The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

On Brouwer's view, then, whatever necessity or apriority belongs to mathematics is not, as it was for Kant, due to any putative inseparability of mathematics from experience that it possesses. Rather, it is the result of certain organizational features of the human intellect; in particular, the fact that the most fundamental of all human intellectual phenomena is the primordial intuition of the two-ity of time (ibid. 70-1) and that mathematics is based on a 'concentration' of the sequences arising from this intuition within the human intellect (ibid. 53). The organizational tools available to the human mind (i.e. the tools available to it for organizing its thoughts and experiences) are those which derive from its ability to arrange things into temporal sequences. The mind is able to extract the order-types of these sequences without immediate thought of applicability, though it none the less has these order-types available to it as possible orderings which might be 'projected into reality' (ibid.) should the opportunity arise.47 A priori judgements are therefore those which reflect the human mind's most basic organizational abilities, and these are intellectual, as opposed to purely sensible, in character. As Brouwer put it ( 1907, 53-4):

Proper to man is a faculty which accompanies all his interactions with nature, namely the faculty of taking a mathematial view of life, of observing in the world repetitions of sequences of events, i.e. of causal systems of time. The basic phenomenon therein is the simple intuition of time, in which repetition is possible in the form: 'thing in time and thing again', as a consequence of which moments of life break up into sequences of things which differ qualitatively. These sequences thereupon concentrate in the intellect into the mathematical sequences, not sensed but observed. . . . after only a little development of the method, the observed sequences will no longer consist solely of phenomena observed independently of man's will, but will be complemented by phenomena evoked by man (deeds without any direct instinctive aim, but executed only to complete the causal system so that it will become more easily manageable).

This in brief, then, is Brouwer's reaction to the developments in nineteenth- century geometry. It constitutes a very considerable departure from Kant's views-- greater, we would say, than that represented by the logicism of either Frege or Dedekind.


REFERENCES

Bernays, P. ( 1933), 'Die Grundgedanken der Fries'schen Philosophie in ihrem Verhältnis zum heutigen Stand der Wissenschaft', Abhandlungen der Fries'schenSchule

____________________
47
Of this departure from instinctive development, Brouwer makes the following interesting remark which connects his views with certain of those developed later by Wittgenstein: 'the non-instinctive nature of this intellectual action renders the certainty that the parts of the sequence really belong together, anything but perfect. Consequently, it can always be falsified, which is observed in the discovery that "the rule no longer applies"' ( Brouwer 1907, 53). This suggests that Brouwer did not regard a priori judgements as invariably possessed of high certainty.

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