The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

as Überzeugung, the 'givenness' of Überredung derives from certain individual characteristics of the judging subject and so is private or subjective in a way that decreases its epistemic value. Überzeugung, on the other hand, is a form of epistemic compulsion which affects everyone (or at least everyone 'in possession of reason' (ibid.)) in the same way. The criterion for deciding whether a given intuitive judgement is a case of 'Überzeugung' or merely a case of 'Überredung' is therefore whether the grounds producing compulsion in the judging subject are such as would compel assent in every being possessing human reason. For such inter-subjectively binding cognitions, Kant believed, there is 'at least the presumption that the ground of the agreement . . . rests upon the common ground, namely upon the object' (ibid.).

We believe that Hilbert had something similar in mind in his insistence on the 'Wiedererkennbarkeit' or repeatable confirmation of finitary intuition (cf. Hilbert 1925G, 171). This would establish a kind of inter- subjectivity of intuition which, in combination with its 'givenness' would give the object of a finitary exhibition an existence which goes beyond the mere thinking of it and, at the same time, insure that this 'givenness', is inter- subjective and not the mere idiosyncracy of the individual exhibitor.

The epistemic asymmetries between creator and non-creator which, as seen earlier, were so important to the intuitionist exhibition, are thus eliminated from finitary exhibition.42 It yields an object which both exhibitor and non-exhibitor alike can, and, indeed, must judge by its inter-subjectively confirmable effects.

Because of these differences between finitary and intuitionistic exhibition, we believe it to be a serious mistake to think of them, as Gödel did, as descendants of a single generic conception of exhibition which merely place different restrictions on the abstractness/concreteness of the objects to be exhibited. Rather, they differ most radically and fundamentally over the very conception of exhibition itself.


5. CONCLUSION

What, ultimately, the differences between the finitist and intuitionist conceptions of exhibition come to, however, is a question that is both too deep and too complex to be adequately dealt with here. Nonetheless, I would

____________________
42
We are assuming here, of course, that there are no laws relating an agent's creative intentions to the effects of her actions that are so certain that, knowing them, a creator could deduce the effects of her creative acts with greater confidence than could be achieved through observation of those effects. Such an assumption seems plausible enough, however, once one has given up on any logical or metaphysical connection between creators' intentions and the effects of their creative acts.

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