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Constructive Reverse Mathematics
by
Douglas Bridges
University of Canterbury
In constructive reverse mathematics we examine theorems either to determine precisely where they fail to be constructive or else to prove, constructively, their equivalence to one of a number of (plausibly) constructive principles, such as versions of Brouwer's fan theorem. This talk deals with a generic form of proof of equivalence to a fan theorem, taking two particular theorems of analysis as illustrations.
Date received: December 3, 2007
Mathematical Problems from the Maine Farmer's Almanac
by
Bruce Burdick
Roger Williams University
A recent paper by Albree and Brown discusses the presence of mathematical problems in The Ladies' Diary (1704-1840), an almanac from Britain. We report on an American version of the same phenomenon. In the nineteenth century and into the twentieth, The Maine Farmer's Almanac made an annual feature out of posing various puzzles, including riddles, anagrams, and mathematical problems. Readers would have a year to work on the problems and then see solutions in the following issue. A few readers would mail in their solutions to the editor for publication. Some of the problems were surprisingly sophisticated for a general readership. We will survey some interesting examples and distribute a list of problems from issues we have seen.
Date received: September 25, 2007
Elegance and insight: what is the link?
by
Philip Catton
Philosophy, University of Canterbury
That elegance links to insight is original to mathematical practice and ineluctable from it. Yet this linkage is not well explained by modern epistemologists of mathematics. Two key tenets of modern epistemology are that a proposition is the content of a declarative assertion, and that we demonstrate a proposition by logically deducing it from other such declarative propositions. Yet in mathematics a proposition is often something that it is proposed to do, and a demonstration often simply the rationally most elegant execution of the proposed task. (Such is how Euclid seeks to work throughout his Elements, for example, and as an indication of this, Euclid expresses himself equally often in the imperative as in the declarative voice, for that way of coaching his reader in practical respects is essential to his understanding of propositions, of demonstrations, and of mathematics itself.) Modern epistemologists look past the practical aspect that Euclid remarks as essential to mathematics, partly because they are much affected in how they view ideal or perfected knowledge by the rigorising and formalising programmes of some nineteenth- and early twentieth-century mathematicians. Yet while those programmes have their point, they also have proved demonstrably limited; and moreover, they orient us in diametrically the wrong way to see the practical connection of mathematical theorising, and therewith the original and ineluctable connection between insight and elegance. In this talk I explore again the classical view, according to which the clear logical ordering of thoughts is not so much foundational for mathematics as a distant and in some ways not fully achievable rational goal for it. Reason according to my conception is not chiefly analytically oriented or logical or symbolic in form; it is chiefly synthetically oriented and intuitive and practical in form. The view that I develop explains the link between elegance and insight appropriately, by associating it with conditions for the very possibility of mathematical thought.
Date received: October 29, 2007
Leonard Euler and the dastardly John Robison
by
Lawrence D'Antonio
Ramapo College of New Jersey
The theory of structures in a very real sense begins with Euler’s research on elasticity. Euler gives an analysis of the shape of bent beams and the buckling of columns. With the onset of the Industrial Revolution engineering practice undergoes rapid changes. The theoretical basis for this practice is provided by a line of research starting with Euler and proceeding through Coulomb, Cauchy and Navier. The Scottish engineer John Robison knew Euler from the time that Robison taught in St. Petersburg. Robison was severely critical of Euler’s theory of elasticity, calling it a “dry mathematical disquisition.” In this talk we show that the subsequent experimental evidence of Hodgkinson and Duleau strongly supports Euler’s theory.
Date received: August 31, 2007
The dark side of constructive reverse mathematics
by
Hannes Diener
University of Canterbury
Basing mathematics on foundations that differ from those used classically can lead to alternate and sometimes strange mathematical universes. One can get some order into these universes by identifying principles that hold in some, but fail in others. In this talk we will discuss a hierarchy of very closely related principles together with their antitheses that impact on the notions of continuity and compactness. Although recent results in constructive reverse mathematics will be presented, the focus of the talk will not be about the logical or analytical details, but how they fit into the grander scheme.
Date received: October 30, 2007
Episodes from the career of the Riemann Hypothesis
by
Hardy Grant
York University, Toronto
I shall survey aspects of the early history of this most celebrated and important of conjectures, focussing on the theoretical and technological advances that enabled extensions of the known range of validity. The account will suggest contemporary perceptions of promising strategies for resolution of the "RH" and contemporary expectations of the eventual outcome.
Date received: September 7, 2007
Limits of solvability: unsolvable problems in Fibonacci's Liber Abbaci
by
John Hannah
University of Canterbury
Leonardo of Pisa (also known as Fibonacci) published his Liber Abbaci at the start of the thirteenth century. It begins with one of the earliest European accounts of arithmetic using the decimal system, but it is mostly devoted to the art of problem solving. Leonardo uses a variety of problem solving strategies (including proportional thinking, false position and al-Khwarizmi's algebra), justifying each of his methods by Euclidean geometrical arguments. He also explores variations on well-known problems (men exchanging money, or finding purses, or buying horses, and so on) investigating the boundaries between solvable and unsolvable problems. Sometimes an unsolvable problem becomes solvable if debts are allowed, but this comes at the cost of violating Leonardo's Euclidean principles. His decisions on when to allow such irregular solutions seem to be guided by whether the resulting scenarios sound sensible in terms of everyday experience.
Date received: September 3, 2007
Indecomposability of the Continuum in Constructive Reverse Mathematics
by
Iris Loeb
University of Canterbury
Different philosophical schools hold different views on the continuum. For example, in contrast to the classical continuum, the intuitionistic continuum cannot be split effectively: it is indecomposable. In this talk we will study some of the consequences of these different philosophical views on the continuum within the programme of Constructive Reverse Mathematics.
Date received: October 29, 2007
Hypsicles of Alexandria and Arithmetical Sequences
by
Clemency Montelle
University of Canterbury
The determination of rising times for the twelve zodiacal signs at a given terrestrial latitude was a challenge for ancient mathematicians and astronomers and many attempts to model this were proposed in antiquity based on the leading mathematical theories and techniques of the day. An important early approach was put forth by the Alexandrian mathematician Hypsicles (fl. ca. 150 BCE (?)) in a work called the Anaphoricos who based his solution on the assumption that rising times increase and decrease strictly linearly with constant difference. Indeed, in an era when the overwhelming success of Ptolemy’s mathematical Syntaxis ensured the redundancy of almost all works that predated it, Hypsicles’s work is not only significant because of the fact that it is a rare glimpse into early Greek mathematical astronomy but also because it invokes some elegant arithmetical mathematical lemmas to solve a practical problem in a scene that was dominated by geometrical ways of thinking. Hypsicles’s presentation is unmistakably Euclidean in style but with some vital differences. This talk will provide a detailed textual, technical, and contextual study of the mathematical content of his work.
Date received: October 29, 2007
Mathematics and observation in Indian astronomical parameters
by
Kim Plofker
Union College
For over two hundred years historians have debated (sometimes with great ferocity) about the methods that medieval Indian mathematical scientists used to derive the parameters for their celestial models. Were the values periodically revised in accordance with obscure but comprehensive observational programs, or were they numerically adjusted in a more ad hoc fashion? This talk examines and attempts to mediate in the latest incarnation of this debate, which pitted the statistical reconstructions of the late Roger Billard against the textual historiography of the late David Pingree.
Date received: September 4, 2007
Probability in Ancient Greek: Moving Beyond the Traditional Narrative
by
Bronwyn Rideout
University of Canterbury, New Zealand
Evidence for a mathematical conception or calculation of probability in Ancient Greece has yet to be uncovered. However, contemporary historians of mathematics interpret this gap in the record to signify either that the practice of such was a trade secret or there was nothing in Greek society to inspire them to engagement within that field. These interpretations are by and large motivated by the reinvigoration of probability via Huygens and Pascal.
Recovery of the hidden history of Greek probability requires the removal of the traditional Huygenian narrative and a reconsideration of Greek thought on probability in its own context. In their mythology, philosophy and art, the Greeks were more than comfortable with the notions of the probable and beating the odds. What is lacking is a transference of that interest into a mathematical mindset and the strongest obstacle to that could be found in philosophy.
A survey of some of the key figures in Greek philosophy over several topics proven integral to probability, i.e. gambling, chance, mathematics etc., will demonstrate that while the Greeks did obtain an understanding of probability akin to its current conception, their beliefs on everything else proved to be a significant barrier.
Date received: October 29, 2007
Mathematical Contributions to The Educational Times from Australia and New Zealand
by
Jim Tattersall
Providence College
Coauthors: Shawnee McMurran, California State University at San Bernardino
A number of significant mathematical journals have included a section devoted to mathematical problems intended to challenge and educate their readers. None has had a more extensive list of contributions and world-wide readership than the monthly periodical The Educational Times. Between 1848 and 1918, there were more than eighteen thousand contributions to the mathematical department from amateur and professional mathematicians. According to the English mathematician William Kingdon Clifford, The Educational Times did more to encourage original mathematical research than any other European periodical in the late nineteenth century. The section devoted to mathematical problems and their solutions was later republished in six-month installments as Mathematical Questions and Their Solutions from the Educational Times. We focus on problems and solutions from Australian and New Zealand contributors. We illustrate the types of problems they submitted and solved in comparison to contributors from other parts of the world.
Date received: August 27, 2007
Algebraic invariant theory and characteristic classes
by
Paul R Wolfson
West Chester University (U.S.A.)
In the middle of the twentieth century, André Weil supplied unity and direction to the rapidly developing theory of characteristic classes of bundles. The Weil homomorphism connected characteristic classes to results from classical algebraic invariant theory. In this talk I shall describe the state of characteristic classes at that time, recall the results from invariant theory, and suggest how the homomorphism opened up new lines of research.
Date received: September 30, 2007