Einstein’s own argument in his “Geometry and Experience.”26 Einstein’s Poincaré rejoins:  In gaining the real definition improved by yourself you have made use of physical laws, the formulation of which presupposes (in this case) Euclidean geometry. The verification, of which you have spoken, refers, therefore, not merely to geometry but to the entire system of physical laws which constitute its foundation. An examination of geometry by itself is consequently not thinkable. – Why should it consequently not be entirely up to me to choose geometry according to my own convenience (i.e., Euclidean) and to fit the remaining (in the usual sense “physical”) laws to this choice in such manner that there can arise no contradiction of the whole with experience?i  What is clear, however, is that, in both works, Einstein understands Poincaré’s argument as an argument from underdetermination, a Duhemian argument. However, while in the earlier work, the consequences of such a holistic view are not fully appreciated, later he comes to terms with its force.ii While both Eddington and Einstein were convinced that Poincaré’s view of geometry was at odds with the picture that the general theory of relativity offers, some philosophers of science, Schlick for instance, iii found Poincaré’s view helpful in dealing with the philosophical questions raised by acceptance of this new picture.iv   i Einstein (1949/1970 p. 677)  ii For an interesting discussion of Duhem’s rather than Poincaré’s influence on Einstein’s philosophy of science, see Howard (1990), to which the discussion of this section is indebted.   iii See Friedman (1955/1999). iv For instance, see Schlick (1915/1979) where he considers the impact of the theory of relativity on both Neo-Kantian’s and empiricist’ view, and for a brief discussion, see Friedman (1983/1999).  108  Schlick’s Poincaré: an epistemic issue? In one of the first attempts to come to terms with the philosophical consequences of the general theory of relativity, Schlick appeals to Poincaré:   Henri Poincaré has shown with convincing clarityi (although Gauss and Helmholtz still thought otherwise) that no experience can compel us to lay down a particular geometrical system, such as Euclid’s, as a basis for depicting the physical regularities of the world. Entirely different systems can actually be chosen for this purpose, though in that case we also have at the same time to adopt other laws of nature. [….] The reason why this choice is possible lies in the fact (already emphasized by Kant) that it is never space itself, but always the spatial behavior of bodies that can become an object of experience, perception and measurement. We are always measuring, as it were, the mere product of two factors, namely the spatial properties of bodies and their physical properties in the narrower sense, and we can assume one of these two factors as we please, so long as we merely take care that the product agrees with experience, which can then be attained by a suitable choice of the other factor. [….] In the case of space, all experience notoriously teaches that it is by far the most convenient thing to base it on Euclidean geometry; [….] It is done, as Poincaré puts it, on the basis of a convention, and his view has therefore been given the name of conventionalism.ii   In this passage, Schlick does not explicitly cite Poincaré’s argument, but plainly equates (CG) with (CPG)iii and bases the latter on the epistemic inaccessibility of space itself to the experience. His exposition and the works cited, iv moreover, make it clear that, as Schlick understands it,  i At this point, Schlick has a footnote and refers to Poincaré (1902/1913), Poincaré (1905/1913) and Poincaré (1907/1913).   ii Schlick (1915/1979, pp. 168-9). iii (CG) claims that the axioms of geometry are conventions, and (CPG) claims that theories of physics could be expressed in whatever geometric framework deemed practically convenient.  iv Following works are cited: Chapter V of the German translation of Science and Hypothesis – where Poincaré’s arguments are, in fact, similar to the argument for the Duhem-Quine thesis – the introduction of chapter III of the German translation of Value of Science – where Poincaré gives a summary of the views he had argued for in Science and Hypothesis and also contains a passage about 109  Poincaré’s position and what motivates it are not different from the general underdetermination thesis.i   Furthermore, like Eddington and Einstein, but unlike Poincaré,ii Schlick seems to be convinced that the inaccessibility of space to experience leads to a holistic solution to the problem of theory choice.iii Schlick (1917/1979), also, offers the following reason for adopting the conventionalist view of Poincaré: We thus see that experience in no wise compels us to make use of an absolute geometry, e.g., that of Euclid, for the physical description of nature. It teaches us only what geometry we must use, if we wish to arrive at the simplest formulae to express the laws of physics. [….] Poincaré has expressed this tersely in the words: “Space itself is amorphous; only the things in it give it a form.”iv   the relativity of space – and a chapter in Science and Method called “The Relativity of Space”.  i Schlick goes on to equate Helmholtz’ view with that of Poincaré. He then quotes a passage from Helmholtz (1876/2005, p. 682) in which Helmholtz appeals to his ‘Mirror World’ parable to argue for the claim that “the axioms of geometry are not concerned with space-relation only but also at the same time with the mechanical deportment of solidest bodies in motion”, which, of course, Schlick takes to be also Poincaré’s point.  ii In last part we will see that Poincaré uses the argument from the inaccessibility of space to experience to argue that geometry is not an empirical science in the first place, and only secondarily in explaining the relationship between geometry and experience. iii See Friedman (1983/1999) where he argues that although Schlick was originally rather a ‘critical realist’, or even to some extent a Neo-Kantian, his engagement with theory of relativity convinced him that the Kantian solution to the problem of theory choice was not tenable, and faced with the problem of theoretical underdetermination, he became attracted to a ‘conventionalist’, or ‘verificationist’ position.   iv Schlick (1917/1979, p. 230). 110  In this passage, geometric conventionalism is equated with (CPG), which Poincaré often refers to by ‘the relativity of space.’i Adolf Grünbaum has also championed an understanding of Poincaré’s geometric conventionalism based on the relativity of space, which he sometimes calls the Poincaré-Riemann conventionality of congruence.ii On his account, the conventionality of metric geometry –  by which he means the possibility of remetrization of space – follows from the fact that space is intrinsically amorphous. This latter fact, Grünbaum believes, is established by Riemann (1854/1999) on the ground that in a homogenous and continuous space, the topological features do not uniquely determine the congruence relations, allowing freedom in adopting different definitions of congruence.27  Other logical positivists, more or less, present Poincaré’s view similarly.iii However, Reichenbach’s interpretation – even though it is not all clear whether it was intended (or even claimed) to be an exposition of Poincaré’s view – has become the most influential one in the literature.iv Accordingly, in the rest of this chapter, I will consider Reichenbach’s exposition in some detail.  i It is discussed in a section called “The Inseparability of Geometry and Physics in Experience,” where Schlick masterfully expresses it in a way that anticipates Wheeler’s motto – that is, “[s]pacetime tells matter how to move; matter tells spacetime how to curve.” Wheeler and Ford (2000, p. 235). ii See Grünbaum (1973). iii For instance, see Carnap (1966, pp. 144-5&157) and Nagel (1961, pp. 254-267).  iv Sklar’s exposition of such interpretation is a case in point, see Sklar (1977, pp. 88-94).  111  Reichenbach’s Poincaré: a semantic issue, after all? For Hans Reichenbach (1891-1953), the question of the geometry of physical space, as is shown by Riemann according to him, is an empirical question that should be answered through practical measurements. If several kinds of geometries were regarded as mathematically equivalent,i the question arose which of these geometries was applicable to physical reality; there is no necessity to single out Euclidean geometry for this purpose. Mathematics shows a variety of possible forms of relations among which physics selects the real one by means of observations and experiments. [….] After the discoveries of non-Euclidean geometries the duality of physical and possible space was recognized. Mathematics reveals the possible spaces; physics decides which among them corresponds to physical space. [….] In a similar way as the inhabitants of a spherical surface can find out its spherical character by taking measurements,ii just as we humans found out about the spherical shape of our earth which we cannot view from the outside, it must be possible to find out, by means of measurements, the geometry of the space in which we live. There is a geodetic method of measuring space analogous to the method of measuring the surface of the earth. However, it would be rash to make this assertion without further qualification.iii   The reason that Reichenbach advises caution is that such an assertion would be true – without qualification – if the question of how the required measurements should be carried out could be answered independently of the issue at hand – that is, the geometry of space. Before such a measurement could be carried out, however, a definition of distance – or congruence – must be  i I am not sure what ‘mathematically equivalent’ means. These geometries are not equivalent up to isomorphism, which is what is usually meant by such a phrase.  I, then, assume that Reichenbach means that ‘as far as logical consistency is concerned’ these geometries are equivalent. ii For example, by calculating the ratio of the circumference of a circle drawn on the sphere to its diameter, and comparing it to π.  iii Reichenbach (1928/1958, pp. 6&11). 112  adopted.i To illustrate this point, Reichenbach considers the problem of the possibility of projecting a non-Euclidean geometry on the Euclidean plane:     Let us imagine [as illustrated above] a big hemisphere made of glass which merges gradually into a huge glass plane; it looks like a surface G consisting of a plane with a hump. Human beings climbing around on this surface would be able to determine its shape by geometrical measurements. They would very soon know that their surface is plane in the outer domains but that it has a hemispherical hump in the middle; they would arrive at this knowledge by noting the differences between their measurements and two-dimensional Euclidean geometry. An opaque plane E is located below the surface G parallel to its plane part. Vertical light rays strike it from above, casting shadows of all objects on the glass surface upon the plane. Every measuring rod which the G-people are using throws a shadow upon the plane; we would say that these shadows suffer deformations in the middle area. The G-people would measure the distances A'B' and B'C' as equal in length, but the corresponding distances of their shadows AB and BC would be called unequal. Let us assume that the plane E is also inhabited by human beings and let us add another strange assumption. On the plane a mysterious force varies the length of all measuring rods moved about in that plane, so that they are always equal in length to the corresponding shadows projected from the surface G. Not only the measuring rods, however, but all objects, such as all the other measuring instruments and the bodies of the people themselves, are affected in the same way; these people, therefore, cannot directly perceive this change. What kind of measurements would the E-people obtain? In the outer areas of the plane nothing would be changed, since the distance P'Q' would be projected in equal length on PQ. But the middle area which lies below the glass hemisphere would not furnish the usual measurements. Obviously the same results would be obtained as those found in the middle region by the  i We will encounter this problem, in terms of the behaviour of measuring apparatus in Poincaré’s ‘Temperature World’ parable, too.  113  G-people. Assume that the two worlds do not know anything about each other, and that there is no outside observer able to look at the surface E -what would the E -people assert about the shape of their surface? They would certainly say the same as the G -people, i.e., that they live on a plane having a hump in the middle. They would not notice the deformation of their measuring rods.i   The conclusion that Reichenbach derives from this argument is that before the factual question of the geometry of space can be meaningfully raised and answered some coordinative definitions must be made, and it has to be done through convention: These considerations raise a strange question. We began by asking for the actual geometry of a real surface. We end with the question: Is it meaningful to assert geometrical differences with respect to real surfaces? This peculiar indeterminacy of the problem of physical geometry is an indication that something was omitted in the formulation of the problem. We forgot that a unique answer can only be found if the question has been stated exhaustively [….] Since the determination of geometry depends on the question whether or not two distances are really equal in length (the distances AB and BC in Fig. 2), we have to know beforehand what it means to say that two distances arc “really equal.” Is really equal a meaningful concept? We have seen that it is impossible to settle this question if we admit universal forces. Let us therefore inquire into the epistemological assumptions of measurement. For this purpose an indispensable concept, which has so far been overlooked by philosophy, must be introduced. The concept of a coordinative definition is essential for the solution of our problem.ii   i Reichenbach (1928/1958, pp. 11-12). ii Reichenbach (1928/1958, pp. 13-14). A universal force – like the force introduced in Reichenbach’s aforementioned argument and Poincaré’s ‘Temperature World’ parable – affects all material equally and cannot be shielded. It is, thus, an in principle undetectable force. Reichenbach’s claim here, however, is not accurate: While the effect of such a universal force on measuring rods is not directly detectable, and accordingly the geometry that E-people and G-people construct based on their measurement are indistinguishable, the E-people readily can notice something weird is going on in the certain region of their world. Their geometry tells them that there is a hump in the middle of their world, but they cannot see it or feel that they are climbing when passing through that region. To make the world of E-people and G-people truly indistinguishable, beside the universal force that Reichenbach introduces here, some other 114   Reichenbach argues that these conventional coordinative definitions, however, do not divorce the system in which they are used from the empirical world. According to Reichenbach, physics, like other empirical sciences, takes advantages of coordination – that is, it defines its concepts not only by other concepts but also by coordinating certain concepts with particular objects in the world. Since these concepts are related by testable relations, these coordinative definitions are not, in general, arbitrary. The methods of coordination cannot, however, get underway unless some preliminary coordinations are already in place. These initial coordinations, or coordinative definitions, are, in a sense, arbitrary: “[t]hey arc arbitrary, like all definitions; on their choice depends the conceptual system which develops with the progress of science.”i The problems encountered when it is tried to determine “the actual geometry of a real surface,” therefore, are not but “pseudo-problems [raised when] we look for truth where definitions are needed.”ii  As I understand it, Reichenbach’s argument, put in a more general context, runs along the following lines. Knowing what a concept means is conceptually prior to formulating questions concerning that concept. No question can be asked, let alone answered unless that concept has already a meaning. When one asks whether a certain apple is red or yellow, for that question to make sense to one, they have to know what “what is the colour of this apple?” means and have  differential (non-universal) forces must be added to the world of E-people: the region below the hemisphere, for example, must have a peculiar index of reflection, a certain gravitational field, both different from the ones elsewhere, etc. Reichenbach’s main point, however, stands. i Reichenbach (1928/1958, p. 14). ii Reichenbach (1928/1958, p. 15). 115  some idea about the concept of colour. Questions about geometrical structures and the geometry of space are not different. Asking such questions entails that the meaning of the concept of ‘congruence’ is already fixed. However, what the mathematician’s definition of the concept of ‘congruence’ – that is, in this case, ‘equal in length’ – provides is a local condition: two line-segments have ‘equal length’ if and only if whenever they are laid down side by side, they coincide. That is all that mathematicians need, and accordingly, their definition could simply be silent about two distant line-segments.   Such a definition, however, is not sufficient for the physicist. To ask and answer geometrical questions about the real world, one has to find something out there – that is, in the real world – which corresponds to the required concepts. Given the mathematician’s definition of ‘congruence,’ the physicist can consider any two objects – of such a dimension that makes them an acceptable material realization of a line-segment – as having ‘equal length’ if and only if they coincide – to an acceptable approximation – whenever they are laid down side by side. Then, provided that it could be assumed that such objects retain their (linear) shapes and lengths – to a reasonable approximation – any such objects could be chosen as the standard for measuring length. That is, an object in the world could thus be assigned to a hitherto merely mathematical concept. Only when such a concept acquires a corresponding ‘object’ in the world via such a coordinative definition, can the question of the geometry of physical space be meaningfully asked and be answered. Transforming the concept of ‘equal in length’ from a geometric one to a physical one comes with a cost. It makes it, in a sense, relative. The geometrical form of a body is no absolute datum of experience, but depends on a preceding coordinative definition; depending on the definition, the same structure may be called a plane or a sphere, or a curved surface.  Just as the measure of the height of a tower does not 116  constitute an absolute number, but depends on the choice of the unit of length, or as the height of a mountain is only defined when the zero level above which the measurements are to be taken is indicated, geometrical shape is determined only after a preceding definition.i   This relativity – in other words, the dependency of the question of how to interpret the results of measurements required for determining the geometry of space on the question of choosing the measuring devices – is the reason why Reichenbach advises caution in dealing with the question of the geometry of physical space, because of which he concludes that the issue at hand  does not concern a matter of cognition but of definition. There is no way of knowing whether a measuring rod retains its length when it is transported to another place; a statement of this kind can only be introduced by a definition. For this purpose a coordinative definition is to be used, because two physical objects distant from each other are defined as equal in length. It is not the concept equality of length which is to be defined, but a real object corresponding to it is to be pointed out. A physical structure is coordinated to the concept equality of length, just as the standard meter is coordinated to the concept unit of length.ii   The conclusion to be derived from this argument, according to Reichenbach, is not that the geometry is conventional, but it is relative.28 From which Reichenbach concludes that:  [A]ll geometries [are] equivalent; [….] It follows that it is meaningless to speak about one geometry as the true geometry. We obtain a statement about physical reality only if in addition to the geometry G of the space its universal field of force F is specified. Only the combination  G+F is a testable statement. [….] Taken alone, the statement that a certain geometry holds for space is therefore meaningless. It acquires meaning only if we add the coordinative definition used in the comparison of widely separated lengths. The same rule holds for the geometrical shape of bodies. The sentence “The earth is a sphere” is an incomplete statement, and resembles the statement “This room is seven units long.” Both statements say something about objective states of affairs only if the assumed coordinative definitions are added, and both statements must be changed if other coordinative definitions are used. These considerations indicate what is meant by relativity of geometry.iii    i Reichenbach (1928/1958, p. 18). ii Reichenbach (1928/1958, p. 16). iii Reichenbach (1928/1958, pp. 33-35). 117  The phenomenon of the relativity of geometry, Reichenbach contends, was what Riemann, Helmholtz, and Poincaré were trying to hint at: This conception […] is essentially the result of the work of Riemann, Helmholtz, and Poincaré and is known as conventionalism. While Riemann prepared the way for an application of geometry to physical reality by his mathematical formulation of the concept of space, Helmholtz laid the philosophical foundations. In particular, he recognized the connection of the problem of geometry with that of rigid bodies and interpreted correctly the possibility of a visual representation of non-Euclidean spaces […]. It is his merit, furthermore, to have clearly stated that Kant's theory of space is untenable in view of recent mathematical developments. Helmholtz' epistemological lectures must therefore be regarded as the source of modern philosophical knowledge of space.i  The true significance of this conception of geometry, Reichenbach believes, was not understood – even by those who originated it – until the theory of general relativity suggested a non-Euclidean geometry for physical space.29 Misled by the name they chose – that is, conventionalism – philosophers, notable among them Poincaré, concluded that the question of the geometry of physical space is meaningless. Unfortunately, the philosophical discussion of conventionalism, misled by its ill-fitting name, did not always present the epistemological aspect of the problem with sufficient clarity. From conventionalism the consequence was derived that it is impossible to make an objective statement about the geometry of physical space, and that we are dealing with subjective arbitrariness only; the concept of geometry of real space was called meaningless.ii  Although Poincaré mistook the relativity of geometry for its conventionality and accordingly failed to realize the possibility of determining the actual geometry of physical space, his genuine contribution was to highlight the importance of the notion of ‘geometric congruence’ underlying a geometry, according to Reichenbach.     i Reichenbach (1928/1958, pp. 35-36). ii Reichenbach (1928/1958, pp. 36-37). 118  This is also true of the expositions by Poincaré, to whom we owe the designation of the geometrical axioms as conventions […] and whose merit it is to have spread the awareness of the definitional character of congruence to a wider audience. He overlooks the possibility of making objective statements about real space in spite of the relativity of geometry and deems it impossible to “discover in geometric empiricism a rational meaning.i  Such an understanding of geometric conventionalism – that is, that of Reichenbach – has become almost the standard one and usually has been taken as what Poincaré’s view of geometry – when streamlined and freed from confusion – was.ii In my opinion, however, Reichenbach does not – and even does not claim to – give an account of Poincaré’s conventionalism. Instead, Reichenbach rejects what he takes to be Poincaré’s version of geometric conventionalism and offers his own.iii His version of geometric conventionalism makes the same claim as does Poincaré’s relativity of space. They are, however, based on different grounds. While the former is based on an essentially semantic argument, the latter, as we will see in part three is grounded on ontological considerations.   In both versions of conventionalism, Poincaré’s and Reichenbach’s, the role of definition is emphasized. This should, however, not mislead us. While in Poincaré’s version, the possibility of having different abstract and mathematical definitions for the notion of ‘distance’ compatible with  i Reichenbach (1928/1958, p. 36; f.3) ii See, for example, Carnap (1966), Sklar (1977), Bunge (1980), Friedman (1983), and Richardson (1998).  iii However, Friedman (1994&1995/1999) argues that while Reichenbach (1920/1965) criticizes Poincaré’s view and rejects geometric conventionalism, convinced by Schlick, Reichenbach in (1928/1958) and his later works endorses geometric conventionalism. For a discussion of Schlick-Reichenbach debate on this regard, see Coffa (1991). 119  the notion of ‘geometric shape’ is at stake, in the other, the possibility of different physical instantiations of those definitions is at issue. In short, Reichenbach’s geometric conventionalism is different from that of Poincaré’s in that the latter is a thesis about geometry as a mathematical science and is based on considerations regarding the nature of geometric object while the former is a thesis about what Reichenbach calls physical geometry, and is based on semantical issues.   Regardless of not being a faithful exposition of Poincaré’s geometric conventionalism, however, it could be still asked whether, and to what extent, these interpretations and the arguments given in their support constitute a tenable thesis worthy of being called geometric conventionalism, which I now briefly consider. The conventionality of physical geometry Despite apparent similarities, Einstein’s and Schlick’s expositions are, in important aspects, different from those of Eddington and Reichenbach. According to the former interpretations, the conventionality of physical geometry follows from the fact that any amount of experimental data can be interpreted in such a way that they conform to incompatible physical geometries. In other words, physical geometry is conventional because it is underdetermined by empirical data. However, as pointed by Friedman and others, being underdetermined by observations has been long seen as a feature of empirical theories.i Thus, it is a very curious move to use the holistic nature of confirmation of theories about a particular structure of the physical world as a reason for  i Even though it is usually called ‘Duhem-Quine thesis’, ‘Helmholtz-Poincaré thesis’ would probably be a more historically accurate name as it probably first appeared in Helmholtz’ and Poincaré’s writings.  120  rejecting their empirical status. I, therefore, submit that Einstein’s and Schlick’s interpretations merely are versions of confirmation holism, not expressions of a genuine conventionalist thesis.  Eddington’s and Reichenbach’s accounts are, however, more subtle. A more detailed analysis, accordingly, is in order.  Despite their differences, Eddington’s and Reichenbach’s accounts of geometric conventionalism are similar in that both take into account the impact of the language, in particular, definitions, used to express scientific theories.i    According to Eddington’s version, the conventionality of physical geometry amounts to nothing more than the freedom in naming certain material bodies either ‘Euclidean solids’ or ‘non-Euclidean solids’ – or to use a much-discussed example, to name the path of light rays ‘Euclidean straights’ or ‘non-Euclidean straights.’ As a consequence of such freedom in naming objects, different linguistic expressions can be used to utter the same facts. One can choose, among many other options, to name panthers ‘panther’ and tigers ‘tiger,’ or vice versa. However, if the latter option is adopted, ‘tigers are spotted, and panthers are striped’ would express the same fact that ‘panthers are spotted and tigers are striped’ would have, had the former option been adopted. Nothing is special about spatial relations and physical geometry in this regard. Although such freedom always seems available and thus trivial, it is not entirely so. There is a small cost to pay – that is, the price for consistency. In other words, any system of assigning names to objects can be chosen as long as that system is consistent. One can start to name objects in a domain as one  i According to Sklar (1977), Reichenbach’s interpretation is only a more elaborated version of Eddington’s.  121  wishes, but as one continues, one has to respect the restrictions that already adopted conventions demand.   Eddington seems to consider different (applied) geometries as different ways of assigning (geometric) names to certain objects in the world.i One can choose to call paths of light rays ‘Euclidean straight lines’ and thus describe their behaviour in the language of Euclidean geometry – as it is usually done. The behaviour of light rays, however, could be described in other languages – say, non-Euclidean geometry. Eddington argues that the choice here is not between two geometries – that is, two different theories about some aspects of the world. It is between two different languages to express such a theory.ii Thus, what is conventional is not the physical geometry, whatever it is supposed to mean, but the language in which it is expressed.  Such a thesis hardly deserves to be called conventionalism except in its most trivial sense. No wonder then that it has been called trivial semantic conventionalism by some. Let us now see whether the more subtle account of Reichenbach can go beyond such trivial semantic conventionalism.   i Also assigning the extensions to (geometric) properties and relations, of course.  ii Poincaré’s misleading language – for instance when he talks about the conventionality of metric systems or expressing the same facts in different languages – certainly encourages such an interpretation; see Norton (1994). However, it should be noted that he uses such a language when talking about (CPG) not (CG). It is also worth noting that the consistency requirement that defines the domain of possible choices in such an interpretation is too weak to restrict the options to geometries of constant curvature which Poincaré insisted on.     122   The problem of coordination, as we have seen, lies at the heart of Reichenbach’s argument. Such a problem, however, arises at a certain stage of theory-building and by no means is peculiar to theories about physical geometry.  When constructing a theoretical system – whether it is a theory about the geometrical relations holding between bodies, their motions, or any other properties and relations studied in physics – certain useful concepts have to be carefully defined and the relations between them studied. To go beyond mere conceptual analysis and to connect a theory to the world, a correspondence between at least some of these theoretically-defined concepts and some objects in the world must be established. If one wishes to study spatial relations among bodies, after formulating a theoretical system which defines concepts that one needs to address the problem at hand – such as, straight lines, the shortest distance between two point-like objects, solid bodies and so on – and in determining the relations that hold among them, one has to find certain objects in the world that, to a reasonable extent, behave/can be consistently assumed to behave, in a way that makes it possible to associate them with some of the theory’s concepts. Consider, for example, the concept of ‘straight lines.’ As it is demonstrated in the parable of imaginary beings in the temperature world, it can be realized by different objects – for instance, either by the trajectory of light assuming the vacuum condition or by the trajectory of light assuming conditions such as those described in the parable.i  i In terms of Reichenbach’s argument, with or without assuming that there is the universal force present. It is worth noting that the theoretically defined concepts on this interpretation, as noticed by Reichenbach himself, are too general to restrict the options only among geometries of constant curvature which Poincaré insisted on.  Not only can these concepts be associated with objects that can be consistently assumed to behave as described in geometries of constant curvature, but also 123   According to Reichenbach’s version, then, the conventionality of physical geometry amounts to the claim that one has to choose conventionally one set of realizations of the concepts of geometry before one goes on to describe the geometrical relations among the rest. Or to put it in terms of Reichenbach’s example of projecting a non-Euclidean geometry on a Euclidean plane, the conventionality of physical geometry amounts to picking a formula to describe the universal force. E-people can choose F = 0 as the formula describing the universal force and accordingly describe their world as a plane with a hump in the middle. Such a choice makes their physics simpler.i However, they also could choose any other geometry and then by comparing the physical geometry obtained by neglecting the universal force with that geometry compute the formula for the universal force that makes their measurements compatible with the geometry of their choice. In particular, if they prefer the simpler Euclidean plane as a description of their world, for example, they can have it provided that they pay the cost of making some aspects of their physics more complicated.   with those described in geometries of variable curvature. It is so because although, in the parable of imaginary beings in the temperature world, it is assumed that the temperature field and the light reflective index vary as a specific function that makes bodies and light rays behave in a way that is consistent with hyperbolic geometry, nothing prevents us to consider more general functions according to which they behave as if they live and move in space with any Riemannian geometry.  i This claim has to be qualified: such a choice will make only certain aspects of E-people’s physics simpler, those that describe certain behaviour of measuring rods. But since it makes their physical geometry more complicated, if other aspects of E-people’s physics depend on the physical geometry of their world, those too will get more complicated. The situation obviously gets even worse if one takes into account the complication raised earlier.  124   But if any choice of physical geometry is possible, in what sense is the chosen physical geometry a physical theory? Does not one’s freedom in choosing any form for the universal force just boil down to one’s freedom to name this or that object a straight line? On this interpretation, too, the conventionality of geometry seems to become just an instance of trivial semantic conventionalism. In fact, even Sklar tacitly admits the charge as he finds no better defence against it but:  Does conventionalism, as we have described it, then, simply reduce to trivial semantic conventionality? Not quite. If we have reached trivial semantic conventionality, we have done so by a very circuitous route, indeed. I am tempted to take refuge in the claim that as far as philosophy is concerned, if the routes differ enough, the end point really cannot be said to be the same. For how you got there is as important as where you arrived.i  Let me, then, conclude this chapter by answering the question of whether the accounts surveyed above express a tenable thesis worthy of being called geometric conventionalism in the negative and move on to another influential interpretation of Poincaré’s geometric conventionalism – that is, that of Michael Friedman.   i Sklar (1969, p. 59).       125  Chapter 6: Michael Friedman’s reception of Poincaré’s geometric conventionalism The great chain of sciences The logical positivists’ exposition of Poincaré’s geometric conventionalism, despite all their shortcomings, was more or less unchallenged for most of the twentieth century and prevailed in the literature before it began to be replaced by another influential interpretation in the last decade of the twentieth century. The appraisal of this exposition of Poincaré’s geometric conventionalism, due to Michael Friedman, is the subject of this chapter.i  Friedman (1983) discusses conventionalism at length. Poincaré’s geometric conventionalism, however, is not its focus; Poincaré’s name is mentioned a few times, but his geometric conventionalism is not distinguished from that of the logical positivists. Friedman presents Poincaré’s geometric conventionalism, and the arguments in its support, as Reichenbach (1928/1958) and Carnap (1966) did.30 Friedman (1995/1999), however, criticizes such an account and offers a new interpretation of Poincaré’s geometric conventionalism and Poincaré’s arguments in its support.  Geometry, physics and their egalitarian relationship  To defend their conception of geometry, the logical positivists, according to Friedman, appealed to both Einstein and Poincaré as “they believed that Poincaré’s philosophical insight had been  i Grünbaum’s works also contain a critique of logical positivists’ exposition of Poincaré’s geometric conventionalism but since, unlike Friedman’s exposition, that of Grünbaum, in addition to being already criticized in the literature, did not become very influential, I chose the former as an example of post logical positivists’ interpretation of Poincaré to investigate in this chapter.  126  realized in Einstein's physical theories.”i Of course, Friedman continues, they were wrong to hold such a belief.31 He argues what misled them was their misunderstanding of Poincaré’s view in that they took it to be a particular case of the more general problem of ‘observational equivalence.’ii However, [T]he logical positivists’ argument from observational equivalence is in no way a good argument for the conventionality of geometry, at least as this was understood by Poincaré himself. For the argument from observational equivalence has no particular relevance to physical geometry and can be applied equally well to any part of our physical theory. The argument shows only that geometry considered in isolation has no empirical consequences: such consequences are only possible if we also add further hypotheses about the behavior of bodies. But this point is completely general and is today well known as the Duhem-Quine thesis: all individual physical hypotheses require further auxiliary hypotheses in order to generate empirical consequences.iii   i Friedman (1995/1999, p. 71). ii In supporting this claim, Friedman quotes passages from Carnap (1966, pp. 144-5) and Schlick (1915/1979, pp. 168-9). Schlick’s and Carnap’s expositions of Poincaré’s geometric conventionalism are obviously vulnerable to Friedman’s objection. Reichenbach’s account, however, is not. For one thing, it is based on the freedom of coordinating certain concepts with objects, rather than observational equivalence of competing theories. That is, Reichenbach’s argument is based on a semantic thesis, rather than an epistemological one. Moreover, Reichenbach’s account requires that any two physical geometries be compatible with all possible observations – that is, the existence of a universal force to render any desired geometry G’ compatible with any observationally inferred one, G. Such requirement is much stronger than the Duhem-Quine thesis according to which any two rival theories are underdetermined by some observations. As we have seen in the previous chapter, it is this strong assumption that makes the two physical geometries in question two different expressions of the same theory rather than the expressions of two competing ones and leads to the trivial semantic conventionalism.     iii Friedman (1995/1999, p. 73). 127  The generality of the Duhem-Quine thesis, according to Friedman, is in sharp contrast to “Poincaré’s own conception [which] involves a very special status for physical geometry.”i   Moreover, Friedman points out that since Poincaré does not deem optics and electrodynamics conventional even though they too are subject to the Duhemian argument, the conventionality of geometry, at least for Poincaré, cannot be a consequence of its being underdetermined by observation. Friedman, thus, concludes that logical positivists misunderstood Poincaré’s grounds for deeming geometry conventional. Friedman, then, claims that Poincaré’s own argument for the conventionality of geometry has two main premises: the special place of geometry in the hierarchy of sciences, and what Friedman calls Poincaré’s essential use of the Helmholtz-Lie solution to the space problem. The latter is a mathematical theorem which, although Friedman does not explain how, plays an indispensable role in Poincaré’s argument. In what follows, I first present Friedman’s argument and then my criticism.  Friedman’s Poincaré: a tale of two hierarchies  Poincaré’s own argument, Friedman claims,  involves two closely related ideas. The first is the already indicated idea that the sciences constitute a series or a hierarchy. This hierarchy begins with the purest a priori science – namely, arithmetic – and continues through the above-mentioned sciences to empirical or experimental physics properly speaking. In the middle of this hierarchy, and thus in a very special place, we find geometry. The second idea, however, is the most interesting and important part of Poincaré’s argument. For Poincaré himself is only able to argue for the conventionality of geometry by making essential use of the Helmholtz-Lie solution to the space problem. This specifically group-theoretic conception of the essence of geometry, that is, is absolutely decisive – and thus unavoidable – in Poincaré’s own argument.ii   i Friedman (1995/1999, p. 73). ii Friedman (1995/1999, p. 74). 128  Friedman’s interpretation of Poincaré’s geometric conventionalism, thus, is based on two theses which, of course, he attributes to Poincaré: (a) A special status of geometry in the hierarchy of sciences, and (b) a group-theoretic conception of geometry in which Lie’s classification of geometries plays an essential role. Let us now see how Friedman spells out and justifies these theses, which he also claims are ‘closely related,’ in turn.  Hierarchy of sciences Friedman apparently infers the special status of physical geometry in the hierarchy of sciences from a passage – to which I refer by (H) – in the introduction to Science and Hypothesis. We will also see that there are various kinds of hypotheses; that some are verifiable and, when once confirmed by experiment, become truths of great fertility; that others, without being able to lead us into error, become useful to us in fixing our ideas, and that the others, finally, are hypotheses in appearance only and reduce to definitions or conventions in disguise.i  In this excerpt, Poincaré states the epistemic ramification of different kinds of hypotheses for “sciences from arithmetic and geometry to mechanics and experimental physics” – that is, the claims that he would argue for in the book. Friedman construes Poincaré’s introductory comments as evidence for his account and writes:  Poincaré then enumerates the sciences where we are involved principally with the free activity of our own mind: arithmetic, the theory of mathematical magnitude, geometry, and the fundamental principles of mechanics. At the end of the series of sciences, however, comes something quite different, namely, experimental physics. Here we are certainly involved with more than our own free activity[.]ii   i Friedman (1995/1999, p. 73); incidentally the reference is misprinted in Friedman (1995/1999) as (1902/1905, p. xii) instead of (p. xxii).  ii Friedman (1995/1999, p. 73). 129  Friedman then draws on Poincaré’s discussions in the rest of Science and Hypothesis to conclude thati   each level in the hierarchy of sciences presupposes all of the preceding levels: we would have no laws of motion if we did not presuppose spatial geometry, no geometry if we did not presuppose the theory of mathematical magnitude, and of course no mathematics at all if we did not presuppose arithmetic.ii  Geometry, measurement, and the Lie group of free motions Friedman expresses his second thesis, which has to do with what he calls Poincaré’s “essential use of the Helmholtz-Lie solution to the space problem,” rather vaguely. From his exposition, however, it seems clear that Friedman believes that (a’) Poincaré’s geometric conventionalism requires the results expressed in what he calls the Helmholtz-Lie theorem – that is, the statement that free motion of an n-dimensional rigid body in an n-dimensional space is possible if and only if that space has a constant curvature. Moreover, Friedman seems to think that (a’) necessitates (b’) adopting a Kleinian conception of geometry. Friedman, then, restricts admissible metrics for the multidimensional continuum to only those that are compatible with what he calls the group of free motions: A multidimensional continuum becomes an object of geometry when one introduces a metric – the idea of measurability – into such a continuum. And, analogously to the case of  i These are the first two chapters of Science and Hypothesis, which are the revised versions of papers written in 1894 and 1893 respectively about arithmetic and analysis and the last two parts of Science and Hypothesis, which are the edited versions of what Poincaré wrote for varied purposes in different occasions mostly between 1899 to 1901 about Newtonian mechanics, the Maxwell-Lorentz theory of electrodynamics, and thermodynamics. For the sources of the chapter of Science and Hypothesis, see Poincaré (2002, pp. 169-171). ii Friedman (1995/1999, p. 73). 130  one-dimensional continua, we achieve this through the introduction of group-theoretical operations. In this case, however, the structure of the operations in question is much more interesting from a mathematical point of view. In the case of a three-dimensional continuum, for example, instead of a continuous, additive semigroup of one dimension, we have a continuous group of free motions (in modern terminology, a Lie group) of six dimensions.i  And goes on to explain how these two theses lead to Poincaré’s thesis of the conventionality of geometry: The metrical properties of physical space are based, as indicated above, on a Lie group of free motions; and the idea of such a group arises, according to Poincaré, from our experience of the motion of our own bodies. [….W]e represent this concept by means of a mathematical group. In this sense – that is, through an idealization – the idea of such a Lie group arises from our experience. At this point, however, a remarkable mathematical theorem comes into play, namely, the Helmholtz-Lie theorem. For, according to his theorem, there are three and only three possibilities for such a group: either it can represent Euclidean geometry […], or it can represent a geometry of constant negative curvature […], or it can represent a geometry of constant positive curvature [....] What is important here, for Poincaré, is that only the idea of such a Lie group can explain the origin of geometry, and, at the same time, this idea drastically restricts the possible forms of geometry. Poincaré, of course, believes that the choice of any one of the three groups is conventional. Whereas experience suggests to us the general idea of a Lie group, it can in no way force us to select a specific group from among the three possibilities. Analogously to the case of the theory of mathematical magnitude, we are here concerned basically with the selection of a standard measure or scale.ii  I refer to these passages as (L1) and (L2), respectively.  Friedman’s Poincaré and mathematical idealizations Realizing that the conception of geometry advocated above is not much different from that of Helmholtz who, according to Friedman, did not believe that axioms of physical geometry were conventional, Friedman explains Poincaré’s different attitude in terms of Poincaré’s different  i Friedman (1995/1999, p. 75).  ii Friedman (1995/1999, pp. 76-7).  131  understanding of mathematical idealization. Unlike Helmholtz, Poincaré […] clearly saw that the idea of the free motion of rigid bodies is itself an idealization: strictly speaking, there are in fact no rigid bodies in nature, for actual bodies are always subject to actual physical forces. It is therefore completely impossible simply to read off, as it were, geometry from the behavior of actual bodies without first formulating theories about physical forces. (In my opinion, the point of the temperature-field example is precisely to make this situation intuitively clear.) And it now follows that geometry cannot depend on the behavior of actual bodies. For, according to the above-described hierarchy of sciences, the determination of particular physical forces presupposes the laws of motion, and the laws of motion in turn presuppose geometry itself: one must first set up a geometry before one can establish a particular theory of physical forces. We have no other choice, therefore, but to select one or another geometry on conventional grounds, which we then can use, so to speak, as a standard measure or scale for the testing and verification of properly empirical or physical theories of force.i  Built on the idea of a hierarchy of sciences, the group-theoretical conception of geometry, the Helmholtz-Lie theorem, and the ideality of rigid bodies, Friedman concludes, Poincaré’s geometric conventionalism amounts to claiming “that establishing mathematical force laws (underlying the physical notion of rigidity) presupposes that we already have a geometry in place in order to make spatial measurements; so we must first choose a particular geometry and then subsequently investigate physical forces.”ii  Friedman argues that given the lesson learnt from the theory of general relativity, logical positivists, obviously, could not have possibly appealed to Poincaré’s geometric conventionalism, and the fact they have done so only shows that they misunderstood Poincaré. Naturally, one might think that Friedman’s interpretation of Poincaré’s geometric conventionalism is a more accurate one. Or is it?  i Friedman (1995/1999, p. 75). ii Friedman (2002, p. 199).  132  Friedman’s Poincaré = Einstein + Klein? The first point to note – and given the detailed discussions of the third part, it needs no more justification –  is that Friedman’s interpretation shares with that of logical positivists the problem of attributing to Poincaré a view about the so-called physical geometry based on arguments radically different from those he used to argue for the conventionality of some axioms of geometry as a mathematical science. The second point is that Friedman’s account is hardly supported by the textual evidence to which he appeals. Finally, his account contains several plainly false assumptions. These points will be considered in what follows.  Dubious textual evidence and false assumptions in Friedman’s account Friedman’s imagined hierarchy of sciences Poincaré, as we will see, explicitly states (CG) and argues for it in his 1891 paper, which is reprinted in the first chapter on geometry in Science and Hypothesis. However, surprisingly enough, in constructing his interpretation, Friedman focuses on the introduction that Poincaré wrote years later for a collection of his articles – that is, Science and Hypothesis. Had he considered more than a few passages in the introduction, Friedman would surely have noticed that the conventionality of geometry could have nothing to do with the thesis of the hierarchy of sciences. This thesis , given the way Friedman formulates it, is not only false but also explicitly rejected by Poincaré.i  As we will see, there is no allusion to the hierarchy of sciences in Poincaré 1891. The only science discussed there is geometry. The well-ordering principle and its equivalent, the principle of mathematical induction, are only mentioned as examples of true synthetic a priori principles,  i See the quotation on page 136. 133  providing a contrast to geometrical axioms. Poincaré’s discussion of the parallax of a distant star comes only after he establishes the thesis of geometric conventionalism, at the end of the paper, as a reply to an anticipated objection.   There is, therefore, no wonder that Friedman attempts to attribute the thesis of the hierarchy of sciences to Poincaré based on some remarks in the introduction to Science and Hypothesis written years later. But, even these remarks do not support Friedman’s claim. He does not explain why and how the passage that he quotes, (H), must be interpreted as he claims. Furthermore, the conclusion that Poincaré draws from this passage is not that the different sciences are structured in a hierarchy according to the status of their hypothetical presuppositions. Instead, he concludes that things in themselves are out of our epistemic reach and we can only hope to know relations holding among them. Some people have been struck by this character of free convention [….] they have forgotten that liberty is not license. Thus they have reached what is called nominalism, and have asked […] if the world [the savant] thinks he discovers is not simply created by his own caprice. [….] If this were so, science would be powerless. Now every day we see it work under our very eyes. That could not be if it taught us nothing of reality. Still, the things themselves are not what it can reach, as the naïve dogmatists think, but only the relations between things. Outside of these relations there is no knowable reality. Such is the conclusion to which we shall come, but for that we must review the series of sciences from arithmetic and geometry to mechanics and experimental physics.i  He, therefore, appeals to several chapters of Science and Hypothesis, all based on works written after 1891. Even if Poincaré had come to believe that sciences come in a hierarchical structure, one might wonder, how he could have based his geometric conventionalism on what he would come to  i Poincaré (1902/1913, p. 28) 134  believe years later? Nevertheless, there is no evidence that Poincaré espoused such a view, at least, in Science and Hypothesis.   Poincaré begins the introduction to Science and Hypothesis by stating a conception of science and mathematics which he wishes to prove faulty:  For a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible, and if the scientists are sometimes mistaken, this is only from their mistaking its rules. “The mathematical verities flow from a small number of self-evident propositions by a chain of impeccable reasonings; they impose themselves not only on us, but on nature itself. They fetter, so to speak, the Creator and only permit him to choose between some relatively few solutions.i  According to Poincaré, what has led to this misguided conception is failing to appreciate the role of hypothesis in science and mathematics, a mistake he plans to amend not by “pronouncing a summary condemnation, [… but by] examin[ing] with care the rôle of hypothesis,”ii revealing “that there are several sorts of hypotheses[.]”iii Once the role of different kinds of hypothesis in science and mathematics is recognized, Poincaré contends, the mistaken belief of the naive scientific realist will be brought to light and the fact that “the things themselves are not what [science] can reach, as the naïve dogmatists think, but only the relations between things [, and o]utside of these relations there is no knowable reality”iv will be illustrated. The plan of the book, accordingly, is to “review the series of sciences from arithmetic and geometry to mechanics and experimental physics”v to  i Poincaré (1902/1913, p. 27). ii Poincaré (1902/1913, p. 27). iii Poincaré (1902/1913, p. 28). iv Poincaré (1902/1913, p. 28). v Poincaré (1902/1913, p. 28). 135  identify the kind and the role of hypotheses in each and reveal the epistemological ramifications they have.  Each part of the book, as reviewing of Science and Hypothesis would reveal, accordingly, is devoted to the question of the possibility of each science given its peculiar characteristics. The first part considers the possibility of mathematics as a certain yet non-analytic science, the second part deals with the possibility