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In his article "On the Foundations of Geometry", Poincaré goes through an extensive discussion to establish that our experience of motion may be properly regarded as displacements. This is requisite to proving that our experience of the world is characteristic of groups (in the mathematical sense of the term). He does this by arguing that an external change A can be undone by an voluntary change A'. For example, if an object moves from the left side to the right of my field of vision, I can voluntarily restore it to its original position on the left by redirecting the direction of my gaze. Proceeding with this line of thought, he goes on to assert the following:

Nor is this all. If two external changes α and α' are regarded as identical on the basis of the convention adopted above, or in other words, are susceptible of being corrected by the same internal change A; if, on the other hand, two other external changes β and β' can be corrected by the same internal change B, and consequently may also be regarded as identical, have we the right to conclude that the two changes α + β and α' + β' are susceptible of being corrected by the same internal change, and are consequently identical? Such a proposition is in no wise evident, and if it be true it cannot be the result of a priori reasoning.

His argument seems to betray a misunderstanding of the issue. If these relations are determined a priori, questioning whether or not they are immediately evident seems to be rather idle. What needs to be asked is whether it's possible for us to implement these internal changes, B + A, which would result in correcting, let's say, α + β, but having some entirely different result from the expected correction of α' + β'. Similarly, it might be asked if it could result in correcting α' + β' but having an unexpected result with respect to α + β.

The nature of a priori truth has nothing to do with how easily it may be verified; rather, it pertains to whether it is determined by our faculties as opposed to being determined by the world represented by those faculties. Kant makes this point clear by pointing out that the understanding has its own particular nature which may not be immediately evident to us:

"[The categories] are merely rules for an understanding... But to show reasons for this peculiar character of our understandings, that it produces unity of apperception a priori only by means of categories, and a certain kind and number thereof, is as impossible as to explain why we are endowed with precisely so many functions of judgement and no more, or why time and space are the only forms of our intuition."

Did Poincaré simply not understand the nature of a priori truth, or is there some other argument that he employs to establish his claim, i.e. an argument that has nothing to do with whether or not things are immediately evident?

  • "Poincaré argues that (metric) geometry is neither a priori nor empirical, but rather conventional." – Mauro ALLEGRANZA Nov 14 '16 at 9:58
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Poincare makes exactly the same point as you in Science and Hypothesis, distinguishing between the "order which is external to us" and the "property of the mind itself". And he explains how experimental laws are converted into a priori principles using it:

"...postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate... [Experimental law] expressed a relation between two real terms, A and B; but it was not rigorously true, it was only approximate. We introduce arbitrarily an intermediate term, C, more or less imaginary, and C is by definition that which has with A exactly the relation expressed by the law. So our law is decomposed into an absolute and rigorous principle which expresses the relation of A to C, and an approximate experimental and revisable law which expresses the relation of C to B."

So it seems there is no misunderstanding on his part. But in the OP quote the relations are not defined a priori, α-s and ß-s are "external changes". Poincare's point seems to be that although α and α' are corrected by A, and ß and ß' by B, there is no telling a priori whether external "adding" will still be "susceptible of being corrected" by A+B for both pairs, no matter how our faculties might "craft" A+B. Even if the constituents are "equivalent" there is no a priori guarantee that their "sums" will be, because establishing "equivalence" is mediated by external input. A good summary of Poincare's views is Brading's Epistemic Structural Realism and Poincare's Philosophy of Science.

A bit of context. Poincare was largely a thoroughgoing Kantian, but not entirely. After the discovery of non-Euclidean geometries the Kantian notion of a priori had to be revised. One revision, by neo-Kantian philosophers, eventually led to what is now called relativized a priori, relativized to evolving "paradigms" and fallible along with them. But 19-th century Kantian scientists, like Helmholtz, Riemann and Poincare, were more conservative. According to them, our "a priori form of outer intuition", space, was not so much revisable as vague. Too vague to single out the 3D Euclidean geometry, as Kant thought, but only a family of geometries (they differed on how broad), leaving the deciding among them beyond a priori. Helmholtz and Riemann thought it was an empirical matter. But Poincare's "spatial a priori" were too vague even for that:

"...there is in all of us an intuitive notion of the continuum of any number of dimensions whatever because we possess the capacity to construct a physical and mathematical continuum; and that this capacity exists in us before any experience... And yet this faculty could be used in different ways; it could enable us to construct a space of four just as well as a space of three dimensions. It is the exterior world, it is experience which induces us to make use of it in one sense rather than in the other... The geometrical axioms are therefore neither synthetic a priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free..."

Finally, what if the relations were determined exclusively by our "faculties", would they then be "immediately evident"? This is reminiscent of Wittgenstein's rule-following paradox as interpreted by Kripke. Suppose so far we performed addition only on numbers less than 57 and now are asked to compute 57+68. "Clearly", the correct answer is 125 because "we mean the addition by +". Kripke's Wittgenstein asks to point out a single fact which certifies that we did not instead "mean" quaddition: it coincides with the regular addition for all numbers less than 57, but returns 5 whenever one of them is 57 or more. And proceeds to show methodically that we can't. References to rules and instructions do not help, the same question can be asked about "meaning" the same rule. If it seems like a stretch replace 57 with some number that exceeds the number of atoms in the universe, is it "immediately evident" what we "mean" by adding them even though they are productions of our faculties?

All our faculties can do when it comes to generalities is rely on a sense of "uniformity", not unlike Humean "uniformity of nature". Of course, it is not the same, productions of our faculties are more accessible and reliable. But the modern view is that we can err even about our own productions. No matter how "immediately evident" it was to Frege that every non-vague property defines a class, or to Cantor that the continuum hypothesis is either true or false, it did not work out that way. Even a priori are fallible. Contra Descartes and Kant nothing is immediately evident, not even "the mind itself", see What are the more complex/interesting examples of synthetic a priori statements?

  • Thanks for the answer! In defense of Kant, non-Euclidean geometry already existed in his day. Perhaps he perceived no conflict with it since a Euclidean frame of reference is always used for describing how we can treat curves as if they were straight. As far as the number of dimensions is concerned, our visual system is specifically wired for three and lacks the physiological equipment for four. – user3017 Nov 15 '16 at 8:35
  • @Pe "non-Euclidean geometry already existed in his day". In Kant's day? – Ram Tobolski Nov 15 '16 at 17:07
  • @RamTobolski. Yes. Theodosius of Bithynia, for example, wrote a book on spherical geometry prior to the first century B.C. – user3017 Nov 15 '16 at 17:41
  • @PédeLeão Yes, spherical geometry is ancient. But it was not a non-Euclidean geometry in the now-familiar sense. It was geometry on a sphere, rather than on a plane. Nobody thought (afaik) about the sphere as a "non Euclidean plane" prior to the 19th century – Ram Tobolski Nov 15 '16 at 19:35
  • @RamTobolski. But it was, in fact, non-Euclidean geometry in every sense of the word. However, it's also true that there was no perceived conflict with these two types of geometry, because it involves redefining commonly used terms, such as straightness and line. For that reason, it's a hollow argument to attribute error to Kant for no other reason than that the ordinary meaning of words could consistently take on a alternative meanings. How are we justified in interpreting him as precluding other metrics when doing so would require the use of radically different language? – user3017 Nov 15 '16 at 20:39

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