[FOM] mathematics as formal

[Vaughan Pratt]

Vladimir Sazonov wrote:
>> From: Vaughan Pratt pratt at cs.stanford.edu
>> I can't accept this as I don't see how it distinguishes an Archimedes, 
>> Euler, Gauss, or Erdos from those who debate logical positivism, 
>> freewill, formalism, etc. I don't consider the latter mathematicians 
>> unless they bring mathematical tools to bear effectively on their subject.
> 
> In my understanding, 
> 
> "bring mathematical tools to bear effectively on their subject" 
> = 
> "taking and *exploiting* the form of human thought seriously 
> and consistently (in mathematical practice)".
> 
> Does not this make the distinguishing you want? Note that 
> "seriously and consistently" above does not mean taking 
> the form of thought by working mathematicians philosophically. 
> They can well have quite queer and probably wrong philosophical 
> views on math. The point is HOW they do mathematics objectively, 
> irrespectively on what they THINK about HOW they do it. 
> That is why I used the word "instinctively" in my previous posting. 
> HOW they do this can be observed and explained by an objective 
> observer, may be psychologist. 

I don't see how "taking and exploiting the form of human thought 
seriously and consistently" distinguishes logical positivists from 
mathematicians.  Adding "(in mathematical practice)" if intended as 
clarification of your notion of mathematics only adds circularity to 
your definition.

My reference to using mathematical tools is not part of my definition 
but enters only as a factor in the question whether logical positivists 
do mathematics.  They might as an adjunct to their work, but the work 
itself lacks the extreme *e uno plures* character of mathematics.  In 
mathematics precise and economical definitions unambiguously yield a 
rich structure.

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.

The boundary between pure and applied mathematics can be drawn according 
to where in the spectrum from simple to motivated the premises sit, and 
likewise between beautiful and useful for the consequences.  The 
boundary itself is not terribly beautiful, and whether it is useful may 
be more a matter of how well the pure and applied mathematicians in a 
department get along than anything else.

Judgment of beauty and utility is subjective, but in either case there 
should be something at stake in the case of mathematical beauty and 
mathematical utility.  That is, a consequence should be interpretable as 
answering a question *about* something.  One of Wolfram's small cellular 
automata may produce a large and beautiful pattern but unless 
propositions can be discerned within the pattern it is hard to see how 
it could count as mathematics, the pattern not being *about* anything in 
particular except its own beauty.  (I'm unclear as to whether Blok and 
Pigozzi's notion of algebraizable logic bears on this, its concerns seem 
narrower.)

Formality may help somewhat, but the example of Euclid's *Elements* 
shows that it is not essential---Euclid's reasoning is close to exact 
and irrefutable without any great formality.  It is a nice question 
whether the picayune logical flaws in the Elements pointed out by 
Proclus seven centuries later would have been exposed or masked by 
greater formality.  Without computer-aided verification, checking every 
symbol of a formal proof is exceedingly tedious making it hard to be 
sure every step has been adequately vetted.

Vaughan Pratt