Re: proof calibration; there are a couple textbooks on proofwriting. I personally used Velleman’s How to Prove It, but another option is Hammack’s Book of Proof, which I haven’t read but appears to cover the same material at approximately equal length. For comparison, Halmos introduces first-order logic on pages 6 and 7 of Naive Set Theory, whereas Velleman spends about 60 pages on the same material.
It doesn’t fit my model of how mathematics works technically or socially that you can really get very confident but wrong about your math knowledge without a lot of self-deception. Exercises provide instant feedback. And according to Terence Tao’s model, students don’t spend most of their education learning whether or not a proof is valid at all, so much as learning how to evaluate longer proofs more quickly without as much conscious thought.
Part of that process is understanding formal things, part of it is understanding how mathematicians’ specialized natural language are shorthand for formal things. E.g. my friend was confused when he read an exercise telling him to prove that a set was “the smallest set” with this property (and perhaps obviously the author didn’t unpack this). What this means formally when expanded is “Prove that this set is a subset of every set with this property.” AFAICT, there’s no way to figure out what this means formally without someone telling you, or (this is unlikely) inventing the formal version yourself because you need it and realizing that ‘smallest set’ is good shorthand and this is probably what was meant. Textbooks are good for fixing this because the authors know that textbooks are where most students will learn how to talk like a mathematician without spelling everything out. I find ProofWiki very useful for having everything spelled out the way I would like it and consistently when I don’t know what the author is trying to say.
Finally, I have a rationalist/adjacent friend who tutored me enough to get to the point where I could verify my own proofs; I haven’t talked to them in a while, but I could try to get in touch and see if they would be interested in checking your proofs. Last time I talked to them, they expressed that the main bottleneck on the number of students they had was students’ willingness to study.
E.g. my friend was confused when he read an exercise telling him to prove that a set was “the smallest set” with this property (and perhaps obviously the author didn’t unpack this). What this means formally when expanded is “Prove that this set is a subset of every set with this property.” AFAICT, there’s no way to figure out what this means formally without someone telling you, or (this is unlikely) inventing the formal version yourself because you need it and realizing that ‘smallest set’ is good shorthand and this is probably what was meant.
Slight nitpick: it means “prove that this set is a subset of every set with this property, and has this property.”
This sort of thing is terrible; I learned most of it from the internet (MathOverflow, Wikipedia, the nLab, blogs), for what it’s worth.
Re: proof calibration; there are a couple textbooks on proofwriting. I personally used Velleman’s How to Prove It, but another option is Hammack’s Book of Proof, which I haven’t read but appears to cover the same material at approximately equal length. For comparison, Halmos introduces first-order logic on pages 6 and 7 of Naive Set Theory, whereas Velleman spends about 60 pages on the same material.
It doesn’t fit my model of how mathematics works technically or socially that you can really get very confident but wrong about your math knowledge without a lot of self-deception. Exercises provide instant feedback. And according to Terence Tao’s model, students don’t spend most of their education learning whether or not a proof is valid at all, so much as learning how to evaluate longer proofs more quickly without as much conscious thought.
Part of that process is understanding formal things, part of it is understanding how mathematicians’ specialized natural language are shorthand for formal things. E.g. my friend was confused when he read an exercise telling him to prove that a set was “the smallest set” with this property (and perhaps obviously the author didn’t unpack this). What this means formally when expanded is “Prove that this set is a subset of every set with this property.” AFAICT, there’s no way to figure out what this means formally without someone telling you, or (this is unlikely) inventing the formal version yourself because you need it and realizing that ‘smallest set’ is good shorthand and this is probably what was meant. Textbooks are good for fixing this because the authors know that textbooks are where most students will learn how to talk like a mathematician without spelling everything out. I find ProofWiki very useful for having everything spelled out the way I would like it and consistently when I don’t know what the author is trying to say.
Finally, I have a rationalist/adjacent friend who tutored me enough to get to the point where I could verify my own proofs; I haven’t talked to them in a while, but I could try to get in touch and see if they would be interested in checking your proofs. Last time I talked to them, they expressed that the main bottleneck on the number of students they had was students’ willingness to study.
Slight nitpick: it means “prove that this set is a subset of every set with this property, and has this property.”
This sort of thing is terrible; I learned most of it from the internet (MathOverflow, Wikipedia, the nLab, blogs), for what it’s worth.
Thanks, that’s very helpful! I appreciate the offer; let me see how I feel after the next book.