Jonathan M. Borwein

Shrum Chair of Science, FRSC

Simon Fraser University
Burnaby BC, V5A 1S6


`` Atelier sur le calcul symbolique
Workshop on Symbolic Computation

     CRM Atelier, September 21--27, 1997



Inverse Symbolic Computation: Empirical Mathematics


Researchers at the Centre for Experimental and Constructive Mathematics (CECM) and various collaborators (including David Broadhurst) have been developing tools and expertise in what we call inverse symbolic computation -- by which we mean methods of inferring symbolic "closed-form" structure from numerical data. In some contexts we also obtain a probability of correctness.

Firstly, I intend to describe, online, how we hunt for identities and infer structure from such data. This often times leads to (close to) certain knowledge without proofs. I will illustrate, in particular, the use of three web-interfaces:

and will emphasize the use of integer relation algorithms.

I also intend to pose for the meeting some questions regarding the appropriate quantification of our inference tools.

Cognate lectures on experimental mathematics and on interactive mathematics are listed below. And since this is a statistics meeting, here are fifty billion random digits.

More sophisticated technology for distributed research and learning/interactive mathematics is being developed in our POLYMATH project within the TeleLearning NCE.

Further examples include a Web interface and:

As time allows I will also look at some related - primarily visualization - tools

Fresh Breezes in the Philosophy of Mathematics

Extracted from an article by Reuben Hersh in:

American Mathematical Monthly,
August - September 1995, 589-594 (Go Back)

Philosophies of Mathematics:

I like to compare philosophy of mathematics today to philosophy of science in the 30's and 40's. That subject was dominated by logical positivists: Rudolf Carnap and his friends of the ``Wiener Kreis'' (Vienna Circle). As a result of taking Bertrand Russell and Ludwig Wittgenstein too seriously, they believed they knew the correct methodology for scientific work: (1) state the axioms; (2) give correspondence rules between words and physical observables; (3) derive the theory, as Euclid derived geometry, or Mach derived mechanics.

It was noticed after a while that what logical positivists said had little in common with what scientists did or wanted to do. New ideas in philosophy of science came from Karl Popper, Tom Kuhn, Imre Lakatos, Paul Feyerabend. These subversives disagreed with each other. But they all thought philosophers of science could think about what scientists actually do, not bring presuppositions and instructions for scientists to ignore.

Philosophy of mathematics is overdue for its Popper, Kuhn, Lakatos, and Feyerabend. It's overdue for analysis of what mathematicians actually do, and the philosophical issues therein. In fact, this turn is taking place. Wittgenstein and Lakatos helped start it. In recent years Michael Polanyi, George Polya, Alfred Renyi, Leslie White, Ray Wilder, Greg Chaitin, Phil Davis, Paul Ernest, Nick Goodman, Phil Kitcher, Penelope Maddy, Michael Resnik, Gian-Carlo Rota, Brian Rotman, Gabriel Stolzenberg, Robert Thomas, Tom Tymoczko, Jean Paul van Bendegem, and Hao Wang have participated.

Here are some ideas some of these people hold.

  1. Mathematics is human. It's part of and fits into human culture. (Not Frege's abstract, timeless, tenseless, objective reality.)

  2. Mathematical knowledge is fallible. Like science, mathematics can advance by making mistakes and then correcting and recorrecting them. (This ``fallibilism'' is brilliantly argued in Lakatos' Proofs and Refutations.)

  3. There are different versions of proof or rigor, depending on time, place, and other things. The use of computers in proofs is a nontraditional version of rigor.

  4. Empirical evidence, numerical experimentation, probabilistic proof all help us decide what to believe in mathematics. Aristotelian logic isn't necessarily always the best way of deciding.

  5. Mathematical objects are a special variety of social- cultural-historical object. We can tell mathematics from literature or religion. Nevertheless, mathematical objects are shared ideas, like Moby Dick in literature, or the Immaculate Conception in religion.

(Go Back)


"Mathematicians are like pilots who maneuver their great lumbering planes into the sky without ever asking how the damn things stay aloft."

... ...

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.

... ...

The existence and nature of mathematics is a more compelling and far deeper problem than any of the problems raised by mathematics itself."


From David Berlinski's somewhat negative review of
The Pleasures of Counting by T. W. Korner (Cambridge, 1996) in The Sciences, July/August 1997, pages 37-41.

Korner is a careful and stimulating writer/teacher whose Fourier Analysis text I have used (interactively!) with great pleasure.

"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

On page 78 of Who got Einstein's Office? by Ed Regis, in his history of the Institute for Advanced Study (Addison-Wesley, 1986).

The answer is Arne Beurling.

The issue of paradigm choice can never be unequivocally settled by logic and experiment alone. ... In these matters neither proof nor error is at issue. The transfer of allegiance from paradigm to paradigm is a conversion experience that cannot be forced.

Thomas Kuhn

On page 216-17 of Who got Einstein's Office?

(Go Back)

Related HTML Lectures

Inverse Symbolic Computation: Empirical Mathematics (Revised)
(September 1998)

Virtual Science: The Changing Face of Mathematical Research
(March 1998)

Inverse Symbolic Computation: Empirical Mathematics
(September 1997)

FPSAC'97: Three Adventures in Symbolic Computing
(July 1997)

A Lecture and Much More About Pi

Doing Mathematics on the Web (Three related talks)
(March 1997)

(Go Back)

SFU / CECM / ~jborwein /
Revised 28/8/98 by Jonathan Borwein