[FOM] Paolo Mancosu's monograph on 17th century mathematics

[Vladimir Sazonov]

Martin Davis wrote:

> I would like to suggest that mathematical practice (and especially the
> formalisms which tend to give it much of its power) is by its nature
> expansive, that concepts and methods tend to "overspill" leading to
> FOUNDATIONAL questions about the status and validity of the new domains
> hesitatingly revealed. The history of mathematics is replete with examples.
> I believe that it is the work on extensions of ZFC, and especially on large
> cardinals that is the main contemporary example of this phenomenon.


I realize that, being not a specialist in large cardinals, 
it is difficult to me to argue. I am rather working in 
applications of (mostly) finite (hyper)sets to database 
theory (semistructured or Web-like databases). Nevertheless, 
I would like to say my opinion on the foundational role of 
the ordinary set theory. 

As I understand, it was mainly the reaction on infinities 
which appeared in mathematics this or other way, essentially 
via Analysis and applications in Astronomy, Physics, Mechanics. 
A clear conceptual and formal foundation was necessary. 

After appearing and formalising, set theory became a part 
of mathematics, but its role for the rest of mathematics 
was always foundational, conceptual. Now, even if it is 
also considered as a branch of mathematics, trying to do 
something on large cardinals is still rather internal 
business of set theory. 

I believe, that another extension of mathematics (besides 
just introducing new concepts and branches of mathematics 
in the traditional framework, essentially of set theory) 
is possible. It will happen trough an INDEPENDENT of set 
theory introducing of new concepts, such new, as it were 
square root of 2 or infinitesimals in the previous times. 
This may be some new kind of infinity or even finity which 
the ordinary set theory is unable to formalize. It may be 
also something else, who knows? Then some new foundations 
will be necessary. This kind of revolutionary extension 
of mathematics seems to me very interesting to anticipate. 
And this would correspond to the lessons of the history 
of mathematics. 

Anyway, where from does it follow that mathematics will 
be always founded on the same (may be extending) formalism 
ZFC? 

That is, I believe that really (radically) new mathematics 
which needs in new foundations will arise not from the old 
foundations, but may be from some needs of the real world, 
say from Computer Science, such as the need in "non-asymptotic" 
version of complexity theory (yet to be well understood) which 
would correspond much better to the real computers and 
computability. 

It is a real challenge to make this idea mathematical and to 
find corresponding foundations. I do not see how the ordinary 
set theory could help. 


Vladimir Sazonov                        V.Sazonov at csc.liv.ac.uk 
Department of Computer Science          tel: (+44) 0151 794-6792
University of Liverpool                 fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K.       http://www.csc.liv.ac.uk/~sazonov


> 
> Martin
> 
>                            Martin Davis
>                     Visiting Scholar UC Berkeley
>                       Professor Emeritus, NYU
>                           martin at eipye.com
>                           (Add 1 and get 0)
>                         http://www.eipye.com
> 
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