Expansion coefficients of size 10^100,000

Since writing An Applied Mathematician’s Apology I find myself noting more examples of how unquantitative mathematicians may be.

Here’s one. A well-known theorem by Müntz asserts that, for example, the function f(x) = x can be approximated arbitrarily closely for 0< x<1 by linear combinations of the functions 1, x2, x4, x6, … What Müntz’s theorem doesn’t tell you is that to do this, say to 6-digit accuracy, you’ll need 140,000 of those terms with coefficients as large as 10100,000 ! Such approximations would be useless in any conceivable application.

You might think that the discovery that a theorem is useless for any application would be of interest to mathematicians. In fact, most would regard this effect as just a curiosity. It doesn’t touch the essential truth or beauty of the theorem, and indeed, perhaps it enhances it by shining a light on the power of rigorous proof.

(I asked a leading expert in the area whether he knew of this 10100,000 effect. No, he responded, “I am not a numbers man.”)

In other words, a mainstream view among mathematicians is that it is not their business to care whether or not a theorem is capable of being applied. This raises the question, if it is not their business, then whose business is it? Engineers? I imagine most mathematicians would feel that no, one need not go so far as that. It is the business of the numerical analysts, who are, after all, mathematicians of a kind.

[27 July 2022]

Items on our bulletin board

The bulletin board nearest my office has seven color posters on it just now:

(1) Oxford Maths Summer Party
(2) Responsible Bystanders: If you see or hear inappropriate behaviour or language….
(3) The Athena Swan Principles
(4) The University does not tolerate any form of harassment
(5) Her Dark Mathematicals Women’s Coffee (for female and trans students and staff)
(6) Mathematrix (upcoming meetings for minorities/women/LGBTQ+/non-binary)
(7) SECTRA – the LGBTQIA+ Staff Network Pride celebration

Item (1) is an invitation to a party the day after tomorrow. All the others are aimed at special groups of people defined mainly by issues related to gender.

[22 June 2022]