FOM: Sazonov on actual infinity

Tue May 7 13:51:56 EDT 2002

Dear Prof. Isles, 

Thanks for your posting. 

> 
> Dear Prof. Sazonov,
>     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)? 

It is only when one expresses oneself through words,
> drawings, actions, etc. that the mathematics becomes accessible. The
> notion of a completely solipsistic mathematician (which seems to be an
> important part of Brouwer's view) only makes sense if you tacitly assume
> the existence of an independent observer who can describe the activities
> of the isolated being. But doing this reintroduces communication between
> agents back into the total picture. To me, anyway, it seems to follow
> that "mathematical concepts" are exactly (certain) communicative
> expressions (words, pictures, actions, etc.) and  that the properties of
> those concepts consists of the relations between those communicative
> expressions and other communicative expressions. Although this viewpoint
> seems to coincide with (what I understand as) "formalism", I believe it
> differs in an important way. Formalism holds that mathematics- or, at
> least, portions of set theory and elementary number theory- consists of
> certain (first-order?) 

Good question mark "?", but not because I am thinking now about second 
order logic. It may be not first order in some other way. It may be just 
arbitrary formal system in any possible language with any possible 
formal rules (and, of course, with any possible intuition behind 
this system). 

> formalisms plus the rules for manipulating them.

Plus corresponding intuitions. Otherwise it would be some caricature 
view on formalism which is unfortunately too widespread. 

> These formalisms themselves are not part of the mathematics being
> considered. This seems incorrect to me: the formalisms, the
> communicative expressions, themselves become objects of study, lead to
> new mathematics, 

I guess you mean METAmathematics. But without any METAmathematics, 
formalisms are used by mathematicians as the air for the breathing, 
even if they do not notice this fact. However, all mathematicians 
know very well that their proofs should be rigorous. This is just 
another wording for the fact that mathematics is based on formalisms 
(or semiformalisms - eventually formalisms). 

> and thus still further formalisms. There is no
> clean-cut separation between the mathematics and the means for
> expressing it..

Of course, without these (formal) expressing means there is no 
mathematics at all. 

>     The problem of  feasibility and the "nonfeasibly hugh" may be
> chimerical.  

I do not understand what is chimerical in the issue of feasibility, 
but I seems to agree with what you write below. 

What usually comes up in these estimates of, for example,
> size of models are functional expressions using, exponentiation,
> hyperexponentiation, Akermann's function, etc. But these are, after all,
> just notations. Nonfeasibility only becomes an issue when one imagines
> that these functions can be evaluated to give a "natural number" (i.e. a
> series of strokes llll...ll) One can avoid this trap by recognizing that
> the "natural numbers" consist of a variety of notations some of which
> may be comparable and others, at a given time, may not be. Comparability
> between different terms seems to be, in general, an empirical matter. 

Yes, and mathematics, as we know, is not foreign to applications. 
Say one of the greatest achievements of mathematics - Analysis of 
Infinitesimals - arose to describe the movement of planets. For me 
the fact that mathematics may be applied seems especially interesting 
and important. Then we cannot ignore feasibility issues (in principle  
related to computational complexity theory). What about a version of 
(f.o.)m. which would be oriented in some way toward these issues, 
unlike the ordinary (f.o.)m. (and even unlike the traditional 
"asymptotic" complexity theory which seemingly forgot why this 
theory ever arose) which ignores these proper feasibility ideas 
almost completely? 

Of
> course if you choose to go this route you have to rethink attitudes
> towards the use of induction in proofs and, in fact, which "proofs" may
> be proofs at all.

Any mathematical intuition (to become mathematical) needs in inventing 
an appropriate formalism. 

>     I hope that these remarks are of some interest.        Yours truly,
> 
> David Isles
> 
> Department of Mathematics
> 
> Tufts University, Medford,
> 
> Mass. 02155 USA
> 
> isles at attbi.com

Best regards! 

-- 
Vladimir Sazonov                        V.Sazonov at csc.liv.ac.uk 
Department of Computer Science          tel: (+44) 0151 794-6792
University of Liverpool                 fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K.       http://www.csc.liv.ac.uk/~sazonov