FOM: December 1 - December 31, 1999

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FOM: GCH for some cardinal nos.



Dear Professor Mycielski,

I like very much your posting to FOM, especially  your notes
on rejecting Platonism and how you do this in the framework of
set theory. I completely agree that

> mathematics is a
> human construction and not a description of an ideal world independent
of
> humanity. A construction which is physical (electrochemical processes
in
> brains, computer computations, and notes on paper) and is as real as
other
> physical objects made by people and machines.

Further, I would like to somewhat reformulate the next phrase:

    Thus, in a real enough
    sense, mathematicians are very much *like* engineers, architects,
    painters or sculptors.

Here mentioning engineers is particularly appropriate.
For example, they construct air planes which strengthen our
ability to move in the space. Mathematicians create somewhat
different "devices" (like rules of multiplication of decimal numbers,
rules of calculating derivatives or integrals in Analysis, logical
modus ponens and reductio ad absurdum rules, etc.) which crucially
strengthen our thought, our ability to describe and understand the
real world. These are not rules *of* thought, but, rather, rules or
levers *for* thought. In this sense mathematics is just a special
kind of engineering. In this sense computers, as levers for thought,
also may be used in (or belong to) mathematics.

The above seems is in coherence with your views.
But I cannot agree that

> mathematicians are *no more* [my emphasis, - VS] formalists than
engineers, architects,
> painters or sculptors.

What is then the crucial difference between mathematics and other
activities? Mathematical rules are formal (rigorous), unlike the
rules which use, say, sculptors in their work. Even lawyers use
not so formal rules as mathematicians, and they hardly could even
hope on the mathematical level of rigor. (Chess game has formal rules
of the same rigor as in Math., but they hardly could be called levers
for thought.)

Formal rules is a *material of which mathematics is made*.
Painters or sculptors use different materials. Actually, mathematicians
need not always reach the fullest formality. But they always (at least
since Euclid) had some ideal of what is rigorous rule or a proof. Each
case when such an ideal was temporary lost was a drama. Contemporary
logic gives good explication of this rigor. (But it could be discussed
whether this  rigor/formality is really fullest possible one.)

Only due to formality the rules of mathematics and logic are so
powerful. Formal - therefore we can use them mechanically and
repeatedly, thinking thereby on more high level entities. (A specific
property of the material, like that of metal for making a knife.)

I have nothing against the term "rationalism". However, it seems
cannot serve as replacement for "formalism". I do not understand
why you do not want to be called a formalist. If your position is
not (rational, reasonable) formalism, what is then formalism?

The fact that formalism was often used in foundations of mathematics
like a shameful brand does not mean that it is so bad. It seems we
should
simply recover the proper meaning of this word. Where from it follows
that formalism deals with formal systems without any respect to their
meaning? Say, engineers, in principle, also can build anything
meaningless/useless by using the same rules and materials by which
they construct a plane. So what?


Best wishes,

Vladimir Sazonov






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