hi
I can see the audience tonight so I can
see also from the side of it which less
time was a blacks lot in front of my
eyes that there must be many of you here
who are not thoroughly familiar with
physics and also a number that are not
too versed in mathematics and I don't
doubt that there are some who know
neither physics nor mathematics very
well that puts a considerable challenge
on a speaker who is going to speak on
the relation of physics and mathematics
a challenge which I however will not
accept I publish say the title of the
talk clear in clear and precise language
and didn't make it sound like it was
something it wasn't it's the relation of
physics and mathematics and if you find
that in some spots it assumes some minor
knowledge of physics or mathematics I
cannot help it this was named in
thinking of the applications of
mathematics to physics it's perfectly
natural that the mathematics will be a
useful when large numbers are involved
in complex situations although for
example if we took a biology the action
of a virus on a bacterium is if you
watch it under the microscope on
mathematical as jiggling virus find some
spot on this odd shaped bacterium and
they're all different shapes and find
some spot maybe it pushes its DNA in and
maybe it doesn't and so forth and yet if
we do the experiment with millions and
billions of bacteria and viruses then we
can learn a great deal about the viruses
by taking averages and working with
large numbers and we can use the
mathematics involved in the averaging we
can see whether the virus is developing
the bacteria some new strains and in
what percentage and so we can study the
genetics the mutation and so forth to
take another more trivial example you
know imagine an enormous board a checker
board to play checkers or draw and
if the board is very large the actual
operation of any one step is not
mathematical it's very simple if it's
mathematical or either goes one side of
the other kind of diagonal or it reaches
and becomes a king and can code
backwards when it reaches the end in
other words the statement the rules are
very simple and do not really involve
any mathematics but you could imagine
that an enormous board with lots and
lots of pieces some analysis of the best
move or good moves or bad news might be
made by a deep kind of reasoning which
would involve somebody having gone off
first and thought about a great great
depth and that becomes mathematics the
substrate for you another example is
switching in computers that you have one
switches either on or off and there's
nothing very mathematical about that
although mathematicians like to start
there with their mathematics but with
all the interconnections and wires to
figure out what a very large system will
do when requires a mathematics and I
would like immediately to say that a
mathematics has its primary application
or as well it has a tremendous
application in physics in a discussion
of the detailed phenomena in complicated
situations granting the fundamental
rules of the game and that is something
which if I were talking only about the
relation of mathematics and physics I
would spend most of my time discussing
but since this is part of a series of
lectures on a character of physical law
I am NOT do not have the time to discuss
the applications of mathematics in
physics to calculating what happens in
complicated situations but we will go
immediately to another question which is
the character of the fundamental law if
we go back to our checker game the
fundamental laws are the rules by which
the checkers move the mathematics may be
applied in a complex situation to figure
out what happens in the next move what's
a good move to make in a complicated set
situation but very little mathematics is
needed in the fundamental simple
character of the basic laws now the
strange thing about physics is that for
the fundamental laws we still need
mathematics for example well I give two
examples one in which we really do not
and one in which we do there's a law in
physics called Faraday
law which says that in an electrolysis
the amount of material which is
deposited is proportional to the current
and to the time that the current is
acting and that means that the amount of
material is proportional to the charge
which goes through the system sounds
very mathematical but what's actually
happening is that electrons going
through the Y each carry one charge and
to take an example a particular example
it may be that if the deposit one atom
requires one electron to come and so the
number of atoms that are deposit is
necessarily proportional to the number
of electrons that flow and thus the
charge goes to the wire so the
mathematically peirong law has as its
basis nothing very deep requiring no
real knowledge of math a night that one
electron is needed for each atom in
order to earth the deposit itself that's
a not a deep that's mathematic I had to
say number one but it's not the kind of
mathematics I'm talking about tonight
now if we take on the other hand
Newton's law for gravitation which has
these aspects which I wrote down last
time just to impress you with the speed
at which mathematical symbols can convey
can carry information we said that the
force of proportional to the product of
the masses of two objects and inversely
as the square of the distance between
them and that bodies react the forces by
changing their speeds or changing their
motions in a direction of the force by
amounts proportional to the force and
inversely proportional to their mass
that sounds that's words all right and I
didn't have to write the equation but
nevertheless it's kind of mathematical
and we would wonder how can this be a
fundamental law how can this planet out
there look what is it dude it looks at
the song and it sees how far away it is
and it decides to calculate on a
internal adding machine the inverse of
the square of the distance and that
tells it how much to move this is
certainly no explanation of the
machinery of gravitation so you might
want to look further and various people
have tried to look further Newton was
originally asked it doesn't mean
anything it doesn't tell assign these as
it tells you how it moves it should be
enough I told you how it moves not one
but people have often unsatisfied
without a mechanism and I would like to
describe one theory which has been
invented for as among others of the type
that you might want that this is the
result of large numbers and that's why
it's mathematical and I give this theory
perhaps you've thought of it yourself
every once a while some student comes
running and he suddenly explains
gravitation suppose that in the world
everywhere there are flying through us
at very high speed a lot of particles
that come equally in all directions
they're just shooting by shooting by
shooting by and once in a while hit
enable body but where we and the Sun are
practically transparent for them nearly
but some hit and so it's not completely
transparent and look what would happen
if the Sun is here and the earth is here
then if the Sun weren't here there would
be particles bombarding from all sides
giving little impulses by the rattle of
these bang bang the few that hit which
would put not shake the earth in any
particular direction because there is
many coming from one side as the other
from top from bottom however when the
Sun is here the particles which are
coming in this direction are partly
absorbed by the Sun because some of them
hit the Sun and don't go through
therefore the number that are coming
from this direction toward the earth is
less than the number that are coming
from the other side because here they
have no opposition from the Sun there
and it's easy to see at the Sun mental
effort that the further the Sun is away
the less in proportion of all of the
particles are being taken out of the
possible directions in which particles
can come solar size appears smaller and
in fact inversely as the square of the
distance so there will therefore be an
impulse toward the Sun on the earth
that's inverse is the square of the
distance and is the result of large
numbers of very simple operations just
just hit one after the other from all
directions and therefore the strangeness
of the mathematical relation will be
very much reduced because the
fundamental operation is much simpler
than calculating the inverse of the
square that this is this machine does
the calculating these particles bounces
only trouble with it is that it doesn't
work for all the reasons every theory
that you make up has to be analyzed
against all the possible consequences
and to see if it predicts anything else
and this predicts something else if the
earth is moving this way more particles
will hit it from the front and from the
back if you're running in the rain more
rain hits you from the front of the face
that in the back of the head because
you're running into the rain and so as
the earth is moving in this direction
it's running into the particles rather
and running away from the ones that are
chasing it from behind so that more
particles hit it from the front and from
the back and there would be of course
also sideways whenever there was any
motion this sideways force would slow
the earth up in the oven and certainly
would not have lasted the at least three
four billion years that has been going
around the Sun so that's the end of that
theory well you say I was a good one
though it got rid of the mathematics for
a while maybe maybe I can invent a
better one and maybe you can because
nobody knows the ultimate butt chewed up
to today from the time of Newton no one
has invented another theoretical
description of the mathematical
machinery behind this law which does
anything else but say the same thing
over a general make the mathematics
harder and at the same time doesn't
produce some wrong phenomena I mean they
have like this model does it but it
produces something which isn't true so
there is no model of the theory of
gravitation today other than the
mathematical form if this were the only
law of this character it would not be so
it would be interesting and rather
annoying but what turns out to be true
is that the more we investigate and the
more laws we find them the deeper we
penetrate nature this disease that every
one of our laws is a purely mathematical
statement in rather complex and abstruse
mathematics this is relatively simple
mathematics it gets more and more
abstruse and more and more difficult as
mathematics and why I haven't the
slightest idea why it is only my purpose
in this lecture to tell you about this
fact in other words it's my purpose in
this lecture to explain really why I
cannot satisfy you if you do not
understand
mathematics to well in trying to explain
nature in any other way it is the burden
of this lecture in fact to just tell you
the fact that is impossible to answer
the really - honestly the challenge of
explaining in a way that a person can
feel the beauties of the laws of nature
without their having some deep
understanding of mathematics I'm sorry
seems to be the case you might tell like
then there's no explanation of the law
at least tell me what the law is why not
tell me in words instead of in the
symbols mathematics is just the language
and I want to be able to translate the
language and in fact I can and with
patience I think I partly did I could go
a little further and explain more
details that this means that if it's
quite as far away the forces one fourth
is much and so on and can convert all
these into words I would be in other
words kind to the laymen as they all sit
hopefully that you will explain them and
various different people are get
different reputations for their skill at
explaining to the laymen in layman's
language these difficult and abstruse
subjects the laymen then searches for
book after book with the hope that he
will avoid the complexity so it
ultimately set in even by the best
exposit for of this type he reads the
things hoping we have one after he finds
as he reads are generally increased
confusion one complicated statement
after the other one difficult to
understand thing after the other all
apparently disconnected from one another
and that becomes a little obscurity
hopes that maybe in some other book
there's some explanation which avoid
which do you mean the man almost made it
you see maybe another fellow makes it
right and I don't think it's possible
because there's another feature
mathematics is not just the language
mathematics is a language plus reasoning
it's like a language plus logic it's
mathematics is a tool for reasoning it's
in fact a big collection of the results
of some person's careful thought and
reasoning by mathematics it is possible
to connect one statement to another for
instance I can say that the force is
directed toward the Sun I can also tell
you
as I did before that the planet moves so
that if I draw a line from the Sun to
the planet and draw another one at some
definite period like three weeks later
the area that swung out by the planet is
exactly the same as it will be the next
three weeks in the next three weeks and
so on as it goes around the Sun now I
can explain both of those statements to
you carefully but I cannot explain why
they're both the same so that if you
don't appreciate the mathematics and you
cannot see that the VAP variety of facts
the enormous apparent complexity of
nature with all its funny laws and rules
each of which have been can't be
explained you are really very closely
interwoven that logic permits you to go
from one or to the other
it may be unbelievable that I can
demonstrate that equal x will be swept
out if the forces are directed toward
the Sun and just to if I may try I will
show you one demonstration to show you
that those two things really are
equivalent and so that you can
appreciate that there's more through
merely the statements of the two laws
that the two laws are connected such
that reasoning alone will bring you from
one to the other and the mathematics is
disorganized reasoning and that it's
good to know how to do that so they will
appreciate the beauty of the
relationship of the statements so I'm
going to prove if I make the
relationship that if the forces are
directed toward the Sun that the equal
areas are swept out in equal times so we
start is a Sun and we imagine that at a
certain time let's say the planet is
here and it's moving in such a way that
let's say one second later a one-hour
pick any time it's a 1 second later it's
moved in such a manner that has gotten
to the position 2 now if the Sun did not
exert any force on it then by a
Galileo's principle it would keep right
on going in a straight line so in the
same interval of time later the next
second it'll move exactly the same
distance in the same straight line to
the position 3 were there no force
all right now first we're going to show
that if there's no force equal areas are
swept out in equal time I remind you
that the area of a triangle is half the
base times the altitude and that the
altitude is the vertical distance to the
base and that if put out if the triangle
is sort of cockeyed there's a name for
it which I forget obtuse obtuse then the
altitude is this vertical height here
see I know about the triangles I just
don't know the names now let us draw the
lines to these two points in the case
that there was no motion whatsoever the
question this doesn't draw very well I'm
not accurate but these two distances are
equal remember question is are these two
areas equal well consider the triangle
made from the Sun and the two points one
and two it's this one
what's its area it's this space
multiplied by this height and what about
the other triangle which is the triangle
in motion from two to three it's this
base times the same altitude the two
triangles have the same altitude and as
I indicated the same base and have the
same area so so far so good
if there were no force from the Sun
equal areas would be swept out in equal
times at two triangles of equal area but
there is a force from the Sun and during
this interval one to two to three
the Sun is pulling and changing the
motion in various directions this way
this way that way to get a good
approximation to that will take the
central position or average position
here and say that the whole effect
during this interval was to change the
motion by some amount in this direction
toward the Sun that means that although
the particles were moving this way and
would have moved this way in the next
second because of the influence of the
Sun the motion is altered by an amount
that's poking in this direction that's
parallel to this exactly parallel these
lines are parallel it's the direction in
which this new motion is the new motion
is a compound of what I wanted to do and
the change that's been endorsed by the
action of the Sun so it doesn't really
end up at position
but rather at position 4 so now we would
like to compare it getting complicated
in the diagram the triangle 2 4 s & 2 3
yes I show you that those are equal
because they have the same base those
two triangles this one here and the one
that happened when we had no force the
one with force and with no force have
the same base and do they have the same
altitude sure because they're included
between parallel lines and so the a this
have the same altitude and thus the area
of the last triangle I drew is the same
as the second one I drew this one free
and that I had proved earlier was the
same as the first one so in the actual
orbital motion of the planet the area of
the first in the first in second and in
the second second are equal so by
reasoning we can see a connection
between the force the fact that the
forces toward the Sun and that the areas
are equal not M genius no this was I
borrowed this from Newton that comes
right out of the Principia diagram and
all the letters are different that's all
because he wrote in Latin these are our
big numerals but incidentally Newton
made his proof geometrical like this and
made all his proofs in his book
geometrical of this type today we don't
use that kind of reasoning we use a kind
of analytic reasoning with symbols which
this requires this kind of reasoning
requires an ingenuity to draw the right
triangles the correct triangles I mean
to notice about the areas and to figure
out how to do it have to be clever but
there have been improvements in the
methods of analysis so that one can be
quite more stupid and I write a much
faster you're much more efficient then
and I want only to show what it looks
like in the notation of the more modern
mathematics where you don't do anything
with right a lot of symbols to figure
out first we would like to talk about
how fast the area changes and we
represent that by area dot and area
changes because of a it's when the
radius is swinging it's the component of
velocity at right angles to the radius
times the radius that tells how fast the
area changes so this is the component of
the radial distance multiplied by the
velocity or rate of
change of the distance now the question
is whether the rate of change of area
itself changes the principle is it's not
supposed to change the rate of change of
area is not supposed to change so we
differentiate so call this again and we
put it'll means a little trick about
putting dots in the right place and that
that's all you have to learn the tricks
no I'm not it's just a series of rules
as people have found out that are very
powerful for such a thing
and this says the component of the
velocity at right angles to the velocity
it is none there is none the velocities
in the same direction as itself and the
acceleration which is this thing the
second derivative or the derivative
alhasan e is the force divided by the
mass so this says that the rate of
change of the rate of change of the area
is the component of force at right
angles to the radius but if the forces
in the direction of the radius as Newton
said then there's no force in at right
angles to the radius and that means that
the change rate of change of area
doesn't change I just want to illustrate
the different kinds of notation now
Newton knew how to do this more or less
the slight different notation but he
wrote everything this way because he
tried to make it possible for people to
read his papers he invented the calculus
which is this kind of mathematics and is
a good illustration of the relation of
mathematics to physics when the problems
and physics get difficult we may often
look to the mathematicians who have
already studied such a thing and have
reasoned about such an item before and
have prepared a line of reasoning for us
to follow on the other hand they may not
have in which case we have to invent our
own line of reasoning which will then
pass back to the mathematicians because
everybody who reasons carefully about
anything
Henri is making a contribution to the
knowledge of what happens when you think
about something and if you abstract it
away and send it to the Department of
Mathematics they put it in the book as a
branch of mathematical
mathematics then is a way of going going
from one set of statements to another
it's evidently useful in physics because
we have all these different ways that we
could speak of things and it permits us
to develop consequences and analyze the
situations and re change the laws in
different ways and to connect all the
various statements so that as a matter
of fact the total amount that a
physicist knows is very little he has
only to remember the rules for getting
from one place to another and he's able
to do that do it then in other words all
of the various statements about equal
times the forces in the direction of the
radius and so on are all interconnected
by reasoning now an interesting question
comes up is there some pattern to it is
there a place to begin fundamental
principles and deduce the whole works or
is there some particular pattern or
order in nature in which we can
understand that these are more
fundamental statements and these are
more consequential space there are two
kinds of ways of looking at mathematics
which for the purpose of this lecture I
will call the Babylonian tradition and
the Greek tradition in Babylonian
schools in mathematics the student would
learn something by doing a large number
of examples until he caught on to the
general rule also a large amount of Jia
Jia matauri for example was known a lot
of properties of circles theorem of
Pythagoras for example formulas for the
areas of cubes and triangles and
everything else and some degree of
argument was available to go from one
thing to another tables of numerical
quantities were available so that you
could solve elaborate equations and so
on everything was prepared for
calculating things out but Euclid
discovered that there was a way in which
all of the theorems of geometry could be
ordered from a set of axioms that were
particularly simple and you're all
familiar without much geometry I'm sure
but the Babylonian attitude was if I
make my my way of talking what I call
Babylonian mathematics is that you know
all these various theorems and many of
the connections in between but you've
never really realized that it could all
come up from a bunch of acts
modern mathematics the most modern
mathematics concentrates on axioms and
demonstrations within a very definite
framework of conventions of what's
acceptable and not acceptable as axioms
for example in geometry it takes
something like Euclid's axioms modified
to be made more perfect and then to show
the deduction of the system for instance
it would not be expected that a theorem
like Pythagoras is that the sum of the
squares of the areas of squares put on
the sides of the triangle will equal the
area of a square on a hypotenuse should
be an axiom on the other hand from
another point of view of method of
geometry that of Descartes the
Pythagorean theorem is an axiom so the
first thing we have to worry about is
that even in mathematics you can't start
in different places because of all these
various theorems are interconnected by
reasoning there isn't any real way to
say well these on the bottom here are
the bottom and these are connected
through logic because if you were told
this one instead or this one you could
also run the logic the other way if you
weren't told that one and work out that
one it's like a bridge with lots of
members and it's over connected if
pieces have dropped out you can
reconnect it another way the
mathematical tradition of today is to
start with some particular ones which
are chosen by come kind of convention to
be axioms and then to build up the
structure from there the babylonian
thing that i'm talking about was not
babylonian but it is to say all i know
happen to know this and i happen to know
that and maybe i know that and i work
everything out from there and nick
tomorrow i forgot that this was true but
i remember that this was true and i
reconstructed again and so on i'm never
quite sure of where I'm supposed to
begin and where I'm supposed to end I
just remember enough all the time so
that has the memory fades and the pieces
fall out by repurchasing back together
again every day the method of starting
from the axioms is not efficient in
obtaining the theorems in working
something out in geometry you're not
very efficient of each time you have to
start back at the action but if you have
to remember a few things in the geometry
you can always get somewhere else it's a
much more efficient to do it the other
way and the what the best axioms are not
exactly the same
not ever the same as the most efficient
way of getting around in the territory
in physics we need the Babylonian method
and not C you could be a Euclidian or
Greek method and I would like to say why
the problem in the Euclidean method is
to make something about the axioms a
little bit more interesting or important
but there the question that we have is
in the case of gravitation is it more
important is it more basic is it more
fundamental is it a better axiom to say
that the forces directed toward the Sun
or to say that equal areas are swept in
equal time well from one point of view
the forces is better because once I
state what the forces are I can deal
with a system with many particles in
which the orbits are no longer ellipses
because of the pull of one on the other
and the theorem about equal areas fails
therefore I think that the force law to
be an axiom instead of the other on the
other hand the principle at equal times
is swept out in equal equal area that
swept out in equal times can be
generalized when there's a system of a
large number of particles to another
theorem which I had prepared to explain
but I see I'm running out of time but
there's another statement which is a
little more general than equal areas in
equal sign well I have to state what it
is it's rather complicated to say and
it's not quite as pretty as this one but
it's obviously the the son of this one I
mean it's the offspring if you look at
all these particles Jupiter Saturn the
Sun and all these things were going
around lots of stars or whatever they
are all interacting with each other and
look at it from far away and projected
on a plane like this picture then
everything everything is moving this
moving this way they're moving that way
and so on then take any point at all at
this point and then calculate how much
each one is changing its area how much
area is being swept out by the radius
every particle and add them all together
but wait those masses which are heavier
count more strongly if this is twice as
heavy as this one than this area it
counts twice as much so
that's doing the sweeping and the total
of all of that is not changing in time
that's the generalization obviously of
the element incidentally the total of
that is called the angular momentum and
this is called the law of conservation
of angular momentum conservation just
means that it doesn't change now one of
the consequences of this is but just to
show what it's good for imagine a lot of
stars falling together to form a nebula
or galaxy as they come closer in if they
were very far out and moving slowly so
there was a little bit of area being
generated but on very long on distances
from the center then if the thing falls
in the distances to the center is
shorter now if all the stars are now
close in the nice radii are smaller and
in order to sweep out the same area they
have to go a lot faster
so as the things come in they swing
swirl around and thus we can lovely
understand the qualitative shape of the
spiral nebulae and also understand in
the same way back the same way way a
skater spins when you start with in
layout as moving slowly and as you pull
the leg in and spins faster because when
the leg is out its contributing when
it's moving slowly a certain amount of
area for a second and then when it comes
in to get the same air you have to go
around faster but I didn't prove it for
the skater the skater uses muscle force
gravity is gret is there a different
force yet it's true for the skater now
we have a problem we can deduce often
from one part of physics like the law of
gravitation a principle which turns out
to be much more valid than the
derivation this doesn't happen in
mathematics that the theorems come out
in places where they not supposed to be
in other words if we were to say that
the postulates of physics were thus law
of gravitation we could deduce the
conservation of angular momentum but
only for gravitation but we discover
experimentally that the conservation of
angular momentum is a much wider thing
now Newton had other PI postulates by
which he could get the more general
conservation law of angular momentum but
Newtonian laws were wrong there's no
forces it's all a lot of baloney the
particles don't have all bits and so on
yes the analogue the exact
transformation of this principle about
the area's the conservation of angular
Menem is true with ionic motions in
quantum mechanics and is still as far as
we can tell today exact so we have these
wide principles would sweep across all
the different laws and if one takes too
seriously is derivation and feels that
this is only valid because this is valid
you cannot understand the
interconnections of the different
branches of physics someday when physics
is complete then maybe with this kind of
argument we know all the laws then we
could start with some axioms and no
doubt somebody will figure out a
particular way of doing it and then all
the do all the deduction will be made
but why we don't know all the laws we
can use some to make guesses of theorems
which extend beyond the proof so in
auditory to understand the physics one
must always have a neat balance and
contain in his head all of the various
propositions and their
interrelationships because the laws
often extend beyond the range of their
deduction this will only have no
importance when all the laws are known
another thing that's interesting in the
relation of mathematics to physics is
this a very strange thing that by
mathematical arguments you can show that
you can start from very many different
apparent starting points and come to the
same thing that's pretty clear if you
have axioms you can use some of the
theorems but actually in the physical
laws are so delicately constructed that
the statements of them have such
qualitatively different character that
is very interesting so if you'll permit
me I'm going to state the law of
gravitation in three different ways all
of which are exactly
that turns out but they sound completely
different one there's a forces between
the objects as described before and each
object when it sees the force on it
accelerates or changes its motion either
at a certain amount per second as I've
described before the regular way I call
it Newton's law now there's a completely
different way that law says that the
force depends on something at a finite
distance away see it has a what we call
non-local quality the force on this
depends on where that one is over there
now you may not like the idea of action
at a distance but it can know what's
going on over there
well then is another way of stating the
laws which are very strange and it's
called the field way of representing the
laws and is so very hard to explain but
I just want to give you some rough idea
of what it's like and it says different
things completely differently that
there's a number at every point in space
I know it's a number it's not a
mechanism it's the trouble with this
whole physics that it must be
mathematical this way there's a number
every point in space here's a number
here's an under thought and the numbers
changing as changes rather when you go
from place to place if an object is
placed at one of these points at
somewhere in space the force on it is in
a direction in which that number I'll
call it the name it's given called a
potential is in a direction which that
potential changes as quick its can and
the force is proportional how fast it
changes as you move
that's one state that's not enough yet
because I have to tell you now how to
determine how the potential varies I
could say the potential varies is one
over the distance from each object but
that's factory action resistance idea
however the forces at a distance but you
can't state the law in another way it
says the following you don't have to
know what's going on anywhere outside of
a little ball if you want to know what's
what the potential is here you tell me
what it is on the surface of any ball no
matter how small you don't have to look
outside you just tell me what it's in
the neighborhood and how much mass there
is in the ball the rule is this that the
potential at the center is equal to the
Pataky average
of the potential on a little ball
surface
- there's constant that's over there and
the other equation divided by twice the
radius of the ball let's suppose the
radius of the ball is it called a and
then multiplied by the mass that's
inside the ball if the ball is small
enough now you see that this flaw is
different than the other one because it
only tells what happens at one point in
terms of what happens very close by
Newton's laws tell what happened at one
time in terms of what happens in MO the
instant if you dip some instance in some
how to work it out but in space it leaps
from place to place but this thing is
both local in time and also local impass
in space because it depends only what's
in the neighborhood and there's another
way of representing that's another way
now there's in a completely different
way that see there's a difference in the
philosophy in qualitative ideas involved
you don't like action in existence you
can get away without it now I'll show
you one which is philosophically the
exact opposite in which there's no
discussion at all about how the thing
works its way from place to place in
which the whole thing is an overall
statement as goes as follows when you
have all the particles around and you
want to know how this one moves from one
place to another you do it as follows
you calculate a certain quantity for you
invent a possible motion that gets from
one given place to some other place that
you're interested in in a given amount
of time say it wants to go from here to
here in an hour
and you want to know by what route it
can get from there to there at an hour
by what curve so what you do is you
calculate a quantity getting the curve
if you try this curve you calculate a
certain number for this quantity I don't
wanted to say what the quantity is but
for those who have heard of these terms
this quantity on this route is the
average of the difference between the
kinetic and potential energy now if you
calculate this quantity for this route
then for another route you'll get of
course different numbers for the answer
but there's one route which gives the
least possible number for that and
that's the route that the particle takes
now we're describing the actual motion
the ellipse by saying something about
the whole curve we have lost the idea of
causality that the particles here it
sees the pull it moves to here point
instead of that in some grand fashion it
smells all the curbs around here and all
of possibility and decides which one to
take but this is an example of the wide
range of beautiful ways of describing
nature and that when people talk that
nature must have causality well you
could talk about it this way nature must
be stated in terms of a minimum
principle well you can talk about it
this way nature must have a local field
view it can do that and so on and the
question is which one is is right now if
these various alternatives are
mathematically not exactly equivalent
and if for certain ones there will be
different consequences than fathers then
it's a variant that's perfectly alright
then because we got only to do the
experiments to find out which way nature
actually chooses to do it mostly people
come along and they argue
philosophically they like this one
better than that one but we have learned
from much experience that all intuitions
about what nature's going to do
philosophically fail it never works one
just has to work out all the
possibilities and just try all the
alternatives now in this particular case
that I'm talking about here these
theories are exactly equivalent the
mathematical consequences and every one
of the different formulations of the
three formulation Newton's laws local
field method and this least this minimum
principle give exactly the same
consequences what do we do then you will
read in all the books that we therefore
cannot decide scientifically on one or
the other that's true they're not
equivalent they're not they are
equivalent
scientifically it is impossible to make
a decision which there's no experiment
the way to distinguish if to all the
consequences is saying psychologically
they're very different in two ways first
philosophically you like them or don't
like training is the only thing you can
do to beat that disease
second psychologically they're different
because they're completely unequivocal
as long as the physics is incomplete and
we're trying to find out the other laws
and understand the other laws then the
different possible formulations give
clues as to what might happen another
circumstance and they've become not
equivalent in a psychologically
suggesting to us the guess as to what
the laws might look like in a more in a
wider situation for instance Einstein
noticed that the law of gravity said
that they he realized that signals
couldn't scrapple propagate faster than
the speed of light for light for
electricity he guessed that it was a
general principle the same print
guessing game as taking this angular
momentum and extending it from one case
where you proved it to the rest of the
universe he guessed that it was true of
everything and he guess that it would be
true of gravitation if the signals can't
go any faster than the speed of light it
turns out that the method of describing
the forces instantaneously is very poor
and in the Einstein generalization or of
gravitation this method of describing
physics is hopelessly inadequate and
enormous ly complicated whereas this one
is neat and simple and so is this one
so we haven't decided between those two
yet in fact it turns out that the
quantum mechanics says that in exactly
as I stated them neither as it is right
but that the fact that a minimum
principle exists is turns out to be a
consequence of the fact that on a small
scale particles obey quantum mechanics
and the fact is is that the best laws at
present understood is really a
combination of the two in which we use
minimum principles plus local force
local laws and the present day believes
that the laws of physics have to have
the local character and also the minimum
principle but we don't really know
so it's this way that if you have a
structure that's only partly accurate
and something is going to fail if you
write it with just the right axioms
maybe only one axiom failed and the rest
remain just changed one little thing but
if you write it with another set of
axiom they all collapse because they all
lean on that but we can't tell ahead of
time without some intuition and
guesswork as to which is the best way to
write it so we find out the new
situation so we must therefore always
keep all of the alternative ways of
looking at the thing in our heads so the
physicists do Babylonian mathematics and
paint our little attention to the
precise reasoning from fixed axioms one
of the amazing characteristics of nature
is this variety of interpretational
schemes which is possible it turns out
that it's only possible because the laws
are just so in special and delicate for
instance that the law is the inverse
square is what permits it to become
local if it with the inverse cube it
couldn't be done this way that the other
end of the equation that the force is
related to the rate of change of the
velocity that's a consequence that
permits this kind of an interval ighting
the laws the minimum principle because
if for instance if the force were
proportional to the rate of change of
position instead of velocity you
couldn't write it in that way but if you
try to modify the laws much you find you
can only write them in a very much fewer
ways I always find that mysterious and I
don't understand the reason why it is
that the laws of physics always seem to
be possible to be expressed in such a
tremendous variety of ways they seem to
be able to get through several wickets
at the same time
now I would like to make a number of
remarks on the relation of mathematics
and physics which a little more general
the first is that the mathematicians
only are dealing with the structure of
the reasoning and they do not really
care about what they're talking they
don't even need to know what they're
talking about or as they themselves say
or whether what they say is true now I
explained that if you state the axioms
is say such-and-such a song in such
constructor so in such and such a so
what then then the logic can be carried
out without knowing what the
such-and-such words mean that is if
there if the statements about the axioms
are true I mean are carefully formulate
and incomplete enough it is not
necessary for the man is doing the
reasoning to have any knowledge of the
meaning of these words and will be able
to deduce in the same language new
concept new conclusions if I use the
word triangle in one of the axiom
there'd be some statement about
triangles in the conclusion whereas the
man who's doing the reason he might not
know what the triangle is but then I can
read his thing back and say oh a
triangle that's just the three side of
what have you that's its own salt and so
I know this new fact in other words
mathematicians prepare abstract
reasoning that's ready to be used if you
will only have a set of axioms about the
real world but the physicist has meaning
to all the phrases and there's a very
important thing that the people who a
lot of people who study physics should
come from mathematics don't appreciate
the physics is not mathematics and
mathematics is not physics one helps the
other but you have to have some
understanding of the connection of the
words with the real world if necessary
to at the end to translate what you
figured out into English into the world
into the blocks of copper and glass that
you're going to do the experiments with
to find out of whether the consequences
are true and this is a problem which is
not a problem of mathematics at all
I've already mentioned one other
relationship that the cause it's obvious
how the mathematical reasonings which
have been developed our great power and
use in for physicists that be on the
other hand sometimes the physicists
reasoning is useful for mathematicians
mathematicians also like to make their
reasoning as general as possible if you
say I have a three-dimensional space of
the ordinary space I want to talk about
ordinary space you know you're in it
that you measure distances and there are
three what numbers you need to tell
where something is you go Brett width
and height three-dimensional space and
you're beginning to ask them about
theorems then they say now look if you
had a space of n dimensions that here
are the theorems
well yeah but I only want the case 3
well substitute N equals 3 and then it
turns out
then it turns out that very many of the
complicated serums they have are much
simpler because it happens to be a
special case
now the physicist is always interested
in a special case
he's never interested in the general
case he does he's talking about
something he's not talking abstractly
about anything he knows what he's
talking about he wants to discuss then
gravity law he doesn't want the
arbitrary force case he wants the
gravity and so there's a certain amount
of reducing because the mathematicians
have prepared these things for a wide
range of problems which is very useful
and later on it always turns out that
the poor physicist has the compactors
excuse me when you wanted to tell me
about this four dimensions
you
now another item that's interesting in
this relationship is the question of how
to do new physics is it important to
have a feeling of kind of into oh I must
mention one other item when you know
what it is you're talking about that
these things are forces and these are
masses and this is inertia and this is
so on then you can use an awful lot of
common-sense seat-of-the-pants feeling
about the world you've seen various
things you know more or less how the
phenomenon is going to behave well the
poor mathematician he translates it into
equations and the symbols don't mean
anything to him and he has no guide but
precise mathematical rigor and care in
the argument whereas the physicist who
knows more or less how the answer can go
is going to come out can sort of get
partway and go right along rather
rapidly they met the mathematical rigor
of great precision is not very useful in
the physics nor is the modern attitude
and mathematics to look at axiom now
mathematicians can do what they want to
do one should not criticize them because
they are not slaves to physics it is not
necessary that just because this we be
useful to you they have to do it that
way they can do what they will it's
their own job and if you want something
else then you work it out yourself
the next point is the question of
whether we should get when we try to get
a new law whether we should use the
seat-of-the-pants feeling and
philosophical principle I don't like a
minimum principle or I do like a minimum
vigil or I don't like action at a
distance or I do like actually the
question is to what extent models help
and it's a very interesting thing very
often models help and most physics
teachers try to teach how to use these
models and get a good physical feel for
how things are going to work out but the
greatest discoveries it always turns out
abstract away from the model it never
did any good Maxwell's discovery of
electrodynamics was first made with a
lot of imaginary wheels on idlers and
everything else in space if you got rid
of all the idlers and everything else in
space the thing was okay
Dirac discovered the correct laws or the
lack of quantum mechanics for relativity
quantum mechanics simply by guessing the
equation and the method of guessing the
equation seems to be a pretty effective
way of guessing new laws this shows
again that mathematics is a deep way of
expressing nature and attempts to
express nature in philosophical
principles or in seat-of-the-pants
mechanical feelings is not an efficient
way I must say that there is possible in
a No I've often made hypothesis that
physics ultimately will not require a
mathematical statement that the
machinery ultimately will be revealed
just to prejudice like one of these
other prejudices it always bothers me
that in spite of all this local business
what goes on in a tiny is naive no
matter how tiny a region of space and no
matter how tiny a region of time
according to the laws as we understand
them today takes the computing machine
an infinite number of logical operations
to figure out now how can all that be
going on in that tiny space that why
should it take an infinite amount of
logic to figure out what one stinky tiny
bit of space-time is going to do and so
I made a hypothesis often that the laws
are going to turn out to be in the end
simple like the checkerboard and that
all the complexity is from size but that
is of the same nature as the other
speculations and other people make it
says I like it you don't like it it's
not good to be too prejudiced about the
thing to summarize I would use the words
of jeans which says that we said that
the great architect seems to be a
mathematician and for you who don't know
mathematics it's really quite difficult
to get a real feeling across up to the
beauty of the deepest beauty of nature
CP snow talked about two cultures I
really think that those two cultures are
people who do and people who will do not
have had this experience of
understanding mathematics well enough
to appreciate nature 1 it's too bad that
it has to be mathematics and mathematics
for some people is hard when one of the
kids reputed I don't know if it's true
that when one of the Kings was trying to
learn geometry from Euclid he complained
that it was difficult and Euclid said
that there's no Royal Road to geometry
and there's no Royal Road it's not the
JA if we cannot as people who look at
this things a physicist cannot convert
this thing to any other language you
have if you want to discuss nature to
learn about nature to appreciate nature
it's necessary to find out the language
that she speaks it she offers her
information only in one form we are not
so on humble as that's a demand that she
change before we pay any attention it
seems to me that that it's like them all
the intellectual arguments that you can
make would not in one in any way or very
very little will communicate to deaf
ears
what music the experience of music
really is and all the intellectual
arguments in the world will not convince
those of the other cultures the
philosophers who tried to teach you by
telling you qualitatively about this
thing me who's trying to describe it to
you which is not getting across because
it's impossible I'm talking talking to
deaf ears and it's when they it's
perhaps that the horizons are limited
which permit such people to imagine that
the center of the universe of interest
is man thank
you