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.