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Show HN: Formalizing Principia Mathematica using Lean

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Re: Show HN: Formalizing Principia Mathematica using Lean

#11

Post by mikrl »

imglorp wrote: post #6
Jtsummers wrote: post #5

Earlier quoted context omitted.

https://www.youtube.com/watch?v=9RD5D4swZfk - Possibly this.

TLDW: Godel's incompleteness theorem is at odds with the goals of Principia.

I remember my Java IDE in undergrad warned me about an infinite loop, and this was before I learned about the diagonalization proof of the non-computability of the halting problem, one of my favourite proofs ever. The fact that not all programs and inputs can be shown to halt did not stop the engineer who wrote that guardrail for the IDE.

Surely the principia and similar efforts will still yield useful results even if they cannot necessarily prove every true statement from the axioms?

Re: Show HN: Formalizing Principia Mathematica using Lean

#12

Post by krick »

jandrese wrote: post #10
krick wrote: post #8

Earlier quoted context omitted.

Thanks. It appears, however, that Dyson considers the whole approach a failure (referring to Gödel as a demolisher of it). So while he is saying it about a book, ironically, it seems hardly applicable in this context anymore. Because with this reasoning, any program in Lean (and the Lean programming language itself) should be seen as "a monumental failure".

This is just my opinion, but reading about Bertrand Russell my impression is that he dedicated his life to Pincipia Mathematica partially because he expected to find God in the foundations of the mathematics, and when that didn't happen it drove him rather insane. And then Gödel shows up and basically knifes him on stage with the Incompleteness Theorm.

I would like if you could refer me to that reading as well. I really know nothing about, uh, any of that, so I cannot judge. But your description strikes me as rather weird: "dedicating his life" seems a bit dramatic, since Principia is a pretty early work of his. He was active for 50-60 more years since he must have been "driven insane", as you say. Most of his famous works were written after that. Also, all of famous results of Gödel were after Russell finished with Principia. Not that he ever finished, but given the fact Second Edition was 15 years after the First one, and mostly contained relatively minor fixes… it seems only logical to conclude that he wasn't pursuing the topic after the first publication, basically, ever since realizing how big of a task would it be to try and formalize all of math like that.

Re: Show HN: Formalizing Principia Mathematica using Lean

#13

Post by gnulinux »

krick wrote: post #3

> Although the Principia is thought to be “a monumental failure”, as said by Prof. Freeman Dyson I'd like some elaboration on that. I failed to find a source.

Principia was written during the naive Logicist era of philosophy of mathematics that couldn't foresee serious foundational decidability issues in logic like Godel's incompleteness theorems, or the Halting Problem. Formalism/Platonism and Constructivism are two streams that came out of Logicism as a way to fix logical issues, and they're (very roughly speaking) the philosophical basis of classical mathematics and constructive mathematics today.

The way formalists (mainstream mathematical community) dealt with the foundational issues was to study them very closely and precisely so that they can ignore it as much as possible. The philosophical justification is that even though a statement P is undecidable, ultimately speaking, within the universe of mathematical truth, it's either true or false and nothing else, even though we may not be able to construct a proof of either.

Constructivists on the other hand took the opposite approach, they equated mathematical truth with provability, therefore undecidable statements P are such that they're neither true nor false, constructively. This means Aristotle's law of excluded middle (for any statement P, P or (not P)) no longer holds and therefore constructivists had to rebuild mathematics from a different logical basis.

The issue with Principia is it doesn't know how to deal with issues like this, so the way it lays out mathematics no longer makes total sense, and its goals (mathematical program) are found to be impossible.

Note: this is an extreme oversimplification. I recommend Stanford Encyclopedia of Philosophy for a more detailed overview. E.g. https://plato.stanford.edu/entries/hilbert-program/

Re: Show HN: Formalizing Principia Mathematica using Lean

#14

Post by psychoslave »

jandrese wrote: post #10
krick wrote: post #8

Earlier quoted context omitted.

Thanks. It appears, however, that Dyson considers the whole approach a failure (referring to Gödel as a demolisher of it). So while he is saying it about a book, ironically, it seems hardly applicable in this context anymore. Because with this reasoning, any program in Lean (and the Lean programming language itself) should be seen as "a monumental failure".

This is just my opinion, but reading about Bertrand Russell my impression is that he dedicated his life to Pincipia Mathematica partially because he expected to find God in the foundations of the mathematics, and when that didn't happen it drove him rather insane. And then Gödel shows up and basically knifes him on stage with the Incompleteness Theorm.

I don't know what you red about Russell, but in my own readings he has always been presented as a fervent atheist, so except with a far stretched interpretation of "neutral monism" as some form of gnoseologic divinity, it's hard to imagine such a character looking for any god.

Also Russel himself ruined the cathedral of Frege with its eponymous paradox, he was clearly among the best to understand how a thing like Godel's incompleteness theorem could come along the way.

And for his relation to madness, his personal life have been felt with many turmoil from an early age. If anything it seems that mathematics saved him, preventing his early desire for suicide.

https://plato.stanford.edu/entries/neutral-monism/

https://en.wikipedia.org/wiki/Copleston%E2%80%93Russell_deba...

Re: Show HN: Formalizing Principia Mathematica using Lean

#17

Post by ndrwnaguib »

hackandthink wrote: post #15

I only see these very initial propositional theorems. Am I missing something, or has the project only just begun? https://github.com/ndrwnaguib/principia/blob/main/Principia/...

You're not missing something. The project begun several months ago (I had to pause while I was writing my thesis). I resumed working on it recently.

Re: Show HN: Formalizing Principia Mathematica using Lean

#19

Post by grandempire »

ndrwnaguib wrote: post #18
grandempire wrote: post #16

It looks like you just have a few pages written. Is that right? Which theorem are you trying to prove?

Yes; the goal is to finish the first volume. I am particularly looking forward to formalizing the well-known 1+1 proof.

My understanding is the first bit follows first order logic fairly close but then diverges as Russel builds different classes of sets etc, do you have line of sight of how it’s going to translate?
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