Bolzano B. comments that Kant’s theory of knowledge seem to promise us with his discovering of a definite and characteristic difference between two main classes of all human a priori knowledge i.e. philosophy and mathematics.
According to Kant mathematical knowledge is capable of adequately presenting all its concepts in a pure intuition, i.e. constructing.
It is also able to demonstrate its theorems; while, on the other hand, philosophical knowledge, devoid of all intuition, must be content with purely discursive concepts.
Consequently the essence of mathematics may be expressed most appropriately by the definition that it is a science of the construction of concepts.
Bolzano suggests that several mathematicians who adhere to the critical philosophy have actually adopted this definition and deserved much credit to Kant’s theory of knowledge for the foundation of pure mathematics.
Kant’s theory of knowledge, as it deserved in his transcendental philosophy, had a distinct work i.e. the “Critique of Pure Reason (1781)”, in which he opens a new epoch in metaphysical thought where far in the history of philosophy the human mind had not been fairly considered.
Thinkers had concerned themselves with the objects of knowledge, not with the mind that knows, while Kant tries to transfer contemplation from the objects that engaged the mind to the mind itself, and thus start philosophy on a new career.
Shabel L. (1998) perceives that epistemologicaly, Kant primarily regards to determine whether the method for obtaining apodictic certainty, that one calls mathematics in the latter science or pure reason in its mathematical use, is identical with that by means of which one seeks the same certainty in philosophy, and that would have to be called as dogmatic.
Prior and after Kant , there are questions left for some writers to get their position about the foundation of mathematics e.g.
Whether or not we have the mathematical ideas that are true of necessity and absolutely?
Are there mathematical ideas that can fairly be pronounced independent in their origin of experience, and out of the reach of experience by their nature?
One party contends that all mathematical knowledge was derived from experience viz. there was nothing in the intellect that had not previously been in the senses.
The opposite party maintains that a portion of mathematical knowledge came from the mind itself viz. the intellect contained powers of its own, and impressed its forms upon the phenomena of sense.
The extreme doctrine of the two schools was represented, on the one side by the materialists, on the other by the mystics.
Between these two extremes Kant might be perceived to offer various degrees of compromise or raging its foundations spuriously.
The ultimately discussion sums up that, in the sphere of Kant’s ‘dogmatic’ notions, his theory of knowledge, in turn, can be said to lead to undogmatization and demythologization of mathematical foundations as well as rages the institutionalization of the research of mathematical foundations in which it encourages the mutual interactions among them.
Smith, N. K. in “A Commentary to Kant’s Critique of Pure Reason”, maintains that some further analytic explanations supporting the claims come from Kant’s claims that there are three possible standpoints in philosophy i.e. the dogmatic, the sceptical, and the critical.
All preceding thinkers come under the first two heads. Kant insists that a dogmatist is one who assumes that human reason can comprehend ultimate reality, and who proceeds upon this assumption; it expresses itself through three factors viz. rationalism, realism, and transcendence.
According to Smith, N. K. , for Kant, Descartes and Leibniz are typical dogmatists.
On the other hand, rationalists held that it is possible to determine from pure a priori principles the ultimate nature of the material universe.
They are realists in that they assert that by human thought, the complete nature of objective reality, can be determined.
However, they also adopt the attitude of transcendence.
Through pure thought , rationalists go out beyond the sensible and determine the supersensuous. Meanwhile , skepticism may similarly be defined through the three terms, empiricism, subjectivism, immanence.
Further, Smith, N. K clarifies that a skeptic can never be a rationalist.
The skeptic must reduce know¬ledge to sense-experience; for this reason also his knowledge is infected by subjective conditions.
Through sensation we cannot hope to determine the nature of the objectively real.
This attitude is also that of immanence and knowledge is limited to the sphere of sense-experience.
Smith, N. K synthesizes that criticism has similarly its three constitutive factors, rationalism, subjectivism, immanence.
Accordingly, it agrees with dogmatism in maintaining that only through a priori principles can true knowledge be obtained.
Such knowledge is, however, subjective in its origin, and for that reason it is also only of immanent application.
Knowledge is possible only in the sphere of sense-experience.
Dogmatist claims that knowledge arises independently of experience and extends beyond it.
Empiricism holds that knowledge arises out of sense-experience and is valid only within it; while, criticism teaches that knowledge arises independently of particular experience but is valid only for experience.
It can be learned from Kant that the skeptic is the taskmaster who constrains the dogmatic reasoner to develop a sound critique of the mathematical understanding and reason.
The skeptical procedure cannot of itself yield any satisfying answer to the questions of mathematical reason, but none the less it prepares the way by awakening its circumspection, and by indicating the radical measures which are adequate to secure it in its legitimate possessions.
Kant develops the method as, the first step, in matters of pure reason, marking its infancy, is dogmatic.
The second step is skeptical to indicate that experience has rendered our judgment wiser and more circumspect.
The third step, is now necessary as it can be taken only by fully matured judgment.
It is not the censorship but the critique of reason whereby not its present bounds but its determinate and necessary limits; not its ignorance on this or that point but is regard to all possible questions of a certain kind.
Mathematical reasons are demonstrated from principles, and not merely arrived at by way of mathematical conjecture.
Skepticism is thus a resting-place for mathematical reason, where it can reflect upon its dogmatic wanderings and make survey of the region in which it finds itself, so that for the future it may be able to choose its part with more certainty.
The role of Kant’s theory of knowledge, in the sense of demythologization of mathematical foundations, refers to history of the mathematical myth from that of Euclid’s to that of contemporary philosophy of mathematics.
The myth of Euclid: "Euclid's Elements contains truths about the universe which are clear and indubitable", however, today advanced student of geometry to learn Euclid's proofs are incomplete and unintelligible. Nevertheless, Euclid's Elements is still upheld as a model of rigorous proof.
The myths of Russell, Brouwer, and Bourbaki - of logicism, intuitionism, and formalism-are treated in The Mathematical Experience.
Contemporary mathematical foundations consist of general myths that:
1. Unity i.e. there is only one mathematics, indivisible now and forever, and it is a single inseparable whole;
2. Universality i.e. the mathematics we know is the only mathematics there can be;
3. Certainty i.e. mathematics has a method, "rigorous proof;" which yields absolutly certain conclusions, given truth of premises;
4. Objectity i.e. mathematical truth is the same for everyone and it doesn't matter who discovers it as well as true whether or not anybody discovers it.
Kant’s theory of knowledge implies to the critical examinations of those myths.
In fact, being a myth doesn't entail its truth or falsity.
Myths validate and support institutions in which their truth may not be determinable.
Those latent mathematical myths are almost universally accepted, but they are not sef-evident or self-¬proving.
From a different perspective, it is possible to question, doubt, or reject them and some people do reject them.
Hersh, R. in “What is Mathematics, Really?” indicates that if mathematics were presented in the style in which it is created, few would believe its universality, unity, certainty, or objectivity.
These myths support the institution of mathematics.
While the purists sometimes even declare applied mathe¬matics is not mathematics.
The clarity and strict necessity of mathematical truth had long provided the rationalists - above all Descartes, Spinoza and Leibniz - with the assurance that, in the world of modern doubt, the human mind had at least one solid basis for attaining certain knowledge.
Kant himself had long been convinced that mathematics could accurately describe the empirical world because mathematical principles neces¬sarily involve a context of space and time.
According to Kant, space and time lay at the basis of all sensory experience i.e. the condition and structure any empirical observation.
For Kant, intuitions are supposed to be eternal and universal features of mind which constitutes all human thinking.
While, for rationalists, mathematics was the main example to confirm their view of the world.
From the three historic schools, the mainstream philosophy of mathematics records only intuitionism pays attention to the construction of mathematics.
Formalists, Logicists, and Platonists sit at a table in the dining room, discussing their rag out as a self-created, autonomous entity.
Smith, N. K. concerns with Kant’s conclusion that there is no dwelling-place for permanent settlement obtained only through perfect certainly in our mathematical knowledge, alike of its objects themselves and of the limits which all our knowledge of object is enclosed.
In other word , Kant’s theory of knowledge implies to un-dogmatization and de-mythologization of mathematical foundations as well as to rage the institutionalization of the research of mathematical foundations.
In term of these perspectives, Kant considers himself as contributing to the further advance of the eighteenth century Enlightenment and in the future prospect of mathematics philosophy.
1Bolzano, B., 1810, “Contributions to a Better-Grounded Presentation of Mathematics” in Ewald, W., 1996, “From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume I”, Oxford: Clarendon Press, p. 175
2 Ibid. p. 175
3 Ibid. p.175
4Immanuel Kant, 1724–1804”, Retrieved 2004
6 Smith, N. K., 2003, “A Commentary to Kant’s Critique of Pure Reason”, New York: Palgrave Macmillan, p. 13
7 Ibid. p. 13
8 Ibid. p. 13
9 Ibid. p. 13
10 Ibid. p.14
11Ibid. p. 14
12Ibid. p. 14
13Ibid. p. 14
14Ibid. p. 14
16Ibid. p. 21
17Ibid. p. 21
19Ibid. p. 14
20Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, p. 37
21 Ibid. p. 38
22Smith, N. K., 2003, “A Commentary to Kant’s Critique of Pure Reason”, New York: Palgrave Macmillan, p. 21