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.2
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.
Euclid’s Stunning 300 BCE Proof of the Infinitude of Prime Numbers
Here is the best—and most ancient as well as most elegant—example of proof by contradiction. It goes back 2,300 years. It is the proof attributed to the Greek mathematician Euclid of Alexandria that there are infinitely many prime numbers. The proof proceeds as follows. Euclid says, “Let’s assume that there are only finitely many prime numbers. Then there must be a largest prime number, after which there are no more primes and all larger number are composite (meaning they are products of prime numbers).” This makes perfect sense, right? If there are only finitely many primes, there must be a largest prime. Let’s call this largest prime p. Now, Euclid says, consider the following number: 2 × 3 x 5 × 7 × 11 × 13 × . . . × p + 1. This is the product of all the prime numbers, 2 through p, plus the number one. Is this new number prime?
If it is, then we have just exhibited a prime number greater than p. And if it isn’t, then it must be divisible (by the definition of nonprime, or composite, numbers) by one of the primes 2 through p. Call that prime number by which 2 × 3 × 5 × 7 × 11 × 13 × . . . × p + 1 is divisible q. But we see that this cannot possibly be true, since such a division will always leave the additional factor of 1 divided by that prime number, q, and 1⁄q could not possibly be an integer. So in either case, we have now exhibited a contradiction, which establishes the theorem.
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.3
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.4
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.”5
This means that, pretty much, Grothendieck felt he had found a generalization that was so powerful that it allowed him to cast mathematics into any mold he pleased. He could view numbers not as mere numerals and abstract entities, but also as geometrical shapes; he could turn shapes into numerical quantities; equally, he could abstract both of them into entities that lived in highly esoteric mathematical realms that only expert mathematicians could visualize or understand—and then “do math” in these weird spaces that few have the ability to imagine. While a number is an abstract concept that can be symbolized by a set of signs (the numerals, arranged in various ways), Grothendieck took that abstraction to a whole new level.
It turns out that the mathematics of a topos, invented by Grothendieck, allows for a mathematically consistent and correct basis that justifies the Eastern kind of logic, as explained by Fred Linton.6 Technically, our strict, either-or logic is necessitated by our reliance on the theory of sets as a basis for mathematics. This gives us the concept of set membership, which is unforgiving: An element is either a member of a set, or it is not; it cannot be both, or neither.
What Grothendieck (with the help of other mathematicians) did was to free mathematics from reliance on set theory and set membership. He employed something called category theory, in which there is no need for sets and membership laws. This freed him to define the topos, within which other logical systems, not requiring the either-or absoluteness, could justifiably exist. Thus it was that through Grothendieck’s work, Linton was able to show that Nagarjuna’s tetralemma had a perfectly valid mathematical basis. The topos places the Eastern logic with its four possibilities on the same solid foundation as our Western logic. And in the Eastern logic, a ruling concept is that of the void, or emptiness, or nothingness: the zero.
In Linton’s topos, applied to the tetralemma, the opposite of “not true” is not the same as “true.” This is the key to the Eastern way of thinking captured by Nagarjuna. In the West, not (not (true)) = true; this allows us to do proofs by contradiction and captures our strict way of thinking. But in Linton’s topos, there is “true,” there is also “not true,” and there is yet a third thing altogether, called “not (not true).” To use Linton’s favorite example, this logic applies perfectly in situations in which we say things like “The coffee wasn’t unsweet”—meaning that it wasn’t sweet but it was not unsweet either.
This reminds me of an occasion when, a few years ago, my publisher mentioned meeting the publicist they had hired to promote my book. He said, “This person wasn’t unattractive.” This is typical of Linton’s in-between logic: My publisher didn’t want to say the person was attractive, but he didn’t want to say unattractive either.
Readers more familiar with fuzzy logic or quantum mechanics, where something can be in a mixture (probabilistic or otherwise) of two seemingly mutually exclusive states, may choose to view the Linton topos in this way. Linton’s work proved that the Eastern logic of Nagarjuna and the tetralemma (catuskoti) has a solid mathematical foundation. Quantum computing, if it becomes a viable reality, may rely on related logical principles that are different from the usual logic we take for granted as the only one—as my friend C. K. Raju, then back at his university post in Malaysia, recently pointed out to me.7 And I believe that the logic of the catuskoti is ultimately what led to the invention of the key numeral of our system, the zero.
7
In India, mathematics and logic—and the intermingling of mathematics and numbers with sex—are very ancient. The earliest known texts in an Indian language are the four collections of religious hymns and rituals, mentioned earlier, known as the Vedas. These were composed in an ancient form of Sanskrit called Vedic Sanskrit, also known as Old Indo-Aryan.1 The Rig-Veda is the oldest of these ancient documents and is believed to have been composed as early as 1100 BCE.2 This text already displays a tendency toward extensive use of numbers, especially powers of ten. Here are some of its verses:
No bad hymns am I offering by exerting my intellect
In praise of Bhavya ruling on the Indus
Who assigned to me a thousand sacrifices,
That incomparable king desirous of fame.
A hundred gold pieces from the fame-seeking king,
Together with a hundred horses as a present have I received,
I, Kakshivant, obtained also a hundred cows from my master
Who exalted thereby his fame immortal up to heaven.3
The historian of India John Keay notes that, in the ending verse of this hymn, “by substituting sexual terms for words like ‘bliss’ and ‘creation,’ it is just possible to grasp” a meaning that made an expert, B. K. Ghosh of Calcutta University, describe this hymn as obscene. We may view it as erotic:
O resplendent lord, with brilliant radiance may you be delighted.
May your own bliss be consummated. Your delightful creation,
The holder of your bliss, is as exhilarating as the bliss itself.
For you, the vigor, equally invigorating is the bliss,
O mighty, giver of a thousand pleasures.4
We find in the Rig-Veda sexual imagery and also extensive use of numbers. According to historian of India John McLeish: “From the time of their earliest civilizations, the inhabitants of the Indian subcontinent had a highly sophisticated awareness of numbers.”5 McLeish further says that the people of Mohenjo Daro—one of the first known cities on the Indian subcontinent, part of the Indus Valley civilization, which flourished some 4,000 years ago—“used a simple decimal system and had methods of counting, weighing and measuring that were far more advanced than those of their contemporaries in Egypt, Babylonia, and Mycenean Greece. Vedic altars had to be built to exacting mathematical prescriptions; the correct dimensions and the right geometry were crucial.”6
It appears that numbers in ancient India were invented for religious purposes very early in human history. While numbers were of a practical concern in the West—a necessity of banking, accounting, and everyday purposes—in the East numbers acquired a spiritual, religious meaning.
I read many sources on Indian mathematics. In the 1925 book on the history of mathematics by David Eugene Smith, I found the following:
The early numerals of India are of various kinds. The earliest known forms are found in inscriptions of King Ashoka, the great patron of Buddhism, who reigned over most of India in the third century BC. The characters are not uniform and vary to meet linguistic conditions in different parts of India. Karosthi numerals are simply vertical marks; the Brahmi characters are more interesting. The Nana Ghat inscriptions, from the Nana Ghat cave, 75 miles from Puna, are a century after Ashoka’s edicts.7
These last numerals from the Nana Ghat inscriptions include a 7 that looks just like our 7, and the 10 looking like the Greek letter alpha. They are shown below.
Numerals from the Nana Ghat cave inscriptions, showing 10 and 7 at center.
These appear to be among the earliest numerals that ultimately evolved into the ones we use today. Buddhist monks inscribed them on the wall of a cave high on a mountain in the Western Ghats. They lived in the cave and used it as a place of worship during the second century BCE. We also know that Buddhist travelers throughout Asia were the main conduit for the eventual spread of the base-10 number system across the continent. To visit the cave, one must make the arduous four-hour climb up the steep incline to the bluffs that hide the entrance to this underground Buddhist site. The Indian government has not done enough to preserve it, and the inscriptions bearing numerals that are the progenitors of our number system are now degraded through vandalism and ne
glect.
But where did the numerals go from there? How did they develop further, after their formulation during the time of Ashoka?
At the National Museum in New Delhi I found a large display explaining the evolution of the letters in Hindi and in other Asian languages. Not far from the display area I saw a working research center. I walked over and began a conversation with two researchers; I was surprised by what one of them said: “We don’t like to admit it, but our written language really originates from Aramaic.” This was unexpected. “Well,” the middle-aged scholar wearing a jacket and bow tie continued, “India had long-standing trade relations with the Middle East and with Greece, and Aramaic—the lingua franca of the ancient Near East—influenced the development of our own script.”
But I assumed that the numerals could not have come from there, since numbers used in the Near East were either the base-60 Babylonian ones or the Greco-Roman letter-kind of numerals. I made this observation to the two researchers, and they nodded in agreement and said that perhaps the numerals were indeed a genuine Indian invention, even if the progenitors of the written script had arrived here from the Near East.
In fact, the earliest numerals ever used were in all likelihood Phoenician letters, from which the Hebrew, Aramaic, and other Semitic alphabets evolved.8 Phoenician is the oldest language in the Near East, and we know that Pythagoras traveled in this region and learned some of his early notions about mathematics from the Phoenicians and the Egyptians and their priests. Our letter A and the Hebrew letter aleph both derive from the Phoenician letter aluf, which means bull and was inspired by a stylized drawing of the head of a bull. This letter once stood for the number 1. As aleph, it is still employed in that role by some religious Jews today. As alpha, it was used for 1 by the ancient Greeks. The Romans then chose to use I for 1, II for 2, and so on, while the Greeks continued in their own alphabet with beta and gamma for 2 and 3, and so on, and the ancient Hebrews with bet and gimmel, and onward.
Finding Zero Page 5
