Fourmilog: None Dare Call It Reason

Reading List: A Force of Nature

Friday, February 20, 2015 21:01

Reeves, Richard. A Force of Nature. New York: W. W. Norton, 2008. ISBN 978-0-393-33369-5.
In 1851, the Crystal Palace Exhibition opened in London. It was a showcase of the wonders of industry and culture of the greatest empire the world had ever seen and attracted a multitude of visitors. Unlike present-day “World's Fair” boondoggles, it made money, and the profits were used to fund good works, including endowing scholarships for talented students from the far reaches of the Empire to study in Britain. In 1895, Ernest Rutherford, hailing from a remote area in New Zealand and recent graduate of Canterbury College in Christchurch, won a scholarship to study at Cambridge. Upon learning of the award in a field of his family's farm, he threw his shovel in the air and exclaimed, “That's the last potato I'll ever dig.” It was.

When he arrived at Cambridge, he could hardly have been more out of place. He and another scholarship winner were the first and only graduate students admitted who were not Cambridge graduates. Cambridge, at the end of the Victorian era, was a clubby, upper-class place, where even those pursuing mathematics were steeped in the classics, hailed from tony public schools, and spoke with refined accents. Rutherford, by contrast, was a rough-edged colonial, bursting with energy and ambition. He spoke with a bizarre accent (which he retained all his life) which blended the Scottish brogue of his ancestors with the curious intonations of the antipodes. He was anything but the ascetic intellectual so common at Cambridge—he had been a fierce competitor at rugby, spoke about three times as loud as was necessary (many years later, when the eminent Rutherford was tapped to make a radio broadcast from Cambridge, England to Cambridge, Massachusetts, one of his associates asked, “Why use radio?”), and spoke vehemently on any and all topics (again, long afterward, when a ceremonial portrait was unveiled, his wife said she was surprised the artist had caught him with his mouth shut).

But it quickly became apparent that this burly, loud, New Zealander was extraordinarily talented, and under the leadership of J.J. Thomson, he began original research in radio, but soon abandoned the field to pursue atomic research, which Thomson had pioneered with his discovery of the electron. In 1898, with Thomson's recommendation, Rutherford accepted a professorship at McGill University in Montreal. While North America was considered a scientific backwater in the era, the generous salary would allow him to marry his fiancée, who he had left behind in New Zealand until he could find a position which would support them.

At McGill, he and his collaborator Frederick Soddy, studying the radioactive decay of thorium, discovered that radioactive decay was characterised by a unique half-life, and was composed of two distinct components which he named alpha and beta radiation. He later named the most penetrating product of nuclear reactions gamma rays. Rutherford was the first to suggest, in 1902, that radioactivity resulted from the transformation of one chemical element into another—something previously thought impossible.

In 1907, Rutherford was offered, and accepted a chair of physics at the University of Manchester, where, with greater laboratory resources than he had had in Canada, pursued the nature of the products of radioactive decay. By 1907, by a clever experiment, he had identified alpha radiation (or particles, as we now call them) with the nuclei of helium atoms—nuclear decay was heavy atoms being spontaneously transformed into a lighter element and a helium nucleus.

Based upon this work, Rutherford won the Nobel Prize in Chemistry in 1908. As a person who considered himself first and foremost an experimental physicist and who was famous for remarking, “All science is either physics or stamp collecting”, winning the Chemistry Nobel had to feel rather odd. He quipped that while he had observed the transmutation of elements in his laboratory, no transmutation was as startling as discovering he had become a chemist. Still, physicist or chemist, his greatest work was yet to come.

In 1909, along with Hans Geiger (later to invent the Geiger counter) and Ernest Marsden, he conducted an experiment where high-energy alpha particles were directed against a very thin sheet of gold foil. The expectation was that few would be deflected and those only slightly. To the astonishment of the experimenters, some alpha particles were found to be deflected through large angles, some bouncing directly back toward the source. Geiger exclaimed, “It was almost as incredible as if you fired a 15-inch [battleship] shell at a piece of tissue paper and it came back and hit you.” It took two years before Rutherford fully understood and published what was going on, and it forever changed the concept of the atom. The only way to explain the scattering results was to replace the early model of the atom with one in which a diffuse cloud of negatively charged electrons surrounded a tiny, extraordinarily dense, positively charged nucleus (that word was not used until 1913). This experimental result fed directly into the development of quantum theory and the elucidation of the force which bound the particles in the nucleus together, which was not fully understood until more than six decades later.

In 1919 Rutherford returned to Cambridge to become the head of the Cavendish Laboratory, the most prestigious position in experimental physics in the world. Continuing his research with alpha emitters, he discovered that bombarding nitrogen gas with alpha particles would transmute nitrogen into oxygen, liberating a proton (the nucleus of hydrogen). Rutherford simultaneously was the first to deliberately transmute one element into another, and also to discover the proton. In 1921, he predicted the existence of the neutron, completing the composition of the nucleus. The neutron was eventually discovered by his associate, James Chadwick, in 1932.

Rutherford's discoveries, all made with benchtop apparatus and a small group of researchers, were the foundation of nuclear physics. He not only discovered the nucleus, he also found or predicted its constituents. He was the first to identify natural nuclear transmutation and the first to produce it on demand in the laboratory. As a teacher and laboratory director his legacy was enormous: eleven of his students and research associates went on to win Nobel prizes. His students John Cockcroft and Ernest Walton built the first particle accelerator and ushered in the era of “big science”. Rutherford not only created the science of nuclear physics, he was the last person to make major discoveries in the field by himself, alone or with a few collaborators, and with simple apparatus made in his own laboratory.

In the heady years between the wars, there were, in the public mind, two great men of physics: Einstein the theoretician and Rutherford the experimenter. (This perception may have understated the contributions of the creators of quantum mechanics, but they were many and less known.) Today, we still revere Einstein, but Rutherford is less remembered (except in New Zealand, where everybody knows his name and achievements). And yet there are few experimentalists who have discovered so much in their lifetimes, with so little funding and the simplest apparatus. Rutherford, that boisterous, loud, and restless colonial, figured out much of what we now know about the atom, largely by himself, through a multitude of tedious experiments which often failed, and he should rightly be regarded as a pillar of 20th century physics.

This is the thousandth book to appear since I began to keep the reading list in January 2001.

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Reading List: Tools for Survival

Wednesday, February 18, 2015 22:14

Rawles, James Wesley. Tools for Survival. New York: Plume, 2014. ISBN 978-0-452-29812-5.
Suppose one day the music stops. We all live, more or less, as part of an intricately-connected web of human society. The water that comes out of the faucet when we open the tap depends (for the vast majority of people) on pumps powered by an electrical grid that spans a continent. So does the removal of sewage when you flush the toilet. The typical city in developed nations has only about three days' supply of food on hand in stores and local warehouses and depends upon a transportation infrastructure as well as computerised inventory and payment systems to function. This system has been optimised over decades to be extremely efficient, but at the same time it has become dangerously fragile against any perturbation. A financial crisis which disrupts just-in-time payments, a large-scale and protracted power outage due to a solar flare or EMP attack, disruption of data networks by malicious attacks, or social unrest can rapidly halt the flow of goods and services upon which hundreds of millions of people depend and rely upon without rarely giving a thought to what life might be like if one day they weren't there.

The author, founder of the essential SurvivalBlog site, has addressed such scenarios in his fiction, which is highly recommended. Here the focus is less speculative, and entirely factual and practical. What are the essential skills and tools one needs to survive in what amounts to a 19th century homestead? If the grid (in all senses) goes down, those who wish to survive the massive disruptions and chaos which will result may find themselves in the position of those on the American frontier in the 1870s: forced into self-reliance for all of the necessities of life, and compelled to use the simple, often manual, tools which their ancestors used—tools which can in many cases be fabricated and repaired on the homestead.

The author does not assume a total collapse to the nineteenth century. He envisions that those who have prepared to ride out a discontinuity in civilisation will have equipped themselves with rudimentary solar electric power and electronic communication systems. But at the same time, people will be largely on their own when it comes to gardening, farming, food preservation, harvesting trees for firewood and lumber, first aid and dental care, self-defence, metalworking, and a multitude of other tasks. As always, the author stresses, it isn't the tools you have but rather the skills between your ears that determine whether you'll survive. You may have the most comprehensive medical kit imaginable, but if nobody knows how to stop the bleeding from a minor injury, disinfect the wound, and suture it, what today is a short trip to the emergency room might be life-threatening.

Here is what I took away from this book. Certainly, you want to have on hand what you need to deal with immediate threats (for example, firefighting when the fire department does not respond, self-defence when there is no sheriff, a supply of water and food so you don't become a refugee if supplies are interrupted, and a knowledge of sanitation so you don't succumb to disease when the toilet doesn't flush). If you have skills in a particular area, for example, if you're a doctor, nurse, or emergency medical technician, by all means lay in a supply of what you need not just to help yourself and your family, but your neighbours. The same goes if you're a welder, carpenter, plumber, shoemaker, or smith. It just isn't reasonable, however, to expect any given family to acquire all the skills and tools (even if they could afford them, where would they put them?) to survive on their own. Far more important is to make the acquaintance of like-minded people in the vicinity who have the diverse set of skills required to survive together. The ability to build and maintain such a community may be the most important survival skill of all.

This book contains a wealth of resources available on the Web (most presented as shortened URLs, not directly linked in the Kindle edition) and a great deal of wisdom about which I find little or nothing to disagree. For the most part the author uses quaint units like inches, pounds, and gallons, but he is writing for a mostly American audience. Please take to heart the safety warnings: it is very easy to kill or gravely injure yourself when woodworking, metal fabricating, welding, doing electrical work, or felling trees and processing lumber. If your goal is to survive and prosper whatever the future may bring, it can ruin your whole plan if you kill yourself acquiring the skills you need to do so.

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Reading List: The Testament of James

Monday, February 9, 2015 23:45

Suprynowicz, Vin. The Testament of James. Pahrump, NV: Mountain Media, 2014. ISBN 978-0-9670259-4-0.
The author is a veteran newspaperman and was arguably the most libertarian writer in the mainstream media during his long career with the Las Vegas Review-Journal. He earlier turned his hand to fiction in 2005's The Black Arrow (May 2005), a delightful libertarian superhero fantasy. In the present volume he tells an engaging tale which weaves together mystery, the origins of Christianity, and the curious subculture of rare book collectors and dealers.

Matthew Hunter is the proprietor of a used book shop in Providence, Rhode Island, dealing both in routine merchandise but also rare volumes obtained from around the world and sold to a network of collectors who trust Hunter's judgement and fair pricing. While Hunter is on a trip to Britain, an employee of the store is found dead under suspicious circumstances, while waiting after hours to receive a visitor from Egypt with a manuscript to be evaluated and sold.

Before long, a series of curious, shady, and downright intimidating people start arriving at the bookshop, all seeking to buy the manuscript which, it appears, was never delivered. The person who was supposed to bring it to the shop has vanished, and his brothers have come to try to find him. Hunter and his friend Chantal Stevens, ex-military who has agreed to help out in the shop, find themselves in the middle of the quest for one of the most legendary, and considered mythical, rare books of all time, The Testament of James, reputed to have been written by James the Just, the (half-)brother of Jesus Christ. (His precise relationship to Jesus is a matter of dispute among Christian sects and scholars.) This Testament (not to be confused with the Epistle of James in the New Testament, also sometimes attributed to James the Just), would have been the most contemporary record of the life of Jesus, well predating the Gospels.

Matthew and Chantal seek to find the book, rescue the seller, and get to the bottom of a mystery dating from the origin of Christianity. Initially dubious such a book might exist, Matthew concludes that so many people would not be trying so hard to lay their hands on it if there weren't something there.

A good part of the book is a charming and often humorous look inside the world of rare books, one with which the author is clearly well-acquainted. There is intrigue, a bit of mysticism, and the occasional libertarian zinger aimed at a deserving target. As the story unfolds, an alternative interpretation of the life and work of Jesus and the history of the early Church emerges, which explains why so many players are so desperately seeking the lost book.

As a mystery, this book works superbly. Its view of “bookmen” (hunters, sellers, and collectors) is a delight. Orthodox Christians (by which I mean those adhering to the main Christian denominations, not just those called “Orthodox”) may find some of the content blasphemous, but before they explode in red-faced sputtering, recall that one can never be sure about the provenance and authenticity of any ancient manuscript. Some of the language and situations are not suitable for young readers, but by the standards of contemporary mass-market fiction, the book is pretty tame. There are essentially no spelling or grammatical errors. To be clear, this is entirely a work of fiction: there is no Testament of James apart from this book, in which it's an invention of the author. A bibliography of works providing alternative (which some will consider heretical) interpretations of the origins of Christianity is provided. You can read an excerpt from the novel at the author's Web log; continue to follow the links in the excerpts to read the first third—20,000 words—of the book for free.

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Reading List: The Case of the Displaced Detective Omnibus

Friday, January 30, 2015 22:36

Osborn, Stephanie. The Case of the Displaced Detective Omnibus. Kingsport, TN: Twilight Times Books, 2013. ASIN B00FOR5LJ4.
This book, available only for the Kindle, collects the first four novels of the author's Displaced Detective series. The individual books included here are The Arrival, At Speed, The Rendlesham Incident, and Endings and Beginnings. Each pair of books, in turn, comprises a single story, the first two The Case of the Displaced Detective and the latter two The Case of the Cosmological Killer. If you read only the first of either pair, it will be obvious that the story has been left in the middle with little resolved. In the trade paperback edition, the four books total more than 1100 pages, so this omnibus edition will keep you busy for a while.

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.

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Reading List: Enlightening Symbols

Saturday, January 10, 2015 22:08

Mazur, Joseph. Enlightening Symbols. Princeton: Princeton University Press, 2014. ISBN 978-0-691-15463-3.
Sometimes an invention is so profound and significant yet apparently obvious in retrospect that it is difficult to imagine how people around the world struggled over millennia to discover it, and how slowly it was to diffuse from its points of origin into general use. Such is the case for our modern decimal system of positional notation for numbers and the notation for algebra and other fields of mathematics which permits rapid calculation and transformation of expressions. This book, written with the extensive source citations of a scholarly work yet accessible to any reader familiar with arithmetic and basic algebra, traces the often murky origins of this essential part of our intellectual heritage.

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:

\begin{eqnarray*}
    (a+b)^2 & = & (a+b)(a+b) \\
    & = & a(a+b)+b(a+b) \\
    & = & aa+ab+ba+bb \\
    & = & a^2+2ab+b^2 \\
    & = & a^2+b^2+2ab
\end{eqnarray*}

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:

\begin{eqnarray*}
    x & = & (\frac{1}{6}+\frac{1}{12}+\frac{1}{7})x+5+y+4 \\
    y & = & \frac{x}{2}
\end{eqnarray*}

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.

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