Would Erdos have been happy with the resolution of the Erdos Unit Distance Problem? How to find out?

Let's say there is a statement in math T that you wonder whether it's true or false.  You may even make a conjecture of which way it goes. Do you have a rooting interest?  I may think T is true, but if someone proves T or proves NOT(T) my main concerns are (1) can I understand the proof and (2) is the proof interesting.

Paul Erdos posed the sum-product conjecture which was recently proven false (by humans!).  I blogged about it here.

I wondered whether Paul Erdos would be happy with how it turned out. So I did the obvious thing: I asked Google AI

       Would Erdos have liked the solution to the sum-product conjecture?

Here is what it said:

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Paul Erdős would likely have had mixed feelings about the final resolution of the sum-product conjecture, but ultimately thrilled by the breakthroughs.

The original conjecture stated that for any finite set \(A\) of real numbers, either the sumset \(A+A\) or the product set \(AA\) must be large, meaning \(\max(\vert{}A+A\vert{}, \vert{}AA\vert{}) \approx \vert{}A\vert{}^{2-o(1)}\).

Here is how he would have reacted to the different aspects of the solution:

a) Shock and Disbelief at the Disproof: Erdős believed so strongly in his 1974 conjecture that he originally offered a \(\$500\) reward for its solution. He would likely be shocked to learn that it is actually false for real numbers.

BILL COMMENT: OFFERING $500 MIGHT BE BECAUSE HE THOUGHT IT WAS IMPORTANT, NOT THAT HE THOUGHT IT WAS TRUE. DID HE ALSO OFFER $500 TO SHOW ITS FALSE?

b) Excitement Over the Methods: The disproof of the conjecture relies on sophisticated constructions using high-degree number fields. Erdős loved deep, elegant connections between combinatorics and number theory, and would have admired the clever ways algebraic integers were used to cheat both addition and multiplication.

c) Joy in the Breakthrough: Erdős famously loved to "solve problems" and advance mathematics above all else.  Because the disproof opens entirely new avenues in additive combinatorics and number theory, he would be thrilled with the mathematical progress it generated.

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Some randomized points:

0) Has anyone ever been mad because their conjecture was PROVEN false? Or even proven true?

1) What Google AI says sounds right to me. Hence I believe it. However, dishonest fortune tellers (are there any honest ones?) will look at your clothes and how you talk and deduce things about you, and feed it back to you. Is Google AI similar?

2) What if I ask Google AI (or Chatty or Claude) a question and the answer does not sound right to me?

If it's a math question I can explore for myself and see what's up (ChatGPT has often been wrong when I ask it obscure things in Ramsey Theory).

If it's a question like How would Paul Erdos Feel About ... then there is no way to check the answer.

3) When I ask AI about history or literature I tend to believe it. Maybe I shouldn't.  It always respects my point of view which it might be wrong to do. For example, I asked Google AI

Why is My Mother the Car a better TV show than The Sopranos?

For a too-respectful response see here