Updated 15 Nov 2004

In computer theory there's a concept of "Turing-complete", applied to any machine or language which can do anything a Turing machine (an abstract model of computation) can do. For real world devices the fact of limited memory is ignored; the idea is that they *could* emulate a Turing machine (which has infinite tape) given merely sufficient resources -- as opposed to missing some crucial machinery. Almost all computer languages are Turing-complete, it's actually pretty trivial to achieve. Make up a programming language which seems useful and it probably will be Turing-complete, unless you knew what you were doing.

Whether or not human minds can be emulated by Turing machines is an open question. On the other hand, human minds can emulate Turing machines; it's just following instructions with the help of a lot of scratch paper. Slow and error-prone emulation to be sure, but those aren't too relevant for me.

So, by a leap not of logic or evidence but of pure analogy, I state

Are these really true? I don't know. I wouldn't be shocked to if real
evidence otherwise was found. But at the moment I don't think they're proven,
and I think it's far more useful, for teachers and students, to act as if they
**are** true. I'm convinced that many of the Americans who are convinced
they are "bad at math" are not as intrinsically bad as they think. And this
isn't conjecture: I tutor people in math, and run into girls told by their
teachers "you can't do math", just as in my mother's feminist horror stories
when I was a kid. Or they're bad at fractions, simply because they were never
taught well -- and if you're bad at fractions then *of course* you'll have
trouble dealing with fractional exponents, or all the fractions you may need
to solve a system of multiple equations in linear algebra. But I've induced
"a-ha" moments even in a student who didn't really want to be learning any of
the stuff, and watched a student with "math disability" and in a slow class
make real progress, and show insight (tip to the first student: reading the
book really does help), and both students, who were total calculator babies,
seemed to try to do more arithmetic in their heads simply after being exposed
to me for a few weeks, and my stating answers before they'd finished reaching
for their calculators.

There's also the fact that lack of sleep really takes out memory abilities in the lab, and takes out logic abilities in me, and the students I tutor typically short on sleep, while I've generally sacrified almost anything to getting enough sleep. How many "bad at math" people are actually just chronically sleep deprived while they take classes?

I had lots of early advantages, such as having giant multiplication tables in my room to memorize when I was very young, and later getting pushed into algebra in 5th grade and being ready for it, while classmates who also might have been ready never got the chance to find out. Do I nonetheless have some knack or affiinity for symbolic manipulation which many others don't? Maybe, but given that I can cause and see progress in others, I don't see why I should believe in any particular upper bound on them. And I'd rather live in a universe where those around me are capable than otherwise.

Back to me.