[FOM] The boundary of objective mathematics
W. Mueckenheim
mueckenh at rz.fh-augsburg.de
Wed Mar 18 06:58:37 EDT 2009
Paul Budnik wrote:
Yet there is increasing scepticism about the objective truth of the
Continuum Hypothesis and similar statements.
Tim Chow wrote:
I'm curious about this "increasing scepticism" that you speak of. Do you
have any statistical evidence of increasing scepticism?
=================================
I have. Let me see whether I'll make it through
the editors' censorship. The scepticism is not
new as we see by old statements of Poincaré,
Brouwer, Weyl, Wittgenstein, or Lorenzen [1]. But
while these statements were lying dormant over
years they become more and more widespread
nowadays, and, as we see by statements of
contemporary university teachers [2], the
scepticism against the actually infinite is growing.
The crucial point has been made by Brouwer and
approved by Weyl: Classical logic was abstracted
from the mathematics of finite sets and their
subsets .... Forgetful of this limited origin,
one afterwards mistook that logic for something
above and prior to all mathematics, and finally
applied it, without justification, to the
mathematics of infinite sets. This is the fall
and original sin of set theory. (H. Weyl)
Concerning the question of statistical evidence
for growing scepticism I can report my personaI
efforts over many years: When I teach Cantor's
diagonal argument, every student understands that
the real numbers are uncountable (because it is
really not hard to understand that argument).
When I represent all the real numbers of the unit
interval by the paths of an infinite binary tree
with a countable number of nodes, every student
understands that there cannot be not more paths
than nodes (because it is really not hard to
understand that argument). No student of mine has
ever argued against that, although that would not
have changed her marks! So we see Weyl's
statement approved: Actual infinity cannot be treated free of contradictions.
In this way I have contributed to increase the
scepticism against transfinite set theory by some
hundreds of heads (that are not below average intelligence).
[1]
-Il n'y a pas d'infini actuel; les Cantoriens
l'ont oublié, et ils sont tombés dans la contradiction. (Henri Poincaré)
-§ 174 Set theory is wrong because it apparently
presupposes a symbolism which doesn't exist
instead of one that does exist (is alone
possible). It builds on a fictitious symbolism,
therefore on nonsense. (Ludwig Wittgenstein)
- In the intellectual framework of our century
the actual infinite appears virtually anachronistic. (Paul Lorenzen).
[2]
- A construction does not exist until it is made;
when something new is made, it is something new
and not a selection from a pre-existing collection. (Edward Nelson)
- When the objects of discussion are linguistic
entities [...] then that collection of entities
may vary as a result of discussion about them. A
consequence of this is that the "natural numbers"
of today are not the same as the "natural numbers" of yesterday. (David Isles)
- Sequences generated by algorithms can be
specified by those algorithms, but what possibly
could it mean to discuss a 'sequence' which is
not generated by such a finite rule? Such an
object would contain an 'infinite amount' of
information, and there are no concrete examples
of such things in the known universe. This is
metaphysics masquerading as mathematics. (Norman Wildberger)
-Cantor's 'paradise' as well as all modern
axiomatic set theory is based on the
(self-contradictory) concept of actual infinity.
Cantor emphasized plainly and constantly that all
transfinite objects of his set theory are based
on the actual infinity. Modern AST-people try to
persuade us to believe that the AST does not use
actual infinity. It is an intentional and blatant
lie, since if infinite sets, X and N, are
potential, then the uncountability of the
continuum becomes unprovable, but without the
notorious uncountablity of continuum the modern
AST as a whole transforms into a long twaddle about nothing. (Alexander Zenkin)
- Herren Geheimrat Hilbert und Prof. Dr. Cantor,
I'd like to be Excused from your "Paradise": It
is a Paradise of Fools, and besides feels more like Hell (Doron Zeilberger).
Regards, WM
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