FOM: Re: Measures of semi algebraic sets

[Harvey Friedman]

(In response to Lou, 8:03AM  12/15/97).

Hi, Lou. Long time, no see.

Interesting. And can the variation of parameters theorem be carried out for
any o-minimal expansion of the field of real numbers? E.g., presumably for
R_an. But what about for (R_an, exp)?

>If one looks at individual semialgebraic sets defined by polynomials
>with rational coefficients, one is soon in big trouble:
>..To decide equality of numbers built up
>polynomially from such quantities seems only a realistic problem
>once there has been earthshaking progress in transcendental number
>theory. In any case, faced with such intractable (at the moment)
>decision problems, the way to make it into a more fruitful question
>is to vary the parameters and study the dependence on parameters.

But what's wrong with the relative decidability program I enunciated in
6:07PM 12/14/97? As you must be aware, this kind of relative decidability
program is the kingpin of complexity theory, where one identifies crucial
problems that are intractable at the moment (and for a while), and then
carries out reductions. This is a huge industry, including "NP-completeness
theory."

You don't like this standard way of proceeding in complexity theory?
Perhaps you want to add "variations of parameters" to complexity theory,
and make it more to your liking! I am sure that this can be appropriately
done, with a fruitful theory. However I am not biased in favor of variation
of parameters: the original complexity problems such as P = NP and P =
PSPACE are crucially important and interesting.

However, one gets the feeling that this is a pervasive phenomenon: it rears
its head in recursion theory with the distinction between boldface and
lightface. There is always an intimate connection between boldface theory
and lightface theory which has never been properly analyzed for formalized.
I suspect something very interesting could be done to elucidate this
intimate connection, at various levels of generality.

Well, now that I have the opportunity, let's get the good old fashioned
wild fom brawls going again. There's 225 people on this list now - many
more than before. What current problems in core mathematics have a
comparable significance to the classic open problems of complexity theory?
And which core mathematicians have made contributions of comparable (or
greater) general intellectual interest to those of Godel this century? Is
current core mathematics a coherent intellectual odyssey with clear stated
intellectually dramatic aims and goals, and clearly stated intellectual
acheivments of general intellectual interest?