Sunday, April 21, 2024

Intelligent Comments on Bill's G.H. Hardy/Avi W post that we did not post.

I posted (see here) about Avi Wigderson being a counterexample to two of G.H. Hardy's opinions:

1) Hardy thought Math was a young man's game. I got some good comments on this. Some agreed and some disagreed.

2) Hardy thought applied math is dull. I got no comments on this one. I assume everyone agreed with my assessment that Hardy was wrong about this.

AND I got the following comment: 

Avi Wigderson's brilliance shatters the false assumptions of  G.H. Hardy, proving that intelligence knows no limits. His groundbreaking ideas challenge the status quo and inspire a new generation of thinkers.

I agree! I was going to post it! Then I realized that 
a) There were two links below it. A bad sign

b) While Avi is a brilliant thinker, is he known for challenging the status quo? The NW result pointed to P=BPP which was not thought to be true at the time. (Micheal Sipser thought P=BPP, see here, but he was an outlier. If my memory is wrong then leave a polite comment.)  Aside from that result  I don't think Avi's  work challenges the status quo. If I am WRONG leave a polite comment correcting my error. 

c) As a side note- I think of challenging the status quo as being over-rated. Wiles proof of FLT did not challenge the status quo. Many great results do not challenge the status quo. If Avi shows P NE NP that would be awesome even though it is the prevailing opinion. 

d) Upon reading the comment carefully,  it looks like it was AI generated. 

e) It fooled me for about 5 minutes. It didn't fool Lance for a second. 

3) But this comment inspired Lance to use ChatGPT to generate comments to my post: one agreeing, and one disagreeing

Avi Wigderson’s recent Turing Award is a compelling moment to reflect on the fluid nature of mathematical careers and the erroneous stereotypes surrounding them. G.H. Hardy’s assertion in *A Mathematician’s Apology* that mathematics is a young man's game is eloquently countered by Avi’s illustrious career and the achievements of others well beyond the supposed prime age. This challenges Hardy's notion and suggests it warrants reevaluation as a mere conjecture rather than a stated fact.

Moreover, Hardy’s dismissal of applied mathematics overlooks the dynamic interplay between various disciplines. Even in his era, the boundaries between pure and applied math, along with physics and computer science, were permeable and productive. Avi Wigderson’s work, though not strictly applied math, beautifully illustrates how applied considerations can drive significant theoretical advancements.

In this light, the recognition of Wigderson’s contributions is not just a celebration of his individual genius but also a testament to the evolving and interconnected landscape of mathematics, which continues to defy the narrow confines set by earlier academic opinions.

This is an intelligent comment; however, it mostly reflects back what I said. Indeed, it is a more eloquent exposition of what I said. Fortunately I can't be fired from blogging and replaced with an AI since I don't get paid anyway. 

While Avi Wigderson's Turing Award certainly highlights his exceptional contributions to theoretical computer science, invoking this as a refutation of G.H. Hardy's views on mathematics as a young man's game might be an oversimplification. Hardy's assertion, though controversial, was not without merit when considering the context of his times and the examples he witnessed firsthand. It's important to remember that exceptions like Wigderson do not necessarily disprove the rule but rather highlight remarkable outliers.

Furthermore, Hardy's critique of applied mathematics as being dull must be understood in the philosophical context of his personal commitment to pure mathematics. While we may not agree with his perspective, dismissing it entirely fails to appreciate the depth of passion that fueled his work and the work of many pure mathematicians. The interplay between disciplines enriches mathematics, indeed, but Hardy’s emphasis on the beauty of pure theory has inspired generations and continues to hold significant value in the mathematical community. 
This does not seem to be just reflecting back what I wrote. 
The argument about Avi is an outlier is reasonable. 
Here is my counterargument for defending Hardy's negative opinion of applied math:  just because the  notion that applied math is dull fueled passion does not make it correct. And one must also see what harm the notion caused.
 If I debated a human on this point they might change their mind, or give better counterarguments. Can ChatGP change its mind? Does that question even make sense? I suspect that ChatGPT could give better or at least different counterarguments.


  1. Terry Tao gave a talk (linked from his blog) at the JMM recently, entitled "Correlations of Multiplicative Functions". In it, he says that for many (most?) number theory problems, mathematicians have a good idea what conjectures are probably true -- but proving them is much much harder. (Shades of the P v. NP problem, I guess.)

    I don't know enough complexity to understand Avi's work (except for having heard of expander graphs). But it sounds like he's come up with brilliant proofs, and this isn't undercut by the proofs being of things which people already expected to be true...

  2. To say that Wiles did not "challenge the status quo" is IMO to take an overly narrow interpretation of the phrase "challenge the status quo." Challenging the status quo is not just about disproving conjectures. Before Wiles, the status quo was, "Fermat's Last Theorem is intractable, and by extension the Shimura-Taniyama-Weil conjecture is intractable." Wiles challenged this conventional wisdom and introduced new ideas, proving a very important special case of STW. Most truly groundbreaking new ideas challenge the status quo by drawing attention to an area that the status quo was neglecting as being unfruitful.

    1. (Bill) Agree. The phrase ``chllenging the status quo'' means different things in different context, however you are of course correct. There is an interesting (well--- I think so anyway) difference between what I meant and what you meant.
      I thought of `challenging the status quo' as more like thinking that FLT was FALSE or that P=NP. You think of it as more like thinking that FLT can be solved soon or that (say) P vs NP can be resolved soon. And YES, my view is to narrow.