Expiremental\Computational Mathematics - Links & Resources

Since the advancement of the computer there has been an active discussion in the math community on the role of computer results in mathematical research. Personal view is that computers are a remarkable tool that can show patterns that would hardly be discovered otherwise.

In the natural sciences (such as physics, chemistry, biology,...) the role of experimentation is unarguable and established and theory usually follows alongside experiment. While in math, experimentation became substantially relevant only recently - the computer serving as a "laboratory". In this page are included a collection of a few links & resources in favor of the benfits of computer use in mathematical research (if you have any suggestions\additions - send to: [email protected] ).

Mathematical Software

If you want to conduct your own mathematical experimentation research you should get mathematical software. You can choose "your weapon of choice" from one of the following: 

Selected Videos

Some talks & videos by Stephan Wolfram:

Some talks by Jonathan Browein: 

​Some talks by Doron Zeilberger:

A talk by John Cremona - on the history of number theory software: 
An insperational interview with Ken Ribet: 
The LMFDB website

One of the remarkable developments of the last few years is the LMFDB website ( http://www.lmfdb.org ). Which is an extensive database of classical mathematical objects arising in Number Theory - computed by the team work of many mathemticians and computer scientists world-wide. 
The following are notes by John Cremona about the project:  https://arxiv.org/pdf/1511.04289.pdf
See also:  https://www.eurekalert.org/pub_releases/2016-05/aiom-etm050316.php
The following is a beautiful clip illustrating the combined world-wide team code-writing efforts:

One of the offical goals of the LMFDB website is to shed new light on the magnum Langlands program. You can see an overview in the following lectures by Edward Frenkel:  

You can see the way the pioneering computational results of the LMFDB website fit in the framework of the Langlands program:  http://www.lmfdb.org/universe
Quotations - mathematics and experiment

"The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it"
(J. Hadamard)

"If I can give an abstract proof of something, I’m reasonably happy. But if I can get a concrete, computational proof and actually produce numbers I’m much happier ... I’m rather an addict of doing things on the computer, because that gives you an explicit criterion of what’s going on. I have a visual way of thinking, and I’m happy if I can see a picture of what I’m working with" (John Milnor)

"If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics" (Kurt Godel, Some Basic Theorems on the Foundations, 1951)

"This new approach to mathematics—the utilization of advanced computing technology in mathematical research—is often called experimental mathematics. The computer provides the mathematician with a “laboratory” in which he or she can perform experiments: analyzing examples, testing out new ideas, or searching for patterns" (Borwein and Bailey - Mathematics by experiment 2004)

"The computer has in turn changed the very nature of mathematical experience, suggesting for the first time that mathematics, like physics, may yet become an empirical discipline, a place where things are discovered because they are seen" (David Berlinski, “Ground Zero: A Review of The Pleasures of Counting, by T. W. Koerner,” 1997)