Outer Limits of Reason (51 page)

In fact, much of the world around us does not fit ever so snugly into the world of mathematics. The beautiful Brooklyn tree outside my window cannot be described by any figure in any geometry textbook. Although 1 + 1 = 2, it is not true that if you take one heap and combine it with another heap you will get two heaps. Rather, you will get only one heap. Years ago I had to buy a package of size 4 diapers. The store was out of that size so instead I bought two packages of size 2 diapers. Needless to say, this did not work well. From this we can conclude that 2 × 2 = 4 does not work in the important realm of diapers.

Another demonstration of the disconnect between the realm of mathematics and the realm of science is that there are vast tracts of mathematics that never get applied to the physical world. Some parts of number theory and set theory have remained “pure” and have never been applied. In fact, I would imagine that the majority of pure mathematical papers never get applied to the real world. Rather than saying that earlier mathematics mysteriously gets used in later physics, say that a later physicist might
choose some parts
of earlier mathematics to describe some of the phenomena she is trying to describe. For example, Kepler used some of the earlier work of Apollonius. But Kepler ignored many other writings of Apollonius. Kepler chose what he needed. It is not that mathematicians create exactly what will be used by physicists. Rather, mathematicians create an immense amount of mathematics and only some of it is chosen by the physicists. This makes much of the mystery go away.

This solution to the unreasonable effectiveness—that the connection between mathematics and science is tenuous—is somewhat problematic. It is true that there are many branches of science whose dependence on mathematics is not direct. Nevertheless, these branches of science are based on other branches that do depend on mathematics. For the most part, sociology is not based on mathematics
³⁸
. But sociology depends on psychology which in turn depends on neurology and cognitive science. These disciplines are closely related to neurochemistry and computer science, which depend heavily on mathematics. While sociologists might not need to learn a lot of mathematics to practice their trade, if they wanted to understand the foundations of sociology, they would have to study much mathematics. As of now, our mathematics is not complicated enough to deal with all the complexities of sociology. Perhaps in the distant future, such mathematics would be possible.
³⁹
Perhaps not. So while many parts of science do not use mathematics, the parts of science that do use mathematics, in a sense,
generate
the foundations of all of science. And for those generating parts, Wigner's mystery remains.

Mathematics Comes from Physics

Probably the most popular answer that researchers give to explain the mystery of Wigner's unreasonable effectiveness is that mathematics comes from observing the physical world. It is not mysterious that we can describe the physical world with mathematics since it was in that same physical world where we learned mathematics.

For example, a child learns that two apples plus three apples equals five apples by looking at sets of apples. She also observes that when two sticks and three sticks are combined they form five sticks. By seeing this over and over, human beings abstract out that two plus three is five, or in symbols, 2 + 3 = 5. This gives human beings the beginnings of the addition operation. This operation is going to work in many places in the physical world. Similarly, many mathematical objects and operations come from seeing different phenomena in the physical world. It is no wonder that these same mathematical objects and operations are used in describing the physical world.

Let us look at an example in a little more depth. If we see 7 boxes and each box has 8 red marbles and 3 blue marbles, then we can just multiply the 7 boxes times 11, the number of total marbles in each box. At the same time we can find the sum total of all marbles by looking at the sum of 7 times 8 red marbles and 7 times 3 blue marbles. After seeing many similar counting arguments we can formalize this symbolically as

7 × (8 + 3) = 7 × 8 + 7 × 3.

This statement is more abstract since it is not about red and blue marbles anymore. It could be about dogs and cats, or boys and girls. We have abstracted out the original content of the statement. Once we see similar rules for many different numbers, mathematicians have further abstracted this statement to

a
× (
b
+
c
) =
a
×
b
+
a
×
c
.

This rule has nothing to do with 7, 8, 3, or any particular number. It is simply a fact that multiplication “distributes” over addition and is true for any numbers. Now, with this rule in hand, we might think of it as a statement of pure mathematics or we might apply this rule to any part of the universe. The fact that this rule can be applied is not mysterious since it was formed by looking at the physical world. Notice that the rule

a
+ (
b
×
c
) = (
a
+
b
) × (
a
+
c
)

will
not
be applied to the physical world simply because this rule—that addition distributes over multiplication—was not seen to be true in the physical world. Rules and mathematical operations that are commonly experienced will be found in the universe and rules that are not seen or experienced will not be found. This must be true.

Let's analyze what happened with our distributive rule. There was a certain phenomenon in the physical world that was observed about marbles. A human being observes it and makes a model of this true fact with numbers. The human then further generalizes it for all numbers. This is a truth that is perhaps shared with other human beings. Many years later, another phenomenon is observed. This second phenomenon also satisfies some type of distributive law and the same model is used to characterize that second phenomenon. Abstract mathematics becomes true on its own and does not relate to the original way it was discovered. Once it is discovered, it can be applied anywhere. It makes sense: the mathematics was learned in the physical world and is applied in the physical world. Where is Wigner's mystery?

This mechanism can help explain some of our historical vignettes at the beginning of this section:

• Humans saw circles and ellipses all over their physical world. Apollonius saw that he could describe and model many of these shapes with conic sections. It is no wonder that Kepler was able to describe another physical phenomenon, the motion of planets, with these same conic sections.

• One can easily work with little strings and the way they interact. If you think about little strings long enough, then you can describe their geometry. If the universe is made of little strings, then it is obvious that we can describe it with the mathematics we learned of little strings.

• Let us examine Euclidean and non-Euclidean geometry. When we are interested in a flat surface, the geometry of Euclid works perfectly. But what happens if we are interested in a curved surface? Consider the longitude lines on a globe. There we have many lines that seem to be parallel with each other but nevertheless meet at the North and South Poles. This is opposite the spirit of Euclid's fifth axiom. It turns out that non-Euclidean geometry works well with curved surfaces. It is not as shocking as we thought that Einstein used non-Euclidean geometry to describe the curvature and shape of space.

It is not strange that physical phenomena are perfectly described by mathematics, since mathematics is an abstraction and a series of generalizations of what is observed in the physical universe. Once we formulate these ideas as mathematics, its connection to the original physical impetus for the discovery is lost. The mathematics becomes abstract and about nothing in particular. Because these concepts are about nothing, they are about everything. We do not care how the ellipses are created, whether they are in the shape of a walnut, whether they come from the intersection of a plane and a cone, or whether they are the path of a planet around a medium-sized star. With an understanding of an ellipse, we know its properties and it can be applied everywhere.

 

Some of these ideas about the symbiotic relationship between mathematics and physics are summarized in an extremely clever piece of art (
figure 8.6
) made by the mathematical physicist Robbert Dijkgraaf.

A few minutes of analysis of this wonderful comic are worthy of our time. The top two sections have calendars dated 1968, while the bottom two take place thirty years later when the same researchers have aged. The left side is in a physics professor's office, while the right side is in a mathematician's office. Now look at what is on the chalkboards.
⁴⁰
What was on the physicist's board in 1968 is being contemplated by later mathematicians. On the other hand, what mathematicians studied in the 1960s is being studied by the physicists in 1998. The path from the upper right to the lower left is what we described at the beginning of this section: the mysterious ability of mathematical ideas to somehow arise in physics. In contrast, the path from the upper left to the lower right is the possible explanation for this mysterious connection: mathematics comes from looking at the physical world.

While the explanation that mathematics comes from the physical world seems to make sense, it is far from perfect. Mathematics comes from the usual world of our everyday experience and yet the mathematics that we formed in such usual circumstances gets applied to places very far from such experiences. (See
figure 8.7
.) For example, special relativity tells us what happens when objects move close to the speed of light. We never travel anywhere near those speeds. Why should our everyday-experience mathematics help us with the strange phenomena of special relativity? Another example is quantum theory. As we saw in
section 7.2
, the quantum world is very different from our world of everyday experience. While walking down the street, we never see objects that are in a superposition or see “spooky action at a distance.” Nevertheless, the mathematics that comes from our everyday experience is very helpful at predicting quantum events. We are back to Wigner's mystery.

Another problem with this refutation of Wigner's unreasonable effectiveness is that not all mathematics is a generalization of everyday experience. Some mathematics is a creative leap away from everyday experience (again, see
figure 8.7
). The simplest example of this is negative numbers. If you have five oranges and you take away eight oranges, how many oranges do you have? There is no such thing as negative three oranges. You cannot take away eight oranges from five oranges. Although they seem obvious to us now, negative numbers were a surprising invention of the Middle Ages. For millennia before medieval times, we counted and traded but did not have concepts like negative numbers. This culturally constructed mathematical set of numbers is nevertheless currently used in every branch of physics. As mentioned, the positron was discovered simply by looking at negative solutions of the Dirac equation. Another example of mathematics that did not spring from everyday experience involves the different notions of infinite sets that we met in
chapter 4
. There does not seem to be any physical example of an infinite set. Nevertheless, the ideas of infinity are used in every physics and engineering textbook. Every building (that remains standing) and every rocket that is created uses notions of infinity in their construction. So mathematics that does not come from everyday experience is still remarkably effective in the physical sciences. Why?

There are some other reasons given for the unreasonable effectiveness, but these are the main ones. One can combine the final two answers—that is, there is a paucity of mathematics in general science and when there is mathematics in the physical world it comes from intuitions learned in the physical world. To me, the combination of these two reasons gives a satisfactory answer to Wigner's mystery.