The Vortex Theory of the Atom

From a review by Martin Gardner of Lee Smolin's and Peter Woit's books attacking String Theory:

In the nineteenth century, a conjecture called the vortex theory of the atom became extremely popular in England and America. Proposed by the famous British physicist Lord Kelvin, it had an uncanny resemblance to string theory. It was widely believed at the time that space was permeated by an in- compressible frictionless fluid called the ether. Atoms, Kelvin suggested, are super-small whirlpools of ether, vaguely similar to smoke rings. They take the form of knots and links. Point particles can’t vibrate. Ether rings can. Their shapes and frequencies determine all the properties of the elements. Vortex theory isn’t mentioned by Woit, although Smolin considers it briefly.

Kelvin published two books defending his conjecture. It was strongly championed in England by J. J. Thomson in his 1907 book The Corpuscular Theory of Matter. Another booster of the theory was Peter Tait, an Irish mathematician. His work, like Witten’s, led to significant advances in knot theory. In the United States, Albert Michelson considered vortex theory so “grand” that “it ought to be true even if it is not.” Hundreds of papers elaborated the theory. Tait predicted it would take generations to develop its elegant mathematics. Alas, beautiful though vortex theory was, it proved to be a glorious road that led nowhere.

I first read about the importance of the vortex theory to the development of knot theory in The Knot Book by Colin Adams.

Angle Error

This afternoon I was assisting in a maths lesson where two children independently made the same error:  they both read the above angle as 160°.  They seemed to be interpreting the curved arc as a leading 1 of the number.  The original diagram on the board, although hand-drawn, seemed quite clear to me, but it obviously wasn't to them.

Unknown Quantity by John Derbyshire and Finding Moonshine by Marcus du Sautoy

Ten years ago I took a series of mathematics courses with the Open University (a distance-learning university based in the UK).  I had always wanted to take my understanding of mathematics beyond what I had done at A-Level, and this seemed to be the best way to do it.  I did much of the course work on the trains to and from my work.  This was back in the bad old days when British trains were subject to frequent delays and cancellations.  However, when my train was late I just used the time to do work that I would have otherwise had to do in the evening or at the weekend, so in a way, I turned the delays to my advantage.

One of my aims in taking these courses was to get a basic understanding of group theory, which I kept coming across in my reading.  The Open University courses did cover group theory, but we ran through it rather quickly so that, although could answer questions on various parts of the subject, I didn't have any overall feel for how the parts fitted together.  I wasn't until I read Unknown Quantity by John Derbyshire and Finding Moonshine by Marcus du Sautoy that my course notes started make sense to me.

Both books are rather similar in the area they cover and in the level at which they cover it: Unknown Quantity is a history of algebra in general, Finding Moonshine is a history of the more restricted field of group theory.  Du Sautoy has the advantage over Derbyshire in that he is an actual practising group theorist (at least until recently) and he supplies some interesting insider details on the Cambridge University group theorists and John Horton Conway in particular.  However, this advantage is counterbalanced a little by Du Sautoy's tendency to get diverted into the arty side of symmetry.  Still, both books are excellent and I would heartily recommend both to anyone studying or intending to study algebra at university undergraduate level.

Taking the Pants from a Highlander

Most of the mathematical logicians who have come after Boole are men who would have stuck at the impossible subtraction in ordinary algebra.  They say virtually, "How can you throw into a heap the same things twice over; and how can you take from a heap things that are not there."  Their great principle is the impossibility of taking the pants from a Highlander.  Their only conception of the analytical processes of addition and subtraction is throwing into a heap and taking out of a heap.  It does not occur to them that the processes of algebra are ideal, and not subject to gross material restrictions.

From "Lectures on ten British mathematicians of the nineteenth century" by Alexander Macfarlane (1916).

(I should perhaps mention, for those mystified by the above reference, that Scotsmen, and presumably Highlanders in particular, are reputed not to wear any pants under their kilts.)

An Irrational Dislike?

From a post by Mark Dominus:

I hate uncountable sets, and I have a fantasy that in the mathematics of the 23rd Century, uncountable sets will be looked back upon as a philosophical confusion of earlier times, like Zeno's paradox, or the luminiferous aether.