Posts Tagged ‘mathematics’
[I write this post with a certain degree of embarrassment, because in the end it turns out (1) to be more simple than I anticipated, and (2) already done before, as I could have found if I had internet access when I did this. :-)]
The so-called “Tupper’s self-referential formula” is the following, due to Jeff Tupper.
Graph the set of all points such that
in the region
where N is the following 544-digit integer:
The result is the following graph:
Whoa. How does this work?
At first sight this is rather too incredible for words.
But after a few moments we can begin to guess what is going on, and see that—while clever—this is perhaps not so extraordinary after all. So let us calmly try to reverse-engineer this feat.
Seeing more should help:
If that’s too easy, how about this?
Both are from Rekhāgaṇita, which is a c. 1720s translation by Jagannatha of Nasir al-Din al-Tusi’s 13th-century Arabic translation of Euclid’s Elements. It seems to be a straightforward translation of the Arabic — it even uses, to label vertices etc., letters in the Arabic order अ ब ज द ह व झ…. The text retains most of the structure and proposition numbers of Euclid, but in fact the Arabic world has considerably elaborated on Euclid. For instance, for the famous first example above, it gives sixteen additional proofs/demonstrations, which are not in the Greek texts.
Notes on the second: some technical vocabulary — a प्रथमाङ्कः is a prime number, and a रूप is a unit (one). The rest of the vocabulary, like “निःशेषं करोति” meaning “divides (without remainder)”, is more or less clear, and also the fact that as in Euclid, numbers are being conceived of as lengths (so दझ and झद mean the same).
It does sound cooler to say “इदमशुद्धम्” than “But this is a contradiction”. :-)
One aspect of mathematicians’ vocabulary that non-mathematicians often find non-intuitive is the word “trivial”. Mathematicians seem to call a great many things “trivial”, most of which are anything but. Here’s a joke, pointed out to me by Mohan:
Two mathematicians are discussing a theorem. The first mathematician says that the theorem is “trivial”. In response to the other’s request for an explanation, he then proceeds with two hours of exposition. At the end of the explanation, the second mathematician agrees that the theorem is trivial.
Like many jokes, this is not far from the truth. This tendency has led others to say, for example, that
In mathematics, there are only two kinds of proofs: Trivial ones, and undiscovered ones.
Or as Feynman liked to say, “mathematicians can prove only trivial theorems, because every theorem that’s proved is trivial”. (The mathematicians to whom Feynman said this do not seem to have liked the statement.)
A little examination, however, shows that all this is not far from reality, and indeed not far from the ideal of mathematics. Consider these excerpts from Gian-Carlo Rota‘s wonderful Indiscrete Thoughts:
“eventually every mathematical problem is proved trivial. The quest for ultimate triviality is characteristic of the mathematical enterprise.” (p.93)
“Every mathematical theorem is eventually proved trivial. The mathematician’s ideal of truth is triviality, and the community of mathematicians will not cease its beaver-like work on a newly discovered result until it has shown to everyone’s satisfaction that all difficulties in the early proofs were spurious, and only an analytic triviality is to be found at the end of the road.” (p. 118, in The Phenomenology of Mathematical Truth)
Are there definitive proofs?
It is an article of faith among mathematicians that after a new theorem is discovered, other simpler proofs of it will be given until a definitive one is found. A cursory inspection of the history of mathematics seems to confirm the mathematician’s faith. The first proof of a great many theorems is needlessly complicated. “Nobody blames a mathematician if the first proof of a new theorem is clumsy”, said Paul Erdős. It takes a long time, from a few decades to centuries, before the facts that are hidden in the first proof are understood, as mathematicians informally say. This gradual bringing out of the significance of a new discovery takes the appearance of a succession of proofs, each one simpler than the preceding. New and simpler versions of a theorem will stop appearing when the facts are finally understood. (p.146, in The Phenomenology of Mathematical Proof, here/here).
For more context, the section titled “Truth and Triviality” is especially worth reading, where he gives the example of proofs of the prime number theorem, starting with Hadamard and de la Vallée Poussin, through Wiener, Erdős and Selberg, and Levinson. Also p. 119, where he feels this is not unique to mathematics:
Any law of physics, when finally ensconced in a proper mathematical setting, turns into a mathematical triviality. The search for a universal law of matter, a search in which physics has been engaged throughout this century, is actually the search for a trivializing principle, for a universal “‘nothing but.” [...] The ideal of all science, not only of mathematics, is to do away with any kind of synthetic a posteriori statement and to leave only analytic trivialities in its wake. Science may be defined as the transformation of synthetic facts of nature into analytic statements of reason.
So to correct the earlier quote, there are three kinds of proofs from the mathematician’s perspective: those that are trivial, those that have not been discovered, and those niggling ones that are not yet trivial.
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:
- 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”.
- 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.)
- 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.
- 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.
For future reference, reminded by a recent webcomic:
- There is no ‘t’ in Cauchy–Schwarz. It is named after Hermann Schwarz (1843–1921), who rediscovered it in 1888. His other work includes some stuff in complex analysis: Schwarz lemma, Schwarz–Christoffel mapping, a theorem about symmetry of second derivatives, etc.
- All other (mathematical) Schwar(t)zes have ‘t’s.
- There’s Jack Schwartz (1930–2009), mathematician–and–computer scientist, who was Gian-Carlo Rota’s advisor and co-wrote the monumental three-volume work on Linear Operators (“Dunford and Schwartz”). And he’s also the name in the Schwartz-Zippel lemma (“Or the Schwartz-Zippel-DeMillo-Lipton Lemma.”)
- There’s Laurent Schwartz (1915–2002), who was at ENS, worked on the theory of distributions, and won the Fields Medal in 1950. (“Schwartz space”)
Those are the major ones, I think.
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 xa+b = xa + xb. A teacher who has emphasized the intrinsic meaning of expressions such as xa will point out that xa+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 xa+b = xa xb. 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.)
If you came here because you were reading Peter Winkler’s “7 Puzzles You Think You Must Not Have Heard Correctly”, the names are supposed to be Eugene Curtin and Max Warshauer, and the paper is called “The locker puzzle”, published in The Mathematical Intelligencer, Volume 28, Number 1 (March 2006), pages 28–31.
[If not, you should read the amazing "7 Puzzles You Think You Must Not Have Heard Correctly", spend several days trying the first problem, read the brilliant solution, and then come back here if you're interested in learning why no other solution can do better.]
The paper is available here if your institution has access. If not, here’s a sketch of the proof that the strategy cannot be improved upon. [Update 2010-01-06: Oliver Nash has a post about the puzzle, explaining both the original solution and the proof of optimality, here. Just the original solution is also worked out by David MacKay here.]
First, let us modify the rules slightly so that each prisoner must continue looking in boxes until he finds the box containing his name. The prisoners win if no prisoner opens more than 50 (i.e., n/2) boxes. This change obviously makes no difference to the outcome. Let’s call this (modified) game Game 1.
A different game involves all the prisoners being in the room at the same time, and is played as follows. The first prisoner opens boxes until he finds his name (i.e., the number “1”). Then, the lowest-numbered prisoner whose name hasn’t been revealed starts opening boxes until he finds his name. Then the next lowest-numbered whose name hasn’t been revealed opens boxes, and so on. The prisoners win if no one opens more than 50 boxes. Call this Game 2.
Let’s say we observe the prisoners as they play Game 2, and record the order in which boxes were revealed. This completely specifies what happened. For example, (with 10 prisoners instead of 100) if we record the list 2,6,1,4,9,7,10,8,3,5, we know that the first prisoner revealed boxes containing 2, 6, 1, then the third (lowest unrevealed) prisoner opened boxes with 4,9,7,10,8,3, then prisoner 5 opened 5, and they lost because the third prisoner opened 6 > 5 boxes.
- Prove: No matter what strategy the prisoners follow, each permutation has the same probability (1/n!) of being the list recorded.
- Prove: “The classical records-to-cycles bijection”. It sends 2,6,1,4,9,7,10,8,3,5 to (2 6 1)(4 9 7 10 8 3)(5), for example.
- So the probability of the prisoners winning Game 2 (no matter what strategy they follow) is exactly the probability that a random permutation has no cycle of length greater than n/2.
- Prove: Any strategy for Game 1 corresponds to a strategy for Game 2 with the same probability. (Trivial: the only change is that you don’t have to open boxes you’ve already seen opened.)
This proves that the pointer-chasing strategy is optimal for Game 1.
Here’s the puzzle as it was originally considered, still open: suppose there are n prisoners and 2n boxes, half of them empty. The prisoners can each examine n lockers. The pointer-chasing strategy doesn’t work as empty boxes point nowhere. Does the probability of winning go to 0 as n→∞?