[FOM] On the Nature of Mathematical Objects

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Tue Aug 10 16:42:23 EDT 2004

This is a prompt reaction with the lack of time and might be written
with more care. Unfortunately, the nearest three weeks I will hardly
be able to participate in any discussions.

Dmytro Taranovsky wrote:
> As Vladimir Sazonov has pointed out, the recent discussion about the scope of
> and formalization in mathematics glossed over what is a mathematical structure.
>  I will try to answer that question.
> The question is what are mathematical objects and what questions about them are
> mathematical.  Mathematical objects are objects whose intrinsic properties are
> absolute, as opposed to being contingent on the state of the world.  

I understand this "as opposed", but I have some doubt on the
absolute nature of mathematical objects which are creatures of
our minds and, as any intuition, are inherently vague (to that
or other degree). I do not want to say that intuition in everyday 
mathematical activity has no value in comparison with mathematical
rigor or formalisms. "Absolute" rigor as complete formalizability
(I think, nobody here has a doubt what does it mean) is only an
ideal which may be never achieved, but, as any ideal (like of a
God - although I do not like to invoke the God in the context of
Science), it determines the whole our life, as well as the life
and the essence of mathematics.

> mathematical question is a rigorous and completely unambiguous question about
> mathematical objects, 

What makes it completely unambiguous? Is not that quite
formal context of the whole mathematics (+ all the intuitions
behind of this context)?

where "about" is interpreted strictly to mean that the
> question cannot invoke non-mathematical objects.  

How to define it? In terms of absolute character of mathematical
objects (what is unclear to me) or in therms of intuitive objects
of our imagination related to our formalisms?

> Questions about fictitious characters, and even about physical reality are
> riddled with ambiguity.  

Same for mathematical objects. They *appear* to us as non-ambiguous
only because we require them to follow quite formal and unambiguous
rules. Let us start thinking on these object slightly differently,
forgetting some of the rules. Say, consider the (well-known paradoxical)
least natural number which cannot be described with less than one
hundred symbols.

I already wrote many times that intuition without the
"skeleton" of a formalism is amoeba like. Analogously,
pure formalism without a (vague) intuition is dead.

> A major development of modern mathematics is the recognition that all
> mathematical questions that have been asked can be rephrased as questions about
> sets, more precisely (if there are other types of sets), well-founded,
> extensional sets that are built from the empty set.  Thus, it seems likely that
> without loss of generality, mathematics could be defined as set theory.

Could not you imagine, at least in principle, that some quite 
mathematical formalism will be found which cannot be grasped
by set theory in its contemporary form? Why should we restrict
ourselves just to ZFC (even if all the contemporary mathematics
have been successfully formalized in ZFC)? It seems much more
safe to say that mathematics deals with various formalisms
( + corresponding intuitions, of course) some of which have
relatively (it is even tempting to say - absolutely - but this
is definitely not true) universal character, like ZFC.

Best wishes,

Vladimir sazonov

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