Archive for the 'Mathematics' Category
Category theory takes a bird’s eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to detect from ground level.
The opening sentences from Tom Leinster : Basic Category Theory. Cambridge University Press. Cambridge Studies in Advanced Mathematics.
This post is a history of the family of Charles Burney FRS (1726-1814), musician and musicologist, and his ancestors and descendants.
Sir MacBurney was one of the 60 Knights who participated in a jousting tournament, supervised by Geoffrey Chaucer on the orders of Richard II, held at Smithfield in London in 1390.
One James Macburney is said to have come south to London from Scotland with King James I and VI in 1603. His descendant (likely a grandson), also James Macburney, was born around 1653 and had a house in Whitehall. His son, also called James Macburney (1678-1749), was born in Great Hanwood, Shropshire, around 1678, and attended Westminster School in London. In 1697, he eloped with Rebecca Ellis, against his father’s wishes. As a consequence, the younger James was not left anything when his father died. The younger man’s stepbrother, Joseph Macburney (born of a second wife) was left the entire estate of their father.
This younger James Macburney (1678-1749) was a dancer, violinist and painter, and was supposedly a wit and bon viveur. He and Rebecca Ellis had 15 children over 20 years, of whom 9 survived into adulthood. By 1720, he had moved to Shrewsbury, and Rebecca had died. He married again, to Ann Cooper, who apparently brought money to the union which helped her somewhat feckless husband. This second marriage produced 5 further children, among whom were Richard Burney (1723-1792) (christened “Berney”). The last two children were twins, Charles Burney (1726-1814) and Susanna (1726-1734?), who died at the age of 8. Their father James had apparently dropped the prefix “Mac” around the time of the birth of the twins.
One of Charles’ half-brothers was James Burney (1710-1789), who was organist at St. Mary’s Church, Shrewsbury, for 54 years, from 1732 to 1786. Charles Burney worked as his assistant from 1742 until 1744.
For a period, Charles Burney and his family lived in Isaac Newton’s former house at 35 St Martin’s Street, Leicester Square, London. Among Charles’ children were:
- Esther Burney (1749-1832), harpsichordist, who married her cousin Charles Rousseau Burney (1747-1819), also a keyboardist and violinist.
- Rear Admiral James Burney RN FRS (1750-1821), naval historian and sailor, who twice sailed around the world with Captain James Cook RN.
- Fanny Burney, later Madame d’Arblay (1752-1840), novelist and playwright.
- Rev. Charles Burney FRS (1757-1817), classical scholar.
- Charlotte Ann Burney, later Mrs Broome (1761-1838), novelist.
- Sarah Harriet Burney (1772-1844), novelist.
Charles’ nephew, Edward Francisco Burney (1760-1848), artist and violinist, was a brother to Charles Rousseau Burney, both sons of Richard Burney (1723-1792), Charles’s elder brother. This is a self-portrait of Edward Francisco Burney (Creative Commons License from National Portrait Gallery, London):
In 1793, Fanny Burney married Alexandre-Jean-Baptiste Piochard D’Arblay (1754-1818), an emigre French aristocrat and soldier, and adjutant-general to Lafayette. Their son, Alexander d’Arblay (1794-1837), was a poet and keen chess-player, and was 10th wrangler in the Mathematics Tripos at Cambridge in 1818, where he was a friend of fellow-student Charles Babbage. He was also a member of Babbage’s Analytical Society (forerunner of the Cambridge Philosophical Society), which sought to introduce modern analysis, including Leibnizian notation for the differential calculus, into mathematics teaching at Cambridge. d’Arblay was ordained and served as founding minister of Camden Town Chapel (later the Greek Orthodox All Saints Camden) from 1824-1837, and then served briefly at Ely Chapel in High Holborn, London. The founding organist at Camden Town Chapel was Samuel Wesley (1766-1837).
Not everyone was a fan of clan Burney. Here is William Hazlitt:
“There are whole families who are born classical, and are entered in the heralds’ college of reputation by the right of consanguinity. Literature, like nobility, runs in the blood. There is the Burney family. There is no end of it or its pretensions. It produces wits, scholars, novelists, musicians, artists in ‘numbers numberless.’ The name is alone a passport to the Temple of Fame. Those who bear it are free of Parnassus by birthright. The founder of it was himself an historian and a musician, but more of a courtier and man of the world than either. The secret of his success may perhaps be discovered in the following passage, where, in alluding to three eminent performers on different instruments, he says: ‘These three illustrious personages were introduced at the Emperor’s court,’ etc.; speaking of them as if they were foreign ambassadors or princes of the blood, and thus magnifying himself and his profession. This overshadowing manner carries nearly everything before it, and mystifies a great many. There is nothing like putting the best face upon things, and leaving others to find out the difference. He who could call three musicians ‘personages’ would himself play a personage through life, and succeed in his leading object. Sir Joshua Reynolds, remarking on this passage, said: ‘No one had a greater respect than he had for his profession, but that he should never think of applying to it epithets that were appropriated merely to external rank and distinction.’ Madame d’Arblay, it must be owned, had cleverness enough to stock a whole family, and to set up her cousin-germans, male and female, for wits and virtuosos to the third and fourth generation. The rest have done nothing, that I know of, but keep up the name.” (On the Aristocracy of Letters, 1822).
K. S. Grant: ” Charles Burney”, Grove Music Online. (Accessed 2006-12-10.)
POST MOST RECENTLY UPDATED: 2014-08-30.
On the categories email list on 5 March 2006, Ronald Brown quoted the following paragraph on mathematical speculation from a 14 June 1983 letter he had received from Alexander Grothendieck:
Talking of his grandfather who had overcome poverty and blindness to become a US Senator, Gore Vidal once wrote that no challenge is finally insurmountable if you mean to prevail. I was reminded of this in reading Edward Frenkel’s superb memoir, Love and Math. Frenkel overcame the widespread and systemic anti-semitism in Soviet Mathematics to establish himself as a world-leading mathematician at a very young age.
Denied entry in 1984 because of his ethnicity to Moscow State University’s (MGU’s) Department of Mechanics and Mathematics (Mekh-Mat), the leading undergraduate mathematics programme in the USSR, he entered instead the mathematics program at Kerosinka, the Moscow Institute of Oil and Gas. Anti-semitism (and anti-Armenianism, anti-Chinese racism, etc) in the admissions process at Mekh-Mat was so widespread, that other Moscow institutions, such as Kerosinka, were able to recruit very good Jewish and minority students. One theory is that this policy was deliberate, since having all the Jewish mathematicians studying in one or two institutions made their monitoring easier for the KGB.
Frenkel had grown up in Kolomna – only 70 miles from Moscow, but well into the provinces – and had not attended a special mathematics school (as did, for example, Vadim Delone at FizMat #2), nor had an opportunity to participate in the mathematical study circles that were widespread in the larger soviet cities. He did have the help of a local mathematician, Evgeny Petrov, a professor at a teacher training college in Kolomna. Frenkel was very fortunate to have such help. I recall my envy on learning on the first day of lectures in my first year at university that some of my fellow students, who had grown up near to the university, had been meeting our professors for years previously for after-school mentoring and coaching. (On the other hand, even the brightest of my fellow students so mentored ended up winning no Fields Medal, nor even becoming a mathematician.)
Good mathematical undergraduates from Kerosinka and other specialized institutes in Moscow literally scaled the fences at MGU to attend, illegally but often with the encouragement of the teachers, lectures at Mekh-Mat. Frenkel did this and was again fortunate in being befriended by some very great mentors: Dmitry Fuchs (now at UC Davis), his student Boris Feigin, and Yakov Khurgin. Their generous mentoring was unpaid, time-intensive, and often brave, given the society they lived in. As a result, Frenkel wrote his first research paper in only his second year as an undegraduate, a paper subsequently published in Israel Gelfand’s famous journal, Functional Analysis and Applications. Gelfand was someone that even my professors, in the 1970s and in faraway Australia, spoke of with awe.
With the opening of perestroika, the Mathematics Department at Harvard University decided to invite some young Soviet mathematicians for research visits, and Frenkel was one of these: He received his invitation in March 1989, before he had even completed his first degree. While at Harvard, he had another Russian mentor, Vladimir Drinfeld (now at University of Chicago), and Frenkel completed his PhD there, in 1 year, under the supervision of another Russian, Joseph Bernstein (now at Tel-Aviv). Frenkel is very generous in his acknowledgement of the support he received from his mentors and from others, and his story warms the heart. Despite the anti-semitism he experienced, he has prevailed in the end, being now a professor at U-Cal Berkeley (and a film-maker). Reading his account, I was reminded repeatedly of the ancient spiritual wisdom: When the disciple is ready, the guru will appear.
Frenkel interleaves his personal story with an account of his changing research focus along the way, a focus which has mostly followed the powerful thread of the Geometric Langlands Programme. His writing is fluent, wise and witty, and he manages to convey well the excitement and pure, joyous exhilaration that mathematical thinking can provide. His writing makes most of the underlying mathematical ideas clear to non-experts. That said, however, the text has a couple of weaknesses, both minor, although both I found irritating. No one who does not already know something of category theory would understand it, even at a high level, from the single paragraph devoted to it on page 156. Another minor criticism is that the text does not always adequately explain the diagrams, or what is being done with them. But then I have particular views about reasoning over diagrams.
In summary, this is a superb book – wise, generous, witty, and heart-warming – and reading it will enlarge your knowledge of mathematics, of the Langlands Program, and of the power of the human spirit. Everyone in the pure mathematical universe should read it.
An index to posts on the Matherati is here.
Edward Frenkel : Love and Math: The Heart of Hidden Reality. New York, NY: Basic Books.
Last week’s Observer carried a debate over the status of string theory by a theoretical physicist, Michael Duff, and a science journalist, James Baggott. Mostly, they talk past each other. There is much in what they say that could provoke comment, but since time is short, I will only comment on one statement.
Duff’s final contribution includes these words:
Finally, you offer no credible alternative. If you don’t like string theory the answer is simple: come up with a better one. “
This is plain wrong for several reasons. First, we would have no scientific progress at all if critics of scientific theories first had to develop an alternative theory before they could advance their criticisms. Indeed, public voicing of criticisms of a theory is one of the key motivations for other scientists to look for alternatives in the first place. So Duff has the horse and the cart backwards here.
Secondly, “come up with a better one“? “better“? What means “better“? Duff has missed precisely the main point of the critics of string theory! We have no way of knowing – not even in principle, let alone in practice – whether string theory is any good or not, nor whether it accurately describes reality. We have no experimental evidence by which to assess it, and most likely (since it posits and models alleged additional dimensions of spacetime that are inaccessible to us) not ever any way to obtain such empirical evidence. As I have argued before, theology has more empirical support – the personal spiritual experiences of religious believers and practitioners – than does string theory. So, suppose we did come up with an alternative theory to string theory: how then could we tell which theory was the better of the two?
Pure mathematicians, like theologians, don’t use empirical evidence as a criterion for evaluating theories. Instead, they use subjective criteria such as beauty, elegance, and self-coherence. There is nothing at all wrong with this. But such criteria ain’t science, which by its nature is a social activity.
From the music critic of The Times, writing in 1952:
At Redbrick [University] they treat mathematics as an instrument of technology; at Cambridge they regard it as an ally of physics and an approach to philosophy; at Oxford they think of it as an art in itself having affinities with music and dancing.”
Cited by Ida Winifred Busbridge, in a 1974 history of mathematics at Oxford University, here.
Oxford University was a strong supporter of Catholicism in Elizabeth I’s time (eg, Thomas Campion), while Cambridge and the Fens, due to their proximity to the Netherlands, was the centre for an extreme Protestant sect, called the Family of Love, or the Familists. Elizabeth I’s religious policy often sought to find a middle ground between these two extremes. These religious differences persisted, so that Oxford was again, in the mid 19th-century, a centre of Catholic, and, within the Anglican Church, Anglo-Catholic (“High Church”) ideas. The Redbrick Universities (Liverpool, Birmingham, Leeds, Victoria University of Manchester, etc), mostly founded in the North and Midlands of England in the late 19th century or early 20th century, were the result of money-raising campaigns by local business people and civic worthies, who were often of a Nonconformist or Jewish religious background. The name Redbrick arose from novels written by a professor of Spanish at the University of Liverpool, Edgar Peers, about a fictional northern university modeled on Liverpool.
Over at the AMS Graduate Student Blog, Jean Joseph wonders what it is that mathematicians do, asking if what they do is to solve problems:
After I heard someone ask about what a mathematician does, I myself wonder what it means to do mathematics if all what one can answer is that mathematicians do mathematics. Solving problems have been considered by some as the main activity of a mathematician, which might then be the answer to the question. But, could reading and writing about mathematics or crafting a new theory be considered as serious mathematical activities or mere extracurricular activities?”
Not all mathematics is problem-solving, as we’ve discussed here before, and I think it would be a great shame if the idea were to take hold that all that mathematicians did was to solve problems. As Joseph says, this view does not account for lots of activities that we know mathematicians engage in which are not anywhere near to problem-solving, such as creating theories, defining concepts, writing expositions, teaching, etc.
I view mathematics (and the related disciplines in the pure mathematical universe) as the rigorous study of structure and relationship. What mathematicians do, then, is to rigorously study structure and relationship. They do this by creating, sharing and jointly manipulating abstract mental models, seeking always to understand the properties and inter-relations of these models.
Some of these models may arise from, or be applied to, particular domains or particular problems, but mathematicians (at least, pure mathematicians) are typically chiefly interested in the abstract models themselves and their formal properties, rather than the applications. In some parts of mathematics (eg, algebra) written documents such as research papers and textbooks provide accurate descriptions of these mental models. In other parts (eg, geometry), the written documents can only approximate the mental models. As mathematician William Thurston once said:
There were published theorems that were generally known to be false, or where the proofs were generally known to be incomplete. Mathematical knowledge and understanding were embedded in the minds and in the social fabric of the community of people thinking about a particular topic. This knowledge was supported by written documents, but the written documents were not really primary.
I think this pattern varies quite a bit from field to field. I was interested in geometric areas of mathematics, where it is often pretty hard to have a document that reflects well the way people actually think. In more algebraic or symbolic fields, this is not necessarily so, and I have the impression that in some areas documents are much closer to carrying the life of the field. But in any field, there is a strong social standard of validity and truth.
. . .
When people are doing mathematics, the flow of ideas and the social standard of validity is much more reliable than formal documents. People are usually not very good in checking formal correctness of proofs, but they are quite good at detecting potential weaknesses or flaws in proofs.”
Computer science typically proceeds by first doing something, and then thinking carefully about it: Engineering usually precedes theory. Some examples:
- The first programmable device in modern times was the Jacquard Loom, a textile loom that could weave different patterns depending on the instruction cards fed into it. This machine dates from the first decade of the 19th century, but we did not have a formal, mathematical theory of programming until the 1960s.
- Charles Babbage designed various machines to undertake automated calculations in the first half of the 19th century, but we did not have a mathematical theory of computation until Alan Turing’s film-projector model a century later.
- We’ve had a fully-functioning, scalable, global network enabling multiple, asynchronous, parallel, sequential and interleaved interactions since Arpanet four decades ago, but we still lack a fully-developed mathematical theory of interaction. In particular, Turing’s film projectors seem inadequate to model interactive computational processes, such as those where new inputs arrive or partial outputs are delivered before processing is complete, or those processes which are infinitely divisible and decentralizable, or nearly so.
- The first mathematical theory of communications (due to Claude Shannon) dates only from the 1940s, and that theory explicitly ignores the meaning of messages. In the half-century since, computer scientists have used speech act theory from the philosophy of language to develop semantic theories of interactive communication. Arguably, however, we still lack a good formal, semantically-rich account of dialogs and utterances about actions. Yet, smoke signals were used for communications in ancient China, in ancient Greece, and in medieval-era southern Africa.
An important consequence of this feature of the discipline is that theory and practice are strongly coupled and symbiotic. We need practice to test and validate our theories, of course. But our theories are not (in general) theories of something found in Nature, but theories of practice and of the objects and processes created by practice. Favouring theory over practice risks creating a sterile, infeasible discipline out of touch with reality – a glass bead game such as string theory or pre-Crash mathematical economics. Favouring practice over theory risks losing the benefits of intelligent thought and modeling about practice, and thus inhibiting our understanding about limits to practice. Neither can exist very long or effectively without the other.
With so many blogs being written by members of the literati, it’s not surprising that a widespread meme involves compiling lists of writers and books. I’ve even succumbed to it myself. Lists of mathematicians are not as common, so I thought I’d present a list of the 20th century greats. Some of these are famous for a small number of contributions, or for work which is only narrow, while others have had impacts across many parts of mathematics.
Each major area of mathematics represented here (eg, category theory, computer science) could equally do with its own list, which perhaps I’ll manage in due course. I’ve included David Hilbert, Felix Hausdorff and Bertrand Russell because their most influential works were published in the 20th century. Although Hilbert reached adulthood in the 19th century, his address to the 1900 International Congress of Mathematicians in Paris greatly influenced the research agenda of mathematicians for much of the 20th century, and his 1899 axiomatization of geometry (following the lead of Mario Pieri) influenced the century’s main style of doing mathematics. For most of the 20th century, mathematics was much more abstract and more general than it had been in the previous two centuries. This abstract style perhaps reached its zenith in the work of Bourbaki, Grothendieck, Eilenberg and Mac Lane, while the mathematics of Thurston, for example, was a throwback to the particularist, even perhaps anti-abstract, style of 19th century mathematics. And Perelman’s major contributions have been in this century, of course.
- David Hilbert (1862-1943)
- Felix Hausdorff (1868-1942)
- Bertrand Russell (1872-1970)
- Henri Lebesgue (1875-1941)
- Godfrey Hardy (1877-1947)
- LEJ (“Bertus”) Brouwer (1881-1966)
- Srinivasa Ramanujan (1887-1920)
- Alonzo Church (1903-1995)
- Andrei Kolmogorov (1903-1987)
- John von Neumann (1903-1957)
- Henri Cartan (1904-2008)
- Kurt Gödel (1906-1978)
- Saunders Mac Lane (1909-2005)
- Leonid Kantorovich (1912-1986)
- Alan Turing (1912-1954)
- Samuel Eilenberg (1913-1998)
- René Thom (1923-2002)
- John Forbes Nash (1928- )
- Alexander Grothendieck (1928- )
- Michael Atiyah (1929- )
- Steven Smale (1930- )
- Paul Cohen (1934-2007)
- Nicolas Bourbaki (1935- )
- Stephen Cook (1939- )
- William Thurston (1946-2012)
- Edward Witten (1951- )
- Andrew Wiles (1953- )
- Richard Borcherds (1959- )
- Grigori Perelman (1966- )
And here’s my list of great mathematical ideas.