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.

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