W. P. Thurston, "How do mathematicians advance understanding of math?"

[Astronuc]

I was searching for information in a response to the OP in the following thread Foundational Mathematics:
https://www.physicsforums.com/threads/foundational-mathematics.1008334/

One of the recommended articles that popped up is William P. Thurston, "On Proof and Progress in Mathematics", Bulletin of the American Mathematical Society, Volume 30, Number 2, April 1994, pp. 161-177, which was written in response to an article by Arthur Jaffe and Frank Quinn, "Theoretical mathematics'': Toward a cultural synthesis of mathematics and theoretical physics," Bulletin of the American Mathematical Society, Volume 29, Number 1, July 1993, Pages 1-13. Reading Thurston's article prompted me to read the article by Jaffe and Quinn. It will take some time.

Thurston - https://arxiv.org/pdf/math/9404236.pdf
Jaffe and Quinn - https://arxiv.org/pdf/math/9307227.pdf

Of Jaffe and Quinn, Thurston writes the "article raises interesting issues that mathematicians should pay more attention to, but it also perpetuates some widely held beliefs and attitudes that need to be questioned and examined.

The article had one paragraph portraying some of my work in a way that diverges from my experience, and it also diverges from the observations of people in the field whom I’ve discussed it with as a reality check.

After some reflection, it seemed to me that what Jaffe and Quinn wrote was an example of the phenomenon that people see what they are tuned to see. Their portrayal of my work resulted from projecting the sociology of mathematics onto a one-dimensional scale (speculation versus rigor) that ignores many basic phenomena."

Thurston asks more or less, "What does one do as a mathematician?" He actually poses, "How do mathematicians advance human understanding of mathematics?" One could also consider, "How do physicists advance human understanding of Physics (or Nature, or the Universe)?"

Jaffe and Quinn start their article, after the abstract, with "Modern mathematics is nearly characterized by the use of rigorous proofs. This practice, the result of literally thousands of years of refinement, has brought to mathematics a clarity and reliability unmatched by any other science. But it also makes mathematics slow and difficult; it is arguably the most disciplined of human intellectual activities."

So, now I am wondering if there are comparable articles since. Are there additional articles reflecting on Thurston's article, or the general question of what we are trying to accomplish as mathematicians and physicists (or chemists, biologists, . . . . ) besides earning a salary or paycheck?

 

[Astronuc]

Where math meets physics - perspective from Penn (University of Pennsylvania)
https://penntoday.upenn.edu/news/where-math-meets-physics
Collaborations between physicists and mathematicians at Penn showcase the importance of research that crosses the traditional boundaries that separate fields of science.

The Penn article is one track I had in mind.

With many complex scientific questions still in need of answers, working across multiple fields is now seen an essential part of research. At Penn, long-running collaborations between the physics and astronomy and the math departments showcase the importance of interdisciplinary research that crosses traditional boundaries. Advances in geometry, string theory, and particle physics, for example, have been made possible by teams of researchers who speak different “languages,” embrace new research cultures, and understand the power of tackling problems through an interdisciplinary approach.

. . . despite their close connections, physics and math research relies on distinct methods. As the systematic study of how matter behaves, physics encompasses the study of both the great and the small, from galaxies and planets to atoms and particles. Questions are addressed using combinations of theories, experiments, models, and observations to either support or refute new ideas about the nature of the universe.

In contrast, math is focused on abstract topics such as quantity (number theory), structure (algebra), and space (geometry). Mathematicians look for patterns and develop new ideas and theories using pure logic and mathematical reasoning. Instead of experiments or observations, mathematicians use proofs to support their ideas.

See the example of mathematical thinking. There is an accompanying statement "Mathematicians look for patterns and ask if that pattern is just a special case (i.e., an exception) or indicative of something deeper." Well, physicists do the same.

With respect to using proofs, see the following:
https://www.maa.org/programs/facult...ch-sampler-8-students-difficulties-with-proof

 

[atyy]

https://arxiv.org/abs/math/9404229
Responses to ``Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics'', by A. Jaffe and F. Quinn
Michael Atiyah, Armand Borel, G. J. Chaitin, Daniel Friedan, James Glimm, Jeremy J. Gray, Morris W. Hirsch, Saunder MacLane, Benoit B. Mandelbrot, David Ruelle, Albert Schwarz, Karen Uhlenbeck, René Thom, Edward Witten, Christopher Zeeman
This article is a collection of letters solicited by the editors of the Bulletin in response to a previous article by Jaffe and Quinn [math.HO/9307227]. The authors discuss the role of rigor in mathematics and the relation between mathematics and theoretical physics.

 

[George Jones]

... the general question of what we are trying to accomplish as mathematicians and physicists (or chemists, biologists, . . . . ) besides earning a salary or paycheck?

Why study and research fundamental physics and mathematics? Why study curved spacetime and general relativity? Quarks and other elementary particles? Topos theory? A selfish answer for many that do is "Because, for me, it's fun and rewarding!" Deeper reasons exist. Science and mathematics, including the fundamental and abstract, is part of who we are as a species. These things are as much part of our culture as music, art and literature. If we lose the desire and ability (possibly through politics) to advance our fundamental knowledge through curiosity, we have failed as humans.