Yogyakarta State University
Kant’s notions of mathematical method can be found in “The Critic Of Pure Reason: Transcendental Doctrine Of Method; Chapter I. The Discipline Of Pure Reason, Section I. The Discipline Of Pure Reason In The Sphere Of Dogmatism”. Kant recites that mathematical method is unattended in the sphere of philosophy by the least advantage that geometry and philosophy are two quite different things, although they go hand in hand in the field of natural science, and, consequently, that the procedure of the one can never be imitated by the other.
According to Kant 1, the evidence of mathematics rests upon definitions, axioms, and demonstrations; however, none of these forms can be employed or imitated in philosophy in the sense in which they are understood by mathematicians. Kant 2 claims that all our mathematical knowledge relates to possible intuitions, for it is these alone that present objects to the mind.
An a priori or non-empirical conception contains either a pure intuition that is it can be constructed; or it contains nothing but the synthesis of possible intuitions, which are not given a priori. Kant 3 sums up that in this latter case, it may help us to form synthetical a priori judgements, but only in the discursive method, by conceptions, not in the intuitive, by means of the construction of conceptions.
On the other hand, Kant 4 explicates that no synthetical principle which is based upon conceptions, can ever be immediately certain, because we require a mediating term to connect the two conceptions of event and cause that is the condition of time-determination in an experience, and we cannot cognize any such principle immediately and from conceptions alone.
Discursive principles are, accordingly, very different from intuitive principles or axioms. In his critic, Kant 5 holds that empirical conception can not be defined, it can only be explained. In a conception of a certain number of marks or signs, which denote a certain class of sensuous objects, we can never be sure that we do not cogitate under the word which.
The science of mathematics alone possesses definitions. According to Kant 6, philosophical definitions are merely expositions of given conceptions and are produced by analysis; while, mathematical definitions are constructions of conceptions originally formed by the mind itself and are produced by a synthesis.
Further, in a mathematical definition 7 the conception is formed; we cannot have a conception prior to the definition. Definition gives us the conception. It must form the commencement of every chain of mathematical reasoning.
In mathematics 8, definition can not be erroneous; it contains only what has been cogitated. However, in term of its form, a mathematical definition may sometimes error due to a want of precision. Kant marks that definition: “Circle is a curved line, every point in which is equally distant from another point called the centre” is faulty, from the fact that the determination indicated by the word curved is superfluous.
For there ought to be a particular theorem, which may be easily proved from the definition, to the effect that every line, which has all its points at equal distances from another point, must be a curved line (see Figure 22.)- that is, that not even the smallest part of it can be straight. 9
Kant (1781) in “The Critic Of Pure Reason: 1. AXIOMS OF INTUITION, The principle of these is: All Intuitions are Extensive Quantities”, illustrates that mathematics have its axioms to express the conditions of sensuous intuition a priori, under which alone the schema of a pure conception of external intuition can exist e.g. "between two points only one straight line is possible", "two straight lines cannot enclose a space," etc.
These 10 are the axioms which properly relate only to quantities as such; but, as regards the quantity of a thing, we have various propositions synthetical and immediately certain (indemonstrabilia) that they are not the axioms. Kant 11 highlights that the propositions: "If equals be added to equals, the wholes are equal"; "If equals be taken from equals, the remainders are equal"; are analytical, because we are immediately conscious of the identity of the production of the one quantity with the production of the other; whereas axioms must be a priori synthetical propositions.
On the other hand 12, the self-evident propositions as to the relation of numbers, are certainly synthetical but not universal, like those of geometry, and for this reason cannot be called axioms, but numerical formulae. Kant 13 proves that 7 + 5 = 12 is not an analytical proposition; for either in the representation of seven, nor of five, nor of the composition of the two numbers; “Do I cogitate the number twelve?” he said.
Although the proposition 14 is synthetical, it is nevertheless only a singular proposition. In so far as regard is here had merely to the synthesis of the homogeneous, it cannot take place except in one manner, although our use of these numbers is afterwards general. Kant then exemplifies the construction of triangle using three lines as the following:
The statement: "A triangle can be constructed with three lines, any two of which taken together are greater than the third" is merely the pure function of the productive imagination, which may draw the lines longer or shorter and construct the angles at its pleasure; therefore, such propositions cannot be called as axioms, but numerical formulae 15
Kant in “The Critic Of Pure Reason: II. Of Pure Reason as the Seat of Transcendental Illusory Appearance, A. OF REASON IN GENERAL”, enumerates that mathematical axioms 16 are general a priori cognitions, and are therefore rightly denominated principles, relatively to the cases which can be subsumed under them. While in “The Critic Of Pure Reason: SECTION III. Of Opinion, Knowledge, and Belief; CHAPTER III.
The Arehitectonic of Pure Reason”, Kant propounds that 17mathematics may possess axioms, because it can always connect the predicates of an object a priori, and without any mediating term, by means of the construction of conceptions in intuition. On the other hand, in “The Critic Of Pure Reason: CHAPTER IV. The History of Pure Reason; SECTION IV. The Discipline of Pure Reason in Relation to Proofs” , Kant designates that in mathematics, all our conclusions may be drawn immediately from pure intuition. Therefore, mathematical proof must demonstrate the possibility of arriving, synthetically and a priori, at a certain knowledge of things, which was not contained in our conceptions of these things.
All 18 the attempts which have been made to prove the principle of sufficient reason, have, according to the universal admission of philosophers, been quite unsuccessful. Before the appearance of transcendental criticism, it was considered better to appeal boldly to the common sense of mankind, rather than attempt to discover new dogmatical proofs. Mathematical proof 19 requires the presentation of instances of certain concepts.
These instances would not function ex¬actly as particulars, for one would not be entitled to assert anything concerning them which did not follow from the general concept. Kant 20 says that mathematical method contains demonstrations because mathematics does not deduce its cognition from conceptions, but from the construction of conceptions, that is, from intuition, which can be given a priori in accordance with conceptions. Ultimately, Kant 21 contends that in algebraic method, the correct answer is deduced by reduction that is a kind of construction; only an apodeictic proof, based upon intuition, can be termed a demonstration.
Kant, I., 1781, “The Critic Of Pure Reason: Transcendental Doctrine Of Method; Chapter I. The Discipline Of Pure Reason, Section I. The Discipline Of Pure Reason In The Sphere Of Dogmatism”, Translated By J. M. D. Meiklejohn, Retrieved 2003
5 Kant, I., 1781, “The Critic Of Pure Reason: Transcendental Doctrine Of Method, Chapter I, Section I .”, Translated By J. M. D. Meiklejohn, Retrieved 2003
10 Kant, I., 1781, “The Critic Of Pure Reason: 1. AXIOMS OF INTUITION, The principle of these is: All Intuitions are Extensive Quantities”, Translated By J. M. D. Meiklejohn, Retrieved 2003
16Kant, I., 1781, “The Critic Of Pure Reason: II. Of Pure Reason as the Seat of Transcendental Illusory Appearance, A. OF REASON IN GENERAL”, Translated By J. M. D. Meiklejohn, Retrieved 2003
17Kant, I., 1781, “The Critic Of Pure Reason: SECTION III. Of Opinion, Knowledge, and Belief; CHAPTER III. The Arehitectonic of Pure Reason” Translated By J. M. D. Meiklejohn, Retrieved 2003
18Kant, I., 1781, “The Critic Of Pure Reason: CHAPTER IV. The History of Pure Reason; SECTION IV. The Discipline of Pure Reason in Relation to Proofs” Translated By J. M. D. Meiklejohn, Retrieved 2003
19Kant in Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York
20Kant, I., 1781, “The Critic Of Pure Reason: Transcendental Doctrine Of Method, Chapter I, Section I .”, Translated By J. M. D. Meiklejohn, Retrieved 2003