I haven’t read more than the introduction yet — a review copy arrived just yesterday — but I imagine it’s good judging by who wrote it. Havil’s book Gamma is my favorite popular math book. (Maybe I should say “semi-popular.” Havil’s books have more mathematical substance than most popular books, but they’re still aimed at a wide audience. I think he strikes a nice balance.) His latest book is a scientific biography, a biography with an unusual number of equations and diagrams.
Napier is best known for his discovery of logarithms. (People debate endlessly whether mathematics is discovered or invented. Logarithms are so natural — pardon the pun — that I say they were discovered. I might describe other mathematical objects, such as Grothendieck’s schemes, as inventions.) He is also known for his work with spherical trigonometry, such as Napier’s mnemonic. Maybe Napier should be known for other things I won’t know about until I finish reading Havil’s book.
Fermat famously claimed to have a proof of his last theorem that he didn’t have room to write down. Mathematicians have speculated ever since what this proof must have been, though everyone is convinced the proof must have been wrong.
The usual argument for Fermat being wrong is that since it took over 350 years, and some very sophisticated mathematics, to prove the theorem, it’s highly unlikely that Fermat had a simple proof. That’s a reasonable argument, but somewhat unsatisfying because it’s risky business to speculate on what a proof must require. Who knows how complex the proof of FLT in The Book is?
André Weil offers what I find to be a more satisfying argument that Fermat did not have a proof, based on our knowledge of Fermat himself. Dana Mackinzie summarizes Weil’s argument as follows.
Fermat repeatedly bragged about the n = 3 and n = 4 cases and posed them as challenges to other mathematicians … But the never mentioned the general case, n = 5 and higher, in any of his letters. Why such restraint? Most likely, Weil argues, because Fermat had realized that his “truly wonderful proof” did not work in those cases.
Every mathematician has had days like this. You think you have a great insight, but then you go out for a walk, or you come back to the problem the next day, and you realize that your great idea has a flaw. Sometimes you can go back and fix it. And sometimes you can’t.
I’ve enjoyed reading The New York Times Book of Physics and Astronomy, ISBN 1402793200, a collection of 129 articles written between 1888 and 2012. Its been much more interesting than its mathematical predecessor. I’m not objective — I have more to learn from a book on physics and astronomy than a book on math — but I think other readers might also find this new book more interesting.
I was surprised by the articles on the bombing of Hiroshima and Nagasaki. New York Times reporter William Lawrence was allowed to go on the mission over Nagasaki. He was not on the plane that dropped the bomb, but was in one of the other B-29 Superfortresses that were part of the mission. Lawrence’s story was published September 9, 1945, exactly one month later. Lawrence was also allowed to tour the ruins of Hiroshima. His article on the experience was published September 5, 1945. I was surprised how candid these articles were and how quickly they were published. Apparently military secrecy evaporated rapidly once WWII was over.
Another thing that surprised me was that some stories were newsworthy more recently than I would have thought. I suppose I underestimated how long it took to work out the consequences of a major discovery. I think we’re also biased to think that whatever we learned as children must have been known for generations, even though the dust may have only settled shortly before we were born.
Paul Halmos divided progress in math into three categories: concepts, explosions, and developments. This was in his 1990 article “Has progress in mathematics slowed down?”. (His conclusion was no.) This three-part classification not limited to math and could be useful in other areas.
Concepts are organizational ideas, frameworks, new vocabulary. Some of his examples were category theory and distributions (generalized functions).
Explosions solve old problems and generate a lot of attention among mathematicians and in the popular press. As Halmos puts is, “hot news not only for the Transactions, but also for the Times for a day, for Time for a week, and for student mathematics clubs for many months.” He cites the solution to the Four Color Theorem as an example. He no doubt would have cited Fermat’s Last Theorem had he written his article five years later.
Developments are “deep and in some cases even breathtaking developments (but not explosions) of the kind that might not make the Times, but could possibly get Fields medals for their discoverers.” One example he gives is the Atiyah-Singer index theorem.
The popular impression of math and science is that progress is all about explosions though it’s more about concepts and developments.
In a nutshell, the authors hope to get some insight into whether a myth is based on fact by seeing whether the social network of characters in the myth looks more like a real social network or like the social network in a work of deliberate fiction. For instance, the social networks of the Iliad and Beowulf look more like actual social networks than does the social network of Harry Potter. Real social networks follow a power law distribution more closely than do social networks in works of fiction.
This could be interesting. For example, the article points out that some scholars believe Beowulf has a basis in historical events, though they don’t believe that Beowulf the character corresponds to a historical person. The network approach lends support to this position: the Beowulf social network looks more realistic when Beowulf himself is removed.
It seems however that an accurate historical account might have a suspicious social network, not because the events in it were made up but because they were filtered according to what the historian thought was important.
Leibniz [1646-1716] actually thought about symbolic logic and why it was powerful and how it could be put to use. He went from that to building a machine that could carry out logical operations on bits. He knew about binary arithmetic. I found that quite startling. Up till then I hadn’t been that well informed about the history of logic and computing. I hadn’t been aware that anyone was thinking about those things so far in the past. I thought it all started with [Alan] Turing. So, I had computers in the 17th century.
I always thought that Fourier was the first to come up with the idea of expressing general functions as infinite sums of sines and cosines. Apparently this isn’t true.
The idea that various functions can be described in terms of Fourier series … was for the first time proposed by Daniel Bernoulli (1700–1782) to solve the one-dimensional wave equation (the equation of motion of a string) about 50 years before Fourier. … However, no one contemporaneous to D. Bernoulli accepted the idea as a general method, and soon the study was forgotten.
The subtitle may be a little misleading. There is a fair amount of math in the book, but the ratio of history to math is pretty high. You might say the book is more about the role of mathematicians than the role of mathematics. As Roger Penrose says on the back cover, the book has “illuminating descriptions and minimal technicality.”
Someone interested in weather prediction but without a strong math background would enjoy reading the book, though someone who knows more math will recognize some familiar names and theorems and will better appreciate how they fit into the narrative.
According to The History of Rome, episode 126, Diocletian increased the size of the Roman imperial bureaucracy from around 15,000 people to around 30,000.
I wanted to compare the size of the bureaucracy that ran the Roman Empire to the size of the bureaucracy that runs Houston, TX. This page suggests that the city of Houston has about 68,000 employees. But far more people work for government in other capacities than work for the city. According to Table 1 of this page, the latest estimate is that 361,800 in the Houston MSA work in the government sector. And about 22 million people work in the government sector nation wide.
Please don’t leave comments saying the Roman Empire and Houston are not directly comparable. Of course they’re not. But still, a very rough comparison is interesting.
Here’s an interesting bit of history from Julian Havil’s new book The Irrationals. In 1593 Francois Vièta discovered the following infinite product for pi:
Havil says this is “the earliest known.” I don’t know whether this is specifically the oldest product representation for pi, or more generally the oldest formula for an infinite sequence of approximations that converge to pi. Vièta’s series is based on the double angle formula for cosine.
The first series for pi I remember seeing comes from evaluating the Taylor series for arc tangent at 1:
I saw this long before I knew what a Taylor series was. I imagine others have had the same experience because the series is fairly common in popular math books. However, this series is completely impractical for computing pi because it converges at a glacial pace. Vièta’s formula, on the other hand, converges fairly quickly. You could see for yourself by running the following Python code:
from math import sqrt
prod = 1.0
radic = 0.0
for i in range(10):
radic = sqrt(2.0 + radic)
prod *= 0.5*radic
After 10 terms, Vièta’s formula is correct to five decimal places.
Related posts: more sophisticated and efficient series for computing pi:
In his essay On Providence, Seneca (4 BC – 65 AD) says the following about tides:
In point of fact, their growth is strictly allotted; at the appropriate day and hour they approach in greater volume or less according as they are attracted by the lunar orb, at whose sway the ocean wells up.
Seneca doesn’t just mention an association between lunar and tidal cycles, but he says tides are attracted by the moon. That sounds awfully Newtonian for someone writing 16 centuries before Newton. The ancients may have understood that gravity wasn’t limited to the pull of the earth, that at least the moon also had a gravitational pull. That’s news to me.
Here’s a perspective on the 1970s I found interesting: The decade was so embarrassing that climbing out of the ’70s was a proud achievement.
The 1970s were America’s low tide. Not since the Depression had the country been so wracked with woe. Never — not even during the Depression — had American pride and self-confidence plunged deeper. But the decade was also, paradoxically, in some ways America’s finest hour. America was afflicted in the 1970s by a systemic crisis analogous to the one that struck Imperial Rome in the middle of the third century A.D. … But unlike the Romans, Americans staggered only briefly before the crisis. They took the blow. For a short time they behaved foolishly, and on one or two occasions, even disgracefully. Then they recouped. They rethought. They reinvented.
I recently received review copies of two books by Benjamin Wardhaugh. Here I will discuss How to Read Historical Mathematics (ISBN 0691140146). The other book is his anthology of historical popular mathematics which I intend to review later.
Here is the key passage, located near the end of How to Read Historical Mathematics, for identifying the author’s perspective.
But not all historical mathematics is significant. And perhaps there is a second kind of significance, where something can be historically significant without being mathematically significant. Some historians (I’m one of them) delight in investigating mathematical writing that contains little or no important or novel mathematics: popular textbooks, self-instruction manuals, … or old almanacs and popular magazines with mathematical news or puzzles in them. These kinds of writing … are certainly significant for a historian who wants to know about popular experiences of mathematics. But they’re not significant in the sense of containing significant mathematics.
Wardhaugh’s perspective is valuable, though it is not one that I share. My interest in historical math is more on the development of the mathematical ideas rather than their social context. I’m interested, for example, in discovering the concrete problems that motivated mathematics that has become more abstract and formal.
I was hoping for something more along the lines of a mapping from historical definitions and notations to their modern counterparts. This book contains a little of that, but it focuses more on how to read historical mathematics as a historian rather than as a mathematician. However, if you are interested in more of the social angle, the book has many good suggestions (and even exercises) for exploring the larger context of historical mathematical writing.