Date: Jan 30, 2013 6:53 AM
Author: fom
Subject: Re: Matheology § 203
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.