FOM: 'constructivism' as 'minimalistic platonism'
Robert.Black at nottingham.ac.uk
Sun Jun 4 11:13:30 EDT 2000
I'm with Steve Simpson on this one.
Peter Schuster says:
>However, any realism powerful enough to `apply across the board' from
>the finite to the infinite has to be a full-blown one, which not only
>assumes all integers to exist but also that `we must know, we will
>know' [Hilbert] everything about them.
and a little later on:
>Classical logic ought rather to be viewed as the logic of what really
>exists AND will entirely be known, sometimes. Because nobody can know
>what some yet unknown knowledge will be like and whether it will
>eventually be known (unless it IS already known), one must invoke
>some strong omniscience principle to rescue classical logic, since
>otherwise it would be restricted to the knowledge of the day.
This sems to me to be completely upside-down. A *realist* (about
arithmetic) is one who thinks that any precisely-formed arithmetical
conjecture (e.g. that there is no odd perfect number) has a truth-value
*whether or not we can discover it in practice or even in principle*. By
contrast, the anti-realist (in Dummett's sense) is one who allows no gap
between the true and the discoverable, one who, as the jargon has it,
rejects the notion of 'verification-transcendent truth'. It is only if you
have already assumed the basic principle of *anti*-realism, namely that for
something to be true it has to be knowably true, that you need (potential)
omniscience to ground LEM. Schuster's 'full-blown realism' is just
*anti*-realism plus a wildly optimistic Hilbertian optimism about what can
>As soon as one takes serious epistemic and/or temporal matters, one
>indeed risks that tertium non datur goes out of the window,
>an observation which could be made also by most realists
>(except the omniscient ones). The question, however, is
>whether mathematicians are willing to let their logic depend on such
>issues; my impression is that most of them are not. By the way, I would
>like to know whether there are attempts to relate intuitionistic logic
>with temporal logic, too, just as with epistemic logic.
Well, it's not really to do with temporal logic as that phrase is usually
understood, but Dummett's paper 'The Reality of the Past', Proc. Arist.
Soc. 1969, pp. 239-58 discusses precisely the claim that if we think that
truth about the past cannot stretch beyond knowable truth about the past
then excluded middle should not in general apply to statements about the
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