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Propagation Networks: A Flexible and Expressive Substrate for Computation
, 2009
"... In this dissertation I propose a shift in the foundations of computation. Modern programming systems are not expressive enough. The traditional image of a single computer that has global effects on a large memory is too restrictive. The propagation paradigm replaces this with computing by networks o ..."
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In this dissertation I propose a shift in the foundations of computation. Modern programming systems are not expressive enough. The traditional image of a single computer that has global effects on a large memory is too restrictive. The propagation paradigm replaces this with computing by networks of local, independent, stateless machines interconnected with stateful storage cells. In so doing, it offers great flexibility and expressive power, and has therefore been much studied, but has not yet been tamed for generalpurpose computation. The novel insight that should finally permit computing with generalpurpose propagation is that a cell should not be seen as storing a value, but as accumulating information about a value. Various forms of the general idea of propagation have been used with great success for various special purposes; perhaps the most immediate example is constraint propagation in constraint satisfaction systems. This success is evidence both
Visual explanations
 Notices of the AMS
, 1999
"... apparently always had a mildly mathematical flavour. In 1975, while at Princeton University, he was asked to teach a seminar on statistics and statistical graphics. This seems to have been a turning point in his career. The eventual outcome was a series of three extremely attractive and intriguing b ..."
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apparently always had a mildly mathematical flavour. In 1975, while at Princeton University, he was asked to teach a seminar on statistics and statistical graphics. This seems to have been a turning point in his career. The eventual outcome was a series of three extremely attractive and intriguing books on what he calls information graphics—The Visual Display of Quantitative Information (1983), Envisioning Information (1990), and Visual Explanations (1997). Tufte was so concerned with quality and cost that he established his own press, which is dedicated exclusively to the publication of these books. Pleasant books to look at, certainly. Not very expensive, considering the quality, which has been improving as new volumes appear, presumably because the endeavour has proven itself financially. And obviously not without relevance at least to some fields of applied mathematics and statistics, because much space in these books is spent discussing how to display large, complicated data sets. That was a major theme in the earlier volumes, and still plays a role in the most recent one. But there is another theme of interest to a wider range of mathematicians. It started off modestly earlier in the series, but has come to be more important in the most recent volume—how to use illustrations to explain complicated, even abstract, ideas effectively. For a mathematician, the most
THINKING LIKE ARCHIMEDES WITH A 3D PRINTER
"... Abstract. We illustrate Archimedes ’ method using models produced with 3D printers. This approach allowed us to create physical proofs of results known to Archimedes and to illustrate ideas of a mathematician who is known both for his mechanical inventions as well as his breakthroughs in geometry an ..."
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Abstract. We illustrate Archimedes ’ method using models produced with 3D printers. This approach allowed us to create physical proofs of results known to Archimedes and to illustrate ideas of a mathematician who is known both for his mechanical inventions as well as his breakthroughs in geometry and new ideas leading to calculus. We use technology from the 21st century to trace intellectual achievements from the 3rd century BC. While we celebrate 2300 years of Archimedes (287212 BC) in 2013, we also live in an exciting time, where 3D printing is becoming popular and affordable. 1.
Rethinking geometrical exactness
 Historia Mathematica
"... A crucial concern of earlymodern geometry was that of fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed in it. According to Bos, this is the exactness concern. I argue that Descartes ’ way to respond to this concern was to suggest a ..."
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A crucial concern of earlymodern geometry was that of fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed in it. According to Bos, this is the exactness concern. I argue that Descartes ’ way to respond to this concern was to suggest an appropriate conservative extension of Euclid’s plane geometry (EPG). In section 1, I outline the exactness concern as, I think, it appeared to Descartes. In section 2, I account for Descartes ’ views on exactness and for his attitude towards the most common sorts of constructions in classical geometry. I also explain in which sense his geometry can be conceived as a conservative extension of EPG. I conclude by briefly discussing some structural similarities and differences between Descartes’ geometry and EPG. Une question cruciale pour la geométrie à l’âge classique fut celle de décider si certains objets, procédures ou arguments devaient ou non être admis au sein de ses limites. Selon Bos, c’est la question de l’exactitude. J’avance que Descartes répondit à cette question en suggérant une extension conservative de la géomètrie plane d’Euclide (EPG). Dans la section 1, je reconstruis la question de l’exactitude ainsi que, selon moi, elle se présentait d’abord aux yeux de Descartes. Dans la section 2, je rends compte des vues de Descartes sur la question de l’exactitude et de son attitude face au types de constructions plus communes dans la geométrie classique. Je montre aussi en quel sens sa geométrie peut se concevoir comme une extension conservative de EPG. Je conclue en discutant brièvement certaines analogies et différences structurales entre la geométrie de Desacrtes et EPG.
Mathematical Models and Reality: a constructivist perspective
 Foundations of Science
, 2010
"... Abstract: To explore the relation between mathematical models and reality, four different levels of reality are distinguished: observerindependent reality (to which there is no direct access), personal reality, social reality and mathematical/formal reality. The concepts of personal and social rea ..."
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Abstract: To explore the relation between mathematical models and reality, four different levels of reality are distinguished: observerindependent reality (to which there is no direct access), personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement about “truth”, but the assignment of mathematics to informal reality is not formally analyzable, and it is dependent on social and personal construction processes. On these levels, absolute agreement cannot be expected. Starting from this point of view, repercussion of mathematical on social and personal reality, the historical development of mathematical modelling, and the role, use and interpretation of mathematical models in scientific practice are discussed. 1
The Bridge Between The Continuous And The Discrete Via Original Sources
"... this paper we summarize the story told through original sources from our chapter on the relationship between the continuous and the discrete, hinging historically on two interlocking themes: the search for formulas for sums of numerical powers, and Euler's development of his summation formula i ..."
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this paper we summarize the story told through original sources from our chapter on the relationship between the continuous and the discrete, hinging historically on two interlocking themes: the search for formulas for sums of numerical powers, and Euler's development of his summation formula in relation to sums of infinite series
The ChurchTuring Thesis as an Immature Form of the ZuseFredkin Thesis (More Arguments in Support of the “Universe as a Cellular Automaton” Idea)
"... In [1] we have shown a strong argument in support of the "Universe as a computer " idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only "some kind of computer", but also a concrete computational ..."
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In [1] we have shown a strong argument in support of the &quot;Universe as a computer &quot; idea. In the current work, we continue our exposition by showing more arguments that reveal why our Universe is not only &quot;some kind of computer&quot;, but also a concrete computational model known as a &quot;cellular automaton&quot;.
Dances between continuous and discrete: Euler's summation formula
, 2002
"... this paper. 9 Large binomials In our last excerpt, Euler applies the summation formula to estimate the size of large binomial coe#cients. I translate just one of his methods here, in which he coalesces two summation series term by term. As a sample application, Euler approximates the ratio 50 ..."
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this paper. 9 Large binomials In our last excerpt, Euler applies the summation formula to estimate the size of large binomial coe#cients. I translate just one of his methods here, in which he coalesces two summation series term by term. As a sample application, Euler approximates the ratio 50 /2 , despite the huge size of its parts, thus closely approximating the probability that if one tosses 100 coins, exactly equal numbers will land heads and tails