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Show HN: Formalizing Principia Mathematica using Lean

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173 points ndrwnaguib | 2 comments | 25 Apr 25 18:49 UTC | HN request time: 0s | source

This project aims to formalize the first volume of Prof. Bertrand Russell’s Principia Mathematica using the Lean theorem prover. Throughout the formalization, I tried to rigorously follow Prof. Russell’s proof, with no or little added statements from my side, which were only necessary for the formalization but not the logical argument. Should you notice any inaccuracy (even if it does not necessarily falsify the proof), please let me know as I would like to proceed with the same spirit of rigour. Before starting this project, I had already found Prof. Elkind’s formalization of the Principia using Rocq (formerly Coq), which is much mature work than this one. However, I still thought it would be fun to do it using Lean4.

https://ndrwnaguib.com/principia/

https://github.com/ndrwnaguib/principia

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krick ◴[25 Apr 25 19:13 UTC] No.43797518[source]▶
> 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.

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Jtsummers ◴[25 Apr 25 19:26 UTC] No.43797657[source]▶
https://www.youtube.com/watch?v=9RD5D4swZfk - Possibly this.
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krick ◴[25 Apr 25 19:52 UTC] No.43797894[source]▶
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".
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jandrese ◴[25 Apr 25 20:06 UTC] No.43798008[source]▶
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.
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1. psychoslave ◴[25 Apr 25 21:08 UTC] No.43798533[source]▶
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...

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2. vixen99 ◴[26 Apr 25 05:55 UTC] No.43801275[source]▶
Incidentally his co-author AN Whitehead was not an atheist as a reading of Science and the Modern World (from lectures at Harvard in 1926 I think.) makes clear.