FOM: Sazonov on actual infinity

[Steve Stevenson]

Vladimir Sazonov writes:
 > >     I am prompted to write to you by your FOM posting of May 2 which I
 > > found very stimulating. I agree with (what I understand as) your
 > > position that our only "grasp" on "actual infinity" is through the very
 > > finite expressions of our formalisms. A related point, which I think is
 > > not emphasized enough in these discussions of foundational ideas, is
 > > that mathematics is a SOCIAL enterprise: What an individual may "think"
 > > or "feel" to herself is, without communication, inaccessible to her
 > > colleagues. 
 > 
 > When we think, we fix our (mathematical) ideas on the paper 
 > (eventually in terms of a formal system). We do this not 
 > only for colleagues but just for ourselves. Otherwise this 
 > would be a philosophy (philosophers also express what they think 
 > but differently) or whatever you want, but not a mathematics. Of 
 > course, now this can be communicated to colleagues, but not necessary, 
 > at least during some time. 
 > 
 > I cannot imagine a mathematician who do not write proofs, 
 > or at least theorems, only "thinks" without fixing anything. 
 > In this sense I cannot understand Brouwer's views on Mathematics.  
 > Was his mathematical behaviour really such one? Was he presenting 
 > non-rigorous proofs (proofs which were impossible to make rigorous)? 

I'm certainly no authority on Brouwer. But given the introduction of
the "social" enterprise, was not Brouwer getting at the psychology of
things? At the times he was formulating his ideas, cognitive
psychology had nowhere the standing it does today. I've heard it
claimed that "problem solving" was never really studied until George
Polya. Consider Ryle's view of things (Gilbert Ryle, *Concept of
Mind*,Hutchinson,1949).  He differentiates between *knowing what*
("logic" or explicit knowledge) and *knowing how* or maybe better
stated as *knowing what ought to be* (implicit knowledge). If you
define *mathematics* with a little "m" as the written artifacts
(explicit knowledge) and *Mathematics* with a *M* as the process of
taking implicit knowledge in the head to the explicit artifactual
knowledge, then it seems to me we are closer to Brouwer's idea of the
"mental constructions." Surely his standing as a true mathematician by
your definition is not in question --- he clearly believed in
artifactual proof.

steve