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What is mathematics for?

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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.”

Written by uncudh

August 24, 2010 at 10:27 pm

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Romantic mathematics

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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.

Written by uncudh

June 26, 2010 at 12:00 pm

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The Loves of the Triangles

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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.

Written by uncudh

December 14, 2009 at 11:32 pm

Posted in literature, math

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Concerning the machines of Archimedes

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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.

Written by uncudh

November 8, 2009 at 3:00 pm

Posted in history, math

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The General Burnside Problem

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

Bounded Burnside Problem. If G is a finitely generated group and there is an integer n such that g^n = 1 for every g\in G, then must G be finite?

The problem also has the following variant:

General Burnside Problem. If G is a finitely generated group and every element of G has finite order, then must G 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 A is a finitely generated algebra over a field F and every element of A is nilpotent, then must A be nilpotent?

(A element a \in A is nilpotent if a^n=0 for some n; and A is itself nilpotent if there is an m such that a_1a_2\cdots a_m = 0 for all a_1,a_2,\ldots,a_m \in A.) The Golod-Shafarevich construction also provides a counter-example to Kurosh’s Problem. We sketch the construction below.

The Golod-Shafarevich Theorem

Let F be a field and let T = F\langle x_1,x_2,\ldots x_d \rangle be the free non-commutative algebra over F generated by the variables x_1,x_2,\ldots,x_d. Let T_n denote the subspace of T consisting of all linear combinations of monomials of degree n. The elements of T_n are the homogeneous elements of degree n. Let I be a two-sided ideal of T generated by a set A of homogeneous elements, each of degree at least 2. Suppose that A has at most r_i elements of degree i for i \geq 2. The following (which we do not prove here) is the main computational result of the Golod-Shafarevich construction:

Theorem. The quotient algebra T/I is infinite dimensional over F if the coefficients in the power series expansion of (1 - dz + \sum_{i=2}^\infty r_i z^i)^{-1} are nonnegative.

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

Counterexample to Kurosh’s Problem

Let T = F\langle x_1,x_2,x_3 \rangle be the free algebra over a countable field F. Let T' be the ideal of T consisting of all elements of T without constant term. Enumerate the elements of T' as t_1, t_2, \ldots. Choose an integer m_1 \geq 2 and write t_1^{m_1} = t_{1,2} + t_{1,3} + \cdots + t_{1,k_1}, where each t_{1,j} \in T_j. Choose another positive integer m_2 sufficiently large so that t_2^{m_2} = t_{2,k_1+1} + t_{2,k_1+2} + \cdots + t_{2,k_2} for some k_2 > k_1. Continue in this way for sufficiently large powers of t_3, t_4, \cdots. Now let I be the ideal generated by the t_{i,j} defined in the process above. Consider the quotient T'/I. The construction of I guarantees that each element of T'/I is nilpotent; but the theorem above ensures that T'/I is infinite dimensional over F (and hence not nilpotent). Thus T'/I 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 p is a prime number and F is the field with p elements. Let T and I be as defined above. Let a_1, a_2, a_3 be the elements x_1 + I, x_2 + I, x_3 + I of the quotient T/I respectively. Let G be the multiplicative semigroup in T/I generated by the elements 1+a_1, 1+a_2, and 1+ a_3. An element of G has the form 1+a for some a \in T'/I. The element a is nilpotent by the construction of T'/I, and so for sufficiently large n, we have a^{p^n} = 0. Since we are in characteristic p we have (1+a)^{p^n} = 1 + a^{p^n} = 1. It follows that 1+a has an inverse, whence G is a group. Moreover every element 1+a of G has finite order (indeed order a power of p). Thus G satisfies the conditions of the General Burnside Problem. It remains to show that G is infinite. If G were finite, then the linear combinations of its elements would form a finite dimensional algebra B over F. Moreover, since both 1 and 1+a_i are in G, the linear combination (1+a_i) - 1 = a_i is in B. Thus, 1, a_1, a_2, a_3 are all in B. But 1, a_1, a_2, a_3 suffice to generate T/I, which was previously shown to be infinite dimensional. The algebra B is therefore also infinite dimensional, a contradiction. Thus G is infinite, as required.

Written by uncudh

September 4, 2009 at 12:20 am

Borges’ Theorem

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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.

Written by uncudh

August 15, 2009 at 1:57 pm

A quotation from Weierstrass

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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.

Written by uncudh

August 8, 2009 at 3:31 pm

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