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  WhitneyLand
  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.”
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  sxzygz
  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.
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  WCSTombs
  It's easy to forget, as we all use digital tools in our day-to-day lives, that the world is fundamentally analog, and there's no way to escape that. Everyone trying to tell you otherwise is just selling snake oil, with one notable exception, which is mathematical rigor in proofs. It's understood now that a rigorous proof in math is exactly one that, in principle, can be digitized and checked automatically. Those are simply the same concept, so introducing a computer there is really a perfect fit of tool and purpose. If we can't use computers to automate the checking of mathematical proofs, then why have computers at all? It's the only serious thing people do that a computer can be literally perfect at!To be clear, there's much more to math than writing down and checking proofs. Some of the most important contributions to math have been simply figuring out the right questions to ask, and also figuring out the useful abstractions. Those are both firmly on the "analog" side of math, and they are every bit as important as writing the proofs. But to say that we have this huge body of rigorous argumentation in math, and then to finally do the work of checking it formally is "taking it too far," is a really bewildering take to me.No, I don't think formalizing proofs in Lean or other proof systems should dominate the practice of math, and no, I don't think every mathematician should have to write formal proofs. Is that really where we're heading, though? I highly doubt it. The article worries about monoculture. It's a legitimate concern, but probably less of one in math than in many other places, since in my experience math people are pretty independent thinkers, and I don't see that changing any time soon.Anyway, the conclusion from all this is that the improved ability for mathematicians to rely on automated tools to verify mathematical reasoning would be a great asset. In my opinion the outcomes of that eventuality would be overwhelmingly good.
• emih
  I like the story in the article, but I think it tries to create some drama where there isn't any.I think it's great that a lot of work is done using proof assistants, because clearly it's working out for researchers; diversity of research and of methods is a great strength of science. I really can't see how you can "push it too far", pen-and-paper proofs are not going anywhere. And as more researchers write machine-checked proofs, new techniques for automating these proofs are invented (which is what my research is about hehe) which will only make it easier for more researchers to join in.> Currently, mathematicians are hoping to formalize all of mathematics using a proof assistant called Lean._Some_ mathematicians are trying to formalize _some_ of mathematics using a proof assistant called Lean. It's not a new development, proof assistants have been used for decades. Lean 4 has definitely been gaining popularity recently compared to others, but other proof assistants are still very popular.> a dedicated group of Lean users is responsible for determining which definitions should go into Lean’s libraryThe article makes it sound like there is a single, universal "The Lean Library" that everyone is restricted to. I assume it refers to mathlib? But at the end of the day it's just code and everyone is writing their own libraries, and they can make their own decisions.
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  jl6
  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.
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  umutisik
  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.
• pfdietz
  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.
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  johnbender
  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?
• zitterbewegung
  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.[1] https://aristotle.harmonic.fun
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  ux266478
  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...
• casey2
  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.
• dbvn
  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
• anthk
  LLM's are not reproducible. Common Lisp, Coq and the like for sure are.
• j45
  Is digitized proofs another way of saying the equivalent of a calculator, when a calculator was new?
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  riverforest
  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.