Outer Limits of Reason (56 page)

The existing evidence does not favor any particular response to our questions about the structure of the universe. We can speculate about all the possible definitions of life and all the possible multiverses, but we have no empirical proof that such life forms or universes actually exist. And so we come to the conclusion that we are in a state of ignorance about the ultimate reasons the universe is the way it is. The universe that created us to have reason and to question everything we see is very coy about revealing its deep secrets in response to these reasonable inquiries.

 

No matter what reason you accept for the anthropic principle, it is hard to avoid the eerie feeling that something strange and wonderful is going on here. The physicist Freeman Dyson said it in its clearest form: “I do not feel like an alien in this universe. The more I examine the universe and study the details of its architecture, the more evidence I find that the universe must in some sense have known that we were coming.”
⁶¹

Further Reading

Section 8.1

There are many fine introductions to the philosophy of science. Some recommendations from easiest to hardest include Okasha 2002, Gorham 2009, Godfrey-Smith 2003, and Losee 2001. Our discussion has also gained much from Rescher 1999. Our presentation of the beauty and the end-of-science issues has borrowed much from the extremely readable Weinberg 1994. We cannot talk about the end-of-science issue without mentioning the controversial—but fun to read—Horgan 1996. Kuhn 1970 is very readable and interesting.

Section 8.2

Mickens 1990 is a collection of nineteen articles that discuss and give possible answers for Wigner's unreasonable effectiveness. Included is Wigner's original article. Many other articles and chapters deal with this issue, including chapter 27 of Klein 1981. Burtt 1932 offers a fascinating historical tour of the ever-increasing role of mathematics in the sciences. One must also include Pickering 1984 for an intriguing perspective.

Section 8.3

The anthropic principle is discussed in many books, including the classic Barrow and Tipler 1986 and Davies 1982, 2008. The second Davies book is a popular book that covers every aspect of the topic and is very deep. Gibbon and Rees 1989 is another popular account of the anthropic principle. There is also a very nice summary article found online by P. C. W. Davies titled “Where Do the Laws of Physics Come From?” (2012).

For more about the multiverse, see Greene 2011. Also see Carr 2007, which contains twenty-eight up-to-date scholarly articles from every different angle. See also Deutsch 1997 as well as Max Tegmark's papers on his web page, 
http://space.mit.edu/home/tegmark
. For a basic introduction to string theory see Musser 2008.

For more about the role of symmetry in formulating the laws of physics, see Eddington 1958, Kilmister 1994, Weyl 1952, Stenger 2006, Cook 1994, and Lederman and Hill 2004.

9

Mathematical Obstructions

As soon as I understood the principles, I relinquished for ever the pursuit of the mathematics; nor can I lament that I desisted, before my mind was hardened by the habit of rigid demonstration, so destructive of the finer feelings of moral evidence, which must, however, determine the actions and opinions of our lives.
¹

—Edward Gibbon (1737–1794)

I had a feeling once about Mathematics, that I saw it all—Depth beyond depth was revealed to me—the Byss and the Abyss. I saw, as one might see the transit of Venus—or even the Lord Mayor's Show, a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened and why the tergiversation was inevitable: and how the one step involved all the others. It was like politics. But it was after dinner and I let it go!
²

—Winston Churchill (1874–1965)

The New Mathematics: standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.
³

—Woody Allen

Mathematics is the epitome of pure reason. Much of science is based on mathematics and logic. In a sense, mathematics is the language of reason. It is the most successful of all human inventions. And yet, as we will see, it too has its limitations.

Section 9.1
introduces some limitations that were dealt with during the classical Greek era.
Section 9.2
contains a short description of Galois theory, which is an entire field of modern mathematics dedicated to showing that certain problems cannot be solved with the usual methods. In
section 9.3
we return to our discussion of solving problems using computers and show that several problems in mathematics are unsolvable by computation—be it mechanical computation or human computation.
Section 9.4
discusses some self-referential aspects of logic including Gödel's famous incompleteness theorem. The chapter closes with a discussion of some technical and philosophical aspects of logic and axiom systems.

9.1  Classical Limits

It was clear to the followers of Pythagoras that Hippasus, a former member of their club, had to be killed. He must be thrown into the sea lest he continue to blaspheme and reveal all of their secrets. Hippasus was being totally irrational!

One of the more interesting characters in the ancient Greek world was Pythagoras of Samos. He lived in the sixth century BC and had a major impact on the ancient and the medieval worlds. Pythagoras—a philosopher, mystic, music connoisseur, and religious leader—was one of the first to study mathematics for its own sake rather than for its applications. This makes him one of the world's first
pure
mathematicians. He also discovered relationships between musical harmonies and mathematics. Pythagoras had an immense influence on the progress of philosophy and mathematics.

One of the central dogmas for Pythagoras and his legion of followers was that the entire world was governed and described by whole numbers or ratios of whole numbers. Such ratios were the only numbers thought to be sane or “rational.” In other words, they did not believe in what we now call
irrational
numbers. To them, rational numbers were the only numbers that existed.

All was well with this worldview until a student of Pythagoras named Hippasus of Metapontum came along. Hippasus contemplated a common square whose sides are of length 1, as in
figure 9.1
.

By Pythagoras's famous theorem for a right triangle we have

x
²
+
y
²
=
z
²

or

In the square in 
figure 9.1
, both
x
and
y
have length 1. That implies that the diagonal,
z,
has length √2. This was fine with everyone, but Hippasus then proceeded to prove that this commonly occurring number, the square root of 2, is not a rational number. This easily described number that is found everywhere was the first known example of an irrational number. This was a shocking result to his contemporaries. In a sense, this was the first violation of a limitation for the mathematics of that era.
⁴

Legend has it that Pythagoras and his followers were upset with Hippasus's finding. They feared that he might reveal his discovery to other people and hence show the inadequacies of their faith. The brotherhood did the only thing they thought they could do: they took Hippasus out to sea and threw him overboard, hoping he would take his secret to a watery grave.

It is not known what proof Hippasus used to show that the square root of 2 is an irrational number. However, a very elegant proof exists that is worthy of our consideration. The proof is geometric and does not contain many of those nasty equations that make people unhappy. It is a proof by contradiction, with which we are already familiar. If we assume (wrongly) that the square root of 2 is a rational number, then we are going to find a contradiction:

The square root of 2 is rational ⇒ contradiction.

If √2 were a rational number, then there would be two positive whole numbers such that their ratio is the square root of 2. Let us assume that the two smallest such numbers are
a
and
b
. That is,

Squaring both sides of this equation gives us

Multiplying both sides by
b
²
gives us

2
b
²
=
a
²
.

Let us look at this equation from a geometric point of view. This means that there is a large square whose sides are length
a
(that has area
a
²
) and two small squares whose sides are length
b
(each has area
b
²
) such that the sum of the areas of the two small squares is exactly the area of the large square, as seen in
figure 9.2
.