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

Retour la page d'accueil

As I read this paper I realized that I have had the same feeling as Hamming about the "unreasonable effectiveness of mathematics" and asked the same question that he was asking in the paper, i.e., "Why is mathematics so unreasonably effective in the sciences?"

ReplyDeleteI have almost completed the first draft of a proposal for developing and teaching a course or series of courses, entitled Research and Knowledge Generation. Interestingly, within this proposal one section deals with the question, "What is mathematical knowledge?" and within that section a subsection deals with Hamming's question, in slightly different words.

The following is that subsection in first draft form.

Sterling Portwood, csp@hawaii.rr.com

Center for Interdisciplinary Science

9) Why does mathematics happen to be the language of science? Discuss.

Actually this is not just a happenstance. In fact it is hard to imagine that it could be any other way. Mathematics is not a free-form language like English; it is a sequence of logically necessary, connected steps from beginning to end; ultimately a tautology. In fact a discourse in (i.e., a branch of) mathematics is a specific instance of the application of logic; it is a subset of logic which is still logic. Therefore mathematics is, in the final analysis, logic.

Determining the value of an unknown in an algebraic equation is simple, but algebra is not absolutely required to attain the result. One could simply start from scratch and use logic alone, rather than the rules of algebra, and still obtain the answer, but with greater difficulty, and as the complexity of the problem increases, the difficulty of the solution using logic alone goes up exponentially. Hence mathematics does not yield a result which transcends logic; it is simply a short hand composed of rules of manipulation (previously derived from a foundational axiom set) which makes it comparatively easy to arrive at a logical implication or consequence.

So, given an appropriate scientific theory, logic alone could eventually lead to an implication flowing from that scientific theory, but often with great difficulty. Hence, as indicated in Subsection 8 above, if the portion of science under consideration satisfies (i.e., is consistent with) the foundations (i.e., the primitives, the definitions, and the axioms) of the mathematical discourse, the logical connection between the beginning of a scientific calculation and the end (i.e., the prediction, the extension, the inference, etc.) of the mathematical manipulation is solid, like an ax handle can be swung at a tree with confidence that the ax head will follow.

With these understandings, it should be clear and reasonable why mathematics is so "unreasonably" effective in the sciences.