FOM: December 1 - December 31, 1999

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Re: FOM: defining "mathematics"

Sam Buss wrote:

> "Mathematics is the study of objects and constructions, or of aspects of
> objects and constructions, which are capable of being fully and completely
> defined.  A defining characteristic of mathematics is that once mathematical
> objects are sufficiently well-specified then mathematical reasoning can be
> carried out with a robust and objective standard of rigor."

I am afraid that these "fully and completely defined objects" would lead
us imperceptible again and again back to Platonism. Also what does it
mean "capable of being fully and completely defined"? I think that this
or other way it is inevitable to explain all of these in terms of appropriate
formal systems.

> One of the distinguishing features of mathematics is the use of
>         proof and of mathematically rigorous reasoning.

I would say "main distinguishing feature", and this is again reducible
to the general concept of formal system.

 > My basic assertion was that the real foundations of mathematics is
 > first-order logic, or the use of rigorous reasoning and
 > mathematical rigor that can be formalized in first logic.  This is
 > in contrast to the usual point of view that set theory is the
 > foundations of mathematics.

I agree, except for the stress on a specific kind of formal systems
based on first-order logic (FOL). Only in contemporary mathematics
FOL (as implicit system of reasoning, even if the most of mathematicians
do not know about this) is prevailing. Who can predict what will be in the
future? Even now we have a lot of other logical systems.

I prefer more general definition of mathematics (related with recent
postings of J. Mycielski; cf. also my reply to him and some other
postings to FOM):

     Mathematics is a kind of *formal engineering*, that is engineering
     of (or by means of, or in terms of) formal systems serving as
     "mechanical devices" accelerating and making powerful the human
     thought and intuition (about anything - abstract or real objects or
     whatever we could imagine and discuss).

Other sciences can use these devices. They even can participate
in their creation and investigation. But only mathematics has
these formalisms as *subject matter* (which was asked by Stephen
Simpson, but he seems had something differend in mind).

Note, that formal systems can be understood in a rather broad sense.
When a "normal" mathematician checks correctness of a proof he
does this rather mechanically, according to some, often implicit but
well "trained" by his previous experience rules.

Finally, I would like to stress that mathematics actually deals
nothing with truth. (Truth about what? Again Platonism?) Of course
we use the words "true", "false" in mathematics very often.
But this is only related with some specific technical features of
FOL. This technical using of "truth" may be *somewhat* related
with the truth in real world. Say, we can imitate or approximate
the real truth. This relation is extremely important for possible
applications. But we cannot say that we discover a proper
"mathematical truth", unlike provability. This formalist point of
view is not related with rejection of intuition behind formal
systems. But the intuition in general is extremely intimate thing
and cannot pretend to be objective. Also intuition is *changing*
simultaneously with its formalization. (Say, recall continuous
and nowhere differentiable functions.) Instead of saying that
a formal system is true it is much more faithful to say that it is
useful or applicable, etc. Some other formalism may be more
useful. There is nothing here on absolute truth.

By the way, as an example of useful and meaningful formal system
I recall *contradictory* Cantorian set theory. (What if in ZFC or
even in PA a contradiction also will be found? This seems
would be a great discovery for the philosophy of mathematics!)

Vladimir Sazonov

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