[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).

-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).

- 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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