In “Proofs and Refutations”, Imre Laktos[1] portrays a socratic discussion between teacher and students as they try to prove Euler’s theorem V+F =E+2. The beauty of this essay is that the discussion mirrors the historical development of the subject, whilst also critiquing the formalist school of thought, the modern agenda of meta-mathematics, and how it doesn’t fit the way mathematics is done in practice. Whilst I’m on board with us not knowing how to formalize mathematical practice, I think it is asolvableproblem. Moreover, I am a staunch ultra-finist believer in reality being computable, which dovetails with a belief in proofs preserving truth. Yet that matters little in the face of such a fantastic exposition on mathematical discovery.[2] Moreover, it functions as a wonderful example of how to alternate between proving and disproving a conjecture, whilst incorporating the insights we gain into our conjecture.
In fact, it was so stimulating that after reading it I came up with two other proofs on the spot, though the core idea is the same as in Laktos’ proof. After that experience, I feel like showing a reader how a proof is generated is a dang good substitute for interacting with a mathematician in real life. Mathematics is one of the few areas where text and images, if read carefully, can transfer most tacit information. We need more essays like this.[3]
Now, if only we could get professor’s to force students to try and prove theorems within the lecture, dialogue with them and transcribe the process. Just think, when the professor comes to the “scrib together lecture material and turn it into a textbook” part of their lifecycle, we’d automatically get beautiful expositions. Please excuse me whilst I go cry in a corner over unattainable dreams.[4]
And how weird things were in the days before Hilbert mastered mathematics and brought rigour to the material world. Listen to this wildly misleading quote: “In the 19th century, geometers, besides finding new proofs of the Euler theorem, were engaged in establishing the exceptions which it suffers under certain conditions.” From p. 36, foot note 1.
Genealized Heat Engine and Lecture 9 of Scott Aaronson’s democritus lectures on QM are two other expositions which are excellent, though not as organic as Proofs and Refutations.
Funnily enough, Euler’s papers contain clear descriptions of how he came to the proofs. At least, those I’ve read. Which is, like, 1⁄10,000th of his material by word count. I’m not kidding.[5][6]
In “Proofs and Refutations”, Imre Laktos[1] portrays a socratic discussion between teacher and students as they try to prove Euler’s theorem V+F =E+2. The beauty of this essay is that the discussion mirrors the historical development of the subject, whilst also critiquing the formalist school of thought, the modern agenda of meta-mathematics, and how it doesn’t fit the way mathematics is done in practice. Whilst I’m on board with us not knowing how to formalize mathematical practice, I think it is a solvable problem. Moreover, I am a staunch
ultra-finistbeliever in reality being computable, which dovetails with a belief in proofs preserving truth. Yet that matters little in the face of such a fantastic exposition on mathematical discovery.[2] Moreover, it functions as a wonderful example of how to alternate between proving and disproving a conjecture, whilst incorporating the insights we gain into our conjecture.In fact, it was so stimulating that after reading it I came up with two other proofs on the spot, though the core idea is the same as in Laktos’ proof. After that experience, I feel like showing a reader how a proof is generated is a dang good substitute for interacting with a mathematician in real life. Mathematics is one of the few areas where text and images, if read carefully, can transfer most tacit information. We need more essays like this.[3]
Now, if only we could get professor’s to force students to try and prove theorems within the lecture, dialogue with them and transcribe the process. Just think, when the professor comes to the “scrib together lecture material and turn it into a textbook” part of their lifecycle, we’d automatically get beautiful expositions. Please excuse me whilst I go cry in a corner over unattainable dreams.[4]
Not a Martian as he wasn’t born in Budapest.
And how weird things were in the days before Hilbert mastered mathematics and brought rigour to the material world. Listen to this wildly misleading quote: “In the 19th century, geometers, besides finding new proofs of the Euler theorem, were engaged in establishing the exceptions which it suffers under certain conditions.” From p. 36, foot note 1.
Genealized Heat Engine and Lecture 9 of Scott Aaronson’s democritus lectures on QM are two other expositions which are excellent, though not as organic as Proofs and Refutations.
Funnily enough, Euler’s papers contain clear descriptions of how he came to the proofs. At least, those I’ve read. Which is, like, 1⁄10,000th of his material by word count. I’m not kidding.[5][6]
http://archive.boston.com/bostonglobe/ideas/brainiac/2012/11/the_100-year_pu.html
http://eulerarchive.maa.org/