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now I would like to make a number of
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remarks on the relation of mathematics
00:00:09
and physics which are a little more
00:00:11
General the first is that the
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mathematicians only are dealing with the
00:00:17
structure of the reasoning and they do
00:00:19
not really care about what they're
00:00:22
talking they don't even need to know
00:00:25
what they're talking about or as they
00:00:26
themselves say or whether what they say
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is true now explain that if you state
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the axioms you say such and such a so
00:00:35
and such and such a so and such and such
00:00:37
a so what then then the logic can be
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carried out without knowing what the
00:00:43
such and such words
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mean that is if they if the statements
00:00:48
about the axim are true I mean are
00:00:51
carefully formulated and complete enough
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it is not necessary for the man who's
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doing the reasoning to have any
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knowledge of the meaning of these words
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and to be able to deduce in the same
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language new con con new conclusions if
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I use the word triangle in one of the
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axioms there'd be some statement about
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triangles in the conclusion whereas the
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man who's doing the reasoning might not
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know what the triangle is but then I can
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read his thing back and say oh a
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triangle that's just a three side of
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what have you this so and so and so I
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know this new fact in other words
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mathematicians prepare abstract
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reasoning that's ready to be used if you
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will only have a set of axioms about the
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real world but the physicist has meaning
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to all the phrases and there's a very
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important thing that the people who a
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lot of people who study Physics that
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come from mathematics don't appreciate
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the physics is not mathematics and
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Mathematics is not physics one helps the
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other but you have to have some
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understanding of the connection of the
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words with the real world it's necessary
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to at the end to translate what you
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figured out into English into the world
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into the blocks of copper and glass that
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you're going to do the experiments with
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to find out whether the consequences are
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true
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and this is a problem which is not a
00:02:02
problem of mathematics at
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all I've already mentioned one other
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relationship that of course it's obvious
00:02:11
how the mathematical reasonings which
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have been developed are of great power
00:02:14
and use in for
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physicist that the on the other hand
00:02:19
sometimes the physicist reasoning is
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useful for
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mathematicians mathematicians also like
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to make their reasoning as general as
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possible if you say I have a three
00:02:29
dimens space the ordinary space I want
00:02:32
to talk about ordinary space you know
00:02:33
you're in it that you measure distances
00:02:35
and there are three numbers you need to
00:02:36
tell where something is you go breadth
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width and
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height threedimensional space and you
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begin to ask them about theorems then
00:02:44
they say now look if you had a space of
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n dimensions then here are the theorems
00:02:48
well I yeah but I only want the case
00:02:50
three well substitute n equals 3 and
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then it turns
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out then it turns out
00:03:00
that very many of the complicated
00:03:02
theorems they have are much simpler
00:03:03
because it happens to be a special case
00:03:05
now the physicist is always interested
00:03:07
in a special case he's never interested
00:03:10
in the general case he does he's talking
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about
00:03:14
something he's not talking abstractly
00:03:17
about any he knows what he's talking
00:03:19
about he wants to discuss the gravity
00:03:22
law he doesn't want the arbitrary Force
00:03:24
case he wants the gravity and so there
00:03:27
is a certain amount of reducing the
00:03:30
mathematicians have prepared these
00:03:31
things for a wide range of problems
00:03:33
which is very useful and later on it
00:03:35
always turns out that the poor physicist
00:03:37
has to come back and say excuse me when
00:03:38
you wanted to tell me about the four
00:03:53
dimensions
00:03:57
now another item that's interesting in
00:03:59
this
00:04:00
relationship is the question of how to
00:04:02
do new
00:04:04
physics is it important to have a
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feeling a kind of in oh I must mentioned
00:04:09
one other item when you know what it is
00:04:11
you're talking about that these things
00:04:12
are forces and these are masses and this
00:04:14
is inertia and this is so on then you
00:04:16
can use an awful lot of common sense
00:04:19
seat of the pants feeling about the
00:04:20
world you've seen various things you
00:04:22
know more or less how the phenomenon is
00:04:24
going to behave well a poor
00:04:26
mathematician he translates it into
00:04:28
equations and the symbols don't mean
00:04:29
mean anything to him and he has no guide
00:04:31
but precise mathematical rigor and Care
00:04:34
in the argument whereas a physicist who
00:04:36
knows more or less how the answer can go
00:04:38
is going to come out can sort of guess
00:04:39
part way and go right along rather
00:04:41
rapidly the ma the mathematical rigor of
00:04:44
great Precision is not very useful in
00:04:47
the physics nor is the modern attitude
00:04:49
in mathematics to look at axians now
00:04:52
mathematicians can do what they want to
00:04:54
do one should not criticize them because
00:04:55
they are not slaves to physics it is not
00:04:58
necessary that just of course this would
00:05:00
be useful to you they have to do it that
00:05:02
way they can do what they will it's
00:05:05
their own job and if you want something
00:05:08
else then you work it out
00:05:11
yourself the next point is the question
00:05:14
of whether we should guess when we try
00:05:16
to get a new law whether we should use
00:05:18
the seat of the pants feeling and
00:05:20
philosophical principles I don't like a
00:05:22
minimum principle or I do like a minimum
00:05:24
principle or I don't like action of the
00:05:27
distance or I do like action the
00:05:29
question is to what extent models help
00:05:33
and it's a very interesting thing very
00:05:36
often models help and most physics
00:05:38
teachers try to teach how to use these
00:05:39
models and get a good physical feel for
00:05:41
how things are going to work
00:05:43
out but the greatest discoveries it
00:05:46
always turns out abstract away from the
00:05:48
model it never did any good Maxwell's
00:05:50
discovery of electrodynamics was first
00:05:52
made with a lot of imaginary wheels and
00:05:54
idlers and everything else in space if
00:05:56
you got rid of all the idlers and
00:05:57
everything else in Space the thing was
00:06:00
okay dur discovered the correct laws of
00:06:03
of quantum mechanics for relativity
00:06:06
quantum mechanics simply by guessing the
00:06:08
equation and the method of guessing the
00:06:10
equation seems to be a pretty effective
00:06:12
way of guessing new laws this shows
00:06:15
again that mathematics is a deep way of
00:06:18
expressing nature and attempts to
00:06:20
express nature in philosophical
00:06:22
principles or in seed of the pants
00:06:24
mechanical feeling is not an efficient
00:06:27
way
00:06:30
I must say that there is possible and I
00:06:32
know I've often made the hypothesis that
00:06:34
physics ultimately will not require a
00:06:36
mathematical statement that the
00:06:38
Machinery ultimately will be revealed
00:06:40
it's just a Prejudice like one of these
00:06:41
other prejudices it always bothers me
00:06:44
that in spite of all this local business
00:06:47
what goes on in a tiny no even no matter
00:06:50
how tiny a region of space and no matter
00:06:52
how tiny a region of time according to
00:06:54
the laws as we understand them today
00:06:56
takes a Computing machine an infinite
00:06:57
number of logical operations to figure
00:06:59
out now how can all that be going on in
00:07:01
that tiny
00:07:04
space that why should it take an
00:07:06
infinite amount of logic to figure out
00:07:08
what one stinky tiny bit of SpaceTime is
00:07:10
going to
00:07:11
do and so I made the hypothesis often
00:07:14
that the laws are going to turn out to
00:07:17
be in the end simple like the
00:07:19
checkerboard and that all the
00:07:21
complexities is from size but that is of
00:07:24
the same nature as the other
00:07:25
speculations that other people make it
00:07:27
say I like it you don't like it it's not
00:07:28
good to be too prejudiced just about the
00:07:32
thing to
00:07:35
summarize I would use the words of genes
00:07:37
which says that who said that uh the
00:07:40
great architect seems to be a
00:07:44
mathematician and for you who don't know
00:07:48
mathematics it's really quite difficult
00:07:51
to get a real feeling across of to the
00:07:53
beauty of the deepest beauty of
00:07:56
nature CP snow talked about two cultures
00:07:59
I really think that those two cultures
00:08:01
are people who do and people who will do
00:08:04
not have had the Su who people who have
00:08:06
had and people who have not had this
00:08:07
experience of understanding mathematics
00:08:09
well enough to appreciate nature
00:08:12
once it's too bad that it has to be
00:08:15
mathematics and that mathematics for
00:08:16
some people is hard when one of the it's
00:08:18
reputed I don't know if it's true that
00:08:20
when one of the Kings was trying to
00:08:21
learn geometry from ukl he complained
00:08:24
that it was difficult and ukl said that
00:08:26
there's no Royal Road to geometry and
00:08:29
there's no Royal Road it's not the JW we
00:08:33
cannot as people who look at this things
00:08:35
as a physicist cannot convert this s to
00:08:37
any other language you have if you want
00:08:40
to discuss nature to learn about nature
00:08:42
and to appreciate nature it's necessary
00:08:45
to find out the language that she speaks
00:08:47
in she offers her information only in
00:08:50
one form we are not so unhumble as to
00:08:53
say theand that she changed before we
00:08:55
pay any
00:08:57
attention it seems to me
00:09:00
that uh that it's like
00:09:03
the all the intellectual arguments that
00:09:06
you can make would not in one in any way
00:09:09
or very very little will communicate to
00:09:11
deaf ears what music the experience of
00:09:14
Music really is and all the intellectual
00:09:17
arguments in the world will not convince
00:09:19
those of the other culture the
00:09:22
philosophers who tried to teach you by
00:09:24
telling you qualitatively about this
00:09:26
thing me who's trying to describe it to
00:09:28
you is it's not getting across it's
00:09:30
impossible I'm talk we talking to deaf
00:09:33
ears and it's when they it's perhaps
00:09:38
that the horizons are limited which
00:09:40
permit such people to imagine that the
00:09:42
center of the universe of interest is
00:09:44
man
