[FOM] mathematics as formal

[Steven Ericsson-Zenith]

On Mar 24, 2008, at 4:03 PM, Vaughan Pratt wrote:
> ... On further reflection I would define mathematics more carefully as
> follows, after first defining logic.
>
> Logic is exact irrefutable argument.
>
> Mathematics is the logical demonstration of many beautiful or useful
> consequences of a few simple or well-motivated premises. ...

I am not happy with Sazonov's argument, but then this does not help us  
either.

It seems to me that "Logic" is first in the business of establishing  
and studying conventions and when the nature of that study is extended  
to include matters of apprehension it is rightly called  
"Semeiotics" (with Peirce and Locke as precedence).

Formalisms are simply rigorous conventions (not limited to algebras).

In contrast "Mathematics" is in the business of quantity and types,  
and identifying the relations between quantities and types. Applied  
mathematics then, is simply the application of Pragmatism to  
mathematics.

To the extent that formalism is required in the discipline of  
mathematics one can say that mathematics uses formal convention but I  
agree with Vaughan Pratt (if I understand his argument correctly) that  
to define mathematics merely as formal is an error. But to include the  
subjective quality of "beauty" and the subjective judgement of "well- 
motivated" (matters of apprehension) in a definition of mathematics I  
really do not understand, unless they are to be quantified by the  
application of Ockham's Razor or allowed by Poetic License.

It follows, of course, that Physics is in the business of measurement.  
Yet Sazonov's rational would appear to require that Physics and  
Mathematics be defined the same, as Formal. It should be clear that  
conventions are necessary in all of our pursuits and the degree to  
which any of them are formal is a matter of pragmatics.

With respect,
Steven

--
Dr. Steven Ericsson-Zenith
Institute for Advanced Science & Engineering
http://iase.info
http://senses.info