fom
Posts:
1,099
Registered:
12/4/12
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Re: Matheology § 203
Posted:
Jan 30, 2013 6:53 AM
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On 1/30/2013 5:29 AM, WM wrote:
> On 30 Jan., 12:02, fom <fomJ...@nyms.net> wrote:
>
>> As for those "logical considerations," I mean that
>> one can develop a hierarchy of definitions that
>> depend on actual infinity. To say that mathematics
>> is "logical" is to concede to such a framework. I
>> do not believe that mathematics is logical at all.
>
> That is a very surprising statement. Why do you think so?
In his papers on algebraic logic, Paul Halmos made
the observation that logicians are concerned with
provability while mathematicians are concerned more
with falsifiability. This is the difference between
a filter and an ideal.
It is also the exact question discussed by Aristotle
when speaking of the relation between definitions and
identity in Topics.
Logical identity, in the modern parlance, is ontological
"self-identity" arising from a combination of Russell's
description theory and Wittgenstein's rejection of
Leibniz' principle of identity of indiscernibles.
Aristotle points out that one can never prove an
assertion of sameness, although one can destroy such
an assertion. The modern logic negates this entire
relationship between identity and definition.
Given the choice, it is better to side with Halmos
and Aristotle (and Frege).
The axiom,
x=x
applies simultaneously to ontology and semantics
and cannot simply be interpreted ontologically as
one must do with Russell and Wittgenstein.
Along similar lines, note that Tarski's paper on
truth in formalized languages specifically excludes
scientific languages built upon definition whereas
Robinson's paper on constrained denotation specifically
includes the relationship between descriptively-defined
names, identity in models, and truth.
And, in Kant, logic is a *negative criterion of truth*.
In other words, one ought not be proving beliefs in
mathematics.
Analysis with synthesis is a circular investigation
of structure. Synthesis without analysis is something
else altogether. When combined with realism, it is
religion.
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