Showing papers in "The Mathematical Intelligencer in 2003"

Probability theory : the logic of science

Abstract: Foreword Preface Part I. Principles and Elementary Applications: 1. Plausible reasoning 2. The quantitative rules 3. Elementary sampling theory 4. Elementary hypothesis testing 5. Queer uses for probability theory 6. Elementary parameter estimation 7. The central, Gaussian or normal distribution 8. Sufficiency, ancillarity, and all that 9. Repetitive experiments, probability and frequency 10. Physics of 'random experiments' Part II. Advanced Applications: 11. Discrete prior probabilities, the entropy principle 12. Ignorance priors and transformation groups 13. Decision theory: historical background 14. Simple applications of decision theory 15. Paradoxes of probability theory 16. Orthodox methods: historical background 17. Principles and pathology of orthodox statistics 18. The Ap distribution and rule of succession 19. Physical measurements 20. Model comparison 21. Outliers and robustness 22. Introduction to communication theory References Appendix A. Other approaches to probability theory Appendix B. Mathematical formalities and style Appendix C. Convolutions and cumulants.

The card game set

Abstract: This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

Consequences of reversing preferences

Abstract: Other than standard election disruptions involving shenanigans, strategic voting, and so forth, it is reasonable to expect that elections are free from difficulties. But this is far from being true; even sincere election outcomes admit all sorts of counterintuitive conclusions.

On Fine’s Partition Theorems, Dyson, Andrews, And Missed Opportunities

Abstract: The history almost never works out the way you want it, especially when you are looking at it after the dust settles. The same is true in mathematics as well. There are times when the solution of a problem is overlooked simply by accident, due to a combination of some unfortunate circumstances. In a celebrated address [D4], Freeman Dyson described several “missed opportunities”, in particular a story of how he never discovered Macdonald’s eta-function identities. We present here the history of Fine’s partition theorems and their combinatorial proofs. As the reader shall see, many of the results could and perhaps should have been discovered a long time ago if not for a number of “missed opportunities”... The central event in our little story is a publication of a short note [F1] by Nathan Fine. To quote George Andrews, “[Fine] announced several elegant and intriguing partition theorems. These results were marked by their simplicity of statement and [...]by the depth of their proof.” [A7] Without taking anything away from the depth and beauty of the results, we will show here that most of them have remarkably simple combinatorial proofs, in a very classical style. Perhaps, that’s exactly how it should be with the important results... Fine’s partition theorems could be split into two (overlapping) categories: those dealing with partitions into odd and distinct parts, à la Euler, and those dealing with Dyson’s rank. We shall separate these two stories as they have relatively little to do with each other. The fortune and misfortune, however, had the same root in both stories, as we are about to discover. Fine’s note [F1] didn’t have any proofs; not even hints on complicated analytic formulae which were used to prove the results. It was published in a National Academy of Science publication, in a journal devoted to all branches of science. Thus the paper was largely overlooked by subsequent investigators. The note contained a promise to have complete proofs published in a journal “devoted entirely to mathematics.” This promise was never fulfilled. What a misfortune! Good news came rather unexpectedly. In the sixties, George Andrews, while a graduate student at the University of Pennsylvania, took a course of Nathan Fine on basic hypergeometric series. As he writes in his mini biography [A8], “His

Autism in mathematicians

Abstract: The cause of autism is mysterious, but genetic factors are important. It takes a variety of forms; the expression autism spectrum, which is often used, gives a false impression that it is just the severity of the disorder that varies. Different people are affected in different ways, but the core problems are impairments of communication, social.

Hermann Weyl, The Reluctant Revolutionary

Abstract: “Brouwer – that is the revolution!” – with these words from his manifesto “On the New Foundations Crisis in Mathematics” (Weyl 1921) Hermann Weyl jumped headlong into ongoing debates concerning the foundations of set theory and analysis His decision to do so was not taken lightly, knowing that this dramatic gesture was bound to have immense repercussions not only for him, but for many others within the fragile and politically fragmented European mathematical community Weyl felt sure that modern mathematics was going to undergo massive changes in the near future By proclaiming a “new” foundations crisis, he implicitly acknowledged that revolutions had transformed mathematics in the past, even uprooting the entire edifice of mathematical knowledge At the same time he drew a parallel with the “ancient” foundations crisis commonly believed to have been occasioned by the discovery of incommensurable magnitudes, a finding that overturned the Pythagorean worldview that was based on the doctrine “all is Number” In the wake of the Great War that changed European life forever, the Zeitgeist appeared ripe for something similar, but even deeper and more pervasive

One hundred prisoners and a lightbulb

Abstract: This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on.

Mathematics and art: mathematical visualization in art and education

Abstract: I’m convinced the title of this book will intrigue most readers of the Mathematical Intelligencer. When you look at the list of contributors and see names like Michele Emmer, Michael Field, George W. Hart, John Hubbard, Richard S. Palais, Konrad Polthier and John Sullivan (to name but a few in alphabetical order), I’m sure you will be even more interested. The book is the proceedings of the Colloquium on Mathematics and Art held in Maubeuge in September 2000, and as soon as I opened the book, I started wishing I had been there. It must have been a killer conference! But does that make for a killer conference proceedings? Mathematics and Art is a very wide term. My background involves teaching a course on Mathematics in Art and Architecture at the National University of Singapore, consulting for an exhibition called “Art Figures: Mathematics in Art” at the Singapore Art Museum, and numerous TV interviews and public lectures at museums, libraries and schools. I personally like to subdivide discussion of mathematics and art into the following four rough categories.

Mathematics and war: An invitation to revisit

Abstract: The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-in-chief, Chandler Davis.

Predicting the future of scholarly publishing

Abstract: What should we conclude? Should we steadfastly maintain the status quo? Do we avoid technology altogether? Of course not.We should experiment; we should try out new things; we should tinker with technology and find better ways to communicate.

Why legendre made a wrong guess about π-(x), and how laguerre’s continued fraction for the logarithmic integral improved it

Abstract: Carl Friedrich Gauβ, in 1792, when he was 15, found by numerical evidence that π(x), the number of primes p such that p ≤ x, goes roughly with x/in x (letter to Encke, 1849). This was, as can be seen from Table 1, a very weak approximation with an error of about 10%. In 1798 and again in 1808,

Parallel Worlds: Escher and Mathematics, Revisited

Abstract: The popularity — and ubiquity — of the graphic work of the Dutch artist M.C. Escher (1898–1972) continues unabated: books on his work remain in print, the public never seems to tire of Escher posters, mugs, T-shirts, calendars, and other paraphernalia, and exhibitions of his work are packed. Over 300,000 visitors attended the six-month “M.C. Escher: A Centennial Tribute” at the National Gallery of Art in Washington in spring, 1998; exhibitions were held in that centennial year in Brazil, Mexico, The Czech Republic, Hong Kong, Great Britain, China, Greece, Italy, Argentina, Canada, Holland, and Peru. “People are attracted like magnets to these works. They come closer and closer and closer, and they stay there an incredible amount of time,” says Jean-Francois Leger of the National Gallery of Canada. “Studies have shown that the average length of time that a gallery visitor will stay in front of a work of art is 17 seconds. But they stay minutes in front of Escher’s, and discuss, and comment, and say ‘Do you see this, have you seen that?’“ What is the magnet, what is the attraction? Is it profound, or is it superficial?

Mathematicians’ visiting Cards

Abstract: This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of “mathematical community” is the broadest. We include “schools” of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

Into the Woods: Norbert Wiener in Maine

Abstract: Wiener would not be remembered on campus until 1953, when his less than flattering description of the University of Maine in his autobiography caused quite a stir among its alumni. One alumna was sufficiently upset to send a transcription of the most damning passages to then-President Hauck, wondering if anything could be done against this “immature and biased judgment.” In his response, Hauck said he had not read the autobiography, but he did know that there were still a few people on campus who remembered Wiener and were quite amused by what he had written. Indeed, a chuckle and a shrug was probably the best reaction.30 With hindsight, it is obvious that Wiener would never have made a good college teacher at a rural university and that his stay in Maine was a stage in his own education more than anything else. Wiener still had many issues to deal with before he would be ready to make his true debut in academia. At best, his stint in Orono was a trial run, as Wiener himself appositely calls the chapter on it in his autobiography.