Outer Limits of Reason (70 page)

It is obvious why such a cubic formula is not taught in high school!

11
. Note that Abel also died young. He, too, lived a tragic life and died in extreme poverty.

12
. I thank Chaim Goodman-Strauss for figures 
9.10
 through 
9.16
.

13
. I am indebted to Chaim Goodman-Strauss for this clever example.

14
. In technical terms, although basic arithmetic is
incomplete
, basic geometry was proven by Hilbert to be
complete
.

15
. Basically this is induction, which we glimpsed in
chapter 3
when we talked about heaps.

16
. Post 1941, in Davis 2004, 343n12.

17
. This is very similar to what we did in 
chapter 6
. There we assigned a unique number to every program. Here we assign a unique number to every proof. The goal is also similar: self-reference. With programs, we wanted programs that deal with numbers to have numbers so that programs can deal with programs. Here we want mathematical statements that deal with numbers to have numbers so that mathematical statements can deal with mathematical statements.

18
. I am ignoring some details here. There was a slight complexity in Gödel's result concerning consistency and something called omega consistency (written ω-consistency). In 1936, John Barkley Rosser (1907–1989) modified Gödel's sentence to get what is now called a
Gödel-Rosser sentence
, showing the result for which we were aiming.

19
. At the end of 
section 6.3
I showed that although computers can perform a countably infinite number of tasks, an uncountably infinite number of tasks remain that computers cannot perform. At the end of 
section 7.1
I presented an argument that science can only deal with a countably infinite number of phenomena, while an uncountably infinite number of phenomena remain that cannot be described by science. Here I present a mathematical analogy to those results.

20
. Along these lines of thought, Leopold Löwenheim (1878–1957) formulated what has come to be known as the
downward Löwenheim-Skolem theorem
. This deep theorem states that what is described cannot be more complicated than the language used to describe it. In detail, statements in mathematics are written with a finite set of symbols. Using these symbols, there is a countably infinite number of possible statements. Now consider a system with an uncountably infinite number of elements. The downward Löwenheim-Skolem theorem states that if there is a consistent way of using a language to talk about such a system, then that language might very well be talking about a system with only a countably infinite number of elements. That is, the axioms might be intended for discussing something uncountably infinite, but we really cannot show that it is has more than a countably infinite number of elements. This gives us a severe limitation on what we can describe.

21
. An interesting result is worth contemplating. As I have said, Peano Arithmetic is weaker than ZFC. In other words, whatever can be proven in Peano Arithmetic can surely be proved in ZFC. But how much weaker is it? It was shown that if you take the axioms of ZFC and leave out the axiom of infinity, then the system left is equipotent with Peano Arithmetic. (By
equipotent
I mean that whatever can be proved with one system can be proved with the other system and vice versa.) This can be written as

ZFC = Peano Arithmetic + Axiom of infinity.

Since ZFC can prove the consistency of Peano Arithmetic (and any other true statement that can be made in Peano Arithmetic), it follows that accepting that Peano Arithmetic is consistent is equivalent to accepting that there is a consistent way of dealing with infinite sets. Alternatively, the above equation points to the fact that if mathematicians would simply give up this belief that infinite sets exist, all of mathematics would be as consistent as basic arithmetic.

22
. Or do we? A central idea in computer science is that all different types of physical computing devices can basically solve the same problems. Some devices can perform them quicker than others, but they all have the same computing ability. What is computable by one device is computable by another. More important for our discussion, what one device can never compute, another device can also never compute. This idea is called the
Church-Turing thesis.
It is a thesis as opposed to a theorem because it can never be proved. There is no way we can prove something about
every
physical device. But perhaps the Church-Turing thesis is wrong. Maybe in the distant future, scientists will develop a device that can solve problems that were previously not solvable. In that case, the unsolvable computer problems described in this book might still be solvable and the computational limitations will be relative, not absolute limitations.

Chapter 10

1
. “Le silence éternel de ces espaces infinis m'effraie” (Pascal,
Pensées,
passage 206).

2
. Nash 1994.

3
. Quoted in Wang 1996, 9.4.3.

4
. I'm reminded of a joke: analogous to “a chicken is an egg's way of making an egg,” I'm stressing that “a scientist is an atom's way of knowing an atom.”

5
. See Yanofsky 2003 for a more comprehensive list of many different self-referential systems.

6
. There are many other self-referential paradoxes that I did not touch on—for example, the
paradox of nirvana
(you can only reach nirvana if you free yourself of all desires . . . including the desire to reach nirvana) and the
paradox of tolerance
(if you want to have a tolerant society you must be intolerant of those who are intolerant). While these paradoxes are fascinating, they fall outside the scope of our discussion.

7
. This is also discussed in chapter 5 of Rescher 2009. The title of the chapter is “More Facts Than Truths.”

8
. I am reminded of Ludwig Wittgenstein's words: “The limits of my language mean the limits of my world” (“Die Grenzen meiner Sprache bedeuten die Grenzen meiner Welt”) (
Tractatus Logico-Philosophicus
5.6).

9
. I briefly touched on the comparison between computer and human abilities in sections 
6.5
and 
9.3
.

10
. There is a quote said to occur in the section on map reading in the Norwegian
Boy Scout Handbook
: “If the terrain differs from the map, believe the terrain.”

11
. In a sense, this is the content of Paul Feyerband's dictum that “anything goes” when it comes to the scientific method.

12
. “Eppur si muove.”

13
. James Randi performed an interesting experiment that highlights something fascinating about human nature. Randi entered a class and asked the students to put their name and some personal information like their birthday and favorite color on a card. He then collected the cards and returned the next day. Attached to every card was a personalized description and horoscope for each student. Randi distributed the papers to the students and asked them to read their own personalized horoscope. Before discussing it with them, he asked them to judge whether their description was accurate. On a scale of “excellent,” “very good,” “good,” or “poor,” the vast majority of the students evaluated the accuracy of their personalized description as “good” or better. Randi then permitted the students to see each other's personalized horoscope and to their shock, all the horoscopes were exactly the same and filled with ambiguous pleasantries. Most students accepted the pleasant horoscope even though it did not say anything about them individually. Randi's experiment can be found online at 
http://www.youtube.com/watch?v=3Dp2Zqk8vHw
.

14
. Wood engraving from Camille Flammarion,
L'Atmosphere: Météorologie Populaire
(Paris, 1888), 163.

15
. While I am criticizing this book, let me add more self-criticism. There is no doubt that I am using the words
limit
and
limitation
in many different ways. The words
reason
and
exist
are also used in diverse ways. In my defense, these words do not have clear definitions.

16
. As opposed to answers to questions such as “How many teeth does a unicorn have?” or “Is the present king of France bald or not?”, for which the information does not exist.

17
. In 
section 6.4
I described a hierarchy of some computer problems that humans and computers cannot solve. Can we make a similar stratification of unknowns here?

18
. The philosophical phrase is “reason is the handmaiden of will and desire.” This is one of the main themes in the writings of Arthur Schopenhauer and David Hume.

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