April 28, 2005

The Books: "The Discoverers: A History of Man's Search to Know His World and Himself" (Daniel J. Boorstin)

Next book in my daily book excerpt from the science and philosophy shelf:

Discoverers.jpgAn excerpt from the MASSIVE book by Daniel Boorstin The Discoverers.

This excerpt comes from the first chapter, which discusses human beings and time - how different cultures have figured out time, and the calendar, in different eras. It's amazing stuff, I tell ya!

EXCERPT FROM The Discoverers, by Daniel Boorstin.

Many of the early Christians, following their own literal interpretation of the Bible, fixed the death of Jesus on a Friday, and the Easter resurrection on the following Sunday. But if the anniversary of the festival was to follow the Jewish lunar calendar, there was no assurance that Easter would occur on a Sunday. The bitter quarrel over the calendar led to one of the earliest schisms between the Eastern Orthodox Church and the Church of Rome. The Eastern Christians, holding to the lunar calendar, continued to observe Easter on the fourteenth day of the lunar month, regardless of the day of the week. At the very first ecumenical (worldwide) council of the Christian Church, held at Nicaea in Asia Minor in 325, one of the world-unifying questions to be decided was the date of Easter. A uniform date was fixed in such a way as both to stay with the traditional lunar calendar and to assure that Easter would always be observed on Sunday.

But this did not quite settle the matter. For community planning someone still had to predict the phases of the moon and locate them on a solar calendar. The Council of Nicaea had left this task to the bishop of Alexandria. In the ancient center of astronomy he was to forecast the phases of the moon for all future years. Disagreement over how to predict those specified cycles led to a division in the Church, with the result that different parts of the world continued to observe Easter on different Sundays.

The reform of the calendar by Pope Gregory XIII was needed because the year that Julius Caeser had borrowed from the Egyptians, and which had ruled Western civilization since then, was not a precise enough measure of the solar cycle. The actual solar year -- the time required for the earth to complete an orbit around the sun -- is 365 days, 5 hours, 48 minutes, and 46 seconds. This was some 11 minutes and 14 seconds less than the 365 1/4 days in the Egyptian year. As a result, dates on the calendar gradually lost their intended relation to solar events and to the seasons. The crucial date, the vernal equinox, from which Easter was calculated, had been fixed by the First Council of Nicaea at March 21. But the accumulating inaccuracy of the Julian calendar meant that by 1582 the vernal equinox was actually occurring on March 11.

Pope Gregory XIII, though notorious now for this public Thanksgiving for the brutal massacre of Protestants in Paris on Saint Bartholomew's Day (1572), was in some matters an energetic reformer. He determined to set the calendar straight. Climaxing a movement for calendar reform which had been developing for at least a century, in 1582 Pope Gregory ordained that October 4 was to be followed by October 15. This meant, too, that in the next year the vernal equinox would occur, as the solar calendar of seasons required, on March 21. In this way the seasonal year was restored to what it had been in 325. The leap years of the old Julian calendar were readjusted. To prevent the accumulationi of another 11-minute-a-year discrepancy, the Gregorian calendar omitted the leap day from years ending in hundreds, unless they were divisible by 400. This produced the modern calendar by which the West still lives.

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April 27, 2005

The Books: "Galileo's Daughter: A Historical Memoir of Science, Faith, and Love" (Dava Sobel)

Next book in my science and philosophy shelf:

GalileosDaughter.jpgAnother book by Dava Sobel - this one called Galileo's Daughter: A Historical Memoir of Science, Faith, and Love. A marvelous book. It tells the story of Galileo's life, as well as tells the story of the life of his illegitimate (but beloved) daughter, whom he put into a convent as a young girl. She had to renounce the world ... yet she and her father remained devoted to one another (even through his trial/inquisition). His letters to her have been lost, sadly, but all of her letters are still around, and were sitting in a library in Rome, I think, collecting dust. Dava Sobel, researching some other project, heard that there was this huge archive of letters from "Suor Marie Celeste" - Galileo's daughter - and they had never been translated into English. Sobel sensed that there was a huge story there, one that had yet to be told, so she went to Italy, translated the letters herself, and wrote this wonderful book. Part science, part biography, part epistolary memoir, it gives an insider's view (through her incredible letters to her famous father) of that world. One of the things I find most moving about her letters to him, is that she never doubted his faith, at a time when he was being treated like a heretic. His discoveries about the universe didn't shake HIS faith, and didn't shake her faith either. Even though she lived in a cloister, she still was aware of his scientific explorations, and discoveries ... and never once did she question him, or back off from him. It can't have been easy for her since she was a nun in the Church that was persecuting him. Judging from her letters, she remained steadfastly supportive, saying that his discoveries merely expanded her own love for God, since he obviously was so much more powerful and imaginative than previously thought. Extraordinary.

It's a really interesting book.

EXCERPT FROM Galileo's Daughter: A Historical Memoir of Science, Faith, and Love by Dava Sobel.

Galileo's daughter, born of his long illicit liaison with the beautiful Marina Gamba of Venice, entered the world in the summer heat of a new century, on August 13, 1600 -- the same year the Dominican friar Giordano Bruno was burned at the stake in Rome for insisting, among his many heresies and blasphemies, that the Earth traveled around the Sun, instead of remaining motionless at the center of the universe. In a world that did not yet know its place, Galileo would engage this same cosmic conflict with the Church, trading a dangerous path between the Heaven he revered as a good Catholic and the heavens he revealed through his telescope.

Galileo christened his daughter Virginia, in honor of his "cherished sister". But because he never married Virginia's mother, he deemed the girl herself unmarriageable. Soon after her thirteenth birthday, he placed her at the Convent of San Matteo in Arcetri, where she lived out her life in poverty and seclusion.

Virginia adopted the name Maria Celeste when she became a nun, in a gesture that acknowledged her father's fascination with the stars. Even after she professed a life of prayer and penance, she remained devoted to Galileo as though to a patron saint. The doting concern evident in her condolence letter [on the occasion of Galileo's sister's death] was only to intensify over the ensuing decade as her father grew old, fell more frequently ill, pursued his singular research nevertheless, and published a book that brought him to trial by the Holy Office of the Inquisition...

Thus Suor Maria Celeste consoled Galileo for being left alone in his world, with daughters cloistered in the separate world of nuns, his son not yet a man, his former mistress dead, his family of origin all deceased or dispersed.

Galileo, now fifty-nine, also stood boldly alone in his worldview, as Suor Maria Celeste knew from reading the books he wrote and the letters he shared with her from colleagues and critics all over Italy, as well as from across the continent beyond the Alps. Although her father had started his career as a professor of mathematics, teaching first at Pisa and then at Padua, every philosopher in Europe tied Galileo's name to the most startling series of astronomical discoveries ever claimed by a single individual.

In 1609, when Suor Maria Celeste was still a child in Padua, Galileo had set a telescope in the garden behind his house and turned it skyward. Never-before-seen stars leaped out of the darkness to enhance familiar constellations; the nebulous Milky Way resolved into a swath of densely packed stars; mountains and valleys pockmarked the storied perfection of the Moon; and a retinue of four attendance bodies traveled regularly around Jupiter like a planetary system in miniature.

"I render infinite thanks to God," Galileo intoned after those nights of wonder, "for being so kind as to make me alone the first observer of marvels kept hidden in obscurity for all previous centuries."

The newfound worlds transformed Galileo's life. He won appointment as chief mathematician and philosopher to the grand duke in 1610, and moved to Florence to assume his position at the court of Cosimo de Medici. He took along wtih him his two daughters, then ten and nine years old, but he left Vincenzio, who was only four when greatness descended on the family, to live a while longer in Padua with Marina.

Galileo found himself lionized as another Columbus for his conquests. Even as he attained the height of his glory, however, he attracted enmity and suspicion. For instead of opening a distant land dominated by heathens, Galileo trespassed on holy ground. Hardly had his first spate of findings stunned the populace of Europe before a new wave followed: He saw dark spots creeping continuously across the face of the Sun, and "the mother of loves," as he called the planet Venus, cycling through phases from full to crescent, just as the Moon did.

All his observations lent credence to the unpopular Sun-centered universe of Nicolaus Copernicus, which had been introduced over half a century previously, but foundered on lack of evidence. Galileo's efforst provided the beginning of a proof. And his flamboyant style of promulgating his ideas -- sometimes in bawdy humorous writings, sometimes loudly at dinner parties and staged debates -- transported the new astronomy from the Latin Quarters of the universities into the public arena. In 1616, a pope and a cardinal inquisitor reprimanded Galileo, warning him to curtail his forays into the supernal realms. The motions of the heavenly bodies, they said, having been touched upon in the Psalms, the Book of Joshua, and elsewhere in the Bible, were matters best left to the Holy Fathers of the Church.

Galileo obeyed their orders, silencing himself on the subject. For seven cautious years he turned his efforts to less perilous pursuits, such as harnessing his Jovian satellites in the service of navigation, to help sailors discover their longitude at sea. He studied poetry and wrote literary criticism. Modifying his telescope, he developed a compound microscope. "I have observed many tiny animals with great admiration," he reported, "among which the flea is quite horrible, the gnat and the moth very beautiful; and with great satisfaction I have seen how flies and other little animals can walk attached to mirrors, upside down."

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April 26, 2005

The Books: "Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time"(Dava Sobel)

Next book on the science and philosophy bookshelf:

Longitude.jpgDava Sobel's wonderful Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time . My mother was the one who made me read this book. She had read it, and found the whole thing intensely inspiring and moving. It is the story of "the longitude problem". In the age of exploration, it was still impossible to calculate the longitude. Latitude was easy, but longitude not so. In order to know your longitude, clocks have to be able to keep time at sea. You have to know what time it is where you are, as well as what time it is back at some fixed point of zero-longitude. But clocks would slow down, at sea, they would get waterlogged, whatever. Sailors did the best they could, but - at least from the story told - catastrophes occurred because of this sailing-blind-without-longitude problem. In 1714, the Parliament in England offered an enormous prize to anybody who could solve this longitude problem.

Along comes a man named John Harrison, who devoted his life to solving the longitude problem. And - like so many other stories of genius - John Harrison was not a scientist, or an astronomer - he had no formal education, he wasn't a Newton or a Galileo. He was a clockmaker. And he also had what it took, in terms of intellectual endurance ... to keep trying, to keep experimenting, until he got it right. It's so so inspiring what he did.

If you haven't read this book, I highly recommend it. Harrison ended up making a series of time-pieces - called H1, H2, H3 ... With each one, he got closer and closer to perfection. H4 is the timepiece that won the prize. H1, H2, and H3 were all heavy, large - After all, these timepieces would need to withstand a storm at sea, would need to keep time steadily throughout the massive up and down motion of the ocean at such times. But H4 is a small and simple pocketwatch. Here is what it looks like.

Here's an excerpt:

EXCERPT FROM Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time , by Dava Sobel.

The active quest for a solution to the problem of longitude persisted over four centuries and across the whole continent of Europe. Most crowned heads of state eventually played a part in the longitude story, notably King George III of England and King Louis XIV of France. Seafaring men such as Captain William Bligh of the Bounty and the great circumnavigator Captain James Cook, who made three long voyages of exploration and experimentation before his violent death in Hawaii, took the more promising methods to sea to test their accuracy and practicability.

Renowned astronomers approached the longitude challenge by appealing to the clockwork universe: Gallileo Galilei, Jean Dominique Cassini, Christiaan Huygens, Sir Isaac Newton, and Edmond Halley, of comet fame, all entreated the moon and stars for help. Palatial observatories were founded at Paris, London, and Berlin, for the express purpose of determining longitude by the heavens. Meanwhile, lesser minds devised schemes that depended on the yelps of wounded dogs, or the cannon blasts of signal ships strategically anchored -- somehow -- on the open ocean.

In the course of their struggle to find longitude, scientists struck upon other discoveries that changed their view of the universe. These include the first accurate determinations of the weight of the Earth, the distance to the stars, and the speed of light.

As time passed and no method proved successful, the search for a solution to the longitude problem assumed legendary proportions, on a par with discovering the Fountain of Youth, the secret of perpetual motion, or the formula for transforming lead into gold. The governments of the great maritime nations -- including Spain, the Netherlands, and certain city-states of Italy -- periodically roiled the fervor by offering jackpot purses for a workable method. The British Parliament, in its famed Longitude Act of 1714, set the highest bounty of all, naming a prize equal to a king's ransom (several million dollars in today's currency) for a "Practicable and Useful" means of determining longitude.

English clockmaker John Harrison, a mechanical genius who pioneered the science of portable precision timekeeping, devoted his life to this quest. He accomplished what Newton had feared was impossible: He invented a clock that would carry the true time for the home port, like an eternal flame, to any remote corner of the world.

Harrison, a man of simple birth and high intelligence, crossed swords with the leading lights of his day. He made a special enemy of the Reverent Nevil Maskelyne, the fifth astronomer royal, who contested his claim to the coveted prize money, and whose tactics at certain junctures can only be described as foul play.

With no formal education or apprenticeship to any watchmaker, Harrison nevertheless constructed a series of virtually friction-free clocks that required no lubrication and no cleaning, that were made from materials impervious to rust, and that kept their moving parts perfectly balanced in relation to one another, regardless of how the world pitched or tossed about them. He did away with the pendulum, and he combined different metals inside his works in such a way that when one component expanded or contracted with changes in temperature, the other counteracted the change and kept the clock's rate constant.

His every success, however, was parried by members of the scientific elite, who distrusted Harrison's magic box. The commissioners charged with awarding the longitude prize -- Nevil Maskelyne among them -- changed the contest rules whenever they saw fit, so as to favor the chances of astronomers over the likes of Harrison and his fellow "mechanics". But the utility and accuracy of Harrison's approach triumphed in the end. His followers shepherded Harrison's intricate, exquisite invention through the design modifications that enabled it to be mass produced and enjoy wide use.

An aged, exhausted Harrison, taken under the wing of King George III, ultimately claimed his rightful monetary reward in 1773 -- after forty struggling years of political intrigue, international warfare, academic backbiting, scientific revolution, and economic upheaval.

All these threads, and more, entwine in the lines of longitude. To unravel them now -- to retrace their story in an age when a network of orbiting satellites can nail down a ship's position within a few feet in just a moment or two -- is to see the globe anew.

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April 25, 2005

The Books: "In Search of Schrodinger's Cat: Quantum Physics and Reality" (John Gribbin)

Next book on the science and philosophy shelf:

SchrodingersCat.jpgThe beautiful little physics book In Search of Schrödinger's Cat: Quantum Physics and Reality, by John Gribbin. I've quoted extensively from his book before. It's one of my favorites:

"The only existing things are atoms and empty space; all else is mere opinion."

Heat is a form of motion ...

a lone voice crying in the wilderness ...

Heisenberg's breakthrough

"At first, I was deeply alarmed."

So here's yet another excerpt from this book: This one has to do with alternative realities, and time travel.

EXCERPT FROM In Search of Schrödinger's Cat: Quantum Physics and Reality, by John Gribbin.

Cosmologists today talk quite happily about events that occurred just after the universe was born in a Big Bang, and they calculate the reactions that occurred when the age of the universe was 10-35 seconds or less. The reactions involve a maelstrom of particles and radiation, pair production and annihilation. The assumptions about how these reactions take place come from a mixture of theory and the observations of the way particles interact in giant accelerators, like the one run by CERN in Geneva. According to these calculations, the laws of physics determined from our puny experiments here on earth can explain in a logical and self-consistent fashion how the universe got from a state of almost infinite density into the state we see it in today. The theories even make a stab at predicting the balance between matter and antimatter in the universe, and between matter and radiation. Everyone interested in science, however mild and passing their interest, has heard of the Big Bang theory origin of the universe. Theorists happily play with numbers describing events that allegedly occurred during split seconds some 15 thousand million years ago. But who today stops to think what these ideas really mean? It is absolutely mind-blowing to attempt to understand the implications of these ideas. Who can appreciate what a number like 10-35 of a second really means, let alone comprehend the nature of the universe when it was 10-35 seconds old? Scientists who deal with such bizarre extremees of nature really should not find it too difficult to stretch their minds to accommodate the concept of parallel worlds.

In face, that felicitous-sounding expression, borrowed from science fiction, is quite inappropriate. The natural image of alternative realities is as alternative branches fanning out from a main stem and running alongside one another through superspace, like the branching lines of a complex railway junction. Like some super-superhighway, with millions of parallel lines, the SF writers imagine all the worlds proceeding side by side through time, our near neighbors almost identical to our own world, but with the differences becoming clearer and more distinct the further we move "sideways in time". This is the image that leads naturally to speculation about the possibility of changing lanes on the superhighway, slipping across into the world next door. Unfortunately, the math isn't quite like this neat picture.

Mathematicians have no trouble handling more dimensions than the familiar three space dimensions so important to our everday lives. The whole of our world, one branch of Everett's many-worlds reality, is described mathematically in four dimensions, three of space and one of time, all at right angles to one another, and the math to describe more dimensions all at right angles to each other and to our own four is routine number juggling. This is where the alternative realities actually lie, not parallel to our own world, but at right angles to it, perpendicular worlds branching off "sideways" through superspace. The pciture is hard to visualize, but it does make it easier to see why slipping sideways into an alternative reality is impossible. If you set off at right angles to our world -- sideways -- you would be creating a new world of your own. Indeed, on the many-worlds theory this is what happens every time the universe is faced with a quantum choice. The only way you could gete in to one of the alternative realities created by such a splitting of the universe as a result of a cat-in-the-box experiment, or a two-holes experiment, would be to go back in time in our own four-dimensional reality to the time of the experiment, and then to go forward in time along the alternative branch, at right angles to our own four-dimensional world.

This might be impossible. Conventional wisdom has it that true time travel must be impossible, because of the paradoxes involved, like the one where you go back in time and kill your own grandfather before your own father has been conceived. On the other hand, at the quantum level particles seem to be involved in time travel all the "time," and Frank Tipler has shown that the equations of general relativity permit time travel. It is possible to conceive of a kind of genuine travel forward and backward in time that does not permit paradoxes, and such a form of time travel depends on the reality of alternative universes. David Gerrold explored these possibilities in an entertaining SF book The Man Who Folded Himself, well worth reading as a guide to the complexities and subtleties of a many-worlds reality. The point is that, taking the classic example, if you back in time and kill your grandfather you are creating, or entering (depending on your point of view) an alternative world branching off at right angles to the world in which you started. In that "new" reality, your father, and yourself, are never born, but there is no paradox because you are still born in the "original" reality, and make the journey back through time and into an alternative branch. Go back again to undo the mischief you have done, and all you do is reenter the original branch of reality, or at least one rather like it.

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April 24, 2005

The Books: "Fermat's Enigma: The Epic Quest to Solive the World's Greatest Mathematical Problem" (Simon Singh)

Next book on the science and philosophy shelf:

FermatsEnigma.jpgA book about 17th century French mathematician Pierre de Fermat and his last theorem. Proving this last theorem turned out to be no easy feat, and mathematicians tried, for 350 years. It has been called "the Holy Grail of mathematics". Obviously, I'm pulling this book down from my "Math and Science for People who Love Math and Science but Don't Understand the Actual Math and Science" shelf. One of my favorite shelves! The book is by Simon Singh and it's called Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem.

The book mostly details the mathematicians throughout history who have struggled to find a proof for Fermat's Last Theorem. But the following excerpt is about Fermat, and his "enigma":

EXCERPT FROM Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem:

One of Fermat's discoveries concerned the so-called friendly numbers, or amicable numbers, closely related to the perfect numbers that had fascinated Pythagoras two thousand years earlier. Friendly numbers are pairs of numbers such that each number is the sum of the divisors of the other number. The Pythagoreans made the extraordinary discovery that 220 and 284 are friendly numbers. The divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, and the sum of all these is 284. On the other hand, the divisors of 284 are 1, 2, 4, 71, 142, and the sum of all these is 220.

The pair 220 and 284 was said to be symbolic of friendship. Martin Gardner's book Mathematical Magic Show tells of talismans sold in the Middle Ages that were inscribed with these numbers on the grounds that wearing the charms would promote love. An Arab numerologist documents the practice of carving 220 on one fruit and 284 on another, and then eating the first one and offering the second one to a lover as a form of mathematical aphrodisiac. Early theologians noted that in Genesis Jacob gave 220 goats to Esau. They believed that the number of goats, one half of a friendly pair, was an expression of Jacob's love for Esau.

No other friendly numbers were identified until 1636, when Fermat discovered the pair 17,296 and 18,416. Although not a profound discovery, it demonstrates Fermat's familiarity with numbers and his love of playing with them. Fermat started a fad for finding friendly numbers; Descartes discovered a thir pair (9,363,584 and 9,437,056), and Leonhard Euler went on to list sixty-two amicable pairs. Curiously they had all overlooked a much smaller pair of friendly numbers. In 1866 a sixteen-year-old Italian, Nicolo Paganini, discovered the pair 1,184 and 1,210.

During the twentieth century mathematicians have extended the idea further and have searched for so-called "sociable numbers", three or more numbers that form a closed loop. For example, in this loop of five numbers (12,496; 14,288; 15,472; 14,536; 14,264) the divisors of the first number add up to the second, the divisors of the second add up to the third, the divisors of the third add up to the fourth, the divisors of the fourth add up to the fifth, and the divisors of the fifth add up to the first. [Note from Sheila: Cool!!!]

Although discovering a new pair of friendly numbers made Fermat something of a celebrity, his reputation was truly confirmed thanks to a series of mathematical challenges. For example, Fermat noticed that 26 is sandwiched between 25 and 17, one of which is a square number (25 = 52 = 5 x 5) and the other is a cube number (27 = 33 = 3 x 3 x 3). He searcherd for other numbers sandwiched between a square and a cube but failed to find any, and suspected that 26 might be unique. After days of strenuous effort he managed to construct an elaborate argument that proved without any doubt that 26 is indeed the only number between a square and a cube. His step-by-step logical proof established that no other numbers could fulfill this criterion.

Fermat announced this unique property of 26 to the mathematical community, and then challenged them to prove that this was the case. He openly admitted that he himself had a proof; the question was, however, did others have the ingenuity to match it? Despite the simplicity of the claim the proof is fiendishly complicated, and Fermat took particular delight in taunting the English mathematicians Wallis and Digby, who eventually had to admit defeat. UYltimately Fermat's greatest claim to fame would turn out to be another challenge to the rest of the world. However, it would be an accidental riddle that was never intended for public discussion.

While studying Book II of the Arithmetica Fermat came upon a whole series of observations, problems, and solutions that concerned Pythagoras's theorem and Pythagorean triples. Fermat was struck by the variety and sheer quantity of Pythagorean triples. He was aware that centuries earlier Euclid had stated a proof which demonstrated that, in fact, there are an infinite number of Pythagorean triples. Fermat must have gazed at Diophantus's detailed exposition of Pythagorean triples and wondered what there was to add to the subject. As he stared at the page he began to play with Pythagoras's equation, trying to discovere something that had evaded the Greeks.

Suddenly, in a moment of genius that would immortalize the Prince of Amateurs, he created an equation that, though very similar to Pythagoras's equation, had no solutions at all...

Instead of considering the equation

x2 + y2 = z2,

Fermat was contemplating a variant of Pythagoras's creation:

x3 + y3 = z3.

As mentioned in the last chapter, Fermat had merely changed the power from 2 to 3, the square to a cube, but his new equation apparently had no whole number solutions whatsoever. Trial and error soon shows the difficulty of finding two cubed numbers that add together to make another cubed number. Could it really be the case that this minor modification turns Pythagoras's equation, one with an infinite number of solutions, into an equation with no solutions?

He altered the equation further by changing the power to numbers bigger than 3, and discovered that finding a solution to each of these equations was equally difficult. According to Fermat there appeared to be no three numbers that would perfectly fit the equation

xn + yn = zn where n represents 3,4,5...

In the margin of his Arithmetica, next to Problem 8, he made a note of his observation:

Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere.

It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a greater power than the second to be written as a sum of two like powers.

Among all the possible numbers there seemed to be no reason why at least one set of solutions could not be found, yet Fermat stated that nowhere in the infinite universe of numbers was there a "Fermatean triple". It was an extraordinary claim, but one that Fermat believed he could prove. After the first marginal note outlining the theory, the mischievous genius jotted down an additional comment that would haunt generations of mathematicians:

Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.

I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.

This was Fermat at his most infuriating. His own words suggest that he was particularly pleased with this "truly marvelous" proof, but he had no intention of bothering to write out the detail of the argument, never mind publishing it. He never told anyone about his proof, and yet, despite the combination of indolence and modesty, Fermat's Last Theorem, as it would later be called, would become famous around the world for centuries to come.

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April 23, 2005

The Books: "Zero: The Biography of a Dangerous Idea" (Charles Seife)

Next book in my science and philosophy section:

Zero.gifA book about the history of the number zero. It is called Zero: The Biography of a Dangerous Idea, and it's by Charles Seife.

Who knew that the number zero could be so eternally controversial?

This book tells the story. And this excerpt deals with the Egyptians, the solar vs. lunar calendar, geometry, and the number zero.

EXCERPT FROM Zero: The Biography of a Dangerous Idea, by Charles Seife:

Though counting abilities were rare in the ancient world, numbers and the fundamentals of counting always developed before writing and reading. When early civilizations started pressing reeds to clay tablets, carving figures in stone, and daubing ink on parchment and on papyrus, number systems had already been well-established. Transcribing the oral number system into written form was a simple task: people just needed to figure out a coding method whereby scribes could set the numbers down in a more permanent form. (Some societies even found a way to do this before they discovered writing. The illiterate Incas, for one, used the quipu, a string of colored, knotted cords, to record calculations.)

The first scribes wrote down numbers in a way that matched their base system, and predictably, did it in the most concise way they could think of. Society had progressed since the time of Gog. Instead of making little groups of marks over and over, the scribes created symbols for each type of grouping; in a quinary system, a scribe might make a certain mark for one, a different symbol for a group of five, yet another mark for a group of 25, and so forth.

The Egyptians did just that. More than 5,000 years ago, before the time of the pyramids, the ancient Egyptians designed a system for transcribing their decimal system, where pictures stood for numbers. A single vertical mark represented a unit, while a heel bone represented 10, a swirly snare stood for 100, and so on. To write down a number with this scheme, all an Egyptian scribe had to do was record groups of these symbols. Instead of having to write down 123 tick marks to denote the number "one hundred and twenty-three", the scribe wrote six symbols: one snare, two heels, and three vertical marks. It was the typical way of doing mathematics in antiquity. And like most other civilizations Egypt did not have -- or need -- a zero.

Yet the ancient Egyptians were quite sophisticated mathematicians. They were master astronomers and timekeepers, which meant that they had to use advanced math, thanks to the wandering nature of the calendar.

Creating a stable calendar was a problem for most ancient peoples, because they generally started out with a lunar calendar: the length of a month was the time between successive full moons. It was a natural choice; the waxing and waning of the moon in the heavens was hard to overlook, and it offered a convenient way of marking periodic cycles of time. But the lunar month is between 29 and 30 days long. No matter how you arrange it, 12 lunar months only add up to about 354 days -- roughly 11 short of the solar year's length. Thirteen lunar months yield roughly 19 days too many. Since it is the solar year, not the lunar year, that determines the time for harvest and planting, the seasons seem to drift when you reckon by an uncorrected lunar year.

Correcting the lunar calendar is a complicated undertaking. A number of modern-day nations, like Israel and Saudi Arabia, still use a modified lunar calendar, but 6,000 years ago the Egyptians came up with a better system. Their method was a much simpler way of keeping track of the passage of the days, producing a calendar that styaed in sync with the seasons for many years. Instead of using the moon to keep track of the passage of time, the Egyptians used the sun, just as most nations do today...

The Egyptians' innovation of the solar calendar was a breakthrough, but they made an even more important mark on history: the invention of the art of geometry. Even without a zero, the Egyptians had quickly become masters of mathematics. They had to, thanks to an angry river. Every year the Nile would overflow its banks and flood the delta. The good news was that the flooding deposited rich, alluvial silt all over the fields, making the Nile delta the richest farmland in the ancient world. The bad news was that the river destroyed many of the boundary markers, erasing all of the landmarks that told farmers which land was theirs to cultivate. (The Egyptians took property rights very seriously. In the Egyptian Book of the Dead, a newly deceased person must swear to the gods that he hasn't cheated his neighbor by stealing his land. It was a sin punishable by having his heart fed to a horrible beast called the devourer. In Egypt, filching your neighbor's land was considered as grave an offense as breaking an oath, murdering somebody, or masturbating in a temple.)

The ancient pharaohs assigned surveyors to assess the damage and reset the boundary markers, and thus geometry was born. These surveyors, or rope stretchers (named for their measuring devices and knotted ropes designed to mark right angles), eventually learned to determine the areas of plots of land by dividing them into rectangles and triangles. The Egyptians also learned how to measure the volumes of objects -- like pyramids. Egyptian mathematics was famed throughout the Mediterranean, and it is likely that the early Greek mathematicians, masters of geometry like Thales and Pythagoras, studied in Egypt. Yet despite the Egyptians' brilliant geometric work, zero was nowhere to be found within Egypt.

This was, in part, because the Egyptians were of a practical bent. They never progressed beyond measuring volumes and counting days and hours. Mathematics wasn't used for anything impractical, except their system of astrology. As a result, their best mathematicians were unable to use the principles of geometry for anything unrelated to real world problems -- they did not take their system of mathematics and turn it into an abstract system of logic. They were also not inclined to put math into their philosophy. The Greeks were different; they embraced the abstract and the philosophical, and brought mathematics to its highest point in ancient times. Yet it was not the Greeks who discovered zero. Zero came from the East, not the West.

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April 22, 2005

The Books: "Driving Mr. Albert: A trip across America with Einstein's brain" (Michael Paterniti)

Next book in my science and philosophy books section:

DrivingAlbert.jpgAnother book about Einstein - this one is called Driving Mr. Albert: A Trip Across America with Einstein's Brain, by Michael Paterniti. This is a fun book. It's a bit of a travelogue - it's a cross-country trip across America, and that is a huge part of the book: describing America, the different states, and what it's like to drive cross-country. It's also a history/biography of the disappearance of Einstein's brain - Thomas Harvey did the autopsy in 1955 and removed the brain and took it home with him. Where it then stayed for over 40 years. Sounds stranger than real-life, but it's true - Einstein's brain disappeared. I can't remember the details of how it was discovered again, and why it needs to be moved to California, but Michael Paterniti has a great idea. He's kind of at a crossroads in his life (the book is also part memoir) - and he needs something new, he needs an adventure. So he proposes to the now 86 year old Dr. Harvey: "Let's drive cross-country with the brain - let's escort it, you and I, to its final destination." The book tells that story. Of what goes through your mind when you have Einstein's BRAIN in the back seat.

Anyway, here's the excerpt - Enjoy:

EXCERPT FROM Driving Mr. Albert: A Trip Across America with Einstein's Brain, by Michael Paterniti

A confession: I want Harvey to sleep. I want him to fall into a deep, blurry, Rip Van Winkle daze, and I want to park the Skylark mother-ship and walk around to the trunk and open it. I want Harvey snoring loudly as I unzip the duffel bag and reach my hands inside, and I want to -- what? -- touch Einstein's brain. I want to touch the brain. Yes, I've admitted it. I want to hold it, coddle it, measure its weight in my palm, handle some of its one hundred billion now-dormant neurons. Does it feel like tofu, sea urchin, bologna? What, exactly? And what does such a desire make me? One of the legion of relic freaks? Or something worse?

The more the idea persists in my head, the more towns slip past outside the window as Harvey gazes into the distant living rooms of happy families, the more I wonder what, in fact, I'd be holding if I held the brain. I mean, it's not really Einstein and it's not really a brain, but disconnected pieces of a brain, just as the passing farms are not really America but parts of a whole, symbols of the thing itself, which is everything and nothing at once.

Still, I'd be touching Einstein the Superstar, immediately recognizable by the electrocuted hair and those mournful mirthful eyes. The man whose American apotheosis is so complete that he's now a coffee mug, a postcard, a T-shirt. A figure of speech, an ad pitchman, a bumper sticker ("I'm hung like Einstein," reads one that I spy on the back of some ironic VW Jetta, "and I'm smart as a horse.") Despite the fact that he was a sixty-one-year-old man when he was naturalized as an American citizen, it's amazing how fully he's been appropriated by this country.

But why? I think the answer is that, more so than anyone else in the last one hundred years, Einstein was not exactly one of us. Even now, he comes back again as both Lear's fool and Tiresias, comically offering his uncanny vision of the future while warning us about the lurking violence of humankind. "I do not know how the third world war will be fought," he is said to have cautioned, "but I do know how the fourth will: with sticks and stones." Because he glimpsed into the workings of the universe and saw an impersonal God -- what he called an "invisible piper" -- and because he greeted the twentieth century by rocketing into the twenty-first with his breakthrough tehories, he assumed a mien of invincibility. And because his sloppy demeanor stood in such stark contrast to what we expect from a white-winged prophet, he seemed both innocent and trustworthy, and thus that much more supernatural.

If we've incorporated the theory of relativity into our scientific view of the universe, as well as our literature, art, music, and culture at large, it's the great scientist's attempt to devise a kind of personal religion -- an intimate spiritual and political manifesto -- that still stands in stark, almost sacred contrast to the Pecksniffian systems of salvation offered by modern society. Einstein's blending of twentieth-century skepticism with nineteenth-century romanticism offers a different kind of hope.

"I am a deeply religious nonbeliever," he said. "This is a somewhat new kind of religion." Pushing further, he sought to marry science and religion by redefining their terms. "I am of the opinion that all the finer speculations in the realm of science spring from a deep religious feeling," he said. "I also believe that this kind of religiousness ... is the only creative religious activity of our time."

To touch Einstein's brain, then, would be to ride a ray of light, as Einstein once dreamed it as a child. To clasp time itself. To feel the warp and wobble of the universe. Einstein claimed that the happiest thought of his life came to him in 1907, during his seven-year tenure at the Patent Office in Bern, when he was twenty-eight and still couldn't find a teaching job. Up to his ears in a worsted-wool suit and patent applications, a voice in his mind whispered, "If a person falls freely, he won't feel his own weight." That became the general theory of relativity. His life and ideas continue to fill thousands of books; even today, scientists are still verifying his work. Recently, a NASA satellite took millions of measurements in space that proved a uniform distribution of primordial temperatures just above absolute zero; that is, the data proved that the universe was in a kind of postcoital afterglow from the big bang, further confirming Einstein's explanation for how the universe began.

It would be good to touch that.

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April 21, 2005

The Books: "Einstein's Dreams" (Alan Lightman)

Next book in my science and philosophy books section:

EinsteinsDream.jpg Einstein's Dreams, by Alan Lightman. A lovely little book: It opens in 1905, with a patent clerk sleeping at his desk. For a couple of months now, he has been having nightly dreams about time. In each dream, time takes a different form. Sometimes it is a circle, sometimes it is water, sometimes it doesn't exist at all. Sometimes time slows waaaayyy down, sometimes it speeds up, sometimes it reverses. And the dreams illuminate to this patent clerk how the world would look if, say, time actually were a circle, or if it speeded up, etc. What's fun about this little book is that - in its own way - each dream is already true. You can recognize elements of our own world in it, our own experience. Sometimes you do think time is "flying", sometimes it does seem as if time goes backwards (deja vu, etc.) ... It's fun to ponder.

Here is one of the patent clerk's dreams:

EXCERPT FROM Einstein's Dreams, by Alan Lightman.

11 June 1905

On the corner of Kramgasse and Theaterplatz there is a small outdoor cafe with six blue tables and a row of blue petunias in the chef's window box, and from this cafe one can see and hear the whole of Berne. People drift through the arcades on Kramgasse, talking and stopping to buy linen or wristwatches or cinnamon; a group of eight-year-old boys, let out for morning recess from the grammar scshool on Kochergasse, follow their teacher in single file through the streets to the banks of the Aare; smoke rises lazily from a mill just over the river; water gurgles from the spouts of the Zahringer Fountain; the giant clock tower on Kramgasse strikes the quarter hour.

If, for the moment, one ignores the sounds and the smells of the city, a remarkable sight will be seen. Two men at the corner of Kochergasse are trying to part but cannot, as if they would never see each other again. They say goodbye, start to walk in opposite directions, then hurry back together and embrace. Nearby, a middle-aged woman sits on the stone rim of a fountain, weeping quietly. She grips the stone with her yellow stained hands, grips it so hard that the blood rushes from her hands, and she stares in despair at the ground. Her loneliness has the permanence of a person who believes she will never see other people again. Two women in sweaters stroll down Kramgasse, arm in arm, laughing with such abandon that they could be thinking no thought of the future.

In fact, this is a world without future. In this world, time is a line that terminates at the present, both in reality and in the mind. In this world, no person can imagine the future. Imagine the future is no more possible than seeing colors beyond violet: the senses cannot conceive what may lie past the visible end of the spectrum. In a world without future, each parting of friends is a death. In a world without future, each laugh is the last laugh. In a world without future, beyond the present lies nothingness, and people cling to the present as if hanging from a cliff.

A person who cannot imagine the future is a person who cannot contemplate the results of his actions. Some are thus paralyzed into inaction. They lie in their beds through the die, wide awake but afraid to put on their clothes. They drink coffee and look at photographs. Others leap out of bed in the morning, unconcerned that each action leads into nothingness, unconcerned that they cannot plan out their lives. They live moment to moment, and each moment is full. Still others substitute the past for the future. They recount each memory, each action taken, each cause and effect, and are fascinated by how events have delivered them to this moment, the last moment of the world, the termination of the line that is time.

In the little cafe with the six outdoor tables and the row of petunias, a young man sits with his coffee and pastry. He has been idly observing the street. He has seen the two laughing women in sweaters, the middle-aged woman at the fountain, the two friends who keep repeating goodbyes. As he sits, a dark rain cloud makes its way over the city. But the young man remains at his table. He can imagine only the present, and at this moment the present is a blackening sky but no rain. As he sips the coffee and eats the pastry, he marvels at how the end of the world is so dark. Still there is no rain, and he squints at his paper in the dwindling light, trying to read the last sentence that he will read in his life. Then, rain. The young man goes inside, takes off his wet jacket, marvels at how the world ends in rain. He discusses food with the chef, but he is not waiting for the rain to stop because he is not waiting for anything. In a world without future, each moment is the end of the world. After twenty minutes, the storm cloud passes, the rain stops, and the sky brightens. The young man returns to his table, marvels that the world ends in sunshine.

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April 20, 2005

The Books: "Sophie's World: A Novel About the History of Philosophy" (Jostein Gaarder)

Moving right along with the excerpt-a-day thing. Top shelf of Bookshelf # 3 in the kitchen (my science and philosophy books)

SophiesWorld.gifNext to Hollywood Babylon we've got the book Sophie's World: A Novel About the History of Philosophy (FSG Classics), by Jostein Gaarder. The book takes the form of mysterious letters left in the mailbox, for a little curious girl to read. She's about 10, and just waking up to the beauty and wonder of the world around her. Letters from a mysterious entity start arriving, and each letter describes for Sophie (the little girl) a different philosopher or philosophical school of thought - a survey course. Because she's a child, the letters are geared towards making things simple for her. (This is a device, obviously, to make things simple for the reader)

Today's excerpt is from the chapter about Socrates (every time I think about Socrates I think about that old Steve Martin sketch ... it's literally the first thing that comes into my mind when I hear the name - heh heh):

EXCERPT FROM Sophie's World: A Novel About the History of Philosophy (FSG Classics), by Jostein Gaarder

Socrates (470 - 399 BC) is possibly the most enigmatic figure in the entire history of philosophy. He never wrote a single line. Yet he is one of the philosophers who has had the greatest influence on European thought, not least because of the dramatic manner of his death.

We know he was born in Athens, and that he spent most of his life in the city squares and marketplaces talking with the people he met there. "The trees in the countryside can teach me nothing," he said. He could also stand lost in thought for hours on end.

Even during his lifetime he was considered somewhat enigmatic, and fairly soon after his death he was held to be the founder of any number of different philisophical schools of thought. The very fact that he was so enigmatic and ambiguous made it possible for widely differing schools of thought to claim him as their own.

We know for a certainty that he was extremely ugly. He was potbellied and had bulging eyes and a snub nose. But inside he was said to be "perfectly delightful". It was also said of him that "You can seek him in the present, you can seek him in the past, but you will never find his equal." Nevertheless he was sentenced to death for his philosopical activities.

The life of Socrates is mainly known to us through the writings of Plato, who was one of his pupils and who became one of the greatest philosophers of all time. Plato wrote a number of Dialogues, or dramatized discussions on philosophy, in which he uses Socrates as his principal character and mouthpiece.

Since Plato is putting his own philosophy in Socrates' mouth, we cannot be sure that the words he speaks in the dialogues were ever actually uttered by him. So it is no easy matter to distinguish between the teachings of Socrates and the philosophy of Plato. Exactly the same problem applies to many other historical persons who left no written accounts. The classic example, of course, is Jesus. We cannot be certain that the "historical" Jesus actually spoke the words that Matthew or Luke ascribed to him. Similarly, what the "historical" Socrates actually said will always be shrouded in mystery.

But who Socrates "really" was is relatively unimportant. It is Plato's portrait of Socrates that has inspired thinkers in the Western world for nearly 2,500 years.

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