Download Doing Mathematics : Convention, Subject, Calculation, by Martin H Krieger PDF

By Martin H Krieger

Doing Mathematics discusses many ways mathematicians and mathematical physicists do their paintings and the topic concerns they discover and model. The conventions they undertake, the topic parts they delimit, what they could end up and calculate in regards to the actual international, and the analogies they detect and hire, all depend upon the math — what is going to figure out and what will not. The instances studied comprise the vital restrict theorem of data, the sound of the form of a drum, the connections among algebra and topology, and the sequence of rigorous proofs of the soundness of subject. the various and sundry suggestions to the two-dimensional Ising version of ferromagnetism make feel as a complete once they are visible in an analogy constructed by means of Richard Dedekind within the Eighteen Eighties to algebraicize Riemann's functionality conception; via Robert Langlands' software in quantity conception and illustration idea; and, through the analogy among one-dimensional quantum mechanics and two-dimensional classical statistical mechanics. In impact, we start to determine "an identification in a manifold presentation of profiles," because the phenomenologists might say.

This moment variation deepens the actual examples; it describe the sensible function of mathematical rigor; it indicates what can be a mathematician's philosophy of arithmetic; and, it indicates how an "ugly" first facts or derivation embodies crucial beneficial properties, basically to be preferred after many next proofs. usual scientists and mathematicians exchange actual versions and summary items, remaking them to fit their wishes, studying new roles for them as within the fresh case of the Painleve transcendents, the Tracy-Widom distribution, and Toeplitz determinants. and arithmetic has supplied the types and analogies, the normal language, for describing the standard global, the constitution of towns, or God's infinitude.

Readership: Mathematicians, physicists, philosophers and historians of technology.

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There are both mathematical and cultural accounts of the convention. The overlap and ramification of those accounts makes a convention appear all but inevitable, just the way the world is. And, in time, in the matter of course, a new convention might arise, and it, too, will be inevitable. How is that air of inevitability maintained? It might be useful to think of ourselves as anthropologists trying to understand the practices of a culture—here, a culture of randomness as practiced in probability, statistics, and decision theory.

Larry Yangster has become a physicist doing statistical mechanics. When Bernie asks Larry to tell him about his current work, Larry mentions that he is working on a statistical counting problem that has three main methods of calculating its partition function, the physicists' way of packaging counting information about a system. The natural logarithm of the partition function is proportional to the thermodynamic free energy that the chemists so value. Bernie remarks that it is amazing that the packaging function could be so related to something a chemist would be interested in.

Now if one were to take, say, Fredrick Mosteller and John Tukey’s 1977 text on statistics as the current gospel, then the alternatives hinted at here and by Huber are in fact fully in place. ”13 Means and variances and Gaussians are seen to be an idiosyncratic and parochial case. 15 Moreover, probability distributions much broader than the Gaussian do play an important role in the natural and the economic sciences. And computation allows for empirical estimates of variation (bootstrap, resampling).

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