[Comp-neuro] Re: Attractors, variability and noise
Dan Goodman
dg.comp-neuro at thesamovar.net
Fri Aug 22 17:35:03 CEST 2008
Hi Jim,
I'm certainly not suggesting that intuition has any mystical status, or
that there can be useful intuitions about a subject that can come from
nowhere without being grounded in something more concrete. What I really
want to say is just that in mathematics, concepts which are not
completely rigorous - and in some cases actually wrong - do appear to be
crucial in our understanding and help drive the subject forward (in the
context of discovery rather than the context of justification). In the
production of a mathematical proof, we're largely driven by high level
ideas which result in a low level proof. What makes this work is to some
extent our training set - we've done lots of proofs in the past and we
have a feel for what sort of high level ideas will work, and how to
generate lower level ideas from high level ones. Saying anything more
than that would be just speculation though.
When I used to supervise undergraduate mathematicians, I was always very
keen on emphasising that you should approach a mathematical problem by
thinking about the meaning and letting the proof follow from that. This
was, I think, important for students because in their lectures they
didn't see this part of the process, they just saw the finished result.
As a consequence, they would often try to prove things by a sort of
random walk in the space of all possible mathematical proofs. Each step
in their reasoning was correct (OK, not actually, they were undergrads),
but it wasn't leading them to the result they were looking for because
they weren't guided by the (intuitive) 'meaning' of the result.
Now whether this applies to neuroscience or not is another question. The
obvious analogy would be that realistic modelling without being guided
by some intuitive sense - or at least a guess - about what the
meaning/function is likely to be is like trying to solve a mathematical
puzzle by taking mathematically correct steps without an idea as to
where they are going. If the analogy were correct, the conclusion would
be that it's important that you CAN generate a realistic model, that you
have some experience with them (and even more importantly, the
underlying biology), but not necessarily important that you DO. But the
analogy is far from exact so I don't want to make a strong claim about that.
Perhaps the simplest summation would be just the famous quote from Einstein:
"Science without religion is lame, religion without science is blind."
In reply to your two questions:
(1) As I said above, mathematical intuition is grounded in experience of
mathematical practice. This does terminate to some extent in a formal
description of mathematical practices, but there are levels of
formality. Certainly, until some time in the early 20th century when
Zermelo-Fraenkel set theory came about, mathematics had no solid formal
basis, so you can obviously get quite far without one. Mathematics only
really developed a formal basis when it needed one, when problems arose.
Most of the development of the subject occurred before the formal basis
was there. There are also some who think that 'premature formalisation'
can slow down the growth of knowledge.
So yes there is a 'physical' structure of mathematics that constrains
and grounds more intuitive mathematics, but it's not clear to me in what
exactly that consists.
(2) Intuitive leaps in mathematics are validated or refuted according to
a few possible mechanisms. Firstly, if you can produce a peer-reviewed
paper, that validates it to some extent. Secondly, if you produce ideas
which other people find useful in producing peer-reviewed papers, that
also validates it (in a second order way). There may be others.
But perhaps what you're getting at is what constitutes the standard of
mathematical knowledge with an eye towards saying this is something very
formal devoid of high level concepts. It's an interesting issue that
changes over time. At the moment, it's peer-reviewed papers which as
I've said, contain big gaps. In the future, it may be completely
formalised mathematical proofs constructed with the aid of computer
assisted proof software. But I would say that the high level concepts
involved in real human proofs constitute mathematical knowledge, even
though they're not in principle necessary for fully formalised proofs. I
would say that even a good choice of notation can constitute
mathematical knowledge, and obviously formally this is irrelevant.
To conclude:
I see two strands in my argument. The first is the important role of
high-level concepts and intuition in the process of discovering new
knowledge. The second is that the high-level concepts, even if strictly
speaking wrong, actually constitute knowledge in themselves, and that
makes them worth producing and studying. I think maybe you would agree
with the first, but not so much with the second?
Dan
james bower wrote:
> Dan,
>
> Again, I do not deny, nor would I ever deny that intuition (what-ever it
> is) plays an important role in advancing human knowledge. I don't
> however, unlike some of my colleagues, believe that intuition is a
> function of "mind", somehow separate from the experience of the brain,
> but rather is more likely to be some manifestation of a deeper and more
> complete consideration of the information the brain has access to, and,
> of course, the structure in which that information is embedded is
> important.
>
> Thus, two related questions:
>
> 1) In what structure is mathematical intuition grounded? In math it is
> in the formal description of (in some sense preexisting) mathematical
> relationships. I suggest this is quite analogous to the preexisting
> structural relationships in neurons and networks (perhaps even directly
> in fact). It would be hard to imagine a mathematician or a whole field
> of mathematics, claiming that one could develop working theories of math
> unrelated to the actual (physical??) structure of mathematics.
>
> 2) Perhaps even more importantly, by what mechanism and with reference
> to what information do 'intuitive leaps' (if there are), get validated
> or refuted. Again, the reference in mathematics is to the physical
> structure (if you will) of mathematical objects.
>
> The reference to more than 2,000 years ago, was, of course, to
> Pythagoras -- who also according to legend picked and chose the
> relationships in math that fit with his own somewhat mystical intuition
> about how things really worked. Pythagoras also (by lore), believed
> that simplicity and regularity were a virtue in and of themselves.
>
> Finally, one has to be very careful, of course, with how scientists and
> mathematicians self describe what they do. There is a long history of
> individuals (Newton Feynman to name two), who, for whatever set of
> reasons, have wanted others to believe that they thought differently
> about things and in particular, relied heavily on an uncommon deep
> intuition. In mathematics as you have pointed out, it is perhaps easier
> to see, or at least discover the contribution of intuition and gaps in
> the formal proof. In models of the brain that have no direct reference
> to the physical structure of the brain itself -- it is much more difficult.
>
> Jim
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