Monday, April 15, 2024

Avi Wigderson is a counterexample to TWO stupid thoughts of G.H. Hardy


1) Avi Wigderson won the Turing Award (See blog posts by Fortnow-here, Scott-here, Lipton-Regan here, and the ACM announcement here).  The last time I could find when Fortnow-Gasarch, Scott, Lipton-Regan all blogged on the same topic was when Goldwasser-Micali won the Turing Award- see the blog entries (here, here,here). We rarely coordinate. For that matter, even Fortnow and Gasarch rarely coordinate.

2) My joint book review of G.H. Hardy's  A Mathematician's Apology (1940) and L.N. Trefethen's An Applied Mathematician's Apology appeared in SIGACT News. 

These two events would seem unrelated. However, I criticize two points in Hardy's book; and those two points relate to Avi.  The book review is here

POINT ONE: Hardy says that Mathematics is a young man's game and that if you are over 40 then you are over the hill. He gives some fair example (Gauss, Newton) and some unfair ones (Galois, Ramanujan who died before they were 40.) Rather than STATE this fact he should have made it a CONJECTURE to be studied. I would make it two conjectures: 

Was it true for math that Hardy would know about? Since most people died younger in those days, there might be to small a sample size. Euler and Leibniz might be counterexamples.

Is it true now? AVI is clearly a counterexample. Other modern counterexamples: Michael Rabin, Leslie  Valiant, Roger Apery (proved zeta(3) irrational at the age of 62), Yitang Zhang (bounded gaps between primes at age 58, which, alas, is not a prime-- would have been really cool if it was a twin prime), Louis de Branges (proof of the Bieberbach conjecture at 51), Andre Weil, Jean-Pierre Serre. Is that enough people to disprove Hardy's conjecture? 

Despite the counterexamples I provided, we have all seen some mathematicians stop producing after a time. I offer two reasons for this

a) (Andrew Gleason told me this one) A mathematician works in a field, and the field dries up. Changing fields is hard since math has so much prereq knowledge.  CS has less of that problem. One can see if in the counterexamples above, and in other counterexamples, the fields they were in didn't dry up. 

b) The Peter Principle: Abosla is a great research so lets make her department chair!

My conjecture: The notion that math is a young mans game is false. 

POINT TWO: Applied Math is dull. Trefethan's book makes a good counter argument to this. I will say something else.

Even in Hardy's time he would have seen (if his head was not so far up his ass) that math, applied math, physics, compute science and perhaps other areas interact with each other. It is common to say that things done in pure math get applied. However, there are also cases where pure math uses a theorem from applied math. Or where Physics MOTIVATES a topic in pure or applied math. The boundaries are rather thin and none of these areas has the intellectual or moral high ground. There is the matter of personal taste, and if G.H. Hardy prefers pure math, that's fine for him. But he should not mistake his tastes for anything global. And is well known, he thought pure math like number theory would never apply to the real world. He was wrong about that of course. But also notice that Cryptography motivated work in number theory.  I am not sure if I would call AVI's work applied math,but it was certainty motivated by applied considerations.


  1. At least Yitang Zhang's age was the average two twin primes (if we allow for Grothendieck's).

  2. Age is a real thing. As we get older, our brain seem to slow down and we focus more on exploiting what we have rather than exploring new things. Both are well-known phenomenons.

    Is 40 the peak point? I don't know. Maybe 50? Maybe 60? Also depends on the person and their life style.

    If you take a distribution of major advancements against their age, historically I think 40 might seem a reasonable peak. Doesn't mean post-40 we cannot do anything, but the likelihood goes down.

    From goals perspectives, people also transition from trying to achieve things themselves to trying to leave a legacy through others (children, student, mentee, ...).

    One of the reasons that Geoffrey Hinton gave recently (at ~75 years old) for focusing more on philosophical and ethical aspects of AI was he is not feeling that he is sharp enough technically as he used to be.

    I think it is a disservice to ourselves to not acknowledge what we lose with age and not cherish what we can do with age. E.g. writing books that share the deep insights one has gained through a lifetime of working in an area as an expert is of great value to the society and next generation.

    1. (Bill) Lets say you lose some sharpness (thats reasonable). But at an older age you KNOW more and have had more experience, so that might compensate.

  3. JP Serre is in his 90s and sill gives technical lectures.. he is the youngest Fields medal winner as well. T. Tao as he is turning 50 next year and he is just getting started on remaking math culture through lean. Once done he will surely move to something else. But I agree.. the chances you get to learn a new field diminishes as you get older since there are lot of barriers and tricks to learn any new field. That includes baking.

  4. Serre's latest arxiv output

    1. (Bill) I didn't know Serre was still alive! He is, at 97. Thanks!

  5. There's a survivor's bias here--the older people we here about are the ones that are active. But the theory conferences back in the day are similar to today--dominated by the young. Where did all those young people go?

  6. Young ones as they become old ... 1. Different fields 2. Not enough prof jobs and decided to change careers 3. Not motivated enough since have life to take care off etc.

  7. Yuly Shipilevsky11:27 PM, April 15, 2024

    My latest efforts to solve P vs NP problem: