Kant on the Basis Validity of the Concept of Arithmetic

By Marsigit
Yogyakarta State University

In his Critic of Pure Reason Kant reveals that arithmetical propositions are synthetical. To show this, Kant convinces it by trying to get a large numbers of evidence that without having recourse to intuition or mere analysis of our conceptions, it is impossible to arrive at the sum total or product. In arithmetic , intuition must therefore here lend its aid only by means of which our synthesis is possible. Arithmetical judgments are therefore synthetical in which we can analyze our concepts without calling visual images to our aid as well as we can never find the arithmetical sum by such mere dissection.


Kant propounds that 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. Both representations , however, are only intuitions because if we omit from the empirical intuitions of bodies and their alterations everything empirical or belonging to sensation, space and time still remain.

According to Kant , arith¬metic produces its concepts of number through successive addition of units in time, and pure mechanics especially can produce its concepts of motion only by means of the representation of time. Kant defines the schema of number in exclusive reference to time; and, as we have noted, it is to this definition that Schulze appeals in support of his view of arithmetic as the science of counting and therefore of time. It at least shows that Kant perceives some form of connection to exist between arithmetic and time.

Kant is aware that arithme¬tic is related closely to the pure categories and to logic. A fully explicit awareness of number goes the successive apprehension of the stages in its construction, so that the structure involved is also rep¬resented by a sequence of moments of time. Time thus provides a realization for any number which can be real¬ized in experience at all. Although this view is plausible enough, it does not seem strictly necessary to preserve the connection with time in the necessary extrapolation be¬yond actual experience.

Kant , as it happens, did not see that arithmetic could be analytic. He explained the following:
Take an example of "7 + 5 = 12" . If "7 + 5" is understood as the subject, and "12" as the predicate, then the concept or meaning of "12" does not occur in the subject; however, intuitively certain that "7 + 5 = 12" cannot be denied without contradiction. In term of the development of propositional logic, proposition like "P or not P" clearly cannot be denied without contradiction, but it is not in a subject-predicate form.

Still, "P or not P" is still clearly about two identical things, the P's, and "7 + 5 = 12" is more complicated than this. But, if "7 + 5 = 12" could be derived directly from logic, without substantive axioms like in geometry, then its analytic nature would be certain.

Hence , thinking of arithmetical construction as a process in time is a useful picture for interpreting problems of the mathematical constructivity. Kant argues that in order to verify "7+5=12", we must consider an instance.

References:

Kant, I., 1787, “The Critic Of Pure Reason: INTRODUCTION: V. In all Theoretical Sciences of Reason, Synthetical Judgements "a priori" are contained as Principles” Translated By J. M. D. Meiklejohn, Retrieved 2003 )
2 Ibid.
3Kant, I, 1783. “Prolegomena to Any Future Metaphysic: Preamble On The Peculiarities Of All Metaphysical Cognition, Sec.2” Trans. Paul Carus.. Retrieved 2003
4Kant, I, 1783. “Prolegomena to Any Future Metaphysic: First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?” Trans. Paul Carus.. Retrieved 2003
5 Ibid.
6Smith, N. K., 2003, “A Commentary to Kant’s Critique of Pure Reason: Kant on Arithmetic,”, New York: Palgrave Macmillan. p. 128
7 Ibid. p. 129
8 Ibid. p. 130
9 Ibid. p. 131
10 Ross, K.L., 2002, “Immanuel Kant (1724-1804)” Retreived 2003
11Ibid.
12Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p. 198