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Death and Strawberries

Posted in Poetry with tags , , , , on August 20, 2010 by telescoper

This week in August 2010 has taken on quite a melancholy mood. Only a few days ago there was the death of physicist Nicola Cabibbo. Yesterday I heard that the great Russian mathematician Vladimir Igorevich Arnold, who did a lot of work of interest to physicists, had also passed away aged 72. And then this morning I was saddened to hear of the death of the wonderful Scottish poet Edwin Morgan, of pneumonia, at the age of 90.

It’s always sad when someone who has contributed so much to their field – whether it’s artistic or scientific – passes away, but the consolation is that each of them in their own way has left a wonderful legacy that remains to be treasured and will also inspire future generations.

Anyway, I thought I’d mark the passing of Edwin Morgan with my favourite poem of his, called Strawberries.

There were never strawberries
like the ones we had
that sultry afternoon
sitting on the step
of the open french window
facing each other
your knees held in mine
the blue plates in our laps
the strawberries glistening
in the hot sunlight
we dipped them in sugar
looking at each other
not hurrying the feast
for one to come
the empty plates
laid on the stone together
with the two forks crossed
and I bent towards you
sweet in that air

in my arms
abandoned like a child
from your eager mouth
the taste of strawberries
in my memory
lean back again
let me love you

let the sun beat
on our forgetfulness
one hour of all
the heat intense
and summer lightning
on the Kilpatrick hills

let the storm wash the plates

It may surprise you to learn that this poem is not written by a man to a woman, but from one man to another. A similar reaction is sometimes provoked by certain of Shakespeare’s Sonnets. It came as a shock to quite a few people when it was finally revealed, in fact, because Edwin Morgan kept to himself for a very long time who this was written about. Actually, it wasn’t until he was 70 that the poet stepped out of the closet, announced that he was gay, and explained that the poem was written about an experience he shared with another man. He maintained that at least part of the reason for him not being open publically was that he didn’t want to be branded as a “gay” poet, and that his poems were intended to be universal, which (in my view) they are but then that depends on what kind of universe you live in.

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Grade Inflation

Posted in Education, Politics with tags , , , on August 12, 2010 by telescoper

Still too busy to post anything too substantial, but since this year’s A-level results are out next week – with the consequent scramble for University places – I thought I’d take a few minutes to share this  graph (taken from an article on the BBC website) which shows the steady dumbing-down improvement of educational standards student performance over the last few decades.

Nowadays, on average, about 27 per cent of students taking an A-level get a grade A. When I took mine (in 1981, if you must ask) the fraction getting an A was about 9%. It’s scary to think that I belong to a generation that must be so much less intelligent than the current one. Or could it be – dare I say it? – that A-level examinations might be getting easier?

Looking at the graph makes it clear that something happened around the mid-1980s that initiated an almost linear growth in the percentage of A-grades. I don’t know what will happen when the results come out next week, but it’s a reasonably safe bet that the trend will continue.

I can’t speak for other subjects, but there’s no question whatsoever that the level of achievement needed to get an A-grade in mathematics is much lower now than it was in the past. This has been proven over and over again. A few years ago, an article in the Times Higher discussed the evidence, including an analysis of the performance of new students on a diagnostic mathematics test they had to take on entering University.  The same test, covering basic algebra, trigonometry and calculus, had been administered every year so provided a good diagnostic of real mathematical ability that could be compared with the A-level grades achieved by the students.  They found, among other things, that students entering university with a grade B in mathematics in 1999 performed at about the same level as students in 1991 who had failed mathematics A-level.

The steadily decreasing level of mathematical training students receive in schools poses great problems not only for mathematics courses, but also for subjects like physics. We have to devote so much more time on the physics equivalent of “basic training” that we struggle to cover all the physics we should be covering in a degree program. Thus the dumbing down of A-levels leads to pressure to dumb down degrees too.

That brings me to the prospect of huge cuts – up to 35% if the stories are true – in government funding for universities, leading to pressure to shorten the traditional three-year Bachelors degree to one that takes only two years to complete. If this goes ahead it won’t be long before a student can get a degree by achieving the same level of knowledge as would have been displayed by an A-level student 30 years ago. Are we supposed to call this progress?

Or perhaps this business about two year degrees all really  does make sense. Maybe we should just accept that universities have to offer such courses because the school system has become broken beyond repair over the last 30 years, and it will be up to certain Higher Education institutions from now on to do the job that school sixth-forms used to do, i.e. teach A-levels.

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(Guest Post) The Emperor’s New Math

Posted in The Universe and Stuff with tags , on April 20, 2010 by telescoper

Time for another guest post from my old chum Anton, this time on the topic of mathematics. I’m not sure any mathematicians reading this piece will be too happy, but if that applies to you then blame him not me. As usual, comments are welcome through the proper channel at the bottom of the page..

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Nowadays a page of mathematics looks to a physicist or engineer like gobbledygook. This was not always so: a century ago a physicist might hope to understand everything in journals of mathematics, and even contribute to them. Fifty years ago a physicist might not be able to understand everything written there, but the mathematics would appear comprehensible in principle. A qualitative change has since taken place.

This change has coincided, roughly, with acceptance of the distinction between ‘pure’ and ‘applied’ (impure??) mathematics, and with the consequent, deliberate, emancipation of ‘pure’ mathematics. This is a new departure: for centuries mathematics evolved side by side with physics, and the mathematics that was studied was the mathematics used in tackling physical problems. Galileo had said (in his work Il Saggiatore, The Assayer) that

…the universe… cannot be understood unless one first learns to comprehend the language… in which it is written. It is written in the language of mathematics.

So the change is recent, and it is huge. I suggest that it is a change for the worse; that in divorcing themselves from physical science pure mathematicians have cut off their air supply; and that the suffocating style of modern pure mathematics is a result. Mathematics was not born in a vacuum, and it will not ultimately flourish in one.

A pure mathematician might respond that I would say that, since I am a physicist. But perhaps an outsider is needed to see the problem; insiders generally adopt the party line. The justification for my stance is this. Mathematicians acknowledge that their subject is the formal study of patterns. And mathematicians think in patterns, not formulae – which are really a highly efficient way to express their thoughts. Crucially, the patterns arising in the natural world are far richer and more diverse than the patterns that even the best pure mathematicians can pull out of their heads by introspection. Even number theory is not an exception, for the positive integers are abstractions – ideals – of the physical realisations of one, two, three etc sheep in a field, or boats on a lake.

The role of pattern explains the “unreasonable effectiveness” of mathematics in physical science (as Eugene Wigner put it), since physics is concerned with relations – correlations – between variables in space and time, and correlation is synonymous with pattern. The theoretical physicist Lev Landau vehemently believed that the best mathematics is the mathematics used in physics. An opposing point of view was taken by the pure mathematician Paul Halmos, in an essay titled Applied Mathematics is Bad Mathematics. Not all mathematicians share Halmos’ view, however. The mathematician Morris Kline was the author of many books about mathematics and its embedding in the cultures which nurtured it. In his book Mathematics: The Loss of Certainty, Kline demonstrated that the history of mathematics in the 20th century has not been the smooth progression that it appears to the outsider; and that arguments about the foundations have led not to resolution, but to schism into differing schools – based on different foundations – that do not talk to each other. Mathematics is not in fact a one-way road running from self-evident axioms to consequences, but is open at both top and bottom.

Already in the 19th century a formal style was developing in the mathematical study of logic, and such distinguished noses as Henri Poincaré (in Science et Methodes, part II) protested as early as 1909 that this tended to hide misleading or negligible content. To no avail: the dominance of the formalistic logical viewpoint led to the adoption of its house style across the whole of mathematics. Below university level, mathematics is still taught today as it used to be, with the emphasis on the understanding of ideas rather than their formal presentation. Freshmen are often shocked when they first meet the new way of doing things, in university lectures given by professional mathematicians. I doubt that the form of modern mathematical writing is governed by its content, for whenever my research has demanded I read some contemporary mathematics, and I have had to translate a piece of modern mathematical writing into something comprehensible to scientists, I have found it difficult to distinguish substantial points from trivia. When, for instance, four axioms are needed to establish a result, they will typically be presented as having equal weight, even if one is the crucial axiom that allows most of the proof to be constructed, and another is used only in closing loopholes. Acknowledging the quality of axioms, as well as the quantity, does not compromise rigour.

When I think of the work of Andrew Wiles and Grigori Perelman, I realise that magnificent work is done today by mathematicians far beyond my own competence. But might mathematicians question whether what they regard as the only way to write mathematics is actually a convention, and not necessarily a good one? If they wrote mathematics as they did fifty years ago, others might be able to see for themselves. More fundamentally, might they also realise what their predecessors understood, that by its abstraction mathematics is given an autonomy of its own, and that to look to the physical world for inspiration is not to make mathematics a slave of physics? The present divorce between mathematics and physics impoverishes everyone.

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My Friend Erdös..

Posted in Biographical with tags , , , , , , on March 28, 2010 by telescoper

After one of my  lectures a few weeks ago, a student came up to me and asked whether I had an Erdős number and, if so, what it was.  I didn’t actually know what he was talking about but was yesterday reminded of it, so tried to find out.

In case you didn’t know, Paul Erdős (who died in 1996) was an eccentric Hungarian mathematician who wrote more than 1000 mathematical papers during his life but never settled in one place for any length of time. He travelled between colleagues and conference, mostly living out of a suitcase, and showed no interest at all in property or possessions. His story is a fascinating one, and his contributions to mathematics were immense and wide-ranging.  The Erdős number is a tiny part of his legacy, but one that seems to have taken hold. Some mathematicians appear to take it very seriously, but most treat it with tongue firmly in cheek, as I certainly do.

So what is the Erdős number?

It’s actually quite simple to define. First, Erdős himself is assigned an Erdős number of zero. Anyone who co-authored a paper with Erdős has an Erdős number of 1. Then anyone who wrote a paper with someone who wrote a paper with Erdős has an Erdős number of 2, and so on. The Erdős number is thus a measure of “collaborative distance”, with lower numbers representing closer connections.

I say it’s quite easy to define, but it’s rather harder to calculate. Or it would be were it not for modern bibliographic databases. In fact there’s a website run by the American Mathematical Society which allows you to calculate your Erdős number as well as a similar measure of collaborative distance with respect to any other mathematician.

A list of individuals with very low Erdős numbers (1, 2 or 3) can be found here.

Given that Erdős was basically a pure mathematician, I didn’t expect first to show up as having any Erdős number at all, since I’m not really a mathematician and I’m certainly not very pure. However, his influence is clearly felt very strongly in  physics and a surprisingly large number of physicists (and astronomers) have a surprisingly small Erdős number. According to the AMS website, mine is 5 – much lower than I would have expected. The path from me to Erdős in this case goes through G.F.R. Ellis, a renowned expert in the mathematics of general relativity (as well as a ridiculous number of other things!). I wrote a paper and a book with George Ellis some time ago.

However, looking at the list I realise that I have another route to Erdős, through the great Russian mathematician Vladimir Arnold, who has an Erdős number of 3. Arnold wrote a paper with Sergei Shandarin with whom I wrote a paper some time ago. That gives me another route to an Erdős number of 5, but I can’t find any paths  shorter than that.

I guess many researchers will have links through their PhD supervisors, so I checked mine – John D. Barrow. It turns out he also has an Erdős number of 5 so a path through him doesn’t lower my number.

I used to work in the School of Mathematical Sciences at Queen Mary, University of London, and it is there that I found some people I know well who have lower Erdős numbers than me. Reza Tavakol, for example, has an Erdős number of 3 but although I’ve known him for 20 years, we’ve never written a paper together. If we did, I could reduce my Erdős number by one. You never know….

This means that anyone I’ve ever written a paper with has an Erdős number no greater than 6. I doubt if it’s very important, but it definitely qualifies as Quite Interesting.

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Killing Vectors

Posted in The Universe and Stuff with tags , , , on February 16, 2010 by telescoper

I’ve been feeling a rant coming for some time now. Since I started teaching again three weeks ago, actually. The target of my vitriol this time is the teaching of Euclidean vectors. Not vectors themselves, of course. I like vectors. They’re great. The trouble is the way we’re forced to write them these days when we use them in introductory level physics classes.

You see, when I was a lad, I was taught to write a geometric vector in the folowing fashion:

\underline{r} =\left(\begin{array}{c} x \\ y \\ z \end{array} \right).

This is a simple column vector, where x,y,z are the components in a three-dimensional cartesian coordinate system. Other kinds of vector, such as those representing states in quantum mechanics, or anywhere else where linear algebra is used, can easily be represented in a similar fashion.

This notation is great because it’s very easy to calculate the scalar (dot) and vector (cross) products of two such objects by writing them in column form next to each other and performing a simple bit of manipulation. For example, the scalar product of the two vectors

\underline{u}=\left(\begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right) and \underline{v}=\left(\begin{array}{c} 1\\ 1 \\ -2 \end{array} \right)

can easily be found by multiplying the corresponding elements of each together and totting them up:

\underline{u}\cdot \underline{v} = (1 \times 1) + (1\times 1) + (1\times -2) =0,

showing immediately that these two vectors are orthogonal. In normalised form, these two particular vectors  appear in other contexts in physics, where they have a more abstract interpretation than simple geometry, such as in the representation of the gluon in particle physics.

Moreover, writing vectors like this makes it a lot easier to transform them via the action of a matrix, by multipying rows in the usual fashion, e.g.

\left(\begin{array}{ccc} \cos \theta & \sin\theta & 0 \\ -\sin\theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array} \right) \left(\begin{array}{c} x \\ y \\ z \end{array} \right) = \left(\begin{array}{c} x\cos \theta + y\sin\theta \\ -x \sin \theta + y\cos \theta \\ z \end{array} \right)

which corresponds to a rotation of the vector in the x-y plane. Transposing a column vector into a row vector is easy too.

Well, that’s how I was taught to do it.

However, somebody, sometime, decided that, in Britain at least, this concise and computationally helpful notation had to be jettisoned and students instead must be forced to write

\underline{r} = x \underline{\hat{i}} + y \underline{\hat{j}} + z \underline{\hat{k}}

Some of you may even be used to doing it that way yourself. Why is this awful? For a start, it’s incredibly clumsy. It is less intuitive, doesn’t lend itself to easy operations on the vectors like I described above, doesn’t translate easily into the more general case of a matrix, and is generally just …well… awful.

Worse still, for the purpose of teaching inexperienced students physics, it offers the possibility of horrible notational confusion. In particular, the unit vector \underline{\hat{i}} is too easily confused with i, the square root of minus one. Introduce a plane wave with a wavevector \underline{k} and it gets even worse, especially when you want to write \exp(i\underline{k}\cdot\underline{x})!

No, give me the row and column notation any day.

I would really like to know is who decided that our schools had to teach the horrible notation, rather than the nice one, and why? I think everyone who teaches physics knows that a clear and user-friendly notation is an enormous help and a bad one is an enormous hindrance.  It doesn’t surprise me that some student struggle with even simple mathematics when its presented in such a silly way. On those grounds, I refuse to play ball, and always use the better notation.

Call me old-fashioned.

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