Re: True Gems of Scientific Epistemology

From: Herman Jurjus (h.jurjus_at_hetnet.nl)
Date: 07/27/04 

Date: Tue, 27 Jul 2004 12:59:49 +0200

Eray Ozkural exa wrote:
> Herman Jurjus <h.jurjus@hetnet.nl> wrote in message news:<2mjqsaFndi9bU1@uni-berlin.de>...
>
>>One more (not a book, but a paper, and with more limited scope):
>>
>>A.S. Yessenin-Volpin, The Ultraintuitionistic Criticism and the
>>Antitraditional Program for Foundations of Mathematics.
>>In: Intuitionism and Proof theory, North-Holland, 1968/1970.
>
>
> A very interesting title. What is ultraintuistionistic criticism?

Well, i'll probably give only a caricature, but i'll have a go.

The intuitionists always claim that they want to restrict to
constructions that can be 'really carried out' (in the mind). But if you
look at the resulting mathematics, intuitionists sometimes also accept
constructions that can only be carried out 'in principle'.

Essenin-Volpin chooses a more radical path. One in which the intuition
'1,2,3,...' does not lead to a unique sequence, because astronomically
large numbers do not exist; but how far you can count is undeterminable
(hence the non-uniqueness).
The result is (a start of) an alternative sort of mathematics in which
complete induction does not hold (0 is feasible, if n is feasible then
so is n+1, but not every natural number is feasible.)

It is quite difficult to turn that into a formal system, since, it has
rather farreaching consequences: after all, a formal language is nothing
but some infinite set, a mathematical thing of some kind. And if you
change how you reason about infinite sets, you must also change the way
you reason about languages, of course.
(Do unfeasibly long sentences count as sensible sentences, for example?
Do unfeasibly long proofs of inconsistency count as inconsistency proofs?) 

-- 
Cheers,
Herman Jurjus

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