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In what way is Principia Mathematica better constructed? The way I learned mathematics seems pretty well constructed. You first learn about sets, then you buil
by jules 16y ago
In what way is Principia Mathematica better constructed?
The way I learned mathematics seems pretty well constructed. You first learn about sets, then you build natural numbers out of sets, then the integers, then rational numbers, from the rational numbers you build real numbers via Cauchy sequences. And you can construct all of this in less time than Principia Mathematica takes to get to 1+1=2.
- ionfish 16y agoSet theory is generally accepted as a better foundation for mathematics than Principia's type theory, but if you wrote all those ZFC proofs with full formality they'd be pretty long too. This answer on Math Overflow is pretty indicative. http://mathoverflow.net/questions/14356/bourbakis-epsilon-calculus-notation/31778#31778 http://mathoverflow.net/questions/14356/bourbakis-epsilon-ca... Usually mathematical arguments are relatively informal; every time you use set comprehension notation, for example, you're skipping a few steps in your proof (and note, of course, that the impredicative definitions one can use with it are open to e.g. Russell's paradox, which is why we have axiomatic set theory in the first place).
- jules 16y agoIf you want to do everything perfectly formally you're better off studying a computer based theorem prover like Coq. Not only if everything proven perfectly formally, the proof is also checked by a computer. This also allows conciser proofs because the theorem prover can often generate parts of your proof.