The Eudemian Summary says that Eudoxus "first increased the number of general theorems, added to the three proportions three more, and raised to a considerable quantity the learning, begun by Plato, on the subject of the section, to which he applied the analytical method." By this 'section' is meant, no doubt, the "golden section" ( sectio aurea ), which cuts a line in extreme and mean ratio. The first five propositions in Euclid XIII. relate to lines cut by this section, and are generally attributed to Eudoxus.
Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras (jabbre and maqabeleh) are geometric facts which are proved by Propositions five and six of Book two of Elements.omar khayyám
Science [is] knowledge of the truth of Propositions and how things are called.thomas hobbes
Literature is based not on life but on Propositions about life, of which this is one.wallace stevens
To make our position clearer, we may formulate it in another way. Let us call a proposition which records an actual or possible observation an experiential proposition. Then we may say that it is the mark of a genuine factual proposition, not that it should be equivalent to an experiential proposition, or any finite number of experiential Propositions, but simply that some experiential Propositions can be deduced from it in conjunction with certain other premises without being deducible from those other premises alone.alfred jules ayer
The principles of logic and mathematics are true simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves, without sinning against the rules which govern the use of language, and so making our utterances self-stultifying. In other words, the truths of logic and mathematics are analytic Propositions or tautologies.alfred jules ayer
No doubt, there are those who believe that judges-and particularly dissenting judges-write to hear themselves say, as it were, I I I. And no doubt, there are also those who believe that judges are, like Joan Didion, primarily engaged in the writing of fiction. I cannot agree with either of those Propositions.brennan, william j., jr.
The cabinet has no Propositions to make, but orders to give.charles de gaulle
I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such Propositions, which one could neither prove nor dispose of.carl friedrich gauss
A theorem is a proposition which is a strict logical consequence of certain definitions and other PropositionsAnatol Rapoport
LOGIC :; The principle governing human intellection. Its nature may be deduced from examining the following Propositions, both of which are held by human beings to be true and often by the same people: “I can’t so you musn’t,” and “I can but you musn’t.”john brunner
Notwithstanding their attacks on the basic conception of rationalism, on synthetic a priori judgments, that is, material Propositions that cannot be contradicted by any experience, the empiricist posits the forms of being as constant.max horkheimer
Leibniz’s theory on the subject as substantia ideans in the sense of a causative agent of decision and acts stands much closer to a materialist interpretation of history than does a philosophy which reduces the thinking subject to the role of subsuming protocol sentences under general Propositions and deducing other sentences from them.max horkheimer
Wherever Mathematics is mixed up with anything, which is outside its field, you will find attempts to demonstrate these merely conventional Propositions a priori, and it will be your task to find out the false deduction in each case.carl gustav jacob jacobi
In obedience to the feeling of reality, we shall insist that, in the analysis of Propositions, nothing "unreal" is to be admitted. But, after all, if there is nothing unreal, how, it may be asked, could we admit anything unreal? The reply is that, in dealing with Propositions, we are dealing in the first instance with symbols, and if we attribute significance to groups of symbols which have no significance, we shall fall into the error of admitting unrealities, in the only sense in which this is possible, namely, as objects described.bertrand russell
The last word remains with Christ and Handel ; and this must stand as the best defence of Tolerance until a better man than I makes a better job of it. Put shortly and undramatically the case is that a civilization cannot progress without criticism, and must therefore, to save itself from stagnation and putrefaction, declare impunity for criticism. This means impunity not only for Propositions which, however novel, seem interesting, statesmanlike, and respectable, but for Propositions that shock the uncritical as obscene, seditious, blasphemous, heretical, and revolutionary.George Bernard Shaw
Life consists Of Propositions about life. The human Revery is a solitude in which We compose these Propositions, torn by dreams, By the terrible incantations of defeats And by the fear that the defeats and the dreams are one. The whole race is a poet that writes down The eccentric Propositions of its fate.wallace stevens
Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity . The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of Propositions, and perhaps geometrical figures or drawings.alan turing
Some philosophers fail to distinguish Propositions from judgements; ... But in the real world it is more important that a proposition be interesting than that it be true. The importance of truth is that it adds to interest.Alfred North Whitehead
You need the "is of identity" to describe conspiracy theories. Korzybski would say that proves that illusions, delusions, and "mental" illnesses require the "is" to perpetuate them. (He often said, "Isness is an illness.") Korzybski also popularized the idea that most sentences, especially the sentences that people quarrel over or even go to war over, do not rank as Propositions in the logical sense, but belong to the category that Bertrand Russell called propositional functions. They do not have one meaning , as a proposition in logic should have; they have several meanings, like an algebraic function.robert anton wilson
There are numerous theorems in economics that rely upon mathematically fallacious Propositions.
A great part of its theories derives an additional charm from the peculiarity that important Propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.
The following are the other extant works generally attributed to Euclid: Phœnomena , a work on spherical geometry and astronomy; Optics , which develops the hypothesis that light proceeds from the eye, and not from the object seen; Catoptrica , containing Propositions on reflections from mirrors; De Divisionibus , a treatise on the division of plane figures into parts having to one another a given ratio; Sectio Canonis , a work on musical intervals.
No doubt, there are those who believe that judges-and particularly dissenting judges-write to hear themselves say, as it were, I I I. And no doubt, there are also those who believe that judges are, like Joan Didion, primarily engaged in the writing of fiction. I cannot agree with either of those Propositions.
The treatises are, without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the Propositions, the stern elimination of everything not immediately relevant to the purpose , the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
Comparatively few of the Propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him.