By Mark Kac

This is a quantity of essays and stories that delightfully discover arithmetic in all its moods — from the sunshine and the witty, and funny to critical, rational, and cerebral. subject matters comprise: common sense, combinatorics, data, economics, man made intelligence, laptop technology, and purposes of arithmetic largely. additionally, you will locate background and philosophy lined, together with dialogue of the paintings of Ulam, Kant, Heidegger between others.

"...these papers consider arithmetic and its effect on human society. they could aid the expert to note what's going round him, they usually could lead informed humans from different domain names to a greater realizing of arithmetic. lots of those papers can suggest educators the right way to shape a contemporary arithmetic schooling, which develops licensed principles and institutions...I respect the stimulating views of the authors."---American Mathematical Society

"‘Mathematicians, like Proust and everybody else, are at their top while writing approximately their old flame’ … they're one of the best possible we've; and their most sensible is superb certainly. … One ways this e-book with excessive hopes. fortunately, one isn't disappointed."--- From The Mathematical Intelligencer

"Mathematics is formed by way of the constant matters and sorts of strong minds —three of that are represented right here. Kac’s paintings is marked through deep dedication and breadth of inquiry. Rota is the best of those authors to read…a satisfaction: witty and urbane, with a transparent and fascinating schedule and an remarkable highbrow variety. To learn him is to be part of a delightful and profitable conversation.—Mathematics Magazine

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**Additional resources for Discrete Thoughts: Essays on Mathematics, Science, and Philosophy**

**Example text**

A n y w a y , there is a sequel to the above story. M a n y years later, I was consulting with an outfit involved with the space p r o g r a m , in particular with automatization of signal detection. T h e p r o b l e m was to pre-teach an a u t o m a t o n so that it can learn to detect signals which it receives when floating a r o u n d in outer space. In this connection it occurred to me to try to see w h e t h e r an a u t o m a t o n exposed to the p r o b l e m of signal detection could discover for itself the N e y m a n - P e a r s o n theory empirically.

Here we may as well imitate our arch rival, the Soviet Union, where mathematical education is almost the same as it was in the Czarist time. They somehow managed to beat us with the sputnik without any New Math and they manage to maintain mathematics on the highest level (certainly comparable to ours) while teaching it with due respect to all of its aspects, as well as to its intimate ties with sister disciplines, notably physics. If in a society so rigidly controlled by an inhumane and pragmatic regime mathematics can flourish, seemingly, at least, free from its internal tensions, then how much better we should be able to do in a free society, provided only that we can find the will.

But, in other cases, the mathematician's habit of making definite his literal-mindedness may have more unfortunate consequences. , convenient points of analytical emphasis, into axioms, and then takes these axioms literally. This brings with it the danger that he may also persuade the scientist to take these axioms literally. The question, central to the scientific investigation but intensely disturbing in the mathematical c o n t e x t - w h a t happens to all this if the axioms are r e l a x e d ?