Date: Jan 31, 2013 2:15 AM
Author: fom
Subject: Re: Matheology § 203

On 1/30/2013 11:19 AM, WM wrote:
> On 30 Jan., 12:53, fom <fomJ...@nyms.net> wrote:
>> 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.

>
> Same is true for physicists. But I had the impression that
> mathematicians are more concerned with proving. I, as a physicist, am
> more concerned with showing counter examples.


Most mathematicians would agree with that
statement on the surface. But, if you think
about how people use mathematics arbitrarily,
you can interpret the activity of mathematics
as establishing the scientific coherence of
these uses. By "scientific," I mean precisely
the epistemological justification for a
demonstrative science given by Aristotle.

What has been lost to the teaching of mathematics
because of the mathematical realism of the late
nineteenth and early twentieth centuries is the
relationship of analysis and synthesis in the
practice of mathematics. Analysis is "reverse
engineering" and synthesis is proof.

There is an inherent circularity in mathematics
that actually belongs there. The best account
of why it belongs there is in Emile Borel's
book "Space and Time." Space has a sense that
can only be made analytic, for example, through
the sign of a determinant.

Analytic philosophy takes that geometric distinction
and turns it into "the True" and "the False".

To see this, get a book on threshold logic. Of
the sixteen basic Boolean functions that represent
the semantics of propositional logic, exactly two
are not linearly separable. Those two are logical
equivalence and exclusive disjunction.

When "logic" finally gets around to a parallel
development with geometry in the cylindrical
algebras of Tarski and Monk, informative
identity, namely

x=y

is equated with two-dimensional subspace through
the origin. For three dimensions, this is a
hyperplane.

Correspondingly, this is exactly why logical
equivalence is not a linearly separable switching
function. Its truth-set and its falsity-set
cannot be separated by a hyperplane.

Analytical philosophy conflates these geometric
relations in its assertions of logical priority.

Quoting Frege,

"The more I have thought the matter over,
the more convinced I have become that arithmetic
and geometry have developed on the same basis --
a geometrical one in fact -- so that mathematics
in its entirety is really geometry. Only on
this view does mathematics present itself as
completely homogeneous in nature. Counting,
which arose psychologically out of the demands
of practical life, has lead the learned astray."

<snip>

>
> All that sounds interesting but is a bit above my level.


Sorry about that.

If you can find Robinson's work ("On Constrained Denotation")
you will appreciate it. In keeping with Frege's original
observation that truth values depend on statements completed
with NAMES, Robinson explains that the diagonal of a
model -- that is, the (x,x) corresponding to x=x -- is
defined by NAMES.

This is largely what you have been saying in your posts.