The urn model is to be the expression of three postulates: (1) the constancy of a probability distribution, ensured by the solidity of the vessel, (2) the randomcharacter of the choice, ensured by the narrowness of the mouth, which is to prevent visibility of the contents and any consciously selective choice, (3) the independence of successive choices, whenever the drawn balls are put back into the urn. Of course in abstract probability and statistics the word " choice " can be avoided and all can be done without any reference to such a model. But as soon as the abstract theory is to be applied, random choice plays an essential role.
Hans FreudenthalRelated topics
abstract choice constancy contents course distribution done draw essential expression independence model mouth narrowness probability random reference role solidity statistics theory three urn vessel visibility word ballsRelated quotes
I have presented the periodic table as a kind of travel guide to an imaginary country, of which the elements are the various regions. This kingdom has a geography: the elements lie in particular juxtaposition to one another, and they are used to produce goods, much as a prairie produces wheat and a lake produces fish. It also has a history. Indeed, it has three kinds of history: the elements were discovered much as the lands of the world were discovered; the kingdom was mapped, just as the world was mapped, and the relative positions of the elements came to take on a great significance; and the elements have their own cosmic history, which can be traced back to the stars.
Peter Atkins
The currently (2010) widely accepted/believed description of the birth and evolution of the universe and of its contents is "Lambda Cold Dark Matter Concordance Cosmological Model" (LCDM CCM) ... My own research was very much confined to the early version of the LCDM CCM (mid-1990's) when I began performing numerical experiments on the satellite galaxies of the Milky Way. I was quite happy with the CCM, as everyone else, and did not bother with the fundamental issues raised by some. With time, however, it became apparent that the LCDM CCM accounts poorly for the properties of the satellite galaxies and their distribution about the Milky Way. Warm dark matter models fared no better.
Pavel Kroupa
In this book I try to show how models of self-organization can be applied to many economic phenomena - how the principle of "order from instability," which explains the growth of hurricanes and embryos, can also explain the formation of cities and business cycles; how the principle of "order from random growth" can explain the strangely simple rules that describe the sizes of earth quakes, meteorites, and metropolitan areas. I believe that the ideas of self-organization theory can add substantially to our understanding of the economy; whatever their ultimate usefulness, these ideas are very exciting, and playing around with them is tremendous fun.
Paul Krugman
Not only is it possible to devise complete models of the economy on hypotheses other than rationality, but in fact virtually every practical theory of macroeconomics is partly so based. The price- and wage- rigidity elements of Keynesian theory are hard to fit into a rational framework, though some valiant efforts have been made. ... But if the Keynesian model is a natural target of criticism by the upholders of universal rationality, it must be added that monetarism is no better. I know of no serious derivation of the demand for money from a rational optimization. ... The use of rationality in these arguments is ritualistic, not essential.
Kenneth Arrow
In relativity, movement is continuous, causally determinate and well defined, while in quantum mechanics it is discontinuous, not causally determinate and not well defined. Each theory is committed to its own notions of essentially static and fragmentary modes of existence (relativity to that of separate events, connectable by signals, and quantum mechanics to a well-defined quantum state). One thus sees that a new kind of theory is needed which drops these basic commitments and at most recovers some essential features of the older theories as abstract forms derived from a deeper reality in which what prevails in unbroken wholeness.
David Bohm
