**From:** Eliezer S. Yudkowsky (*sentience@pobox.com*)

**Date:** Mon Nov 27 2000 - 15:51:30 MST

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Ben Goertzel wrote:

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*> > > Cognitive science will not become a real science as the result of math.
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*> > > It will not advance as a result of advances in mathematics.
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*>
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*> I'm just curious, what's your math background?
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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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