The King of Infinite Space (2 page)

The circumstances under which Euclid composed his masterpiece, the
Elements
, remain, like the details of his life, largely unknown. There is some evidence that Euclid taught at the great library in Alexandria founded by Ptolemy I. The Euclid of the
Elements
is stern, logical, unrelenting, a man able to concentrate the powers of his mind on what is abstract and remote. It would be fascinating to know the details of his life in Alexandria, to
see
Euclid toddling off to the baths or with a sense that he has let things get out of hand, submitting to having his eyebrows trimmed. There are suggestions here and there that as a teacher, Euclid was urbane, helpful, and kind. Among its other virtues, the
Elements
is a great textbook; perhaps Euclid read aloud from
his masterpiece as the warm sunlit air slitted through the dancing motes of dust, his students unaware that they were the first to hear a lesson that others would hear so many times and from so many other voices.

As a mathematician, Euclid took from his predecessors, men such as Eudoxus and Theaetetus, and gave to his successors, Apollonius and Archimedes. He summarized; he adjusted and refined; he was a living synthetic force and in very short order a monument—all this we know from what we can guess and from later commentaries, but the man himself remains invisible, his influence conveyed by his industry, a magnificent mole in the history of thought, a great digger of tunnels.

He must have been a man of heavy architecture, and at some point in his concourse with those endlessly gabbling philosophers, he gathered up his robes and, with a dawning sense of his powers, determined that he had something to offer that they had not seen and could not express.

F
OR MORE THAN
two thousand years, geometry has meant Euclidean geometry, and Euclidean geometry, Euclid's
Elements.
It is the oldest complete text in the Western mathematical tradition and the most influential of its textbooks.

The first book of Euclid's
Elements
contains 48 propositions, the second, 14. There are in all thirteen books comprising
467 propositions, and two more books of uncertain authorship, which are donkey-tailed onto older editions of the
Elements.

The propositions in Books I through IV are concerned with points, straight lines, circles, squares, right angles, triangles, and rectangles, the stable shapes of art and architecture. Books V through IX develop a theory of magnitudes, proportions, and numbers. The remaining books are devoted to solid geometry. Every book of the
Elements
is compelling, but the Euclid of myth and memory is the Euclid of the first four books of his treatise.

In every generation, a few students have found themselves ravished by the
Elements.
“At the age of eleven,” Bertrand Russell recalls in his autobiography, “I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined that there was anything so delicious in the world.”

A course in Euclidean geometry has long been a part of the universal curriculum of mankind. Those not ravished by its study have very often remarked that the Euclidean discipline did them good nevertheless. It improved their mental hygiene. Students study algebra at roughly the same time that they study geometry, and curiously enough, they rarely remark on the improvement that it confers.

Algebra, students complain, is just nasty.

N
OTHING FROM
E
UCLID
'
S
hand survives into the twenty-first century. We know Euclid only from copies of copies, these passing through the mangler of translation from Greek to Latin and then to Arabic and back to Greek and finally to medieval Latin. Modern versions of Euclid are based on a tenth-century Greek manuscript identified in the eighteenth century by the French scholar François Peyrard. There is a poignant distinction between the solidity of Euclid's thoughts and the perishable papyrus he used to express them. Long before Euclid, the Babylonians wrote laboriously on tablets.
Plop
went the wet clay on a long table. Inscriptions by means of a curved stylus—
chiff, chiff, chiff.
The oven of the sun. And thereafter, immortality. We can see their words as well as their works. Euclid himself we cannot see at all.

If Euclid imposed order on his subject by making it a system, it was an order so severe as to force geometry into a fixed shape until at least the Italian Renaissance in the sixteenth century. Thereafter, a long and confusing process followed in which the Euclidean monument was variously chipped away, until in the nineteenth century, mathematicians discovered
non
-Euclidean geometries, Euclidean geometry becoming one among many, mathematicians half-mad with possibilities absorbing themselves with spaces that bulged like basketballs, or curved like saddlebacks, or that went on forever without getting anywhere.

Euclid's
Elements
represents the great achievement of the Greek mathematical tradition. Archimedes was a more brilliant mathematician than Euclid. He gave to the world what great mathematicians always give, and that is a record of his genius, but in the idea of an axiomatic system, Euclid gave to mathematics something even more enduring, and that was a way of life. It was a way of life invisible to the people who preceded the Greeks, and it was invisible as well to the Chinese, the masters of a subtle technological culture.

And as one might expect, it is invisible to everyone else as well—now, today, even so—and must as a result be taught like any other artifact of civilization.

 

1
.
Tony Judt, “The Glory of the Rails,”
New York Review of Books
, December 23, 2010.

A
N AXIOMATIC SYSTEM
is a stylized organization of intellectual life, an abstraction from the gabble. Euclid conceived of an axiomatic system in order to fulfill an ambition that had before Euclid gone unconceived and so unexpressed: to derive all of the propositions about geometry from a handful of assumptions. The Egyptians who built the pyramids surely knew something about pyramids. They were not unsophisticated. They had a feel for measurements and mensuration. But what they knew, they knew incompletely. They took what they needed; they had no grasp of the whole. Euclid believed that there is a form of unity beneath the diversity of experience, and it is this that marks the difference between Euclid and the Egyptian mathematicians, men of the lash.

Euclid required a double insight before he could strike for immortality. The first: that the various propositions of geometry
could
be organized into a single structure; and the second: that the principle of organization binding geometric propositions
must
be logical, and so alien to geometry itself.

These are radically counterintuitive ideas, Pharaonic in their audacity.

Euclid's assumptions are commonly called axioms, and sometimes postulates; his conclusions, theorems. A proof is a chain connecting the axioms to the theorems in logically unassailable links. Euclid assumed five axioms, and from these he derived 467 theorems.

A sense of this intellectual power, its grandeur—this is the Euclidean gift. The Pythagoreans before Euclid were men consumed by the rapture of mathematics. They communed with the numbers, and they were often tempted by gross intellectual follies. They took pleasure in mumbo jumbo. Euclid is by comparison imperturbable. There is no rapture in the
Elements
, but neither is there anything insane. The structure that Euclid created is intellectually accessible to anyone capable of following an argument.

Like the pyramids, an axiomatic system is a public work.

E
UCLIDEAN GEOMETRY IS
the study of shapes in space. Shapes are not bound to the wheel of time. There is no
place where the Euclidean triangle resides, and no time at which it arrived there. Plato argued that the shapes are a part of the Kingdom of Forms, caves and cavemen, shadows and the ecstatic sun. No philosopher since Plato has been entirely satisfied with the kingdom. Existing in the great beyond, the Platonic forms have no obvious causal powers. Yet if they have no obvious causal powers, they have obvious causal effects. Euclid reached his conclusions about triangles by reasoning about the form of the triangle, the essential thing.

If the Platonic forms are difficult to accept, they are impossible to avoid. There is no escaping them. Mathematicians often draw a distinction between concrete and abstract models of Euclidean geometry. In the abstract models of Euclidean geometry, shapes enjoy a pure Platonic existence. The concrete models are in the physical world. Freeways masquerade as straight lines, ink drops as points, amphitheaters as circles, and planetary orbits as ellipses. Mathematicians have often supposed that the concrete models of Euclidean geometry have a degree of vitality denied the Platonic models. “One must be able to say at all times,” the German mathematician David Hilbert remarked, “instead of points, straight lines, and planes—tables, beer mugs, and chairs.” These words convey a reassuring impression of ordinary life. Tables, beer mugs, and chairs! What could be more down-to-earth? But the
phrase
instead of
prompts a reservation. The shortest distance between two beers is a straight line in time
or
in space. Yes, that is certainly true. But the shortest distance between two beers is a straight line
because
the shortest distance between two points is a straight line. Nothing is instead of anything.

Without the Platonic models, the concrete models would have no interest. Euclid does not, after all, invite his readers to consider
more or less
straight lines. How much more, how much less? And if there are no purely straight lines, what would the comparison come to?

The concrete models of Euclidean geometry include the tables, chairs, and beer mugs. They are where they have always been, and that is in the barroom.

The Platonic models of Euclidean geometry include the points, lines, and planes. They are where they have always been as well, and that is God knows where.

I
F THE THEOREMS
of an axiomatic system follow from its axioms, it is reasonable to ask what
following from
might mean. What
does
it mean? The image is physical, as when a bruise follows a blow, but the connection is metaphorical. The relationship between the axioms and the theorems of an axiomatic system is, when metaphors are discarded, remarkably recondite, invisible for this reason to all of the ancient civilizations but the Greek.

The men of the ancient Near East no doubt knew what arguments were. They had so many of them. What they knew, they knew imperfectly. They lacked words to make clear the distinctions that they sensed. Why assess an argument when it was so much easier to end it by either violence or indifference? This point of view has never completely lost favor. It was the Greeks who did the assessment and forced the very idea of an inference into consciousness, asking patiently for an account of its nature, the way it controlled the movement of the mind, and where in the catalog of human powers it belonged.

At roughly the same time that Euclid composed the
Elements
, Aristotle provided a subtle and refined analysis of syllogistic inference, the pattern in argument that takes Socrates—and the rest of us, alas—to his death by virtue of the fact that he is a man and we are mortal. Born in 384
BC
and dying in 322
BC
(another victim of his own syllogism), Aristotle might conceivably have known Euclid when Euclid was still a young man, perhaps even palpating his togaed shoulder. Far too little is known about the circumstances of Euclid's life to say just whose hand he might have shaken. The two men worked hand in hand all the same.

A
RGUMENTS
, A
RISTOTLE ARGUED
, may be divided into those that are good and those that are not. In the syllogism, two premises resolve themselves in one conclusion:

Good.

Bad.

Shame.

As these examples might indicate, good arguments are good by virtue of their form and not their content. The logician is indifferent to the distinction between
all dogs are mammals
and
all men are mortal
; both cases are swallowed whole by
all A's are B.
This is the Aristotelian insight, and logicians have accepted it ever since. The conclusion of a valid argument is entrained by its premises. Truth plays an ancillary role. If the premises of a valid argument are true, then their conclusion must be true, but
whether
they are true is a matter on which the logician has little to say; an argument may be good even though its premises are false, and bad even though its premises are true.

It is tempting to imagine a fraternal give-and-take between Euclid and Aristotle, Euclid taking, Aristotle giving, with Euclid advancing proofs and arguments that Aristotle had antecedently assessed and classified. This is not quite so. The
Elements
is a work of great logical sophistication, but it is not a work of logical self-consciousness. Euclid's subject is geometry, his business is proof, and Euclid was not a mathematician disposed to step back to catch himself in the act of stepping back. That his arguments were valid, he had no doubt, but in the question of what made them valid, he had no interest.