Re: Grothendieck toposes

[Marta Bunge <martabunge@hotmail.com>]

Dear Andre,


Of course I agree with you in that the natural world is only partially explained by science. I also believe that much of mathematics has been inspired  by a desire to apply it to the natural sciences. Notice, however, that I said "much" and not "all",  and here is where mathematics and the natural sciences differ. Mathematicians have a freedom not afforded to scientists.  It is this freedom which allows the invention of objects such as the complex  numbers or of the infinitesimals. Now, is it a mystery that such products of the human mind find applications in scientific theories, or is it rather  that the latter themselves are also the product of the human mind? After all, it is only through rational thinking (including intuition) that we are able to (believe we) understand the natural world. Now, what about actual applications of science? Those are not just the product of the theories themselves, but also of experimentation and of successive approximations. It is  for this reason that I see no mystery in that certain scientific theories can sometimes be succesfully applied. Whether this is or is not a metaphysical point of view it is not for me to say.


Best regards,

Marta




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Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics
McGill University
Montreal, QC, Canada H3A 2K6
Home: (514) 935-3618
marta.bunge@mcgill.ca
http://www.math.mcgill.ca/people/bunge
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________________________________
From: Joyal, André <joyal.andre@uqam.ca>
Sent: November 2, 2016 1:50:08 PM
To: Marta Bunge; categories@mta.ca
Cc: Steve Vickers; Patrik Eklund
Subject: RE: categories: Re: Grothendieck toposes

Dear Marta,

Mathematics and science are very often regarded
as the pure product of human rationality.
I can agree with the importance of rationality, except that humanity
is as much the product of nature as it is of rational choices.
You will agree that the natural world is only partially explained by science.
The rest is a big mystery. Not that the mystery is absolutly impenetrable.
I feel compelled to recognize the presence of mysteries even
in mathematics. The history of complex numbers, from the
discovery by Cardano to their applications in quantum physics is bewildering.
They belong to this universe as much as the electron and the human mind.
The fact that we human can understand  complex numbers may
have a metaphysical meaning.
What is it?

Best,
André



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