Comment by chowells
1 day ago
If you can't understand what the machine has done, math is not being done. Math is not pushing symbols around to find sequences that follow all the rules. Math is the process of understanding how those sequences of symbols map to meanings. If you can't follow the proof but have validated it's correct, all you know is that it's a proof of something. Is it a proof of what's claimed in the adjacent plaintext? Who knows!
Math is about clear communication between people first and foremost. That's why Shinichi Mochizuki's claimed proof of the ABC conjecture isn't accepted - he's unable to communicate it to enough mathematicians to get a consensus on its validity. LLMs generating Lean proofs are basically following that same schema. Sure, something has been proven. But what? Where's the clear communication explaining it to the satisfaction of other mathematicians?
> Math is about clear communication between people first and foremost.
I can't imagine I'm the first one to point out that this is perhaps the greatest unsolved problem in mathematics.
> Where's the clear communication explaining it to the satisfaction of other mathematicians?
Coming in about a year for $20/month, sure seems like.
I mean it's just patently absurd. It is trendy to use this hubristic, and frankly bureaucratic, definition or requirement on math.
People can do math alone. They don't have to share it with others. Mochizuki was doing math (with errors). The aliens from Arrival could do math and we wouldn't necessarily be entitled to that. It doesn't make these things not valid math.
The only benefit of defining math through this vulgar social reductivism is to benefit the status quo. So you see some experts and expert-adjacent commentators using these math slogans but never critically justifying those words. Case in point, the above commenter asserts "...math is foremost about communication.", but the very next sentence does not explain why it is foremost, it just uses the sentence as a premise for the following sentence. Guess what, that is exactly how ideologues argue, by question begging fallacy. Ironic that STEM people who ought know better fail to catch this simple rational mistake.
Math that isn't communicated is valueless - by definition.
Applies to STEM generally, because the foundation of STEM is peer review and independent testing and confirmation.
Even if the practice never quite lives up to the theory (true) that's no excuse to hand-wave away the requirement as if it doesn't matter.
Proofs that can't be confirmed are valueless because no one learns anything from them. The value of a proof isn't just in proving something but in inventing new techniques and insights that can be reused for other applications.
Brute force searches are fundamentally less valuable, because the creative foundation of math is discovering leaps of logic that solve problems with insightful shortcuts that can be generalised and reused for other applications.
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Human understanding is important, but I don’t think it’s the end all be all. There’s no reason to expect humans to be the best mathematical reasoners in the universe. And if math is important, beyond being a human art form, I think we should explore that math beyond our understanding.
What part of "math is the process of humans understanding math" did you not understand or agree with?
I'm on the side of human understanding being fundamental to mathematics, but I'll try arguing from the other side.
If we consider the reductionist view that "math is just a tool" to solve problems, and machines can generate new tools that are actually useful, then it can be argued that we don't need to understand how those tools work internally for us to benefit from its usefulness.
My hope is that such a view is given its proper place as a worthy pursuit but that it does not consume the entirety of the field of mathematics as a human activity with culture, philosophy, and art. And I feel the same for the field of software.
"Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding."
https://mathoverflow.net/questions/43690/whats-a-mathematici...
Does math exist without humans?
I mean, I could not understand 95% of all the math out there due to skill issue, and it’s still valid. If machines can understand the math and humans can’t, we can’t validate it, but that doesn’t make the math itself invalid.
It could be logically valid but irrelevant? Allowing for the atrophying of all our skills is dangerous? While simultaneously building ever more complicated edifices that control our lives? Not a doomer but you really just want to shrug and say it ain't no thing?
Perhaps humans have peaked due to the fundamental biological nature of our brains and AI will eventually surpass us in mathematical intelligence.
That has been the ultimate goal of creating AI -- not to simply match human intelligence but to surpass it. If it surpasses us, does that make it irrelevant? Everything around us is powered by mathematics, most of it I don't understand, and soon perhaps everything will be powered by mathematics that everybody doesn't understand.
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> If machines can understand the math and humans can’t, we can’t validate it, but that doesn’t make the math itself invalid.
How can humans tell whether machines understand the math if they themselves don't? Math is a human language for quantitative reasoning.
If large sectors of the population decline in their ability to read and write <natural language>, how will they be able to evaluate machine-generated <natural language>?
Perhaps, but two things: 1. most people don't understand this level of maths either 2. math, unlike most natural language, can be (dis)proven without a person understanding.
The second one is very important. While it is a matter of pride that some do understand, and (given enough time, in my opinion) will continue to understand, it is sufficient to be merely correct even if one does not understand why. This is a nice feature of math: it remains true regardless of understanding.
The obvious caveat is the machine needs to prove it in the end, even if we can not.
Does math already exist and is discovered? Or is it a thing invented by people?
The underlying tautologies always existed. Making them accessible to us as truth and usable for one or more functions is what's invented by people (and potentially by AI) via insightful proofs and narratives.
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