FOM: Fiction and non-fiction
Torkel Franzen
torkel at sm.luth.se
Sat Dec 6 06:35:17 EST 1997
John Steel says:
>The difference of opinion does matter, in grant reviews, in hiring
>committees, in what you advise students to do, and elsewhere.
No doubt, but this is equally true of differences of opinion
regarding the interest, prospects or importance of resolutely
non-foundational mathematics. No philosophical or foundational
opposition is implied by this.
>On this view, it would seem that set theorists should submit their
>grant applications to the NEH, not the NSF, and be hired by English
>departments rather than Mathematics departments.
Well, many people find it possible to do mathematics without assuming
that our mathematical understanding or intuition implicitly contains the
answer to every mathematical question. Your comment is very much in
the spirit of Hilbert, though.
Lee Stanley comments on our intuitions about sets:
>(which are admittedly somewhat duller than our intuitions
>about natural numbers, but as a matter of degree, not
>as a matter of kind, so that in my view, they are both
>dismissed as fictions, or both admitted as some sort of
>non-fictions)
This is reasonable, but it remains that our attitude towards questions
about the fictititious natural numbers is in many cases different from
our attitude towards questions abut the fictitious infinite sets.
As an illustration of this, take the following philosophical aside
in a paper on set theory:
Of course, as with any axiom, an initial act of faith is required
concerning the consistency: we assume
that the existence of, say, an inaccessible cardinal does not
lead to a contradiction with ZFC... It is the
author's viewpoint that consistency is the only point at issue
here, and that the question as to the
"existence" of inaccessible cardinals is totally meaningless. To
us, large cardinal theory is a
(worthwhile) structure theory, no more.
What is striking in this passage is the difference in attitude
towards the question of consistency and the question of existence. The
latter is described as "meaningless", which we may assume at any rate
implies that there isn't any fact of the matter. It wouldn't make
sense, for example, to say that even if inaccessible cardinals do
exist we may never be able to establish this with any degree of
certainty, or that even if they don't exist, it need not be possible
to demonstrate this, if the assumption of their existence is a
consistent one.
The question of consistency, on the other hand, is assumed to be a
question of fact. This emerges, not in an emphatic assertion of the
corresponding instance of the law of excluded middle, but in the
presentation of the assumption of consistency as an act of faith: we
are assuming that the theory is "in fact" consistent, even though we
may not be able to prove this.
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