Is God a Mathematician? (25 page)

Figure 48

Central to Frege’s philosophy was the assertion that truth is independent of human judgment. In his
Basic Laws of Arithmetic
he writes: “Being true is different from being taken to be true, whether by one or many or everybody, and in no case is it to be reduced to it. There is no contradiction in something’s being true which everybody takes to be false. I understand by ‘laws of logic’ not psychological laws of takings-to-be-true, but laws of truth…they [the laws of truth] are boundary stones set in an eternal foundation, which our thought can overflow, but never displace.”

Frege’s logical axioms generally take the form “for all…if…then…” For instance, one of the axioms reads: “for all
p,
if not-(not-
p
) then
p
.” This axiom basically states that if a proposition that is contradictory to the one under discussion is false, then the proposition is true. For instance, if it is not true that you do not have to stop your
car at a stop sign, then you definitely do have to stop at a stop sign. To actually develop a logical “language,” Frege supplemented the set of axioms with an important new feature. He replaced the traditional subject/predicate style of classical logic by concepts borrowed from the mathematical theory of functions. Let me briefly explain. When one writes in mathematics expressions such as:
f(x)
3
x
1, this means that
f
is a function of the variable
x
and that the value of the function can be obtained by multiplying the value of the variable by three and then adding one. Frege defined what he called
concepts
as functions. For example, suppose you want to discuss the concept “eats meat.” This concept would be denoted symbolically by a function “
F(x)
,” and the value of this function would be “true” if
x
lion, and “false” if
x
deer. Similarly, with respect to numbers, the concept (function) “being smaller than 7” would map every number equal to or larger than 7 to “false” and all numbers smaller than 7 to “true.” Frege referred to objects for which a certain concept gave the value of “true” as “falling under” that concept.

As I noted above, Frege firmly believed that every proposition concerning the natural numbers was knowable and derivable solely from logical definitions and laws. Accordingly, he started his exposition of the subject of natural numbers without requiring any prior understanding of the notion of “number.” For instance, in Frege’s logical language, two concepts are
equinumerous
(that is, they have the same number associated with them) if there is a one-to-one correspondence between the objects “falling under” one concept and the objects “falling under” the other. That is, garbage can lids are equinumerous with the garbage cans themselves (if every can has a lid), and this definition does not require any mention of numbers. Frege then introduced an ingenious logical definition of the number 0. Imagine a concept
F
defined by “not identical to itself.” Since every object has to be identical to itself, no objects fall under
F.
In other words, for any object
x, F(x)
false. Frege defined the common number zero as being the “number of the concept
F.
” He then went on to define all the natural numbers in terms of entities he called
extensions.
The extension of a concept was the class of all the objects that fall under that concept. While this definition may not be the easiest to digest for
the nonlogician, it is really quite simple. The extension of the concept “woman,” for instance, was the class of all women. Note that the extension of “woman” is not in itself a woman.

You may wonder how this abstract logical definition helped to define, say, the number 4. According to Frege, the number 4 was the extension (or class) of all the concepts that have four objects falling under them. So, the concept “being a leg of a particular dog named Snoopy” belongs to that class (and therefore to the number 4), as does the concept “being a grandparent of Gottlob Frege.”

Frege’s program was extraordinarily impressive, but it also suffered from some serious drawbacks. On one hand, the idea of using concepts—the bread and butter of thinking—to construct arithmetic, was pure genius. On the other, Frege did not detect some crucial inconsistencies in his formalism. In particular, one of his axioms—known as
Basic Law V
—proved to lead to a contradiction and was therefore fatally flawed.

The law itself stated innocently enough that the extension of a concept
F
is identical to the extension of concept
G
if and only if
F
and
G
have the same objects under them. But the bomb was dropped on June 16, 1902, when Bertrand Russell (figure 49) wrote a letter to
Frege, pointing out to him a certain paradox that showed Basic Law V to be inconsistent. As fate would have it, Russell’s letter arrived just as the second volume of Frege’s
Basic Laws of Arithmetic
was going to press. The shocked Frege hastened to add to the manuscript the frank admission: “A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.” To Russell himself, Frege graciously wrote: “Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic.”

Figure 49

The fact that one paradox could have such a devastating effect on an entire program aimed at creating the bedrock of mathematics may sound surprising at first, but as Harvard University logician W. V. O. Quine once noted: “More than once in history the discovery of paradox has been the occasion for major reconstruction at the foundation of thought.” Russell’s paradox provided for precisely such an occasion.

Russell’s Paradox

The person who essentially single-handedly founded the theory of sets was the German mathematician Georg Cantor. Sets, or classes, quickly proved to be so fundamental and so intertwined with logic that any attempt to build mathematics on the foundation of logic necessarily implied that one was building it on the axiomatic foundation of set theory.

A class or a set is simply a collection of objects. The objects don’t have to be related in any way. You can speak of one class containing all of the following items: the soap operas that aired in 2003, Napoleon’s white horse, and the concept of true love. The elements that belong to a certain class are called
members
of that class.

Most classes of objects you are likely to come up with are not members of themselves. For instance, the class of all snowflakes is not in itself a snowflake; the class of all antique watches is not an antique watch, and so on. But some classes actually are members of themselves. For example, the class of “everything that is not an antique watch” is
a member of itself, since this class is definitely not an antique watch. Similarly, the class of all classes is a member of itself since obviously it is a class. How about, however, the class of “all of those classes that are not members of themselves”? Let’s call that class
R.
Is
R
a member of itself (of
R
) or not? Clearly
R
cannot belong to
R,
because if it did, it would violate the definition of the
R
membership. But if
R
does not belong to itself, then according to the definition it must be a member of
R
. Similar to the situation with the village barber, we therefore find that the class
R
both belongs and does not belong to
R,
which is a logical contradiction. This was the paradox that Russell sent to Frege. Since this antinomy undermined the entire process by which classes or sets could be determined, the blow to Frege’s program was deadly. While Frege did make some desperate attempts to remedy his axiom system, he was unsuccessful. The conclusion appeared to be disastrous—rather than being more solid than mathematics, formal logic appeared to be more vulnerable to paralyzing inconsistencies.

Around the same time that Frege was developing his logicist program, the Italian mathematician and logician Giuseppe Peano was attempting a somewhat different approach. Peano wanted to base arithmetic on an axiomatic foundation. Consequently, his starting point was the formulation of a concise and simple set of axioms. For instance, his first three axioms read:

1. Zero is a number.
   
2. The successor to any number is also a number.
   
3. No two numbers have the same successor.
   

The problem was that while Peano’s axiomatic system could indeed reproduce the known laws of arithmetic (when additional definitions had been introduced), there was nothing about it that uniquely identified the natural numbers.

The next step was taken by Bertrand Russell. Russell maintained that Frege’s original idea—that of deriving arithmetic from logic—was still the right way to go. In response to this tall order, Russell produced, together with Alfred North Whitehead (figure 50), an incredible logical masterpiece—the landmark three-volume
Principia
Mathematica
. With the possible exception of Aristotle’s
Organon,
this has probably been the most influential work in the history of logic (figure 51 shows the title page of the first edition).

Figure 50

In the
Principia,
Russell and Whitehead defended the view that mathematics was basically an elaboration of the laws of logic, with no clear demarcation between them. To achieve a self-consistent description, however, they still had to somehow bring the antinomies or paradoxes (additional ones to Russell’s paradox had been discovered) under control. This required some skillful logical juggling. Russell argued that those paradoxes arose only because of a “vicious circle” in which one was defining entities in terms of a class of objects that in itself contained the defined entity. In Russell’s words: “If I say ‘Napoleon had all the qualities that make a great general,’ I must define ‘qualities’ in such a way that it will not include what I am now saying, i.e. ‘having all the qualities that make a great general’ must not be itself a quality in the sense supposed.”

To avoid the paradox, Russell proposed a
theory of types,
in which a class (or set) belongs to a higher logical type than that to which its members belong. For instance, all the individual players of the Dallas Cowboys football team would be of type 0. The Dallas Cow
boys team itself, which is a class of players, would be of type 1. The National Football League, which is a class of teams, would be of type 2; a collection of leagues (if one existed) would be of type 3, and so on. In this scheme, the mere notion of “a class that is a member of itself” is neither true nor false, but simply meaningless. Consequently, paradoxes of the kind of Russell’s paradox are never encountered.