Philosophy of Mathematics: Selected Readings by

Language: eng
Format: azw3
Publisher: Cambridge University Press
Published: 1984-01-26T16:00:00+00:00

(3)

If all pairs n,m, … [same as in (3)], then in no case would the product be found to equal 10100 + 1.

(4)

expresses a necessary truth, although it may be physically impossible to discover which one. Yet this same mathematician or philosopher, who is quite happy in this context with the notion of mathematical possibility (and who does not ask for any nominalistic reduction) and who treats mathematical necessity as well defined in this case, for a reason which is essentially circular, regards it as ‘platonistic’ to suppose that the continuum hypothesis has a truth value.1 I, realize that this is an ad hominem argument, but still – if there is such an intellectual sin as ‘platonism’ (and it is remarkably unclear what this supposed sin consists of), why is it not already to commit it, if one supposes that ‘10100 + 1 is a prime number’ has a truth value, even if no nominalistic reduction of this statement can be offered? (When one is defending a commonsense position, very often the only argument is ad hominem – for one has to keep throwing the burden of the argument back to the other side, by asking to be told precisely what is ‘unclear’ about the notions being attacked, or why a ‘reduction’ of the kind being demanded is necessary, or why a ‘foundation’ for the science in question is needed.)

In passing, I should like to remark that the following two principles, which many people seem to accept, can be shown to be inconsistent, by applying the Gödel theorem:

(I) That, even if some arithmetical (or set-theoretical) statements have no truth value, still, to say of any arithmetical (or set-theoretical) statement that it has (or lacks) a truth value is itself always either true or false (i.e. the statement either has a truth value or it doesn’t).

(II) All and only the decidable statements have a truth value.

For the statement that a mathematical statement S is decidable may itself be undecidable. Then, by (II), it has no truth value to say ‘S is decidable’. But, by (I), it has a truth value to say ‘S has a truth value’ (in fact, falsity; since if S has a truth value, then S is decidable, by (II), and, if S is decidable, then ‘S is decidable’ is also decidable). Since it is false (by the previous parenthetical remark) to say ‘S has a truth value’ and since we accept the equivalence of ‘S has a truth value’ and ‘S is decidable’, then it must also be false to say ‘S is decidable’. But it has no truth value to say ‘S is decidable’. Contradiction.

The significance of the antinomies

The most difficult question in the philosophy of mathematics is, perhaps, the question raised by the antinomies and by the plurality of conflicting set theories. Part of the paradox is this: the antinomies do not at all seem to affect the notion ‘set of sets of integers’, etc. Yet they do seem to affect the notion ‘all sets’.

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