good evening everyone good evening thank
you for coming I'm Geri sab laughs the
president of the Santa Fe Institute and
delighted to see such a terrific crowd
welcome to the our annual rule on
lectures they were inaugurated in 1994
so for all of us who can almost count
this is the 19th annual Coulomb lecture
these lectures are were named in honor
of the great mathematician the late
stanislaw Coulomb who was you know as
part of the Manhattan Project and
subsequently lived here in Santa Fe as
part of his legacy his library now forms
the core of SF eyes scientific library
today as always we gratefully
acknowledge the support of the Los
Alamos National Bank which underwrites
our entire public lecture program it's
my great privilege this evening to
introduce our 2012
Gulam lecturer Bob May Bob is a member
of the SFI Science Board and I think
it's just a terrific example of an SFI
scientist with his wide-ranging
interdisciplinary interest and his
concern with complex adaptive systems I
think without a doubt he's one of the
most important eminent and influential
scientists in the world today although
his ph.d is in theoretical physics from
the University of Sydney in Australia he
currently holds a professorship in the
department of zoology at Oxford nice
example right there of his
interdisciplinary interest he has held a
number of other important positions
including chief scientific adviser to
the UK government and president of the
Royal Society of Britain he was knighted
in 1996 and in 2001 he was raised to the
House of Lords in Britain as Baron may
of Oxford he has won numerous honors and
prizes as well as a host of honorary
degrees the list the listing on the
website takes up a considerable amount
of space
trust me that he has been honored by the
most importance of scientific societies
around the world he has written and
edited a number of widely cited books in
ecology and biology and a
number of scholarly articles including
such recent key pieces as uses and
abuses of mathematics and biology and
the famous ecology for bankers that had
SFI science board members Simon Levin as
one of its co-authors among Bob's
current interests is research on the
factors that influence the diversity and
abundance of plants and animals and the
nature of their extinction Bob will be
giving three lectures over the next
three nights tomorrow night he will
speak on the topic of and obviously of
great importance of literally what is
stability in today's complex financial
systems a far better choice I would
argue to you to listen to than the
presidential debate where where you
already know what both of them are going
to say whereas you don't know what Bob
will say so this this would be a much
more adventurous choice
to rejoin us tomorrow night on Thursday
night Bob will speak on topic of people
and tomorrow's to small world tonight he
will examine the intriguing subject of
beauty and truth in mathematics and
society he will consider the fascinating
question of the role and influence of
aesthetics in mathematical thinking and
the pursuit of truth for the equation
folks in the audience and I know you're
there I urge you not to be put off by
the appearance of the occasional
equation on the screen because you'll
find I think that Bob's explanation will
be both fun and beautiful so please join
me in welcoming our distinguished
speaker tonight Bob may I give it upon
myself to extend the remit to discuss
the role of elegance and ultimately
beauty in mathematics in its role of
helping us simply to advance the quest
that is as old as humanity to try and
understand the workings of the world
around us and the talk I'm going to give
is going essentially to have three parts
I'm going to begin with the narrower
remit
of suggesting to you some of the reasons
why I think mathematics really is both
can be amazing in ways that I find
aesthetically pleasing but I'm going to
do that as a background to then giving
some examples of how questions of
aesthetic satisfaction have shaped
really important advances in science and
I'm going to conclude with a brief look
forward to some of the larger questions
that this opens which I'm going to
explore in the next two lectures and the
banking lecture is partly because it's
something I've been drawn into but
partly because it's something that I
think illustrates very interestingly the
way we suffer from an enterprise which
is what most of the financial system is
which where the mode of discourse is
still pre enlightenment and would be
much more familiar to Socrates Athens
than it would to a post enlightenment
scientist okay so here we go fasten your
seat belts and I would begin by
observing that the primary the most
important thing that distinguishes
humanity from the rest of the
fascinating array of animals that
populate our world is our clear and
deliberate quest for an understanding of
how we got here and how the world works
and the first stirrings of that are lost
in the mists of myths and mysticism that
leave their relics in stone circles and
wonderful paintings of enigmatic meaning
in caves any in so far as the early
quest did have a foundation it was in
practical descriptive things looking at
the movement of the stars which in those
days without streetlights you could see
pondering the mystic resonance between
the period of one month of the moon and
the resonance with the menstrual period
pondering things with learning by doing
of medicinal use of plants and nutrient
values of plants in a way that
prefigured and still persists although
often in rather silly ways today and
from that emerged the foundations of
mathematics which are more recent and
have an immense appeal because they're
about things that you really can prove
with no ifs and buts like two sides of a
triangle or equal then the two angles in
the triangle would be equal and there
you are you're not going nobody can mess
around with that so let me take that as
the first example I remember when I
entered high school started first became
acquainted with geometry at the age of
12 and we started to go through Euclid's
book of theorems and the first one is
this that says if you've got an
isosceles triangle so that a B equals a
C then prove that two angles are equal
and the teacher went ahead then and did
one I what Pythagoras did and what I saw
what's his name the bloke Euclid did and
you drop a perpendicular and then you
say the side Abe in the Tri the side a B
equals the side AC and the side going
from a down to the bottom of the
perpendicular is common to the two
triangles so they've got two sides equal
and the same right angle at the bottom
therefore they're congruent
therefore the angles are equal and I at
that time early in the first term in
school set a pattern from which I was
become to become unpopular
by saying surely it's just completely
bloody obvious that by symmetry that
they have the sides are equal the angles
are equal but I was assured no no we're
going to prove things rigorously and I
was thus pleased one of the first things
I learned when I came as a postdoc to
Harvard in 1959 I was in the division of
Engineering and Applied Physics and
there will be people in the audience who
would have known people there in
particularly George carrier and I
remember one of our first lunches he
told me of something that had happened a
few years earlier in the beginnings of
computers and artificial intelligence
some group of people that had the very
nice idea of testing their artificial
intelligence program by giving it
Euclid's theorems and they particularly
thought this would be a good one because
you've got to make an imaginative extra
construction in order and I want to go
backward here you've got to put
something in to prove it
what did the computer do the computer
said the triangle ABC and the triangle a
CB are congruent a B and a C it's a the
side a B equals the side AC and the side
AC equals the side a B and the angle is
common therefore those two triangles are
congruent so the exit which is a
refinement of the symmetry argument and
I think that's beautiful I give you
another thing which I think is one of
the most magic things in mathematics and
miss some will be familiar and some will
find it simply reinforcing their belief
that some of it is silly of the
definition and I'm doing this not likely
because it's leading me somewhere else
of complex numbers we start with the
equation where
looking for the unknown number X which
has the property then when you multiply
it by itself x squared add to it twice
itself 2x + 2 you get 0 and what's the
number well you can go about this
another way you can say well x squared
plus 2x plus 1 plus 1 equals 0
but x squared plus 2x plus 1 is just X
plus 1 squared so you've got X plus 1
squared equals 1 or X plus 1 squared
equals -1 then now we'll take the square
root X plus 1 is the square root of -1
the square root of -1 a number that when
you multiply it by itself gives you
minus 1 I mean that's a nonsense if your
mother 2 minuses give you a plus there
is no way it's it's an imaginary number
well let us be bold we will say the
solution is x equals minus 1 plus or
minus this imaginary number and we'll
christen it we'll call it I an imaginary
number then this sounds like a lot of
loonies stuff but look where it leads
we've now in a sense got a curious way
of playing with numbers in a
two-dimensional way one of the
dimensions is the real axis of sensible
real numbers that you could use with the
butcher or the supermarket the other is
another line perpendicular to it of
these so-called imaginary numbers which
are a real number multiplied by the
square root of -1 so why are we doing
this well we've constructed a rather
elegant and sometimes useful ways I'll
show you in a moment of doing
complicated things in two dimensions
that take us in some interesting
directions one of the things we can do
and I'm not going to go into this in
detail one of the things we could also
describe this complex number Z which is
the real number X plus the imaginary
number iy an ordinary number y times I
we could also measure it by it's a
distance from where the horizontal and
the vertical line intersect the origin
and the distance from that to the point
and the angle that that line makes polar
coordinates and if we did that and did
some fancy stuff we would end up one of
the most magical formula in science it's
a formula that makes alchemy look
ordinary it it's genuine mysticism in a
sense it's got two fundamental constants
e the base of natural logarithms this is
a number it's a number that emerges from
asking what is the mathematical function
or thing which describes things where
the rate at which they change is
proportional to their magnitude
something if you've had for example
something that was putting on weight and
the faster it put on weight the faster
it put on more weight and that do that
mathematically and we'll give you this
fundamental constant pi is the ratio of
the circumference to the diameter of a
circle I is this ridiculous seeming
thing and you put them all together and
you get something ordinary like minus
one of course some people have a more
pragmatic attitude to this one of my
favorite things of some of the
astonishing things that have happened in
this wonderful country is it toward the
end of the 1800s the Wisconsin
Legislature got a bit fed up with the
fact that pi was such a awkward number
and actually legislated to redefine pi
to be 3.2
I kid you not it was done on behalf of
the construction industry because it
made carpentry easier and the thing I
find puzzling about it actually there's
not that they did it but that if you're
going to make it simpler why not make it
three for God's sake and I have to agree
to tell you that it never made its way
into law because the governor vetoed it
I'm now going to move on to something
weirder and with more practical
applications and it is the following
thing this is the so call this is
something that I came across and I was
by no means the first when I
accidentally got interested and moved
from my chair in theoretical physics at
Sydney to Princeton's biology department
and I got accidentally interested in
questions of ecology as part of part of
essentially the troubled times of the
late 1950s and Vietnam and caring about
the environment and I've tried started
trying to find out what I was really
caring about and I came across this
notion of what regulates natural
populations and a very oversimplified
metaphor is this sort of little equation
here that says the number of say
knapweed gourd golf lies in white and
woods outside Oxford next year T plus 1
is going to be equal to the number of
animals there with their this year X of
T times the number of offspring they had
which will let be a constant a but it
can't go growing indefinitely we've got
to recognize in the real world if X gets
to be too big too many animals then
either food limitations or attracting
predators or various other things will
tend to cut that back and they're not
all their offspring will survive so a
metaphor for that a deliberately
oversimplified version is to
say the next X is the X we got at the
moment times one minus the x times the
constant a ten-year-old could iterate
that on a hand calculator what's going
to happen to a population like that well
the firm curve there shows in terms of
how many there were this year along the
x-axis
how many they'll be next year and the
dashed line represents a population that
doesn't change and you'll see if it's a
levels less than that the population
will grow and if it's levels bigger than
that it'll decay but the basic question
is how does that equation then behave if
you let it just run and the answer is as
many in the audience by this time we'll
know but as was not at all fully
understood or even appreciated that the
phenomenon existed if you indeed go back
to the late 1950s if the constant a is
if it's less than 1 then the population
just dies out because it doesn't leave
enough descendants in each generation if
it's between 1 and 3 what happens is
what you'd into it it settles to some
steady value held there by the density
dependence it says if you're too big
you're decrease to small you increase
once you go past 3 the population starts
regular oscillations up and down once
you go about beyond about three point
six or three point seven it starts
behaving apparently almost randomly if
you were to plot the fixed points now by
fixed points we mean the points where
the actual map intersects nothing
changing the dotted line so the of
course these fixed points may or may not
be stable points they may be as it were
like marbles in the bottom of a cup the
bottom of the cup is a stable fixed
point
they may be like a marble poised on a
tip of a pool cue be an unstable fixed
point and what you see here is the
initial value of a steady value flips to
an up-down cycle and then that
bifurcates to give you the 4-point cycle
and so on and so forth
and I had rediscovered all that for
myself
and was giving a seminar at Maryland and
I said and finally it comes to a point
of accumulation where you can just get
anything and it just looks like random I
don't know what's going on and Jim York
at Maryland had just published his paper
period 3 implies chaos he hadn't he was
unaware of the period doubling
phenomenon that took you there but he'd
proved and that other vacant area toward
the end that's where you get a period 3
first appearing and he had shown that if
you have one of these nonlinear
difference equations that will give you
a period three orbit then as you change
the parameters you will be able to find
every integer period but the really
interesting thing about all this is not
it turns out various other people had
discovered the mathematics but they were
all very pure mathematicians the first
person who did it
published in Finnish in nineteen the
late 1940s I think Lam here along with
metropolis and Stein had stumbled on the
phenomenon but not quite got it it's
very unusual for Stan and there has been
a little bit of retrospective history
writing here but I'd say he got very
close but Jim York says we weren't the
first to find it the boy we were the
last to find it because we jumped up and
down and said what it meant and this is
what it really meant not only is this
look messy in the chaotic region but
it's so sensitive
the initial conditions that you come
even though you know the rules and
there's nothing random in them it's all
deterministic you still can't make
predictions look at this here's the
quadratic map with a equals 3.8 to the
next x 3.8 X 1 minus X let's start with
X is not 0.3 that's the solid line let's
make a mistake of one part in a thousand
let's start with X is not 0.301
for the first few iterates you can't
tell the difference but by the time
you're about 10 or 12 it or it's out
you've got totally different predictions
this is a really it's not just weird but
this is the end of the Newtonian dream I
mean when I was a graduate student it
was thought that with increasing
computer power we would get better and
better weather predictions because we
knew the equations navier-stokes and we
could put more realistic models of the
earth anon but to summarize it the best
account of this the best summary of the
change this maker
I think is found in Tom Stoppard's
Arcadia where one of the characters is
working on actually I'm very fond of
this particular play some will be
familiar with it some not my most read
publication is the program notes about
chaos for Arcadia and by two orders of
magnitude that's my most cited work
except it has no citations then it
illustrates the nonsense they're judging
people scientific productivity by
counting the citations but I digress but
in that play Valentine says we're better
at predicting events at the edge of the
galaxy or inside the nucleus of an atom
than whether it'll rain on auntie's
garden party three Sundays from now we
can't even predict the next drip from
the dripping tap when it gets irregular
each day
sets up the conditions for the next the
smallest variation blows prediction
apart the weather is unpredictable in
the same way will always be
unpredictable we'll probably never be
able to do local weather beyond about 20
days
mark you climate change is something
different that's not weather and saying
you don't because you can't trust the
weather report you can't trust climate
change is a bit like saying I can't tell
when the next wave is going to break on
Bondi Beach so I don't believe in tides
again some of the most colorful
characters in the early days of chaos
after it came center stage were Santa Fe
join farmer and the Santa Cruz kids
wonderfully imaginative colorful people
but if you thought that was weird
this is even better most people think
for those who are familiar with the
Mandelbrot set and fractals and stuff
don't associate it with chaos but it's
basically it's just two-dimensional
chaos suppose we put together after this
I'm going to become less mathematical
suppose we put together on the one hand
two complex numbers and on the other
hand what we've just seen with the
one-dimensional quadratic map and look
at let the number in the next X is a X 1
minus X let that X to be Z a complex
number in effect all that does is take
our one-dimensional system into two
dimensions and so now we've got a more
complicated pair of equations which I
won't go into detail about we've got
them more complicated next X which
depends on both X and y and I'm going to
have to explore what's going to happen
in two dimensions rather than along a
line and the value what the value a is
doing and this is just don't you close
your eyes if you wish on this bit
most people don't realize the intimate
association between what I'm about to
show you and the stuff on chaos and the
quadratic map because it's usually
written as the next complex Z to Z
squared plus C we just had a very good
program on chaos at the BBC and the chap
who made it treated these this is a
completely different topic from what
I've just been talking about it's the
same topic and you can see that because
you just got to redefine what Z is
that's what that's about but the really
interesting thing is what happens I'm
going to show you what happens if you
look I showed you the bifurcation
diagram the fixed points of the
one-dimensional thing now I'm going to
show you the fixed points of that
two-dimensional system and these are the
things that benoit mandelbrot publicized
interestingly among mathematicians he
never got much credit for he was a sort
of pushy bloke and people said but Julia
looked at this set so you're not the
first point is Julia just saw it as a
mathematical curiosity whereas benoit
mandelbrot again wasn't the first person
to find it but he was the last in this
area and the prize that was given the
key a big Kyoto prize same scale as the
Nobel but less status that was given for
complex systems a few years ago in my
opinion went to the right two people
jim york and benoit mandelbrot what he
did is he plotted the fixed points of
that pair of equations of two
dimensional quadratic map complex
numbers in with the compas real numbers
X and the complex numbers with the real
part Y so those are the fixed points
that's what the boundary looks like it's
a figure which actually with the value
of a that he chose has a total area that
can be enclosed in a circle of radius
two point five about the origin which is
somewhere here and that itself is
sort of complicated enough but if you a
lot of the fine detail is just fuzz
doubt there if you look into more of the
detail the closer you look the more you
find and you can go into one of the
twiddles and any around the fringe and
blow it up and you get stuff like this
and you can go into one of these
twiddles like this and blow it up and
you get stuff like this is absolutely
enchantingly gorgeous but not only is it
beautiful it is saying something really
very profound more profound than this
this is it crops this is something
there's a great craze for crop circles
in Britain and there are lots of people
in Britain who believe that this has all
come from outer space yes I suppress the
adlib there what we realize much better
than we used to and I'm coming now into
the second and they're not all of equal
length part of the talk about the role
of mathematics in actually understanding
the world we can go back to Galileo and
the early days of mathematics in science
and when he enunciated his belief that
the grand book wasn't too pleasing to
the relevant pope who thought the grand
book was written essentially by his
apparatus apparatus the grand book is
written in the language of mathematics
and its characters are triangles circles
and other geometric objects post
Mandelbrot we realize the great book is
written in mathematics yes but the
objects aren't triangles and circles and
squares instead of the triangles we have
fractal geometries instead of the nice
regular orbits we have
strange attractors instead of regular
cycles we have messy things going right
back to this also we're introduced to a
novel idea that is important I said the
area of this gorgeous set of fixed
points is finite but the boundary which
autumn is a line ultimately is infinite
and it introduced us to this word
fractal geometries of geometries that
where the lines aren't necessarily
one-dimensional the coastlines of most
countries have fractal geometries now
all that was by way of background to
other lectures that will come in the
fullness of time about how much we know
the journey we're still embarked on and
what guides us Galileo is not that
really all that different from Keats
enunciated the view that elegance is
part is a reliable guide in the quest
for understanding the real world the
Enlightenment says something in many
ways a bit different in the real our
progress up to the Enlightenment is one
that really was heavily focused on ideas
of things being beautiful and even
Newton I mean the idea of the big
mathematics in that is important most
people I think don't realize the first
application of calculus and in Dague the
motivating thing for calculus is because
he wanted to
ferm his intuition that in doing
gravitational calculations when you had
a sphere like the earth or the Sun you
could validly replace it by a point at
the center and to do that and put it
into the gravitational equation requires
calculus but the Enlightenment well went
well beyond that it because it said you
don't appeal the beauty and truth you
also want experiments and things like
that I think nonetheless while that is
true I don't think I mean there's a
personal belief that you may not share
I believe that elegance has is an
inherent part of the quest and I'm going
to give you two of the really notable
illustrations of that the first of which
will be familiar to you and the second
one perhaps less the first one is
special relativity and its implications
and that all comes as you know from a
really important experiment that was
done which was an experiment to try and
show just how fast we were moving
through spaces that were true and find
out what was the speed of light and how
it depended on movement and it produced
the ridiculous answer that the speed of
light here anything consistent with the
experiment was that the speed of light
was an absolute constant independent of
whether the way the observer was moving
I mean if you're trying to measure the
speed of a car that's going along the
interstate at 60 miles an hour it
doesn't matter whether you're standing
beside it if you're standing beside it
you'll see it go at 60 miles an hour if
you're driving alongside it at 50 miles
an hour then you'll see its relative
speed is 10 miles an hour but if the car
was moving at the speed of light it
wouldn't move whether you matter whether
you are moving at half the speed of
light which but
you couldn't do because the observations
make it clear the speed of light is the
speed of light independent of its motion
the lights motion relative to the
observer and that just means the laws of
physics were fundamentally in error it
meant that what if speed of light is
constant it must mean that the
fundamentally intuitive things that you
clearly you would think of as just
independent verities like the length of
a meter stick or the time from your time
you were born and your twin was born
must be dependent on circumstance and it
wasn't in its time and extraordinary
leap that einstein undertook what he
effectively did wrong page of my notes
and I don't know what I did with the
other one with the page I want and it
doesn't really matter except I would
like to find it so I want to put it
precisely is what he said was I'm going
to put this in a framework that assumes
speed of light is constant and the laws
of physics are the same at all times in
all places that means I'm going to have
to redefine what I mean by the
coordinates of space x y&z in the
three-dimensional space we inhabit and
time these four variables and I'm going
to have to construct a set of equations
a way of thinking about this in an
elegant and self consistent way that is
respectful for the crucial experiment so
it's a mixture of elegance and a
surprising fact and see how it all works
and I'm not going to try and take you
through that in detail I'm just going to
say that one of the consequences that
our mint merges is
it means that whereas in elementary
physics you'd think of momentum of a
particle is its mass times its velocity
but it's velocity is now involving
measurements and times which are not
absolute and when you put that
correction in in a way that's consistent
with the absolute constancy of the speed
of light you find that you've got to
modify the definition of momentum from
mass times velocity which you observe to
mass times velocity divided by the
square root of 1 minus the square on the
velocity over the square of the velocity
of light but that has wider implications
so if you're going to put together
equation now for energy you find in the
scheme that Einstein put forward that
you've hit upon the notion that there
must be an inherent rest energy
associated with any mass and that that
rest energy corresponds to its mass
times the square on the speed of light
and it comes from preserving the
symmetries in the equations and it
appears really again ludicrous until you
begin to see if you can test the
conclusions of course one of the
conclusions is if mass is lost as it is
in nuclear fusion in nuclear fission
when things atom splits apart when the
mass is lost that Energy's got to appear
in some other form however
counterintuitive that may have appeared
that Beauty inspired formulation found
its proof in the blossoming of the first
mushroom cloud not that far away from
here in the deserts of Nevada and we
fully appreciate today how such lost
mass exists and how devastating these
other forms can be of course in everyday
life you don't see much of this actually
if you want to get into the
technicalities the Einsteinian
formulation gives you the top equation
there and you fiddle around with it and
you end up with the definition that the
energy in special relativity is going to
be the mass times C squared divided by
that factor thing and if you then
approximate that for V over C very tiny
you recover classical physics apart from
the rest energy which is a sense not
observable as long as it's not been
messed around with you just get a years
a half the mass times the velocity
squared so it's consistent with watching
you but it has huge implications on the
other hand important though this is and
wildly counterintuitive though it might
be it pales in comparison with my second
example which is Dirac this is Dirac's
Memorial in Westminster Abbey and it
contains in the simplest form of the
equation that came from his thinking
which I will now outline in broad
principle Dirac set himself the task of
unifying two things he wanted to take
quantum mechanics which was also a new
thing of 1920's 1930's the recognition
that when you get down to the molecular
atomic level conventional notions of
space and time are also interrupted
there and you instead of describing
things deterministically you have an
equation the Schrodinger equation which
gives you probability distributions for
what particles will do but this was all
otherwise in within the framework of
classical physics and then he wanted to
put it
together with maked with relativity have
a unified framework there that would be
relevant at very high velocities the
equation he got is this clear away the
confusing detail and he written like
this it's not nearly as simple as it
looks
gamma represents a four by four matrix
it's a four by four matrix operating on
the four the vector quantity with four
components which are the three
dimensions of space XY and Z and that I
mention of time and this thing Delta is
a vector operator that differ it tells
you the rate at which in response to
things going on around it each of these
components XYZ and time are changing
it's essentially it's the derivative of
this thing
sy and sy is essentially the Schrodinger
wave equation it's the thing that tells
you the probability distribution of
these four quantities of space-time and
that's the relation between them and or
lots of numerical constants and things
have been factored out of this but it
also has just illustrating its mystical
quantity the thing I began with the
imaginary number managed to get into
this equation and Westminster Abbey but
the thing that's really uh Turley and to
my mind almost unbelievable about this
direct direct of course wanted to
understand now I've got this equation
and it's predicting in effect it's
telling me about the dynamics in
relation to the four dimensions of space
and time of four quantities which are
the other part of the four times four
matrix gamma so what are these four
fundamental particles
two of them are electrons
which everyone's familiar with their
electrons of two kinds those with spins
up those with spins down that their
electrons positive relativistic rest
energy negative charge what are the
other two they've got to be things with
spin up and one with spin up and one
spin down they've got to have negative
rest energy and positive charge and at
first direct hope they might be protons
the electrons companions in the atom but
the protons at 2,000 times heavier the
whole symmetry is destroyed and clearly
wasn't them so what he saw as the way
out of this
I think the boldest leap in science he
said let me assume that the entire
cosmos is such that it is completely
filled with these two missing kinds of
stuff so we can't see it because it's
just there everywhere not doing anything
the only way we can see it is when there
are some holes where a bits missing and
then we'll see the hole because the hole
will have the property of having
negative rest energy positive charge
spin up or spin down and let's call it a
positron the cousin never seen and only
there in this scheme by virtue of the
fact that our entire universe is
permeated by filling up all the
positrons slots and what we're seeing
are the holes it's a theory of holes for
positrons I mean the wildest flight of
mathematics and beauty counterintuitive
but with elegance as a guide and I
imagine if one had lived at that time
and wanting to take a bet on it you
would have got very
odds that this was all nonsense but it
did have a virtue it had a virtue would
actually string theory
today's analog of pursuit of beauty and
trying to understand the world does not
have it had a testable prediction so it
was while guided in this more aesthetic
way it was in the idiom of the
Enlightenment as I think you have bit of
trouble defending string theory as being
in the idiom of the Enlightenment but I
digress
the prediction was it made a specific
prediction a that you might find these
positrons ins holes and also it made
specific predictions about the magnetic
moment of electrons and a couple of
years later in 1932 these were verifying
and pleasingly he got the Nobel Prize
with amazing rapidity I mean it took him
20 years to give Einstein his if they
gave him his next year and he really
deserved it it's interesting that
everyone's heard of Einstein and very
few people have heard of Dirac there's a
wonderful book just been written about
him called the strangest man by Gram
family which I commend Einstein courted
publicity Dirac shunned it and the I
find fascinating not just the two most
brilliant examples of elegance and
imagination but at the same time directs
much better into such different people
okay in conclusion I'd just like to
foreshadow very quickly for because I
know not everybody be able to make it
tomorrow hello my recommendation of
course would be you should record the
debate but um
I'm going to go on a bit about the
aspects of the banking problem and the
way we're going about dealing with it
but what I'm really going to try and
focus on is that many of the questions
that confront us today and specifically
in banking and more generally in some of
other things actually lie outside the
domain of tomatoe of today's science
some of them even though they're
squarely within the idiom of science
like climate change and some of which
are essentially outside the idiom of
science but are still rather atavistic
alee rooted in belief systems and
appeals to Authority and maybe as we
learn more about neuroscience and the
evilly ecology and evolution of behavior
of individuals and communities this will
change but I'm going to take the liberty
of speculating in the concrete context
of banking things about some of these
issues and then more generally in the
way the problems that we face in
tomorrow's world we are not addressing
well as a result of our failure to
understand much of what we need to
understand about ourselves and I think
on that note I should stop because it's
about time
incidentally people who have to leave
should leave uninhibited by the fact
that we're having questions I'd put it
somewhere into I mean I repeat the
question known I talked about special
relativity and then Dirac was working on
reconciling quantum mechanics or special
relativity the question was Einstein
went on to what was I mean it the
special relativity is anchored in the
fact that this velocity of life is
universal constant going on to general
relativity you enter into domains which
are guided much more by an aesthetic
feeling for and I would rank it in that
cut in that sense it's really
significantly higher than special
relativity but still nowhere near the
head Dirac the guts to suggest that the
thing that's missing that you're
predicted and it isn't there is isn't
there because it's everywhere and all
you're going to see the holes that took
courage ah well I mean we start making a
rank ordered list of of the great ah the
requesting again where would I rate
right Maxwell who took a set of
phenomena that for example electricity
and magnetism which people thought were
separate things and Maxwell said let let
me think about this and I can put this
all together into again into a unified
framework where I get a beautiful set of
equations that explain all the knowing
facts about electricity and magnetism
but
also have some interesting suggestions
about applications we make that we
haven't thought of yet I would rank it
very highly
I'm reluctant to get too far in this
game of ranking things this way because
the the Giants upon whose shoulders we
stand did so many remarkable things but
I do up personally because I myself
would be AI incapable of the sorts of
things that directed whereas I'm not
sure that given the right circumstances
and cultural things the systematic
codification that someone like max will
did I can see how you could do it
and that doesn't mean it's better or
worse it's just that I'm more bowled
over by Dirac
I'm going to ask you to start again my
hearings not as good as it could be
there's a really great question and if I
may rephrase it the question is as
building on observations increases which
increasingly showed that the things in
the heavens didn't move round in circles
Ptolemy put together this more
complicated thing which however was
still constrained by the fact that even
though there were circles within circles
everything had to be circles and going a
bit further than that there's a one that
there are lots of things that one does
learn in school which are so firmly
embedded that that's what's always
taught but just don't happen to be true
and one of them is that Copernicus
realized that the earth moved around the
Sun rather than the Sun round the earth
the story is much more complicated what
Copernicus recognize what Ptolemy had
done by putting Epis in what having the
planets move around the Sun the earth
rather than everything move around the
earth and in more-or-less in circles but
then there were little epicycle circles
what he had done was construct for the
Sun round the earth and it doesn't
matter whether it's the Sun round the
earth of the earth round the Sun anyhow
because it's just a relative motion what
he'd constructed with his epicycles was
an ellipse which we actually have
correct a second order in the
eccentricity of the ellipse and what
Copernicus did was not necessarily say
the earth goes round the Sun of the Sun
round the earth because it's relative
motion and both are right or wrong but
what he did he displaced and he made it
more convenient and incurred the wrath
of descended on him by making the Sun at
the center and but the main thing he did
was displace the Sun from the center so
that the earth went in a circle round
was displaced by the distance that he
displaced the Sun from the centre of
Circle and that also just like Ptolemy
gave you an ellipse to second order in
the early eccentricity so there were two
different approximations to what
eventually when Kepler put it all
together and realized that equal arcs
were swept out in equal times and then
uten realized that implied an inverse
square law of gravitation but
nonetheless it was an example of being
trapped not just using beauty as a guide
but being ptolemy being trapped in a
world that was so caught up with rather
dogmatic views about what you meant by
beauty that everything had to be in
circles and Copernicus broke that and in
breaking it because he chose to put the
Sun at the middle of rather he could put
the earth at the center and displaced it
and had the Sun and got all the same
predictions he's misrepresent he did
indeed suggest maybe that the earth
moved around the Sun but because I say
they're moving relatively that's not the
really the big deal its presented
I missed I missed the that sentence do I
really believe that inherently well I'd
never think about the question that way
mathematics is on the one-handed toolkit
but on the other hand I just find it an
aesthetically pleasing toolkit and
that's yeah I mean I am what what
mathematics is and I'd meant to say this
actually on the way through was one of
the bits where I realized I skipped over
something and didn't go back what
mathematics really is it is a tool for
thinking clearly
nothing more nothing less
it's a powerful tool for thinking
analytically dispassionately rigorously
putting down what you've assumed in an
unambiguous way and when you say that it
makes it sound dead boring but it is a
hugely important tool on the other hand
if you take it to the extent that you
think that is the only lock you need
it's the only keyhole you need a key to
then you'll be trapped in what in the
kind of Socratic dialogues that preceded
the invention of the scientific method
where yes you think clearly about what
your ideas mean but then you go on and
try and test it rather than have the
argument with someone who disagrees
about whether your belief system is
better than their belief system so I see
the mathematics itself as I say as a way
of thinking clearly and can be very
pleasing but it's not going to make much
progress
in helping you understand how the world
really works unless you also get your
hands dirty basically I do tend to like
things that I find pleasing and elegant
at much of mathematics but I also like
things that make me go away feeling I
understand something better than I did
before I started and it that
understanding can come in ways that are
really very straightforward well that's
ok too and I guess I'm quirky enough
that I like it even better if the
understanding comes in a way that I
learn something from the journey I come
close yeah well I was going well now in
be I went when I was introduced to I
think you referred to use and abuse of
mathematics and biology um yeah now I'm
happy to expand on that when I first got
into a college II I brought with me I'm
I'm going to talk about some of this
stuff tomorrow at the beginning then
what I did is and it was this is how I'm
going to start tomorrow but it was at a
time when ecology is still a very young
subject smelly hundred years of the
words only about a hundred years old
first fifty years were very descriptive
and it was beginning to acquire a
conceptual base and it had some ideas
rather romantic ideas about the balance
of nature and stuff and I looked at them
and I thought they were a bit silly and
I made a little model that made more
explicit what these ideas were and
showed they were wrong it's very
interesting there were people who tended
to be among the more interesting people
of that day Robert MacArthur at
Princeton particularly who'd just
learned that he had less than a year to
live when I was on the sabbatical
in this country at the Institute of
Advanced Study with John Bacall him
again some of you will have known and I
had just got interested in this stuff
and I had one of my friends had written
to Robert MacArthur and said you might
like to talk and I went and talked to
Robert about it and I was quite an
amazing story it wouldn't happen today
here was this person he'd never met
before and he'd lent he he at the end of
ten minutes he's one of his colleagues
came and called him said you wanted on
the phone then he came back we talked
for about an hour then he said you know
I'm not well I'm going to be dead within
less than a year and I'm tired now I'm
going to go home but he said I'm looking
for my successor at Princeton and I was
hoped to get Jared Diamond but he wants
to stay on the west coast would you be
interested in coming here as professor
in the zoology department absolutely
gobsmacking
yeah and I I I said I was very
flattering but now I'm happy and I won't
going to the story anymore but so the
reaction in short from some of the
people and obviously these are the
people I respected was this is
interesting stuff and it has a part to
play along with then here's a way of
testing it but the majority of people in
ecology in those early days the attitude
was purely hostile many of them were
people and you still get it in areas
I've as you go into newer areas when Roy
and I Anderson and I started going into
infectious disease much of there more of
them reaction was positive but some but
was irrationally hostile and my theory
for some of it and so I shouldn't be
confessing in front of a larger audience
is that many of these people were people
who wanted to do science really
committed to it but they couldn't do
they were no good at math some of us are
good at something some are good at other
things and they felt resented the notion
that math had anything to do with it
because they were doing this kind of
science precisely because you didn't
need any
Malick's that my feeling is that there
are all sorts of different people and
the hardside the so-called hard sciences
are the easy sciences physical sciences
there are invariance principles there
are conservation laws hamiltonians and
stuff it makes it easy the life sciences
are the same basic physical science but
now elaborated into functioning
organisms and it's so much more
complicated that's getting there but
it's a and the most important sciences I
mean I'm on our one of our we have
legislation about climate change which
I'll also talk about briefly in the last
lecture in and what was I going to say
loss I lost the thread there sorry um
yeah oh I'm in the habit of saying for
climate change the most important
science is the behavioral science but
more important than the physics and
stuff and that is got all the
complications of the biological sciences
and then on top of it all in addition to
all that complexity the things you're
philosophizing about here what you're
saying and it affects what they do right
so you you're trying to think about the
banking crisis not only you've got to
try and understand what's going on in it
and what you might do but when you do do
something the people into are going to
react to what you did in ways which are
not necessarily coupled in any rational
way with the dynamics of the system so
there's a big hierarchy I think the hard
sciences are easy biological sciences
are difficult but we're getting there
and the social sciences which is the
most important
for our collective future we're way
behind
