alt Hacker NewsIn math, rigor is vital, but are digitized proofs taking it too far?
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The problem with this ambition is that it turns mathematics into software development. There’s absolutely nothing wrong with this per se, however what happens is that, as in software, certain ideas get ossified. That’s why, for example, every OS has a POSIX layer even though technically the process/namespace/security model could be radically reimagined possibly to create more easily engineered, correct software.
Mathematics is going through a huge, quiet, upheaval. The litmus test will be when, if ever, someone wins a Fields using a proof-assistant in an essential way.
With sufficient automation, there shouldn't really be a trade-off between rigor and anything else. The goal should be to automate as much as possible so that whatever well-defined useful thing can come out theory can come out faster and more easily. Formal proofs make sense as part of this goal.
Imagine a future where proofs are discovered autonomously and proved rigorously by machines, and the work of the human mathematician becomes to articulate the most compelling motivations, the clearest explanations, and the most useful maps between intuitions, theorems, and applications. Mathematicians as illuminators and bards of their craft.
In the long run creating a certificate that guarantees a certain probability of correctness will take much less energy. Right now we can run miller-rabin and show with 1-(1/10^100) certainty that the number is/isn't prime. Similar for hash collisions, after a certain point these can't happen in reality. If Anthropic can get their uptime from 1 9 to 9 9s (software isn't the bottleneck for 9 9s) then we don't need formally checked proofs.
I’m confused by the calculus example and I’m hoping someone here can clarify why one can’t state the needed assumptions for roughed out theory that still need to be proven? That is, I’m curious if the critical concern the article is highlighting the requirement to “prove all assumptions before use” or instead the idea that sometimes we can’t even define the blind spots as assumptions in a theory before we use it?
I think the future of having lean as a tool is mathematicians using this or similar software and have it create a corresponding lean code. [1] This is an LLM that outputs Lean code given a mathematical paper. It can also reason within lean projects and enhance or fix lean code.
A few comments:
(1) Math journals are being flooded with AI slop papers loaded with errors. I can see a time when they will require papers to be accompanied by formal proofs of the results. This will enable much of the slop to be filtered out.
(2) Formalization enables AI to do extensive search while staying grounded.
(3) Formalization of the historical math literature (about 3.5M papers) will allow all those results to become available for training and mining, to a greater extent that if they're just given as plain text input to LLMs.
Rigor was never vital to mathematics. ZFC was explicitly pushed as the foundation for mathematics because Type Theory was too rigorous and demanding. I think that mathematicians are coming around to TT is a bit of funny irony lost on many. Now we just need to restore Logicism...
There's no such thing as being too rigorous when you're talking about proofs in math. It either proves it or it doesn't. You get as rigorous as you need to
LLM's are not reproducible. Common Lisp, Coq and the like for sure are.
Is digitized proofs another way of saying the equivalent of a calculator, when a calculator was new?
Rigor is the whole point of math. The moment you start asking if there is too much of it you are solving a different problem.
Great quote from Hilbert, I think it’s also a useful thought for software development.
“The edifice of science is not raised like a dwelling, in which the foundations are first firmly laid and only then one proceeds to construct and to enlarge the rooms,” the great mathematician David Hilbert wrote in 1905 (opens a new tab). Rather, scientists should first find “comfortable spaces to wander around and only subsequently, when signs appear here and there that the loose foundations are not able to sustain the expansion of the rooms, [should they] support and fortify them.”