The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

We assume that ψ and χ preserve intuitability. Then we have for any numerical term s the inference from CI(s) to CI[ψ(s)] and by replacement to CI[φ(0, s)]. Suppose we can infer CI[φ(b, s)] from CI(s). Then since χ preserves intuitability, we have CI[χ(b, φ(b, s))],43 and hence by replacement CI[φ(Sb, s)]. Hence we have by induction ∀x{NxCI[φ(x, s)]}. By UII, for any term t such that Nt and CI(t), CI[φ(t, s)]. That is, φ preserves intuitability.

But this is enough to show that CI(t) holds for any term of PRA.

If we allow more complicated inductions involving the predicate CI, we can prove that functors defined by more complex recursions preserve intuitability, for example Ackermann-type functors introduced by nested double recursion. Particularly since these proofs will still be intuitionistic, it is not obvious that they should be ruled out. But that they are possible does indicate that the mathematical possibility of intuition is being conceived rather generously.

Does this give rise to an argument for Hilberts thesis? Suppose s and t are such that I can intuit a string of length s, and I can intuit a string of length t. Then, surely, by comparing them I can determine whether s = t is true. It thus seems that any closed formula of PRA can be decided in an intuitive way. But what follows about intuitive knowledge of generalizav tions? The most this argument shows is that if a formula of PRA is true for all values of its variables, then in each particular case this can be known intuitively. But this does not yield intuitive knowledge of the generalization. Even if this obstacle can be circumvented, I see no way to get around the fact that what these considerations yield is the possibility of intuitive knowledge, according to a rather liberal kind of possibility. But Hilberts thesis concerns actual intuitive knowledge, at least given a proof in PRA. Even if the questions about the ideas involved can be resolved, the case for Hilberts thesis does not seem to be materially advanced. More generally, the relevance to questions about what we know intuitively of questions about what intuitions are possible in principle is not clear.


APPENDIX

For the conventional formulation of PRA, we can assume that the logic allowed is minimal propositional logic, with ¬A defined as A → 0 = 1. For to obtain intuitionistic logic, it suffices to derive 0 = 1 → A for any A. If A is atomic it is an equation s = t. We introduce by primitive recursion a function symbol φ satisfying φ(0) = s

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43
Note that CI(b) follows from the assumption Nb, since we have accepted the induction that yields ∀x[NxCI(x)].

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