[FOM] NIM Challenges

[Walt Read]

Harvey:

Thanks for the thoughtful and detailed response. Lots of interesting ideas.

> For you and other investigators to work out.

And thanks for your vote of confidence but it's a little late for me.
I've been retired for some years now. (And, yes, I do like reminding
people who still have one of those "jobs", no matter how much my wife
glares at me!)

What I'd really like to do here is to try to elucidate these ideas
around "replace mathematicians" or "behave like a real mathematician"
or "do as we do" or what constitutes "doing math". I think many people
have the same reaction to the "uncanny" nature of a lot of these
systems and in some cases, at least, a simple problem-solving system
isn't acceptable. We have to consider levels of description, whether
we're talking about internal (thinking) or external (speaking,
writing) activity and sometimes even the nature of the input. Let me
be a little more concrete.

Suppose Kevin gives Miranda a quadratic equation and she responds with
the roots. This is the most basic level (level 0?) of doing or
performing. To Kevin, Miranda is a black box with a public interface,
just like many AI systems. But at a certain level of description the
process is as opaque to Miranda as it is to the rest of us. At a
certain level - a few steps above the really basic levels - she is
probably using something similar to an artificial neural net, except
orders of magnitude more complex. If we were given a wiring diagram
and an input pattern for Miranda's brain - or better a device
implementing it - we would probably have a sense of that uncanniness
and be tempted to say "that's not how humans solve these" even if it
exactly duplicated how Miranda solves these.

So before we call this performance "like a real mathematician" or
"doing math" we might want Miranda to be able to explain her result.
"Oh, I use this formula, plug in values and carry out the arithmetic."
Or maybe to be really real, we would want her to be able to prove that
her method always worked. The acceptable level depends on a number of
factors. For example, musicians regularly work at level 0. Someone
presents a description (e.g., score) of the sounds they want from an
instrument and the musician, through activation of neurons in memory,
executive function, sensory-motor cortex, etc. produces those sounds.
Such a person would certainly be considered a real musician. Sometimes
the "why" (level 1) is critical. Medical assistant software should
always be able to explain it's recommendations in terms doctors can
understand. (Nobody wants to be sitting in a courtroom saying "I just
did what the software told me, even though it didn't make sense to
me".) And it might depend on the question. If instead of an equation,
Kevin gave Miranda a statement and she returned a good proof, nobody
would expect her to be able to articulate a general proof-finding
procedure to be declared a real mathematician.

Think of it as a kind of "math Turing test". If Miranda asked to join
this group, how could we know we're dealing with a human being? (Can
we be sure even now?)

-Walt