[FOM] Formalization Thesis
Aatu Koskensilta
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
More information about the FOM
mailing list