FOM: Sazonov on actual infinity

[David Isles]

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. 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?) formalisms plus the rules for manipulating them.
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, and thus still further formalisms. There is no
clean-cut separation between the mathematics and the means for
expressing it..
    The problem of  feasibility and the "nonfeasibly hugh" may be
chimerical.  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. 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.
    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