[FOM] Falsify Platonism
W.Taylor@math.canterbury.ac.nz
W.Taylor at math.canterbury.ac.nz
Sun Apr 25 06:17:31 EDT 2010
Quoting hendrick:
> So far, whenever anyone derives a contradiction from set theory,
> the response has not been to discard set-theoretic Platonism,
> but to change set theory.
That may well be so, and is an indicator of the slipperiness of set theory,
at least when compared to number theory and some other studies (e.g.
geometry).
However, a word of caution! So from from being "whenever", as you say above,
it may not have ever happened, as yet!
Popularizers DO like to speak of "the paradoxes", and their allegedly
devastating effect. But the only really famous paradox (Russell vs Frege)
came in logic, not set theory; the set theoretical paradoxes are virtually
all identical (whether framed as Cantor's, Burali-Forti's, Russell's or
whatever), and were seen immediately and IN ADVANCE by Cantor himself,
during the pre-formal stage of set theory, one might almost call it
the pre-mathematical stage. Cantor airily dismissed them with a wave
of the hand about "inconsistent totalities".
True set theory began with Zermelo's 1908 axiomatization, and was designed
NOT to counter paradoxes, but to clarify what people were talking about
and in particular find a decent proof of the well-ordering theorem.
This was all achieved, and any lingering whiff of paradox banished
simultaneously, and has never been seriously challenged since. Indeed, it
has been a real struggle to INTRODUCE paradoxes (!) by means of super-large
cardinals, alternative axiomatizations and whatnot.
So, the whole "disastrous paradoxes" thing is really just an academic
folklore.
The situation is not unlike the introduction of imaginary numbers, or
non-Euclidean geomtries, before set theory. Popularisers love to harp
on them, but mathematicians just get on with the job, an important part
of which is formalization, which is more or less Platonization.
-- Bill of writes.
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