Outer Limits of Reason (50 page)

Hundreds of years later, when physicists were trying to describe the strange world of the quantum, they found that they needed these very strange complex numbers in an essential way. It turns out that the superposition, which is at the core of quantum theory, is described by these complex numbers. More specifically, the many positions of a quantum state are indexed by complex numbers. Those strange curiosities are needed to describe our world.

Abstract Algebra and Quantum Theory II: Noncommutative Operators

Irish mathematician William Rowan Hamilton (1805–1865) was looking at complex numbers and saw the way they extended the real numbers to two dimensions. Hamilton wondered if there was a way to extend the real numbers to three dimensions. In 1843, he was able to form
quaternions
or
Hamiltonian
numbers. Rather than just looking at
i
as in complex numbers, Hamilton postulated
i
,
j
, and
k
as special numbers. The quaternions are then numbers

a
+
bi
+
cj
+
dk

where
a
,
b
,
c
, and
d
are real numbers. He required that we not only have
i
²
= –1, as with complex numbers, but also
j
²
= –1,
k
²
= –1, and
ijk
= –1. While Hamilton was working out the properties he noticed that such numbers do not satisfy one of the normal properties of numbers: they are not commutative. That is, for all real or complex numbers
x
and
y
, it is a fact that

xy
=
yx
.

We say that such numbers are commutative. However, Hamilton noticed that there are quaternion numbers
x
and
y
such that

xy
≠
yx
.

This operation of multiplying such numbers is
noncommutative
. This is very strange; after all, nearly every child knows that when you multiply regular numbers, it does not matter what order the numbers are multiplied in. Such noncommutative operations remained mathematical curiosities for many decades. Examples and properties of such operations were worked out by mathematicians such as Hamilton, Hermann Günther Grassmann (1809–1877), and Arthur Cayley (1821–1895) but ignored by physicists and regular people.

In the early twentieth century, when physicists were trying to formulate Heisenberg's famous uncertainty principle (see 
section 7.2
), they found this idea of noncommutative operations very helpful. In particular, imagine having two different properties of a quantum system that you want to measure. Call them
X
and
Y
. Measuring
X
and then measuring
Y
will give you different answers if you measure
Y
first and then measure
X
. That is, essentially the results of
XY
are different from
YX
. In symbols
XY
≠
YX
.

Abstract Algebra and Quantum Theory III: Group Theory

A final short example of the use of abstract algebra in quantum mechanics is group theory.
³²
In the middle of the nineteenth century, mathematicians studying whether or not a polynomial equation has a solution formulated the notion of a group. This is a mathematical object that describes certain symmetries. Many years later, when physicists were working to understand quantum theory, they found group theory was invaluable because it describes the workings of all subatomic particles.

In all three of these cases—complex numbers, noncommutative operations, and group theory—mathematicians were defining different structures that seemed to have nothing to do with the physical world. They were using these structures to deal with their own mathematical problems. And in all three cases, these structures are now used by physicists to make sense of the quantum universe.

 

In all these historical vignettes, mathematicians were playing little mind games with mathematical curiosities that later turned out to be useful for physicists dealing with the physical world. This is the core of Wigner's unreasonable-effectiveness question. Why should this be? Why does science follow mathematics in such a way? Steven Weinberg put it this way in his wonderful book
Dreams of a Final Theory
:

It is very strange that mathematicians are led by their sense of mathematical beauty to develop formal structures that physicists only later find useful, even where the mathematician had no such goal in mind. . . . Physicists generally find the ability of mathematicians to anticipate the mathematics needed in the theories of physics quite uncanny. It is as if Neil Armstrong in 1969 when he first set foot on the surface of the moon had found in the lunar dust the footsteps of Jules Verne.
³³

Scientists and philosophers have given many different answers to explain the apparent mystery of the connection between the realms of mathematics and the sciences. Let us look at several of them.

A Deity

One of the oldest answers to this question is that a deity exists who set up the universe in this way. The universe was created with perfect laws and these laws were written in a perfect mathematical language. This mathematical language is made to be understandable to human beings. Johannes Kepler stated this clearly and succinctly: “The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.”

Pope Benedict XVI, echoes these ideas:

Was it not the Pisan scientist [Galileo] who maintained that God wrote the book of nature in the language of mathematics? Yet the human mind invented mathematics in order to understand creation; but if nature is really structured with a mathematical language and mathematics invented by man can manage to understand it, this demonstrates something extraordinary. The objective structure of the universe and the intellectual structure of the human being coincide; the subjective reason and the objectified reason in nature are identical. In the end it is “one” reason that links both and invites us to look to a unique creative Intelligence.
³⁴

In other words, science follows mathematics because they both emerge from the mind of a divinity. Neptune follows the fixed laws of motion that were set up by a divinity. Leverrier was able to calculate the exact position of Neptune because the same divinity that set up the planet's motion also set up the mathematics of calculus discovered (not invented) by Newton. Group theory works perfectly for quantum mechanics because the divinity set up quantum mechanics using group theory. These laws of mathematics and physics are timeless and hence there is no mystery as to why one occurs before the other. They are all part of one divine mind.

While this solution is gratifying for one who already believes in a deity, for those who do not, it is unsatisfying. First, it does not banish the mystery. In fact, a deity or a divine intelligence is more mysterious than Wigner's unreasonable effectiveness. Scientists looking for a scientific explanation of the connection between mathematics and science will find the existence of a deity beyond the scope of their reasoning. They prefer a solution that is less metaphysical and more testable.

A Platonic Realm

A slightly less metaphysical explanation for Wigner's unreasonable effectiveness is a solution that goes back thousands of years. The Pythagoreans comprised an ancient Greek school that held that numbers and relationships between numbers had some mystical control over the physical world. To them, the essence of the universe was mathematics. Parts of this ideology were taken on by Plato and came to be known as Platonism. To Plato and his followers through the centuries, abstract entities such as mathematical objects and physical laws exist in some platonic realm. The physical world was a meager shadow of the real world, which exists and could be called “Plato's attic.” All the laws of planetary motion, general relativity, and quantum mechanics are neatly found in Plato's attic waiting for inquisitive human beings to discover them. To a Platonist, mathematics is not a human invention. Rather mathematics exists independent of human beings within this perfect platonic realm. In this attic, all the theorems of Apollonius about ellipses, all the properties of non-Euclidean geometry, and all the features of complex numbers are perfectly laid out. And in this realm, the physical laws are all perfectly stated in the language of mathematics.

One of the founders of electromagnetic theory, Heinrich Hertz (1857–1894), put it like this: “One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.”
³⁵
The writer Martin Gardner (1914–2010) gave a solid defense of Platonism, saying that “. . . if two dinosaurs met two other dinosaurs in a clearing there would have been four there even if no humans were around to observe them. The equation 2 + 2 = 4 is a timeless truth.”
³⁶
In general, Platonists find non-Platonists frustrating. If mathematics does not conform to anything “out there,” then it is just scribbles on a paper. Why should different people scribbling on different papers separated by thousands of miles and in different ages agree? Why should their scribbles miraculously not contradict each other? Platonists answer that the equation 6 × 7 = 42 is inherently true. It is not something that a bunch of people just seem to agree on. A perfect circle really exists in Plato's attic and in this—and only in this—perfect circle, the ratio of the circumference to the diameter is pi.

For a Platonist, Wigner's challenge is answered with ease. The reason why the universe follows mathematics is that the natural laws are set in this platonic realm, and mathematicians learn their trade by peeking into this very same realm. The above examples of mathematics being formulated before physical laws are simply times when mathematicians peeked a bit earlier than physicists.

While many people accept Platonism as the correct dogma, there are some problems with this purported solution. The first problem is: How do you know that such a realm exists? As with all metaphysical presuppositions, we must be suspicious of all specious claims. If we can explain our world without this platonic realm, why invoke it? Occam's razor places the existence of Plato's attic in doubt. However, even if we give a Platonist the benefit of the doubt and accept that this mystical attic exists, there are many other mysteries to contend with. Who set up this wonderful magical realm? How do mathematicians peek into this realm? What is the mechanism that physicists learn from this realm? How does this platonic realm have control over our physical world? In short, one can see the problem as in
figure 8.5
. Wigner is trying to understand the connection between the physical world and the mathematical world. Platonists invoke a
third
platonic world to solve the problem. Now we have to deal with the connections between three worlds as opposed to the previous two worlds. This is more of a mystery. Not less.

While science cannot prove or disprove that a platonic realm exists or that a deity set up the universe, it does try to find other, more scientific, explanations.

A Paucity of Mathematics

One of the more intriguing explanations for the mysterious connections between science and mathematics asserts that the connection is, in fact, rather questionable. Most physical phenomena cannot be described by our mathematics. As we saw in
section 7.1
, the vast majority of physical systems cannot be formulated in mathematics. What will the clouds look like tomorrow? Why were this week's lottery numbers the way they were? Who will win the next presidential election? All of these are legitimate questions about physical phenomena to which no mathematician in the world can give definitive answers.

There are many branches of science that study and aim to predict physical phenomena but find much of mathematics unhelpful. Israel M. Gelfand (1913–2009), a world-famous mathematician who worked in biomathematics and molecular biology, has been quoted as saying:

Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.

Other branches of science such as sociology, psychology, and anthropology also study physical phenomena but do not use mathematics in an extensive way. A physicist would protest and say that those disciplines are not “real” sciences (“After all, those disciplines do not use mathematics”). However, those studying the “soft sciences” do study physical phenomena. They would legitimately counter that the mathematics that exists today cannot help them with the complicated phenomena that they want to study. Mathematics is only good for helping with the predictable behavior of balls rolling down ramps or subatomic particles passing through one of two slits. In contrast, how a crowd would react to a certain event, or how a human would react to a relationship, is far too complicated for our mathematics. Mathematics does not predict all phenomena. It only helps with predictable phenomena. Or, as it is slightly humorously phrased, “God gave the easy problems to the physicists.”
³⁷