Theorem Quotes - page 2
I know well that many of my readers do not think as I do. This also is most natural and confirms the theorem. For although my opinion turn out erroneous, there will always remain the fact that many of those dissentient readers have never given five minutes' thought to this complex matter. How are they going to think as I do? But by believing that they have a right to an opinion on the matter without previous effort to work one out for themselves, they prove patently that they belong to that absurd type of human being which I have called the "rebel mass." It is precisely what I mean by having one's soul obliterated, hermetically closed. Here it would be the special case of intellectual hermetism.
José Ortega y Gasset
It has been generally believed that only the complex numbers could legitimately be used as the ground field in discussing quantum-mechanical operators. Over the complex field, Frobenius' theorem is of course not valid; the only division algebra over the complex field is formed by the complex numbers themselves. However, Frobenius' theorem is relevant precisely because the appropriate ground field for much of quantum mechanics is real rather than complex.
Freeman Dyson
Gödel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.
Freeman Dyson
One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics.
Pierre Louis Maupertuis
Growing up in Gary Indiana gave me, I think, a distinct advantage over many of my classmates who had grown up in affluent suburbs. They could read articles that argued that in competitive equilibrium, there could not be discrimination, so long as there are some non-discriminatory individuals or firms, since it would pay any such firm to hire the lower wage discriminated-against individuals, and take them seriously. I knew that discrimination existed, even though there were many individuals who were not prejudiced. To me, the theorem simply proved that one or more of the assumptions that went into the theory was wrong.
Joseph Stiglitz
There is little doubt that the observation that quality may depend on price (productivity on wages; default probability on interest rates) has provided a rich mine for economic theorists: A simple modification of the basic assumptions results in a profound alteration of many of the basic conclusions of the standard paradigm. The Law of Supply and Demand has been repealed. The Law of the Single Price has been repealed. The Fundamental Theorem of Welfare Economics has been shown not to be valid.
More than that, the theories that we describe here provide the basis of progress toward a unification of macroeconomics and microeconomics. They pro vide an explanation of unemployment and credit rationing, derived from basic microeconomic principles. It is a theory in which the extensive idleness that periodically confronts society's resources, human and capital, is seen as but the most obvious example of market failures that prevasively and persistently distort the allocation of resources.
Joseph Stiglitz
My interest in economics had the following roots. Like many of my generation I considered the heavy unemployment in the United Kingdom in the inter-war period as both stupid and wicked. Moreover, I knew the cure for this evil, because I had become a disciple of the monetary crank, Major C. H. Douglas, to whose works I had been introduced by a much loved but somewhat eccentric maiden aunt. But my shift to the serious study of economics gradually weakened my belief in Major Douglas's A+B theorem, which was replaced in my thought by the expression MV = PT.
James Meade
We completely ignore the human value of the information. A selection of 100 letters is given a certain information value, and we do not investigate whether it makes sense in English, and, if so, whether the meaning of the sentence is of any practical importance. According to our definition, a set of 100 letters selected at random (according to the rules of Table 1.1), a sentence of 100 letters from a newspaper, a piece of Shakespeare or a theorem of Einstein are given exactly the same informational value.
Léon Brillouin
There are two general Methods made use of in the Mathematicks, viz. Synthesis and Analysis, which we shall explain, after having acquainted the Reader, that the Method we make use of to resolve a Mathematical Problem, is called Zetetick; and that that Method which determines when, and by what way, and how many different ways a Problem may be resolved, is called Poristick. But in treating of Methods, we will first premise, that in general, a Method is the Art of disposing a Train of Arguments or Consequences in a right Order, either to discover the Truth of a Theorem, which we would find out, or to demonstrate it to others, when found.
Jacques Ozanam
The definition of random in terms of a physical operation is notoriously without effect on the mathematical operations of statistical theory because so far as these mathematical operations are concerned random is purely and simply an undefined term. The formal and abstract mathematical theory has an independent and sometimes lonely existence of its own. But when an undefined mathematical term such as random is given a definite operational meaning in physical terms, it takes on empirical and practical significance. Every mathematical theorem involving this mathematically undefined concept can then be given the following predictive form: If you do so and so, then such and such will happen.
Walter A. Shewhart
The construction of hypotheses is a creative act of inspiration, intuition, invention; its essence is the vision of something new in familiar material. The process must be discussed in psychological, not logical, categories; studied in autobiographies and biographies, not treatises on scientific method; and promoted by maxim and example, not syllogism or theorem.
Milton Friedman
My book is written, as befits such easy material, in merciless telegram style ("Axiom," "Definition," "Theorem," "Proof," occasionally "Preliminary Remark")... I hope I have written this book in such a way that a normal student can read it in two days. And then (since he already knows the formal rules from school) he may forget its contents.
Edmund Landau