Derek Crozier – Irish Artist

Derek Crozier, cruciverbalist, died on 3rd April 2010. There have been warm tributes paid to his life and work.

Crozier composed the “Crosaire” cryptic crossword in the Irish Times for 67 years. The obituary tells a nice story of the column’s creation – his wife was a crossword fanatic and often complained about how easy she was finding her crosswords, how they no longer constituted a challenge. Nag, nag, nag.

So he set about designing a crossword that would cause her more trouble, a truly cryptic crossword. She was stumped (and impressed). A few days later, Christmas Eve in Emergency Dublin, Crozier met Big Bertie Smyllie, legendary editor of the Irish Times, in the Palace Bar on Westmoreland St. Crozier mentioned his crossword, Smylie had a look and was equally impressed and found him some space for a daily column. And so it began.

I won’t pretend I could solve these brain-teasers. If ever we had consecutive day’s newspapers lying around the house, I’d read the clues, have a superficial think, and then look at the solutions in the following day’s papers and marvel, for there is no other word, at the ingenuity of the clues and answers.

To celebrate his life, the Irish Times published a booklet with a selection of his most brilliant crosswords (with, thankfully, solutions), and it truly shows how talented, and original, was this man’s work. Try these:

• Country Rot, I See (6). (Rustic)
• May be in the form of arcs (4). (Scar)
• The buck still goes to the bad when he gets the girl (9). (Stagnancy)
• It’s ridiculous to have anything so light with nothing on the end (7). (Lampoon)
• Does Garry sound as if he needs a wig? That takes the biscuit! (9) (Garibaldi)
• With him you’re got to buy your own drinks (7). (Vintner)
• ETSETSETSETSETSETSETSETSETSETS ones holds (6). (Tenets) – To explain this one; there are ten “ETS” here, hence “tenets”, principles  one holds.

The booklet is good enough to list some Crosaire conventions:

• clues can usually be broken into three parts
• Clues often employ anagrams, synonyms, homophones, palindromes or letter plays.

An example of the reasoning given: “That miserable accountant is worth his weight in diamonds (5)” Solution: The solution is CARAT – miserable = RAT, accountant = CA (chartered accountant); RAT+CA (anagram) = CARAT (diamond weight).

Reading through these, you really have to marvel at the ingenuity of this man’s mind!

In all the acclaim following his death, there were suggestions that the column should be continued forever, starting over from column number one when the last one is published. I wonder has anyone considered whether this man’s work should not be studied as literature?

To me, he seems the natural successor to Joyce; think of the wordplay and classical learning of Finnegans Wake, the multi-lingual, multi-level puns, the use of words to transcend utilitarianism. And if he would be candidate to succeed Joyce, then he fits naturally in the same bracket as Flann O’Brien, playful and irreverent, a multi-generational Irish Times institution?

He fits so neatly into the canon, that it would seem a shame not to light the fuse and at least see what sort of fanfare is loosed.

The only objections to such a course that I can think of  are:
Can crosswords be considered literature? In anguished voice: “But is it Art?” Would Crozier have called it Art himself? Was it ever meant as Art?

Had Crozier artistic purpose? Joyce embarked on his fiendish and unreadable masterpiece with an aim. I’ve read the first and the last paragraphs of Finnegans Wake, and this is enough to say with confidence that he had a schema, an outline of how it was to be and what it was to say. This is enough to qualify as a work of art. \it could be said that Crozier made crossword puzzles, clever granted, but mere diversion for Dublin’s polymath eggheads with too much time on their hands. I don’t doubt that Crozier, by all accounts a modest man with no pretensions to anything, would have himself considered his work Art.

But for something to be labelled Art, does it necessarily have to have been meant as Art by its creator? Obviously it usually is, but i can think of many examples where some work has been acclaimed as High Art and the original creator would have been both gobsmacked and tickled pink to know about it. For example, listed architectural structures like the Georgian redbricks in town. These were built for a utilitarian purpose – yet in time became a symbol of an era, and a mark of distinction to the venerable capital among European capitals, like a cragged face who claims with pride to his equally battered peers that he still has some of his original teeth in his head. Another example is of the original comic books, Spiderman or Batman, that now sell for hundreds of thousands of dollars – a genre that dubbed itself “pulp fiction” but is now recognised as, culturally, a major influence on the second half of the twentieth century. These originals now reside in private collections alongside Picassos and Van Goghs, or in serious museums worldwide. Their original creators had no such pretensions; they were paid by the box, and the emphasis was on quantity, not quality. Graphic novels being produced now are the most innovative advance in literature since Joyce. More: the prehistoric cave painters in Lascaux were just simple cavemen. Outsider art?

No, lack of artistic intent does not disqualify a work from being called Art. “All art is quite useless” said Wilde. On this count too, Crozier’s crosswords qualify. Really, Art is Art when someone, anyone, says it is Art and appreciates it so. It needn’t be the creator. Ivan Goncharov disowned “Oblomov” thinking it was a piece of twaddle, and, indeed, threw the only manuscript of it into the fire. His horrified friends dived in after it, saving most of it, at their own skin’s expense. Its now considered a classic of Russian literature.

I, personally, look forward to a time when Irish students have a choice of Shakespeare or Crozier to study for the Leaving.

Ethical Investment

I am interested in atheism; what it means, what it entails and what it is. If you are interested in atheism and actually engage with it as a subject, then you are automatically interested in religion. It is related to religion, in that it defines itself as a rejection of this mode of thinking. It is not a system of belief itself. There was a letter to the Irish Times recently (from Colm FitzPatrick - an actuary, I know him) that said that atheism was as much a structured system of belief as religion is and that prominent atheists were as much “religious fundamentalists” as the religious variety. A perceptive letter of reply contradicted this opinion elegantly, saying that you could not claim “darkness” as a “thing”, rather it has to be defined as an absence of light. It is a fine metaphor – except for the unfortunate association of “darkness” with atheism, and the Biblical light/darkness association with good/evil – which is exactly how religious folk will interpret the allusion, missing the main point of the statement… I’ve been racking my brains to think of another physical relationship in the universe whereby one thing exists because of the absence of another, but does not exist in itself… the only one I can think of is a minor re-statement – light/shade. Love/hate? Sight/blindness? Too subjective.

Anyway, in addition to thinking about things like this, I am also partial to spending Saturday mornings over a long, lingering breakfast of coffee, fruit and scones and reading the Irish Times and Financial Times. It is one of my chief pleasures in life. I enjoy the book reviews in both; Breda O’Brien’s columns which never fail to annoy me, but whom, and whose opinions, I respect; the “what’s hot, what’s not” column; the always hilarious restaurant review;  Ian O’Riordan’s athletics column; Tyler Brulée and Harry Eyres on the back page of he FT Weekend – often I suspect neither person actually exists: I think they may both be joke caricatures of a way of life, one impossibly racy and materialistic, the other impossibly virtuous and balanced; finally, I enjoy the FT Money section, as I can pretend to be super-rich and spend my times in fine cafés deciding where to invest my pile of money.

Anyways, pulling these themes together: the FT Money section reviews the universe of investment opportunities – half of the twenty page supplement is raw statistics, on fund performances, commodities, share prices, yield curves, currencies, exchange traded funds, investment trusts, unit trusts, venture capital funds… Its intoxicating, and a little overwhelming. The bones of the rest of the supplement are the pontifications of financial journalists on two things:

• What investments will perform in the future?
• What investments performed in the past?

As to the first question, no-one really knows and the fun is in reading the different, often contradictory, opinions.

The second touches upon the old adage about economists; an economist is someone who will tell you tomorrow exactly why the prediction he made today was wrong.

Reent issues of the FT Money have been pretty unanimous as to what has performed well in the last five years (“well” meaning least bad, in the context of a global meltdown): obviously gold, some currencies, some bond funds. In the mutual fund sector, the clear leader over five years are “ethical funds”.

These are a new phenomenon, so the data doesn’t go back too far, and we can’t say for certainty how they perform in the long-term, but it is clear that of these “ethical” funds, the funds that have done the best are the “religious ethical” funds. This is a Middle-East and American phenomenon – in Europe we have “ethical” funds, but they tend to screen investments on environmental or relativistic/secular moral grounds. In contrast, religious ethical investments screen the set of investment opportunities for those that are permissible by a set of religious codes – for instance, the most established religious ethical investing practice is the “Sharia compliant” investment, investing with reference to what is allowed by the “Sharia Law”, that prescribed by the Koran. This, for instance, bans the receipt of interest payments, or investing in alcohol companies, since alcohol is prohibited under Sharia Law. The American versions are along the same lines, except they tend to vary much more dramatically; there are Catholic funds, Lutheran funds, Methodist funds, Unitarian funds, Scientology funds, Mormon funds…more funds than you could shake a stick at.

They vary widely in what they can and can’t invest in, but they generally avoid the “vices”: gambling, pornography, prostitution, armaments, alcohol, tobacco, oil, mining, pharmaceutical companies making birth control etc. There may or may not be an environmental element within this.

It sounds inherently contradictory – does it make sense to piously avoid gambling stocks when, fundamentally, buying equities are a form of gambling?

But the statistics say that, incontrovertibly, these funds have performed creditably over the last five years, most of them well into the upper quartile of active managed mutual funds.

Should we atheists repent? Is God’s law actually profitable to follow? Is God guiding the markets in favour of his commandments? Should I sell short companies producing meat every Friday and buy them back on a Saturday?

Some theologians have claimed that it is, indeed, justification of their religious beliefs – that the performance of these funds somehow supports the view that we should live our lives according to the tenets of a particular faith (and investment). Maybe they are right. (But to be logically consistent with this viewpoint, should we not all pitch in our lot with the best performing religion – since these ones will have the most correct set of principles that God approves of? (Islam, say the league tables, followed by the Catholic funds). Or if an investment really tanks, should I abandon the religion altogether, since God clearly doesn’t approve of what it is saying? Theologians love taking credit where it is not due, and then not following through their lines of thinking.)

But I suggest not. One, these funds are new; there is about five year’s worth of creditable data, so the jury is still very much out. Two, the five year period in particular corresponds to the worst recession since 1929 – ethical funds, investing in primarily defensive sectors, will naturally do well in this climate. Sectors like oil, commodities, tobacco are much more cyclical – they rise as the global economy rises, and vice versa.

Rather than viewing the outperformance of religious funds as divine approval, I’d be more inclined to attribute he performances to human nature. In a contracting economy, people hoard money – they cut back on the smokes, they drink less (maybe more, but bootleg), they cycle instead of using the car, they don’t fly around the world on holidays twice a year. Some vices are more inelastic than others – consumption of tobacco, being highly addictive, inevitably does not fall as much as, say, oil. But all fall.

When the economy waxes again, and rashers again grease the pan, then, reassuringly, good old human failings roar to the fore again. Disposable income is wasted on the usual vices, and conventional stocks rise in valuation. As simple as. But maybe I’m wearing my economist’s hat here. Who knows?

I’ll keep my Sabbath ritual of the IT, FT, coffee, and avoidance of all semblance of work, and I will keep betting on the predictability of mankinds’ weaknesses; fear and greed, its what we all share.

That’s my two cents invested.

On the teaching of mathematics

I must be something of a geek. I’ve lately been reading popular maths books for no other reason than for enjoyment (and to look cool on the bus). In particular, the books of Ian Stewart – I just finished “Taming the Infinite: The Story of Mathematics from the first Numbers to Chaos Theory”, and can recommend it highly.

This book is exactly as it says on the tin; a roughly chronological history of the developments of mathematics, from ancient Arabia to frontiers of modern maths. What made the book so enjoyable is the way the history of each branch of mathematics is related. Stewart doesn't flit from one dry area to another, all equations and definitions – he tells of each development through the biographies of the people who developed them, the historical and geographical context in which the discoveries were made, and – crucially – relates these discoveries to how they are applied in the real world.

One of my chief issues with the way maths is taught in schools and universities is the “abstractness” of it all. I don’t refer to the maths itself – for instance, complex analysis or higher-dimensional topography, which is pretty abstract by nature. I refer instead to the lack of context provided along with the theory, at all levels, from primary school to third level.

It starts well; in primary school we are taught the numbers, then addition and subtraction by relating them to physical things. “The number two represents two apples. If I receive another two apples, then I have four apples. Likewise, if I eat two of these apples, two are subtracted and I am back to two apples.” This is intuitive. Likewise, multiplication and division were related (to me anyway) by the concepts of lots of apples. “If a packet of apples contains four apples, and I buy four packets of these, then I now have sixteen apples.” Obviously, the simplistic storyline was soon abandoned and the theory of long multiplication and division introduced, but always in the background, what you were doing was manipulating apples. From the humble introduction you were aware of what this arcane manipulation of numbers could be used for in real life, in the real world.

It is only in secondary school that it all goes wrong really. The syllabus wades into trigonometry with nary an introduction. “If this triangle has an angle of 30 degrees, an adjacent side of length 10, what are the other two angles in the triangle if the hypotenuse has length twice the size of the opposite?” You’d be forgiven for answering: “Who cares?”

But given the three layers of context (historical, personal, applications) would, I suggest, make it much more interesting and relevant. On trigonometry, Stewart writes of the origins of this subject, Greek astronomers around 300BC used basic trigonometry to estimate, accurately enough, the distance from the Earth to the Moon. Ptolemy of Alexandria (of map-making fame), around 150AD, used trigonometry to develop a model of the planetary cycles in the night sky. The major advances in trigonometry were by Arabian and Hindu mathematicians in the first millennium AD – where they advanced trigonometric concepts to spherical geometry. These ideas came into their own around 1400AD when they were adapted for use in navigation – the Astrolabe was the principle tool by which ships set sail from Europe and were able to navigate their way to far-flung corners of the globe. Trade flourished, knowledge accumulated, slaves abducted, cultures met, and the world would never be the same again. In modern times, trigonometric principles are used in the engineering of buildings, in mobile communications technology, modern weaponry and GPS systems.

If such a brief prologue were given, by imaginative teachers, before every new topic was broached – before the nitty-gritty of the theory and the tools – then these subjects would be far easier to study. There is a joy in solving such problems, but I believe it is augmented further if the pupil is satisfied that the learning and exercises are not just for the sake of it - not just abstract logical walkabouts, like brain training - but can, and has been used in the great wide world to achieve amazing things, moral or not.

The list goes on. I’d suggest three contexts, the historical, the biographical and the applications, that should preface or underlie each branch of mathematics.

The historical:

The flourishing of maths in the Greek empire; the development of numbers and impact on commerce (without numbers there can be no economy worthy of the name – no debts, no fiat currencies, no credits); the use of maths to design Greek temples (think of the columns of the acropolis); the incredible strides made of Arabian empires in the first millennium and then the sudden stagnancy in the sciences in these regions as Islam closed the minds of its best and brightest (“algebra” is named after the Arabic phrase “Al-Jabr”);  the golden age of exploration and their trigonometric navigation aids; Galileo's heliocentric theory and victimisation by the Inquisition (the pinnacle of the European dark ages); the Enlightenment – when a broad flourishing of maths began, and hasn’t yet ended – when Newton, Gauss, Leibniz all threw light on light, gravity, the language of the physical world; where maths was subsumed into design, and the Industrial Revolution was brought into being; where this design was bought into the world of weaponry, where Colt mastered his designs using Newton’s calculus, where Alfred Nobel cracked the chemical compounds needed to cause explosions; where the two World Wars used maths as a tool for more ingenious and deadly advances – the fluid dynamics of submarines, the aerodynamics of spitfires, the Brownian motion of diffusing mustard gas, the development of nuclear weaponry, the development of ever more elaborate coding and encryption systems, and the means to crack them;  the twin forces of military might and economic girth heralding the era of the superpowers; Einstein’s relativity and the era of space exploration; the maths of medicine cracked polio, partial differential equations are used to model the spreads of epidemics, the world saw an unheralded advance in life expectancy, the use of statistics in evaluation effects of new drugs; with increased longevity came materialism, maths-based techniques raised efficiency in production to new levels, sustaining (most of) the word’s population in plenty and concurrent environmental destruction; the Information revolution, buttressed by the physical sciences, arguably making the world smaller, the end of the era of the nation…

The biography:

Talk of Pythagoras, the first expressed belief that numbers and maths are the language of the universe; the Pythagoreans who believed they had found a rigorous mathematical theory of the universe – one of these, Hippasus, discovered the existence of the square root of two which completely destroyed their theory – he announced is discovery to them on a boat sailing across the Mediterranean, they were so incensed they threw him overboard and he drowned; Archimedes, foiling the Roman army almost singlehandedly (using his “Law of the Lever” he invented the catapult, he also used the geometry of optical reflection to direct the suns rays onto Roman boats so that they caught fire); running through the streets naked shouting “Eureka!” when he discovered his method for determining the volume of irregular shaped objects in the bath; talk of Hypatia of Alexandria (~400AD), the first great female mathematician – depressingly she was inevitably accused of witchcraft and was murdered by a mob of Christians, wielding, strangely, oyster shells, with which they hacked her to death; Leonardo da Vinci, who introduced the base-10 Arabic numerical system to Europe (in use today*) and, among dozens of ingenious inventions, somehow invented the helicopter and contact lenses back in 1200; the intense unpleasantness of Isaac Newton, and his unsurpassed genius; the eight successive generations of the Swiss Bernoulli family, all mathematicians of note; Johann Kepler, who refined Copernicus's idea that the Earth revolves around the sun, and who’s mother was accused of being a witch, but was released because the authorities didn’t follow the correct procedures for torture; Galileo, who proved Copernicus's helio-centric theory, and who died under house arrest imposed by the Inquisition; Sofia Vasilyevna Kosalevskaya, another genius and female, whose father – in order to save money – wallpapered her nursery with pages from old calculus books, who taught herself calculus from her bedroom walls, who was never admitted to any university to study because of her gender, who entered an essay to a competition of the Academy of Sciences in 1886 and won, the jury finding the essay so brilliant that they increased the prize money; the artist Maurits Escher, who created astonishing art from his studies of hyperbolic geometry and studies of infinity, and was referenced by the Flight of the Conchords in a song “Inner City Pressure” - “Your perspective’s all messed up, like a painting by Escher”; Andrew Wiles, who proved the four hundred year old problem known as “Fermat’s Last Theorem”; the Irish mathematician, William Rowan Hamilton, who, walking with his wife along the Royal canal in Dublin in 1843, had the sudden inspiration for “Quaternions”, four dimensional complex numbers, and immediately carved the his profound equation into the stone of Brougham Bridge (still there today) – his inspiration (and tens years of subsequent perspiration) ultimately led to modern computer games, among other things; Godel, whose Incompleteness Theorem changed the philosophical discourse on maths, whose mentor Mortitz Schlick was murdered by a student for being Jewish (Germany, 1930s), who left Nazi Germany for the USA, who had two breakdowns, and in a terminal paranoia about being poisoned, starved himself to death…

Finally, the most interesting context, the applications:

Trigonometry – navigation, GPS, mobile phones.

Number theory – commerce, currency, debt, credit, trade, codes and cryptography (modular numbers).

Geometry – industrial design, astronomy, CGI graphics, 3-D cinema, art (the development of portraits in Renaissance Italy was founded upon a study of the geometries of the human body).

Logic – Boolean logic in internet search engines, computer programming languages, the scientific method

Algebra – everything! Apropos Stewart: “Algebra…is the mathematics of symbolic expressions…equations can be solved to represent unknown quantities in terms of known ones.” As such we use algebra in every facet of science, it is the fundamental tool in manipulating the world.

Calculus – the design of airplanes and spacecraft, design of bridges, meteorological predictions, design of formula one cars, studies of populations, construction of bridges, computer aided design, nuclear power and weapons.

Probability – winning at poker (game theory), avoiding nuclear holocausts (game theory again), medical trials of new drugs (statistics is a branch of probability), social sciences and anthropology (statistics again)

This is just the three contexts I'd suggest for teaching maths in secondary school. Maths is a human-invented tool, not unlike language, for manipulating the physical world and advancing our understanding of the universe and our place in it. We do not teach languages (say English) by focusing relentlessly on the tools of language – vowels, syllables, letters, syntax, verbal constructs – while important, we focus on studying great works in the various branches of language (drama, essays, novels, poetry) and are encouraged to use the tools of language to comment upon or create our own such works. While the tools are important, it is always in the penumbra of the great human achievements that we study these. Maths should be no different.

It should be more applied, with more context given (in the areas I suggest above – historical, biographical and areas of application). Having read Ian Stewart’s book and others like it, I am convinced that the development of mathematics is inextricably bound with the development of civilisation. To this student of maths, this was unfortunately never conveyed to me or my classmates.

Some links:

Scary stuff

www.irishexaminer.com/ireland/half-of-maths-teachers-unqualified-112420.html

Mind-bending paintings by Escher:

http://en.wikipedia.org/wiki/File:Escher%27s_Relativity.jpg

 

http://en.wikipedia.org/wiki/Base_10#Alternative_bases

* The Base-10 Hindu-Arabic number system (representing all numbers with 10 numerals, 0,1,2,3,4,5,6,7,8,9) that Leonardo popularised seems very natural to us now, indeed it seems inevitable; but it wasn’t. Base-10 is thought to have developed, obviously enough, because we have ten fingers and used to count on our fingers. It is thought that the Maya civilisations used base-20, counting on fingers and toes. Other civilisations had different bases, 2, 4, 5 and 12 being prevalent. The Babylonians used base-60, and this is why 60 is such a prevalent number in mathematics today – 60 minutes in an hour, 60 seconds in a minute, 360 degrees in a circle… Indeed it is sheer coincidence that the base-10 prevailed, although it does seem the most efficient counting system but that might be wisdom after the event. I often wondered (as I said, I may be a geek) whether there were any patriarchal societies with base-21 counting systems, counting on fingers, toes and another appendage?