What is the project? What are you trying to do?

What we are doing is best explained by a picture: *2.37: the original *Principia* proof, the `Coq` proof, and the rewritten proof side-by-side-by-side We are taking Whitehead and Russell’s original Principia proof on the left, which is really a proof sketch omitting a lot of steps, and filling out the missing steps. For this we are using the interactive theorem prover Coq to ensure each proof step is a valid step according to Principia’s axioms and that no steps are skipped, even by accident. Finally, we are using the Coq encoding to typeset the full, gapless proof as it would appear in Principia’s notation.

Why are you doing all that?

Because it is fun. We also want academics and the public to engage more with Whitehead and Russell’s landmark three-volume work on logic and the foundations of mathematics. The project aims to facilitate that by producing a scholarly edition of Principia that is pleasingly typeset and searchable.

Are those the only reasons?

No. We also think that computational methods, and especially interactive theorem provers that help one rationally reconstruct arguments, can be of some use to historians of philosophy. We aim to prove this - that computational tools can augment traditional methods in history of philosophy - by giving an example.

Are there any other reasons?

Yes. Principia has been criticized by famous logicians like Kurt Gödel as being a “step backwards” from Frege and for being generally sloppy. We think that some of these charges are definitely overstated, and we want to contribute to the scholarly literature by rebutting some of them.

Is that all you are doing?

No. Because we want to typeset Principia pleasingly, we are also creating a LaTeX package for typesetting Principia’s idiosyncratic notation, which is no longer included in standard typesetting engines. We have already published a principia package on CTAN that covers all notations occurring in Volume I of Principia. We will extend it to Volumes II and III in due course.

Is there anything else you are doing?

We also want to create a subway-style map of Principia that allows one to click a theorem and see where it is used, what is used to prove it, and so on. One will be able to survey the text of Principia for the first time. So we are also marking up Principia’s theorems and proof sketches, and the Coq ones, in XML. Then we will use Python to create the interactive maps.

That sounds like a lot. What is your timeline?

It is definitely a lot. We would be very happy to get Volume I done by 2025, which will be exactly one hundred years since Principia’s second edition was published. So we might give a rough estimate of one volume every five years.

Who is doing this?

The principal investigator is your resident logical atomist, Landon D. C. Elkind (@LogicalAtomist), who is affiliated with the Department of Philosophy at the University of Alberta. I began and am currently undertaking this project with the support of an Izaak Walton Killam Postdoctoral Fellowship.

Updates

The Principia Rewrite project has a GitHub repository. Regular updates and the source code are made available there. Project milestones will be announced below.

May 1, 2021

Happy International Workers’ Day! The source code for the verification of Principia’s propositional logic proof sketches in Coq has been updated again. Now all the proofs are forward-directed - steps in the proofs involve only transformations of premises - rather than backward-directed - steps in the proofs do not involve transformations of the theorem (or proof goal). Also, the conjunction Ltac Conj and the equivalence Ltac Equiv have both been updated to abbreviate the proofs. The code is now only 4,400 lines.

April 20, 2021

The source code for the verification of Principia’s propositional logic proof sketches has been updated. Many proofs have been rewritten with the Coq standard library’s modules on classical logic and classical logic facts: notably, the axioms of Principia have also been proven from these standard libraries. So assuming the Coq standard libraries and kernel are kosher, then all the propositional logic proofs are verified on that background system. See the Github repository for the updated code.

February 15, 2021

The project has formally verified Principia’s propositional logic proof sketches - for all 189 theorems in *1-to-*5 - for the very first time. This improves significantly on the earlier computer verification work of Allen Newell et al., Hao Wang, and Daniel O’Leary. In all three cases, either some theorems were not verified or their proofs were not reconstructed according to Principia’s proof sketches. Our project in contrast reconstructs all the proofs in a way that is faithful to the text: each completion of Principia’s proof in Coq makes use of every theorem cited in Principia’s proof sketch.

Digital publications

As of February 15, 2021, the Principia Rewrite project has also produced:

a LaTeX package, principia, for typesetting any symbol in Principia’s Volume I
a Coq file of Principia’s propositional logic (*1-*5), running about 4,700 lines of code / 113 pages
a pleasingly typeset edition of Principia’s first 131 pages in LaTeX, going from its introduction through the end of its propositional logic (up to *5).

There are further developments to come: a high priority for the project is to build a map of the first five chapters (“starred numbers”) of Principia - because maps are fun and cool, besides being informative when done well.

Contact

If you have any questions, criticisms, or thoughts related to the Principia Rewrite project, write to elkind at ualberta dot ca.

*Principia* & `Coq` side-by-side Sample page of *Principia* digitized with `LaTeX` using the `principia` package

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