Polymath

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# Wiki Contributions

The Parable Of The Talents

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" diagnostic test I just made up (but which is informed by many years of teaching and tutoring students at many different levels of math understanding).

1) Can you explain why a real number has a decimal expansion that eventually repeats cyclically if and only if it is a rational number? (If you can't do both directions confidently, your understanding of rational and real arithmetic isn't good enough for calculus.)

2) Could you explain why the Pythagorean Theorem is true in a way that would completely satisfy the curiosity of a typical intelligent 12-year-old? (If you can't, your understanding of plane geometry and area isn't good enough for calculus.)

3) Can you state a version of the Binomial Theorem completely precisely? (Only state, not prove -- if you can't, your understanding of polynomial algebra and math notation isn't good enough for calculus.)

All these things are definitely PRE calculus mathematics, and they are taught in high school, but rarely taught well enough that students could comfortably answer all three questions.

The Parable Of The Talents

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 of people 1 SD above averag, but pretty much no one in the left half of the bell curve), but it might require a significant amount of "unlearning" and then relearning the prerequisite math topics in the right way. There are undoubtedly limits to your math potential, but it seems quite unlikely to me that your mind would fail to grasp Calculus if it were properly trained, given your good score on the Math SAT and obvious logical ability. However, it's not the kind of thing you could do by self-study, you'd need an experienced tutor who could figure out where your prior understanding of math was blocking you.

Logical Pinpointing

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 true facts about integers, so the criticism of second-order logic boils down to saying that you don't like using the word "logic" to be applied to any system powerful enough to EXPRESS quantified statements about the integers, but only to systems weak enough that all their consequences can be enumerated.

This demand is unreasonable. Even if logic is only about "correct reasoning", the usual framework given by SOL does not presume any dubious principles of reasoning and ZF proves its consistency. The existence of propositions which are not deductively settled by that framework but which can be given mathematical interpretations means nothing more than that our repertoire of "techniques of correct reasoning", which has grown over the centuries, isn't necessarily finalized.

Second-Order Logic: The Controversy

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 inadequate for results which depend on Frankel's Replacement Axiom).

2) First-order logic has better syntactic properties than Second-order logic, in particular it satisfies Godel's Completeness Theorem which guarantees that the syntax is adequate for the semantics. Second-order logic has better semantic properties than first-order logic, in particular avoiding nonstandard models of the integers and of set-theoretical universes V_k for various ordinals k.

3) The critics of second-order logic say that they don't understand the concept of "finite integer" and "all subsets" and that the people using second-order logic are making meaningless noises, and all we really can be sure about is ZFC and its consequences, and since the ZFC axioms actually allow us to prove more theorems about objects we care about than the usual formalizations of second-order logic (such as, for example, the consistency of full Zermelo set theory), the fans of second-order logic should just quit and use ZFC instead

4) the fans of second-order logic reply that nothing we actually prove and translate into the language of set theory is false and our reasoning is sound by your standards so stop insulting us by claiming that since YOU can't decide what's true about sets, OUR notion of "property" or "subset" is vague and incoherent. We say that CH is either true of false but we don't know which because we don't have a deductive calculus that settles it which we are confident in, you say that CH is vague and meaningless, but this is a pseudo-problem until either you come up with new set-theoretical axioms which decide CH or we come up with a stonger deductive system which decides it. We can still prove "ZF proves Con(Z)", we just think that the theorems we can derive in pure second-order logic are epistemologically better supported than theorems you need the full power of ZFC to prove (because they already follow from "logical" rather than "mathematical" axioms) and you shouldn't mind us making this distinction.

5) The best theories of physics we have need real infinities and finitely many iterations of the Powerset axiom but Second Order Logic or "Type Theory" is completely adequate for them and it is hard to conceive that questions about higher infinites that require infinitely many iterations of the Powerset operation can be relevant to physics.

6) It is also hard to conceive that any form of mathematical finitism is adequate for making actual progress in theoretical physics, even if results that are proved using standard infinitary mathematics can be reformulated as finitary statements about integers and computations, these nominalistic reductions are so cumbersome that the self-imposed avoidance of infinitary ontology cripples physical insight.

Hindsight Devalues Science

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.