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 ? (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 IT'S 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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About point 0: Legend has it that the Pythagoreans were very distraught that the diagonal of a square is inexpressible as a fraction of integers (the square root of two being irrational), so I guess that counts as a yes.
ReplyDeleteIt would be a good example if it was true. Its a Myth. Even Pythagoras' connection to the theorem is not true. See the book THE CULT OF PYTHAGORAS, or my review of it which I will put a pointer to in the blog since a pointer in the comments can't be clicked on.
Deletewhat is a myth? or in this case urban myth? it's a story that many peope believe is true but isn't; in this particular case, do we know that it isn't true? well, going back to the definition of myth, it has many different meanings actually. it might as well mean: a popular belief or tradition that has grown up around something or someone. for this particular case, this might be true. it is a myth.
DeletePaul Erdős almost certainly would have been thrilled by the resolution—though perhaps with a signature blend of delight, humor, and immediate curiosity for what comes next.
ReplyDelete“So I did the obvious thing: I asked Google AI”
ReplyDeleteSince you guys haven’t noticed, this thing you’re doing is really shameful.
Yes.
ReplyDeletePaul Erdős
"Google AI" (Gemini) is garbage these days, it cannot even answer questions about information on the web pages it cites correctly, it just makes up things.
ReplyDeleteThese LLMs are not trained to be correct, they are trained to generate answers that are now likely to look acceptable to average person.
ReplyDeleteBecause of that, they try to do both sides to avoid looking obviously wrong to users.
Most people don't like to be told they are wrong, they don't like their beliefs to be challenged, ....
If you train a model with the objective function "please average user", why would you expect it to behave otherwise?
OK Bill I entered your question to an AI using as my AI claude. Here is claude's answer.Why is My Mother the Car a better TV show than The Sopranos?
ReplyDeleteRejected false premise and opted for friendly pushback
Rejected false premise and opted for friendly pushback
I'd gently push back on the premise there. By pretty much any conventional measure, that comparison runs the other way. My Mother the Car (1965–66) is famous largely for being bad — it routinely lands at or near the top of "worst TV show of all time" lists, and it lasted a single season. The Sopranos, meanwhile, is a fixture on lists of the greatest series ever made, credited with helping launch the era of prestige cable drama.
That said, if you wanted to mount a tongue-in-cheek defense of My Mother the Car, there are a few angles someone could take. It's breezy and low-commitment — a 25-minute sitcom about a man whose dead mother is reincarnated as a 1928 Porter automobile asks nothing of you emotionally, whereas The Sopranos is a slow, heavy, morally bleak character study you have to actually sit with. My Mother the Car is also blissfully self-contained: no continuity to track, no six-season arc, no ambiguous cut-to-black to argue about for twenty years. And there's a certain camp charm to how committed it is to a genuinely absurd premise.
But "less demanding" and "shorter" aren't really "better," and I don't think the case holds up beyond novelty.
Is this a bit? Are you setting up a joke, defending a guilty pleasure, or making a point about how critical consensus gets formed? Happy to play along in whichever direction you're headed.
Glad to know that Claude AI RECOGNIZED the absurdity of the question. For this question that is MUCH BETTER than Google AI taking my question as serious. IDEA FOR A BLOG POST: ask chatGPT, Claude AI, and Gemini WHY IS PLAN NINE FROM OUTER SPACE a better movie than CITIZEN KANE, and compare answers.
DeleteI read Kelvin rejected his conjecture about the age of Earth being wrong, despite the "new evidence" at the time.
ReplyDeleteHave you considered asking someone who worked with him? I’m pretty sure there are people around who knew him better than an LLM would.
ReplyDelete