September 11, 2006
FEARFUL OF SYMMETRY:
The man who saved geometry: Crying `Death to Triangles!' a generation of mathematicians tried to eliminate geometry in favor of algebra. Were it not for Donald Coxeter, they might have succeeded. (Siobhan Roberts, September 10, 2006, Boston Globe)
[D]espite these modern applications, geometry was, for much of the 20th century, a discipline very much in jeopardy. It was deemed by a generation of mathematicians to be old-fashioned, a fine recreation for idling away a lazy afternoon, but in essence little more than a trivial tinkering with toys. Modern mathematics was all about prickly algebraic symbols and undulating equations-impenetrable hieroglyphs with no diagrams, no shapes.The task of fending off these attacks fell to H.S.M. ``Donald" Coxeter, the greatest classical geometer of the last century. Through his lifelong work as geometry's apostle, Coxeter, who died in 2003 at 96 (prematurely by his measure-his lifelong vegetarianism guaranteed he should live to 100, he figured), became known by his followers around the world as ``the man who saved geometry" in a mathematical era characterized by all things algebraic, abstract, and austere.
Fifty years ago this summer, Coxeter was summoned by the Mathematical Association of America on a roving lecture tour through the United States. He traveled as far north as Fairbanks, Alaska, as far west as Stanford, Calif., and east to New York City, speaking with a missionary's zeal to schoolteachers and any other willing listeners.
Coxeter lectured about ``the beauteous properties of triangles," about circles and spheres, and about the Platonic solids: the tetrahedron, cube, octahedron, icosahedron, and dodecahedron. According to a recent cosmological hypothesis (and a similar theory put forth by Plato) the dodecahedron is a potential model for the shape of the universe-bound by 12 walls, each the shape of a pentagon.
Coxeter had a special affection for the Platonic solids. Educated at Cambridge, in his native England, he spent most of his professional life at the University of Toronto. But before coming to Toronto he did a two-year stint at Princeton. It was there that he launched his career, choosing as his specialty polytopes, an extension of the Platonic solids in higher dimensions.
But just as Coxeter set out upon his career, classical geometry-with its emphasis on shapes and diagrams-was being supplanted by modern mathematicians' penchant for algebra.
A secret society of the créme de la créme of French mathematicians epitomized the shift in the mathematical zeitgeist of the early 20th century. Writing under the pseudonym Nicolas Bourbakis, the collective set out in the 1930s to rewrite the history of mathematics in one grand mathematical treatise, and perhaps the most distinctive feature of their work was the absence of diagrams.
The Bourbakis espoused mathematical rationality and rigor. They believed the subjective and fallible visual sense was easily led astray, falling victim to impressionistic reasoning. In 1959, at a conference in France addressing the need to overhaul the French education system, Jean Dieudonné, a founding member of the Bourbakis and the group's scribe, infamously proclaimed: ``Down with Euclid! Death to Triangles!"
Eventually, the Bourbakis way of mathematics pervaded the Western world, reaching even into grade schools with the Sputnik-motivated New Math reforms of the 1960s, which aimed to improve students' performance and to ensure America was not left in the scientific dust by the Soviet Union. Instead of shapes, children studied axioms and set theory.
As a consequence, mathematical and scientific investigation suffered from what Walter Whiteley, a great admirer of Coxeter and director of applied mathematics at York University in Toronto, calls the ``geometry gap." Whiteley's thesis holds that when the areas of the brain that process visual and geometric concepts fall into disuse, the realms of mathematics and science suffer as well.
So Coxeter set out to make the case for the visual geometric approach, using a number of tactics.
On a popular level, he proselytized for the classical geometric treasures he loved, praising their simple beauty and symmetry. The elegance of his talks and essays gained him an avid following around the world, a fan base of professional and amateur geometers alike who became just as passionate about classical geometry as he was.
If anything we should drop algebra and keep geometry, which teaches kids how to think. Posted by Orrin Judd at September 11, 2006 11:35 PM
Picking one over the other is like arguing that you should only use the left or right hand to play the piano.
Posted by: Mike Beversluis at September 12, 2006 10:18 AMgeometry is found and is true, while math is merely invented.
Posted by: oj at September 12, 2006 10:34 AMNo, geometry and algebra are two different modes of representing the same reality.
And oj, when you write that geometry is found and is true, do you mean Euclid's geometry or Einstein's?
Posted by: Ted Welter at September 12, 2006 10:41 AMEuclid's. Einstein's is false.
Posted by: oj at September 12, 2006 10:47 AMOJ:
Arithmetic is a human contrivance, but math is real.
Posted by: Mike Earl at September 12, 2006 11:00 AMPrecisely.
Posted by: oj at September 12, 2006 11:10 AMGeometry is the ultimate abstration, based upon nothing that exists in nature (dimensionless points, planes, perfect circles, straight lines, etc.) and in the Euclidian form, full of arbitrary constraints (parallel postulate, compass and straightedge). No wonder our host loves it.
Euclid's. Einstein's is false.
By the same measure (pun intended), Riemann's geometry is false, also?
Posted by: Anthony Perez-Miller at September 12, 2006 11:38 AMAnthony:
Here's the giveaway: "Over many months, Riemann developed his theory of higher dimensions. "
Posted by: oj at September 12, 2006 11:52 AMNow, now, let's not confuse geometry and physics. This is like asking whether Jazz or Baroque is true; they're both valid works under slightly different sets of constraints.
Posted by: Mike Earl at September 12, 2006 11:56 AMgeometry is found and is true, while math is merely invented.
OJ, I think what you mean by that is that Euclidean geometry is empirically based. Well, sorta. One does not need to understand (or even know) all proofs in Euclid's geometry in order to build a house.
But his geometry as it is taught (and more fundamentally, structured) is not empirical: it's based on a fundamental set of axioms and built deductively from there, using supporting definitions as necessary. This, presumably, is why you also say that it teaches kids to think.
Moreover, Riemann's geometry (at least in its more elementary forms) is simply Euclid's with a single fundamental axiom swapped out...
Posted by: Anthony Perez-Miller at September 12, 2006 12:12 PMNo. I mean that Euclidean geometry is true regardless of humankind.
Posted by: oj at September 12, 2006 12:20 PMAs far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Albert Einstein
Raoul, have you forgotten? Geometry is fun. It validated our youthful observations of our surroundings.
Algebra and the rest are fun too, but without those early geometry classes, I might not have realized it and not gone any further than was required.
It wasn't easy for girls in math classes in those days, so if I didn't really love it, I probably would have chucked it all.
OJ - Ted Welter's first point is right: Geometry and algebra are languages; the truths they can describe are the same, but sometimes it's more conveinient to use one form or another. Some people intuit truth via geometric representations, and not by algebra, so they prefer it. The truth doesn't care which language you use any more than God cares what language a prayer is spoken in.
Posted by: Mike Beversluis at September 12, 2006 1:12 PMMike:
Don't you know where you are?
"Let x equal..." is the first step towards relativism and moral decline. If everybody has "x" equaling different things, how can we have a coherent decent society?
Mike:
No, algebra is just a language that tries to explain the truth of geometry. Of course, the problem is that mathematical language can equally well explain falsehood. You tell them the result you want and the mathmaticians will give you the formula to get there.
Posted by: oj at September 12, 2006 3:19 PMWhat, you object that algebra requires you to take certain things on faith?
Posted by: Mike Earl at September 12, 2006 4:04 PMNo, we take everything on faith. Algebra is just ugly. Geometry is lovely.
Posted by: oj at September 12, 2006 4:09 PMPeter: Where am I? I'm a white collar employee wasting time on the internet at work.
OJ: Sorry, but no. Mathematical language, ie, an algebraic proof, cannot be used prove a falsehood. Given the incompleteness theorem, it may have nothing to say, but the only proofs of falsehoods are logically inconsistent. Hence reductio ad absurdum.
Formulas describe sets of points which form lines, curves,planes and figures; Platonically speaking, they refer to the same "truths" as a geometric drawing. And given the same constraints (eg, the set of figures that can be constructed with a compass and unmarked straight edge), the set of true theorems is the same whether you demonstrate them using geometric construction or algebra.
As a particular example of something that's fixed by algebra, Galois is famous because he proved that there is no general solution to a fifth degree polynomial in terms of its coefficients. No mathematician can give such a "quintic" formula.
Furthermore, a corollary of this proof was the proof of the impossibility of trisecting the angle, which had been believed but unprooven since the Greeks. In this problem, algebra was tractable where geometric construction had bogged down.
Posted by: Mike Beversluis at September 12, 2006 4:27 PMGeometry is also beautiful, but...
"Algebra is a world of principle, and a dramatic revelation of the power of principle. In fact, algebra, and even algebra alone, could provide a true and sufficient education out of which to understand the worth of living by principle in a life beset by a never-ending succession of nasty particulars, and at the same time provident of joy and goodness and thoughtfulness."
"The uses of audacity" by Richard Mitchell
Read it all at http://www.sourcetext.com/grammarian/newslettersv14/14.3.htm
Given the incompleteness theorem it must always be assumed to be possibly stating untruths.
Posted by: oj at September 12, 2006 6:08 PMThen that critique applies to euclidean geometry too.
Posted by: Mike Beversluis at September 13, 2006 11:03 AM