Principia was an attempt to derive math from logical ground up, motivated by ideas about the foundations of mathematics and logic (wrong ones, it turned out). Knuth is more like A Magical Mystery Tour of Algorithms, wherein Knuth is both tour guide and man behind the curtain.
"Wrong" is too strong. The fundamental bases it used are not generally used today, but it was the first of its kind and inspired much. Many of its details are still fine.
If you are interested in the underlying goal of Principia Mathematica, I urge you to check out the Metamath Proof Explorer (MPE):
https://us.metamath.org/mpeuni/mmset.html
By itself, Metamath doesn't have built-in axioms. MPE uses Metamath to first state a small set of widely-accepted axioms, namely classical logic and ZFC set theory, and then proves a tremendous amount of things, building up to a lot of mathematics from careful formal proofs that rigorously prove every step.
Some things cannot be proven, but that doesn't mean that proof is dead.
Yes, it was a tad tendentious, but I don’t think anyone really buys the logicist program anymore.
Don’t get me wrong, PM is marvelous and there’s no gainsaying its enormous historical impact.
If your characterization of Metamath is correct, I don’t think that’s in the spirit of PM at all. One of the major problems PM had was the rejection of (what later became) the Axiom of Choice in favor of Russell’s convoluted theory of types. Indeed, that set theory (ZFC) is needed to get the rest of the way R&W were trying to go is one of the signal failures of the logicist program behind PM.
If you believe the big advantage of Principia Mathematica was that it starts with a very few axioms and then manages to formally and exactly prove many things, then MPE is a worthy successor. I'm in that camp.
However, if you think the main point of Principia Mathematica was the very specific set of axioms that they chose, then that's different. The PM authors chose to use a "ramified" theory of types, which is complex. It does have sets, it's just not ZFC sets. Few like its complexities. Later on Quine found a simplification of their approach and explained it in "New Foundations for Mathematical Logic". There's a Metamath database for that "New Foundations" axiom set as well (it's not as popular, but it certainly exists):
https://us.metamath.org/index.html
Strictly speaking that's true for any system that can handle arithmetic (as proved by Goedel). You can show an inconsistency, but you can't prove consistency. No one's found an inconsistency.
Yes, I was too loose with me words again. I was gesturing at relative consistency (especially, at least, when compared to the lack of serious doubt about ZF/C’s consistency), as well as the work of folks like RB Jensen and Randall Holmes.
The biggest problem NF has is, as usual, a social one: there just ain’t a lot of people working on, or interested in working on, NF compared to other set theories.
Also check out the Principia Rewrite project, which aims to use 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.
It's a bit extreme imo to call Principia wrong. It isn't wrong so much as it will never completely succeed. These are two different things. The logical perspective presented in Principia is still sound and a relatively useful framework for understanding most of mathematics, and it's a monumental and impressive piece of work. I find it hard to believe anyone who has read it wouldn't leave with a better overall comprehension of what exactly it is that we are doing when we do mathematics.
I didn’t say PM was wrong. I said the ideas behind it were—-although I should’ve said idea (singular) insofar as I was thinking in particular of the strong form of logicism motivating Russell.
My understanding of the flaws of PM is in its attempts to avoid self-reference, which was sort of folly from the beginning as proven by Godel. I learned this from I Am A Strange Loop, and I'm not sure how accurate it is historically. But Godel's Incompleteness Theorem is one of the most interesting things I've ever read about.
The principal flaw of PM if you were to read it now is that it is an evolutionary dead end.
The elementary vernacular foundations of modern mathematics is (more or less) naive set theory; the starting tools of serious foundational work (as arcane as it is even within maths as a whole) are logic and more rigorous set theory, perhaps with some computability mixed in; the main tool of a mathematician who wants to reason in great generality is category theory (with some handwaving in the direction of the previous point about universe hierarchies and whatnot). There are some signs of convergence between these (and of mainstream mathematicians starting to once again take foundations seriously), but at the basic level those are what you’ll be dealing with.
None of them existed in their current form at the time PM was written, even logic (no Kripke semantics! no forcing! and no Gödel of course). Some did not yet exist at all. Some changed quite drastically in direct response to PM. And of course PM is the origin of (embryonic) type theory, which is the inspiration of the unified approach I referenced above. So as a historically important text, sure, if that’s what you want, but as a gateway to understanding more interesting maths it’d be terribly inefficient.
In that respect TAoCP was uniquely lucky. It was also a self-obsoleting book: it ceased to be comprehensive months after it was published, exactly because it told you everything there was to know about algorithms to date. Yet none of the stuff that’s in it is itself obsolete, there’s just immesurably more stuff now. PM, on the other hand, was attempt at “rationalization” in the 19th-century sense, and mostly a failed one except for serving as fertilizer for all of the later ones.
It wasn't a folly from the beginning because it seemed like it could be done when Russel and Whitehead started on the PM. It was only after it was published that Gödel proved it to have been a folly.
