Euler, pirates, and the discovery of America

Is there nothing Euler wasn’t involved in?!

That rhetorical question is independent of the following two, which are exceedingly weak connections.

“Connections” to piracy: Very tenuous connections, of course, but briefly, summarising from the article:

1. Maupertuis: President of the Berlin Academy for much of the time Euler was there. His father got a license from the French king to attack English ships, made a fortune, and retired. Maupertuis is known for formulating the Principle of Least Action (but maybe it was Euler), and best known for taking measurements showing the Earth bulges at the equator as Newton had predicted, thus “The Man Who Flattened the Earth”.
2. Henry Watson: English privateer living in India, lost a fortune to the scheming British East India Company. Wanted to be a pirate, but wasn’t actually one. Known for: translated Euler’s Théorie complette [E426] from its original French: A complete theory of the construction and properties of vessels: with practical conclusions for the management of ships, made easy to navigators. (Yes, Euler wrote that.)
3. Kenelm Digby: Not connected to Euler actually, just the recipient of a letter by Fermat in which a problem that was later solved by Euler was discussed. Distinguished alchemist, one of the founders of the Royal Society, did some pirating (once) and was knighted for it.
4. Another guy, nevermind.

Moral: The fundamental interconnectedness of all things. Or, connections don’t mean a thing.

The discovery of America: Columbus never set foot on the mainland of America, and died thinking he had found a shorter route to India and China, not whole new continents that were in the way. The question remained whether these new lands were part of Asia (thus, “Very Far East”) or not. The czar of Russia (centuries later) sent Bering to determine the bounds of Russia, and the Bering Strait separating the two continents was discovered and reported back: America was not part of Russia. At about this time, there were riots in Russia, there was nobody to make the announcement, and “Making the announcement fell to Leonhard Euler, still the preeminent member of the St. Petersburg Academy, and really the only member who was still taking his responsibilities seriously.” As the man in charge of drawing the geography of Russia, Euler knew a little, and wrote a letter to Wetstein, member of the Royal Society in London. So it was only through Euler that the world knew that the America that was discovered was new. This letter [E107], with others, is about the only work of Euler in English. That Euler knew English (surprisingly!) is otherwise evident from the fact that he translated and “annotated” a book on ballistics by the Englishman Benjamin Robins. The original was 150 pages long; with Euler’s comments added, it was 720. [E77, translated back into English as New principles of gunnery.]

Most or all of the above is from Ed Sandifer’s monthly column How Euler Did It.

The works of Leonhard Euler online has pages for all 866 of his works; 132 of them are available in English, including the translations from the Latin posted by graduate student Jordan Bell on the arXiv. They are very readable.

This includes his Letters to a German Princess on various topics in physics and philosophy [E343,E344,E417], which were bestsellers when reprinted as science books for a general audience. It includes his textbook, Elements of Algebra [E387,E388]. Find others on Google Books. The translations do not seem to include (among his other books) his classic textbook Introductio in analysin infinitorum [E101,E102, “the foremost textbook of modern times”], though there are French and German translations available.

Apparently, Euler’s Latin is (relatively) not too hard to follow.

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Portrait of the “lazy” student

[I had some thoughts, but I didn’t write anything here and this turned into one of those drafts lost in time. At least it’s a dump of two links for now.]

Found today Doron Zeilberger’s What About “Quarter Einsteins”?, written in 1967 (emphasis mine):

There is yet another kind of talented students, whose natural curiosity lead them, already from a young age, to read and look at more advanced material, in order to satisfy their natural curiosity.

When such a student enters high school (and in fact, already in the higher grades of elementary school) he sees that the material that he has already studied on his own presented in a different way. The learning is induced through severe disciple (all the system of examinations and grades), and the material is taught the same way as in animal training. The fascinating science of Chemistry turns into a boring list of dry formulas, that he has to learn by heart, and the threats and the incentives practiced in school badly offend him. As though out of spite, he does not listen to the commands of his teachers, but instead studies on his own material that is not included in the curriculum. Obviously, even the most talented student can not learn from just sitting in class, (and even during class he often studies other material), and so starts the “tragedy” described in your article.

[…]

[Einstein] managed somehow to find his way in life (but it wasn’t easy, even for him). But besides him there are “half and quarter Einsteins”, and just plain talented students that had the potential to contribute to society, but […] flunked out of high school and their “genius” did not help them find their proper place in modern society […].

Mark Tarver wrote a post in 2006 called the The Bipolar Lisp Programmer, which many readers found to be a frighteningly accurate description. (Reproduced below.) Again, please ignore the remarks about “outstanding brilliance” etc. That is not the point; the point is the tragedy of talented (even if only very moderately so) students failing to cope with the system.

The Bipolar Lisp Programmer

Any lecturer who serves his time will probably graduate hundreds, if not thousands of students. Mostly they merge into a blur; like those paintings of crowd scenes where the leading faces are clearly picked out and the rest just have iconic representations. This anonymity can be embarrassing when some past student hails you by name and you really haven’t got the foggiest idea of who he or she is. It both nice to be remembered and also toe curlingly embarrassing to admit that you cannot recognise who you are talking to.

But some faces you do remember; students who did a project under you. Also two other categories – the very good and the very bad. Brilliance and abject failure both stick in the mind. And one of the oddest things, and really why I’m writing this short essay, is that there are some students who actually fall into both camps. Here’s another confession. I’ve always liked these students and had a strong sympathy for them.

Now abject failure is nothing new in life. Quite often I’ve had students who have failed miserably for no other reason than they had very little ability. This is nothing new. What is new is that in the UK, we now graduate a lot of students like that. But, hey, that’s a different story and I’m not going down that route.

No I want to look at the brilliant failures. Because brilliance amd failure are so often mixed together and our initial reaction is it shouldn’t be. But it happens and it happens a lot. Why?

Well, to understand that, we have to go back before university. Lets go back to high school and look at a brilliant failure in the making. Those of you who have seen the film “Donnie Darko” will know exactly the kind of student I’m talking about. But if you haven’t, don’t worry, because you’ll soon recognise the kind of person I’m talking about. Almost every high school has one every other year or so.

Generally what we’re talking about here is a student of outstanding brilliance. Someone who is used to acing most of his assignments; of doing things at the last minute but still doing pretty well at them. At some level he doesn’t take the whole shebang all that seriously; because, when you get down to it, a lot of the rules at school are pretty damned stupid. In fact a lot of the things in our world don’t make a lot of sense, if you really look at them with a fresh mind. And generally our man does have a fresh mind and a very sharp one.

So we have two aspects to this guy; intellectual acuteness and not taking things seriously. The not taking things seriously goes with finding it all pretty easy and a bit dull. But also it goes with realising that a lot of human activity is really pretty pointless, and when you realise that and internalise it then you become cynical and also a bit sad – because you yourself are caught up in this machine and you have to play along if you want to get on. Teenagers are really good at spotting this kind of phony nonsense. Its also the seed of an illness; a melancholia that can deepen in later life into full blown depression.

Another feature about this guy is his low threshold of boredom. He’ll pick up on a task and work frantically at it, accomplishing wonders in a short time and then get bored and drop it before its properly finished. He’ll do nothing but strum his guitar and lie around in bed for several days after. Thats also part of the pattern too; periods of frenetic activity followed by periods of melancholia, withdrawal and inactivity. This is a bipolar personality.

Alright so far? OK, well lets graduate this guy and see him go to university. What happens to him then?

Here we have two stories; a light story and a dark one.

The light story is that he’s really turned on by what he chooses and he goes on to graduate summa cum laude, vindicating his natural brilliance.

But that’s not the story I want to look at. I want to look at the dark story. The one where brilliance and failure get mixed together.

This is where this student begins by recognising that university, like school, is also fairly phony in many ways. What saves university is generally the beauty of the subject as built by great minds. But if you just look at the professors and don’t see past their narrow obsession with their pointless and largely unread (and unreadable) publications to the great invisible university of the mind, you will probably conclude its as phony as anything else. Which it is.

But lets stick to this guy’s story.

Now the big difference between school and university for the fresher is FREEDOM. Freedom from mom and dad, freedom to do your own thing. Freedom in fact to screw up in a major way. So our hero begins a new life and finds he can do all he wants. Get drunk, stumble in at 3.00 AM. So he goes to town and he relies on his natural brilliance to carry him through because, hey, it worked at school. And it does work for a time.

But brilliance is not enough. You need application too, because the material is harder at university. So pretty soon our man is getting B+, then Bs and then Cs for his assignments. He experiences alternating feelings of failure cutting through his usual self assurance. He can still stay up to 5.00AM and hand in his assignment before the 9.00AM deadline, but what he hands in is not so great. Or perhaps he doesn’t get into beer, but into some mental digression from his official studies that takes him too far away from the main syllabus.

This sort of student used to pass my way every now and then, Riding on the bottom of the class. One of them had Bored> as his UNIX prompt. If I spotted one I used to connect well with them. (In fact I rescued one and now he’s a professor and miserable because he’s surrounded by phonies – but hey, what can you do?). Generally he would come alive in the final year project when he could do his own thing and hand in something really really good. Something that would show (shock, horror) originality. And a lot of professors wouldn’t give it a fair mark for that very reason – and because the student was known to be scraping along the bottom.

Often this kind of student never makes it to the end. He flunks himself by dropping out. He ends on a soda fountain or doing yard work, but all the time reading and studying because a good mind is always hungry.

(Rest of essay follows, with the Lisp parts gratuitously excised, but you can read the original post.)

[…] the peculiar strengths and weaknesses of the brilliant bipolar mind (BBM).

[…] He can see far; further than in fact his strength allows him to travel. He conceives of brilliant ambitious projects requiring great resources, and he embarks on them only to run out of steam. Its not that he’s lazy; its just that his resources are insufficient.

[…] not just the strengths but also the weaknesses of the BBM.

One of these is the inability to finish things off properly. The phrase ‘throw-away design’ is absolutely made for the BBM […]

[…] And he is, unlike the rank and file, unprepared to compromise. And this leads to many things.

[…]

And this brings me to the last feature of the BBM. The flip side of all that energy and intelligence – the sadness, melancholia and loss of self during a down phase. The intelligence is directed inwards in mournful contemplation of the inadequacies […]. The problems are soluble […], but when you’re down everything seems insoluble. […]

[…]

So what’s the problem with Lisp? Basically, there is no problem with Lisp, because Lisp is, like life, what you make of it.

Is it important to “understand” first?

The Oxford University Press has been publishing a book series known as “Very Short Introductions”. These slim volumes are an excellent idea, and cover over 200 topics already. The volume Mathematics: A Very Short Introduction is written by Timothy Gowers.

Gowers is one of the leading mathematicians today, and a winner of the Fields Medal (in 1998). In addition to his research work, he has also done an amazing amount of service to mathematics in other ways. He edited the 1000-page Princeton Companion to Mathematics, getting the best experts to write, and writing many articles himself. He also started the Polymath project and the Tricki, the “tricks wiki”. You can watch his talk on The Importance of Mathematics (with slides) (transcript), and read his illuminating mathematical discusssions, and his blog. His great article The Two Cultures of Mathematics is on the “theory builders and problem solvers” theme, and is a paper every mathematician should read.

Needless to say, “Mathematics: A Very Short Introduction” is a very good read. Unlike many books aimed at non-mathematicians, Gowers is quite clear that he does “presuppose some interest on the part of the reader rather than trying to drum it up myself. For this reason I have done without anecdotes, cartoons, exclamation marks, jokey chapter titles, or pictures of the Mandelbrot set. I have also avoided topics such as chaos theory and Godel’s theorem, which have a hold on the public imagination out of proportion to their impact on current mathematical research”. What follows is a great book that particularly excels at describing what it is that mathematicians do. Some parts of the book, being Gowers’s personal views on the philosophy of mathematics, might not work very well when directed at laypersons, not because they require advanced knowledge, but assume a culture of mathematics. Doron Zeilberger thinks that this book “should be recommended reading to everyone and required reading to mathematicians”.

Its last chapter, “Some frequently asked questions”, carries Gowers’s thoughts on some interesting questions. With whole-hearted apologies for inserting my own misleading “summaries” of the answers in brackets, they are the following: “1.Is it true that mathematicians are past it by the time they are 30?” (no), “2. Why are there so few women mathematicians?” (puzzling and regrettable), “3. Do mathematics and music go together?” (not really), “4. Why do so many people positively dislike mathematics?” (more on this below), “5. Do mathematicians use computers in their work?” (not yet), “6. How is research in mathematics possible?” (if you have read this book you won’t ask), “7. Are famous mathematical problems ever solved by amateurs?” (not really), “8. Why do mathematicians refer to some theorems and proofs as beautiful?” (already discussed. Also, “One difference is that […] a mathematician is more anonymous than an artist. […] it is, in the end, the mathematics itself that delights us”.) As I said, you should read the book itself, not my summaries.

The interesting one is (4).

4. Why do so many people positively dislike mathematics?

One does not often hear people saying that they have never liked biology, or English literature. To be sure, not everybody is excited by these subjects, but those who are not tend to understand perfectly well that others are. By contrast, mathematics, and subjects with a high mathematical content such as physics, seem to provoke not just indifference but actual antipathy. What is it that causes many people to give mathematical subjects up as soon as they possibly can and remember them with dread for the rest of their lives?

Probably it is not so much mathematics itself that people find unappealing as the experience of mathematics lessons, and this is easier to understand. Because mathematics continually builds on itself, it is important to keep up when learning it. For example, if you are not reasonably adept at multiplying two-digit numbers together,then you probably won’t have a good intuitive feel for the distributive law (discussed in Chapter 2). Without this, you are unlikely to be comfortable with multiplying out the brackets in an expression such as , and then you will not be able to understand quadratic equations properly. And if you do not understand quadratic equations, then you will not understand why the golden ratio is .

There are many chains of this kind, but there is more to keeping up with mathematics than just maintaining technical fluency. Every so often, a new idea is introduced which is very important and markedly more sophisticated than those that have come before, and each one provides an opportunity to fall behind. An obvious example is the use of letters to stand for numbers, which many find confusing but which is fundamental to all mathematics above a certain level. Other examples are negative numbers, complex numbers, trigonometry, raising to powers, logarithms, and the beginnings of calculus. Those who are not ready to make the necessary conceptual leap when they meet one of these ideas will feel insecure about all the mathematics that builds on it. Gradually they will get used to only half understanding what their mathematics teachers say, and after a few more missed leaps they will find that even half is an overestimate. Meanwhile, they will see others in their class who are keeping up with no difficulty at all. It is no wonder that mathematics lessons become, for many people, something of an ordeal.

This seems to be exactly the right reason. No one would enjoy being put through drudgery that they were not competent at, and without the beauty at the end of the pursuit being apparent. (I hated my drawing classes in school, too.) See also Lockhart’s Lament, another article that everyone — even, or especially, non-mathematicians — should read.

As noted earlier, Gowers has some things to say about the philosophy of mathematics. As is evident from his talk “Does mathematics need a philosophy?” (also typeset as essay 10 of 18 Unconventional Essays on the Nature of Mathematics), he has rejected the Platonic philosophy (≈ mathematical truths exist, and we’re discovering them) in favour of a formalist one (≈ it’s all just manipulating expressions and symbols, just stuff we do). The argument is interesting and convincing, but I find myself unwilling to change my attitude. Yuri Manin says in a recent interview that “I am an emotional Platonist (not a rational one: there are no rational arguments in favor of Platonism)”, so it’s perhaps just as well.

Anyway, the anti-Platonist / formalist idea of Gowers is evident throughout the book, and of course it has its great side: “a mathematical object is what it does” is his slogan, and most of us can agree that “one should learn to think abstractly, because by doing so many philosophical difficulties disappear” , etc. The only controversial suggestion, perhaps, follows the excerpt quoted above (of “Why do so many people positively dislike mathematics?”):

Is this a necessary state of affairs? Are some people just doomed to dislike mathematics at school? Or might it be possible to teach the subject differently in such a way that far fewer people are excluded from it? I am convinced that any child who is given one-to-one tuition in mathematics from an early age by a good and enthusiastic teacher will grow up liking it. This, of course, does not immediately suggest a feasible educational policy, but it does at least indicate that there might be room for improvement in how mathematics is taught.

One recommendation follows from the ideas I have emphasized in this book. Above, I implicitly drew a contrast between being technically fluent and understanding difficult concepts, but it seems that almost everybody who is good at one is good at the other. And indeed, if understanding a mathematical object is largely a question of learning the rules it obeys rather than grasping its essence, then this is exactly what one would expect — the distinction between technical fluency and mathematical understanding is less clear-cut than one might imagine.

How should this observation influence classroom practice? I do not advocate any revolutionary change — mathematics has suffered from too many of them already — but a small change in emphasis could pay dividends. For example, suppose that a pupil makes the common mistake of thinking that x^a+b = x^a + x^b. A teacher who has emphasized the intrinsic meaning of expressions such as x^a will point out that x^a+b means a+b xs all multiplied together, which is clearly the same as a of them multiplied together multiplied by b of them multiplied together. Unfortunately, many children find this argument too complicated to take in, and anyhow it ceases to be valid if a and b are not positive integers.

Such children might benefit from a more abstract approach. As I pointed out in Chapter 2, everything one needs to know about powers can be deduced from a few very simple rules, of which the most important is x^a+b = x^a x^b. If this rule has been emphasized, then not only is the above mistake less likely in the first place, but it is also easier to correct: those who make the mistake can simply be told that they have forgotten to apply the right rule. Of course, it is important to be familiar with basic facts such as that x³ means x times x times x, but these can be presented as consequences of the rules rather than as justifications for them.

I do not wish to suggest that one should try to explain to children what the abstract approach is, but merely that teachers should be aware of its implications. The main one is that it is quite possible to learn to use mathematical concepts correctly without being able to say exactly what they mean. This might sound a bad idea, but the use is often easier to teach, and a deeper understanding of the meaning, if there is any meaning over and above the use, often follows of its own accord.

Of course, there is an instinctive reason to immediately reject such a proposal — as the MAA review by Fernando Q. Gouvêa observes, ‘I suspect, however, that there is far too much “that’s the rule” teaching, and far too little explaining of reasons in elementary mathematics teaching. Such a focus on rules can easily lead to students having to remember a huge list of unrelated rules. I fear Gowers’ suggestion here may in fact be counterproductive.’ Nevertheless, the idea that technical fluency may precede and lead to mathematical understanding is worth pondering.

(Unfortunately, even though true, it may not actually help with teaching: in practice, drilling-in “mere” technical fluency can be as unsuccessful as imparting understanding.)

The Decline and Fall of The Decline and Fall

(Yes, this post is written just for the title. More details would be received gratefully.)

Over a period of 17 years from 1770 to 1787, Edward Gibbon wrote The History of the Decline and Fall of the Roman Empire. It was, among other things, a mammoth history (6 volumes, 71 chapters) of the last days of Rome, which for Gibbon apparently meant several centuries. (The book covers over thirteen centuries of history; here’s an outline.)

The work received instant praise. Adam Smith’s letter to Gibbon is typical:

“I cannot express to you the pleasure it gives me to find that by the universal consent of every man of taste and learning whom I either know or correspond with, it sets you at the very head of the whole literary tribe at present existing in Europe.”

The Decline and Fall became the model for all historians that followed — including its pessimism (history as “little more than the register of the crimes, follies, and misfortunes of mankind”), its overarching narrative, and its indictment of religion.

It became a literary monument of the 18th century, and one of the works that every educated man was expected to have read, a part of every bookshelf. Churchill (“I devoured Gibbon. […] I rode triumphantly through it from end to end and enjoyed it all”), Carlyle (“how gorgeously does it swing across the gloomy and tumultuous chasm of these barbarous centuries”), Virginia Woolf (“not merely a master of the pageant and the story; he is also the critic and the historian of the mind […] We seem as we read him raised above the tumult and the chaos into a clear and rational air”)… everyone read The Decline and Fall and spoke of it in the highest terms. (Gandhi read it in jail, and considered it an inferior version of the Mahabharata.) It was read by doctors, politicians, lawyers, novelists, even Sanskrit professors.

But then times began to change. Education stopped being the reading of “classics“, and became the learning of “subjects”. Today, no one I know has read The Decline And Fall, nor considers it worth the time.