Re: True Gems of Scientific Epistemology

From: Eray Ozkural exa (erayo_at_bilkent.edu.tr)
Date: 07/27/04 

Date: 27 Jul 2004 11:20:13 -0700

Herman Jurjus <h.jurjus@hetnet.nl> wrote in message news:<2mmr09FoqaetU1@uni-berlin.de>...
> 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'.

Okay, but they do not restrict the constructions to effective procedures?

> 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).

Okay. Undeterminable in which sense? Like undecidable?

> 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.)

Ok.

> 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?)

I see. Thanks for the information.

Regards, 

--
Eray Ozkural

Relevant Pages

  • Re: True Gems of Scientific Epistemology
    ... > look at the resulting mathematics, intuitionists sometimes also accept ... but they do not restrict the constructions to effective procedures? ... > (Do unfeasibly long sentences count as sensible sentences, ... > Do unfeasibly long proofs of inconsistency count as inconsistency proofs?) ...
    (sci.math)
  • Re: True Gems of Scientific Epistemology
    ... > look at the resulting mathematics, intuitionists sometimes also accept ... but they do not restrict the constructions to effective procedures? ... > (Do unfeasibly long sentences count as sensible sentences, ... > Do unfeasibly long proofs of inconsistency count as inconsistency proofs?) ...
    (sci.physics.relativity)
  • Re: True Gems of Scientific Epistemology
    ... intuitionists sometimes also accept ... but they do not restrict the constructions to effective procedures? ... >>but some infinite set, a mathematical thing of some kind. ... >>Do unfeasibly long proofs of inconsistency count as inconsistency proofs?) ...
    (sci.math)
  • Re: True Gems of Scientific Epistemology
    ... intuitionists sometimes also accept ... but they do not restrict the constructions to effective procedures? ... >>but some infinite set, a mathematical thing of some kind. ... >>Do unfeasibly long proofs of inconsistency count as inconsistency proofs?) ...
    (sci.cognitive)
  • Re: True Gems of Scientific Epistemology
    ... intuitionists sometimes also accept ... but they do not restrict the constructions to effective procedures? ... >>but some infinite set, a mathematical thing of some kind. ... >>Do unfeasibly long proofs of inconsistency count as inconsistency proofs?) ...
    (sci.physics.relativity)