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Background: I found this interesting translation of Godel's On formally undecidable propositions of Principia Mathematica and related systems I that, along with translating it into English, uses more modern and understandable symbols, includes some hyper text links throughout the paper, and, if I'm not mistaken, makes the language a little more readable. This seems like a good and noble undertaking: taking an important paper and revamping it to make it more accessible, or at least less intimidating for modern readers.

So while looking through it, I couldn't help but think of another work that could benefit from such a treatment, Russell and Whitehead's Principia Mathematica. Every review I have read about it is something like "It lead to a lot of important developments in logic, but is mostly a historical curiosity. And besides, it is incredibly difficult to read, so putting forth the necessary effort is probably not worth it."

Here is the question: does such a thing exist, and would such a paper be worthwhile? Is there anything in the Principia that would be of interest to modern mathematicians. (Finding some points of interest and reproducing them seems like a better idea than trying to reproduce all three volumes.) Godel's theorem and his proof are still interesting, in spite of the fact that there are many other (better?) ways of getting to the Incompleteness theorems.

Another example of this sort of revision is The Annotated Turing.

I'd appreciate any knowledge of such a source, or any insight into why one wouldn't be too valuable, or really any thoughts on looking at Principia Mathematica at all.

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"but is mostly a historical curiosity" - this is about right. There are other ways of doing mathematical foundations that R&W didn't have (ZFC, for example, not to mention topos theory) that are much easier and more relevant to modern mathematics. Goedel's paper is still relevant today - that is why it is able to be translated into modern symbols. –  David Roberts Nov 24 '11 at 4:47
It might be preferable to do a comparison with other attempts at mechanizing (formalizing at a low level). I would be more inspired to check out PM if I knew that there was a direct correlation to, say, the foundational theory used by Mizar, and what some of the differences were. On the other hand, if a proof theorist gave a reasonable explanation why learning LISP would be better, I could be swayed away from looking at PM. Gerhard "Doesn't Know Lambda Calculus Either" Paseman, 2011.11.23 –  Gerhard Paseman Nov 24 '11 at 6:10
In addition to what David wrote, Godel's paper is much shorter that 3 volumes of Principia. The difficulty in reading Principia is because it is mostly mathematical statements and proofs in too much detail, not because of their notation, one can pick up the notation after a few pages. The difficulty of reading is, I think, more similar to difficulty of reading full small details of a formal proof. If you don't enjoy reading the full details of a proof in say Coq you won't enjoy reading Principia even if someone replaces their notation with a modern one. –  Kaveh Nov 24 '11 at 6:18
Some years ago, Cambridge University Press produced a soft-cover edition of Principia Mathematica (the beginning, to *56). The introduction (95 pages) is still quite interesting. And then there is a sample (the remaining 300 pages) of the main symbolic development ... you can see why it would be a chore to read. But if you like you can read up to the famous result: 1+1=2 . –  Gerald Edgar Nov 24 '11 at 14:10
You might be interested in checking out MetaMath us.metamath.org/index.html . It is proof validation software with a collection of statements written in MetaMath which is easily readable by humans and inspired by Principia Mathematica. I believe that some of the documentation for some of the MetaMath theorems references their occurrence in PM. But MetaMath also discusses some of the alternative sets of axioms that it has considered to base it's core system on. –  Daniel Geisler Nov 29 '11 at 16:08

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In his Stanford Encyclopedia of Philosophy article "The Notation of Principia Mathematica", Bernard Linsky makes the following claim:

This translation is offered as an aid to learning the original notation, which itself is a subject of scholarly dispute, and embodies substantive logical doctrines so that it cannot simply be replaced by contemporary symbolism. Learning the notation, then, is a first step to learning the distinctive logical doctrines of Principia Mathematica.

The point is that Principia was not intended simply to be a development of mathematics in type theory: it was intended to make a philosophical argument that mathematics could be carried out using only "logic". Thus translating PM so that the underlying mathematical principles are more clearly described would miss the point that there are not supposed to be any underlying mathematical principles, only "logical" ones. This differs sharply from Gödel's paper, which was intended to be mathematical (the result can be viewed as just a particular type of combinatorics or number theory) rather than philosophical.

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