Kant on mathematics

I recently learned about Kant’s theory that mathematics and philosophy are both a priori, but philosophy proceeds through concepts, while mathematics proceeds through the construction of concepts in “intuition” – which, in this context, I believe to be Kant’s name for what I call imagination. This made me think the following thoughts about it. I may have more thoughts later on, or revise these thoughts; I have not finished reading any of the relevant works by Kant.

0. Contents

1. Comparison of Kant’s theory with Ptolemy’s views on mathematical science
2. Interpretation of Kant’s theory, with respect to the necessity of theorems
3. Illustration, drawn from Poincaré, of the foregoing interpretation
4. Comparison of Kant’s theory with Poincaré’s views on mathematical induction
5. Appendix – translation of Lalande, “tautology”, sense B and observations

1. Comparison of Kant’s theory with Ptolemy’s views on mathematical science

Kant’s theory reminded me of the preface that the ancient astronomer Ptolemy wrote to his Almagest, as it may be read in G.J. Toomer’s translation. For copyright reasons, I quote only part of it. Ptolemy comments on Aristotle’s division of theoretical philosophy into physics, mathematics, and theology, and explains his own preference for mathematical studies:

The division of theoretical philosophy which investigates material and ever-moving nature, and which concerns itself with ‘white’, ‘hot’, ‘sweet’, ‘soft’ and suchlike qualities one may call ‘physics’; such an order of being is situated (for the most part) amongst corruptible bodies and below the lunar sphere. That division of theoretical philosophy which determines the nature involved in forms and motion from place to place, and which serves to investigate shape, number, size, and place, time and suchlike, one may define as ‘mathematics’. Its subject-matter falls as it were in the middle between the other two, since, firstly, it can be conceived of both with and without the aid of the senses, and, secondly, it is an attribute of all existing things without exception, both mortal and immortal: for those things which are perpetually changing in their inseparable form, it changes with them, while for eternal things which have an aethereal nature, it keeps their unchanging form unchanged.

From all this we concluded: that the first two divisions of theoretical philosophy should rather be called guesswork than knowledge, theology because of its completely invisible and ungraspable nature, physics because of the unstable and unclear nature of matter; hence there is no hope that philosophers will ever be agreed about them; and that only mathematics can provide sure and unshakeable knowledge to its devotees, provided one approaches it rigorously. For its kind of proof proceeds by indisputable methods, namely arithmetic and geometry.

For a better parallel with Kant, and without violence to the thought of any of these three authors, we may refer to what Aristotle and Ptolemy called “theology” as metaphysics. As seen in this passage, then, Ptolemy seems to have been aware, like Kant, that mathematics and metaphysics are both a priori, unlike physics, which can only be conceived of with the aid of the senses. He also notes, similarly to Kant, that mathematics, unlike metaphysics, treats of visible things, and therefore can make use of the senses. Though predating Kant’s distinctions, he seems even to have come to Kant’s conclusion that mathematics is more evidently an actual body of synthetic a priori knowledge than metaphysics is.

2. Interpretation of Kant’s theory, with respect to the necessity of theorems

Since the imagination is a sensitive power, distinct and separate from reason, I would interpret Kant’s theory as meaning that mathematical judgments are not necessarily true of their objects, but only of their objects as they may be constructed through the forms of human sensible intuition.

These forms, in turn, are essential to human beings, so that mathematical judgments really are necessary truths – but they are necessary to the human nature, not to the nature of the mathematical objects that they involve. A being with different forms of sensible intuition is conceivable, and such a being could make different, and equally true, mathematical judgments about the same mathematical objects.

The interpretation is made clearer by thinking of different axiomatic mathematical systems, which give different interpretations of such human concepts as lines and planes. You could, conceivably, write the same definition of triangle in a non-Euclidean system of geometry that you would use in the Euclidean one, but because the systems interpret the terms differently, it would have different consequences, such as, for instance, its angles not adding up to 180 degrees. This is similar to how a being with different forms of sensible intuition would construct different relations between the same mathematical concepts.

Only the discoveries of deduction and analysis are absolutely necessary to their objects. What we find through our senses is, at best, necessary to our senses – even if we find it a priori.

3. Illustration, drawn from Poincaré, of the foregoing interpretation

Henri Poincaré, in Science and Hypothesis, provides a helpful illustration of the possibility of rational beings with different forms of sensible intuition. He does so by imagining that they live in a world under certain peculiar conditions. Since the work is in the public domain, I quote the relevant section in full:

Suppose, for example, a world enclosed in a large sphere and subject to the following laws:

1. The temperature is not uniform; it is greatest at the centre, and gradually decreases as we move towards the circumference of the sphere, where it is absolute zero. The law of this temperature is as follows:—If R be the radius of the sphere, and r the distance of the point considered from the centre, the absolute temperature will be proportional to R²−r².
2. Further, I shall suppose that in this world all bodies have the same co-efficient of dilatation, so that the linear dilatation of any body is proportional to its absolute temperature.
3. Finally, I shall assume that a body transported from one point to another of different temperature is instantaneously in thermal equilibrium with its new environment.

There is nothing in these hypotheses either contradictory or unimaginable. A moving object will become smaller and smaller as it approaches the circumference of the sphere. Let us observe, in the first place, that although from the point of view of our ordinary geometry this world is finite, to its inhabitants it will appear infinite. As they approach the surface of the sphere they become colder, and at the same time smaller and smaller. The steps they take are therefore also smaller and smaller, so that they can never reach the boundary of the sphere. If to us geometry is only the study of the laws according to which invariable solids move, to these imaginary beings it will be the study of the laws of motion of solids deformed by the differences of temperature alluded to.

No doubt, in our world, natural solids also experience variations of form and volume due to differences of temperature. But in laying the foundations of geometry we neglect these variations; for besides being but small they are irregular, and consequently appear to us to be accidental. In our hypothetical world this will no longer be the case, the variations will obey very simple and regular laws. On the other hand, the different solid parts of which the bodies of these inhabitants are composed will undergo the same variations of form and volume.  Let me make another hypothesis: suppose that light passes through media of different refractive indices, such that the index of refraction is inversely proportional to R²−r². Under these conditions it is clear that the rays of light will no longer be rectilinear but circular.

To justify what has been said, we have to prove that certain changes in the position of external objects may be corrected by correlative movements of the beings which inhabit this imaginary world; and in such a way as to restore the primitive aggregate of the impressions experienced by these sentient beings. Suppose, for example, that an object is displaced and deformed, not like an invariable solid, but like a solid subjected to unequal dilatations in exact conformity with the law of temperature assumed above. To use an abbreviation, we shall call such a movement a non-Euclidean displacement. 

If a sentient being be in the neighbourhood of such a displacement of the object, his impressions will be modified; but by moving in a suitable manner, he may reconstruct them. For this purpose, all that is required is that the aggregate of the sentient being and the object, considered as forming a single body, shall experience one of those special displacements which I have just called non-Euclidean. This is possible if we suppose that the limbs of these beings dilate according to the same laws as the other bodies of the world they inhabit. 

Although from the point of view of our ordinary geometry there is a deformation of the bodies in this displacement, and although their different parts are no longer in the same relative position, nevertheless we shall see that the impressions of the sentient being remain the same as before; in fact, though the mutual distances of the different parts have varied, yet the parts which at first were in contact are still in contact. It follows that tactile impressions will be unchanged. On the other hand, from the hypothesis as to refraction and the curvature of the rays of light, visual impressions will also be unchanged.

These imaginary beings will therefore be led to classify the phenomena they observe, and to distinguish among them the “changes of position,” which may be corrected by a voluntary correlative movement, just as we do. If they construct a geometry, it will not be like ours, which is the study of the movements of our invariable solids; it will be the study of the changes of position which they will have thus distinguished, and will be “non-Euclidean displacements,” and this will be non-Euclidean geometry. So that beings like ourselves, educated in such a world, will not have the same geometry as ours.

4. Comparison of Kant’s theory with Poincaré’s views on mathematical induction

Poincaré also provides a worthwhile comparison with Kant regarding the very main topic of this post, i.e., the type of cognition involved in mathematical reasoning.

Like Kant, Poincaré sought to establish the status of mathematical cognition as synthetic a priori. He did not do so, however, by claiming that mathematics proceeds through construction of concepts in intuition. Indeed, throughout Science and Hypothesis, he seems to use the word intuition in the Cartesian sense, i.e., an indubitable, clear and distinct conception. (This is my interpretation, since he uses the word without ever defining it.) In line with this usage, he actually shares the Cartesian assumption that most mathematical judgments are, in fact, “a series of purely analytical deductions”.

Instead of sharing Kant’s theory, Poincaré defends mathematics as synthetic a priori by interpreting mathematical inductions as being synthetic a priori cognitions, distinct both from deduction and from ordinary (“physical”) induction. As I first explained it in 2020-07-19, in a private Facebook group, Poincaré believed that mathematical induction is distinct in this way for the following reasons.

• It is distinct from deduction because it proceeds from the particular to the general; from seeing that a theorem holds true for a series of a few numbers, we can proceed to the more general theorem. A recursive definition “contains, condensed, so to speak, in a single formula, an infinite number of syllogisms” – while “analytical verification will always be possible” for any particular case, the general case cannot be arrived at purely deductively, because “to reach it we should require an infinite number of syllogisms”.

• It is distinct from physical induction because it is more certain. “Induction applied to the physical sciences is always uncertain, because it is based on the belief in a general order of the universe, an order which is external to us. Mathematical induction—i.e., proof by recurrence—is, on the contrary, necessarily imposed on us, because it is only the affirmation of a property of the mind itself.”

The latter point, of course, amounts simply to saying that it is a priori, which no one doubts; the idea that states of the mind are more certain than sensory impressions also adds to his apparent Cartesianism, which I attribute to his being French.

As I said in the same 2020 post, defending this conclusion was important to Poincaré for this reason:

• This was important to Poincaré because he believed that “the syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle.” If this were true of mathematics, then mathematics could not be a science, according to him. “There is no science but the science of the general” – in order for mathematics to be a science, it must arrive at new general truths which are not reducible to its axioms. By distinguishing mathematical induction in this way, he was able to affirm that mathematics, while maintaining “that perfect rigour which is challenged by none”, is yet not “reduced to a gigantic tautology”.

Poincaré thought, then, that by its reliance on mathematical induction at several key points, the system of mathematics secures the fruitful nature of the science.

I disagree with this characterization of analysis and deduction as fruitless and “tautological” – on this, see the appendix to this post. And at this time, I would say that I agree with Kant’s theory, as I have interpreted it, so that all properly mathematical cognitions rely on constructions of concepts in the imagination. Poincaré’s interpretation of mathematical induction seems to follow as a corollary, then, but I disagree with his Cartesian interpretation of other mathematical judgments.

Poincaré’s point about mathematical induction seems particularly helpful, however, insofar as it seems to show us that any purportedly “purely analytic” axiomatic mathematical systems should not be able to make use of mathematical induction at any point. 

5. Appendix – translation of Lalande, “tautology”, sense B and observations

I made this translation of sense B of the term “tautology”, from André Lalande’s Vocabulaire technique et critique de la philosophie, and posted it to a Facebook group in 2021-07-02. I do not understand French. If I recall correctly, I made it by machine-translating the original French and then hand-correcting it according to an edition in Portuguese. (If this seems unreliable to you, then go read the original – there are no professional English translations available.)

TAUTOLOGY, sense B

(Recent sense.) WITTGENSTEIN proposed to call “tautology” any complex proposition which remains true by virtue of its form alone, whatever the truth-value of the propositions which compose it. Tractatus logico-philosophicus (1922), nos. 4, 46 et seq., p. 96.

This usage was widely spread by the Vienna School, in particular by R. CARNAP; see in particular Die alte und die neue Logik, § VII, in Erkenntnis, vol. 1 (1930), p. 2; French translation by the general VOUILLEMIN, Actualitées scientifiques, no. 76, p. 29.

And as, according to the same authors, all the propositions of logic and mathematics present the character of being thus purely formal, and of teaching us nothing about reality, these sciences are called by them “tautological”. See Observations.

Observations

Whether we admit or not the unity of Logic and Mathematics, and the opinion that all the propositions of these sciences are true by virtue only of the definition of their terms, the words tautology and tautological seem to us ill-chosen to express that character. This choice is no doubt explained: first, by the opposition that one wants to mark between the facts of experience, which increase the matter of knowledge, and pure reasoning, which develops its content; second, by the idea, correct in my opinion, that identity is the ideal of logic. But tautology, in fact, implies equivalence, reversibility; now implication, which is the fundamental relation through which mathematics progresses, is a non-reversible, non-symmetrical relation: a ⊃ b does not necessarily entail b ⊃ a ; and when both are true at the same time, this is vi materiae. The result of a deduction is, therefore, far from being identical, even in pure theory, either to the set of its premises, or to any one of them. It cannot even be said, in many cases, that there is “partial identity” between them. See “Logique et Logistique”, Revue Philosophique, January 1945, p. 76. (A. L.)

On the other hand, G. Bachelard writes, “if mathematics is tautological, why is it so varied, so difficult, so interesting?” cf. “Logique, mathématiques, et connaissance de la réalité”, in the Recherches philosophiques, 1935–1936, p. 450.

Same observations from A. Bridoux, A. Burloud, G. Davy, Ed. Le Roy, D. Parodi.