The physicist needs a facility in looking at problems from several points of view. The exact analysis of real physical problems is usually quite complicated, and any particular physical situation may be too complicated to analyze directly by solving the differential equation. But one can still get a very good idea of the behavior of a system if one has some feel for the character of the solution in different circumstances. Ideas such as the field lines, capacitance, resistance, and inductance are, for such purposes, very useful. ... On the other hand, none of the heuristic models, such as field lines, is really adequate and accurate for all situations. There is only one precise way of presenting the laws, and that is by means of differential equations. They have the advantage of being fundamental and, so far as we know, precise. If you have learned the differential equations you can always go back to them. There is nothing to unlearn.
Richard FeynmanRelated topics
advantage character different differential equation facility far field fundamental good heuristic idea looking needs none nothing physicist quite real situation solving view way means circumstances Laws pointsRelated quotes
One can conclude that certain essential, or fundamental, rights should exist in any just society. It does not follow that each of those essential rights is one that we as judges can enforce under the written Constitution. The Due Process Clause is not a guarantee of every right that should inhere in an ideal system. Many argue that a just society grants a right to engage in homosexual conduct. If that view is accepted, the Bowers decision in effect says the State of Georgia has the right to make a wrong decision - wrong in the sense that it violates some people's views of rights in a just society. We can extend that slightly to say that Georgia's right to be wrong in matters not specifically controlled by the Constitution is a necessary component of its own political processes. Its citizens have the political liberty to direct the governmental process to make decisions that might be wrong in the ideal sense, subject to correction in the ordinary political process.
Anthony Kennedy
I say that things are useful whenever they can be put to any use at all; whenever they are seen to be capable of satisfying a want. In this connection, there is no need to consider the subtle shades of meaning classified in ordinary language under terms ranging from the necessary to the useful, from the useful to the agreeable, from the agreeable to the superfluous. For present purposes, necessary, useful, agreeable and superfluous simply mean more or less useful. Furthermore, we need not concern ourselves with the morality or immorality of any desire which a useful thing answers or serves to satisfy. From other points of view the question of whether a drug is wanted by a doctor to cure a patient, or by a murderer to kill his family is a very serious matter, but from our point of view, it is totally irrelevant. So far as we are concerned, the drug is useful in both cases, and may even be more so in the latter case than in the former.
Léon Walras
I would summarize the Keynesian view in terms of four points:1. Economies sometimes produce much less than they could, and employ many fewer workers than they should, because there just isn't enough spending. Such episodes can happen for a variety of reasons; the question is how to respond.
2. There are normally forces that tend to push the economy back toward full employment. But they work slowly; a hands-off policy toward depressed economies means accepting a long, unnecessary period of pain.
3. It is often possible to drastically shorten this period of pain and greatly reduce the human and financial losses by "printing money”, using the central bank's power of currency creation to push interest rates down.
4. Sometimes, however, monetary policy loses its effectiveness, especially when rates are close to zero. In that case temporary deficit spending can provide a useful boost. And conversely, fiscal austerity in a depressed economy imposes large economic losses.
Paul Krugman
Had it merely called to our attention the existence and exact nature of certain fundamental gaps in economic theory, the Theory of Economic Behavior by von Neumann and Morgenstern would have been a book of outstanding importance. But it does more than that. It is essentially constructive: where existing theory is considered to be inadequate, the authors put in its place a highly novel analytical apparatus designed to cope with the problem.
It would be doing the authors an injustice to say that theirs is a contribution to economics only. The scope of the book is much broader. The techniques applied by the authors in tackling economic problems are of sufficient generality to be valid in political science, sociology, or even military strategy. The applicability to games proper (chess and poker) is obvious from the title. Moreover, the book is of considerable interest from a purely mathematical point of view.
Leonid Hurwicz
In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius.
Carl Friedrich Gauss