- 1 I have written about both of these examples at greater length in prior publications [Stump 1989, 1 (...)
1I have studied Poincaré for a long time and one thing that has always troubled me is how to respond to the bewildering number of interpretations of Poincaré that exist in the literature. His œuvre is vast and spans many different fields. He played many roles in his life, from mine inspector (briefly), to academic—mathematician, physicist and philosopher—to his work in the bureau of telegraph and wireless and the commission on longitude, to expert witness in the Dreyfus Affair. Poincaré became a national representative of science and its defender in the popular press of his time, where he is mentioned frequently. Poincaré wrote a lot and the secondary literature interpreting and criticizing Poincaré is exponentially larger, indeed, overwhelmingly vast. It spans more than one hundred years, several languages and many genres of philosophy. If we included all the mathematicians and scientists who mention Poincaré in some context, we get an even larger literature. People working on philosophy of mathematics or on spacetime physics obviously have something to say about Poincaré, as do most who study general issues in the philosophy of science. Furthermore, conventionalism in a broad sense has a large literature outside of the philosophy of science and even though Poincaré’s conventionalism is narrow, he still appears in much of this literature and is frequently taken to have promoted a global kind of conventionalism. These different groups of philosophers tend to ignore each other’s work, so there is little to constrain interpretations that can be in conflict with each other. Given this situation, it is perhaps not surprising that interpretations of Poincaré can diverge wildly. As in the classic Indian parable of the blind men and the elephant, each interpreter of Poincaré investigates one small part of his works and comes up with an impression that is radically incorrect as a picture of the whole. Furthermore, some interpretations were very influential and were repeated frequently, which amplified the misunderstanding of Poincaré. I will present two examples of influential (and incorrect) interpretations of Poincaré, pinpointing their errors and documenting some of their spread through the literature.1 I will then reflect on whether it is possible to avoid fragmentation in the interpretation of Poincaré, using Jeremy Gray’s admirable attempt at completeness in this intellectual biography of Poincaré as an example. Finally, I will propose one possible new direction for study of Poincaré by making use of the Archives Henri-Poincaré, one that might enrich our view of him substantially. Current, better interpretations of Poincaré have been enabled by the work of the Archive and its founder, Gerhard Heinzmann. I believe that we can go further and that there is still work to be done.
2Schlick was a strict empiricist. There is some dispute in the literature as to whether he started out as a neo-Kantian and then changed his view to empiricism, or whether he was always a strict empiricist, but in any case, the relevant point here is that he interpreted Poincaré in keeping with his empiricist viewpoint. Crucial to the rejection of Kant and embrace of empiricism is Schlick’s denial of the existence of a synthetic a priori. All statements in science are either definitions and hence analytic, or else empirical and hence synthetic. For Schlick, conventionalism applies to analytic statements. We can change geometries by adjusting definitions that are part of our physical theory. Discussing Poincaré’s conventionalism, Schlick argues that we must make changes in our physics when we change geometries, which he claims is an extension of Poincaré’s view that makes it more plausible [Schlick 1917], [Schlick 1979, 224–227]. In a later article in which he rejects the idea that natural laws are conventions, Schlick says that in the case of geometry:
One and the same physical world can be described in terms of different geometries, if only the formulation of the physical laws is each time adapted to the special geometry used. [Schlick 1935, 442]
Thus, on Schlick’s view, conventionalism relies on something like the Duhem-Quine thesis or holism. If we get new evidence that seems to refute our hypothesis, we can change other aspects of our background theory in order to make the evidence consistent with our hypothesis. In the case of metric conventionalism, we can change our geometry if we make compensating changes in our physics. This interpretation of the basis for metric conventionalism has been called the epistemological interpretation in the spacetime literature. However, if we interpret Poincaré as using such an argument to defend his thesis of the conventionality of metric geometry, it is hard to explain how Poincaré can limit his conventionalism, as he clearly does. Indeed, given that all empirical theories, and not just physical geometry, are underdetermined in the Duhemian sense, the major interpretive problem for those who hold the epistemological interpretation of conventionalism is to explain why Poincaré holds that only metric geometry and a few principles of physics are conventional [Stump 1989, 348], [Friedman 1995], [Friedman 1999, 73]. If Poincaré were using this argument, we should expect him to claim that conventions are very widespread in science, however, this is a position that Poincaré emphatically rejects in his criticism of Édouard Le Roy, who espoused such a view [Poincaré 1902c].
3We can start to see the influence that Schlick had by first considering his impact on Hans Reichenbach. After Reichenbach had published Relativitätstheorie und Erkenntnis Apriori [1920], Schlick complained to him in a series of letters that rather than modifying a priori knowledge to maintain a Kantian viewpoint, he should jettison both Kant and the a priori and adopt an empiricism augmented with what Schlick took to be Poincaré’s conventionalism. In other words, rather than accounting for the existence of non-Euclidean geometries with a theory of the changing a priori, we should account for them as conventions. Reichenbach had famously distinguished between two elements of the Kantian a priori, the a priori as necessary and the a priori as constitutive. He had argued that we should give up on necessity, given that the existence of non-Euclidean geometries showed that there were alternatives, but that we should maintain the idea of the constitutive a priori as preconditions for further inquiry. We thus end up with a relative a priori, which has recently been developed by Michael Friedman [2001] and others.
- 2 Friedman argues that Schlick changed his view as well on account of the letters, taking convention (...)
4Reichenbach’s views on the relative a priori changed after his exchange of letters with Schlick.2 He dropped the idea of a relative a priori and held a kind of conventionalism instead, developed in his book on spacetime [Reichenbach 1928]. Reichenbach discusses his changed view in print, noting that he had taken Schlick to be ignoring the constitutive significance of some concepts used in science, but later saw their dispute as terminological [Reichenbach 1922], [Reichenbach & Cohen 1978, 37–39]. Friedman rejects the idea that the dispute between Reichenbach and Schlick is merely terminological, and instead describes it as follows:
Schlick argues that we should no longer characterize constitutive principles—for example, and especially, the principles of geometry—as a priori at all: we should rather characterize them as conventions in the sense of Henri Poincaré. [Friedman 1991], [Friedman 1999, 63]
5The phrase “conventions in the sense of Henri Poincaré” is taken directly from Schlick and is, of course, predicated on Schlick’s interpretation of Poincaré, which is an especially linguistic interpretation having to do with how we define the terms we use in a scientific theory.
From Hilbert, Schlick adopts the idea that the axioms of geometry “implicitly define” the primitive terms of that science. This explains why the axioms of geometry are both nonempirical and conventional: Alternative systems of geometry—Euclidean or non-Euclidean—simply count as different definitions of “point,” “line,” “between,” and so on. Hilbert’s view thus accounts for the nonempirical status of pure geometry. But we need to add Poincaré’s ideas to account for applied geometry. Here Schlick reasons as follows: In applying such a purely formal system of implicit definitions to our actual experience of nature, no merely empirical considerations can force us to adopt one system rather than another; rather, only experience plus the requirement of overall simplicity of the laws of nature yield a determinate such system. [Friedman 1991], [Friedman 1999, 64]
- 3 See [Massey 1964, chap. 3] for a survey of the literature and compare [Grünbaum 1968, 106 ff.].
6Friedman agrees that, on Schlick’s interpretation, conventionalism is equivalent to the Duhem-Quine thesis and that it is more general than Poincaré’s conventionalist thesis. Reichenbach’s discussion of his theorem in his Philosophie der Raum-Zeit-Lehre [1928] seems to follow exactly the line of argument in the Duhem-Quine thesis. He says that our theory of space is comprised of a geometry plus a physics. We can change geometries by adjusting our physics [Reichenbach 1928, 1958, 33]. In his early work, Friedman discusses this so-called epistemological argument for conventionalism especially with regard to Reichenbach, but considers the argument as a standard line used by everyone including Poincaré [Friedman 1983, 296–301]. Indeed, in the early Anglo-American literature on spacetime, almost everyone except Adolf Grünbaum interprets Poincaré as using the epistemological argument for conventionalism.3 Further evidence that the epistemological interpretation of Poincaré’s conventionalism was commonplace can be found in the fact that Carnap casually equates Duhem and Poincaré in a comment in Logische Syntax der Sprache [1934], showing how much the link between Duhem and Poincaré was taken for granted:
Thus the test applies, at bottom, not to a single hypothesis but to the whole system of physics as a system of hypotheses (Duhem, Poincaré). [Carnap 1934, 1937, 318]
7Unfortunately, Schlick’s interpretation of Poincaré leads directly to a refutation of conventionalism [Schlick 1935]. Interpreted as so-called “implicit definitions” conventions are either merely changes in definitions of words—this was dubbed Trivial Semantic Conventionalism in the literature—or they are empirical and hence not conventional at all. It is worth noting that Poincaré explicitly denies the trivial reading of conventionalism:
If therefore, during an eclipse, it is asked: Is it growing dark? Everyone will answer yes. Doubtless those speaking a language where bright was called dark, and dark bright, would answer no. But of what importance is that? [Poincaré 1913, 328, translation slightly modified]
There must be more to Poincaré’s conventionalism that a simple change of one word for another.
8My point here is that strict empiricism makes conventionalism impossible to understand. We have a strict dichotomy between merely verbal changes and changes in physical theory. Conventions fit in neither category. As Friedman points out, Reichenbach’s early critique of conventionalism in his 1920 book on the relative a priori is important and is lost in his later work where he adopts Schlick’s point of view. Reichenbach’s early point is that the general theory of relativity has shown that metric conventionalism is false.
Reichenbach cannot accept Poincaré’s (and Schlick’s) conventionalism as a general philosophical doctrine about geometry as such, independently of any specific theoretical context, and it is for precisely this reason Reichenbach explicitly rejects conventionalism in the Introduction to his book... For Schlick, by contrast, geometry remains conventional or nonempirical in the context of the general theory of relativity. Duhemian holistic considerations still apply, and all that general relativity actually shows is that the simplest total system of natural laws employs non-Euclidean geometry: Euclidean geometry thus remains an equally “correct” option. Moreover, Reichenbach [1922, 1978] agrees with Schlick on this crucial point. [Friedman 1991], [Friedman 1999, 66–67]
9Although the underlying mathematical structure of the general theory of relativity does not have a determinate metric, metric is determined empirically in the general theory of relativity by the distribution of mass, a point that was not widely understood until the 1970s and 1980s, that is, well after the Logical Empiricist discussion of spacetime theories. Writing in 1929, Reichenbach clearly expresses the standard Logical Empiricist view that there is no a priori knowledge. He refers to his early work [1920], but is now fully critical of Kant, saying that logic and definitions are presupposed but “empty,” and that “physics contains no principles that are a priori in the sense of ‘independent of experience.’ We hold to even the most general principles of knowledge solely because they prove themselves in experience” [1929, 172].
10To sum up what we can learn from this example, we see that interpreting Poincaré from a particular point of view leads to seeing him as holding a view which not only does not fit with what he says about conventionalism, but also self-destructs. Conventionalism becomes true only in a trivial sense. Fortunately, the current literature on Poincaré has moved past this interpretation.
11Poincaré uses a series of analogies to explain his conventionalist view of the geometry used in physics. He says that geometry is the language of physics and that we can translate a physical theory that uses Euclidean geometry into one which uses non-Euclidean geometry in much the same way as we might translate a French text on physics into German [Poincaré 1900, 79], [Poincaré 1913, 93, 338]. Alternatively, he claims that in physics, Euclidean and non-Euclidean geometries are as interchangeable as Cartesian and Polar coordinates or, as interchangeable as metric and British units of measure, or as interchangeable as the Fahrenheit and Reaumur temperature scales [Poincaré 1886-1887, 90–91], [Poincaré 1899, 266] and [Poincaré 1913, 82, 238, respectively]. These passages unfortunately give credence to critics who say that Poincaré’s conventionalism is based on changes in coordinates or in systems of measurement and true only in a trivial sense. There is clearly more to Poincaré’s translation thesis, but passages such as these leave Poincaré’s translation thesis quite vague and there have been repeated attempts in the literature to specify what he could mean. I will show how the literature on Poincaré’s translation thesis goes astray.
12In a well-known article, Max Black interprets Poincaré’s translation thesis as following from the existence of Euclidean models of Lobachevskiian geometry and says that the existence of these models is the main basis for Poincaré’s conventionalism [Black 1942, 341]. In fact, the models show the relative consistency of Lobachevskiian geometry and say nothing about why geometry is conventional, that is, why either Euclidean or a non-Euclidean geometry could be used in physics. Black interprets Poincaré’s argument for the conventionality of metric geometry as a term-by-term translation of Euclidean and non-Euclidean geometry, a view introduced by Louis Rougier, whom Black cites [Rougier 1920, 114].
A mere outline of the procedure involved will suffice. It consists, in effect, of supplying a “dictionary” which so defines every term of H [Lobachevskii’s geometry] by means of terms drawn exclusively from E [Euclidean geometry] that when all the terms are replaced by their definientia every sentence of H becomes transformed into a sentence of E [...]. We may add that the converse is true, each of the geometries E, H and R [Riemann geometry] being convertible in the manner described into each of the others.
Poincaré prefers to summarize these remarkable relations by saying that hyperbolic geometry (H) has been translated into Euclidean geometry, with the implication that the apparently diverse formulations of pure geometries are merely alternative formulations of the very same system of abstract relationships. [Black 1942, 341–342, Black’s emphasis]
- 4 See [Tuller 1967, 165 ff.] and [Torretti 1978, 136–137] for more on Poincaré’s model, [Milnor 1982 (...)
13Black claims that reciprocal translations between geometries of different curvatures is possible, but as Gerald Massey points out, in the passage upon which Black bases his interpretation, Poincaré never claims to have given a reciprocal translation of Euclidean and non-Euclidean geometries [Massey 1964, 103–104]. In the section of Science and Hypothesis discussed by Black, Poincaré is defending Lobachevskiian geometry from a priori objections that Euclidean geometry is the only possible geometry. To this end he introduces a model, which he does call a “translation,” in order to show that Lobachevskii’s geometry is consistent relative to Euclidean geometry [Poincaré 1913, 59–60].4 Poincaré shows that the theorems of Lobachevskiian geometry can be interpreted as a subclass of the theorems of Euclidean geometry, but this does not support Black’s claim of reciprocal translation [Massey 1964, 105].
14Even though Massey is correct that in the section of Science and Hypothesis discussed by Black, Poincaré does not claim that Euclidean geometry is translatable into non-Euclidean geometry, it seems wrong to say that Poincaré never makes such claims [Massey 1964, 113, 123]. For example, in Science and Method Poincaré tries once again to explain what he means by his claim that Euclidean geometry and non-Euclidean geometries are conventional alternatives in physical theory:
[...] we have fixed rules which permit us to translate French enunciations into German, and inversely. It is for that that grammars and dictionaries have been made. There are also fixed rules for translating the Euclidean language into the non-Euclidean language, or, if there are not, they could be made. [Poincaré 1913, 338]
15I do not think that Black overstates Poincaré’s thesis. However, this passage also provides clear evidence that Poincaré does not consider his model to be a reciprocal translation. If he did, he would refer to the model published several years earlier rather than merely saying that the rules for translating the Euclidean language into the non-Euclidean language could be made. It remains unclear what Poincaré means by translation of Euclidean geometry and non-Euclidean geometry in this context.
16Ernst Nagel also takes Poincaré’s translation claim to be essential to his conventionalism and he furthers the discussion by considering whether Euclidean and non-Euclidean geometries would be translatable if these geometries were mutually isomorphic, have the same class of models, or were notational variants of each other [Nagel 1979, chap. 9]. These ideas have been further elaborated and criticized by Massey [1964, chaps. 4 and 5], Roberto Torretti [1978, 325–327] and Peter Spirtes [1981, chap. 3 and appendix B] who show that Black and Nagel’s claims are either false or else true in such a way that they would not support the conventionality of metric. For example, Massey correctly argues that Euclidean geometry and Lobachevskiian geometry are not mutually isomorphic in a logical sense:
Two deductive systems, S1 and S2, are isomorphic (Carnap) if there is a one-to-one mapping f of the primitive symbols of S1 onto those of S2 which maps wffs [well-formed formulas] into wffs and which preserves consequence relations among wffs [...] it is evident that [Poincaré’s] dictionary does not purport to provide a one-to-one correspondence between the primitive symbols of H and E. [Massey 1964, 127]. See [Carnap 1934, 1937, § 61] and [Nagel 1979, 252]
17Torretti argues that one part of Black’s interpretation, namely that axiomatic geometries are translatable into their contraries, is false and that even if it were true, it would not support metric conventionalism [1978, 325–327].
18We also should be suspicious of Black and Nagel’s interpretation of Poincaré because it is anachronistic. Poincaré did not consider geometries as logically isomorphic axiomatic systems, which are required to understand Nagel’s interpretation of Poincaré’s translation thesis. To understand Poincaré’s metric conventionalism this way, it must be possible to express physical theory as an axiomatic system, but it is clear that Poincaré never does so. Like others in their generation, Black, Nagel, Massey, and Spirtes all assume that it is possible to express physical theory as a formal axiomatic system and use the tools of meta-mathematics to discuss these theories. Given that these do not apply to Poincaré, we are no closer to understanding what he means by translation. When Poincaré introduces a term-by-term translation of the primitives of Lobachevskiian geometry in his model, the point of his argument is that Lobachevskiian geometry is consistent relative to Euclidean geometry. His argument for considering Euclidean and non-Euclidean geometries as alternatives in physics has a completely different basis. We must look at Poincaré’s arguments for conventionalism in physical geometry to see what translation means in this context.
19Given that the literature discussing the standard interpretation of Poincaré’s translation thesis ultimately refutes a position which he never held, we would do better to follow Grünbaum when he suggests that Poincaré’s translation thesis simply means that it should be possible to develop physics using either Euclidean or a non-Euclidean geometry [Grünbaum 1968, 107–108]. In other words, we should be able to rewrite a physical theory using Euclidean geometry as one using a non-Euclidean geometry, and vice versa, and express the same facts. The central point in Poincaré’s conventionalism is the claim that we may choose any metric geometry in physics and our choice will not be right or wrong, a claim that he expresses in an early article in terms of a choice of a group within the theory of transformations.
- 5 Torretti’s translation [1978, 335]. Giedymin [1982], Johnson [1981, 87, n. 5] and Torretti have al (...)
Among all possible groups, we have chosen one in particular, in order to refer to it all physical phenomena, just as we choose three coordinate axes in order to refer to them a geometrical figure.5 [Poincaré 1886-1887, 91]
- 6 It is worth noting that Schlick takes up this precise case when explaining Poincaré’s conventional (...)
20Notice that Poincaré uses one of the analogies he used in expressing the translation thesis—we choose a metric geometry much as we would choose a coordinate system. I claim that Poincaré’s study of the theory of group transformations (and later topology) is central to understanding his thesis that Euclidean and non-Euclidean geometries can be used interchangeably in physics. Felix Klein had shown that Euclidean, Lobachevskiian, and elliptic groups of motions are subgroups of the projective group [Torretti 1978, chap. 2.3.8]. After reviewing Hilbert’s Grundlagen der Geometrie in 1902, Poincaré changed from a group-theoretic to a topological approach [Johnson 1979, 103–104], [Poincaré 1902a] and I will adopt this in my exposition. Poincaré develops the idea of alternative descriptions of physical phenomena by considering and extending the case of a uniformly expanding universe, which he says would be undetectable.6 He uses this thought experiment to support his claim that either Euclidean or a non-Euclidean geometry may be used in physics because both can account for the same information. Furthermore, the point transformations show that a Euclidean framework can be transformed into a non-Euclidean one, and vice versa.
[...] worlds will be indistinguishable not only if they are equal or similar, that is, if we can pass from one to the other by changing the axes of coordinates, or by changing the scale to which lengths are referred; but they will still be indistinguishable if we can pass from one to the other by any “point-transformation” whatever. I will explain my meaning. I suppose that to each point of one corresponds one point of the other and only one, and inversely; and besides that, the coordinates of a point are continuous functions, otherwise altogether arbitrary, of the corresponding point. I suppose besides that to each object of the first world corresponds in the second an object of the same nature placed precisely at the corresponding point. I suppose finally that this correspondence fulfilled at the initial instant is maintained indefinitely. We should have no means of distinguishing these two worlds one from the other. The relativity of space is not ordinarily understood in so broad a sense; it is thus, however, that it would be proper to understand it. [Poincaré 1913, 237–238, Poincaré’s emphasis]. Also [1963, 16, 27]
21In the case of a uniformly expanding universe, we consider a constant function such that each point (x, y, z) in R3 is associated with another point (ax, ay, az), where is a constant. Poincaré asks us to consider the case where each point (x, y, z) in R3 is associated with another point (f(x), g(y), h(z)), where f, g and h are arbitrary continuous one-to-one functions. Poincaré ’s thesis that Euclidean and non-Euclidean geometries are translatable follows from his arguments that Euclidean and Lobachevskiian spaces can account for the same information, are indistinguishable topologically, and point transformations can get us from one space to another. As Torretti points out in a discussion of the early statement of Poincaré’s conventionalism (quoted above, [Poincaré 1886-1887, 91]), it is clear that the underlying manifolds of Euclidean and Lobachevskiian geometries are homeomorphic.
The Euclidean group and the BL [Bolyai-Lobachevskii] group are topological groups which can be assumed to act transitively and effectively on one and the same topological space. Each is indeed isomorphic with a different subgroup of the group of continuous transformations of R3. Obviously, every item of information concerning figures in R3 can be conveyed in terms of the invariants of either group. Consequently, if, as in classical mechanics, physical space is assumed to be homeomorphic with R3, both Euclidean and BL geometry can be used as a framework for the geometrical description of physical phenomena. [Torretti 1978, 336]
22In the section of The Value of Science where Poincaré discusses the expanding universe, the issue is precisely whether it is possible to choose either Euclidean or a non-Euclidean geometry in physics. Poincaré is committed to the view that there is no sense to the claim that the world is either Euclidean or non-Euclidean because he holds that metric is conventional—we must choose which geometry to use in physics. He further claims that any experiment which seems to prove that the world is non-Euclidean can be rewritten in a Euclidean framework and his arbitrary continuous transformation argument gives him the basis for doing so, given that the two spaces can express the same information. Poincaré holds that science is cumulative and that its (true) claims are never overturned. That is his motivation for saying, notoriously, that Euclidean geometry will always remain the preferred geometry in physics [Poincaré 1913, 65]. The passage quoted above on point transformations is Poincaré’s most specific and least metaphorical explanation of what he means by the claim that Euclidean and non-Euclidean geometries are translatable, that is, his claim that Euclidean and non-Euclidean geometries are conventional alternatives in physics [Poincaré 1913, 238]. Both geometries can account for the same information, which is why they are indistinguishable. Furthermore, one can be transformed into the other, which is the basis of the translation thesis.
- 7 Johnson points out that, contrary to standard usage, Poincaré seems to refer to both spherical and (...)
23Another reason to interpret Poincaré’s translation thesis as following from the fact that the underlying manifolds of Euclidean and Lobachevskiian geometries are homeomorphic is that it explains his restriction of his thesis to Euclidean and Lobachevskiian geometries in many important passages. Euclidean and Lobachevskiian geometries are topologically equivalent globally, while other geometries can be considered topologically equivalent only from a local perspective, that is, patchwise [Torretti 1978, 336]. While Poincaré uses the general term “non-Euclidean geometry,” in the quote I gave above, he refers to “triangles the sum of whose angles is less than two right angles” as the alternative to Euclidean triangles in the same section of the text, without considering the possibility of triangles the sum of whose angles is greater than two right angles [Poincaré 1913, 235]. Poincaré also limits his claim of translatability to Euclidean and Lobachevskiian geometries in the early statement of his conventionalist thesis which I quoted above [Poincaré 1886-1887, 90] and in his discussion of a non-Euclidean body which I treat elsewhere [Stump 1991, 651–654]. Torretti claims that Poincaré might have mistakenly generalized his arbitrary continuous transformation argument to spherical geometry as well as Euclidean and Lobachevskiian geometries [Torretti 1978, 336]. I suggest instead that Poincaré restricts his translation thesis—using the term “non-Euclidean” as synonymous with “Lobachevskiian” in this context. Of course, his general conventionalist thesis that non-Euclidean geometries could be used in physics is broader. For example, in his famous discussion of the parallax experiment, Poincaré includes Riemannian geometry [1913, 81].7 In any case, Poincaré’s translation thesis depends on his broader conventionalism since arbitrary continuous point-transformations would only be possible in physics if there is a physical theory in which metric relations were not factual. Such a theory would be possible if Poincaré could develop a purely relational dynamics so that only topological features of space are non-conventional. I have argued elsewhere that his conventionalism must be seen as part of a program for such a purely relational theory [Stump 1989]. Poincaré’s statement that Euclidean and Lobachevskiian geometries are translatable can then be seen as part of his claim that physics can be developed using either of these geometries. The translation thesis gives Poincaré a method for discounting experiments which show that space is Lobachevskiian. Assuming his conventionalism is correct, he would be able to rewrite any such experimental data in the Euclidean framework of classical mechanics. Thus, we can understand Poincaré’s confidence that Euclidean geometry will never be overturned as following from his relationalism and general topology. Of course, Poincaré’s relational and conventional program for theories of space is incomplete, but it is a more consistent and well-grounded program than many of his critics have allowed.
24We have now seen two examples of influential interpretations of Poincaré that go astray. Both interpretations ignore the context in which Poincaré was working by taking his views to represent a position in another context—empiricism versus neo-Kantianism in the case of Schlick and formal axiomatic systems in the case of Black and Nagel. We are no doubt more sensitive to historical context now and have done better at interpreting Poincaré, but there is more work to be done. I will now consider an admirable recent attempt at capturing Poincaré’s thought.
25The breadth of Gray’s intellectual biography [2013] is quite amazing and makes one realize even more how amazing Poincaré’s breadth was. Gray manages to discuss Poincaré’s work in pure mathematics, in mathematical physics, and in philosophy, all at a very high level. Gray made a decision to study each area of Poincaré’s inquiry separately, following a chronology within each area. This was undoubtably a good choice, because otherwise the reader would be so swamped in detail and the transitions would be so swift that most readers would simply get lost. By contrast, Gray develops a narrative for each area of inquiry that allows the reader to see the development of Poincaré’s thought in each. The narrative gains considerably by maintaining the cohesiveness of Poincaré’s work in mathematics, in philosophy and in mathematical physics respectively. Given that Poincaré worked to a remarkable extent in diverse areas simultaneously, Gray’s choice to arrange by topic functions to create a coherence that may not exist in a comprehensive chronology. Thus, there is a trade off in Gray’s division of areas of inquiry into separate chapters. We lose the possibility of seeing connections between what Poincaré was doing in separate fields at any given time—how or whether the mathematics that he was working on at a particular time affected the philosophy of that time and vice-versa, or how Poincaré’s understanding of physical theory was informed by his work on mathematics or philosophy. My concern is that Gray’s division into separate areas of study eliminates even the possibility of there being such a connection. Of course, many have noted how Poincaré’s early work on Fuchsian functions and group theory influenced his philosophy of geometry, but that is a connection that spans a good deal of time. I am thinking here of the possibility of temporally contiguous and much more specific connections.
- 8 For a complete list of the works that Poincaré published in 1904, see the Website Henri Poincaré P (...)
26We should examine whether there are such specific connections and see what we can find. However, a complete chronological study of all that Poincaré did, year by year or even moment by moment seems an impossible task. Poincaré accomplished all that he did in his rather short lifetime, but it might require several lifetimes to follow his train of thought. What I would like to propose is the study of all the areas of inquiry within which Poincaré worked, but in a narrow time frame. For example, what was Poincaré working on in 1904, when he took a trip to America to give a speech at the St. Louis World’s Fair (a speech in which he first used the term “principle of relativity”), put together his book La Valeur de la science (published 1905) and published twenty two works, including a book on electricity, articles on astronomy, physics and mathematics (such as the 5 complement on Analysis Situs), a nomination of Hilbert for the Lobachevskii Prize for his Grundlagen der Geometrie as well as several philosophical and popular articles.8 How do these disparate works relate to each other, besides being written by one individual at more or less the same time? I believe that this kind of study has not yet been attempted.
27Thanks to the groundwork at the Archives Henri Poincaré, such a study should be possible, but the works to be considered are quite diverse. To extend consideration of a few of the works that I have mentioned, in Poincaré’s St. Louis address, we see reflections on the current status of physical theory, with clear indications that Poincaré thinks that great changes will take place in physics soon [Poincaré 1904a]. In his nomination for Hilbert, Poincaré presents the advances that Hilbert made, saying that these alone are enough to justify the awarding of the Lobachevskii prize, and that any further commentary would be useless [Poincaré 1904b, 39]. Interestingly, while the 1904 nomination follows closely the review of Hilbert that Poincaré had published two years earlier [Poincaré 1902a], the comments critical of Hilbert at the end of the review are not repeated in 1904. The Value of Science is, of course, largely a defense of science and a repudiation of Le Roy’s radical interpretation of Poincaré’s conventionalism (Poincaré labels Le Roy’s view “nominalism”). On the face of it, these three works are totally independent of one another, and yet, Poincaré was working on them at more or less the same time. However, we might consider the idea of radical change in science as a common element between two of the publications, given that it is mentioned in the St. Louis address and in Poincaré’s defense of science in The Value of Science [1905a, 23], [1913, 140]. Furthermore, Poincaré’s lectures in physics are in large part presentations of recent work being done, so this could be a source of the idea of continuing radical change in science. Even his review of Hilbert shows that he is open to changing ideas in mathematics and science. Of course, part of Poincaré’s defense of science is an argument that something remains constant even when science changes radically. Established evidence is conserved through changes in theory, expressed in the relations between objects. Poincaré sees both constancy and change in science.
28As Ferdinand Verhulst notes in the preface to his biography, it is hard for us to keep up with Poincaré, even a century after his death [Verhulst 2012, xx]. We have, however, made much progress towards a better and more comprehensive understanding of Poincaré’s work. There is no doubt that the work will continue and build on the foundation that has been put in place.