Universality in Kant's Geometry

In this paper, I argue that the key to Kant’s ability to justify synthetic a priori knowledge of mathematical composition hinges on his argument for the derivability of universality from a particular intuition. I am specifically focusing on Kant’s analysis of Euclidean geometry, in the way that it was largely understood in his own time. Although I consider counter arguments from later mathematicians and philosophers, such as Bertrand Russell, the scope of this paper is simply to recontextualize the metaphysical framework of universality in Kant’s geometric reasoning. In particular, I argue that Kant requires a conception of a concrete universal in order to be able to defend his move from particular to universal, and that, furthermore, this construct may already be implicit in his system.

In his Critique of Pure Reason, Kant introduces his argument that mathematical judgements constitute synthetic a priori knowledge. This argument follows that mathematical judgements must be a priori, and never empirical, because “they carry necessity with them, which cannot be derived from experience.”(Kant B15) What is important to note here is that necessity cannot be derived from experience, but the actual act of mathematical inference still originates from intuition, because “although all these principles, and the representation of the object with which this science occupies itself, are generated in the mind completely a priori, they would still not signify anything at all if we could not exhibit their significance in appearance.”(B299) Geometry, for Kant, cannot be purely formal, even in terms of its axioms, if it is to mean anything, a position which is in accordance with his famous maxim, that “thoughts without content are empty, intuitions without concepts are blind.”(B76) Secondly, mathematics must be synthetic. To this end, Kant gives the example of the statement, “7+5 = 12.”(B16) This statement must be synthetic because, Kant argues, though “one might initially think that the proposition ‘7+5=12’ is a merely analytic proposition that follows from the concept of a sum  of seven and five in accordance with the principle of contradiction… the concept of twelve is by no means already thought merely by my thinking of that unification of seven and five.”(B16) Kant establishes that the difference between analytic and synthetic judgements follows that analytic judgements are “Judgements of Clarification”(B11) and synthetic judgements are “Judgements of Amplification”(B11). As such, we can distinguish one from the other based on the fact that in an analytic judgement “I do not need to go beyond the concept.”(B11) In Kant’s example “All bodies are extended,”(B11) bodies already contain the predicate, extended, so I am simply repeating what was said on the left side of the sentence. If I say 4 = 4, that too is an analytic judgement because the right side of the equation is already contained in the left side of the equation. Synthetic judgements, on the other hand, are judgments in which “the predicate is something entirely different from that which I think in the mere concept of a body in general,”(B12) therefore, the statement 5 + 7 = 12 seems to be synthetic according to Kant’s formulation of the concept. As such, Kant believes that he has established that there are mathematical propositions which are indeed synthetic a priori.

Mathematics then, is something which, while originating in appearances, is grounded a priori, and which is necessary, that is, universal. As such, the truth of Kant’s claim that mathematical knowledge is indeed synthetic a priori hinges largely on the universality of mathematical statements, as something that can be necessary if and only if its failure to occur given sufficient conditions for its occurrence is not possible. Kant, however, holds that such necessary knowledge is given via proofs of concepts in concreto. The area in which we can best examine this exercise is in his treatment of geometry. Kant argues that given the concept of a triangle, the geometer, “through a chain of inferences that is always guided by intuition, he arrives at a fully illuminating and at the same time general solution of the question.”(B745) Kant here states that knowledge of the universal not only can, but must be reached, or at least ‘guided,’ by concrete particulars. Indeed, he endorses the view that an instance of a concept would be able to stand in for a universal, “Thus I construct a triangle by exhibiting an object corresponding to this concept, either through mere imagination, in pure intuition, or on paper, in empirical intuition… The individual drawn figure is empirical, and nevertheless serves to express the concept without damage to its universality.”(B742) This exposes Kant to a serious problem: how can any universal knowledge be derived from particular intuitions?

The problem, then, is to justify how sensibility, which has to do with the immediate relationship between the subject and appearance, is able to grant us general, universal knowledge. In his analysis, Kant claims that “mathematical cognition [is] that from the construction of concepts. .. For the construction of a concept, therefore, a non-empirical intuition is required…”(Kant B742) This leaves open the question of what precisely intuition adds to the conceptual understanding of a mathematical, or, more specifically, geometric object. Indeed, in his Introduction to Mathematical Philosophy, Russell criticized Kant along precisely these lines, “Kant, having observed that the geometers of his day could not prove their theorems by unaided argument, but required an appeal to the figure, invented a theory of mathematical reasoning according to which the inference is never strictly logical, but always requires the support of what is called ‘intuition.’”(Russell 145) At face value, a simple rebuttal could be that Kant is simply not concerned with providing an account of the structure of a rigorous proof, but only of how a subject comes to know a geometric concept. However, Russell does not stop there, he specifically relates the ability to know something to the ability to prove something: “The things in the mathematics of Kant’s day which cannot be proved, cannot be known - for example, the axiom of parallels. What can be known, in mathematics and by mathematical methods, is what can be deduced from pure logic.”(145) Therefore, for Russell, our access to a mathematical concept is justified only if we have a rigorous, deductive proof of that concept. 

Kant defends his position by giving an example of how one would prove that the sum of a triangle’s angles is equal to 180 degrees, in which he argues that casting a particular, concrete example, to stand in for a universal is unproblematic because in the construction of this triangle, “many determinations… are entirely indifferent, and we have abstracted from these differences, which do not alter the concept of the triangle.”(Kant B742) That is to say, when justifying knowledge of a general mathematical concept, what is essential is that there be certain essential properties which remain unchanged for the purpose of the proof, but that the accidental qualities of the triangle are what is subject to variety. From this example we can better understand Kant’s initial claim that mathematical cognition requires intuition. Although Kant allows that such an intuition only exists in the imagination, what he has in mind is the actualization of a general concept in the world as such. That is to say, a concept which, though it may seem prima facie to only exist in abstraction, such as the ideal triangle, is brought into actuality as a particular, but which still continues to be capable of representing all other instances of that same triangle. This does not seem to make much sense: if a triangle is brought over from the purely abstract into the actual, how can it still continue to have any validity over other triangles? Why would the right handed triangle which I draw on a piece of paper have any relation to other right handed triangles beyond both simply having the property of right handedness. Why would this right hand triangle itself be a thing of which those other right hand triangles are members? 

I believe that the best way to make sense of this metaphysical impasse is via a categorical reframing borrowed from Hegel. For Hegel there exist two different categories of universality: abstract and concrete. The former is what is typically thought of by the term, something which exists not as an individual, but as a mode of a certain individual’s being. If this is the way by which we try to understand Kant’s claim, it would seem to be rather confused, such a universality can by no means, as Russell adequately pointed out, be known in virtue of anything other than logical form. The latter category consists of a universal which actually exists in the world, and it does this via its instantiation through the entirety of all existing, and indeed all possible, particulars contained in it. So when I actualize a right handed triangle, this right handed triangle is one of many other right handed triangles which are related in that they instantiate the universal category of a right handed triangle. My right handed triangle is the right handed triangle, just as your right handed triangle is also the right handed triangle. And, barring the particular quantities which constitute this triangle, such as the length of its sides, which are accidental properties; I am able to know things about the essential properties of your right handed triangle, such as the fact that the square of its hypotenuse is equal to the square of its two other sides by the fact that I know this, and am able to prove this, about my own triangle. An immediate question arises here of how I am able to distinguish between what are essential properties and what are particular properties: why would the pythagorean theorem be part of the essence of a right handed triangle, and not merely particular and accidental? Because, for Kant, “every particular idea as distinguished from general concepts is an intuition,”(Hintikka 23) whereas a general concept, under our framework, subsists through these intuitions but is not itself an intuition, it is something for which we can provide a general logical form via our cognition and understanding, as Russell seems to desire. The Pythagorean theorem is not itself an intuition, as Kant points out, a philosopher could “try to find out in his way how the sum of its angles might be related to a right angle. He has nothing but the concept of a figure enclosed by three straight lines, and in it the concept of equal many angles. Now he may reflect on this concept as long as he wants, yet he will never produce anything new.”(B 745) So there needs to be a conceptual framework which actually persists through the particulars, but cannot be conflated with the particulars. The Pythagorean theorem is just one such general framework which actually exists, and which can be observed from the particulars, but which is separate from the particular determinations. 

Indeed, such a notion of the concrete universal may already be found in Kant’s transcendental idealism. Universal experience in Kant, as described by  Nick Stang in his paper Who’s Afraid of Double Affection? functions in much the same way as the Concrete Universal. Within universal experience, as described by Stang, “all subjects’ perceptual experiences constitute the set whose contents ground the content of universal experience.”(Stang 17)  That is to say, universal experience actually exists, and is in fact even what allows us to determine “whether the moon is inhabited,” as it “depends upon whether universal experience represents it as inhabited.”(11) This universal experience is by definition actualized by being instantiated in each of its particulars. Both your experience and my experience are parts of universal experience which by their being parts actually are the universal which is made actual by its simply being the collection of experience, as indicated by the fact that we utilize it to figure out and distinguish basic facts about our actual reality. Indeed, “everything is actual that stands in one context with a perception in accordance with the laws of the empirical progression. Thus they are real when they stand in empirical connection with my real consciousness, although they are not therefore real in themselves, i. e. outside this progress of experience.”(Kant B521) Here we can see how universal experience is founded upon a collection of singular experiences, which all contribute to and derive from this universal experience. That is why my not experiencing a purple elephant in the corner of the room has any bearing over someone else’s experiencing of the purple elephant and vice versa. In much the same way, my particular right handed triangle has bearing over your particular right triangle and vice versa.

Given that such a notion seems to already exist in the most basic elements of Kant’s account of how the mind interfaces with intuition, it by no means seems all that far fetched to think that a similar dynamic is at play in his claim about the universality of a geometrical object, such as a triangle. My particular right handed triangle itself instantiates the universal right handed triangle in concreto, and this particular right handed triangle, in conjunction with all other particular right handed triangles, makes up the universal concept of a right handed triangle. It is for this reason that it is permissible to use an instantiation of a particular triangle to prove things about all other triangles which are like it. However, this is still a far cry from Kant’s initial claim. Originally, Kant claimed that “a chain of inferences... is always guided by intuition.”(Kant B742) Even if we take it that it has indeed been established that, via the existence of a universal in concreto, the conceptual can be worked out in terms of the concrete, Russell still seems to have the upper hand due to our not yet having shown that we must have intuition in conjunction with the geometric concepts in order to work out a geometrical demonstration. Why wouldn’t a concept on its own function just as well sans intuition? Because we are focusing specifically on 

Euclidean geometry, and not modern pure geometry, arithmetic or algebra, in which case it seems that mathematical reasoning can just as easily be purely derived from logic; we can observe that objects for Kant are themselves magnitudes, that magnitude is not a property which objects have in contradistinction to our contemporary conception of magnitude. Therefore, for Kant, a geometric proof does not deal with abstract properties of objects, but it deals with the actual objects which instantiate the universal about which you are proving things. When I prove something about the magnitude of a triangle, I am not proving something about a property, or about the modality of other objects, but I must actually be proving things about the object itself. In order to be able to prove something about an object, I must, to some extent, have an idea or an intuition of that object. When I prove something about a right handed triangle, I must be able to intuit the right handed triangle as an actually existing thing, because magnitude seems to be itself actual, that is, it is objects existing in the world which are represented to me through the intuition. If we step outside of geometry, however, and look towards pure arithmetic or algebra, this argument becomes far less satisfactory, particularly given the level of abstraction involved in those two fields. Contained to a specific sort of euclidean geometry, this framework in which universality both exists through, by being the totality of, its particulars, makes sense. By proving something about the ratio of magnitudes, that is, the conceptual structure of an actual triangle, in such a way that it elucidates an actually existing universal, I am able to apply this proof to all other triangles which stand under the same universality, and thus indeed satisfy the condition for knowing general geometric propositions.

In conclusion, Kant’s understanding of mathematical cognition, when viewed in terms of the Euclidean geometry of Kant’s day, is entirely defensible and intelligible. When viewed in terms of a modified understanding of universality, in which universality can itself actually exist and be instantiated as an object, we are able to better understand that Kant’s seemingly wrongheaded claim of moving from particular to universal is an imprecise formulation. Rather, what Kant seems to have in mind, is the instantiation of a universal as an actual object, which is capable of both being one of, while standing over, all other particulars, that is, a concrete universal.

Hintikka, Jaakko. “Kant on the Mathematical Method.” Kant’s Philosophy of Mathematics, 1992, pp. 21–42., https://doi.org/10.1007/978-94-015-8046-5_2. 

Stang, Nicholas. “Who's Afraid of Double Affection.” Philosophers' Imprint, vol. 15, no. 18, 2015. 

Kant, Immanuel, and Paul Guyer. Critique of Pure Reason. Cambridge Univ. Press, 2009. 

Russell, Bertrand. Introduction to Mathematical Philosophy. Allen & Unwin, 1950. 

Gris, Juan. Harlequin with a Guitar.