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10.1 Introduction

Poincaré’s works extend to many fields of mathematics, such as analysis, algebra, arithmetic, geometry, topology, and celestial mechanics. Among these, geometry and topology, on which we focus in this paper, constitute very important parts. Poincaré interpreted non-Euclidean geometry, first discovered by Lobatchevsky and Bolyai, using a conformal model which was already mentioned by Riemann in his Habilitationsvorstrag [40] and is now called Poincaré model, and studied the isometry groups of hyperbolic spaces. This, along with his other papers on Fuchsian groups, is an important part of his work on geometry. As for topology, Poincaré’s work started with a small piece entitled “Sur l’Analysis Situs”, which appeared in 1892 [29], and culminated in a famous series of papers “Analysis Situs” and its five complements [30,31,32,33,34,35]. This series includes, among others, a formal definition of manifold, which is a concept invented by Riemann [40], the notion of Poincaré duality, the invention of fundamental groups, and the “Poincaré conjecture ” which was solved more than 100 years later.

Poincaré wrote several papers whose topics would be labelled as philosophy or philosophy of science. Most of them were eventually included in his four books, La science et l’hypothèse, La valeur de la science, Science et méthode, and Les dernières pensées [36,37,38,39]. Reading these books, we can easily see that Poincaré’s philosophy was motivated and influenced by his work on geometry and topology. In particular, it is evident that his work on non-Euclidean geometry was an important factor leading him to his geometric “conventionalism”. His view on the relationship between the “real world” and mathematics was quite radical compared to that of his contemporaries, and was very far from the Kantian or neo-Kantian worldview which was influential in his time.

In this paper, we shall first review Poincaré’s work on geometry and topology, particularly focusing on hyperbolic geometry and topology, “analysis situs” in his time. Then we shall look at his philosophical works, focusing on those concerning his view on space and time. We shall see how his work on and understanding of geometry and topology influenced his philosophical view.

Poincaré’s philosophy would certainly be one of what Althusser called “spontaneous philosophy of scientists” [1]. Althusser’s claim was that spontaneous philosophy of scientists, influenced by the ideology of their time, would always have idealistic elements, and that the philosophers’ role should be to discern such elements and draw a “line of demarcation”. Still, as far as we can see, Poincaré’s epistemology was much freer from an idealistic worldview and more profound than most of the philosophers of his days. We shall explain this more in the last section.

The author would like to express his gratitude to Athanase Papadopoulos for inviting him to write this paper. He is also grateful to the anonymous referee for his/her valuable suggestions and drawing the author’s attention to a magnificent book by Gray [12].

10.2 Hyperbolic Geometry

Non-Euclidean geometry usually means geometry for which Euclid’s fifth postulate does not hold, and instead, its negation holds. The fifth postulate is equivalent to what is called Playfair’s axiom, saying that for any straight line ℓ and a point P outside ℓ, there is a unique straight line ℓ′ passing through P which is parallel to ℓ. Two kinds of non-Euclidean geometry can be considered: one is spherical geometry, where two “straight lines” always intersect, and the other is hyperbolic geometry, where there are more than one “straight lines” passing through P parallel to ℓ in the setting above.

Spherical geometry in dimension 2 is known to have been already studied in ancient Greek, and was extensively studied by Euler, but was regarded by the latter as being part of Euclidean space geometry rather than non-Euclidean geometry. (See Papadopoulos [24, 25].) In contrast, hyperbolic geometry in dimension 2, which was first found by Lobatchevsky [20] and Bolyai, was born from the beginning as a trial to construct a geometry without Euclid’s fifth postulate. These two authors showed that they could prove non-Euclidean versions of many theorems in elementary geometry including trigonometry under the second type of the negation of the fifth postulate: the existence of more than one parallel lines as was described above. Still, at this stage, it was not clear that such a geometry is realisable without contradictions.

Riemann introduced the general notion of metric for spaces in all dimensions for the first time in his Habilitationsvorstrag [40]. Relying on Riemann’s idea, Beltrami gave a concrete geometric object having a non-Euclidean geometry in all dimensions. This in particular settled the problem of realisability for hyperbolic geometry in dimension 2. Let us explain this in more detail.

Suppose that on an n-dimensional space \(\mathbb E^n\), a Cartesian coordinate system (x 1, …, x n) is given. By the Pythagorean law, the distance between two points (x 1, …, x n) and (y 1, …, y n) is expressed as \(\sqrt {\sum _{j=1}^n (x_j - y_j)^2}\). Riemann in [40] regards its infinitesimal form: \(ds=\sqrt {\sum _{j=1}^n dx_j^2}\) as a metric defining the Euclidean space. For a curve γ in \(\mathbb E^n\), its length is defined to be ∫γds, and for two points in \(\mathbb E^n\), the infimum of the lengths of arcs joining them coincides with their distance, which is realised by a straight segment joining them. Riemann noticed that by changing this infinitesimal form ds, we can get a different metric on the space. In particular, he mentioned that a form \(\displaystyle ds=\frac {1}{1+\frac {\alpha }{4} \sum _{j =1}^n x_j^2}\sqrt {\sum _{j=1}^n dx_j^2}\) gives a metric of “curvature” α. Although his definition of curvature is rather intuitive, what he meant was the sectional curvature in modern terminology.

Developing this idea of Riemann, Beltrami considered the n-dimensional space with curvature − 1 in [4]. He showed that if we consider the upper half-space {(x 1, …, x n)∣x n > 0} and equip it with a metric given by \(\displaystyle ds=\frac {\sqrt {\sum _{j=1}^n dx_j^2}}{x_n}\), we get a model of n-dimensional hyperbolic geometry. Mapping the upper half-space to the open unit ball \(\{(x_1, \dots , x_n) \mid \sum _{j=1}^n x_j^2 < 1\}\) conformally, he also got the same expression as Riemann’s for α = −1 on the open unit ball. In Beltrami’s upper-half space model of hyperbolic geometry, for any two distinct points, the shortest path connecting them is a part of either a line or a circle intersecting the hyperplane {(x 1, …, x n)∣x n = 0} perpendicularly. In the open unit ball model, the shortest path is a part of a circle intersecting the unit sphere perpendicularly. A different model of hyperbolic geometry was given in another paper by Beltrami [6], and also by Klein [18] from the viewpoint of projective geometry. The base space of Klein’s model is also the open unit ball, but the shortest path between two distinct points is a Euclidean segment connecting them.

Poincaré used the same model as Beltrami’s to study hyperbolic geometry, although it is not certain whether Poincaré was aware of Beltrami’s work. This choice of model was important for his argument regarding “conventionalism”. In fact, in Chapitre 4 of [36], he considered a universe where the metric is deformed conformally by temperature. Poincaré determined the isometry group of the hyperbolic plane in [28], which turned out to be the group of linear fractional transformations with real coefficients. Furthermore, Poincaré studied discrete groups of linear fractional transformations, which he called Fuchsian groups, using fundamental domains in the hyperbolic plane. His work in particular shows that every closed orientable surface of genus greater than 1 has a hyperbolic metric. This is very important in the history of hyperbolic geometry: by proving that hyperbolic geometry is a natural geometry on closed surfaces of high genera, Poincaré showed that hyperbolic geometry has the same right as spherical geometry, which is a natural geometry for genus-0 surface, and as Euclidean geometry, a natural geometry for genus-1 surface. This anticipated the same kind of naturalness of hyperbolic geometry in dimension 3, which would be formulated by Thurston much later, in the 1980s (see [47]).

In Chapitre 3 in Livre I of [38], Poincaré described the process in which he found Fuchsian groups to be isometry groups of the hyperbolic plane. According to his description there, his invention of Fuchsian groups did not arise from his study of non-Euclidean geometry, but from his study of Fuchsian functions, i.e. automorphic functions. He called groups preserving Fuchsian functions Fuchsian groups. He found that Fuchsian groups are also isometry groups of the hyperbolic plane only afterward, while he was taking a “course géologique” planned by the École des Mines, forgetting his mathematics on the surface of consciousness, but probably thinking about it subconsciously.

10.3 Topology-Analysis Situs

Poincaré wrote nine papers on topology, two short papers with the same title “Sur l’Analysis Situs”, published in 1892 and 1901 respectively, “Analysis Situs” , published in 1895, a one-page paper entitled “Sur les nombres de Betti”, published in 1899, “Sur la connection des surfaces algébriques”, published in 1901, and five complements to the “Analysis Situs”, published in 1899, 1900, two in 1902, and 1904 respectively. The short papers “Sur les nombres de Betti”, the second “Sur l’Analysis Situs”, and “Sur la connection des surfaces algébriques” are just announcements of the results whose details are contained in the first, the third and the fourth complements respectively. Therefore we have seven papers to discuss here. (There is one more paper entitled “Sur un théorème de géométrie” in the section on topology of Poincaré’s collected works, which is an unfinished work and whose topics is not closely related to the subject of the present paper.) The fifth complement to the “Analysis Situs” contains a very famous conjecture, which is now called the Poincaré conjecture, saying that every simply connected closed 3-manifold should be homeomorphic to the 3-sphere. The conjecture was finally resolved by Perelman about 100 years after the conjecture was raised. The Poincaré conjecture has been one of the strongest driving forces in research of topology throughout this 100 years. (See Berevstokii’s paper in this volume [5] for more on the Poincaré conjecture .) We fully admit the importance of this conjecture, but we must never forget other works of Poincaré on topology, which also contain many important topics. We refer the reader to Sarkaria’s paper [43] for more detailed accounts about Poincaré’s papers on topology.

In the first “Sur l’Analysis Situs”, Poincaré considered a question asking whether the Betti numbers determine the homeomorphism type of manifolds. This is the first paper in history which deals with a long-standing question of topology, from which modern topology has developed and is still developing: “ how do algebraic invariants determine homeomorphism types of manifolds?” In [6], Betti considered manifolds embedded in a Euclidean space, and introduced the notion called “l’ordine di connessione”, which should be interpreted as the real dimension of the k-th homology plus one, i.e. one greater than what we call the k-th Betti number today. His definition does not involve simplices or cycles in the way we use to define homology groups in modern textbooks. He just considered k-dimensional submanifolds embedded in a given manifold, and counted how many independent k-dimensional submanifolds there are. There is a subtle problem in the way to define the independence of submanifolds, but anyway, this work of Betti gave birth to homology theory.

Poincaré asked in this paper whether two manifolds having the same Betti numbers can be deformed from one to the other continuously, and showed that the answer is no, by giving examples of 3-manifolds having the same Betti numbers without being homeomorphic. His examples are torus bundles over the circle. He claimed that a torus bundle over the circle with monodromy \(A \in {\mathrm {SL}_2 (\mathbb Z)}\) has first Betti number 3, which he called “quadruple connection” using Betti’s term, if and only if A = E; 2, which he called “triple connection”, if and only if \(\operatorname {Tr} A=2\); and 1, which he called “double connection”, otherwise. Poincaré also claimed that two torus bundles, with monodromies A and B, are homeomorphic if and only if A and B are conjugate in \({\mathrm {GL}_2 (\mathbb Z)}\). We note that it is not so simple to prove the last claim, even using modern techniques in low-dimensional topology. Poincaré discussed this example again in the following paper “Analysis Situs” and gave a “proof” of it. We are not sure whether a rigorous proof could be given using only the techniques known in Poincaré’s time.

Poincaré’s second paper on topology, “Analysis Situs”, was published in the Journal de l’École Polytechnique in 1895. This paper, consisting of 121 pages, gave foundations for the theory of manifolds, first invented by Riemann in his Habilitationsvortrag [40], and for the homology theory of manifolds. Poincaré gave two definitions of manifold. In the first of them, a manifold is defined to be a set of points (x 1, …, x n) in the n-dimensional Euclidean space \(\mathbb E^n\) satisfying a system of p (differentiable) equations F 1, …, F p and q inequalities such that the rank of the Jacobian matrix of F 1, …, F p has rank p. In other words, Poincaré’s first definition of a manifold is that of what we call an (n − p)-dimensional submanifold in the n-dimensional Euclidean space defined by implicit functions. Local charts or local coordinate systems which we use in defining manifolds today did not appear in this definition. This is quite similar to the definition of “spazio” of dimension (n − p) by Betti in [6] although in Betti’s paper the condition for the independence of equations is quite obscure.

In the second definition, Poincaré considered m-dimensional sets in \(\mathbb E^n\) parameterised by m variables in such a way that two parameterisations are transformed from one to the other by analytic functions at their intersection. In this definition, for the first time, a construction of a manifold by patching up local coordinate systems appeared, which would lead to a later definition of manifolds using local charts due to Hilbert, Weyl, Kneser, and Veblen-Whitehead. (See Scholz [44] and Ohshika [23].)

Poincaré then introduced the notion of homology, clarifying and refining Betti’s work. He considered a q-dimensional manifold W with boundary in a p-dimensional manifold V , and he regarded the boundary components of W, which are (q − 1)-dimensional submanifolds of V , as being related by a homology. This definition is more like that of cobordism in today’s terminology. He then defined the (q − 1)-th Betti number to be the maximal number of linearly independent (q − 1)-dimensional submanifolds with regard to homologies. Here he took into account the possibility that more than one of the boundary components represent parallel copies of the same (q − 1)-dimensional manifold. This makes his definition slightly different from Betti’s original definition. This difference concerns the topic of the first complement. Poincaré also showed the duality of Betti numbers using intersection number: for a closed orientable manifold of dimension n, the p-th Betti number is equal to the (n − p)-th Betti number .

In the latter part of the paper, Poincaré studied 3-dimensional manifolds, systematically constructing examples from polyhedra. This can be regarded as a generalisation of his construction of torus bundles in the previous paper. Poincaré in particular observed the following from this construction.

  • 3-manifolds can be obtained from properly discontinuous actions of groups on Euclidean space just like Fuchsian groups.

  • There are distinct 3-manifolds with the same Betti numbers , as was already mentioned in “Sur l’Analysis Situs”, such that the groups corresponding to these manifolds (fundamental groups) are different.

  • There are 3-manifolds having finite fundamental groups.

Poincaré also posed the following essential problems:

  1. (1)

    Etant donné un groupe G défini par un certain nombre d’équivalences fondamentales, peut-il donner naissance à une variété fermée à n dimensions ? Comment doit-on s’y prendre pour former cette variété ?

    (Given a group G defined by a certain number of fundamental equivalences, can it give birth to a closed n-dimensional manifold? How should we proceed to form this manifold?)Footnote 1

  2. (2)

    Deux variétés d’un même nombre de dimensions, qui ont même groupe G, sont-elles toujours homéomorphes ?

    (Are two manifolds of the same dimension which have the same group G always homeomorphic?)

The condition for G to be defined by a certain number of fundamental equivalences should mean that it is finitely generated. We know today that for the first problem, we need to add the assumption that G is finitely presented, but with this condition, this has turned out to be true for n ≥ 4, but false for n = 3. For the second problem, the answer is no for every dimension n ≥ 3. What is amazing is that Poincaré was so far-sighted that he could consider such a problem which would decide the direction of research in topology for a long time to come and could be resolved only much later in the twentieth century.

In the first complement to the Analysis Situs, Poincaré clarified the duality of Betti numbers which he stated in his “Analysis Situs”, responding to a criticism by Heegaard [14], and gave another proof, which is more rigorous than the one contained in the “Analysis Situs”. It is interesting for us to see that in those days, results appearing in papers were sometimes incomplete, and could be rectified after discussion in other papers. The new proof is nearer to what we can find in a textbook on simplicial homology theory today. Poincaré defined Betti numbers using polyhedra and boundary operators. To prove the duality, he used the dual cell complex as is done in most textbooks today. In the second complement, using the same line of argument as he did in the first complement, Poincaré observed the existence of a torsion invariant, which corresponds to the torsion part of homology groups. At the end of the paper, Poincaré said:

Tout polyhèdre qui a tous ses nombres de Betti égaux à 1 et tous ses tableaux T q bilatères est simplement connexe, c’est-à -dire homéomorphe à l’hypersphère.

(Every polyhedron all of whose Betti numbers are equal to 1 and all of whose charts T q are bilateral is simply connected, i.e. is homeomorphic to the 3-sphere.)

In other words, Poincaré said that every combinatorial 3-manifold with trivial first homology group is homeomorphic to S 3. This is of course false, and indeed Poincaré himself realised that this is not the case as was to be shown in the fifth complement.

In the third complement, Poincaré studied specific 3-manifolds. First, he considered an algebraic surface in \(\mathbb C^3\) expressed as \(z=\sqrt {F(x,y)}\), where F is a polynomial. It was assumed that for every y except for finitely many singular points, F(x, y) = 0 has 2p + 2 simple roots as an equation in x. He then considered a loop on the complex y-plane going around singularities, and the set of corresponding points in the algebraic surface. This becomes a 3-manifold which is a closed surface bundle over the circle. Poincaré showed how to compute the fundamental groups of such 3-manifolds.

In the fourth complement, Poincaré studied the topology of algebraic surfaces defined by f(x, y, z) = 0 when f is a polynomial. The study of such surfaces was started by Picard in [26]. Poincaré calculated the Betti numbers of such surfaces. He also showed that there is a duality between the first and the third Betti numbers for 4-manifolds, which was later termed Poincaré duality.

The fifth complement is very famous for the fact that the Poincaré conjecture was formulated for the first time there. The paper also contains many other important results. In the first part of the paper, Poincaré introduced what is now called the Morse theory. We should have in mind that this paper was published in 1904 (dated on 3 November 1903), which is more than 20 years before Morse, to whom the invention of the “Morse theory” is usually attributed, started writing a series of paper on this topic. (See Morse [22] for his work.) Although Poincaré’s setting is very special, it is interesting that this case is the same as the one which is shown as an example of an application of Morse theory in most textbooks. He considered an m-dimensional manifold V  embedded in a higher-dimensional Euclidean space \(\mathbb E^k\), and a function ϕ defined on \(\mathbb E^k\). Then he considered to slice V  at ϕ(x 1, …, x k) = t, which gives a family of m − 1-dimensional manifolds W(t) (allowing singularities for some t), and observed that the topological type of W(t) changes only at points t where W(t) has singularities. He also analysed how W(t) changes before and after t passes a singular value, and how this affects the topology of V . As the first application of this theory, he gave an alternative proof of the topological classification of closed surfaces. The second application, which is the most important part of this paper, is for 3-manifolds. Poincaré showed that there exists a closed 3-manifold with trivial Betti numbers and trivial torsions which is not homeomorphic to the 3-sphere. In other words, he constructed a homology 3-sphere. In the case of dimension 3, slicing V  into W(t) corresponds to a Heegaard splitting of V , i.e. a decomposition of V  into two handlebodies, which was first studied by Heegaard in [14]. Poincaré constructed a concrete Heegaard splitting, which gives a homology sphere, and calculated its fundamental group to show that the resulting 3-manifold is not homeomorphic to the 3-sphere. In the last page of the paper he posed the following question, which is the very famous Poincaré conjecture :

Est-il possible que le group fondamental de V  se réduise à la substitution identique et que pourtant V  ne soit pas simplement connexe?

(Is it possible that the fundamental group of V  is reduced to the identical substitution and that nevertheless V  is not simply connected?)

We should recall that Poincaré called a 3-manifold simply connected when it is homeomorphic to the 3-sphere. The paper is concluded with this sentence:

Mais cette question nous entraînerait trop loin.

(But this question would take us too far.)

Looking back at later development of topology, we see that the Poincaré conjecture was solved in higher dimensions by Smale [46] in a way extrapolated from Poincaré’s own thinking. In fact, using Morse theory and handle decomposition to prove the h-cobordism theorem is a natural extension of Poincaré’s idea of slicing a manifold to get a decomposition into a family of codimension-1 submanifolds. In dimension 4, Freedman [7] proved the conjecture in the topological category, using Casson handles instead of ordinary handle decomposition in higher dimensions. (See also Poénaru’s paper in this volume [27] for related topics.) On the other hand, the original Poincaré conjecture , i.e. for dimension 3, was solved in a way which should have been quite unexpected to Poincaré. To get to the final solution by Perelman, it was necessary to pass through the geometric understanding of 3-manifolds by Thurston [47], and the deformation of Riemannian metrics by Ricci flows due to Hamilton (see Morgan-Tian [21]).

10.4 Epistemology of Space and Time

When we talk about epistemology in the nineteenth and the early twentieth centuries, we cannot ignore the strong impact of the work of Kant and his apriorism. From the ancient Greek period on, Euclidean geometry was considered to be a precise reflection of the reality. For empiricists, geometry should also be derived from experience. Kant’s position is quite different from this. He maintained clearly that “space is not an empirical concept that has been drawn from outer experiences” in his major work “Kritik der reinen Vernunft” [17]. He regarded the notions of space and time as “synthetic a priori”. This means that these notions are not what we construct based on our experience, but are foundations upon which all other recognitions should be built. This view of Kant was quite influential among (continental) European philosophers, in particular those who adhered to German idealism, and scientists, whether they agree on it or not. For instance, Herbart, an impact of whose philosophy can be also found in Riemann’s Habilitaionsvorstrag, emphasised the aspect of the notion of space as a human construction, which can be regarded as a criticism of Kantian apriorism ([15], see also Banks [3]). Fries, while keeping the principle of apriorism by Kant, reexamined Kantian transcendental logic introducing psychologic basis of knowledge [10]. Still, we can say that these two philosophers are within a Kantian paradigm: their approaches are guided by problems posed by Kant, and their works are responses to the writings of Kant. (See §2 of Gray [12] for a more detailed account on these post-Kantian philosophers and their influences on mathematics.)

On the other hand, modern empiricists such as J-S. Mill developed ideas opposed to this Kantian apriorism. In his book entitled “A System of Logic, Ratiocinative and Inductive”, which is famous for his emphasis on the importance of induction as a logic supporting science, he insisted that geometry is also a part of experimental science whose validity should be justified by induction. For him, even Euclidean geometry can be justified by experience.

Poincaré’s view, which is often referred to under the name of (geometric) conventionalism, is quite different from both Kantianism and Mill’s version of empiricism. By the discovery of non-Euclidean geometry, the apriorism concerning Euclidean geometry such as Kant proposed broke down. Still one could expect that it was possible to prove that Euclidean geometry is the right one in the universe where we live, through either experiments or observations. Poincaré did not think that this was the case. He further thought even properties of space common to both Euclidean and non-Euclidean geometry, such as the dimension of space, neither are given a priori nor can be proved by experiments or observations.

Poincaré published three books of a philosophical nature, La science et l’hypothèse, La valeur de la science, and Science et méthode. Another book was published posthumously under the title Les dernières pensées. Poincaré discussed epistemology of space and time throughout these four books. Here, we are going to examine what is expressed in the first two of his books.

The second part of his first book La science et l’hypothèse is entitled “ l’espace” and contains three chapters “les géométries non-euclidiennes”, “ l’espace et la géométrie”, and “l’expérience et la géométrie”. In the first chapter of these three, Poincaré first explains how non-Euclidean geometries, spherical one and hyperbolic one, were born, through the work of Riemann, Lobatchevsky and Beltrami, and why these geometries are natural in the same way as Euclidean geometry. He then analyses what should be geometry in general. He emphasises the importance of the existence of a group acting on a space preserving the shapes of things lying there to make it possible to consider geometry. This view, which Poincaré attributes to S. Lie for its invention, is also a precursor of today’s definition of geometric structures, which are also called (G, X)-structures. (The reader may refer to Goldman’s paper [11] in this volume for more on (G, X)-structures.) We should recall here that for Poincaré, the definition of fundamental group is different from what we find today in textbooks: in modern terminology, his fundamental group is the covering translation group acting on the universal cover. Therefore, considering the group consisting of possible motions in each geometry should be very natural for him.

We should also note that Poincaré argues, contrary to Kant’s view, that the axioms of geometry do not constitute synthetic a priori judgements. For Poincaré, the principle of induction is, for instance, a synthetic a priori judgement, for which he considered it was impossible to consider an alternative arithmetic. Since it is quite possible to consider the world in which non-Euclidean geometry holds instead of Euclidean geometry, the axioms of geometry could not be thrown into this category. On the other hand, Poincaré admits that Euclidean geometry reflects motions of solid bodies in the real world and that projective geometry is an abstraction of behaviours of light. Still he rejects the view that geometry is an experimental science, for it is not an object which is subject to revision. As a result, he concludes:

Les axiomes géométriques ne sont donc ni des jugements synthétiques à priori ni des faits expérimentaux.

(The geometric axioms are therefore neither synthetic a priori judgements nor experimental facts.)

Ce sont des conventions; notre choix, parmi toutes les conventions possibles, est guidé par des faits expérimentaux: mais il reste libre et n’est limité que par la nécessité d’éviter toute contradiction.

(They are conventions; our choice, among all possible conventions, is guided by the experimental facts, but it remains free and is limited only by the necessity to avoid any contradiction.)

He then insists that it is meaningless to ask if between two geometries, for example Euclidean and non-Euclidean geometries, one is truer than the other, and that what we can say is just one is more convenient than the other.

In the next chapter, “l’espace et la géométrie”, Poincaré goes deeper into an epistemological aspect of our cognition of space and its relation to geometry. He characterises space as we understand it by the following five properties: (1) it is continuous; (2) it is infinite; (3) it has three dimensions; (4) it is homogenous; and (5) it is isotropic. He then analyses how these properties are derived from our “experience”. He observes that our notion of space is derived from three different spaces which our senses directly perceive: visual space, tactile space, and motor space. He points out that the visual space has two dimensions, and observes that only by converging the views from two eyes and accommodating them, we get a sense of the third dimension: the distance. For the tactile and motor spaces, the process to get hold of three dimensional space is more complicated. Furthermore, Poincaré argues that our space and its geometry are not mere consequences of integrating these three kinds of spaces which we perceive. It is only through displacement of solid objects, causing changes in view or in touch, and their recoveries through our movement, we form our sense of geometry. This final point is very important in his epistemology of space. He says:

Aucune de nos sensations, isolée, n’aurait pu nous conduire à l’idée de l’espace, nous y sommes amenés seulement en étudiant les lois suivant lesquelles ces sensations se succèdent.

(None of our sensations, if they were isolated, would lead us to the idea of space. We are brought there only by studying the laws following which these sensations succeed to one another.)

Therefore, our space and time are products of our reasoning, not what was given a priori.

In the last of the three chapters, Poincaré discusses if geometry can be derived from experiments or observations. His answer is definitely no. For instance, he asks whether it is possible to determine if the space we are living in is either Euclidean or spherical or hyperbolic (Lobatchevskian in his words). Some may think that this is possible for instance by measuring parallaxes of stars. Poincaré says that this is not the case. To make this kind of idea acceptable, we must assume that light always proceeds along a geodesic. Since there is no way to prove this, Poincaré says, using parallaxes to measure the curvature is just shifting a problem to another one. More generally, Poincaré says,

Aucune expérience ne sera jamais en contradiction avec le postulatum d’Euclide; en revanche aucune expérience ne sera jamais en contradiction avec postulatum de Lobatchevsky.

(No experiment will ever lead to contradiction with the postulate of Euclid, on the other hand, no experiment will ever lead to contradiction with the postulate of Lobatchevsky.)

One may think that the existence of Euclidean solids would prove the “flatness” of space, but Poincaré points out that this is not true, for Euclidean solids can be realised also in a hyperbolic space as solids bounded by parts of horospherical surfaces. This observation is very shrewd, and reminds us of the fact that Lobatchevsky called his geometry “pangeometry” [20].

In the second book La valeur de la science, Poincaré takes up the epistemology of space again in two chapters, which are entitled “la notion d’espace” and “l’espace et ses trois dimensions”. After reviewing what he showed in La science et l’hypothèse, Poincaré goes on to study the subject more deeply. Recall that both Euclidean geometry and (hyperbolic) non-Euclidean geometry presuppose three-dimensional space on which their metrics are defined. This space without any metric is called “amorphous continuum” by Poincaré. This continuum has some properties, which can be studied by “l’analysis situs”, i.e. if we use modern terminology, topological properties. Poincaré poses the same questions for these topological properties as those which he posed for the geometry of space: whether they are given a priori or whether they can be verified by experiments and so forth. As one of the most important topological properties of space, he chooses its dimension and considers whether it can be determined a priori or by experiments. First, Poincaré asks how we can define the dimension of an amorphous continuum in which we live. In the real existing physical space, if two points are close enough, we cannot distinguish them. Therefore Poincaré thinks of the continuum not as a mere point-set, but as a set on which the relation of distinguishability is defined for any two points. We should note that at the time when Poincaré wrote this book, there was no notion of topological space. This notion was first introduced by Hausdorff in 1914, but Poincaré’s idea is very similar to that of neighbourhood which appeared in Hausdorff’s book [13] for the first time.

Poincaré considers what he called “coupures”, i.e. cuts, for a continuum C. Cuts are a collection of either elements of C or continua contained in C. Suppose that one proceeds from one element A in C to another element B in C, by which we mean that there is a sequence of elements E 1, …, E n of C such that A is indistinguishable from E 1; E j is indistinguishable from E j+1 for each j; B is indistinguishable from E n; and two non-adjacent E i and E j are distinguishable. If there are cuts {e k} of C such that for any E 1, …, E n as above, some E j is indistinguishable from some of e k, then we say that the cuts {e k} divide C. Poincaré defines that C is one-dimensional if there are cuts consisting elements of C which divide C. Then inductively he defines C has n dimensions if there are cuts of C consisting of (n − 1)-dimensional continua which divide C.

Having defined dimension for continua in this way, Poincaré returns to the original question. He first analyses how we can apply the above definition of dimension for the space in which we live. Then in the same way as in “l’espace et la géométrie” in the previous book, he analyses how we get the sense of three-dimensionality from our visual, tactile, and motor senses. Again, Poincaré says, we derive this from displacement of objects causing changes of senses and their recoveries by our movement. He concludes that our experience alone cannot prove that our space has three dimensions: in fact it may have more than three dimensions, but we choose three-dimensionality for its convenience. The experience has only guided this choice. He writes:

Quel est alors le rôle de l’expérience ? C’est elle qui lui (l’esprit) donne les indications d’après lesquelles il fait son choix.

(What is then the role of experience? It is the experience that gives the mind the indications following which it makes choice.)

10.5 Spontaneous Philosophy: Conclusion

Althusser, arguably one of the most influential Marxist philosophers in the twentieth century, gave an inaugural lecture of a course on philosophy for scientists in 1967 at l’École Normale Supérieure. This lecture was later published as a book entitled Philosophie et philosophie spontanée des savants in 1974 [1]. Throughout the book, he described what he thought philosophy was and what role philosophy should play. What is relevant to the present article is his view on spontaneous philosophy of scientists, which is succinctly explained in his Thèse 25 as follows:

Dans leur pratique scientifique, les spécialistes des différentes disciplines reconnaissent « spontanément » l’existence de la philosophie, et le rapport privilégié de la philosophie aux sciences. Cette reconnaissance est généralement inconsciente : elle peut devenir, en certaines circonstances, partiellement consciente. Mais elle reste alors enveloppée dans les formes propres de la reconnaissance inconsciente : ces formes constituent la « philosophie spontanée des scientifiques » (P.S.S.).

(In the practice of science, the specialists of different disciplines recognise “spontaneously” the existence of philosophy, and the special relation of philosophy to science. This recognition is generally unconscious, and in some occasions it may become partially conscious. But then it remains covered in its own forms of unconscious recognition, and these forms constitute the “spontaneous philosophy of scientists” (P.S.S.).)

Althusser observed that P.S.S. contains often idealistic ideas coming from the outside of science, which he called “Élément 2”, and that it constitutes an ideological aspect of P.S.S. For Althusser, the principal role of philosophers (as himself) with regard to P.S.S. is to draw a line of demarcation between science, which must be materialism, and ideology, which consists of idealism. We should have in mind that, as can been seen in [2], for Althusser a role model of such an intervention on the part of philosophers is Lenin’s disparagement of empiriocriticism as one can find in [19].

It is evident that Poincaré’s epistemology was born out of his work on geometry and topology. The progress in geometry and topology in Poincaré’s day made it necessary to change the classical view on space and time as was formulated by Kant. In this sense, from Althusser’s viewpoint, Poincaré’s epistemology should be regarded as an example of P.S.S. (An analysis of Poincaré’s philosophy in this line can be found in Rollet [41].) However, what we want to emphasise is that Poincaré’s epistemology, which can be regarded as a precursor of empiriocriticism along with E. Mach, was quite ahead of his time, and that compared to naïve materialism such as the one propounded by Lenin in [19], it was based on far more profound insights into how our recognition of space and time is formed, as we saw in the previous section.

Putting aside a rather amateurish approach of Lenin, let us compare Poincaré’s philosophy with his contemporary professional philosophers. Among them, we can in particular think of Frege, Husserl and Russell as those who studied seriously epistemological problem of space and time. All of them had an academic training in mathematics. Frege, who is often regarded as a precursor of logical positivism, studied the foundation of mathematics, as in [8], which inspired both Wittgenstein and the famous work of Russell-Whitehead, Principia Mathematica. Frege also had good knowledge on the work of Riemann. Still, his attitude toward the non-Euclidean geometries and epistemological problem of space can be regarded as staying within a Kantian framework: Frege seemed to believe that our space of intuition should be the Euclidean three-dimensional space, as can be seen in his correspondence with Hilbert and his review on Hilbert’s book, both of which can be found in [9] (see also Shipley [45] for a more detailed account on Frege’s epistemological view on geometry and space).

Husserl, who is now regarded as a precursor of phenomenology, also worked for philosophical aspects of mathematics. He gave a course on Riemann’s theory of geometry at University of Halle in 1889 [16], where he dealt with Riemann’s notion of metric critically. In particular, after criticising Riemann’s definition of curvature, Husserl claimed that since Euclidean space is necessary to define Gaussian curvature, Euclidean space is a special entity which is given to us by pure intuition in Kantian sense. If we scrutinise his argument from the viewpoint of modern geometry, we notice that Husserl was confusing intrinsic and extrinsic natures of space. He did not seem to pay attention to the fact that spheres and Euclidean planes can be realised even in hyperbolic space. (Recall that this is why Lobachevsky coined the term “Pangeometry” as we noted in §4.)

Russell, who belongs to a younger generation than the other two, published in his twenties a book entitled An essay on foundation of geometry [42]. In this book, he criticised Kantian apriorism in geometry, and observed that there are two elements in geometry, what is a priori and what is empirical. Russell rejected the a-priority of Euclidean geometry, and claimed that experience can decide which of Euclidean or non-Euclidean geometry is valid. He further insisted that their common ground, which is projective geometry, should be a priori. This means that Russell embraced some part of conventionalism, but retained a weak version of Kantian apriorism. This position of Russell was harshly criticised by Poincaré in his review of this book.

Thus we have seen that even these three famous philosophers who were familiar with Riemann’s work could not be as radical as Poincaré in abandoning Kantianism, and this seems to be caused by the fact that without experience of research on geometry and topology, they could not reach the level of Poincaré’s understanding of space. In the latter half of the twentieth century, there appeared philosophers such as Merleau-Ponty, who analysed how our perception leads to our recognition of space, but in Poincaré’s day no professional philosopher could approach the epistemology of space as deeply as Poincaré. According to Althusser, as we have explained above, the role of philosophers would be to draw a line of demarcation between science and ideology in spontaneous philosophy by scientists. However for this to be possible, philosophers need to have a deep understanding on what constitutes the core of spontaneous philosophy. In Poincaré’s case, the core is nothing but his geometric conventionalism, which in turn depends on his research in geometry and topology. In his time, it was very difficult for other philosophers to reach this level for playing the role to draw such a line of demarcation in Poincaré’s epistemology.