All of Polymath's Comments + Replies

I realize you have probably had people tell you this already, and their attempts to explain Calculus to you have left you frustrated and more firmly convinced that it was beyond your capacity. But I don't propose to demonstrate my hypothesis in that way. Rather, I can try to double-crux this by showing you that your understanding of regular school mathematics isn't really what it should be and is what might actually be holding you back. In the spirit of your 3-question cognitive abilities quiz, here is a 3-question "ready for Calculus"

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6Dweomite
I attended a university where the median entering student had a perfect score on the math SAT and all students were required to take several calculus classes regardless of major (the first class involved epsilon-delta proofs).  I never felt I had any particular problem with calculus; mostly got As.  Majored in a math-adjacent field (though not one that uses calculus) and graduated with honors. I'm not sure I could answer ANY of your 3 questions.  Possibly if I spent a considerable time carefully thinking about them. It's possible that I could have answered them at the time and have since forgotten, but I don't feel like that's the case.

The disparity between your verbal and math abilities is unusually large, even more so because you are also very good at logic. I hypothesize that, even though you might not have enough innate math talent to get a college degree in math, your difficulty passing calculus is the result of an accumulation of misunderstandings related to mediocre or poor teaching. As an experienced math teacher and tutor, my judgment is that anyone with an IQ two SDs or so above average is capable of fully understanding math up through 1-variable calculus (as are the majority

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I agree with this. Like Scott, I sucked at math and excelled at English without any effort. At the age of thirty I decided to study physics at university. I was in a class with brilliant school leavers, top of their class in double A level math. I didn't have a fraction of their background and had forgotten what little calculus I had learned (which turned out to be an advantage.) We had a math professor who taught us calculus from scratch, as if we were looking over Newton's shoulder as he developed it to make sense of laboratory data. Together w... (read more)

This is a facepalm "Duh" moment, I hear this criticism all the time but it does not mean that "logic" depends on "set theory". There is a confusion here between what can be STATED and what can be KNOWN. The criticism only has any force if you think that all "logical truths" ought to be recognizable so that they can be effectively enumerated. But the critics don't mind that for any effective enumeration of theorems of arithmetic, there are true statements about integers that won't be included -- we can't KNOW all the ... (read more)

There is a lot of psuedo-arguing going on here. I'm a mathematician who specializes in logic and foundations, I think I can settle most of the controversy with the following summary:

1) Second-order logic is a perfectly acceptable way to formalize mathematics-in-general, stronger than Peano Arithmetic but somewhat weaker than ZFC (comparable in proof-theoretic strength to "Type Theory" of "Maclane Set Theory" or "Bounded Zermelo Set Theory", which means sufficient for almost all mainstream mathematical research, though inadequa... (read more)

6Wei Dai
If sets with cardinalities equal to these higher infinities exist as mathematical entities, then some of them will be logically dependent on us, on what we do. For example those sets may contain intelligent beings who are simulating our universe and taking different actions based on what they observe. I don't know if this is actually the case, or if it is, how much we ought to care, but it would be nice if an FAI could consider and try to answer questions like this, and not be arbitrarily limited in its reasoning by the choice of mathematical language we make when building it.

(Since the average LW reader may not know whether to trust that this commenter is a mathematician specializing in logic and foundations, I remark that the above summary sounds much like the papers I've read on second-order logic. Though 'pseudo-arguing' is an odd way to describe the ancient expositional tradition of dialogue.)

-1[anonymous]
Best mathematical comment on LessWrong in four years. I applaud you, sir or madam.

I smelled a rat immediately and decided to evaluate all five statements as if they had been randomly replaced with their opposites, or not. All five sounded wrong to me, I could think of rationalizations on each side but the rationalizations for the way they were actually presented sounded more forced.