[tlringer]

It does. And if classical logic is what they like, they can use Isabelle/HOL just as nicely. It is classical. It has wonderful automation. Kevin gets attention because he is bold, not because he is correct.

Anonymous comments are cowardly; please show yourself. If you stay anonymous I am not aware of power dynamics, and I cannot adjust my response accordingly.

[throwaway45434]

I agree with you that it's not obvious that Lean is the only or best option for formalizing "all" or at least most of mathematics. But I didn't get the impression that Kevin thinks that it's the best tool imaginable but rather the best tool that is currently available.

But that wasn't the point: He's right in that constructive/intuitionistic mathematics is so niche that it can be safely ignored when discussing formalization of the mathematics he and almost every other mathematician or scientist is interested in.

Kevin gets attention because he's an accomplished mathematician who tries to formalize his work, or at least what it's based on. I find it ridiculous to dismiss his opinion the way you do.

[tlringer]

Lean is not classical. I don't know where the idea that Lean is classical is coming from. Lean is constructive. It is consistent with classical axioms, just like Coq is, and just like Cubical is in the propositional fragment. Isabelle/HOL on the other hand is classical, and is quite a natural choice if classical logic is what you want, which is why mathematicians have been using it for decades now.

I know Kevin. He does good work. He also heckled every talk at ITP, including my own, in the middle to express his opinions. I have never met another researcher in my life who does that. Media outlets interview him and not the other 50 years of researchers doing work on proof assistants and using them both for mathematics and for verifying systems. I wrote a whole book about the history of program verification and for sure nobody has ever interviewed me about it. Kevin is loud and bold, that is why his opinions spread like fire. But they are just that, opinions.

[lakecresva]

It's because the mathlib authors don't really care about being constructive and frequently use things like choice/LEM, in contrast to Coq/math-comp. If there's a constructive and non-constructive version of something, the mathlib people will use whatever's easier or nicer to work with. There's some info here about how Lean does realizability with pervasive use of choice: https://github.com/coq/coq/issues/10871#issuecomment-5466412...

[tlringer]

That makes sense! I just wish discussion on this end would stay nuanced: In that case Lean has great library support for classical mathematics. This is an important distinction because it is something that the authors of other constructive proof assistants can focus on if they want to reach more mathematicians.

[kmill]

Sort of the official policy is that inside proofs (i.e., terms of types in Prop) constructibility doesn't matter, but inside definitions (i.e., terms of types in Type 0 and above) try to be constructive. For example, it is ok to define a function constructively but with a nonconstructive proof of termination.

LEM is used everywhere because it's actually a theorem. It follows from functional extentionality and the existence of quotient types. https://github.com/leanprover/lean/blob/master/library/init/...

The axiom of choice is an actual axiom, and it too ends up being used everywhere, but mostly to introduce decidable instances to permit things like double negation elimination inside proofs. An illustration: [1] introduces these instances to the file and [2] implicitly uses the instance.

[1] https://github.com/leanprover-community/mathlib/blob/master/... [2] https://github.com/leanprover-community/mathlib/blob/master/...

[tlringer]

It is my understanding that functional extensionality in Lean itself follows from the axiom propositional extensionality, so in that sense LEM is still a consequence of an axiom. The core theory of Lean is constructive.

[kmill]

I'm not an expert on the foundations of Lean, but I do know that funext in the source code follows only from the axioms of quotient types and the eliminator for equality. There are a number of axioms that are implicit in the type checker itself, so you might say that proof irrelevance is built-in in that way. Propositional extensionality (propext) is that propositions that imply each other are equal, and that's an axiom in the library, but funext does not depend on propext.

[zozbot234]

LEM and double-negation elimination is very poor style from a constructive POV, though. If you need to prove something non-constructively, just stick to the negative fragment.

[kmill]

I think your comment illustrates a difference in cultures -- mathematicians generally don't care about whether it's bad style in the constructive POV. Sure, you can 'just' stick to the negative fragment, but that 'just' is sweeping a lot of questions of ergonomics under the rug. One thing that attracted me to Lean was that I could formalize mathematics as I knew it, more or less, and there was a vibrant community with a fairly well developed library of basic classical mathematics.

I'm not saying these concerns about constructibility aren't unimportant. I do appreciate both POVs. I just don't think it's something you have to force mathematicians to worry about if all they're interested in is formalizing their math on a computer.

Something that's interesting about Lean is that double-negation elimination is used in Prop but not generally in Type and above. You have to explicitly reach for the axiom of choice if you want to "construct" a term of a type that isn't empty.

[tlringer]

I have learned more about Kevin and feel bad for bringing up the heckling thing now. I would ignore that part of what I said.

[withtypes]

Saying Lean "rules out many kinds of mathematics" and not giving a single example, sounds more sensationalist to me. I also don't see what's wrong with anonymous comments as long as they are constructive and people are nice.

I don't know Kevin as well as you do (apparently) but maybe he is just ignorant on the topics you know so much about. Why something gets attention of media (Quanta Magazine?) is complicated. From your comments, you also seem very "bold".

[tlringer]

I'm a woman though, and women rarely get positive attention for anything in CS. I'm bold because as a woman in the field you need to be bold to survive as a researcher. In the type theory and proof assistant worlds there are only a handful of us. A typical ratio at a conference for proof assistant research is 1 woman for every 40 men.

Anonymous comments scare me because there are (a few, thankfully not many) abusive people in the PL community I am afraid of engaging with. I don't want to accidentally find myself arguing with one of them. I need to know when to exit the conversation.

Proof relevant mathematics, higher category theory, lots of topology. Sorry. I don't spend all of time on HackerNews, I sometimes don't get around to responding to things. I am talking to Kevin about this framing so we can figure out how to make it healthier in the future. Some of my comments about Kevin were likewise out of line because I interpreted some of our previous interactions as rooted in a gender-based power dynamic when they were actually just rooted in something Kevin finds difficult and wants to do better at, and that is my honest mistake.