[FOM] Resources on the empirical foundations of mathematics
Alexander Zenkin
alexzen at alexzen.info
Fri Jan 13 03:29:58 EST 2006
On Tue, Jan 10, 2006 at 03:02:12PM -0800, Richard Haney wrote:
BEGIN of quotation
I am interested in studying the *empirical* foundations of mathematics.
[. . .]
So [. . .] I would like [. . .] to find some really good resources for
research in this area.
Can anyone help me out with this?
Richard Haney
END of quotation
Dear Richard Haney,
I would like to recommend you two very deep and good resources for
research in the area of real foundations of Mathematics.
1. DORON ZEILBERGER, "Real" analysis is a degenerate case of discrete
analysis. -
See at:
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/real.html
Citation from the paper: "The main thrust of that article was the
concept of `symbolic discretization', akin to, but much more powerful
than, `numeric discretization'. I believe that this crazy idea has a
great potential.
But, even more important, it suggests a truly rigorous and
honest foundation for the whole of mathematics.
Towards a FINITE (and hence RIGOROUS) Foundation of Mathematics
(i) The mathematical (and physical) universe is a huge (but FINITE)
DIGITAL computer."
And also at least two Doron ZEILBERGER's famous opinions:
Doron Zeilberger's OPINION 57: The Unbearable Non-Triviality of
"Trivial" Mathematics, and the even more Unbearable Triviality of
"Non-Trivial Mathematics"
See at: http://www.math.rutgers.edu/~zeilberg/Opinion57.html
Main ideas.
"Let's try to understand why he (G.H.Hardy) thought that Chess is
trivial, while Number Theory is not.
I call it `infinite-fixation'. Most mathematicians think that
whatever is `finite' is trivial, and reducing a problem to `checking
finitely many cases' renders it trivial, and makes the actual checking
unnecessary. On the other hand a real theorem incorporates infinitely
many facts, hence is `transcendental' and `deep'.
But this infinite is the biggest fiction ever invented by
humans. We are all finite creatures who live in a finite world, and even
our mathematical world is finite. [for more details, see my paper "Real"
Analysis is a Degenerate Case of Discrete Analysis ].
< http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/real.html
>"
Doron Zeilberger's OPINION 68: Herren Geheimrat Hilbert und Prof. Dr.
Cantor, I'd like to be Excused from your "Paradise": It is a Paradise of
Fools, and besides feels more like Hell. -
See at: http://www.math.rutgers.edu/~zeilberg/Opinion68.html
Main ideas. - They are not in need of any explanation here.
2. TIM GOWERS:
"Mathematical discussions contents page.
These pages are of various kinds, but they are nearly all attempts to
show how mathematical ideas arise naturally, in the hope that some
people will find them a useful supplement to university mathematics
courses. Often they contain ideas that I have come across in one way or
another and wish I had been told as an undergraduate. (Probably I was
told several of them, and just wasn't concentrating enough to take them
in at the time.)"
See at http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html
Tim Gowers's WEB-page: http://www.dpmms.cam.ac.uk/~wtg10/
You can also cast a glance at http://alexzen.by.ru/
Take pleasure in reading these 'resources' and best regards,
Alexander Zenkin
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