Mokka mit Schlag » Math

I’m going to respond to Michael Xue here because, as is all too common, the comment form simply isn’t working.

Spoken like a true mathematician. Harvard Reform might indeed have left a generation of students less prepared for advanced mathematics than ever before, but it’s not as if there’s any shortage of advanced mathematicians today, and more importantly it’s not like future mathematicians were then or now the primary audience for calculus classes. I’d estimate that well more than 90% of students who actually use calculus do so in physics, engineering, economics, and other scientific fields. Mathematics for mathematics’s sake was always a small minority. I learned calculus the old way, but I never understood it until I taught it the new way.

Newton, Leibniz, Euler, and Gauss didn’t need epsilon delta proofs. Neither do students of calculus, with the sole exception of the small minority who are trying to become pure mathematicians, and they can pick it up in their first real analysis course.

Bertrand Russell once wrote to the eminent mathematician and philosopher
Gottlob Frege to ask him how his system would handle the set of all sets that are not members of themselves.
Frege famously wrote this in the epilogue to the second volume of Basic Laws of Arithmetic:

A scientific writer can scarcely encounter anything more undesirable than, after completing a work, to have one of the foundations shaken. I became aware of this situation through a letter from Mr. Bertrand Russell as the printing of this volume neared completion.

In fact, today Frege is perhaps better remembered for this failure than for all his other work combined. At least this was an interesting mistake that taught mathematicians something. It’s much worse to publish a two-volume 500 page proof only to have an undergraduate notice an obvious mistake on page 2.

Russell’s paradox is a simple way of expressing the statement, “This statement is false” into the set theory Frege had laid out. It’s commonly responded to by removing the Principle of Unrestricted Comprehension, as well as some other changes eventually made in Zermelo-Fraenkel set theory instead of what mathematicians now call naive set theory. How successful this is I’m not sure since Godel’s Theorem still applies, even with ZF. Regardless, today I found myself thinking of an alternate problem with the Principle of Unrestricted Comprehension. What about the set of all sets that do contain themselves?
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I can’t believe I got this far in mathematics without noticing that not only can the parallel postulate be proved, but that it’s been proved for hundreds of years. For the last hundred or so, it’s been completely provable, along with the rest of Euclidean geometry, in Zermelo-Frankel set theory. That is, since ZF was invented. You don’t even need the Axiom of Choice. It’s provable with naive set theory too, or just with the Peano postulates and basic algebra.

Are you surprised? I was. What about non-Euclidean geometry? It turns out we can prove that too, and it’s all consistent (assuming ZF is consistent). How about two thousand years of mathematicians trying (and failing) to prove the parallel postulate? Did they just miss it? In one sense, yes, but in one sense no. Let’s dig a little deeper.
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Lately I’ve been thinking about ZFC, naive set theory, and some other things; and increasingly I find myself wondering, is any of this at all important? Does set theory actually matter? For anything? What, exactly, is the point of sets? Math got a very long way before naive set theory was invented, much less ZFC. Let’s explore.
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A quick proof of something that bothered me in basic topology. Assume the standard topology on ℝⁿ based on open balls. What about an open box? I.e. all points in ℝⁿ such that a₁ < x₁ < b₁; a₂ < x₂ < b₂;…;a_n < x_n < b_n. Is this an open set? I.e. can you build it up out of a union of open balls? Or, more colloquially, can you pack a square hole with round pegs without leaving any gaps?

Short answer: yes, if the balls can overlap and you have infinitely many of them. Long answer:
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C|Net accuses Apple of favoring iTunes songs over CD-ripped songs in iTunes random playlists. Unfortunately they don’t have the statistical chops to prove anything or do any real analysis:

It’s obviously difficult to tell whether back-room marketing deals or just dumb luck were responsible for the results we saw, but it appears that we can safely lend credence to the suspicions of myriad iPod users around the world. When it comes to choosing songs, ‘random’ clearly is relative.

Actually folks, it’s totally possible to figure out whether your results are random luck or not. For one thing, try repeating the experiment. But what you really need are better statistics. In particular try calculating the chance your results would occur by pure randomness. You haven’t published the raw data, so I can’t do it for you; but this should be well within the reach of anyone whose taken a couple of undergraduate courses in statistics. In fact, it would make a very nice final project for a statistics course. I don’t think it quite rises to the level of an undergraduate thesis though.
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