Archive for the ‘Mathematics’ Category
I have always known that mathematics teaching in high schools is generally disastrous. I suspect that almost all non-specialists and even a majority of specialists could leave a high-school mathematics course with the idea that most mathematicians are dead, those who are alive spend their time doing arithmetic of enormous complexity and that mathematics was complete sometime around the time of Pythagoras (say about 1850). When one points out that most mathematicians who have ever lived are still alive, that one can do a mathematics degree without meeting a number other than 0 and 1 and that there are important mathematical problems that have remained unsolved since the 19th century there is a feeling of incredulity. And, I hope, a feeling of having been cheated by the school system.
Well, over the past few months I’ve had a similar road to Damascus experience with history and feel a great sense of having been cheated. I’ve recently read two sources:
- “Anaximander and the Origins of Greek Cosmology” by Charles H Kahn (ISBN 978-0-87220-255-9)
- “The Official History of Heraclius’ Persian Campaigns” by James Howard-Johnston (included in his book “East Rome, Sasanian Persia and the End of Antiquity”, ISBN 978-0-8607-8992-5)
What the two books have in common is that they both deduce the existence and contents of documents that have been lost. As Kahn says: “Since the written work of Anaximander is known to us only by a single brief citation in a late author for whom the original was already lost, it may well seem an act of folly to undertake a detailed study of his thought”. But by a process somewhat akin to ded reckoning (“ded” for “deduced” is a term used by pilots) Kahn shews how the analysis of the output of later authors presupposes a common origin for much of the cosmological thought and traces that origin back to Anaximander. A fascinating intellectual journey built on a remarkable knowledge of the later literature.
Howard-Johnston makes a similar journey to deduce the existence of a lost prose/poem created by George of Pisidia from Heraclius’ official dispatches from the battlefield back to the citizens of Constantinople. His deductions start from Theophanes’ Chronographia and work backwards, demonstrating the links in the chain and holding them up for the reader to test their strength, to George of Pisidia. Another intellectual tour de force.
I had a good history teacher at school. All teachers were expected to read some inspirational text in our morning assemblies and I still remember his (he was not invited to do so again and was eventually sacked for his behaviour with a 6th form girl at the end-of-term dance and we had a collection to buy him a crate of beer): “Cromwell said, ‘put your trust in God; but mind to keep your powder dry'”. He was good (I remember his graphic account of what the crowd did with the bodies of Mussolini and his mistress) but not good enough to tell me that history was also the type of deduction that Kahn and Howard-Johnston were doing.
And if we’re misled about mathematics and history, what about the other subjects? Perhaps even biology isn’t totally mind-numbingly boring. Although that seems unlikely.
Browsing in our excellent, local second-hand bookshop a week or so ago I came across a book I should have known but didn’t: The Mathematical Experience (Study Edition) by Davis, Hersh and Marchisotto. Browsing in that book today I came across a theorem I should have known but didn’t: the “pancake theorem”, the two-dimensional version of the “ham sandwich theorem”.
The pancake theorem says that, given any two closed curves in the plane, there is a single straight line that bisects the area of both of them. The proof is straight-forward if you accept the concept of continuous functions and I see from the link above that it also appears in Courant and Robbins, an old favourite of mine.
However, the point of demonstrating the proof was that there is one step at which students tend to balk. They eat the lemma and the definition of two functions p(theta) and q(theta) and then the next step is “define r(theta) = p(theta) – q(theta)” and this is where they suddenly dig in their heels.
Of course, at heart you can’t argue with a definition. If I want to define r to be (p – q) then that’s what it is. What fascinates the author of the article is why the students block at that point. And, of course, it’s the arbitrariness of it in what he calls “proof by coercion” that causes the sticking point. As it says in the Mathematician’s Miscellany: “but please Sir, what if x is not the number of sheep in the field?”
The author then describes a different proof that avoids the arbitrary definition and reports on much greater success with students when using it.
I think that it’s a slightly different issue. Although I’d never seen the original proof (the one with r = (p – q)) before, I found that, when I was about 25% of the way through, I could see where it was going and only needed to skim-read the rest before I felt comfortable I could reproduce it: it fell into a well-known pattern and I didn’t need the details. For a student coming to this type of proof for the first time, the “let r = p – q” must feel like another item to be pushed onto an already over-loaded stack.
When a good pianist comes to sight-read a page of music she knows immediately what notes can be left out while retaining the musicality of the whole. I can’t remember ever having been taught how to read a proof—it was always assumed that one started at the top and worked through line by line until one reached the end. And arbitrary things like “let r = p – q” were meaningless and annoying: “WHY!?”. My basic mathematical education took place a long time ago and the book originally came out in 1981. Perhaps things are a lot better in secondary school mathematics these days.
Anyway, I think the term “study edition” in the title means that the book includes questions. And there’re a couple at the back that would reward some thought:
- “Suppose you were able to travel back in time to meet some like Polya, Descartes or Pythagoras. Suppose he was discouraged and was considering giving up the pursuit of mathematics. How would you encourage him?”
- “Discuss the relationship between belief and information.”
My wife has already written a blog posting about our trip to the tents in the park last Friday. It has taken me a little longer to work out the implications. For those of you not familiar with my wife’s blog, I should say that for three nights last week some 70 tents were erected on Major’s Hill Park in central Ottawa. Each tent was lit from the inside and each told the story of a person with an intellectual disability and displayed one of his or her works of art. One could browse around the tents, read the stories and at least look at, and in some cases interact with, the works of art.
In general the works of art were dire. This is not a reflexion on the intellectual abilities of the artists—most works of art are dire. I speak as a person with no ability in the art of drawing and painting but someone who, albeit limited by my red/green colour deficiency, can understand the excitement of seeing a great painting or drawing. Points of revelation and understanding in my life have included paintings—mixed in with reading Gödel’s incompleteness proof, hearing Ligeti’s first quartet, watching Richard II, reading Dermot Healy’s When They Want to Know What We Were Like and so on for the first time, I have also been changed by Turner’s Rain, Steam, Speed, Dürer’s Oswald Krell and several of Rembrandt’s self portraits.
My understanding is that a great painting requires:
- a significant idea,
- an impulse to express it,
- an original manner of portraying it and
- technical competence to execute it.
Ideas I have a-plenty. And the drive to express them. What I lack is the intellectual creativity to find a form in which to portray them (step 3) and the technical competence to put the portrayal on paper (step 4).
And this brings me back to the works we saw last weekend. They were generally dire not because of the execution and presumably not because of the idea (after all, they were produced by people trying to express their (largely justified) anger and frustration). The hardest step is to find a way of portraying the idea and that was the missing link. And this set me thinking further. I have never cared for or worked with a person with intellectual disabilities so am way out of my depth here but I always seem to hear of such people being told to use drawing, painting and sometimes music to express themselves and I wonder whether this is too limiting.
Compared with Littlewood or William of Ockham I am intellectually disabled. But that does not stop me creating (meta-)mathematics and philosophy. Littlewood and Bill would justifiably say that my scratchings are trivial but that neither dissuades me from creating them nor reduces my pleasure from the creation. And mathematical and philosphical ideas, however trivial, may be right or wrong but are never dire.
I wonder whether the people who created those works of art that I saw last weekend would get pleasure from creating mathematics; could they obtain pleasure from rediscovering some simple theorems, already well-known in the literature but not to them? I suspect that they are not given the opportunity because art is somehow seen as “suitable” whereas mathematics is not.
The intellectual engagement required to create mathematics beyond your current knowledge is no greater than that required to move across my step 3 above to produce a good painting. However, the difference would be one of quality—all mathematical proofs are good, however simple. Most paintings are bad, however complex. And that must affect the satisfaction of the creator.
I’ve been giving more thought to the “average speed camera” concept that I described in my last post. For those who haven’t read it, the concept is superficially simple: in the UK a car is timed entering a section of road at the point A and again leaving it at the point B. The distance between A and B has been measured and the car’s average speed is calculated. If the average speed is greater than the speed limit then the motorist is prosecuted because, somewhere between A and B he must have exceeded the speed limit. This is a rather subtle existence proof and I have been thinking about the exchange in court.
Sir Ethelred Carrotcake Q.C.: Tell me, where exactly was my client when the alleged offence took place?
P.C. Fred Smith: Well, I don’t actually know but it was somewhere between A and B.
Q.C.: And how far is it between A and B?
P.C.: Six miles.
Q.C.: Six miles! So you are saying that the alleged offence took place somewhere within a six mile section of road but you cannot say where. And how fast was my client alleged to have been driving?
P.C.: I don’t know, but it must have been more than the speed limit: 60 miles per hour.
Q.C.: You don’t know! So are you expecting the jury to convict my client because you think he may have driven at some speed greater than the speed limit (but don’t know by how much) at some point between A and B (but you don’t know where)?
P.C.: But his average speed was 70 miles per hour! So he must have been doing more than 60 somewhere.
Q.C.: And what makes you say that?
P.C.: But it’s obvious.
Q.C.: [turning to the jury] Members of the jury, I will be calling 10 expert witnesses this afternoon, all pure mathematicians, who will dazzle you with theorems and convince you that, far from being obvious, this law apparently relies on a very subtle mathematical theorem. But, let us test this P.C.’s understanding of the obviousness of average speeds. [turning to the witness] Now, let us assume for the sake of argument that the speed limit for the first three miles of this road was 60 miles per hour and for the second three miles 40 miles per hour. At what average speed would my client have had to travel before getting into trouble with you?
P.C.: 50 miles per hour, of course: the average of 60 and 40.
Q.C.: Would it surprise you to learn that all ten of my expert witnesses will say that the correct answer is 48?
P.C.: That can’t be true….
Q.C.: It certainly can’t be true if your simple idea of average speed is correct, can it? But [turning to the jury], let me summarise where we stand at the end of this witness’ evidence: he cannot say where my client is alleged to have committed this offence, he was not present when the alleged offence took place, he cannot say what offence my client is alleged to have committed since he makes no pretence to know the speed at which my client is alleged to have been travelling and he has a completely erroneous understanding of the concept of average speed, the very concept on which he is basing this flimsy case.
At this point the judge, convinced by Sir Ethelred’s argument and fearing the parade of mathematical expert witnesses, stopped the case and directed the jury to find the accused innocent so we will never know what the final outcome would have been.
The lacuna has been caused by my being on the (literally) high seas crossing the North Atlantic from Montreal to Liverpool as one of four passengers in a container ship. The full, unexpurgated version of this trip (including The Big Wave) will appear on my wife’s blog in due course. Here I wanted to give my first impressions of my native land, England, after a gap of several years. Both impressions are concerned with the law.
- In Liverpool I was horrified to see three cars parked outside the police station in the town centre. Large notices stated that these cars had been seized by the police because they were found to have no insurance. The message was clear: get insurance for your car or the police will seize it. I hope that everyone reacts with the same revulsion as me to this evidence of a police state. Note that these cars were not seized by the courts (perhaps using the police as their agents) and displayed outside the court building: they were displayed outside a police station and had been seized by the police. This is an erosion of the legal system far beyond even that for which Private Eye had prepared me.
- The second issue is perhaps even more profound. On the M4 motorway I saw a number of notices saying “Average Speed Cameras”. My daughter explained that this was not a comment on the quality of the speed cameras but a different way of checking whether cars are breaking the speed limit. Apparently the time at which the car passes a certain point. A, is recorded and the time at which it passes another point, B, a known distance away is also recorded. By this means the time average of the car’s speed can be calculated. Using a fairly obscure piece of mathematics the police (?) or court (?) apparently argues that if the average speed is x m/s then, at some point in the journey, the car must have been travelling at x m/s or faster. This seems to be the sort of argument around which any competent lawyer could run rings: “And precisely which theorem are you using to imply that? What assumptions does that make about the continuity of the speed/time curve and its differentials?”. Perhaps more importantly, this is the only law I know based on an existence proof. We don’t know where you broke the speed limit between A and B, we don’t know at what speed you were travelling at that time nor for how long you held that speed (is it, for example, illegal to travel at very high speed for 1 microsecond, thereby increasing the average?) but we have an existence proof based on some dubious assumptions about continuity and a little-known mathematical theorem that says you must have broken the speed limit somewhere. So we’ll prosecute you.
Coming, as I do, from the fens of East Anglia, a place well-known for its lawlessness and Hereward the Wake, I cannot understand how the English have allowed this type of state to arise.
Sometimes I dream about having time. I read books by people who have obviously had time to bake their ideas and put them down on paper. Mine rarely reach the half-baked stage and normally end up at most 10% baked. Later this month I have 8 or so Internet- and telephone-free days on a freighter ploughing the North Atlantic from Montréal to Liverpool. Who knows what might emerge. In the meantime, here’s what I have 5% baked at the moment, building on a couple of recent posts, particularly the one about whether or not mathematics just a collection of tautologies.
First thread. In his book Rationality in Action, John Searle considers deciding for whom to vote in a political election. He argues that there must be a rational choice for you (one where you would expect to gain the most) but that the reasons don’t act on you, you select a reason for voting for person X.
Second thread. At work, I’m sponsoring some research at the University of Waterloo related to proximate systems: systems where for one reason or another, an exact output (decision) cannot be made. Perhaps the output is known to depend of the value of some variables for which we cannot get precise or accurate values. Perhaps the computation is infeasible. Perhaps there is no correct outcome.
It is generally agreed that, while mathematics may be tautological, for practical purposes it provides endless hours of fun for mathematicians who find things that, once found, are obvious (because any intelligent deity would have deduced them immediately from the axioms).
Deciding for whom I should vote in the next Canadian election is presumably also obvious to that intelligent deity. My views about income tax, the price of cigarettes, the seal hunt, the war in Afghanistan, immigration and all the other topics with which politicians concern themselves are known to that deity. The promises of the politicians and their track records for keeping promises are known and so there is only one possible candidate for whom I should vote. This is a proximate system only because I can’t fully express my views on those topics precisely even to myself and I certainly don’t have the time (or inclination) to study the promise-keeping of the politicians. But, in principle, I could find out.
So, the tautological resolution of Riemann’s Hypothesis and the resolution of the manner in which I should vote at the next election are not that far apart. They are both hidden precisely because of my limited capacity for rational thought. The question is whether they are intrinsically so.
I tried to differentiate, a few postings ago, between things that can’t be known and things that we can’t know. A number of people (including my wife) have incorrectly said that this is a distinction without a difference. Clearly, something that cannot be known is also something we cannot know but I do not believe that the two sets are equal. A complete arithmetic, for example, cannot be known and we cannot know it. But there is another element in those things we cannot know: us. If the computational theory of mind is even an approximation to our reasoning apparatus, then our minds are digital and finite. Given this, there must obviously exist things that can be known but which we cannot know: things that are too big or which fall through the digital cracks. My meaning, however, was deeper. I believe that, leaving aside the finite nature of our brains, bringing a human into the process introduces unknowable entities.
As a total digression, there is another interesting observation in Searle’s book. I have spent a reasonable part of my life building systems that rely, to some extent, on rational behaviour by participants: behaviour to maximise their expected utility. Searle considers the case of a 25 cent coin (or whatever is the equivalent in the UK: a fifty pence coin?)—a coin that is not particuarly valuable but which I would probably take the trouble to bend down and pick up if I passed it in the street. That I would be willing to stoop to pick it up indicates that it has some worth to me. I should therefore be willing to accept a bet, at some odds, to gain 25 cents against losing my life. Of course, there are no such odds. At 10:1 it’s laughable and at 1,000,000:1 I still wouldn’t take the bet. This is, of course, irrational behaviour.
I remember the quotation that “God cannot do pure mathematics” from somewhere but unfortunately Google doesn’t, so I can’t provide a reference. The idea, of course, is that once the axioms of any system are written down, an omniscient intellect would immediately see all the consequences. The reason we need to do pure mathematics is that, with our limited intellects, we get the fun of solving the puzzles.
There is, of course, an underlying assumption that may rest on an incorrect idea of the way mathematics is done. We teach children in high-school science that a scientist uses what is known as the “scientific method”: the scientist creates a hypothesis, devises experiments to test that hypothesis, carries out the experiments and then accepts or rejects the hypothesis. Tell that to Kuhn and Popper! Similarly we teach (taught? I may be out of date here) children that pure mathematics is done by laying down axioms:
1. Things which are equal to the same thing are equal to each other.
2. If equals be added to equals, the wholes are equal.
down to the more worrisome
12. If a straight line meet two straight lines so as to make the interior angles on one side of it together less than two right angles, these straight lines will meet if continually produced on the side on which the angles which are together less than two right angles.
and then deducing things therefrom. This has always seemed a process as unlikely as that of the “scientific method” but it took section 7.5 of David Corfield’s Towards a Philosophy of Real Mathematics to give me a structure in which I could understand why.
I’ve been wondering about applying the scientific method to the question of whether pure mathematics is tautological. It is said (although not by me) that over the past centuries scientists have made progress while philosophers are still discussing, without much success, the same problems that Plato wrote about 2500 years ago. No scientist spends much time today worrying about the number of epicycles in the Ptolemaic theory of the solar system but many philosophers are still engaged in questions of free will, the reality of the material world, etc. So there must be something in that scientific method after all. Mustn’t there?
So, we need to devise a deterministic experiment to test whether mathematics is tautological: whether, for example, Pythagoras’ Theorem was discovered or invented. When we’ve carried out the experiment, we could devise another one to determine the question of free will. An inability to devise such experiments could be taken by some people to argue that the very question is invalid.