[FOM] Formalization Thesis
aatu.koskensilta at xortec.fi
Sat Jan 12 02:40:08 EST 2008
Timothy Y. Chow wrote:
> (1) Is the Formalization Thesis, as I've formulated it, approximately as
> precise as the Church-Turing Thesis? If not, can the precision problem be
> fixed easily with a simple rewording?
Not as far as I can see. To express the objection I and, I think,
Torkel, presented when you proposed this thesis in sci.logic, succintly:
the problem is that "correctly capturing" is not at all as clear and
unambiguous notion as the notion of extensional equality for functions.
The post <slrnf6am4g.q15.aatu.koskensilta at localhost.localdomain>
(http://groups.google.com/group/sci.logic/msg/1cf3026be617d644) might be
of some relevance here. I write, in particular, that
> To recapitulate: a sentence P formalises, or expresses, a mathematical
> statement if it's truth
> is equivalent to the statement using trivial mathematical reasoning,
> and a
> formula R(x1, ..., xn) formalises, or expresses, a mathematical
> relation P
> if it can be established, using trivial mathematical reasoning, that
> for all a1, ..., an R[num(a1)/x1, ..., num(an)/xn] is true iff P(a1,
> ..., an).
If we take "trivial mathematical reasoning" -- presumably reasoning
formalisable in some weak base theory, perhaps determined by context --
modulo some coding, it seems very plausible that all statements of
"ordinary mathematics" are expressible in the language of set theory in
this sense; and if we take ordinary mathematical talk to be
set-theoretical, all with presentations of this and that in terms of
sets, we needn't even bother with coding. But it's a stretch to think of
ordinary mathematical talk in terms of such a reduction, and then we
need to consider all sorts of intensional questions, e.g. whether facts
about the coding itself are relevant, etc.
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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