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0 pointskmill5y ago0 comments

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.

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zozbot2345y ago· 2 in thread

> Sure, you can 'just' stick to the negative fragment, but that 'just' is sweeping a lot of questions of ergonomics under the rug.

With all due respect, that seems a bit vague to me. What are the "questions of ergonomics" you're thinking of here? Choosing the negative form of a goal whenever you have to prove something non-constructively seems even easier ergonomically than, e.g. adding LEM as an axiom. You basically don't have to change your proof methods as long as the statement you're proving is of the appropriate form.

kmillOP5y ago

Ergonomics to me means "what is the moment-to-moment experience of a user of the language, in particular frictions between what the user wants to say and how it must be said." For someone who does not care much about constructibility, it is better not having to do anything differently depending on whether or not something is constructive. The workaround for such people could be to always include at least one negation for every statement, but that's a pervasive reminder of questions of constructibility! If practitioners are going to be using this transformation by default, then the language would be more ergonomic if it was just automatic.

I don't think having to think about constructibility all the time is more ergonomic than accepting that the system has a well-accepted axiom. There doesn't need to be a one-true system, of course, and for people interested in foundations, reverse mathematics, constructive mathematics, higher category theory, and homotopy type theory, there are other systems. Most mathematicians don't study these things, and accepting LEM is natural for them.

zozbot2345y ago

> For someone who does not care much about constructibility, it is better not having to do anything differently depending on whether or not something is constructive.

Well, there are many possible constructive 'versions' of any non-constructive statement, so 'not having to do anything or be reminde[d] of questions of constructivity', strictly speaking, leads to just sticking to the non-constructive version by default. Since it's easy, even in constructive logic, to derive the non-constructive 'version' of some statement from a constructive one, this transformation (properly understood) is as good as automatic.

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