FOM: Platonism and social constructivism
Vladimir Sazonov
sazonov at logic.botik.ru
Wed Mar 25 10:30:36 EST 1998
Vaughan Pratt wrote:
> It seems
> to me that Hersh, Machover, and I are in essential agreement.
It is interesting that I also got analogous impression in
connection with positions of Moshe' Machover and mine own after
reading his posting on platonism and social constructivism. His
words on "illusion of platonism" which "enables mathematicians
to think and reason *as though* mathematical objects exist
independently of anyone's consciousness" seems to accumulates the
main idea of his posting which seems even to coincide with my view.
However, there are some points where it could be possible to object
or to rewrite somewhat differently. But, I believe, the reason is
mainly in "difference of emphasis", as Reuben Hersh wrote, or in
unfortunate misunderstanding, which often arise in a polemic
discussion.
Thus, I would say that mathematical objects exist *only*
together with and via some *formal* axioms, rules and
principles. Say, natural numbers exist in our minds together
with the *iteration rule* according to which we have + as a
*total* operation by iterating the successor, * by iterating +,
exponential by iterating *, etc. (Anyone may recall his/her
school lessons in arithmetic where this rule -- primitive
recursive schema -- was explicitly presented by these
examples.) Only via these (and may be other) rules we understand
(explicitly or not) this concept. There may not be *absolutely
pure* understanding prior any formalism. Without such rules as
above we would get somewhat different concept of natural numbers
(say, feasible numbers with *partial recursive* multiplication
or exponential functions). This is not an "absolutely
relativistic attitude to mathematical truth" [the words of
Moshe' Machover] or free play with axioms, but rather the
ordinary healthy relativism wrt a subject matter under
consideration.
Moshe' Machover in a later posting:
> one of the most profound problems of the philosophy of maths only
> *begins*: how to account for the ineluctable, necessary, objective nature
> of mathematical propositions.
I am not sure that I properly understand the point of this question.
It is formulated in too general terms.
If we decided to consider such and such mathematical idealized
objects satisfying such and such axioms and governed by such and
such rules, just follow these axioms and rules by using
corresponding intuition. If the axioms and rules reflect correctly
the intended concepts then the theorems will be also correct. We
have some kind of reality and some formal system which describes it
adequately. What is the problem? What else kind of objectivity we
need? May be you ask on a mechanism by which creation of
mathematical concepts and corresponding formalisms is done? That
there are some lows of mathematical thought which govern this
process? No doubts they are and it is most important to discover
(discuss) them.
Vladimir Sazonov
--
Program Systems Institute, | Tel. +7-08535-98945 (Inst.),
Russian Acad. of Sci. | Fax. +7-08535-20566
Pereslavl-Zalessky, | e-mail: sazonov at logic.botik.ru
152140, RUSSIA | http://www.botik.ru/~logic/SAZONOV/
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