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Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of ..."
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
Cyclotomy and Ramanujan sums in quantum phase locking
, 2003
"... Abstract. Phaselocking governs the phase noise in classical clocks through effects described in precise mathematical terms. We seek here a quantum counterpart of these effects by working in a finite Hilbert space. We use a coprimality condition to define phaselocked quantum states and the correspo ..."
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Abstract. Phaselocking governs the phase noise in classical clocks through effects described in precise mathematical terms. We seek here a quantum counterpart of these effects by working in a finite Hilbert space. We use a coprimality condition to define phaselocked quantum states and the corresponding PeggBarnett type phase operator. Cyclotomic symmetries in matrix elements are revealed and related to Ramanujan sums in the theory of prime numbers. The phasenumber commutator vanishes as in the classical case, but a new type of quantum phase noise emerges in expectation values of phase and phase variance. The employed mathematical procedures also emphasize the isomorphism between algebraic number theory and the theory of quantum entanglement. Time and phase are amongst the most federal concepts of science. Hersh mentions Kant putting time before number and making the intuition of time the origin of arithmetics [1]. The present work can be considered as an extension of a longstanding effort to model phase noise and phaselocking effects that are found in highly stable classical oscillators. It was unambiguously demonstrated that the observed variability (i.e., the 1/f frequency noise of such oscillators) is related to the finite dynamics of states during the measurement process and to the
Implications of Experimental Mathematics for the Philosophy of Mathematics,” chapter to appear
 Current Issues in the Philosophy of Mathematics From the Viewpoint of Mathematicians and Teachers of Mathematics, 2006. [Ddrive Preprint 280
"... Christopher Koch [34] accurately captures a great scientific distaste for philosophizing: “Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation depends on particular philosophical background assumptions. This hidden influence is an acute emba ..."
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Christopher Koch [34] accurately captures a great scientific distaste for philosophizing: “Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation depends on particular philosophical background assumptions. This hidden influence is an acute embarrassment to many researchers, and it is therefore not often acknowledged. ” (Christopher Koch, 2004) That acknowledged, I am of the opinion that mathematical philosophy matters more now than it has in nearly a century. The power of modern computers matched with that of modern mathematical software and the sophistication of current mathematics is changing the way we do mathematics. In my view it is now both necessary and possible to admit quasiempirical inductive methods fully into mathematical argument. In doing so carefully we will enrich mathematics and yet preserve the mathematical literature’s deserved reputation for reliability—even as the methods and criteria change. What do I mean by reliability? Well, research mathematicians still consult Euler or Riemann to be informed, anatomists only consult Harvey 3 for historical reasons. Mathematicians happily quote old papers as core steps of arguments, physical scientists expect to have to confirm results with another experiment. 1 Mathematical Knowledge as I View It Somewhat unusually, I can exactly place the day at registration that I became a mathematician and I recall the reason why. I was about to deposit my punch cards in the ‘honours history bin’. I remember thinking “If I do study history, in ten years I shall have forgotten how to use the calculus properly. If I take mathematics, I shall still be able to read competently about the War of 1812 or the Papal schism. ” (Jonathan Borwein, 1968) The inescapable reality of objective mathematical knowledge is still with me. Nonetheless, my view then of the edifice I was entering is not that close to my view of the one I inhabit forty years later. 1 The companion web site is at www.experimentalmath.info
TopDown and BottomUp Philosophy of Mathematics
"... Abstract. The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way ’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed ..."
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Abstract. The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way ’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a topdown approach, that is, an approach which explains the nature of mathematics in terms of some general unproven assumption. As an alternative, a bottomup approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the topdown approach.
Exploratory Experimentation: Digitallyassisted Discovery and Proof
 Chapter in ICMI Study 19: On Proof and Proving in Mathematics Education. In press
, 2010
"... Abstract The mathematical community (appropriately defined) faces a great challenge to reevaluate the role of proof in light of the power of current computer systems, the sophistication of modern mathematical computing packages, and the growing capacity to datamine on the internet. Added to those ..."
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Abstract The mathematical community (appropriately defined) faces a great challenge to reevaluate the role of proof in light of the power of current computer systems, the sophistication of modern mathematical computing packages, and the growing capacity to datamine on the internet. Added to those are the enormous complexity of many modern mathematical results such as the Poincaré conjecture, Fermat’s last theorem, and the classification of finite simple groups. With great challenges come great opportunities. Here, I survey the current challenges and opportunities for the learning and doing of mathematics. As the prospects for inductive mathematics blossom, the need to ensure that the role of proof is properly founded remains undiminished. Much of this material was presented as a plenary talk in May
EMPIRICISM, CONTINGENCY AND EVOLUTIONARY METAPHORS: GETTING BEYOND THE “MATH WARS”
"... As a middle school mathematics teacher, I was frequently frustrated by what went on in the classroom. Theorists and practitioners in other subject areas have worked to explicitly link the role of human agency to their respective disciplines and to find ways to apply school knowledge in reasonably re ..."
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As a middle school mathematics teacher, I was frequently frustrated by what went on in the classroom. Theorists and practitioners in other subject areas have worked to explicitly link the role of human agency to their respective disciplines and to find ways to apply school knowledge in reasonably realistic contexts. In language arts, science, and history, emphasis on
Probability And Statistics Ideas In The Classroom – Lessons From History
"... When examining how the history of probability and statistics can be useful in the classroom, it is first useful to examine the styles in which the history of the subjects is written. These styles may be divided into internalist (those working within the area) and externalist (those outside the area) ..."
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When examining how the history of probability and statistics can be useful in the classroom, it is first useful to examine the styles in which the history of the subjects is written. These styles may be divided into internalist (those working within the area) and externalist (those outside the area) approaches. It is natural for teachers of probability and statistics to follow an internalist approach for classroom discussion. In order to discover what principles apply in transferring the lessons of history to the classroom, the work of William Sealy Gosset (Student) is discussed as a case study. What follows from this case study is that the most important historical lesson to convey is the motivation for an individual’s work. This lesson is illustrated further in discussions of the solution to the problem of points or division of stakes and of the FisherNeyman dispute over their approaches to statistical inference. 1.
Joint Statistical Meetings 20032005: D.R. Bellhouse Probability and Statistics Ideas in the Classroom – Lessons from History
"... Almost any introductory statistics textbook is a compendium of the history of elementary probability since the Middle Ages and statistical methods since the seventeenth century. Of course, more modern developments obtained throughout the twentieth century are also included in these texts. Dicing pro ..."
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Almost any introductory statistics textbook is a compendium of the history of elementary probability since the Middle Ages and statistical methods since the seventeenth century. Of course, more modern developments obtained throughout the twentieth century are also included in these texts. Dicing probabilities, sometimes given as problems to solve in these texts, first