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Actually, I disgree. My discussion of Cantor's work and which infinity is larger was addressed to those who do NOT have a math background. Matching up fingers and toes is hardly esoteric, no?
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I missed the discussion - did you use the diagonalization argument to show aleph_naught < 2^aleph_naught? That is one of the nicest examples of showing what mathematics is really about and why it is beautiful -- that mathematics is not about calculation, but about thinking. (The difficulty in showing the diagonalizaqtion argument usually arises usually in getting a non-mathematics trained mind to understand that a nonterminating decimal sequence of nines is the same as 1.0.) I can't tell you how many times I've seen people get bogged down in that to the exclusion of the cleverness of Cantor's diagonalization argument. The extension of the notion of cardinaltiy to infinite sets is not as simple, however, as matching fingers on hands. The critical sttudent may well question what is usually handled as a hand waving, it works for finite sets so lets use it for infinite sets (infiunite being defined in the beg-the-question manner through cardinality). There is room for philosophical (though not, perhaps technical) debate on the reasonbleness of an extension of a definition based on intuitions about finite sets to sets on which we do not have the same intuition. But anyway...
What most college graduates think of as mathematics ie, Calculus, ODEs (with the exception of the FTC related theorems, limits, and existence and uniqueness for ODEs) for the most boil down to advanced bookeeping and skills more akin to memorizing multiplication tables (the derivative of this is that, the integral of this is that). Likewise with linear algebra as it's often taught -- most classes reduce it to calculation of eigenvalues etc. as opposed to introducing inner product spaces, etc. As a side observation, some of the people with the greatest difficulty in understanding concepts such as limits have been, in my experience, computer programmers (not computer scientists who must master "Big-O" type concepts) but people who through programming see everything as finite loop caculations. These people think of decimal expansions as an ongoing set of calculations which they handle implicitly as being in finite time. I can't even remember how many engineers I encountered who said that they don't care how anything was proved, that they just wanted to know how to do the calculations.
What you said about explaining complex ideas I believe is true but in a very limited sense. That is, some of the most gifted mathematicians I have met could not and did not explain anything worth a damn. Perhaps it was because their intuitions were informed by a different ability to see geometrically, etc. which was so internalized as to be of little value to others. Others, (perhaps not surprisingly for the greater part foundations people) could give you a wonderful feeling that you truly understood after a lecture -- a feeling that rapidly dissipated after exposure to hard problems. The truth is that one of the frustrating aspects of mathematics is that it requires you to spend a good deal of time by yourself actually thinking. This is in a perverse sense also the source of its rewards. This is as opposed to grinding out solutions to calc problems etc. Few people make good mathematicians who don't have a healthy dose of scepticism and definite show-me inclinations. Most students are unwilling to put in the effort. I've stood in front of large classes of pre-meds/pre business, etc. students who were taking integral calculus, trying to make them understand why limits are interesting. They want to know not if an exponential function dominates a polynomial, but what will be on the quiz. Epsilon, delta proofs, relying heavily on limit concepts are forever beyond them, much less philosophico-historic arguments about why mathematics has to use limits as opposed to infinitessmals.
The ignorance is pervasive. I remember sitting in a graduate macro seminar in a PhD programm in economics that was at that time tied for number one in which the instructor blithley evaluated an integral as zero because it contained a dt^2 term. According to this instructor, dt was small so dt^2 was zero and the integral of f by zero was zero. Having spent an entire semester, carefully extending the reals in a nonstandard analysis/model theory class, I informed him he was in no position to do what he had done. Another mathematics PhD also protested -- but we get the correct derivation was the reply... This instructor likely had had a measure theoretic introduction to probability theory, but he clearly had not learned mathematics, rather he had spent his time grinding out solutions numeric or otherwise and being trained how to grind out solutions rather than to understand what he was really doing. Thus, better questions than calculate the solution to this or that would be even simple constructive proofs such as prove the existence of a function continuous at only one point or construct an everywhere continuous, but nowhere differentiable function, or give three proofs for the Cauchy Schwartz, etc., etc.
Ahh, sorry, rant off. /ubbthreads/images/graemlins/grin.gif
