Gödel and the nature of mathematical truth

A talk with Rebecca Goldstein

Mathematicians and physicists are just as guided by principles of elegance and beauty as novelists and musicians are. Einstein told the philosopher of science Hans Reichenbach that he'd known even before the solar eclipse of 1918 supported his general theory of relativity that the theory must be true because it was so beautiful. And Hermann Weyl, who worked on both relativity theory and quantum mechanics, said "My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful."

Kunstspaziergänger - am Freitag, 24. Juni 2005, 07:51 - Rubrik: Klassische Probleme

mazzola meinte am 13. Jul, 10:55:

truth and beauty

I sent this email to: Rebecca.Goldstein@trincoll.eduDear Rebecca Goldstein,

I found your fascinating interview about Gödel, maths, and beauty on the web. Very much in my own spirit. So let you know these informations:

I am a University mathematician, first specialized in algebraic geometry, now specialized in mathematical music theory and music informatics. I have been working for 25 years on this field. My last book about mathematical music theory is

The Topos of Music—Geometric Logic of Concepts, Theory, and Performance. Birkhäuser, Basel-Bostono 2002.

To learn more, go to

http://www.encyclospace.org

In this book and in my recent talks, I have strongly supported the thesis that modern Mathemaical Logic in its shape given by Topos Theory, allows for identification of truth and beauty, i.e., the same logical objects deal with logical and aesthetic categories.

I have recently given two talks, one at the famous Ecole normale supérieure in Paris (http://www.encyclospace.org/talks/mamusem.ppt), one at the Zurich conservatory (http://www.encyclospace.org/talks/topolog.ppt), both dealing with this subject.

Perhaps this could interest you?

sincerely yours

Guerino Mazzola

-----

PD Dr. habil. Guerino Mazzola

CSD/U Zurich

Office 2.28

Andreasstrasse 15

CH-8050 Zürich

Voice:

Office +41 44 635 67 46

Private +41 44 821 98 56

Mobile +41 79 752 20 18

Fax:

Office +411 635 45 07

Private +411 821 98 51

Emails:

guerino@mazzola.ch

mazzola@ifi.unizh.ch

Web: www.encyclospace.org

phunk meinte am 6. Feb, 17:10:

art vs. science

wonderful article. again this rises the question of the difference between art and science. Inspired by the american novelist Theodore Sturgeon (1918-1985):

The goal of an artist and a scientist is the same: Seek for an utopian truth. Biologists, physicians or sociologist; finally they all want to find an ultimate truth just in different terms like - i am guessing here - evolution, energy or society. Philosophers and mathematicians do the same - just from a more abstract, more ubiquitious level.

So what is the difference? Sturgeon said it is simply the method of this search. While an artist rather tries to find the 'ultimate masterpiece' in an unsystematic, punctiform way, a scientist clearly defines steps towards this goal (= rather gradual search). To define and check these steps gramma) and vocabulary (= formalisms) allow greater expressiveness. I guess gramma/syntax might be more important for mathematics (rather: trivial to non-trivial expressions) whereas vocabulary/semantics is more intensly used in philosophy (rather: non-trivial to non-trivial expressions). (edit: could be plain stupid...)

But what is the use of strong formalisms as in mathematics if you can follow the truth in an unsystematic way as well? In my opinion - coming from an artist perspective and now trying to learn some more rigid formal methods - the unsystematic approach is just extremely hard to follow consistently: it is hard to keep on working. An example might be the very short period of Dadaism in Zurich (1915-~1922). This also reflects my way of thinking about modern art (and btw also a about bit modern science): pseudo-pluralist-liberal materialistic short-term thinking: For example take a 'modern artist' that does painting by randomly slapping on a coat of paint. If he does during all its life I am impressed but often his performance is accompanied by huge publication efforts and it is necessary to speak so much about the work because the work does not speak for itself.

Adding formal methods inproves our thinking itself; they are not constraints rather than "boost-packs" for new ideas. Nietzsche said: "Den Stil zu verbessern heisst den Gedanken zu verbessern" (to improve style means to improve thought). Yet as I learned in Mrs. Goldstein's article it should not be done too excessive as Gödel did. I hold to the position of critical constructivism, inspired by Sartre: If it is true that only I construct the world and I, the subject, cannot prove that anything else but lifeless objects exist (Descartes' solipsistic cogito ergo sum) how can I disprove that there are objects that can do the same vice-versa (= another person looking at me and construct my entitiy as I do construct his entity)? Only the reflection of thought towards others can assure reality. Reality does not exist a priori but it is a social construct of what it possible (= I can only look at others to know what exists). Nonetheless there are great (isolated) individuals that can extend what is possible by climbing a high mountain or writing a book that sets new constraints, meaning, contradicting my logical constructs. Simply by knowing it has been done it keeps me going. Martin Cohen collected some of the philosophical riddles that had astonished the old Greeks (Achilles-turtle race, I love that story!).

Maybe the human species is not about fulfilling a target goal (as explained in (truly esoteric) micro-theoretical economics) or to fulfill a fitness function of surviving of the fittest as in Darwinian "prose" but to stay in constant but always changing contradiction.

a loudmouth-student (sorry for disturbing.....it was good to release some thoughts....and: sorry for the polemic issues, hehe)