2015 |
(It should be noted that the only account we have of Dirac's childhood experience comes from himself, much later in life. He made no attempt to conceal the extent he despised his father [who was respected by his colleagues and acquaintances in Bristol], and there is no way to know whether Paul exaggerated or embroidered upon the circumstances of his childhood.)
After a primary education in which he was regarded as a sound but not exceptional pupil, Paul followed his brother Felix into the Merchant Venturers' School, a Bristol technical school ranked among the finest in the country. There he quickly distinguished himself, ranking near the top in most subjects. The instruction was intensely practical, eschewing Latin, Greek, and music in favour of mathematics, science, geometric and mechanical drawing, and practical skills such as operating machine tools. Dirac learned physics and mathematics with the engineer's eye to “getting the answer out” as opposed to finding the most elegant solution to the problem. He then pursued his engineering studies at Bristol University, where he excelled in mathematics but struggled with experiments.
Dirac graduated with a first-class honours degree in engineering, only to find the British economy in a terrible post-war depression, the worst economic downturn since the start of the Industrial Revolution. Unable to find employment as an engineer, he returned to Bristol University to do a second degree in mathematics, where it was arranged he could skip the first year of the program and pay no tuition fees. Dirac quickly established himself as the star of the mathematics programme, and also attended lectures about the enigmatic quantum theory.
His father had been working in the background to secure a position at Cambridge for Paul, and after cobbling together scholarships and a gift from his father, Dirac arrived at the university in October 1923 to pursue a doctorate in theoretical physics. Dirac would already have seemed strange to his fellow students. While most were scions of the upper class, classically trained, with plummy accents, Dirac knew no Latin or Greek, spoke with a Bristol accent, and approached problems as an engineer or mathematician, not a physicist. He had hoped to study Einstein's general relativity, the discovery of which had first interested him in theoretical physics, but his supervisor was interested in quantum mechanics and directed his work into that field.
It was an auspicious time for a talented researcher to undertake work in quantum theory. The “old quantum theory”, elaborated in the early years of the 20th century, had explained puzzles like the distribution of energy in heat radiation and the photoelectric effect, but by the 1920s it was clear that nature was much more subtle. For example, the original quantum theory failed to explain even the spectral lines of hydrogen, the simplest atom. Dirac began working on modest questions related to quantum theory, but his life was changed when he read Heisenberg's 1925 paper which is now considered one of the pillars of the new quantum mechanics. After initially dismissing the paper as overly complicated and artificial, he came to believe that it pointed the way forward, dismissing Bohr's concept of atoms like little solar systems in favour of a probability density function which gives the probability an electron will be observed in a given position. This represented not just a change in the model of the atom but the discarding entirely of models in favour of a mathematical formulation which permitted calculating what could be observed without providing any mechanism whatsoever explaining how it worked.
After reading and fully appreciating the significance of Heisenberg's work, Dirac embarked on one of the most productive bursts of discovery in the history of modern physics. Between 1925 and 1933 he published one foundational paper after another. His Ph.D. in 1926, the first granted by Cambridge for work in quantum mechanics, linked Heisenberg's theory to the classical mechanics he had learned as an engineer and provided a framework which made Heisenberg's work more accessible. Scholarly writing did not come easily to Dirac, but he mastered the art to such an extent that his papers are still read today as examples of pellucid exposition. At a time when many contributions to quantum mechanics were rough-edged and difficult to understand even by specialists, Dirac's papers were, in the words of Freeman Dyson, “like exquisitely carved marble statues falling out of the sky, one after another.”
In 1928, Dirac took the first step to unify quantum mechanics and special relativity in the Dirac equation. The consequences of this equation led Dirac to predict the existence of a positively-charged electron, which had never been observed. This was the first time a theoretical physicist had predicted the existence of a new particle. This “positron” was observed in debris from cosmic ray collisions in 1932. The Dirac equation also interpreted the spin (angular momentum) of particles as a relativistic phenomenon.
Dirac, along with Enrico Fermi, elaborated the statistics of particles with half-integral spin (now called “fermions”). The behaviour of ensembles of one such particle, the electron, is essential to the devices you use to read this article. He took the first steps toward a relativistic theory of light and matter and coined the name, “quantum electrodynamics”, for the field, but never found a theory sufficiently simple and beautiful to satisfy himself. He published The Principles of Quantum Mechanics in 1930, for many years the standard textbook on the subject and still read today. He worked out the theory of magnetic monopoles (not detected to this date) and speculated on the origin and possible links between large numbers in physics and cosmology.
The significance of Dirac's work was recognised at the time. He was elected a Fellow of the Royal Society in 1930, became the Lucasian Professor of Mathematics (Newton's chair) at Cambridge in 1932, and shared the Nobel Prize in Physics for 1933 with Erwin Schrödinger. After rejecting a knighthood because he disliked being addressed by his first name, he was awarded the Order of Merit in 1973. He is commemorated by a plaque in Westminster Abbey, close to that of Newton; the plaque bears his name and the Dirac equation, the only equation so honoured.
Many physicists consider Dirac the second greatest theoretical physicist of the 20th century, after Einstein. While Einstein produced great leaps of intellectual achievement in fields neglected by others, Dirac, working alone, contributed to the grand edifice of quantum mechanics, which occupied many of the most talented theorists of a generation. You have to dig a bit deeper into the history of quantum mechanics to fully appreciate Dirac's achievement, which probably accounts for his name not being as well known as it deserves.
There is much more to Dirac, all described in this extensively-documented scientific biography. While declining to join the British atomic weapons project during World War II because he refused to work as part of a collaboration, he spent much of the war doing consulting work for the project on his own, including inventing a new technique for isotope separation. (Dirac's process proved less efficient that those eventually chosen by the Manhattan project and was not used.) As an extreme introvert, nobody expected him to ever marry, and he astonished even his closest associates when he married the sister of his fellow physicist Eugene Wigner, Manci, a Hungarian divorcée with two children by her first husband. Manci was as extroverted as Dirac was reserved, and their marriage in 1937 lasted until Dirac's death in 1984. They had two daughters together, and lived a remarkably normal family life. Dirac, who disdained philosophy in his early years, became intensely interested in the philosophy of science later in life, even arguing that mathematical beauty, not experimental results, could best guide theorists to the best expression of the laws of nature.
Paul Dirac was a very complicated man, and this is a complicated and occasionally self-contradictory biography (but the contradiction is in the subject's life, not the fault of the biographer). This book provides a glimpse of a unique intellect whom even many of his closest associates never really felt they completely knew.
From prehistoric times humans have had the need to count things, for example, the number of sheep in a field. This could be done by establishing a one-to-one correspondence between the sheep and something else more portable such as one's fingers (for a small flock), or pebbles kept in a sack. To determine whether a sheep was missing, just remove a pebble for each sheep and if any remained in the sack, that indicates how many are absent. At a slightly more abstract level, one could make tally marks on a piece of bark or clay tablet, one for each sheep. But all of this does not imply number as an abstraction independent of individual items of some kind or another. Ancestral humans don't seem to have required more than the simplest notion of numbers: until the middle of the 20th century several tribes of Australian aborigines had no words for numbers in their languages at all, but counted things by making marks in the sand. Anthropologists discovered tribes in remote areas of the Americas, Pacific Islands, and Australia whose languages had no words for numbers greater than four.
With the emergence of settled human populations and the increasingly complex interactions of trade between villages and eventually cities, a more sophisticated notion of numbers was required. A merchant might need to compute how many kinds of one good to exchange for another and to keep records of his inventory of various items. The earliest known written records of numerical writing are Sumerian cuneiform clay tablets dating from around 3400 B.C. These tablets show number symbols formed from two distinct kinds of marks pressed into wet clay with a stylus. While the smaller numbers seem clearly evolved from tally marks, larger numbers are formed by complicated combinations of the two symbols representing numbers from 1 to 59. Larger numbers were written as groups of powers of 60 separated by spaces. This was the first known instance of a positional number system, but there is no evidence it was used for complicated calculations—just as a means of recording quantities.
Ancient civilisations: Egypt, Hebrew, Greece, China, Rome, and the Aztecs and Mayas in the Western Hemisphere all invented ways of writing numbers, some sophisticated and capable of representing large quantities. Many of these systems were additive: they used symbols, sometimes derived from letters in their alphabets, and composed numbers by writing symbols which summed to the total. To write the number 563, a Greek would write “φξγ”, where φ=500, ξ=60, and γ=3. By convention, numbers were written with letters in descending order of the value they represented, but the system was not positional. This made the system clumsy for representing large numbers, reusing letters with accent marks to represent thousands and an entirely different convention for ten thousands.
How did such advanced civilisations get along using number systems in which it is almost impossible to compute? Just imagine a Roman faced with multiplying MDXLIX by XLVII (1549 × 47)—where do you start? You don't: all of these civilisations used some form of mechanical computational aid: an abacus, counting rods, stones in grooves, and so on to actually manipulate numbers. The Sun Zi Suan Jing, dating from fifth century China, provides instructions (algorithms) for multiplication, division, and square and cube root extraction using bamboo counting sticks (or written symbols representing them). The result of the computation was then written using the numerals of the language. The written language was thus a way to represent numbers, but not compute with them.
Many of the various forms of numbers and especially computational tools such as the abacus came ever-so-close to stumbling on the place value system, but it was in India, probably before the third century B.C. that a positional decimal number system including zero as a place holder, with digit forms recognisably ancestral to those we use today emerged. This was a breakthrough in two regards. Now, by memorising tables of addition, subtraction, multiplication, and division and simple algorithms once learned by schoolchildren before calculators supplanted that part of their brains, it was possible to directly compute from written numbers. (Despite this, the abacus remained in common use.) But, more profoundly, this was a universal representation of whole numbers. Earlier number systems (with the possible exception of that invented by Archimedes in The Sand Reckoner [but never used practically]) either had a limit on the largest number they could represent or required cumbersome and/or lengthy conventions for large numbers. The Indian number system needed only ten symbols to represent any non-negative number, and only the single convention that each digit in a number represented how many of that power of ten depending on its position.
Knowledge diffused slowly in antiquity, and despite India being on active trade routes, it was not until the 13th century A.D. that Fibonacci introduced the new number system, which had been transmitted via Islamic scholars writing in Arabic, to Europe in his Liber Abaci. This book not only introduced the new number system, it provided instructions for a variety of practical computations and applications to higher mathematics. As revolutionary as this book was, in an era of hand-copied manuscripts, its influence spread very slowly, and it was not until the 16th century that the new numbers became almost universally used. The author describes this protracted process, about which a great deal of controversy remains to the present day.
Just as the decimal positional number system was becoming established in Europe, another revolution in notation began which would transform mathematics, how it was done, and our understanding of the meaning of numbers. Algebra, as we now understand it, was known in antiquity, but it was expressed in a rhetorical way—in words. For example, proposition 7 of book 2 of Euclid's Elements states:
If a straight line be cut at random, the square of the whole is equal to the squares on the segments and twice the rectangle contained by the segments.
Now, given such a problem, Euclid or any of those following in his tradition would draw a diagram and proceed to prove from the axioms of plane geometry the correctness of the statement. But it isn't obvious how to apply this identity to other problems, or how it illustrates the behaviour of general numbers. Today, we'd express the problem and proceed as follows:
Once again, faced with the word problem, it's difficult to know where to begin, but once expressed in symbolic form, it can be solved by applying rules of algebra which many master before reaching high school. Indeed, the process of simplifying such an equation is so mechanical that computer tools are readily available to do so.
Or consider the following brain-twister posed in the 7th century A.D. about the Greek mathematician and father of algebra Diophantus: how many years did he live?
“Here lies Diophantus,” the wonder behold.
Through art algebraic, the stone tells how old;
“God gave him his boyhood one-sixth of his life,
One twelfth more as youth while whiskers grew rife;
And then one-seventh ere marriage begun;
In five years there came a bounding new son.
Alas, the dear child of master and sage
After attaining half the measure of his father's life chill fate took him.
After consoling his fate by the science of numbers for four years, he ended his life.”
Oh, go ahead, give it a try before reading on!
Today, we'd read through the problem and write a system of two simultaneous equations, where x is the age of Diophantus at his death and y the number of years his son lived. Then:
Plug the second equation into the first, do a little algebraic symbol twiddling, and the answer, 84, pops right out. Note that not only are the rules for solving this equation the same as for any other, with a little practice it is easy to read the word problem and write down the equations ready to solve. Go back and re-read the original problem and the equations and you'll see how straightforwardly they follow.
Once you have transformed a mass of words into symbols, they invite you to discover new ways in which they apply. What is the solution of the equation x+4=0? In antiquity many would have said the equation is meaningless: there is no number you can add to four to get zero. But that's because their conception of number was too limited: negative numbers such as −4 are completely valid and obey all the laws of algebra. By admitting them, we discovered we'd overlooked half of the real numbers. What about the solution to the equation x² + 4 = 0? This was again considered ill-formed, or imaginary, since the square of any real number, positive or negative, is positive. Another leap of imagination, admitting the square root of minus one to the family of numbers, expanded the number line into the complex plane, yielding the answer 2i as we'd now express it, and extending our concept of number into one which is now fundamental not only in abstract mathematics but also science and engineering. And in recognising negative and complex numbers, we'd come closer to unifying algebra and geometry by bringing rotation into the family of numbers.
This book explores the groping over centuries toward a symbolic representation of mathematics which hid the specifics while revealing the commonality underlying them. As one who learned mathematics during the height of the “new math” craze, I can't recall a time when I didn't think of mathematics as a game of symbolic transformation of expressions which may or may not have any connection with the real world. But what one discovers in reading this book is that while this is a concept very easy to brainwash into a 7th grader, it was extraordinarily difficult for even some of the most brilliant humans ever to have lived to grasp in the first place. When Newton invented calculus, for example, he always expressed his “fluxions” as derivatives of time, and did not write of the general derivative of a function of arbitrary variables.
Also, notation is important. Writing something in a more expressive and easily manipulated way can reveal new insights about it. We benefit not just from the discoveries of those in the past, but from those who created the symbolic language in which we now express them.
This book is a treasure chest of information about how the language of science came to be. We encounter a host of characters along the way, not just great mathematicians and scientists, but scoundrels, master forgers, chauvinists, those who preserved precious manuscripts and those who burned them, all leading to the symbolic language in which we so effortlessly write and do mathematics today.
Dr. Skye Chadwick is a hyperspatial physicist and chief scientist of Project Tesseract. Research into the multiverse and brane world solutions of string theory has revealed that our continuum—all of the spacetime we inhabit—is just one of an unknown number adjacent to one another in a higher dimensional membrane (“brane”), and that while every continuum is different, those close to one another in the hyperdimensional space tend to be similar. Project Tesseract, a highly classified military project operating from an underground laboratory in Colorado, is developing hardware based on advanced particle physics which allows passively observing or even interacting with these other continua (or parallel universes).
The researchers are amazed to discover that in some continua characters which are fictional in our world actually exist, much as they were described in literature. Perhaps Heinlein and Borges were right in speculating that fiction exists in parallel universes, and maybe that's where some of authors' ideas come from. In any case, exploration of Continuum 114 has revealed it to be one of those in which Sherlock Holmes is a living, breathing man. Chadwick and her team decide to investigate one of the pivotal and enigmatic episodes in the Holmes literature, the fight at Reichenbach Falls. As Holmes and Moriarty battle, it is apparent that both will fall to their death. Chadwick acts impulsively and pulls Holmes from the brink of the cliff, back through the Tesseract, into our continuum. In an instant, Sherlock Holmes, consulting detective of 1891 London, finds himself in twenty-first century Colorado, where he previously existed only in the stories of Arthur Conan Doyle.
Holmes finds much to adapt to in this often bewildering world, but then he was always a shrewd observer and master of disguise, so few people would be as well equipped. At the same time, the Tesseract project faces a crisis, as a disaster and subsequent investigation reveals the possibility of sabotage and an espionage ring operating within the project. A trusted, outside investigator with no ties to the project is needed, and who better than Holmes, who owes his life to it? With Chadwick at his side, they dig into the mystery surrounding the project.
As they work together, they find themselves increasingly attracted to one another, and Holmes must confront his fear that emotional involvement will impair the logical functioning of his mind upon which his career is founded. Chadwick, learning to become a talented investigator in her own right, fears that a deeper than professional involvement with Holmes will harm her own emerging talents.
I found that this long story started out just fine, and indeed I recommended it to several people after finishing the first of the four novels collected here. To me, it began to run off the rails in the second book and didn't get any better in the remaining two (which begin with Holmes and Chadwick an established detective team, summoned to help with a perplexing mystery in Britain which may have consequences for all of the myriad contunua in the multiverse). The fundamental problem is that these books are trying to do too much all at the same time. They can't decide whether they're science fiction, mystery, detective procedural, or romance, and as they jump back and forth among the genres, so little happens in the ones being neglected at the moment that the parallel story lines develop at a glacial pace. My estimation is that an editor with a sharp red pencil could cut this material by 50–60% and end up with a better book, omitting nothing central to the story and transforming what often seemed a tedious slog into a page-turner.
Sherlock Holmes is truly one of the great timeless characters in literature. He can be dropped into any epoch, any location, and, in this case, anywhere in the multiverse, and rapidly start to get to the bottom of the situation while entertaining the reader looking over his shoulder. There is nothing wrong with the premise of these books and there are interesting ideas and characters in them, but the execution just isn't up to the potential of the concept. The science fiction part sometimes sinks to the techno-babble level of Star Trek (“Higgs boson injection beginning…”). I am no prude, but I found the repeated and explicit sex scenes a bit much (tedious, actually), and they make the books unsuitable for younger readers for whom the original Sherlock Holmes stories are a pure delight. If you're interested in the idea, I'd suggest buying just the first book separately and see how you like it before deciding to proceed, bearing in mind that I found it the best of the four.