Yogyakarta State University
In his Critic of Pure Reason (1787) Kant elaborates that geometry is based upon the pure intuition of space; and, arithmetic accomplishes its concept of number by the successive addition of units in time; and pure mechanics especially cannot attain its concepts of motion without employing the representation of time. Kant stresses that both representations, however, are only intuitions; for if we omit from the empirical intuitions of bodies and their alterations (motion) everything empirical, or belonging to sensation, space and time still remain.
Therefore, Kant concludes that pure mathematics is synthetical cognition a priori. Pure mathematics is only possible by referring to no other objects than those of the senses, in which, at the basis of their empirical intuition lies a pure intuition of space and time which is a priori. Kant illustrates, that in ordinary and necessary procedure of geometers, all proofs of the complete congruence of two given figures come ultimately to to coincide; which is evidently nothing else than a synthetical proposition resting upon immediate intuition.
This intuition must be pure or given a priori, otherwise the proposition could not rank as apodictically certain, but would have empirical certainty only. Kant further claims that everywhere space has three dimensions. This claim is based on the proposition that not more than three lines can intersect at right angles in one point.
Kant argues that drawing the line to infinity and representing the series of changes e.g. spaces travers by motion can only attach to intuition, then he concludes that the basis of mathematics actually are pure intuitions; while the transcendental deduction of the notions of space and of time explains the possibility of pure mathematics.
Kant defines that geometry is a science which determines the properties of space synthetically, and yet a priori. What, then, must be our representation of space, in order that such a cognition of it may be possible? Kant explains that it must be originally intuition, for from a mere conception, no propositions can be deduced which go out beyond the conception, and yet this happens in geometry. But this intuition must be found in the mind a priori, that is, before any perception of objects, consequently must be pure, not empirical, intuition.
According to Kant , geometrical principles are always apodeictic, that is, united with the consciousness of their necessity; however, propositions as "space has only three dimensions", cannot be empirical judgments nor conclusions from them. Kant claims that it is only by means of our explanation that the possibility of geometry, as a synthetical science a priori, becomes comprehensible.
As the propositions of geometry are cognized synthetically a priori, and with apodeictic certainty. According to Kant , all principles of geometry are no less analytical; and it based upon the pure intuition of space. However, the space of the geometer would be considered a mere fiction, and it would not be credited with objective validity, because we cannot see how things must of necessity agree with an image of them, which we make spontaneously and previous to our acquaintance with them.
But if the image is the essential property of our sensibility and if this sensibility represents not things in themselves, we shall easily comprehend that all external objects of our world of sense must necessarily coincide in the most rigorous way with the propositions of geometry. The space of the geometer is exactly the form of sensuous intuition which we find a priori and contains the ground of the possibility of all external appearances.
In his own remarks on geometry, Kant regularly cites Euclid’s angle-sum theorem as a paradigm example of a synthetic a priori judgment derived via the constructive procedure that he takes to be unique to mathematical reasoning. Kant describes the sort of procedure that leads the geometer to a priori cognition of the necessary and universal truth of the angle-sum theorem as:
The object of the theorem—the constructed triangle—is in this case “determined in accordance with the conditions of…pure intuition.” The triangle is then “assessed in concreto” in pure intuition and the resulting cognition is pure and a priori, thus rational and properly mathematical. To illustrate, I turn to Euclid’s demonstration of the angle-sum theorem, a paradigm case of what Kant considered a priori reasoning based on the ostensive but pure construction of mathematical concepts.
Euclid reasons as follows: given a triangle ABC , extend the base BC to D. Then construct a line through C to E such that CE is parallel to AB. Since AB is parallel to CE and AC is a transversal, angle 1 is equal to angle 1'. Likewise, since BD is a transversal, angle 2
For Kant , the axioms or principles that ground the constructions of Euclidean geometry comprise the features of space that are cognitively accessible to us immediately and uniquely, and which precede the actual practice of geometry. Kant said that space is three dimensional; two straight lines cannot enclose a space; a triangle cannot be constructed except on the condition that any two of its sides are together longer than the third.
Kant takes the procedure of describing geometrical space to be pure, or a priori, since it is performed by means of a prior pure intuition of space itself. According to Kant, our cognition of individual spatial regions is a priori since they are cognized in, or as limitations on, the essentially single and all encompassing space itself.
Of the truths of geometry e.g. in performing the geometric proof on a triangle that the sum of the angles of any triangle is 180°, it would seem that our constructed imaginary triangle is operated on in such a way as to ensure complete independence from any particular empirical content.
So, in term of geometric truths, Kant might suggest that they are necessary truths or are they contingent viz. it being possible to imagine otherwise. Kant argues that geometric truth in general relies on human intuition, and requires a synthetic addition of information from our pure intuition of space, which is a three-dimensional Euclidean space. Kant does not claim that the idea of such intuition can be reduced out to make the truth analytic.
In the Prolegomena, Kant gives an everyday example of a geometric necessary truth for humans that a left and right hand are incongruent. The notion of "hand" here need not be understood as the empirical object hand. According to Kant, we can assume that our pure intuition filter has adequately abstracted our hand-experience into something detached from its empirical component, so we are merely dealing with a three-dimensional geometric figure shaped like a hand.
By “incongruent", the geometer simply means that no matter how we move one figure around in relation to the other, we cannot get the two figures to coincide, to match up perfectly. Kant points out, there is still something true about the 3-D Euclidean case that has some kind of priority over the other cases. Synthetically, it is necessarily true that the figures are incongruent, since the choice of view point in point of fact no choice at all.
Kant, I, 1783. “Prolegomena to Any Future Metaphysic: , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible? Sect.10, p. 34
2Ibid. p. 35
3 Kant, I., 1787, “The Critic Of Pure Reason: SS 9 General Remarks on Transcendental Aesthetic.” Translated By J. M. D. Meiklejohn, Retrieved 2003
12Kant, I, 1783, “Prolegomena to Any Future Metaphysic: REMARK 1” Trans. Paul Carus.. Retrieved 2003
15Shabel, L., 1998, “Kant’s “Argument from Geometry”, Journal of the History of Philosophy, The Ohio State University, p.24
16Ibid. p. 28
20 …., 1987, “Geometry: Analytic, Synthetic A Priori, or Synthetic A Posteriori?”, Encyclopedic Dictionary of Mathematics, Vol. I., "Geometry", , The MIT Press, p. 685
21Ibid. p. 686
22Ibid. p. 689