I have come across two thought-provoking notices over the last few days.

I am informed, by a normally reliable witness, that the first appears in the women’s toilet in the Chapters Bookshop on the corner of Rideau Street and Sussex Drive. It is inscribed on the wall and reads

I think, therefore I am. Aristotle.

The second appears more openly in the window of a hairdresser on Beechwood (a street in Vanier, Ottawa rather than a bosky grove). It says

Say ‘NO’ to thermal abuse.

I have no recollection of ever having said “yes” to thermal abuse and, as a glider pilot, I have abused many. I remember one of my instructors saying that it was incredible how I had been able to fly across the floor of a valley in central Wales, missing every one of the thermals. The problem, of course, is that if you catch the edge of a thermal with your wing, it lifts the wing and tends to throw you away from the thermal. The trick is to detect the tiny upward movement of the wing and immediately turn into the thermal. Thereby gaining lift.

It it still a puzzle why anyone would be that worried about thermals but the quotation from Aristotle is even more thought-provoking.

I assume that the versions of Aristotle that we have, translated and interpreted by people like Abū ‘Alī al-Ḥusayn ibn ‘Abd Allāh ibn Sīnā’ (OK, I looked Avicenna’s real name up in Wikipedia), have had his sceptical side suppressed. It is good to see that someone, and someone who is female and who frequents Chapters, has a more accurate version.

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:

1. “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?”
2. “Discuss the relationship between belief and information.”

I imagine everyone who reads this blog (indeed everyone who can read) has by now heard of the terrible injustice being done to Simon Singh in Britain’s law courts by the British Chiropractic Association. If not, there are plenty of articles about the case on almost every science-based blog and even some of the better newspapers (e.g., Discover and The Guardian). I have nothing but outrage to add but wanted to post this so that I could include the button:

Well, it appears that I angered the new gods yesterday.

We’ve had a very dry Spring in Ottawa this year with cloudless skies for days on end. Last night I went to a student’s home to do some private groundschool teaching and, unexpectedly, it started to pour with rain. When I left 90 minutes later it was still pouring and even now, 14 hours later, it’s still raining.

I was wondering what had caused this when I realised my terrible sin: in an attempt to find my student’s house I had asked Google for directions. And then I didn’t follow them. I thought up a better route for myself and thereby displeased Google.

I walked out of a cafe about half an hour ago after lunch, climbed into my car and turned on the radio. On CBC radio 2 the opera from the New York Metropolitan Opera House is broadcast on Saturday afternoons and on radio 1 a programme entitled “Definitely Not The Opera” runs in parallel. My radio was tuned to radio 1 and I was pitched into the middle of an interview with an artist from Winnipeg.

He said that he had been thinking about frozen rivers (it’s that time of the year when I begin to wonder whether there is any other sort but the dynamiting of the Rideau River outside our house did begin today—the harbinger of Spring) and how the ice was just frozen water. The river had effectively been stopped (frozen?) in time. He said that books contained ideas frozen in time and he had therefore constructed a work of art combining the two. I missed the part of the interview where the art work was described but it presumably included books on the Assiniboine River.

That statement about books containing ideas frozen in time jarred with me. Many, many years ago I read a book (by Anthony Hopkins?) on music that described Sonata Form. He likened the form to walking through a house that you don’t know but are thinking of buying. You enter the hall and get a first impression. You then tour the rest of the house and end up back in the hall. You see the hall again (the recapitulation) but because you now know the rest of the house you see it in a different way. In a sonata you hear the theme again during the recapitulation but, because of the journey you’ve made in the meantime, it means more to you: the notes are the same (more-or-less) but the context is different.

Surely books are the same: their ideas are not frozen because ideas have to be interpreted. At the moment I’m rereading Gerald Priestland’s autobiography, “Something Understood” which I last read at least 15 years ago and Dimitri Gutas’ “Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement in Baghdad and Early Abbasid Society” which I last read about 6 months ago.

In each case, the ideas I read during this recapitulation are new, not because the words have changed, but because I have been through other rooms and can now see these books in a new light. The words are the same but the context is different.

And it’s not just the ideas that the books contain. Sometimes I pick up a book from my shelves and find the unmistakable signs that, at some time in the past, I dropped it in the bath (I spend hours lying in the bath reading and sometimes books suffer). That too is part of the context that I bring to the next reading.

At the risk of being accused (by one particular colleague whom I won’t name) of being a relativist, the idea that books are ideas frozen in time really seems indefensible.

I was thinking about linguistic inertia—the way in which the world moves on and the language doesn’t. My grandson today was shewing me his “steam roller” which was clearly actually a “diesel roller”. And I flew this afternoon in the “cockpit” of my aircraft. I suspect that very few of us have ever been to a cock fight and even those who have would not have recognised the term’s path, through maritime terminology, to an aircraft.

Anyway, the word that was interesting me was “newspaper”. There is a local newspaper in Ottawa called the Ottawa Citizen and it can be found at http://www.ottawacitizen.com/. We were rung the other day by someone offering to print the web pages and deliver the resulting paper copy to our door each day, a sort of printed Kindle I suppose. I thought that this was a quaint idea—in principle one could, I suppose, offer to print almost any web pages and deliver them on paper. I think that the person offering the service was still tied up with the “paper” part of “newspaper”.

Imagine my surprise then, on coming home from work last week to find two large volumes sitting on the door step. These contained paper printouts of telephone numbers and had been distributed by a company called Bell. Actually, even had I felt it useful to have a list of people’s contact numbers on paper, this one wouldn’t have been the one I needed. It contained no one’s cellular telephone number and no one’s Skype user name. In fact, as far as I could see, it contained only telephone landline numbers, perhaps the least useful contact information for people. What was really strange was that it associated the landline number with the name of a person as though it were a cellular telephone number or Skype name. Telephone landlines, of course, are not associated with people, rather with houses or offices. But I should not be critical, it was a nice thought. Someone at Bell, obviously confused by the linguistic inertia in the phrase “Telephone Book”, felt that it would be useful to print out this list as a “book” and deliver it to me personally. I’m sure that the cellular telephone numbers and Skype identities will appear in the next issue.

Two things have been niggling at me for a week or so. I was thinking about Anaximander’s opposites at war with one another: hot and cold, wet and dry and so on. The thought of someone living in Miletus 2500 years ago realising that “wet” and “dry” are somehow ends of a spectrum does not surprise me: presumably he saw clothes drying in the wind and water evaporating from puddles.

Today we think of hot and cold as “opposites” in some sense without giving it a great deal of thought: we associate cold with low numbers on the thermometer and hot with high numbers. But what about someone living in Miletus (average yearly temperature span only 18 Cdeg spread gently over 6 months), 2200 years before the first thermometer, 2300 years before we gained a reasonable understanding of the distinction between heat and temperature and 2400 years before the first refrigerator? I can’t see how hot and cold would be obvious ends of the same spectrum. I would have been less surprised to see “cold” at one end and, say, “soft” at the other (since colder things tend to be stiffer).

As I understand it, the only practical experiment that Anaximader carried out was to demonstrate by blowing on his hand that compressed air is cooler than non-compressed air. A totally erroneous result based on ignoring the cooling effect of the evaporation of sweat from his hand.

This raises two questions that have been niggling at me:

1. What led Anaximader to believe that “hot” and “cold” were opposites?
2. Did we develop a temperature scale which has cold things at one end and hot things at the other because we saw “hot” and “cold” as opposites or is that scale natural in some way?

On an unrelated note, I also learned recently that Celsius designated the freezing point of water as 100 degrees and the boiling point as 0 degrees when he introduced his scale. This might indicate that at least our association of “hotter” with larger numbers on the thermometer is fairly recent.

Lying abed this morning I finished reading Ivar Ekeland’s book “The Best of All Possible Worlds: Mathematics and Destiny” and, while brushing my teeth, alighted on Leah Price’s review of William Sherman’s book “Used Books: Marking Readers in Renaissance England” in the 9th October 2008 edition of the London Review of Books.

I have written before about sometimes being present at the crossroads of the universe when two ideas arrive from different directions, coalesce and drive off together up the hill. Let me try to explain my surprise this time.

Ekeland’s book is really two books. Both are good and well worth reading and, by buying “The Best of All Possible Worlds” (C$14.00 when I bought my copy just before Christmas) you get the two bound into one volume. The first book is a fascinating exploration of the least-action principle from Pierre Louis Maupertuis, a chap of whom I had never heard but who was, apparently, the origin of Voltaire’s Doctor Pangloss, to Poincaré. This book requires no mathematical maturity from the reader and the couple of theorems needed for the argument (including a geometric argument for the existence of a second (non-maximal) diameter of a convex closed curve) are relegated to appendices. It also raises the concept of playing billiards on non-rectangular tables. I have always thought billiards and snooker fairly boring to play and extremely boring to watch but the idea of having different (but convex) table shapes would certainly make them more interesting. Imagine snooker competitions where the competitors didn’t know the (irregular but convex) shape of the table until the start of the match.

Ekeland’s second book deals with human behaviour. He writes in the final chapter “There is a general feeling that science has given us longer and better lives, but has not taught us how to live them.” The last two or three chapters should be compulsory reading for all politicians and university administrators.

Science seems to raise more questions than it provides answers, but human beings are in quest of certainties, and if science will not provide them, then others will—religions and ideologies. And indeed, the first half of the last century was the era of ideologies, which ended with the bloody clash of fascism and communism, while the second half has seen religions emerge as the main actors, and may yet lead us to another conflagration between the Abrahamic creeds…

I find a confusion in this part of the book between science and technology (I don’t, for example, think that there was much science beyond Newton’s Laws of Motion involved in getting men to the moon) but this does not detract from Ekeland’s cri de coeur for sanity and rationality in our thinking. We are in the classical games theory dilemma of needing everyone to act rationally for the good of all. Ekeland believes that the application of mathematics and logic to human and well as scientific problems could, indeed, result in the “best of all possible worlds” in the sense that Voltaire mocked.

The two parts of Ekeland’s book are bound together by that phrase and play on the word “best”: in the first part in its scientific sense selecting our universe from all of those available in the multiverse and in the second part in its societal sense. This is, of course, the very confusion between what “Dr Pangloss” was really saying and what Voltaire was (deliberately?) confusing.

So, where does the article from the London Review of Books come in? The book being reviewed deals with the things that people wrote in the margins of books. Apparently 17th century marginalia now adds to, rather than detracts from, the value of old books. However, the part that interested me was

Even after the invention of printing, aids like tables of contents and indexes were added by hand by individuals. Readers would interleave blank pages, rearrange sections, and could even combine sections from different volumes. The printed page was seen as merely a starting place…

If this was so in the 16th century, how much more should it be the case now. With more and more books becoming available in soft copy (pdf) and with really useful tools like pdftk capable of splitting and recombining pdf files in all sorts of ways (as it says in the pdftk documentation “If PDF is electronic paper, then pdftk is an electronic staple-remover, hole-punch, binder, secret-decoder-ring and X-Ray-glasses”), we are free to build precisely the books we need. If I had a soft copy of Ekeland’s book, I could split it into two and combine the first part with some sections from Voltaire’s Candide (perhaps the world’s most tedious book in its entirety but OK in chunks) and thereby build a new, personalised book. I could then take the second part of Ekeland’s book and combine it, perhaps, with some of Grayling’s essays to build another new book personalised for me.

As an author, I would actually feel quite comfortable about people doing this with my books. I wonder about other authors.

There are pictures which, depending on how your brain interprets them, switch between two women’s faces and a vase. Or a cube that comes out of, or goes into, the page. Switching between the two can be almost instantaneous. I had a similar experience yesterday over the comments on a Globe and Mail article (as I always say on these occasions, for non-Canadian readers I should explain that the Globe and Mail is a Toronto local newspaper distributed Canada-wide in paper form and world-wide on the internet). When I first accessed the article, it spoke about a woman giving birth to twins while “…laying in the snow in the street”.

I obviously left a comment saying that my first reaction to the use of the transitive verb “to lay” in this context was that it was a grammatical error, possibly a substitution for the intransitive verb “to lie”: perhaps the sentence should have read “…lying in the snow in the street”. On reflexion I thought that, given the nature of the story, “laying” might have been a clever choice on the part of the reporter.

This comment led to a thread discussing whether women “lay” babies or not and, after a while, the verb was changed in the original story (indicating that someone from the Globe and Mail is monitoring the comments even on Christmas Eve!). So far, so normal.

And then a woman added a comment saying something like “I don’t know what you people are talking about but I hope that you’re not being disrespectful to the mother who gave birth in the street”. That was a moment when the picture switched. Of course, we were being disrepectful to a grammar-challenged reporter, not the mother, but I suddenly realised that the story to me is a constructed list of words; to someone else it is a representation of a woman in a dire situation. It is possible (from the “I don’t know what you people are talking about….”) that the person leaving that comment did not understand the difference between laying in the street and lying in the street, and all she heard was the disrespectful and condescending attitude we were adopting.

This brings me back to something about which I blogged many months ago. I had created a CD of photographs that my wife was to take to the local photography shop for printing. I labelled the CD “Photo’s for Printing” and my wife queried the apostrophe. I explained that “Photo” was a short form of “Photograph” and that the apostrophe indicated that letters were missing. She was concerned that ignorant people might think that I had incorrectly used the apostrophe to represent plurality or possession. In effect, she wanted me to make a mistake so that ignorant people would not think me ignorant.

I wonder whether a lot of the controversy we hear about religious people objecting to comments being made about their irrational beliefs fall into the same catagory. I am not particularly sympathetic to the views of people who believe in the tooth fairy, Father Christmas, honest politicians or the Christian/Islamic/Jewish/Druidical gods but I do suddenly have pause to think whether we have a right to dismay the faithful. I think we do.

I’m sure that most people reading this posting will also have read the numerous news items and other blog postings regarding the Oxford University Press’ decision to remove certain words from its Children’s Dictionary.

I would like to add my voice to the chorus of dissenters. It is important that our younger generation, when they reach the age of discernment, engage properly in political debate. How on earth can they do this without words like “weasel” and “adder”? How can we instill pride of country into them without “empire”? Losing the religious terms (“holly”, “ivy”, “mistletoe”) will turn out a generation without the basic words to discuss Druidism when the Druids come knocking on the door on a Sunday morning.

I must say that I’m not a ferret lover (probably the world’s most evil animal) and losing that word doesn’t worry me much. Nor the words to do with sex and sexuality that I’m sure the previous generation of schoolchildren giggled over: “oyster”, “vicar”, “thrush”, “gooseberry”, “oats” and “pansy”.

But what about the words that are being added? I suppose the most promising is “square number” and the least “creep”. That “creep” has a Uriah Heep flavour to it and I hope that the dictionary already contains “obsequious” so that it can be cross-referenced.

All in all (which it never is), the children seem to have lost more than they have gained. A good lesson for them in these times of plunging stock markets.