## Archive for the ‘**math**’ Category

## What is mathematics for?

A while back, Underwood Dudley wrote this essay in the* Notices of the American Mathematical Society*. His main point (which many may well disagree with) is that mathematics is very seldom useful in the workplace. His concluding paragraph summarizes his argument:

What mathematics education is for is not for jobs. It is to teach the race to reason. It does not, heaven knows, always succeed, but it is the best method that we have. It is not the only road to the goal, but there is none better. Furthermore, it is worth teaching. Were I given to hyperbole I would say that mathematics is the most glorious creation of the human intellect, but I am not given to hyperbole so I will not say that. However, when I am before a bar of judgment, heavenly or otherwise, and asked to justify my life, I will draw myself up proudly and say, “I was one of the stewards of mathematics, and it came to no harm in my care.” I will not say, “I helped people get jobs.”

## Romantic mathematics

Here is a short quotation from the excellent book *Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics*, by Amir Alexander:

The iconic tale of the tragic romantic mathematician, it seems clear, was both novel and exclusive in the early nineteenth century—novel because it ran counter to the images that had prevailed only a few years earlier, when mathematicians were more likely to be viewed as simple natural men than as striving romantic heroes; exclusive because the new story was reserved, among the sciences, to mathematics alone. Only mathematics came to be viewed as a quest for pure sublime truth, only mathematics was perceived as a creative art rather than a science, and only mathematicians became tragic romantic strivers in the manner of contemporary poets, painters, and musicians. It can well be said that in the early nineteenth century mathematics took leave of the natural sciences, which had been its companions for millennia, and sought a place for itself instead with the creative arts.

## The Loves of the Triangles

One of the most bizarre poems in the English language is surely “The Loves of the Triangles”, which is a parody of an almost equally bizarre poem, “The Loves of the Plants“, by Erasmus Darwin. The following excerpt describes the erotic tendencies of the different conic sections: parabolas, hyperbolas, and ellipses. I imagine that if parents knew just how dirty geometry can be, we would not be allowed to teach the subject in high schools!

And first, the fair PARABOLA behold,

Her timid arms, with virgin blush, unfold!

Though, on one *focus* fix’d, her eyes betray

A heart that glows with love’s resistless sway,

Though, climbing oft, she strive with bolder grace

Round his tall neck to clasp her fond embrace,

Still e’er she reach it from his polish’d side

Her trembling hands in devious *Tangents* glide.

Not thus HYPERBOLA:—with subtlest art

The blue-eyed wanton plays her changeful part;

Quick as her *conjugated axes* move

Through every posture of luxurious love,

Her sportive limbs with easiest grace expand;

Her charms unveil’d provoke the lover’s hand:—

Unveil’d except in many a filmy ray

Where light *Asymptotes* o’er her bosom play,

Nor touch her glowing skin, nor intercept the day.

Yet why, ELLIPSIS, at thy fate repine?

More lasting bliss, securer joys are thine.

Though to each fair his treacherous wish may stray,

Though each in turn, may seize a transient sway,

‘Tis thine with mild coercion to restrain,

Twine round his struggling heart, and bind with endless chain.

The full text can be found in *The Poetry of the Anti-Jacobin*, which is available for free download on Google Books.

## Concerning the machines of Archimedes

Below is an excerpt from Plutarch’s *Life of Marcellus* (the famous Dryden translation) concerning the mechanical work of Archimedes in devising war machines for the defense of Syracuse. What is notable to me is the mention of Plato’s indignation at the application of mathematics to practical ends.

These machines he had designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with King Hiero’s desire and request, some little time before, that he should reduce to practice some part of his admirable speculation in science, and by accommodating the theoretic truth to sensation and ordinary use, bring it more within the appreciation of the people in general. Eudoxus and Archytas had been the first originators of this far-famed and highly-prized art of mechanics, which they employed as an elegant illustration of geometrical truths, and as means of sustaining experimentally, to the satisfaction of the senses, conclusions too intricate for proof by words and diagrams. As, for example, to solve the problem, so often required in constructing geometrical figures, given the two extremes, to find the two mean lines of a proportion, both these mathematicians had recourse to the aid of instruments, adapting to their purpose certain curves and sections of lines. **But what with Plato’s indignation at it, and his invectives against it as the mere corruption and annihilation of the one good of geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence to recur to sensation, and to ask help (not to be obtained without base supervisions and depravation) from matter; so it was that mechanics came to be separated from geometry, and, repudiated and neglected by philosophers, took its place as a military art.** Archimedes, however, in writing to King Hiero, whose friend and near relation he was, had stated that given the force, any given weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could remove this. Hiero being struck with amazement at this, and entreating him to make good this problem by actual experiment, and show some great weight moved by a small engine, he fixed accordingly upon a ship of burden out of the king’s arsenal, which could not be drawn out of the dock without great labour and many men; and, loading her with many passengers and a full freight, sitting himself the while far off, with no great endeavour, but only holding the head of the pulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, as smoothly and evenly as if she had been in the sea. The king, astonished at this, and convinced of the power of the art, prevailed upon Archimedes to make him engines accommodated to all the purposes, offensive and defensive, of a siege. These the king himself never made use of, because he spent almost all his life in a profound quiet and the highest affluence. But the apparatus was, in most opportune time, ready at hand for the Syracusans, and with it also the engineer himself.

The manner of the death of Archimedes is quite famous, and is described by Plutarch (*Marcellus*):

But nothing afflicted Marcellus so much as the death of Archimedes, who was then, as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow to Marcellus; which he declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword and ran him through. Others write that a Roman soldier, running upon him with a drawn sword, offered to kill him; and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was then at work upon inconclusive and imperfect; but the soldier, nothing moved by his entreaty, instantly killed him. Others again relate that, as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him. Certain it is that his death was very afflicting to Marcellus; and that Marcellus ever after regarded him that killed him as a murderer; and that he sought for his kindred and honoured them with signal favours.

## The General Burnside Problem

In 1902 William Burnside posed the following question concerning finitely generated groups:

**Bounded Burnside Problem**. If is a finitely generated group and there is an integer such that for every , then must be finite?

The problem also has the following variant:

**General Burnside Problem**. If is a finitely generated group and every element of has finite order, then must be finite?

The answer to both questions was expected to be “yes”’; the solution to the General Burnside Problem was therefore anticipated to be somewhat harder than that of the Bounded Burnside Problem. However, the answer to both problems turned out to be “no”. A counterexample to the General Burnside problem was given by a beautiful and elegant construction of Golod and Shafarevich in 1964; and a counterexample to the Bounded Burnside Problem was given by Novikov and Adjan in 1968.

The Burnside Problem has the following ring-theoretic analogue:

**Kurosh’s Problem**. If is a finitely generated algebra over a field and every element of is nilpotent, then must be nilpotent?

(A element is nilpotent if for some ; and is itself nilpotent if there is an such that for all .) The Golod-Shafarevich construction also provides a counter-example to Kurosh’s Problem. We sketch the construction below.

**The Golod-Shafarevich Theorem
**

Let be a field and let be the free non-commutative algebra over generated by the variables . Let denote the subspace of consisting of all linear combinations of monomials of degree . The elements of are the homogeneous elements of degree . Let be a two-sided ideal of generated by a set of homogeneous elements, each of degree at least 2. Suppose that has at most elements of degree for . The following (which we do not prove here) is the main computational result of the Golod-Shafarevich construction:

**Theorem**. The quotient algebra is infinite dimensional over if the coefficients in the power series expansion of are nonnegative.

Using this theorem, one constructs a counterexample to Kurosh’s Problem as follows.

**Counterexample to Kurosh’s Problem**

Let be the free algebra over a countable field . Let be the ideal of consisting of all elements of without constant term. Enumerate the elements of as . Choose an integer and write , where each . Choose another positive integer sufficiently large so that for some . Continue in this way for sufficiently large powers of . Now let be the ideal generated by the defined in the process above. Consider the quotient . The construction of guarantees that each element of is nilpotent; but the theorem above ensures that is infinite dimensional over (and hence not nilpotent). Thus is a counterexample to Kurosh’s Problem. From this construction, we can in turn build a counterexample to the General Burnside Problem.

**Counterexample to the General Burnside Problem**

Let us suppose now that is a prime number and is the field with elements. Let and be as defined above. Let be the elements of the quotient respectively. Let be the multiplicative semigroup in generated by the elements , , and . An element of has the form for some . The element is nilpotent by the construction of , and so for sufficiently large , we have . Since we are in characteristic we have . It follows that has an inverse, whence is a group. Moreover every element of has finite order (indeed order a power of ). Thus satisfies the conditions of the General Burnside Problem. It remains to show that is infinite. If were finite, then the linear combinations of its elements would form a finite dimensional algebra over . Moreover, since both 1 and are in , the linear combination is in . Thus, are all in . But suffice to generate , which was previously shown to be infinite dimensional. The algebra is therefore also infinite dimensional, a contradiction. Thus is infinite, as required.

## Borges’ Theorem

No doubt one of the most delightful parts of writing a book of mathematics is the search for clever literary quotations to place at the beginning of chapters or to scatter throughout the book as appropriate (perhaps also, in the eyes of some, one of the greatest wastes of time). Flajolet and Sedgewick, in Analytic Combinatorics, make reference to the following result, which they call “Borges’ Theorem”, namely that given any finite set P of words, a random text of length n will contain every word of P with probability tending to 1 exponentially fast as n tends to infinity. The reason for the appellation “Borges’ Theorem” is due to Borges’ story The Library of Babel, which describes a library that contains

Everything: the minutely detailed history of the future, the archangels’ autobiographies, the faithful catalogues of the Library, thousands and thousands of false catalogues, the demonstration of the fallacy of those catalogues, the demonstration of the fallacy of the true catalogue, the Gnostic gospel of Basilides, the commentary on that gospel, the commentary on the commentary on that gospel, the true story of your death, the translation of every book in all languages, the interpolations of every book in all books.

## A quotation from Weierstrass

I recently came across the following quotation from Weierstrass, which I found to be interesting.

*A mathematician who is not also something of a poet will never be a complete mathematician.*

It reminded me a little bit of the following quotations from Hardy.

*A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.*

*Beauty is the first test: there is no permanent place in the world for ugly mathematics.*