Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers by Joseph Mazur

Disorderly presentation of a complex subject

My attraction to arcane knowledge sometimes leads me to read a book that’s somewhat over my head. Such is the case with Joseph Mazur’s Enlightening Symbols, published in 2014. Mazur is a mathematician and considers himself a “mathematics journalist.” He has published several books on mathematics that appear to be aimed at a general trade audience, but this book may be for a more specialized audience.

Enlightening Symbols is a history of mathematics, in particular the history of how symbols have been created and adopted for various mathematical operations, functions, and concepts. As the subtitle indicates, much of the book is concerned with notation—such as, where did we get +, -, ×, and ÷—but that’s not the whole story. In the early portions of the book, Mazur talks about counting with fingers, sticks, and stones, and the use of the abacus. Ancient mathematicians didn’t even have numerals, at least not widely accepted ones. Early math texts, such as Euclid’s Elements, are written entirely in words. Mazur details how Arabic numerals spread and came to be accepted in Europe—except they’re not really Arabic numerals; they’re of Indian or Hindu origin. The book then goes into the history of algebra and how certain Greek and Latin letters came to be used in equations. Who was the first to decide the mathematical meaning of pi (π), for example?

I read quite a few books on the history of science, which has led to an interest in the history of mathematics. I’m also quite interested in different written languages, to which the subject of mathematical notation pertains. I wish this book would have stuck a little closer to its subtitle and focused more on notation. Mazur, however, keeps venturing into symbology and semiotics—how we process and interpret symbols. Towards the end of the book, he summarizes various psychological experiments concerning mathematical thought, and he gives his own epistemological theories about how the mind works. It’s a subject that clearly fascinates him, but that psychological content was far less interesting to me than the math history.

As a history, however, Enlightening Symbols does have its problems. While the table of contents indicates the book is organized roughly chronologically, Mazur jumps all over the place temporally. No matter what century he’s talking about, he constantly flashes back to earlier figures like Brahmagupta (c. 598–c. 668) and Al-Khwarizmi (c. 789–c. 850), creating a tangled narrative that’s really difficult to follow. In examining who was the first to introduce Indian numerals to Western Europe, Mazur seems to propose about 17 answers to that question. Some he disproves, but I couldn’t even tell you whom he finally settled on. I kept hoping for a chart, but when I finally got one, halfway through the book, it was still confusing. A simple timeline in list or table form would have been most helpful.

You don’t really need to know a whole lot of math to appreciate the history in Enlightening Symbols. I’m not up on calculus, but I could still roughly follow what Mazur had to say about the history of calculus notation. Some interest in linguistics helps. I’ll admit that I’m probably not qualified to review this book. If you’re reading this review at my blog, however, rather than reading a review in the Journal of Mathematics, then you’re probably in the same boat I am. I provide my layman’s perspective of the book, for what it’s worth, to help you decide if you want to read it.

Captain Blood by Rafael Sabatini

The unentertaining oxymoron of the gentleman pirate

Rafael Sabatini (1875–1950) was an Italian-British writer of adventure novels. His books often feature swashbuckling heroes who would go on to be played by the likes of Errol Flynn, Tyrone Power, and Stewart Granger in Hollywood film adaptations. One such novel (and a film role for Flynn) is Captain Blood, published in 1922.

The auspiciously named Peter Blood is a middle-aged Irish-born ex-soldier and -adventurer who now plies the gentler trade of physician in the English town of Bridgewater. In the 1680s, when rebels supporting James Scott, the Duke of Monmouth, rose up against the English King James II, Blood could care less about the politics, but one of his neighbors, a Monmouth rebel, is injured in the fighting, and Blood dutifully tends his wounds. For this service, Blood is arrested by royal forces for aiding and abetting the rebellion. Under normal circumstances, he would be hanged, but instead James sentences a number of prisoners, Blood included, to slavery on the English plantations of Barbados. Even as a slave, Blood manages to gain a reputation as a physician in Barbados, but needless to say, slavery is not to his liking. He escapes his captors, commandeers a ship, and assumes the role of a pirate.

As a pirate story, Captain Blood has its enjoyable moments. The scenes of naval battles and sword duels are well-done and exciting but few and far between. The problem with Captain Blood is that Sabatini constantly wants to emphasize that Blood is a gentleman and only a reluctant pirate. This preoccupation with gentlemanliness, so common to English literature of the late 19th and early 20th centuries, takes almost all the fun out of the story. One might hope for a Han Solo-type scoundrel-with-a-heart-of-gold, but no, that’s not what you’re getting here. Instead, Captain Blood is a dull, smug, self-righteous prig.

Sabatini clearly wants to hold Blood up as the exemplar of manhood, and in doing so he makes the character practically flawless. Blood is the smartest mind in every argument, the most skillful sword in every fight, and the wiliest strategician in every naval engagement. When you think about memorable heroes, they often lose before they can win. We like to see them struggle to earn their victories, but Blood never struggles for anything because he’s perfection incarnate. The only area in which Sabatini allows the character any vulnerability is his heart. Blood develops an unwise and unrealistic infatuation for a beautiful young woman who happens to be the daughter of the slavemaster who owns him. Contrary to his martial heroics, in emotional matters Blood is an absolute milquetoast, and it’s difficult to admire his abjectly submissive way with women.

In between the few action scenes, Captain Blood consists mostly of conversations in which honor is debated at length. Each character asserts: I live by a code of honor. My actions are honorable; yours are not. These are the qualities of a gentleman; you sir, are not a gentleman. In addition, there are the familiar tropes of sworn enemies who “give their word” that they won’t attack their adversaries or try to escape captivity—as usual, that comes across as unrealistic and silly. Again, this English obsession with nobility and gentlemanliness. Credit to the French where it’s due: you won’t find these tedious debates in The Three Musketeers. Instead of wasting so many words on honor, just show us the definition of honor through your character’s behaviors. Alas, Captain Blood could have been a much more compelling book without all this sanctimonious posing. The plot of this novel would have made a good adventure novel along the lines of Alexandre Dumas’s or Robert Louis Stevenson’s work, but the telling is tiresome and reduces this story to a middling pirate tale at best.

Marthe by Joris-Karl Huysmans

Nana’s little sister

French author Joris-Karl Huysmans (1848–1907) published about two dozen books over the course of his literary career, but less than half of those have been translated into English. His first novel, Marthe: The Story of a Girl, was published in 1876. Huysmans writes in a naturalist style, a form of anti-romantic social realism that stresses unflinching descriptive detail. The Naturalist school of literature was founded by Emile Zola, who, upon reading Marthe, took Huysmans under his wing and mentored his early career. Of all of Zola’s protégés and imitators, Huysmans is the writer whose style most closely matches the master, at least for the first several years of his career. Judging by this impressive debut novel, Huysmans had already found his mature naturalist literary style before availing himself of Zola’s mentorship.

Marthe is born in Paris, the daughter of a painter and a factory worker. By the age of 15, both of her parents have died, leaving Marthe to care for herself. Like her mother before her, she works in an artificial pearl factory, crafting the fake jewels from powdered fish scales and glue. The pay is poor, the job is bad for her health, and she is surrounded by young women coworkers who regale her with tales of their dissolute lifestyles and licentious activities. She is discovered by a cabaret impresario who invites her to join the showgirls at a sleazy local theatre, and it is not long before Marthe learns that she can make a living by selling her body. From there, Marthe’s career runs the sexworker gamut from burlesque dancer to literal prostitute to a kept woman, the mistress of a sugar daddy. Along the way she meets Léo, a struggling poet and journalist, and the pair embark on a roller-coaster romance. Portions of the book are based on Huysmans’s own youthful love affair with a lady of the night.

Not long after Marthe, Zola published his own novel about a Parisian harlot, the classic Nana (1880). Zola’s novel is much more epic in scale and grander in scope than Marthe, which is a very short novel and intimate in its focus, similar to Stephen Crane’s Maggie: A Girl of the Streets (1893). Zola makes Nana the high-class of whore of Babylon who has all the powerful men of Paris wrapped around her finger, while Marthe is a more realistic garden-variety prostitute, the girl next door gone bad like so many poverty-stricken urban women of her age. Despite Naturalism’s pretensions of brutally scientific realism, Zola would exaggerate elements of his novels for dramatic effect. Huysmans, on the other hand, would exaggerate elements of his novels for repulsive effect. In the three books I’ve read by him, he seems to go out of his way to disgust, put off, and offend his audience. In Marthe, for example, Huysmans’s descriptions of a low-rent flophouse and a sleazy back alley are so loaded with sordid detail you might want to put the Kindle down and sanitize your hands. There are also some pretty racy sex scenes in this book, not too graphic by today’s standards but enough to be considered pornographic in the 1870s. Not surprisingly, the book was banned. Charged with obscenity in France, Huysmans had the book published in the more free-wheeling Belgium.

Among English-language audiences, Huysmans is better-known for his novels Au Rebours (Against Nature or Against the Grain) and Là-Bas (Down There or The Damned, a novel about Satan worship), but I found Marthe superior to them both. As an unromanticized depiction of social ills, this is a superb naturalist novel. Huysmans pulls no punches, and his prose is strikingly brilliant (at least in English translation). This is not a pleasant story, but Huysmans succeeds in viscerally immersing the reader into this sordid and squalid world. Zola is my favorite author, and I’m always look for other writers of a similar philosophy and style. Marthe is one novel in which one of the so-called lesser naturalists actually gives Maître Zola a run for his money.

The Man Who Sold the Moon by Robert A. Heinlein

Surprisingly preoccupied with business matters

The Man Who Sold the Moon is a science-fiction novella by American author Robert A. Heinlein that was published in 1950. It is also the title of a collection of short fiction published that same year, in which that novella is included. It is this latter collection to which this review pertains. In addition to its title selection, the collection entitled The Man Who Sold the Moon also contains five short stories originally published from 1939 to 1940. All six pieces are works in Heinlein’s Future History series, a large assemblage of novels, novellas, and stories that all take place in the same fictional future. The works collected in this volume represent the beginning of that series; that is to say, they are the earliest narratives in the fictional chronology.

The Man Who Sold the Moon (the novella) is an extended prequel to the much shorter story entitled “Requiem.” With the exception of that pair, the stories in this volume are only loosely related to one another. Although all united by the Future History scheme, they feature different characters and are set in different decades. Occasionally a character or his technological achievements may be mentioned in a later story. The half dozen stories in The Man Who Sold the Moon don’t seem to be presented in any logical order—neither in order of publication nor in order of Heinlein’s fictional timeline. This book includes a chart, put together by Heinlein, outlining the chronological order of the series. These being the earliest stories in the timeline, they take place roughly from the present day (the time of publication being late 1930s) to the fictional 1980s.

Heinlein’s first published story, “Life-Line,” is about an inventor who creates a device that can predict the future by “reading” timelines. It has the fanciful and thought-provoking qualities of a good Twilight Zone episode. “Let There Be Light” is about two scientists who pursue a more efficient form of lighting, and it’s about as exciting as that sounds. In “The Roads Must Roll,” America’s highways have been replaced by a cockamamie system of conveyer belts—not Heinlein’s brightest idea. “Blowups Happen” is a speculative story about nuclear power, written before nuclear reactors were a reality.

The centerpiece of this collection, The Man Who Sold the Moon, chronicles the endeavours of D. D. Harriman, a successful businessman, high-dollar huckster, and big-time negotiator who takes it upon himself to spearhead mankind’s first trip to the moon. In order to bring this dream to reality, Harriman must employ some questionable ethics, navigate legal loopholes, and exercise much financial chicanery. The plot is a chronicle of all this wheeling and dealing and not so much about the actual space travel. Harriman has elements of P. T. Barnum, Donald Trump, and Elon Musk, but more likeable and sympathetic than the latter two. “Requiem,” the sequel, catches up with Harriman later in life.

Given Heinlein’s reputation as the greatest science fiction writer of the 20th century, or at least one of the big three (along with Isaac Asimov and Arthur C. Clarke), I expected more from this volume. Heinlein’s Future History series is highly acclaimed, but this is still very early in that history. Space travel is just getting started, and we haven’t yet gotten anywhere near the colonization of other worlds. The result is a book full of surprisingly down-to-earth stories that have more to do with prosaic matters of business and industry than with astrophysics or space exploration. I’m surprised that the readers of Astounding Science Fiction magazine would have found fascinating these stories about government regulation and bureaucratic red tape. In just about every story here, Heinlein pushes some kind of libertarian agenda. This is most evident in the title selection, in which space travel and exploration are best left to private interests, preferably unregulated. Governments are merely obstacles in Heinlein’s vision of the future.

Despite my tepid review, there is enough here to build a reasonably satisfying foundation for a fictional universe, and I’m sure as Heinlein ventures farther from Earth, his narratives will become more deserving of the adjective “astounding.” I’m intrigued enough to follow this future timeline into the next volume of collected stories.

Stories in this collection

“Let There Be Light”

The Roads Must Roll

The Man Who Sold the Moon

Requiem

Life-Line

Blowups Happen

Introduction to Mathematical Philosophy by Bertrand Russell

Like reading a glossary for another book

Bertrand Russell won the 1950 Nobel Prize in Literature, though he was a philosopher, mathematician, and public intellectual, not a literary writer. (I suppose any nonfiction, however, could be considered essays.) As near as I can tell, Russell explained his mathematical system of philosophy in three works: In The Principles of Mathematics, published in 1903, he spelled out the fundamentals of his system of thought. He then built upon that foundation to create his magnum opus, together with Alfred North Whitehead, entitled Principia Mathematica, which was published in three volumes from 1910 to 1913. Neither of those books is intended for a general audience. Presumably one would need a PhD in mathematics to understand them. Russell did write many books for general readers, however, and in 1919 he released Introduction to Mathematical Philosophy, a book intended for lay readers (like myself).

Most of us, when we do math, take for granted the fact that 1 + 1 = 2 or 2 x 2 = 4. As a philosopher, however, Russell takes nothing for granted. Like Rene Descartes or Baruch Spinoza, he starts with first principles, assuming nothing, and systematically builds his argument from the ground up. It’s not enough to accept that 1 + 1 = 2 , one must ask WHY 1 + 1 = 2. Indeed, what exactly is addition, or for that matter, what is a number? That is what this book is about. Russell essentially defines the basic terms and functions of mathematics from a foundation of pure logical reasoning. Entire chapters are devoted to defining terms like rational numbers, real numbers, cardinal numbers, or differentiating between finite and infinite series. Reading this Introduction is much like reading an appendix or glossary to some other book, and Russell even admits as much. Introduction to Mathematical Philosophy merely provides the basic Rosetta Stone for understanding more advanced Russell books, like The Principles of Mathematics or the Principia.

You may not need a PhD in mathematics to read this Introduction, but you might very well need a master’s degree. Even if you’re good with numbers, however, that may not help you here. Leaning towards the philosopher rather than the mathematician, Russell writes all the mathematical formulae in paragraph prose, often using confusing, serpentine syntax. In order to express these concepts in textual form, Russell has to give new mathematical definitions to everyday words, which requires you to learn and memorize this new complex vocabulary.

Around chapter 14 (out of 18), however, the book finally turns in the direction of what I expected when I chose to read it; that is, how we use all these mathematical foundations to make logical deductions. From this point onward, the book is really quite interesting and can actually be understood by non-mathematicians. I question, however, whether I really needed to know many of those definitions that Russell spent so much time on. Once you get past chapter 13, many of those terms aren’t even used, and the text reads more like common sense.

What’s the point of all this? I suspect that Russell is a modern version of Pythagoras. He thinks mathematics is not merely a part of reality; mathematics is reality. Or rather, the logical rules that govern mathematics are the same that govern the movements of atoms or the thoughts in our heads. Russell sees no separation between mathematics and logic; they both operate on the same principles, whether the propositions are expressed through number or word. I believe Russell would assert that all human thought is fundamentally mathematical. We collect empirical sensory data, group it together in a complicated system of set theory, and then assess and organize that data with logical principles to create knowledge.

I was lost for much of the first half of this book, but I did find the latter half rewarding. Overall, I don’t think this Introduction is very effective in accomplishing what Russell set out to do, which is to provide the general public with an on-ramp to his school of thought. To get a handle on this subject, you’d be better off reading a Logic 101 textbook for college undergrads.

My Life with the Eskimo by Vilhjálmur Stéfansson

Traveling to hunt, hunting to travel

Vilhjálmur Stéfansson (1879–1962) was a Canadian-American explorer and ethnographer. He was born in Manitoba, the son of Icelandic immigrants. Shortly after his birth, his family moved to the Dakota Territory of the United States. He was educated at the universities of North Dakota, Iowa, and Harvard. As an explorer, his area of interest was the Arctic, and as an ethnographer, his expertise was the Inuit or Eskimo (the former is the more proper term, but the latter was more widely used in Stefansson’s lifetime). From 1908 to 1912, Stéfansson lived among the Eskimo on the north coast of Alaska and Canada, studying their ways and collecting artifacts and data for the American Museum of Natural History, which afterwards published his scientific findings. In 1912, Stéfansson also published a popular account of his expedition, intended for general readers, entitled My Life with the Eskimo. At that time, Stéfansson’s more than three year sojourn in the frozen North was the longest Arctic expedition on record.

But did he use all that time wisely? If this book were all you had to go on, it wouldn’t seem so. Stéfansson’s stated aim was to travel eastward to find bands of Inuit who have previously had little or no contact with White people. He bounces around the Alaskan coast for two years, however, and 162 pages of the book go by before he even gets started towards that goal. Even then, the book makes it unclear what exactly he’s trying to accomplish. The bulk of the text is about hunting caribou and seals. Stéfansson needs to hunt to feed himself, his assistants, and his sled dog team. He travels hundreds of miles to shoot the food necessary to travel hundreds of more miles to shoot more food to travel hundreds of more miles, and so on. Almost every animal they see, they kill, either for the meat, the skin, or as a specimen for the AMNH. Also a member of the expedition is a zoologist named Dr. Anderson, but he and Stéfansson are never in the same place, and the two are constantly traveling hundreds of miles to look for one another.

While all this is going on, Stéfansson is conducting scientific research, but he doesn’t tell you much about it. He’ll say, “I collected folklore from the people in this village.” That’s great. Would you like to tell us a little about that? No, let’s get back to another hunting story. Not until after page 360 does he finally discuss some of the archaeological digs, ethnological research, and linguistic studies that he conducted. These topics, much more interesting than all the caribou hunting, are relegated to the last few chapters of the book, treated as if they were appendices. If the entire book were more like these last 70 pages, My Life with the Eskimo would have been a much more satisfying read.

Reminiscent of another popular expedition narrative, Thor Heyerdahl’s Kon-Tiki (1948), Stéfansson also has an ethnological thesis to argue in My Life with the Eskimo. In his travels, he encounters what he calls “Blonde Eskimo”—Inuit with light hair, light skin, and blue eyes. Stéfansson theorizes that these may be the descendants of Danish and/or Norwegian settlers who disappeared from the west coast of Greenland centuries ago and perhaps drifted westward. Unlike Kon-Tiki, Stéfansson does not belabor the point. This theory is not the book’s raison d’etre, just one matter discussed among many. Though Stéfansson’s hypothesis of the Blonde Eskimo’s Norse origins does not seem so farfetched, as with Heyerdahl’s theory, 21st-century DNA testing has since disproved it.

I read quite a few explorer memoirs, and this one is just mediocre. Stéfansson is clearly very knowledgeable about the Inuit, but writing for the general public does not seem to have been his forte. Again, the last 70 pages of this book are quite good, but up to that point there’s a fair amount of boredom between nuggets of interest.

Freethinkers: A History of American Secularism by Susan Jacoby

The disheartening saga of the separation of church and state

Susan Jacoby is a former journalist and an outspoken atheist who has published a few books on the history of atheism and freethought, among other topics. Her book Freethinkers: A History of American Secularism was published in 2004. “Freethinker” is a blanket term that could be used to describe those who don’t subscribe wholeheartedly to any formal organized religion—perhaps anyone from a hardcore atheist to a liberal protestant. In American history, the term may refer to many of the founding fathers who were deists (e.g. John Adams, Thomas Jefferson, James Madison, Benjamin Franklin, Thomas Paine) or to the atheist and agnostic activists of the late 19th and 20th centuries (e.g. Robert Ingersoll, Elizabeth Cady Stanton, Clarence Darrow, Madalyn Murray O’Hair). Was Abraham Lincoln a freethinker? Jacoby looks into that too. Basically anyone who believes in the separation of church and state is covered here. The book essentially charts the changing status of that separation from the drafting of the U.S. Constitution to the end of the 20th century.

This is not an impartial, objective history. Jacoby certainly has an axe to grind, but that’s part of her point. The real history of freethought has been suppressed by American religious factions, and the only person who would write a book like this is a true nonbeliever. Many of America’s founders were freethinkers, yet today a myth persists that the U.S. was established as a Christian nation. Historically, many White Christian clergy in America were pro-slavery, anti-women’s rights, and anti-civil rights, and encouraged their congregations to toe those party lines, yet somehow when you read today’s history textbooks you’d think that all those problems were solved by the God-fearing faithful singing “Amazing Grace.” Jacoby sets the record straight from a freethinker’s perspective.

This book often reads less like a history of freethinkers than a history of freethought’s adversaries. It’s as if Jacoby defines freethinkers by their challenges, and their persecutors seem to get more coverage than the secular saints they’re persecuting. The book includes entire chapters on Catholics and their exertion of influence on American government. Jews, on the other hand, have generally been political allies of freethought who often found themselves grouped with atheists as alleged radicals and communists. Jacoby discusses the movements to abolish slavery, win women’s suffrage, teach evolution in schools, and make abortion legal, all of which were notably supported by freethinkers.

For an American freethinker such as myself, this book can be really depressing. Among the events chronicled, there are few freethought victories here. The one high point is the popularity of Robert Ingersoll in the 1880s and ‘90s. Thomas Paine is another hallowed freethought luminary, but sadly he ended his life more reviled than revered. By the time you get to the late 20th century, you’re convinced that the United States is one step away from becoming the Christian equivalent of the authoritarian Islamic regimes of the Ayatollahs or the Taliban. Jacoby repeatedly and bluntly points out how Christian Conservatives have subverted the Constitution for their own theistical agenda. This book ends with the George W. Bush administration, which Jacoby considers the low point in the ongoing battle for the separation of church and state. Things have only gotten worse since then—not because Donald Trump is more religious than Bush, but because Trump’s administration cozies up to religious interests to further his conservative aims. I’m glad somebody told this story, and Jacoby tells it well, but the truth hurts.