Abstract
Mathematical intuitionism, for which Kant on the one hand and investigators such as H. Poincare on the other prepared the way, was systematically developed for the first time by L. E. J. Brouwer and his students; it means a totally new trend of thought in the investigation of the foundations of mathematics. If a tentative description of the basic ideas of intuitionism be desired, I would like to give it in the form of the following propositions: (1) One cannot separate the inquiry into the foundations of mathematics from the discussion of the conditions by which the mathematical activity of the mind is brought about. (2) Investigations which do not refer to these conditions teach us nothing about the essence of mathematics, but only something concerning its outward form of appearance, its language. Here the contrast with Frege’s standpoint is interesting. Frege stresses that one must abstract from the subjective processes, through which the mathematical activities of the mind are brought about, in order to penetrate to the essence of mathematics, since mathematics is objectively accessible to logical investigation only insofar as it has been embodied in the modes of expression provided by mathematical language. Brouwer is of the opinion that the outward form of appearance impedes rather than advances the penetration into the essence of mathematics; that is the reason why he wants to abstract precisely from this form of appearance. (3) Mathematics is independent of logic: logic rather rests upon mathematics; in the construction of mathematics, the principles of logic are not of unlimited application. (4) Not only the current conceptions about mathematics, but even the practical cultivation of mathematics makes an appeal to unproven, possibly incorrect, presuppositions regarding the essence of mathematical entities and the mathematical activity of the mind. (5) It is desirable to accomplish the construction of mathematics independently of any suppositions of this kind; for one can pass judgment on the correctness of such suppositions only after the construction has been accomplished.
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