The King of Infinite Space (11 page)

The hypotenuse of a right triangle whose two sides are both 1 is, by the Pythagorean theorem, the square root of 2. The square root of 2 is neither a natural number nor the ratio of natural numbers. A proof simpler than Euclid's own proceeds by contradiction. Suppose that the square root of 2
could
be represented as the ratio of two integers so that √2 =
a
/
b
. Squaring both sides of this little equation: 2 =
a
²
/
b
²
. Cross-cutting:
a
²
= 2
b
²
.

Now the fundamental theorem of arithmetic affirms that every positive integer can be represented as a unique product of positive prime numbers. A prime number is a number divisible only by itself and the number 1. Known
to the Greeks, this theorem was known to Euclid. It was widely known; it had gotten around.

The little equation
a
²
= 2
b
²
is shortly to undergo a bad accident. Whatever the number of prime factors in
a
, there must be an
even
number of them in
a
²
. There are twice as many. Ditto for
b
²
. But the number 2
b
²
has an
odd
number of prime factors. The number 2 is, after all, prime. Either the square root of 2 is not a number, or some numbers cannot be expressed as natural numbers or as the ratio of natural numbers.

This is the bad accident.

The consequences are obvious. If the Euclidean line does not contain a point corresponding to the square root of 2, how can the Cantor-Dedekind axiom be true, and if it does, how can the line be Euclidean?

T
HE NATURAL NUMBERS
1, 2, 3, . . . constitute the smallest set of numbers whose existence cannot be denied without commonly being thought insane. Whatever their nature, nineteenth-century mathematicians discovered how numbers beyond them might be defined and so made useful. A single analytical tool is at work. New numbers arise as they are needed to solve equations that cannot be solved using numbers that are old. Zero is the number that results when any positive number is taken from itself and so appears as the solution to equations of the form
x − x
=
z
. The negative
numbers provide solutions to equations of the form
x–y
=
z
, where
y
is greater than
x
. The common fractions, numerator riding shotgun on top, denominator ridden below, are solutions in style to any equation of the form
x
÷
y
=
z
.

There remained equations such as
x
²
= 2. The equation is there in plain sight. What, then, is
x
? The answer proved difficult to contrive. The Greeks endeavored to find a sense suitable to the square root of 2, but they did not entirely succeed, and beyond the imperative of solving this equation, mathematicians had no common currency with which they could easily pay for its solution. They had nothing in experience.

In the late nineteenth century, Richard Dedekind defined the irrational numbers—how did numbers that are not rational come to be
ir
rational?—in terms of
cuts
, a partitioning of the integers into two classes, A and B. Every number in A, Dedekind affirmed, is less than any number in B, and what is more, there is no greatest number in A. The cuts themselves he counted as new numbers, the enigmatic square root of 2 corresponding to the cuts A and B in which all numbers less than the square root of 2 are in A, and all those greater than the square root of 2 are in B. Dedekind's cuts are not the sort of animals one is apt to find in an ordinary zoo. Dedekind's cuts are, it must be admitted, transgendered, their identity as numbers at odds with their appearance as classes. Dedekind demonstrated, nevertheless, that they were what they did not seem to be,
and that is number-like in nature. They could be added and multiplied together; they could be divided and subtracted from one another. They took a lot of abuse. They were fine. They were, in any event, more appealing than the supposition that where there really should be a number answering to the square root of 2, there was no number at all.

The formal introduction of the real numbers in the nineteenth century brought to a close an arithmetical saga, one in which numbers that had once inspired unease acquired their own, their sovereign, identity. The positive integers, zero, the negative integers, the fractions, and the real numbers were all in place. They had acquired an indubitable existence in the minds of mathematicians. The system had a kind of abstract integrity. It held together under scrutiny. It was not adventitious.

T
HE REAL NUMBER
system represented the confluence of two triumphs: the triumph of arithmetic and the triumph of algebra. The triumph of arithmetic is obvious. The real number system is a system of real
numbers
. The triumph of algebra, less so. The real numbers satisfy the axioms for an identifiable algebraic structure, what mathematicians call a
field
. The great achievement of nineteenth- and early twentieth-century mathematics was by a python-like compression of concepts to detach the structure from its examples. Writing in 1910, the German mathematician Ernst
Steinitz proposed to make use of fields in an “
abstrakten und algmeinen Weise
”—in an abstract and general sense. A field, he wrote, is a system of elements with two operations: addition and multiplication. That is all that it is. Steinitz then introduced the distinctively new, entirely modern note, the one that marks a decisive promotion of an interesting idea into an independent idea. Never mind the question, the field of
what
? The abstract concept of a field is itself at the
mittelpunkt
of his interests. The examples dwindle away and disappear. The field remains. It becomes itself.
¹

T
HE AXIOMS FOR
a field bind its various far-flung properties together.
²
Their exposition calls to mind the lawyers in
Bleak House
rising to make a point.

—A field is a set of elements,
M'Lud
. . .

—Elements,
M'lud
, anything really.

—Feel it my duty to add,
M'lud
, that there are two operations on these elements . . .

—Beg pardon? Any two distinct operations,
M'lud
.

—Feel it my duty to add that there is 0 somewhere,
M'lud
. Yes, here it is.

—Do? It does nothing
M'lud: a
+ 0 is always
a
.

—There is a 1, too,
M'lud
. Yes, I have it here. Beg pardon? Nothing. It does nothing
M'lud
: 1
a
is always
a
.

—Feel it my duty to add a word about inverses,
M'lud
. I have them here.

—Beg pardon? Do? They invert,
M'lud
. Any element plus its inverse is 0, and any element times its inverse is 1.

There is no need to pursue this particular courtroom drama beyond the judge's demand that his attorneys sit down. A field is an abstract object, and so above it all. Still, it is an abstract object whose most compelling example is the ordinary real numbers. An associative law holds force:
a
+ (
b
+
c
) = (
a
+
b
) +
c
. And so does a distributive law:
a
(
b
+
c
) =
ab
+
ac
. Identities in 0 and 1, and inverses in the negative numbers and fractions, make possible the recovery of subtraction and division. It is, as lawyers say, familiar fare. A last lawyer rises to remind the judge that the real numbers are ordered. It is always one number before the other, or after the other; it is always, as the judge mutters, one thing or another.

No matter the lawyers, this idea has been a triumph, the second, after the definition of the real numbers themselves. This prompts the obvious question: a triumph over what?

I
N
1899, D
AVID
Hilbert published a slender treatise titled
Grundlagen der Geometrie
(The foundations of geometry). Having for many years lost himself in abstractions, a great mathematician had chosen to revisit his roots. Over the next thirty years, Hilbert would revise his book, changing its emphasis slightly, fiddling, never perfectly satisfied. The
Grundlagen—
the German word has an earthiness lacking in English—is a moving book, at once a gesture of historical respect and an achievement in self-consciousness. In writing about Euclidean geometry, Hilbert was sensitive to the anxieties running through nineteenth-century thought. Well hidden beneath the exuberant development of various non-Euclidean geometries, the anxieties could often seem arcane. But what mathematicians had suppressed was a concern, sometimes amounting to a doubt, that in geometry, the monumental aspect of Euclid's system might all along have disguised the fact that none of it made any sense.

Were the axioms of Euclidean geometry consistent? Or was there buried in the dark flood of their consequences propositions that together with their negations could
both
be demonstrated? To imagine that Euclidean geometry might be
in
consistent would be to place in doubt more than an axiomatic system, but the way of life that it engendered. Hilbert's
Grundlagen
did not answer this question completely because it cannot be completely answered. Hilbert
showed that geometry is consistent
if
arithmetic is consistent, an achievement a little like demonstrating that one building is tall if another is taller, but an achievement nonetheless.

Hilbert undertook the reformation of Euclidean geometry by expanding to twenty Euclid's original list of five axioms. In a remark of some cheekiness, Thom described Hilbert's system as a work of “tedious complexity.” The details
are
onerous. Hilbert had found and then corrected a number of logical lapses in Euclid; he was fastidious. Hilbert accepted, as Euclid did, points, lines, and planes as fundamental, bringing them explicitly into existence by assumption. He had already outlined his method in an essay titled “
Uber den Zahlbegriff
” (On the concept of number): “One begins by assuming the existence of all elements (that is one assumes at the beginning three different systems of things: points, lines and planes) and one puts these elements into certain relations to one another by means of certain axioms, in particular the axioms of connection, order, congruence and continuity.”

Having set out twenty axioms, Hilbert then steps back to cast a cool, appraising eye on what he has done. There is a change in emphasis, a heightened sense of explicitness. Euclid's analysis is directed toward the world of shapes, but Hilbert has begun to think about the analysis itself, his patient, as so often happens, left droning on the leather couch.

The subtle distinctions needed to make these issues immediate did not exist at the beginning of the twentieth century. Logicians required time to develop them. Hilbert was careful; he made no mistakes in his treatise, but he was not up to date.

A
theory
, logicians now say, consists of a set of axioms together with its logical consequences. Euclidean geometry is a theory, the first in human history. A
model
of a theory consists of the structures in which it is satisfied, a mathematical world, a place in which a theory is at home. Euclidean geometry is satisfied in the Euclidean plane. The simple idea in which theories are juxtaposed to their models makes it possible to ask what models make theories true and whether one theory could be expressed within the alembic of another. It is this idea of re-expression or reinterpretation that Hilbert advanced in his treatise, the tool that he developed.

H
ILBERT
'
S
G
RUNDLAGEN
IS
a work with divided purposes. It is, among other things, a defense of classical analytical geometry.

In thinking about the numbers, Hilbert considered two axioms, the so-called Archimedean axiom, and the so- called completeness axiom. No two axioms have ever been so many times so-called. The first axiom may be found in Eudoxus; it is an implicit aspect of his theory of proportions.
The axiom has a simple, powerful intuitive meaning.
²
When it comes to certain numbers, there is no greatest among them and no least among them either. The axiom is satisfied by the rational numbers. The axiom went far in the ancient world, but it did not go far enough. It did not suffice to characterize the real numbers, and for this, the completeness axiom was required.

“To a system of points, straight lines, and planes,” Hilbert wrote, “it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms.” There are as many points on the line as there are real numbers. There are enough to go around. This is not the Cantor-Dedekind axiom, which speaks to a
correspondence
between points and numbers. The completeness axiom is of the enforcement variety. It establishes the existence of those points. It brings them about. It guarantees them. The guarantee makes possible, if not plausible, the techniques of analytic geometry.

But Hilbert's completeness axiom is not an axiom of geometry. The objects that the axiom introduces to complete the points on the Euclidean line are not Euclidean: they are not geometric. They belong to arithmetic and they come from afar.

H
AVING OFFERED AN
aggressive defense of analytical geometry—here they are, the real numbers, take them or leave them—Hilbert at once revised his tone and tome in order to argue peacefully in favor of a version of Euclidean geometry requiring no direct concourse whatsoever with the arithmetical side of things.