Russell and mathematical certainty

 

 “I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure then the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.”

“Mathematics is, I believe,” says Russell, “the chief source of the belief in eternal and exact truth, as well as in a super-sensible intelligible world. Geometry deals with exact circles, but no sensible object is exactly circular; however carefully we may use our compasses, there will be some imperfections and irregularities. This suggests the view that all exact reasoning applies to ideal as opposed to sensible objects; it is natural to go further, and to argue that thought is nobler than sense, and the objects of thought more real than those of sense-perception. Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for mathematical objects, such as number, if real at all, are eternal and not in time. Such eternal objects can be conceived as God’s thoughts. Hence Plato’s doctrine that God is a geometer, and Sir James Jeans’ belief that He is addicted to arithmetic. Rationalistic as opposed to apocalyptic religion has been, ever since Pythagoras, and notably ever since Plato, very completely dominated by mathematics and mathematical method.

Why Mathematicians Can’t Find the Hay in a Haystack

By Kevin Hartnett

In math, sometimes the most common things are the hardest to find. 

The first time I heard a mathematician use the phrase, I was sure he’d misspoken. We were on the phone, talking about the search for shapes with certain properties, and he said, “It’s like looking for hay in a haystack.”

“Don’t you mean a needle?” I almost interjected. Then he said it again.

In mathematics, it turns out, conventional modes of thought sometimes get turned on their head. The mathematician I was speaking with, Dave Jensen of the University of Kentucky, really did mean “hay in a haystack.” By it, he was expressing a strange fact about mathematical research:  Sometimes the most common things are the hardest to find.

“In many areas of mathematics you’re looking for examples of something, and examples are really abundant, but somehow any time you try to write down an example, you get it wrong,” said Jensen.

The hay-in-a-haystack phenomenon is at work in one of the first objects that kids encounter in mathematics: the number line. Points on the number line include the positive and negative integers (such as 2 and –29), rational numbers (ratios of integers like 3/2 and1/137) and all irrational numbers — those numbers, like pi or root 2, that can’t be expressed as a ratio.

Irrational numbers occupy the vast, vast majority of space on a number line — so vast, in fact, that if you were to pick a number on the number line at random, there is literally a 100 percent chance that it will be irrational.*

* The reason it’s 100 percent and not 99.9999 percent has to do with the strange things that happen once you start thinking about the probabilities when infinity is involved. Around the turn of the 20th century, the mathematician Henri Lebesque helped to develop “measure theory,” which provided a mathematically rigorous way for calculating the probabilities when infinite outcomes are possible.

Yet despite their overwhelming presence, we almost never encounter irrational numbers in our daily lives. Instead we count with whole numbers and follow recipes with fractions. The numbers we know best are the extremely rare numbers, the special numbers — the needles in the haystack.

The hay is hard to find precisely because it’s so unexceptional. Rational numbers have the distinctive property that it’s possible to write them down. This calls them to our attention. Irrational numbers have an infinite decimal expansion. You couldn’t write one down even with an endless amount of time. That these numbers lack the exceptional property of “write-down-able-ness” is what makes them nearly invisible to our way of seeing.

“We’re looking with a magnet, and you’re not going to find hay with magnet; you’re only going to find needles,” said Dhruv Ranganathan, a mathematician who is in the midst of a move from the Massachusetts Institute of Technology to the University of Cambridge.

The search for hay in a haystack characterizes many different areas of math, including the subject of my most recent Quanta article, “Tinkertoy Models Produce New Geometric Insights.” There I wrote about mathematicians who are investigating the relationship between geometric shapes and the equations used to describe them. In rare cases, objects can be expressed by simple equations. These are the needles, the shapes we know best: lines, parabolas, circles, spheres.

The overwhelming preponderance of shapes resist such elegant formulation. They may be everywhere, but because you can’t write down the equations that describe them, it’s hard to establish that even a single one of them exists.

In my article, I explained how techniques from a field called “tropical geometry” serve as an especially sly way of deducing the existence of these ubiquitous geometric objects — the ones that, like the irrational numbers, are everywhere, even if you can’t write them down.

In mathematics it often happens that either something doesn’t exist, or it exists in abundance. The nature of those abundant objects might make them hard to detect, but if you’re a mathematician and you believe they’re there, and you believe they constitute almost all of everything, your task is straightforward: Find just one.

It’s as if you were convinced the oceans were filled with water, said Ranganathan, but every time you took a sample, you came up with something else — a shell, a rock, a plant. Yet to start to believe your hypothesis was correct, you’d hardly need to empty the sea.

“All you have to do is find any water,” he said. “One droplet of water will do.”

Quanta Magazine

Restrictions Birth Creativity

Human laws are flexible, subject to repeal and renegotiation. The drinking age is 21 in the US, 18 in Britain, 16 in Cuba, “never” in Afghanistan, and “step right up” in Cambodia. Any country can stiffen or relax its drinking laws at a whim (and of course, the stiffer your drink and the laxer your laws, the more whim-prone you become). Geometry’s laws are not like that. There’s no wiggle room: no president to issue pardons, no jury to acquit, no officer to let you off with a warning. Math’s rules are self-enforcing, unbreakable by their nature.

Yet as we’ve seen, and shall see many times again, that’s not a bad thing. Restrictions birth creativity. The laws on what shapes can’t do come packaged with case studies illuminating what they can. In design projects ranging from sturdy buildings to useful paper to planet-destroying space stations, geometry inspires even as it constrains.

Islamic Art and Mathematics

The Greeks were not the only people intrigued by the wonders of geometrical shapes. The most sacred object in Islam is a Platonic solid: the Ka’ba, or Cube, is the black palladium at the centre of Mecca’s Sacred Mosque, around which pilgrims walk anticlockwise during the Hajj. (In fact, its dimensions make it just off a perfect cube.) The Ka’ba also marks the point that worshippers must face during daily prayer, wherever they are in the world. Mathematics plays more of a role in Islam than in any other major religion. More than a millennium before the advent of GPS technology, the requirement to face Mecca relied on complicated astronomical calculations—which is one reason why Islamic science was unequalled for almost a thousand years.

Islamic art is epitomized by the ingenious geometrical mosaic arrangements on the walls, ceilings and floors of its sacred buildings, a consequence of the ban on images of people and animals in holy sites. Geometry was thought to express truth beyond what was merely human, much in keeping with the Pythagorean position that the universe reveals itself through mathematics. The symmetries and endless loops that Islamic craftsmen created in their patterns were an allegory of the Infinite, an expression of the sacred, mathematical order of the world.

**

The discovery of nonperiodic tiling was an exciting breakthrough for maths, but not as exciting as it would later be for physics and chemistry. In the 1980s researchers were amazed to discover a type of crystal that they did not believe existed. The tiny structure displayed a nonperiodic pattern, behaving in three dimensions just like Penrose’s tiles did in two. The existence of these structures—called quasicrystals —changed the way scientists understood the nature of matter, since it contradicted classical theory that all crystals must have symmetrical lattices derived from the Platonic solids. Penrose may have invented his tiles for fun, but they were unduly prophetic about the natural world.

Half a millennium ago, Islamic geometers might also have understood about non-periodic tessellations. In 2007 Peter J. Lu from Harvard University and Paul J. Steinhardt from Princeton claimed that their studies of mosaics in Uzbekistan, Afghanistan, Iran, Iraq and Turkey showed that the craftsmen had made ‘nearly perfect quasi-crystalline Penrose patterns, five centuries before discovery in the West’. It is possible, therefore, that Islamic mathematics may have been even more advanced than historians of science have traditionally thought.

Numerology and Pythagorean BrotherHood

This, he explained, showed which numbers corresponded to which letters. He took his fingers to the first column: ‘Letters that equal one are A, J, S. Allah, Jehovah, Jesus, Saviour, Salvation. Two is the number of diplomats, ambassadors. Two gives good advice, two you love, you’re a team player, that’s B, K and T, that’s why if you go to a Burger King you can have it your way. Number three controls radio, TV, entertainment and numerology. C, L, U. Of course, you go into radio and television, you don’t have a clue.’ He gave me an ironic wink. ‘But if you learn numerology, it will open you up to the clue of life. Number four: D, M, V. How many wheels on a car? Where do you get the licence? The Department of Motor Vehicles. Five is halfway between 1 and 10: E, N and W. Five is the number of change. If you scramble the letters you get “new”. Six is the number of Venus, love, family, community. When you see a beautiful woman, what do you see? A FOX. Seven is the number of spirituality. Jesus was born on the twenty-fifth, 2 and 5 equals 7. Eight is the number of business, finance, commerce, money. Where do you keep the money? In the headquarters. Nine is the only one that has two letters. I and R. You ever talked to a Jamaican? Everything is irie, man.’

On conclusion he put down his pen and looked me full in the face: ‘This,’ he said, ‘is Jerome Carter’s method of the Pythagorean system.’

Pythagoras is the most famous name in mathematics, entirely due to his theorem about triangles. (More about that later.) He is credited with other contributions, though, such as the discovery of ‘square numbers’.

**

Pythagoras was entranced by the numerical patterns he found in nature, believing that the secrets of the universe could be understood only through mathematics. Yet rather than seeing maths merely as a tool to describe nature, he saw numbers as somehow the essence of nature—and he tutored his flock to revere them. For Pythagoras was not just a scholar. He was the charismatic leader of a mystical sect devoted to philosophical and mathematical contemplation, the Pythagorean Brotherhood, which was a combination of health farm, boot camp and ashram. Disciples had to obey strict rules, such as never urinating towards the sun, never marrying a woman who wears gold jewellery, and never passing an ass lying in the street. So select was the group that those wishing to join the Brotherhood had to go through a five-year probationary period, during which they were allowed to see Pythagoras only from behind a curtain.

In the Pythagorean spiritual cosmos, ten was divine not for any reason to do with fingers or toes, but because it was the sum of the first four numbers (1 + 2 + 3 + 4 = 10), each of which symbolized one of the four elements: fire, air, water and earth. The number 2 was female, 3 was male, and 5 – their union—was sacred. The crest of the Brotherhood was the pentagram, or five-pointed star. While the idea of worshipping numbers may now seem bizarre, it perhaps reflects the scale of wonderment at the discovery of the first fragments of abstract mathematical knowledge. The excitement of learning that there is order in nature, when previously you were not aware that there was any at all, must have felt like a religious awakening.

Pythagoras’s spiritual teachings were more than just numerological. They included a belief in reincarnation, and he was probably a vegetarian. In fact, his dietary requirements have been hotly debated for more than 2000 years. The Brotherhood famously forbade ingestion of the small, round, black fava bean, and one account of Pythagoras’s death has him fleeing attackers when he came to a field of fava beans. As the story goes, he preferred to be captured and killed rather than tread on them. The reason the beans were sacred, according to one ancient source, was that they sprouted from the same primordial muck as humans did. Pythagoras had proved this by showing that if you chew up a bean, crush it with your teeth, and then put it for a short while in the sun, it will begin to smell like semen. A more recent hypothesis was that the Brotherhood was just a colony for those with hereditary fava-bean allergies.

Pythagoras lived in the sixth century BC. He did not write any books. All we know about him was written many years after he died. Though the Brotherhood was lampooned in ancient Athenian comic theatre, by the beginning of the Christian Era Pythagoras himself was seen in a rather favourable light, viewed as being a unique genius; his mathematical insights making him the intellectual forefather of the great Greek philosophers. Miracles were attributed to him, and some authors, rather oddly, claimed that he had a thigh made of gold. Others wrote that he once walked across a river, and the river called out to him, loud enough for all to hear, ‘Greetings, Pythagoras’. This posthumous myth-making has parallels with the story of another Mediterranean spiritual leader and, in fact, Pythagoras and Jesus were temporarily religious rivals. As Christianity was taking root in Rome in the second century CE, the empress Julia Domna encouraged her citizens to worship Apollonius of Tyna, who claimed he was Pythagoras reincarnated.

Pythagoras has a dual and contradictory legacy: his mathematics and his anti-mathematics. Maybe, in fact, as some academics now suggest, the only ideas that can be correctly attributed to him are the mystical ones. Pythagorean esotericism has been a constant presence in Western thought since antiquity, but was especially in vogue during the Renaissance, thanks to the rediscovery of a poem of ‘self-help’ maxims written around the fourth century BC called The Golden Verses of Pythagoras. The Pythagorean Brotherhood was the model for many occult secret societies and influenced the creation of freemasonry, a fraternal organization with elaborate rituals that is believed to have almost half a million members in the UK alone. Pythagoras also inspired the ‘founding mother’ of Western numerology, Mrs L. Dow Balliett, an Atlantic City housewife who wrote the book The Philosophy of Numbers in 1908. ‘Pythagoras said the Heavens and Earth vibrate to the single numbers or digits of numbers,’ she wrote, and she proposed a system of fortune-telling in which each letter of the alphabet corresponded with a number from 1 to 9. By adding up the numbers of the letters in a name, she asserted, personality traits could be divined. I tested this idea on myself. ‘Alex’ is 1 + 3 + 5 + 6 = 15. I completed the process by adding the two digits of the answer, getting 1 + 5 = 6. This gives me a name vibration of six, which means that I ‘should always be dressed with care and precision; be fond of dainty effects and colors, lifting your especial colors of orange, scarlet and heliotrope into their lighter shades, yet always keeping their true tones’. My gems are topaz, diamond, onyx and jasper, while my mineral is borax, and my flowers are tuberose, laurel and chrysanthemum. My odour is japonica.

Numerology, of course, is now an established dish on the buffet of modern mysticism, with no shortage of gurus willing to advise on lottery numbers or speculate on the portent of a prospective date. It sounds like harmless fun—and I enjoyed my conversation with Jerome Carter immensely—yet giving numbers spiritual significance can also have sinister consequences. In 1987, for example, the military government in Burma issued new banknotes with a face value divisible by nine—for the sole reason that nine was the ruling general’s favourite number. The new notes helped precipitate an economic crisis, which led to an uprising on 8 August 1988 – the eighth of the eighth of ’88. (Eight was the anti-dictatorship movement’s favourite number.) The protest was violently put down, however, on 18 September: in the ninth month, on a day divisible by nine.

**

In some commentaries about Pythagoras it is said that before he founded the Brotherhood he travelled on a fact-finding mission to Egypt. If he had spent any time on an Egyptian building site he would have seen that the labourers used a trick to create a right angle that was an application of the theorem that would later gain his name. A rope was marked with knots spread out at a distance of 3, 4 and 5 units. Since 32 + 42 = 52, when the rope was stretched around three posts, with a knot at each post, it formed a triangle with one right angle.

**

Rope-stretching was the most convenient way to achieve right angles, which were needed so that bricks, or giant stone blocks such as those used to construct the Pyramids, could be stacked next to and on top of each other. (The word hypotenuse comes from the Greek for ‘stretched against’.) The Egyptians could have used many other numbers in addition to 3, 4 and 5 to get real right angles. In fact, there is an infinite number of numbers a, b and c such that a² + b² = c². They could have marked out their rope into sections of 5, 12 and 13, for example since 25 + 144 = 169, or 8, 15 and 17, since 64 + 225 = 289, or even 2772, 9605 and 9997 (7,683,984 + 92,256,025 = 99,940,009) though that would hardly have been practical. The numbers 3, 4, 5 are best suited to the task. As well as being the triple with the lowest value, it is also the only one whose digits are consecutive integers. Due to its rope-stretching heritage, the right-angled triangle with sides that are in the ratio 3:4:5 is known as an Egyptian triangle.

In a Word

• Words are also extremely important. If we do not know the word for a thing, we cannot easily talk about that thing—without using the word “thing” or pointing or describing what the object is used for. We’re forced to resort either to gesture or circumlocution (a lengthy way of expressing something). We cannot be ignorant of words connected with international relations or computers or the national economy or the life of the mind. If we are, we will find ourselves unable to take part in discussions of matters important to ourselves and to society.

• SCHOOL “School” traces its roots to a Latin noun with the same meaning, schola. The Romans borrowed the Greek noun schole, referring to a group of young men who studied with a philosopher. The Greeks chose schole, meaning “leisure,” because only those who did not have to work could afford to study.

• BRAZIL Early European explorers of the Americas discovered that the native people knew how to make a dye extract that produced bright red and deep purple colors. The dye used by the native Americans came from a common tree, known as the brasil. European merchants soon developed a great trade in brasilwood. As a result of the Treaty of Tordesillas in 1494, the Portuguese claimed the land where these trees grew and named the area Terra de Brasil or “Land of Red-dye Wood.” Soon map makers and others began to refer to the land as Brasil. English later adopted the name and its pronunciation, only changing the spelling to include a “z” instead of an “s.”

• ALGEBRA A long time ago, the Arabic term al jabr was used to refer to the medical operation used to set broken bones. It means the reuniting of something broken. References in old texts also show that al jabr later came to refer to the process of putting numbers together. In the 16th century, the two words appeared in English texts as algeber. Within a century, algeber came to be used in its present sense, as the name of the branch of mathematics that includes equations and the relationships between numbers.

• GEOMETRY Geometry means “to measure the earth.” It was formed by joining two Greek words, ge, meaning “earth,” and metrein, meaning “to measure.” Such a choice was appropriate, since this science was first used to measure the land.

• MATHEMATICS Because the early Greek thinkers saw numbers as a way to help them explain relationships, they spent much time developing formulas and equations. As a result, the Greeks used their adjective mathematikos, meaning “eager to learn,” to name this new science. Mathematikos itself was a derivative of the Greek verb manthanein, “to learn.”

• The Black Death was a plague that reached Genoa, Italy, on New Year’s Eve 1347 and then spread across Europe and into England, Wales, Ireland, and parts of Scandinavia before entering Russia in 1353. It then spread into portions of Asia. Overall, the Black Death killed anywhere from one-third to one-half of all people living in these areas.

• The actions of anyone walking or talking on the deck of a vessel are usually clearly visible. Below deck, however, in the passageways and dark corners, mutinies, smuggling schemes, and other plots can be hatched unseen. Thus, the expression “to be aboveboard” came to denote being honest, open-minded, and not concealing anything.

• “Tartar,” however, may have been chosen because of its resemblance to “Tatar,” the name used to refer to several tribes that the renowned Mongol leader Genghis Khan conquered early in his career. According to tradition, when an Irish soldier was in battle with a Turkish soldier some time in the 1800s, the Irish soldier called out to his officer that he had “caught a Tartar.” When ordered to bring him in, the soldier replied, “I can’t. This Tartar won’t let me.” The story was repeated, and the expression “to catch a Tartar” came to represent a situation in which a person wins more than he or she can control.

• WHITE ELEPHANT In the southeast Asian country of Thailand (formerly called Siam), it was a national tradition to present the emperor with any white elephant that was captured. These rare elephants were sacred to the emperor. It was commonly believed that if the emperor became dissatisfied with one of his nobles, he would order a white elephant sent to him. Unable to use or destroy the sacred animal, the noble was forced to bear the cost of maintaining the huge beast. In 1629, the emperor of Siam sent King Charles I of England an elephant (records do not say whether it was white). The cost of its maintenance became a great drain on the royal treasury. Gradually, the phrase “white elephant” came to designate possessions that are expensive but useless.

• MARCH In the early Roman calendar, March was the first month of the year and the time when military campaigns began. (Around 45 b.c., Julius Caesar revised the calendar, making January the first month and March the third month.) Since March was when fair weather returned to the Roman world, it had always been considered the perfect time to go to battle. The Romans had named the month in honor of their god of war, Mars.

• HAZARD “Risk,” “chance,” and “danger”—these three words all define the English term “hazard.” Its roots do the same, since hazard traces its origin to the Arabic article al (“the”) and the Arabic noun zahr (“die”). Throwing dice (the plural of “die”) involves chance and the unknown. Other languages also adapted al-zahr. For example, in Spanish it became azar, “an unexpected accident.”