Theorem Quotes - page 3
![First then I will set out the very first theorem which became known to me by means of mechanics, namely that
Any segment of a section of a right angled cone (i.e., a parabola) is four-thirds of the triangle which has the same base and equal height,
and after this I will give each of the other theorems investigated by the same method. Then at the end of the book I will give the geometrical [proofs of the propositions]... (Archimedes)](https://cdn.quotesdtb.com/img/quotes_images_webp/24/archimedes-base-book-cone-421524.webp)
A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
Lotfi A. Zadeh
As far as the use of mathematics in economics is concerned, there is an abundance of formulas where such are not needed. They are frequently introduced, one fears, in order to show off. The more difficult the mathematical theorem, the more esoteric the name of the mathematician quoted, the better.
Oskar Morgenstern
Descartes... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish.... historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do.
Morris Kline
The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858.
Richard Dedekind
Calculation establishes that a quantum positive energy theorem may be possible using a representation based on the Ashtekar connection. Left open is a key question of whether this use of the Ashtekar connection is necessary or whether a positive quantum energy result can be achieved for representations based on other connections, i. e., for [other] values of the Immirzi parameter.
Lee Smolin
I took a break from acting for four years to get a degree in mathematics at UCLA, and during that time I had the rare opportunity to actually do research as an undergraduate. And myself and two other people co-authored a new theorem: Percolation and Gibbs States Multiplicity for Ferromagnetic Ashkin-Teller Models on Two Dimensions, or Z2.
Danica McKellar
Theorem II. Any feel which feeling has not informed me of, is unknown to me. Comments. 1. Words are sensibly intelligent to a man of only such words as he has experienced. 2. The intellectual signification of words discriminated from the sensible signification. 3. Intellectual intimations discriminated from sensible revelations.
Alexander Bryan Johnson
Arrow's Impossibility Theorem is quite surprising. It shows that three very plausible and desirable features of a social decision mechanism are inconsistent with democracy: there is no "perfect” way to make social decisions. There is no perfect way to "aggregate” individual preferences to make one social preference. If we want to find a way to aggregate individual preferences to form social preferences, we will have to give up one of the properties of a social decision mechanism described in Arrow's theorem.
Hal Varian
But not in my family. My mother and father (with immigrant pragmatism) assumed the American tongue would reinvent their children. Just so did several immigrant Hispanic mothers in Southern California recently remark their children's reluctance to join America. These mothers feared their children were not swimming in the American current-not in the swifts and not in the depths; not even in the pop. They blamed "bilingual education,” a leaky boat theorem ostensibly designed to sink into the American current. (In fact, the theorem became a bureacracy preoccupied with prolonging itself.) These few mothers organized an opposition to bilingual education and eventually they sank the Armada in California. Theirs was an American impulse: to engage the American flow directly and to let their children be taken by it.
Richard Rodriguez