Finding Zero (6 page)

This, like the Khajuraho magic square constructed almost six centuries earlier, is a “normal” magic square, meaning that all numbers from 1 to 16 must appear on it, and its sums are all 34. But while the Khajuraho magic square is surrounded by smiling naked or seminude figures engaged in carnal pleasures, Dürer's magic square is placed next to a melancholic, solitary, and fully clothed female figure. This is one more example of the differences I perceived in logic—and outlook—between East and West.

The Khajuraho inscription not only shows that the tenth-century Indians were adept at this kind of magic-square arithmetic, but it also showcases the numerals they used at that time (shown in the picture of the Parsvanatha magic square) and the correspondence to our numerals. Which numbers are the same and which different? How did the Hindu numerals become our own? And how did they change?

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There are different kinds of logic—not just the Western, “linear” kind of logic. Though—very broadly speaking—in the West religion often seems to be antithetical to sex, in the East religion and sex are part of one grand celebration of life and all its pleasures. Mathematics is linked to both sex and religion, as the placement of the Khajuraho magic square and the erotic imagery in religious temples suggests. In fact, sex embodies perhaps the greatest mystery of life, and mathematics—the abstraction of logically based processes we perceive in nature around us—is arguably the greatest intellectual mystery.

I wondered whether the ancient Jains and Hindus who built these temples pondered such deep mysteries. How could they not have, given the evidence they left us? Why are our lives so deeply ruled by sex? And why is the universe fundamentally ruled by mathematics? What is the secret to desire? And why do numbers behave in such curious ways, as evident in a magic square and in the remarkable way arithmetic works? These may well have been some of the questions that Eastern peoples asked themselves in antiquity, and their answers might have led them to belief, and hence
to the establishment of their religions and the invention of their gods and the construction of places of worship, where they placed symbols of their greatest mysteries: sex and mathematics. This, at least, was my conjecture. Was I on the right path?

The erotic statues of Khajuraho reminded me of the previous time I had seen ancient erotic art. It was in Pompeii, during another cruise I took aboard my father's ship when I was 14. Laci couldn't accompany me this time because my father had asked him to supervise the loading of specially roasted Italian coffee. The company insisted on serving its passengers only the highest-quality Italian coffee, so this commodity was always loaded onto the ship at an Italian port. My mother and sister went shopping, and my father stayed aboard. So the wife of the chief engineer, Ruth Chet, an attractive and sophisticated 32-year-old, accompanied me on the visit to Pompeii.

We arrived at the archaeological site and visited the antiquities, and then entered the special exhibits area, where erotic statues and frescoes found in the ruined city were on display. But the Italians had a strange, sexist rule in those days: Men of any age could visit the exhibit, but no women. I was naturally curious about this art and went in. My young age was no issue, but Mrs. Chet was barred, and despite her loud protestations, pleading, begging, and threatening, the guard would not let her pass.

In the exhibit hall I saw a statue of a small man with a giant erect penis in his hands, reaching almost to his neck, and couples on beds copulating in various positions. The women's breasts were often covered with strapless bras. These statues and frescoes were all pre-Christian, as Pompeii was destroyed in 79 CE, although
the covered breasts might demonstrate a degree of modesty even in sexual situations. Once Christianity was adopted in the West, the use of erotic imagery declined drastically. This is in contrast with what was happening at the same time in India. As a 14-year-old, I was deeply curious about the subject and, naturally, also very embarrassed by it. And Ruth Chet, being barred from the display, took her frustrations out on me. As soon as we returned to the ship, she ran to my father. “Your son has a dirty mind!” she cried. “He went into the pornographic exhibit—and they wouldn't allow me to go in.” My father laughed.

At 14, I was intensely shy looking at the Roman art. But now, at Khajuraho, I was a mature adult on a mission to discover ancient numerals. The mysterious suggestive statues of Khajuraho were in a sense like the mathematical objects that hid in their midst. Thinking about the similarities and the contrasts between the two assemblages of sensual art, I came to the conclusion that Eastern peoples of the tenth century had no hang-ups about sex and sexuality. The freedom exhibited by the Khajuraho statuary evidences such openness and sheer excitement about life and its pleasures that I felt certain it pointed to a fundamental difference between East and West. I wondered whether this disparity of views was somehow connected with the fact that Eastern logic is different from the usual Western way of thinking, and whether both relate somehow to the ability to abstract numerals out of the void and thus create a number system so powerful that it would one day take over the world. In the East, sex and logic and math seemed to be related.

We tend to think that our Western logic is the only valid kind of logic. A few years ago, I became frustrated that my sister,
Ilana, who had been diagnosed with breast cancer, was not making what I believed to be logical decisions. She shunned Western medicine in favor of Chinese qigong as her only treatment for the disease. In desperation, trying to understand how anyone could be so “illogical,” I bought the book
Logic for Dummies
by Mark Zegarelli. He says that Aristotle was the true founder of classical logic. I read,

For example, here's Aristotle's most famous syllogism:

Premises: All men are mortal

Socrates is a man

Conclusion: Socrates is mortal.
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So far, so good. But then he continues:

The Square of Oppositions . . .

A:
All cats are sleeping

O:
Not all cats are sleeping

Aristotle noticed relationships among all of these types of statements. The most important of these relationships is the
contradictory
relationship between those statements that are diagonal from each other. With contradictory pairs, one statement is true and the other false.

Clearly, if every cat in the world is sleeping at the moment, then
A
is true and
O
is false; otherwise, the situation is reversed.
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But this disagrees with the Buddhist idea expressed by Nagarjuna: Anything is true, or false, or both true and false, or neither true nor false. His statement implies that there could be situations where the opposite of an assertion could be as true as the assertion itself. How is this possible?

To a Western mind, Nagarjuna's “true or not true or both or neither” may seem like absolute nonsense. True and not true are mutually exclusive and exhaustive states for any proposition. If something is true, then there is no way that it can be
not
true. In fact, this idea underlies what in mathematics we call the
law of the excluded middle
—which says that there is no middle ground between true and not true. In mathematics, the law of the excluded middle—the fact that true and not true are mutually exclusive and exhaustive states—forms the basis for much of the mainstream approach in proof theory. Proofs can be constructive, building step by step to a final, positive conclusion. But most often, we prove theorems by contradiction (because it is much easier and is often the only way we see how to do it). If we try to prove that something is true, we first assume that it is not true
and then show that this assumption leads to a contradiction. That contradiction establishes the truth of the original proposition.

But the entire structure of proof by contradiction assumes that nothing
in the universe can be both true and untrue, or neither true nor untrue. So if we disallow the law of the excluded middle, proof by contradiction would not hold, and many theorems in mathematics would be unproved and undecided. So, what's behind this puzzling statement attributed to the Buddha? And why should
we care? The reason I cared about this question was that I was convinced that all of this was tightly bound with the appearance of numbers—the mystery that had drawn me to the East.

While researching Buddha's logic—since I was so sure that it had something deep to do with the invention of zero and infinity—I came across an intriguing article by the American logician Fred
Linton of Wesleyan University. His curious paper actually explained the Buddhist idea of the four logical possibilities (they are called the
tetralemma
in Greek or the
catuskoti
in Sanskrit, meaning four corners) in the verse by Nagarjuna in a rational, mathematical way. Let's look at some everyday examples that Linton provides for situations where the additional two logical possibilities may hold: both true and untrue, and neither true nor untrue.

If you have a student, Linton writes, who is brilliant in mathematics but also has a knack for getting arrested in campus demonstrations, you might rightly say that he is both very bright and not very bright. A cup of coffee with just a small amount of sugar, Linton points out, could very well be described as neither sweet nor unsweet. Such examples abound.
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Apparently, Eastern thinking is more in tune with such gradations of truth and falsity, so the law of the excluded middle doesn't apply. In a sense, the strict interpretation that anything must be either true or not true may well represent a Western bias in thinking about nature and life. In an e-mail message to me, Linton provided more examples, of the opposite kind, which are obviously Western in their strict
either-or
bias: “You are either with me or against me”; “If you're not part of the solution, you're part of the problem”; and “Which will you have—tea or coffee?” Then Linton added, “I blame Aristotle for that!”

In fact, our Western logic does go back to Aristotle, who is famous for logical deductive statements such as the one above. But there are other kinds of logic as well, and they may apply in other situations and contexts. Eastern thinking modes seem
to be more likely to accept differing ways of understanding the universe. But the question arises: Isn't it true that mathematics brings us only to the Western either-or kind of logic? Surprisingly, the answer is no.

Alexander Grothendieck (born in 1928) is one of the brightest and yet most troubled mathematicians of all time. Grothendieck had a penetrating vision in many areas of mathematics, including: the theory of measure—how we measure things even in the most complicated, abstract settings; topology—the theory of spaces and continuous mappings from one space to another; and algebraic geometry—the realm in which algebra and geometry merge, so that numerical information can be understood through geometrical forms. Grothendieck's entire oeuvre was motivated and driven by his quest to understand the meaning of numbers—including the deep concepts zero
and infinity—as evident also in the formulation of the ten mysterious numerals that rule our world. His quest led him far afield, and he became the most celebrated mathematician of our time.

Then, at the peak of his career, during the 1968 student riots in Paris, he went a little crazy. The American war in Vietnam was at its height, and Grothendieck became so fervently antiwar that he traveled to Vietnam in protest. From then on, he was almost exclusively involved with political and environmental activism.

When asked to give a talk about mathematics, he would surprise his audience by refusing to speak about the intended topic, and instead turn his lectern into an antiwar, pro-environment pulpit. While most of his listeners were politically on his side, they also felt cheated: They had come to hear a mathematics lecture
and not a political sermon, and they became disappointed with the man they had once admired.

Grothendieck then began to disappear for long periods of time and finally, sometime in the 1990s, made his final break from society. He is still living in hiding in the French Pyrenees, having withdrawn from the world. Reportedly, he is obsessed with good and evil and believes that the Devil rules the universe and has deliberately corrupted the speed of light from the nice round number of 300,000 kilometers per second to 299,792.458 kilometers per second.
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But long before he disappeared in his mountain hideaway, Grothendieck completely recast the field of algebraic geometry, as mentioned, and as part of that brilliant undertaking he invented a new concept: the
topos.
The topos is the ultimate generalization of the concept of space. Only Grothendieck could have the audacity, and the incredible facility with mathematics, to dare propose such a bold idea. Then, according to Pierre Cartier, a longtime member of the secret French mathematical association named Nicolas Bourbaki and a friend of Grothendieck (although he says that he has not seen him since his disappearance): “Grothendieck
claimed the right to transcribe mathematics into any topos whatever.
”
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