Is God a Mathematician? (23 page)

First, the introduction of abstract geometric spaces and of the notion of infinity (in both geometry and the theory of sets) had blurred the meaning of “quantity” and of “measurement” beyond recognition. Second, the rapidly multiplying studies of mathematical abstractions helped to distance mathematics even further from physical reality, while breathing life and “existence” into the abstractions themselves.

Georg Cantor (1845–1918), the creator of
set theory,
characterized the newly found spirit of freedom of mathematics by the following “declaration of independence”: “Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and estab
lished.” To which algebraist Richard Dedekind (1831–1916) added six years later: “I consider the number concept entirely independent of the notions or intuitions of space and time…Numbers are free creations of the human mind.” That is, both Cantor and Dedekind viewed mathematics as an abstract, conceptual investigation, constrained only by the requirement of consistency, with no obligations whatsoever toward either calculation or the language of physical reality. As Cantor has summarized it: “The
essence of mathematics
lies entirely in its
freedom
.”

By the end of the nineteenth century most mathematicians accepted Cantor’s and Dedekind’s views on the freedom of mathematics. The objective of mathematics changed from being the search for truths about nature to the construction of abstract structures—systems of axioms—and the pursuit of all the logical consequences of those axioms.

One might have thought that this would put an end to all the agonizing over the question of whether mathematics was discovered or invented. If mathematics was nothing more than a game, albeit a complex one, played with arbitrarily invented rules, then clearly there was no point in believing in the reality of mathematical concepts, was there?

Surprisingly, the breaking away from physical reality infused some mathematicians with precisely the opposite sentiment. Rather than concluding that mathematics was a human invention, they returned to the original Platonic notion of mathematics as an independent world of truths, whose existence was as real as that of the physical universe. The attempts to connect mathematics with physics were treated by these “neo-Platonists” as dabbling in
applied
mathematics, as opposed to the
pure
mathematics that was supposed to be indifferent to anything physical. Here is how the French mathematician Charles Hermite (1822–1901) put it in a letter written to the Dutch mathematician Thomas Joannes Stieltjes (1856–94) on May 13, 1894: “My dear friend,” he wrote,

The English mathematician G. H. Hardy, himself a practitioner of pure mathematics, was one of the most outspoken modern Platonists. In an eloquent address to the British Association for the Advancement of Science on September 7, 1922, he pronounced:

Clearly, even with the contemporary evidence pointing to the arbitrary nature of mathematics, the die-hard Platonists were not about to lay down their arms. Quite the contrary, they found the opportunity to delve into, in Hardy’s words, “their reality,” even more exciting than to continue to explore the ties to physical reality. Irrespective, however, of the opinions on the metaphysical reality of mathematics, one thing was becoming obvious. Even with the seemingly unbridled freedom of mathematics, one constraint remained unchanging and unshakable—that of logical consistency. Mathematicians and philosophers were becoming more aware than ever that the umbilical
cord between mathematics and logic could not be cut. This gave birth to another idea: Could all of mathematics be built on a single logical foundation? And if it could, was that the secret of its effectiveness? Or conversely, could mathematical methods be used in the study of reasoning in general? In which case, mathematics would become not just the language of nature, but also the language of human thought.

The sign outside a barber shop in one village reads: “I shave all and only those men in the village who do not shave themselves.” Sounds perfectly reasonable, right? Clearly, the men who shave themselves do not need the services of the barber, and it is only natural for the barber to shave everyone else. But, ask yourself, who shaves the barber? If he shaves himself, then according to the sign he should be one of those he does not shave. On the other hand, if he does not shave himself, then again according to the sign he should be one of those he does shave! So does he or doesn’t he? Much lesser questions have historically resulted in serious family feuds. This paradox was introduced by Bertrand Russell (1872–1970), one of the most prominent logicians and philosophers of the twentieth century, simply to demonstrate that human logical intuition is fallible. Paradoxes or
antinomies
reflect situations in which apparently acceptable premises lead to unacceptable conclusions. In the example above, the village barber both shaves and doesn’t shave himself. Can this particular paradox be resolved? One possible resolution to the paradox, strictly as stated above, is simple: The barber is a woman! On the other hand, were we told in advance that the barber had to be a man, then the absurd conclusion would have been the result of accepting the premise in the first place. In other words, such a barber simply cannot exist. But what does any of this have to do with mathematics? As it turns out, mathematics and logic are intimately related. Here is how Russell himself described the linkage:

Russell holds here that, largely,
mathematics can be reduced to logic
. In other words, that the basic concepts of mathematics, even objects such as numbers, can in fact be defined in terms of the fundamental laws of reasoning. Furthermore, Russell would later argue that one can use those definitions in conjunction with logical principles to give birth to the theorems of mathematics.

Originally, this view of the nature of mathematics (known as
logicism
) had received the blessing of both those who regarded mathematics as nothing but a human-invented, elaborate game (the
formalists
), and the troubled Platonists. The former were initially happy to see a collection of seemingly unrelated “games” coalesce into one “mother of all games.” The latter saw a ray of hope in the idea that the whole of mathematics could have stemmed from one indubitable source. In the Platonists’ eyes, this enhanced the probability of a single metaphysical origin. Needless to say, a single root of mathematics could have also helped, in principle at least, to identify the cause for its powers.

For completeness, I should note that there was one school of thought—
intuitionism
—that was vehemently opposed to both logicism and formalism. The torch-bearer of this school was the rather fanatical Dutch mathematician Luitzen E. J. Brouwer (1881–1966). Brouwer believed that the natural numbers derive from a human intuition of time and of discrete moments in our experience. To him, there was no question that mathematics was a result of human thought, and he therefore saw no need for universal logical laws of the type that Russell envisioned. Brouwer did go much further, however, and declared that the only meaningful mathematical enti
ties were those that could be explicitly constructed on the basis of the natural numbers, using a finite number of steps. Consequently, he rejected large parts of mathematics for which constructive proofs were not possible. Another logical concept denied by Brouwer was the
principle of the excluded middle
—the stipulation that any statement is either true or false. Instead, he allowed for statements to linger in a third limbo state in which they were “undecided.” These, and a few other intuitionist limiting constraints, somewhat marginalized this school of thought. Nevertheless, intuitionist ideas did anticipate some of the findings of cognitive scientists concerning the question of how humans actually acquire mathematical knowledge (a topic to be discussed in chapter 9), and they also informed the discussions of some modern philosophers of mathematics (such as Michael Dummett). Dummett’s approach is basically linguistic, stating forcefully that “the meaning of a mathematical statement determines and is exhaustively determined by its
use
.”

But how did such a close partnership between mathematics and logic develop? And was the logicist program at all viable? Let me briefly review a few of the milestones of the last four centuries.

Logic and Mathematics

Traditionally, logic dealt with the relationships between concepts and propositions and with the processes by which valid inferences could be distilled from those relationships. As a simple example, inferences of the general form “every
X
is a
Y
; some
Z
’s are
X
’s; therefore some
Z
’s are
Y
’s” are constructed so as to automatically ensure the truth of the conclusion, as long as the premises are true. For instance, “every biographer is an author; some politicians are biographers; therefore some politicians are authors” produces a true conclusion. On the other hand, inferences of the general form “every
X
is a
Y
; some
Z
’s are
Y
’s; therefore, some
Z
’s are
X
’s” are not valid, since one can find examples where in spite of the premises being true, the conclusion is false. For example: “every man is a mammal; some horned animals are mammals; therefore, some horned animals are men.”

As long as some rules are being followed, the validity of an
argument does not depend on the subjects of the statements. For instance:

produces a valid deduction. The soundness of this argument does not rely at all on our opinion of the butler or on the relationship between the millionaire and his daughter. The validity here is ensured by the fact that propositions of the general form “if either
p
or
q,
and not
q,
then
p
” yield logical truth.

You may have noticed that in the first two examples
X, Y,
and
Z
play roles very similar to those of the variables in mathematical equations—they mark the place where expressions can be inserted, in the same way that numerical values are inserted for variables in algebra. Similarly, the truth in the inference “if either
p
or
q,
and not
q,
then
p
” is reminiscent of the axioms in Euclid’s geometry. Still, nearly two millennia of contemplation of logic had to pass before mathematicians took this analogy to heart.

The first person to have attempted to combine the two disciplines of logic and mathematics into one “universal mathematics” was the German mathematician and rationalist philosopher Gottfried Wilhelm Leibniz (1646–1716). Leibniz, whose formal training was in law, did most of his work on mathematics, physics, and philosophy in his spare time. During his lifetime, he was best known for formulating independently of (and almost simultaneously with) Newton the foundations of calculus (and for the ensuing bitter dispute between them on priority). In an essay conceived almost entirely at age sixteen, Leibniz envisaged a universal language of reasoning, or
characteristica universalis,
which he regarded as the ultimate thinking tool. His plan was to represent simple notions and ideas by symbols, more complex ones by appropriate combinations of those basic signs. Leibniz hoped to be able to literally compute the truth of any statement, in any scientific discipline, by mere algebraic operations. He prophesied
that with the proper logical calculus, debates in philosophy would be resolved by calculation. Unfortunately, Leibniz did not get very far in actually developing his algebra of logic. In addition to the general principle of an “alphabet of thought,” his two main contributions have been a clear statement about when we should view two things as equal and the somewhat obvious recognition that no statement can be true and false at the same time. Consequently, even though Leibniz’s ideas were scintillating, they went almost entirely unnoticed.

Logic became more in vogue again in the middle of the nineteenth century, and the sudden surge in interest produced important works, first by Augustus De Morgan (1806–71) and later by George Boole (1815–64), Gottlob Frege (1848–1925), and Giuseppe Peano (1858–1932).

De Morgan was an incredibly prolific writer who published literally thousands of articles and books on a variety of topics in mathematics, the history of mathematics, and philosophy. His more unusual work included an almanac of full moons (covering millennia) and a compendium of eccentric mathematics. When asked once about his age he replied: “I was
x
years old in the year
x
²
.” You can check that the only number that, when squared, gives a number between 1806 and 1871 (the years of De Morgan’s birth and death) is 43. De Morgan’s most original contributions were still probably in the field of logic, where he both considerably expanded the scope of Aristotle’s syllogisms and rehearsed an algebraic approach to reasoning. De Morgan looked at logic with the eyes of an algebraist and at algebra with the eyes of a logician. In one of his articles he described this visionary perspective: “It is to algebra that we must look for the most habitual use of logical forms…the algebraist was living in the higher atmosphere of syllogism, the unceasing composition of relation, before it was admitted that such an atmosphere existed.”

One of De Morgan’s most important contributions to logic is known as
quantification of the predicate.
This is a somewhat bombastic name for what one might view as a surprising oversight on the part of the logicians of the classical period. Aristotelians correctly realized that from premises such as “some
Z
’s are
X
’s” and “some
Z
’s are
Y
’s” no conclusion of necessity can be reached about the relation between
the
X
’s and the
Y
’s. For instance, the phrases “some people eat bread” and “some people eat apples” permit no decisive conclusions about the relation between the apple eaters and the bread eaters. Until the nineteenth century, logicians also assumed that for any relation between the
X
’s and the
Y
’s to follow of necessity, the middle term (“Z” above) must be “universal” in one of the premises. That is, the phrase must include “all Z’s.” De Morgan showed this assumption to be wrong. In his book
Formal Logic
(published in 1847), he pointed out that from premises such as “most
Z
’s are
X
’s” and “most
Z
’s are
Y
’s” it necessarily follows that “some
X
’s are
Y
’s.” For instance, the phrases “most people eat bread” and “most people eat apples” inevitably imply that “some people eat both bread and apples.” De Morgan went even further and put his new syllogism in precise quantitative form. Imagine that the total number of
Z
’s is
z,
the number of
Z
’s that are also
X
’s is
x,
and the number of
Z
’s that are also
Y
’s is
y.
In the above example, there could be 100 people in total (
z
100), of which 57 eat bread (
x
57) and 69 eat apples (
y
69). Then, De Morgan noticed, there must be at least (
x y z
)
X
’s that are also
Y
’s. At least 26 people (obtained from 57 69 100 26) eat both bread and apples.

Unfortunately, this clever method of quantifying the predicate dragged De Morgan into an unpleasant public dispute. The Scottish philosopher William Hamilton (1788–1856)—not to be confused with the Irish mathematician William Rowan Hamilton—accused De Morgan of plagiarism, because Hamilton had published somewhat related (but much less accurate) ideas a few years before De Morgan. Hamilton’s attack was not at all surprising, given his general attitude toward mathematics and mathematicians. He once said: “An excessive study of mathematics absolutely incapacitates the mind for those intellectual energies which philosophy and life require.” The flurry of acrimonious letters that followed Hamilton’s accusation produced one positive, if totally unintended, result: It guided algebraist George Boole to logic. Boole later recounted in
The Mathematical Analysis of Logic:

These humble words describe the initiation of what was to become a seminal effort in symbolic logic.