Although many people appreciate mathematics as a discipline exhibiting sound reasoning with a universal appeal, the mere fact that this academic discipline has a history shows that there must have been different and even clashing schools of thought present in it. Greek mathematics advanced under the Pythagorean flag of an �Arithmetica universalis� but soon switched to a spatial perspective owing to the discovery of irrational numbers as an effect of this first foundational crisis of mathematics. The second crisis centred in the limit concept which prompted mathematicians to employ the actual infinite. However, this move gave rise to the third foundational crisis when Russell and Zermelo discovered that Cantor�s set concept entailed antinomies. This caused the rise of the intuitionistic mathematics of Brouwer and his followers as well as the logicism of Frege, Dedekind and Russell. At once this brought Greek and medieval conceptions to the fore, such as the infinite divisibility of a continuum. The dominant understanding of infinity restricted its meaning to what Kant called the �successive infinite� (the potential infinite) � which can never be understood as an infinite totality given at once, i.e., as the at once infinite (actual infinite). However, restricting mathematics to the successive infinite gave rise to intuitionistic mathematics which arrived at results that find, according to Beth, �no counterpart in classical mathematics�. Hilbert followed Kant in accepting certain extralogical concrete objects which are intuited as directly experienced prior to all thinking. But although he accepts the successive infinite with Kant, Hilbert still wants to inhabit the paradise created for us by Cantor. At the background of Kant�s thought the tension between nature and freedom lurks, also surfacing in the distinction between �Ding-an-sich� and appearance. Interestingly, Hilbert made an appeal to Kant�s conception of transcendental ideas (concepts of reason) in order to justify his employment of the actual infinite (at once infinite). But he did not realize that Kant�s view of transcendental ideas entails the idea of infinite totalities. Hilbert had to use the at once infinite in order to justify the use of the at once infinite. The final irony of his firm belief that it would be possible to prove the consistency of mathematics, is found in the famous 1931 article of G�del on this issue. In 1946 Hermann Weyl remarked, with a ring to the successive infinite as embedded in human intuition: �It must have been hard on Hilbert, the axiomatist, to acknowledge that the insight of consistency is rather to be attained by intuitive reasoning which is based on evidence and not on axioms.� G�del has shown that a formal mathematical system is consistent or complete. Roos pointed out that Hilbert never recovered from this blow.

Username
Password