I have a feature article in this week’s New Scientist magazine, about the Continuum Hypothesis, set theory, and Hugh Woodin’s Ultimate L. It’s in the shops, or here. [£]
the confused heap of destructive disinformation known as “the mathematics curriculum”
then I strongly encourage you to put that right.
The focus of Lockhart’s ire is mathematics in the US school system, but it translates without difficulty to the UK and probably a great many other places.
Lockhart doesn’t diagnose the cause of this mathematical malady, but I will offer one thought: the take-over of exam results as the be-all and end-all of educational attainment.
If you fancy discussing this issue further, there’s a webinar on the topic over at MathFuture this evening, hosted by David Wees and Richard DeMerchant.
It was hosted over at Mathfuture, by Maria Droujkova. My aim in the talk was to give a very brisk overview of how several different families of wonderful, complex shapes all arise from juggling a very small number of simple criteria. I’m separately uploading the slides for my presentation here [pdf]. They are quite rough and ready, without any detailed explanations, or even any pictures – I used Stella for those. But it does sketch the central story (which I also covered in this blogpost). I may spruce them up one day, if I give the same talk again.
I found the whole thing a thoroughly enjoyable experience, and the Elluminate technology worked extremely smoothly. The format allowed me to talk while sharing my whole desktop with the audience, with the optimal result of people being able to hear my voice and watch everything I was doing, without having to endure looking at my face. And we could all do it from the comfort of our living rooms! This is sort of thing the internet was intended for, isn’t it?
|Naming Infinity by Loren Graham & Jean-Michel Kantor tells the story of the beginnings of the mathematical subject called descriptive set theory. The backdrop to the story is the pioneering work performed by Georg Cantor in the late 19th century.
Cantor’s great revelation was that infinity is not a single entity, but comes in a variety of flavours (an infinite variety, in fact). With Pandora’s box opened, there then followed an almighty scramble throughout the mathematical world, to make sense of this extraordinary revelation.
A particularly pressing question was how the two most familiar levels of infinity relate to each other. The first of these is countable infinity, that is the infinity of the natural numbers: 1,2,3,4,… The second is the larger infinity of the real numbers (meaning all possible infinite decimal strings). This level of infinity is known as the continuum. A major question left unanswered by Cantor (to his considerable distress) was whether there are any levels sitting between these two. The assertion that there are no such intermediate infinities is known as the continuum hypothesis.
A solution to the continuum hypothesis was a major goal in the early days of set theory. One angle of attack was to try to understand the possible subsets of the collection of real numbers, and to determine their sizes, as well as understand the many other phenomena which emerged during this investigation. This was descriptive set theory, where Cantor’s abstract modern subject met the more mature areas of calculus, continuity, and analysis.
Although the continuum hypothesis was a continuing motivation to these mathematicians, with hindsight we can see that the story rather deviates at this stage. The enduring importance of the work that followed was to begin cataloguing and classifying the immense variety of possible sets and functions of real numbers, using set theory to enrich our understanding of analysis, and to dig into the logical foundations of the real numbers. (The continuum hypothesis was eventually settled by Paul Cohen in 1963, through essentially different means.)
Transfinite Numbers and Name Worshipping
The early running in descriptive set theory was made in France, notably by the trio of Émile Borel, Henri Lebesgue, and René Baire. Great as their achievements were, they were reluctant to embrace the full power of Cantor’s transfinite numbers, and in the narrative of Naming Infinity, the French trio “lost their nerve”.
The baton then passed to Russia, where the intellectual climate was better suited to handle the conceptual challenges posed by the new theory. This fortuitous match of cultures is the main focus of Naming Infinity.
The trouble was that the higher rungs of Cantor’s infinite ladder corresponded to nothing else in the history of human thought. This caused a division in the mathematical world. Many felt that a mathematician’s job was to analyse objects whose roots were in the physical universe (albeit abstracted and idealised), rather than disappearing into a world of weird concoctions of their own invention. Naming Infinity provides a concise account of this crisis in mathematics, tracing its roots to the difference between the old Aristotelian and Platonic schools of thought.
In Moscow, set theory found fertile soil among thinkers whose religious beliefs allowed them to embrace Cantor’s radical new ideas. The mathematical heavyweights were Dmitri Egorov and Nikolai Luzin, while the group’s religious mentor was the priest Pavel Florensky. These mathematicians were inspired by a Christian mystic movement known as Name Worshipping. The crux of Name Worshipping is expressed at the beginning of John’s Gospel (my emphasis): “In the beginning was the Word, and the Word was with God, and the Word was God.”
For these mystics, the act of naming had a profound meaning. To name something was to grasp its essence, and to utter the name of God, in particular, was to commune with him at the very deepest level. The name-worshippers developed their own sacred practices, consisting of repeating the names of Jesus and God over and over again, like a meditative mantra. These practices were judged heretical by many in the Russian Orthodox Church.
The emphasis on naming gelled well with Cantor’s set theory, in which the transfinite numbers were named in turn, as the ladder was ascended: the countable level was denoted ℵ0 (ℵ being the aleph, first letter of the Hebrew alphabet). Then come ℵ1, ℵ2, and so on. For Egorov, Luzin, and Florensky, the fact that such concepts could be pinned down enough to be named leant them a deep legitimacy, irrespective of their seeming unworldliness.
The heart of the Naming Infinity is the fascinating story of this meeting of mathematics and mysticism. The authors are at pains not to overstate their case, as they say: “when we emphasize the importance of Name Worshipping to men like Luzin, Egorov, and Florensky, we are not claiming a unique or necessary relationship… It could have happened another way; but it did not.”
The Moscow School of Mathematics
The story can be read as the birth of not only of descriptive set theory, but of the Moscow School of Mathematics, whose influence was extensive throughout the 20th century and continues today. Among other legacies, this school gave us (through Alexandrov) point-set topology and (through Kolmogorov) the modern axiomatic approach to probability. Its founding fathers were Luzin and Egorov, and the circle of young men and women who worked with them: the Lusitania.
It was not just the world of mathematics which was turned upside-down during this period. This was the time of the Russian revolution, and the turmoil and terror that followed. Naming Infinity tells extraordinary stories of the dedication with which these mathematicians worked, continuing their seminars through terrible poverty and hunger, and even bitter cold.
In times of political oppression, as in Stalin’s Russia, it is comforting to think of mathematicians and scientists as the innocent victims of the political ideologues. So the most troubling chapter of the book concerns the so-called Luzin affair. The mathematician Nikolai Luzin found himself not only pursued by the Soviet authorities for his supposed counter-revolutionary activity, but also denounced and betrayed by his colleagues and pupils. To modern eyes, it seems almost incomprehensible that Aristotle and Plato’s debate about the foundations of mathematics could be dragged into the the swirl of political and ideological intolerance, and all the accompanying suspicion and hatred. But it was, all the same. The Moscow School of Mathematics was condemned for its supposed beourgois and anti-Soviet tendencies, while the Name Worshippers, having already been declared heretical by the Church, now found themselves pursued by the State.
History and Mathematics
Naming Infinity is a short book about the history of mathematics, and as such it is excellent. It relates with great clarity and humanity a chapter in the history of our subject which was previously unfamiliar to me. The interplay of mathematical and religious ideas is fascinating, and the story of mathematics flourishing in such adverse conditions is at once tragic and inspiring.
In purely historical terms, the book takes a terrifying slice through Stalin’s Russia. It describes the lives of just a few individuals, behind whom can be imagined hundreds of thousands of others, whose stories will very likely never been heard. The history is told with a respect for the main characters, neither indulgently excusing their crimes, nor wrenching them from their historical context.
I do have a quibble though, and that is in the book’s account of the mathematics itself. I cannot say that I really got a sense of what the breakthroughs of the principle characters really were. To my mind, too little time is spent on that side of the tale. In the end, I took to reading it in parallel with some old lecture notes on descriptive set theory. That was a hugely satisfying combination, but not one I can suggest to my non-mathematical friends and family, much as I would like to recommend them this book.
Perhaps I am wrong, but I can’t help feeling that a non-mathematical reader might be rather bemused by some of the technical terms in the book, which tend to come and go, without ever being fully fleshed out. The continuum hypothesis, for instance, begins centre stage, but then disappears from view over the course of the book. While mathematically understandable, I can’t help worrying that this might perplex the reader who wishes to understand the many intellectual triumphs which lie behind the human tragedy.
[Update: I’m updating this post with more math raps as I find them, so scroll to the bottom for newer material.]
I can see the future. And what I see is math-rapping. So here’s a celebration of today’s trail-blazers:
First up, TRM:
Next we have the smooth-flowing Mr Mc E=MC2:
Now a massive youtube viral hit, WYKAMATH‘s What you know about math? (part 1):
And Part 2:
E=Mc2‘s The Math Rap
Stepping up the technicality is Essiness with One Geometry (The Poincaré Conjecture Rap)
Also by Essiness is Down With That (The Bolzano-Weierstrass Rap)
Am I missing any good ones?
UPDATE! Yes I am. The Kellers’ Pythagoras rap, worth watching as much for the video as the song:
…and an Eminem-style video from Brigham Young University’s Mathletics Team
I’ve got an article about the smooth 4-dimensional Poincaré conjecture, and Yog-Sothoth, over at Plus magazine.
Alexandrov said [pdf] of their relationship:
“in 1979 this friendship celebrated its fiftieth anniversary and over the whole of this half century there was not only never any breach in it, there was also never any quarrel, in all this time there was never any misunderstanding between us on any question, no matter how important for our lives and our philosophy; even when our opinions on one of these questions differed, we showed complete understanding and sympathy for the views of each other.”
No quarrels, no misunderstandings, complete understanding and symapthy for 50 years…. that really sets the bar for the rest of us.
Dick Lipton has an interesting post about the history of mathematical notation, and how it affects the thinking of those who use it. During the discussion he references a mathematician I had never heard of, one “Johann Gauss”. Well, it turns out that was indeed the great man’s name!
UPDATE: it occurs to me that this post might be incomprehensible to the non-geek community. The point is that Gauss – one of the greatest of all mathematicians – is near universally known as Carl Friedrich Gauss. I am quite surprised that I have managed to get this far through my life without knowing his full name.
I just ran into Tom Henderson’s Punk Math Manifesto:
The video’s an appeal for funds taken from Kickstarter, but it looks like the target’s already been reached. (Not that a few more pennies would go unappreciated, I’m sure.)
I definitely dig the philosophy, fleshed out in more detail in this interview. So it’ll be good to see the project develop.
Having said all that, punk’s not really my genre. Maybe I should try experimenting with some Jazz Geometry, or Death Metal Model Theory.
Circle packing is a classical topic in discrete geometry. As Axel Thue and László Fejes Tóth showed, if you want to fit as many identical circular coins on a table as possible (all sitting side by side, no piling up or overlapping), the best you can achieve is for around 90.7% of the table to be covered. This is done by arranging the coins along a hexagonal lattice.
That’s fine, but we can pose the same problem, using coins which are not circular. Now here is an interesting question: which shape is the worst packer?
Then the answer is conjectured to be the smoothed octagon, with a maximum packing density of around 90.2%.
The smoothing is done by rounding off each corner with a hyperpola which is tangent to the two meeting sides, and which asymptotically approaches the two sides beyond those.
[Image from Wikipedia]