[Comp-neuro] Re: Attractors, variability and noise
Dan Goodman
dg.comp-neuro at thesamovar.net
Wed Aug 20 14:42:36 CEST 2008
(I sent this email a little while ago but it was rejected because I sent
it from the wrong email address. Apologies for it being slightly out of
sync with the discussion.)
Hi Jim,
Thanks!
I agree that there's a big difference, a huge difference, between how
we should be reasoning about maths and how we should be reasoning about
the brain. Actually I suspect we'll need quite new types of reasoning
to understand the brain, probably quite different to any sort of
reasoning we've seen before. It seems likely that computer simulation
(at many different levels of detail) will be involved in that. But
let's not abandon the successful examples of reasoning we already know
about too soon - and intuitive, simplified concepts play a big role in
those.
"In this sense isn't it fair to say that what mathematicians
fundamentally do is build the equivalent of realistic models in
biology?"
That's an interesting analogy (!), I see what you're getting at and
it's a useful way to look at mathematics, but it's not the whole story.
Apologies for the following slightly long digression about maths. In a
modern mathematical proof, we don't give all the steps in the reasoning
because it's practically impossible. As a personal example, in a paper
I wrote there are two equations with "from this it follows that" in
between them - to actually do that calculation took 12 pages of
algebra. I believe this is pretty standard in mathematical papers, and
in fact many papers would require significantly more work to fill in
the steps than that. Recently, the Mizar project found a complete proof
of the Jordan Curve Theorem (a very simple mathematical theorem that
basically says if you draw a wiggly circle that doesn't intersect
itself it will have an inside and outside, i.e. the sort of thing
normal people would just assume without proof), this was the result of
14 years of in-depth work involving lots of people and computers
working together. The final proof was 200,000 lines of mathematical
reasoning. That was to prove that wiggly circles have insides and
outsides! So obviously, proofs of serious mathematical results cannot
possibly give all the steps because nobody would read them. Instead,
mathematics relies on the judgement and training of mathematicians -
they are trained to know which statements can be made 'safely' and
which can't. The idea is that 'in principle' a mathematician could
write out the complete proof, but it's almost never done, and the
Jordan Curve Theorem example shows just how difficult that is. Real
mathematics absolutely relies on intuitive concepts, it would be
impossible without them.
But there's more even than that, because sometimes mathematicians rely
on intuitions that are actually wrong! In the "golden age" of
mathematics (in the 18th century), the great mathematician Leonhard
Euler proved extraordinary results using infinite series. He's the one
responsible for the infinite series for pi, that pi^2/6 is the sum of
the reciprocals of the squares of the integers 1+1/4+1/9+... The
trouble was that in order to "prove" this result, and many others like
it, he had to use series that diverged as intermediate steps. It's
known that he was aware of this problem - so we can't just say he made
a mistake - but that his intuition told him that it was somehow
meaningful anyway. It turns out he's right, one can take his ideas and
make them rigorous by developing and overlaying new concepts on them.
His 'understanding' of numbers and infinite series went beyond his
ability to 'realistically model' them, so to speak. If you're
interested in this you can read more about it in Morris Kline's
excellent "Mathematical thought from ancient to modern times", volume
2, chapter 20, section 7, page 460 on the version that is free to read
on Google Book Search.
In modern times, there are more examples. The contemporary
mathematician William Thurston is partly famous for his "Geometrisation
Conjecture". Thurston didn't manage to prove this theorem himself (it
was just recently proved by Russian Mathematician Grigori Perelman
making him eligible for the $1m Clay Mathematics Institute prize), but
his enormous insights into the nature of three-dimensional geometry
meant that this conjecture essentially determined the course of most of
the work in 3D geometry since 1982, when he made the conjecture, until
now. All of his insights have proved correct, and have been invaluable
in the development of that subject (which was my field).
I'm not terribly knowledgeable about theoretical physics, but I'm told
there are people in that subject who are in a similar position: they
have incredible insights and intuitions about their work that go beyond
our ability to make them formal (yet!) but that they are hugely
successful in practice. I believe Edward Witten is the person I have in
mind. Or for that matter, there's no mathematically consistent theory
of Feynmann's path integrals, but they too are incredibly useful.
So in conclusion...
"Didn't they give up on the kind of "gut" intuition about the meaning
of math sometime around two thousand years ago?"
No! :-) (Although they'd like you to believe that, for some reason.)
In regards to your second point, you're absolutely right that these
intuitions don't come from nowhere, and that intuition about intuition
guiding our intuition is a dangerous place to be in. But I don't think
anyone is arguing that mathematicians and physicists working in the
field should be ignorant about this stuff are they? Certainly for my
part I feel there's an awful lot more biology I need to know.
On your third point, I agree that in both physics and maths often the
elegant theories came from an attempt to make elegant a mess of complex
and seemingly inconsistent results. But at some point, hopefully we'll
find that elegant theory, and who's to say we're not yet in a position
to do that? I would imagine that when this insight comes, if it does,
the nature of it will be a big surprise (if not, we would already know
it). The point is, we shouldn't rule out one way or another because we
don't yet know which way will be the successful one. (Is there a
history of unsuccessful ideas in physics and maths? There should be. I
know that Popper wrote some interesting stuff about pre-Socratic
science in "The World of Parmenides" but apart from a few cases of
really well known things like the 'phlogiston theory', I don't know of
anything more recent that talks about this.)
I'll just add that I'd imagine you agree with that, but that it's still
important to make this point explicit because opinions about where the
answers are likely to be found affect where the funding goes, and this
is pretty important.
Dan
jim bower wrote:
> Dan
>
> Great post, and I probably should have been a bit more careful in
my wording. but a couple of things to say.
>
> First, human intuition when applied to mathematics is quite a
different thing than human intuition applied to the system (the brain)
that generates human intuition. The discussion of free will and our
deep belief in the primacy of individual-based learning being an
example. Intuition in mathematics is generated in the context of a
formal description related to discovered mathematical relationships
themselves pushing you guys into rather bizare musings . In this sense
isn't it fair to say that what mathematicians fundamentally do is build
the equivalent of realistic models in biology? Didn't they give up on
the kind of "gut" intuition about the meaning of math sometime around
two thousand years ago?
>
>
> Second, I would be the very last to suggest that intuition is not
a key element in human progress. One reason I have expected every
physicist entering my laboratory for years to do experiments or at
least build realistic models is to give them a basis in biology for
their intuitions - the question is- what is the characteristic of the
tools that guide that intuition. If it is intuition about intuiton that
is guiding intuition, everything becomes rather circular.
>
> Third, while I belive it will be at least very hard, and for sure
require new tools, I have no objection in principle to theories that
are easier to understand. I am happy to hope this will be the case for
biology as well (although I am skeptical for all the reasons mentioned
previously). But the critical question for me is where those theories
came from. Physics and math are full of examples where something the
data pushes to be very complex and hard to understand is replaced or
captured in something more eligent and even intuitive. . It was the
complexty of the earth centric models that lead compernicus to propose
a simpler (and widely less accurate) sun centered model, coupled with a
necessary fudge in the earth centric models (the equint) which violated
intuition. The question is what you do first and what you do second. It
is my contention that the chances that more abstract top down models
will discover brain function, are much less than the possibility that
detailed realistic models will provide the kind of brain structure
directed intuition that could lead to some more formal elegent or
understandable general principle. In almost all cases now (although
there are exceptions) , more abstract modelers base their assumptions
on descriptions of the nervous system often less sophisticated than
those found in our textbooks (and they are very much lacking). In my
view, it is likely to be more useful to start with a big complex
realistic model (there are a bunch of them in model-DB and we are happy
to give you any of ours you want) and work from there.
>
> This is what Sharon Crook, Bard and I were starting to do 15 years
ago in the paper I referenced previously.
>
> But what I was railing against was a field that holds as its
highest achievment a one and a half page publication in Science or
Nature that makes a clear profound (and "understandable") point, and
whose leading journal (journal of neuroscience) limits all publications
regardless of the subject matter to 1500 words (and you can see my
personal problem with that. ;-). The drive for short format, easu to
understand, sound-byte science just seems so completely out of step
with the structure we are trying to figure out.
>
> Jim bower
>
>
> Sent via BlackBerry by AT&T
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