From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Mon Nov 27 2000 - 15:51:30 MST
Ben Goertzel wrote:
>
> > > Cognitive science will not become a real science as the result of math.
> > > It will not advance as a result of advances in mathematics.
>
> I'm just curious, what's your math background?
Well, there was a point where I knew what a partial differential equation
was and could even solve some of the simpler ones, but I'm not even sure I
could recognize one anymore. Beginner's calculus remains mine forever,
though - enough for me to read an allegedly mathematical psychology paper
and decode some impressive-looking equation as actually meaning "A affects
B in some way". The most advanced course I've taken so far was a college
course at Northwestern University on linear algebra, where I was fortunate
enough to get a teacher who'd I'd actually heard of previously... I'm
afraid that not much of that has stayed with me either, though.
I am sure you will be dismayed but not surprised to learn that every
single bit of formal mathematical training I have ever received has been
focused on continuous variables, with a few very minor detours into
geometry, rings versus fields, and so on. In fact, I already knew about
fractions when my first-grade math teacher tried to get me to say "two,
remainder one", like it said in the textbook, instead of "two point
five". (I refused, of course.) Perhaps this has given me a bad
impression, but I did once try reading a book on computer-science
mathematics - "Concrete Mathematics" (by Knuth, of course) - and what I
saw there was amazingly awkward, confirming completely my initial
impression that mathematics was simply not an appropriate description for
computational systems.
If-then-else - not just the specific token-level IFJMP, but higher-level
processes with major transitions of an if-then-else nature - are poison to
mathematical descriptions. To be specific, complex structures built from
if-then-else components, or from if-then-else operations acting to return
different *structures* instead of different atomic elements, usually
render any attempt at mathematical description completely useless. Not
impossible, but "useless", because even if you come up with a description,
it takes a form which cannot be analyzed using any of the mathematical
tools you have available.
When you have a programmatic function that returns a smooth curve as a
function of input, you can apply all kinds of tools. As soon as you, for
some reason, hack that function to return -3 if the input is between 4.5
and 5.5, the only way to amend the mathematical description is by
incorporating incredibly awkward mathematical symbols that say, verbatim,
"except that this function returns -3 if the input is between 4.5 and
5.5". The function would be easier to think about if you just scrawled
that note in English. If you were building any higher mental structures
of mathematics from the smooth function you started out with, those
structures are permanently wrecked; this is what I mean when I say a
mathematical description would be "useless", although you can sometimes
get one if you're willing to torture the notation by adding in every
single if-then-else element separately.
Programming - thinking about structures with if-then-else nature - is
*not* more (or less!) "artistic" than mathematics, but it does use a
different kind of thinking.
-- -- -- -- --
Eliezer S. Yudkowsky http://intelligence.org/
Research Fellow, Singularity Institute for Artificial Intelligence
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