The Unreasonable Effectiveness of Mathematics in the Natural Sciences: "
Prologue. It is evident from the title that this is a philosophical
discussion. I shall not apologize for the philosophy, though I am well
aware that most scientists, engineers, and mathematicians have little
regard for it; instead, I shall give this short prologue to justify the
approach.
Man, so far as we know, has always wondered about himself, the world
around him, and what life is all about. We have many myths from the past
that tell how and why God, or the gods, made man and the universe. These
I shall call theological explanations. They have one principal
characteristic in common-there is little point in asking why things are
the way they are, since we are given mainly a description of the
creation as the gods chose to do it.
Philosophy started when man began to wonder about the world outside of
this theological framework. An early example is the description by the
philosophers that the world is made of earth, fire, water, and air. No
doubt they were told at the time that the gods made things that way and
to stop worrying about it.
From these early attempts to explain things slowly came philosophy as
well as our present science. Not that science explains 'why' things are
as they are-gravitation does not explain why things fall-but science
gives so many details of 'how' that we have the feeling we understand
'why.' Let us be clear about this point; it is by the sea of
interrelated details that science seems to say 'why' the universe is as
it is.
Our main tool for carrying out the long chains of tight reasoning
required by science is mathematics. Indeed, mathematics might be defined
as being the mental tool designed for this purpose. Many people through
the ages have asked the question I am effectively asking in the title,
'Why is mathematics so unreasonably effective?' In asking this we are
merely looking more at the logical side and less at the material side of
what the universe is and how it works.
Mathematicians working in the foundations of mathematics are concerned
mainly with the self-consistency and limitations of the system. They
seem not to concern themselves with why the world apparently admits of a
logical explanation. In a sense I am in the position of the early Greek
philosophers who wondered about the material side, and my answers on the
logical side are probably not much better than theirs were in their
time. But we must begin somewhere and sometime to explain the phenomenon
that the world seems to be organized in a logical pattern that parallels
much of mathematics, that mathematics is the language of science and
engineering.
Once I had organized the main outline, I had then to consider how best
to communicate my ideas and opinions to others. Experience shows that I
am not always successful in this matter. It finally occurred to me that
the following preliminary remarks would help.
In some respects this discussion is highly theoretical. I have to
mention, at least slightly, various theories of the general activity
called mathematics, as well as touch on selected parts of it.
Furthermore, there are various theories of applications. Thus, to some
extent, this leads to a theory of theories. What may surprise you is
that I shall take the experimentalist's approach in discussing things.
Never mind what the theories are supposed to be, or what you think they
should be, or even what the experts in the field assert they are; let us
take the scientific attitude and look at what they are. I am well aware
that much of what I say, especially about the nature of mathematics,
will annoy many mathematicians. My experimental approach is quite
foreign to their mentality and preconceived beliefs. So be it!
The inspiration for this article came from the similarly entitled
article, 'The Unreasonable Effectiveness of Mathematics in the Natural
Sciences'
[1. E. P. Wigner, The unreasonable effectiveness of mathematics in the
natural sciences, Comm. Pure Appl. Math., 13 (Feb. 1960).],
by E. P. Wigner. It will be noticed that I have left out
part of the title, and by those who have already read it that I do not
duplicate much of his material (I do not feel I can improve on his
presentation). On the other hand, I shall spend relatively more time
trying to explain the implied question of the title. But when all my
explanations are over, the residue is still so large as to leave the
question essentially unanswered.
The Effectiveness of Mathematics. In his paper, Wigner gives a large
number of examples of the effectiveness of mathematics in the physical
sciences. Let me, therefore, draw on my own experiences that are closer
to engineering. My first real experience in the use of mathematics to
predict things in the real world was in connection with the design of
atomic bombs during the Second World War. How was it that the numbers we
so patiently computed on the primitive relay computers agreed so well
with what happened on the first test shot at Almagordo? There were, and
could be, no small-scale experiments to check the computations directly.
Later experience with guided missiles showed me that this was not an
isolated phenomenon - constantly what we predict from the manipulation
of mathematical symbols is realized in the real world. Naturally,
working as I did for the Bell System, I did many telephone computations
and other mathematical work on such varied things as traveling wave
tubes, the equalization of television lines, the stability of complex
communication systems, the blocking of calls through a telephone central
office, to name but a few. For glamour, I can cite transistor research,
space flight, and computer design, but almost all of science and
engineering has used extensive mathematical manipulations with
remarkable successes.
Many of you know the story of Maxwell's equations, how to some extent
for reasons of symmetry he put in a certain term, and in time the radio
waves that the theory predicted were found by Hertz. Many other examples
of successfully predicting unknown physical effects from a mathematical
formulation are well known and need not be repeated here.
The fundamental role of invariance is stressed by Wigner. It is basic to
much of mathematics as well as to science. It was the lack of invariance
of Newton's equations (the need for an absolute frame of reference for
velocities) that drove Lorentz, Fitzgerald, Poincare, and Einstein to
the special theory of relativity.
Wigner also observes that the same mathematical concepts turn up in
entirely unexpected connections. For example, the trigonometric
functions which occur in Ptolemy's astronomy turn out to be the
functions which are invariant with respect to translation (time
invariance). They are also the appropriate functions for linear systems.
The enormous usefulness of the same pieces of mathematics in widely
different situations has no rational explanation (as yet).
Furthermore, the simplicity of mathematics has long been held to be the
key to applications in physics. Einstein is the most famous exponent of
this belief. But even in mathematics itself the simplicity is
remarkable, at least to me; the simplest algebraic equations, linear and
quadratic, correspond to the simplest geometric entities, straight
lines, circles, and conics. This makes analytic geometry possible in a
practical way. How can it be that simple mathematics, being after all a
product of the human mind, can be so remarkably useful in so many widely
different situations?
Because of these successes of mathematics there is at present a strong
trend toward making each of the sciences mathematical. It is usually
regarded as a goal to be achieved, if not today, then tomorrow. For this
audience I will stick to physics and astronomy for further examples.
Pythagoras is the first man to be recorded who clearly stated that
'Mathematics is the way to understand the universe.' He said it both
loudly and clearly, 'Number is the measure of all things.'
Kepler is another famous example of this attitude. He passionately
believed that God's handiwork could be understood only through
mathematics. After twenty years of tedious computations, he found his
famous three laws of planetary motion-three comparatively simple
mathematical expressions that described the apparently complex motions
of the planets.
It was Galileo who said, 'The laws of Nature are written in the language
of mathematics.' Newton used the results of both Kepler and Galileo to
deduce the famous Newtonian laws of motion, which together with the law
of gravitation are perhaps the most famous example of the unreasonable
effectiveness of mathematics in science. They not only predicted where
the known planets would be but successfully predicted the positions of
unknown planets, the motions of distant stars, tides, and so forth.
Science is composed of laws which were originally based on a small,
carefully selected set of observations, often not very accurately
measured originally; but the laws have later been found to apply over
much wider ranges of observations and much more accurately than the
original data justified. Not always, to be sure, but often enough to
require explanation.
During my thirty years of practicing mathematics in industry, I often
worried about the predictions I made. From the mathematics that I did in
my office I confidently (at least to others) predicted some future
events-if you do so and so, you will see such and such-and it usually
turned out that I was right. How could the phenomena know what I had
predicted (based on human-made mathematics) so that it could support my
predictions? It is ridiculous to think that is the way things go. No, it
is that mathematics provides, somehow, a reliable model for much of what
happens in the universe. And since I am able to do only comparatively
simple mathematics, how can it be that simple mathematics suffices to
predict so much?
I could go on citing more examples illustrating the unreasonable
effectiveness of mathematics, but it would only be boring. Indeed, I
suspect that many of you know examples that I do not. Let me, therefore,
assume that you grant me a very long list of successes, many of them as
spectacular as the prediction of a new planet, of a new physical
phenomenon, of a new artifact. With limited time, I want to spend it
attempting to do what I think Wigner evaded-to give at least some
partial answers to the implied question of the title.
What is Mathematics? Having looked at the effectiveness of mathematics,
we need to look at the question,'What is Mathematics?' This is the title
of a famous book by Courant and Robbins
[2. R. Courant and H. Robbins, What Is Mathematics? Oxford University
Press, 1941.].
In it they do not attempt
to give a formal definition, rather they are content to show what
mathematics is by giving many examples. Similarly, I shall not give a
comprehensive definition. But I will come closer than they did to
discussing certain salient features of mathematics as I see them.
Perhaps the best way to approach the question of what mathematics is, is
to start at the beginning. In the far distant prehistoric past, where we
must look for the beginnings of mathematics, there were already four
major faces of mathematics. First, there was the ability to carry on the
long chains of close reasoning that to this day characterize much of
mathematics. Second, there was geometry, leading through the concept of
continuity to topology and beyond. Third, there was number, leading to
arithmetic, algebra, and beyond. Finally there was artistic taste, which
plays so large a role in modern mathematics. There are, of course, many
different kinds of beauty in mathematics. In number theory it seems to
be mainly the beauty of the almost infinite detail; in abstract algebra
the beauty is mainly in the generality. Various areas of mathematics
thus have various standards of aesthetics.
The earliest history of mathematics must, of course, be all speculation,
since there is not now, nor does there ever seem likely to be, any
actual, convincing evidence. It seems, however, that in the very
foundations of primitive life there was built in, for survival purposes
if for nothing else, an understanding of cause and effect. Once this
trait is built up beyond a single observation to a sequence of, 'If
this, then that, and then it follows still further that . . . ,' we are
on the path of the first feature of mathematics I mentioned, long chains
of close reasoning. But it is hard for me to see how simple Darwinian
survival of the fittest would select for the ability to do the long
chains that mathematics and science seem to require.
Geometry seems to have arisen from the problems of decorating the human
body for various purposes, such as religious rites, social affairs, and
attracting the opposite sex, as well as from the problems of decorating
the surfaces of walls, pots, utensils and clothing. This also implies
the fourth aspect I mentioned, aesthetic taste, and this is one of the
deep foundations of mathematics. Most textbooks repeat the Greeks and
say that geometry arose from the needs of the Egyptians to survey the
land after each flooding by the Nile River, but I attribute much more to
aesthetics than do most historians of mathematics and correspondingly
less to immediately utility.
The third aspect of mathematics, numbers, arose from counting. So basic
are numbers that a famous mathematician once said, 'God made the
integers, man did the rest'
[3. L. Kronecker, Item 1634. in On Mathematics and Mathematicians, by R E
Moritz.].
The integers seem to us to be so
fundamental that we expect to find them wherever we find intelligent
life in the universe. I have tried, with little success, to get some of
my friends to understand my amazement that the abstraction of integers
for counting is both possible and useful. Is it not remarkable that 6
sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13
stones? Is it not a miracle that the universe is so constructed that
such a simple abstraction as a number is possible? To me this is one of
the strongest examples of the unreasonable effectiveness of mathematics.
Indeed, l find it both strange and unexplainable.
In the development of numbers, we next come to the fact that these
counting numbers, the integers, were used successfully in measuring how
many times a standard length can be used to exhaust the desired length
that is being measured. But it must have soon happened, comparatively
speaking, that a whole number of units did not exactly fit the length
being measured, and the measurers were driven to the fractions-the extra
piece that was left over was used to measure the standard length.
Fractions are not counting numbers; they are measuring numbers. Because
of their common use in measuring, the fractions were, by a suitable
extension of ideas, soon found to obey the same rules for manipulations
as did the integers, with the added benefit that they made division
possible in all cases (I have not yet come to the number zero). Some
acquaintance with the fractions soon reveals that between any two
fractions you can put as many more as you please and that in some sense
they are homogeneously dense everywhere. But when we extend the concept
of number to include the fractions, we have to give up the idea of the
next number,
This brings us again to Pythagoras, who is reputed to be the first man
to prove that the diagonal of a square and the side of the square have
no common measure-that they are irrationally related. This observation
apparently produced a profound upheaval in Greek: mathematics. Up to
that time the discrete number system and the continuous geometry
flourished side by side with little conflict. The crisis of
incommensurability tripped off the Euclidean approach to mathematics. It
is a curious fact that the early Greeks attempted to make mathematics
rigorous by replacing the uncertainties of numbers by what they felt was
the more certain geometry (due to Eudoxus). It was a major event to
Euclid, and as a result you find in The Elements
[4. Euclid, Euclid's Elements, T. E. Heath, Dover Publications, New York,
1956.]
a lot of what we
now consider number theory and algebra cast in the form of geometry.
Opposed to the early Greeks, who doubted the existence of the real
number system, we have decided that there should be a number that
measures the length of the diagonal of a unit square (though we need not
do so), and that is more or less how we extended the rational number
system to include the algebraic numbers. It was the simple desire to
measure lengths that did it. How can anyone deny that there is a number
to measure the length of any straight line segment?
The algebraic numbers, which are roots of polynomials with integer,
fractional, and, as was later proved, even algebraic numbers as
coefficients, were soon under control by simply extending the same
operations that were used on the simpler system of numbers.
However, the measurement of the circumference of a circle with respect
to its diameter soon forced us to consider the ratio called pi. This is
not an algebraic number, since no linear combination of the power of pi
with integer coefficients will exactly vanish. One length, the
circumference, being a curved line, and the other length, the diameter,
being a straight line, make the existence of the ratio less certain than
is the ratio of the diagonal of a square to its side; but since it seems
that there ought to be such a number, the transcendental numbers
gradually got into the number system. Thus by a further suitable
extension of the earlier ideas of numbers, the transcendental numbers
were admitted consistently into the number system, though few students
are at all comfortable with the technical apparatus we conventionally
use to show the consistency.
Further tinkering with the number system brought both the number zero
and the negative numbers. This time the extension required that we
abandon the division for the single number zero. This seems to round out
the real number system for us (as long as we confine ourselves to the
process of taking limits of sequences of numbers and do not admit still
further operations) -not that we have to this day a firm, logical,
simple, foundation for them; but they say that familiarity breeds
contempt, and we are all more or less familiar with the real number
system. Very few of us in our saner moments believe that the particular
postulates that some logicians have dreamed up create the numbers - no,
most of us believe that the real numbers are simply there and that it
has been an interesting, amusing, and important game to try to find a
nice set of postulates to account for them. But let us not confuse
ourselves-Zeno's paradoxes are still, even after 2,000 years, too fresh
in our minds to delude ourselves that we understand all that we wish we
did about the relationship between the discrete number system and the
continuous line we want to model. We know, from nonstandard analysis if
from no other place, that logicians can make postulates that put still
further entities on the real line, but so far few of us have wanted to
go down that path. It is only fair to mention that there are some
mathematicians who doubt the existence of the conventional real number
system. A few computer theoreticians admit the existense of only 'the
computable numbers.'
The next step in the discussion is the complex number system. As I read
history, it was Cardan who was the first to understand them in any real
sense. In his The Great Art or Rules of Algebra
[5. G. Cardano, The Great Art or Rules of Algebra, transl. by T. R.
Witmer, MIT Press, 1968, pp. 219-220]
he says, 'Putting
aside the mental tortures involved multiply (5 + sqrt 15) by (5 - sqrt
-15) making 25-(-15) ....' Thus he clearly recognized that the same
formal operations on the symbols for complex numbers would give
meaningful results. In this way the real number system was gradually
extended to the complex number system, except that this time the
extension required giving up the property of ordering the numbers-the
complex numbers cannot be ordered in the usual sense.
Cauchy was apparently led to the theory of complex variables by the
problem of integrating real functions along the real line. He found that
by bending the path of integration into the complex plane he could solve
real integration problems.
A few years ago I had the pleasure of teaching a course in complex
variables. As always happens when I become involved in the topic, I
again came away with the feeling that 'God made the universe out of
complex numbers.' Clearly, they play a central role in quantum
mechanics. They are a natural tool in many other areas of application,
such as electric circuits, fields, and so on.
To summarize, from simple counting using the God-given integers, we made
various extensions of the ideas of numbers to include more things.
Sometimes the extensions were made for what amounted to aesthetic
reasons, and often we gave up some property of the earlier number
system. Thus we came to a number system that is unreasonably effective
even in mathematics itself; witness the way we have solved many number
theory problems of the original highly discrete counting system by using
a complex variable.
From the above we see that one of the main strands of mathematics is the
extension, the generalization, the abstraction - they are all more or
less the same thing-of well-known concepts to new situations. But note
that in the very process the definitions themselves are subtly altered.
Therefore, what is not so widely recognized, old proofs of theorems may
become false proofs. The old proofs no longer cover the newly defined
things. The miracle is that almost always the theorems are still true;
it is merely a matter of fixing up the proofs. The classic example of
this fixing up is Euclid's The Elements [4]. We have found it necessary
to add quite a few new postulates (or axioms, if you wish, since we no
longer care to distinguish between them) in order to meet current
standards of proof. Yet how does it happen that no theorem in all the
thirteen books is now false? Not one theorem has been found to be false,
though often the proofs given by Euclid seem now to be false. And this
phenomenon is not confined to the past. It is claimed that an ex-editor
of Mathematical Reviews once said that over half of the new theorems
published these days are essentially true though the published proofs
are false. How can this be if mathematics is the rigorous deduction of
theorems from assumed postulates and earlier results? Well, it is
obvious to anyone who is not blinded by authority that mathematics is
not what the elementary teachers said it was. It is clearly something
else.
What is this 'else'? Once you start to look you find that if you were
confined to the axioms and postulates then you could deduce very little.
The first major step is to introduce new concepts derived from the
assumptions, concepts such as triangles. The search for proper concepts
and definitions is one of the main features of doing great mathematics.
While on the topic of proofs, classical geometry begins with the theorem
and tries to find a proof. Apparently it was only in the 1850's or so
that it was clearly recognized that the opposite approach is also valid
(it must have been occasionally used before then). Often it is the proof
that generates the theorem. We see what we can prove and then examine
the proof to see what we have proved! These are often called 'proof
generated theorems'
[6. Imre Lakatos, Proofs and Refutations; Cambridge University Press,
1976, p. 33.].
A classic example is the concept of uniform
convergence. Cauchy had proved that a convergent series of terms, each
of which is continuous, converges to a continuous function. At the same
time there were known to be Fourier series of continuous functions that
converged to a discontinuous limit. By a careful examination of Cauchy's
proof, the error was found and fixed up by changing the hypothesis of
the theorem to read, 'a uniformly convergent series.'
More recently, we have had an intense study of what is called the
foundations of mathematics-which in my opinion should be regarded as the
top battlements of mathematics and not the foundations. It is an
interesting field, but the main results of mathematics are impervious to
what is found there-we simply will not abandon much of mathematics no
matter how illogical it is made to appear by research in the
foundations.
I hope that I have shown that mathematics is not the thing it is often
assumed to be, that mathematics is constantly changing and hence even if
I did succeed in defining it today the definition would not be
appropriate tomorrow. Similarly with the idea of rigor-we have a
changing standard. The dominant attitude in science is that we are not
the center of the universe, that we are not uniquely placed, etc., and
similarly it is difficult for me to believe that we have now reached the
ultimate of rigor. Thus we cannot be sure of the current proofs of our
theorems. Indeed it seems to me:
The Postulates of Mathematics Were Not on the Stone Tablets that Moses
Brought Down from Mt. Sinai.
It is necessary to emphasize this. We begin with a vague concept in our
minds, then we create various sets of postulates, and gradually we
settle down to one particular set. In the rigorous postulational
approach the original concept is now replaced by what the postulates
define. This makes further evolution of the concept rather difficult and
as a result tends to slow down the evolution of mathematics. It is not
that the postulation approach is wrong, only that its arbitrariness
should be clearly recognized, and we should be prepared to change
postulates when the need becomes apparent.
Mathematics has been made by man and therefore is apt to be altered
rather continuously by him. Perhaps the original sources of mathematics
were forced on us, but as in the example I have used we see that in the
development of so simple a concept as number we have made choices for
the extensions that were only partly controlled by necessity and often,
it seems to me, more by aesthetics. We have tried to make mathematics a
consistent, beautiful thing, and by so doing we have had an amazing
number of successful applications to the real world.
The idea that theorems follow from the postulates does not correspond to
simple observation. If the Pythagorean theorem were found to not follow
from the postulates, we would again search for a way to alter the
postulates until it was true. Euclid's postulates came from the
Pythagorean theorem, not the other way. For over thirty years I have
been making the remark that if you came into my office and showed me a
proof that Cauchy's theorem was false I would be very interested, but I
believe that in the final analysis we would alter the assumptions until
the theorem was true. Thus there are many results in mathematics that
are independent of the assumptions and the proof.
How do we decide in a 'crisis' what parts of mathematics to keep and
what parts to abandon? Usefulness is one main criterion, but often it is
usefulness in creating more mathematics rather than in the applications
to the real world! So much for my discussion of mathematics.
Some Partial Explanations. I will arrange my explanations of the
unreasonable effectiveness of mathematics under four headings.
1. We see what we look for. No one is surprised if after putting on blue
tinted glasses the world appears bluish. I propose to show some examples
of how much this is true in current science. To do this I am again going
to violate a lot of widely, passionately held beliefs. But hear me out.
I picked the example of scientists in the earlier part for a good
reason. Pythagoras is to my mind the first great physicist. It was he
who found that we live in what the mathematicians call L2-the sum of the
squares of the two sides of a right triangle gives the square of the
hypotenuse. As I said before, this is not a result of the postulates of
geometry-this is one of the results that shaped the postulates.
Let us next consider Galileo. Not too long ago I was trying to put
myself in Galileo's shoes, as it were, so that I might feel how he came
to discover the law of falling bodies. I try to do this kind of thing so
that I can learn to think like the masters did-I deliberately try to
think as they might have done.
Well, Galileo was a well-educated man and a master of scholastic
arguments. He well knew how to argue the number of angels on the head of
a pin, how to argue both sides of any question. He was trained in these
arts far better than any of us these days. I picture him sitting one day
with a light and a heavy ball, one in each hand, and tossing them
gently. He says, hefting them, 'It is obvious to anyone that heavy
objects fall faster than light ones-and, anyway, Aristotle says so.'
'But suppose,' he says to himself, having that kind of a mind, 'that in
falling the body broke into two pieces. Of course the two pieces would
immediately slow down to their appropriate speeds. But suppose further
that one piece happened to touch the other one. Would they now be one
piece and both speed up? Suppose I tied the two pieces together. How
tightly must I do it to make them one piece? A light string? A rope?
Glue? When are two pieces one?'
The more he thought about it-and the more you think about it-the more
unreasonable becomes the question of when two bodies are one. There is
simply no reasonable answer to the question of how a body knows how
heavy it is-if it is one piece, or two, or many. Since falling bodies do
something, the only possible thing is that they all fall at the same
speed-unless interfered with by other forces. There's nothing else they
can do. He may have later made some experiments, but I strongly suspect
that something like what I imagined actually happened. I later found a
similar story in a book by Polya
[7. G. Polya, Mathematical Methods in Science, MAA, 1963, pp. 83-85.].
Galileo found his law not by
experimenting but by simple, plain thinking, by scholastic reasoning.
I know that the textbooks often present the falling body law as an
experimental observation; I am claiming that it is a logical law, a
consequence of how we tend to think.
Newton, as you read in books, deduced the inverse square law from
Kepler's laws, though they often present it the other way; from the
inverse square law the textbooks deduce Kepler's laws. But if you
believe in anything like the conservation of energy and think that we
live in a three-dimensional Euclidean space, then how else could a
symmetric central-force field fall off? Measurements of the exponent by
doing experiments are to a great extent attempts to find out if we live
in a Euclidean space, and not a test of the inverse square law at all.
But if you do not like these two examples, let me turn to the most
highly touted law of recent times, the uncertainty principle. It happens
that recently I became involved in writing a book on Digital Filters
[8. R. W. Hamming, Digital Filters, Prentice-Hall, Englewood Cliffs, NJ.,
1977.]
when I knew very little about the topic. As a result I early asked the
question, 'Why should I do all the analysis in terms of Fourier
integrals? Why are they the natural tools for the problem?' I soon found
out, as many of you already know, that the eigenfunctions of translation
are the complex exponentials. If you want time invariance, and certainly
physicists and engineers do (so that an experiment done today or
tomorrow will give the same results), then you are led to these
functions. Similarly, if you believe in linearity then they are again
the eigenfunctions. In quantum mechanics the quantum states are
absolutely additive; they are not just a convenient linear
approximation. Thus the trigonometric functions are the eigenfunctions
one needs in both digital filter theory and quantum mechanics, to name
but two places.
Now when you use these eigenfunctions you are naturally led to
representing various functions, first as a countable number and then as
a non-countable number of them-namely, the Fourier series and the
Fourier integral. Well, it is a theorem in the theory of Fourier
integrals that the variability of the function multiplied by the
variability of its transform exceeds a fixed constant, in one notation
l/2pi. This says to me that in any linear, time invariant system you
must find an uncertainty principle. The size of Planck's constant is a
matter of the detailed identification of the variables with integrals,
but the inequality must occur.
As another example of what has often been thought to be a physical
discovery but which turns out to have been put in there by ourselves, I
turn to the well-known fact that the distribution of physical constants
is not uniform; rather the probability of a random physical constant
having a leading digit of 1. 2, or 3 is approximately 60%, and of course
the leading digits of 5, 6, 7, 8, and 9 occur in total only about 40% of
the time. This distribution applies to many types of numbers, including
the distribution of the coefficients of a power series having only one
singularity on the circle of convergence. A close examination of this
phenomenon shows that it is mainly an artifact of the way we use
numbers.
Having given four widely different examples of nontrivial situations
where it turns out that the original phenomenon arises from the
mathematical tools we use and not from the real world, I am ready to
strongly suggest that a lot of what we see comes from the glasses we put
on. Of course this goes against much of what you have been taught, but
consider the arguments carefully. You can say that it was the experiment
that forced the model on us, but I suggest that the more you think about
the four examples the more uncomfortable you are apt to become. They are
not arbitrary theories that I have selected, but ones which are central
to physics,
In recent years it was Einstein who most loudly proclaimed the
simplicity of the laws of physics, who used mathematics so exclusively
as to be popularly known as a mathematician. When examining his special
theory of relativity paper
[9. G. Holton Thematic Origins of Scientific Thought, Kepler to Einstein,
Harvard University Press, 1973.]
one has the feeling that one is dealing
with a scholastic philosopher's approach. He knew in advance what the
theory should look like. and he explored the theories with mathematical
tools, not actual experiments. He was so confident of the rightness of
the relativity theories that, when experiments were done to check them,
he was not much interested in the outcomes, saying that they had to come
out that way or else the experiments were wrong. And many people believe
that the two relativity theories rest more on philosophical grounds than
on actual experiments.
Thus my first answer to the implied question about the unreasonable
effectiveness of mathematics is that we approach the situations with an
intellectual apparatus so that we can only find what we do in many
cases. It is both that simple, and that awful. What we were taught about
the basis of science being experiments in the real world is only
partially true. Eddington went further than this; he claimed that a
sufficiently wise mind could deduce all of physics. I am only suggesting
that a surprising amount can be so deduced. Eddington gave a lovely
parable to illustrate this point. He said, 'Some men went fishing in the
sea with a net, and upon examining what they caught they concluded that
there was a minimum size to the fish in the sea.'
2. We select the kind of mathematics to use. Mathematics does not always
work. When we found that scalars did not work for forces, we invented a
new mathematics, vectors. And going further we have invented tensors. In
a book I have recently written
[10. R. W. Hamming, Coding and Information Theory, Prentice-Hall,
Englewood Cliffs, NJ., 1980.]
conventional integers are used for
labels, and real numbers are used for probabilities; but otherwise all
the arithmetic and algebra that occurs in the book, and there is a lot
of both, has the rule that
1+1=0.
Thus my second explanation is that we select the mathematics to fit the
situation, and it is simply not true that the same mathematics works
every place.
3. Science in fact answers comparatively few problems. We have the
illusion that science has answers to most of our questions, but this is
not so. From the earliest of times man must have pondered over what
Truth, Beauty, and Justice are. But so far as I can see science has
contributed nothing to the answers, nor does it seem to me that science
will do much in the near future. So long as we use a mathematics in
which the whole is the sum of the parts we are not likely to have
mathematics as a major tool in examining these famous three questions.
Indeed, to generalize, almost all of our experiences in this world do
not fall under the domain of science or mathematics. Furthermore, we
know (at least we think we do) that from Godel's theorem there are
definite limits to what pure logical manipulation of symbols can do,
there are limits to the domain of mathematics. It has been an act of
faith on the part of scientists that the world can be explained in the
simple terms that mathematics handles. When you consider how much
science has not answered then you see that our successes are not so
impressive as they might otherwise appear.
4. The evolution of man provided the model. I have already touched on
the matter of the evolution of man. I remarked that in the earliest
forms of life there must have been the seeds of our current ability to
create and follow long chains of close reasoning. Some people
[11. H. Mohr, Structure and Significance of Science, Springer-Verlag,
1977.] have
further claimed that Darwinian evolution would naturally select for
survival those competing forms of life which had the best models of
reality in their minds-'best' meaning best for surviving and
propagating. There is no doubt that there is some truth in this. We
find, for example, that we can cope with thinking about the world when
it is of comparable size to ourselves and our raw unaided senses, but
that when we go to the very small or the very large then our thinking
has great trouble. We seem not to be able to think appropriately about
the extremes beyond normal size.
Just as there are odors that dogs can smell and we cannot, as well as
sounds that dogs can hear and we cannot, so too there are wavelengths of
light we cannot see and flavors we cannot taste. Why then, given our
brains wired the way they are, does the remark 'Perhaps there are
thoughts we cannot think,' surprise you? Evolution, so far, may possibly
have blocked us from being able to think in some directions; there could
be unthinkable thoughts.
If you recall that modern science is only about 400 years old, and that
there have been from 3 to 5 generations per century, then there have
been at most 20 generations since Newton and Galileo. If you pick 4,000
years for the age of science, generally, then you get an upper bound of
200 generations. Considering the effects of evolution we are looking for
via selection of small chance variations, it does not seem to me that
evolution can explain more than a small part of the unreasonable
effectiveness of mathematics.
Conclusion. From all of this I am forced to conclude both that
mathematics is unreasonably effective and that all of the explanations I
have given when added together simply are not enough to explain what I
set out to account for. I think that we-meaning you, mainly-must
continue to try to explain why the logical side of science-meaning
mathematics, mainly-is the proper tool for exploring the universe as we
perceive it at present. I suspect that my explanations are hardly as
good as those of the early Greeks, who said for the material side of the
question that the nature of the universe is earth, fire, water, and air.
The logical side of the nature of the universe requires further
exploration.
I (Larry Frazier, who (with R. Hamming's permission) scanned this and
put it online) was pleased to note that 58 people visited this essay in
a recent 2-month period. I assume most of you are finding this from a
pointer in the Gutenberg Project hierarchy.
On the other hand, I feel like thousands of people should be reading
this. It is the most profound essay I have seen regarding philosophy of
science; important, significant, in fact, for our whole understanding of
thought, of knowing, or reality.
Drop me a note if you have any comments. Larry Frazier
Merci W. Cooper et Larry Frazier
Histoire et philosophie des mathmatiques
Le quasi-empirisme en philosophie des mathématiques. Une presentation
Liens mathmatiques en relation indirecte avec le quasi-empirisme
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