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.
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.
Closely related to the problem of the parallel postulate is the problem of whether physical space is infinite. Euclid assumes in Postulate 2 that a straight-line segment can be extended as far as necessary; he uses this fact, but only to find a larger finite length for example in Book I, Propositions 11, 16, and 20. For these proofs Heron gave new proofs that avoided extending the lines, in order to meet the objection of anyone who would deny that the space was available for the extension.
Propositions show what they say: tautologies and contradictions show that they say nothing.
The central Propositions [of Descartes]?are these: There is a path that leads to the truth so surely that any one who will follow it must needs reach the goal? And there is one guiding rule by which a man mayalways find this path?give unqualified assent to no Propositions but those the truth of which is so clear and distinct that they cannot be doubted.
If an angel were ever to tell us anything of his philosophy I believe many Propositions would sound like 2 times 2 equals13.Georg Christoph Lichtenberg
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
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 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.carl friedrich gauss
I was trying to show colour, but I realized at the private view that the public were prisoners of a preconceived point of view and that, confronted with all these surfaces of different colours, they responded far more to the inter-relationship of the different Propositions, they reconstituted the elements of a decorative polychromy.yves klein
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
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
The progressive historical role of capitalism may be summed up in two brief Propositions: increase in the productive forces of social labour, and the socialisation of that labour. But both these facts manifest themselves in extremely diverse processes in different branches of the national economy.
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
There are numerous theorems in economics that rely upon mathematically fallacious Propositions.
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.
Intuitive cognition is such that when some things are cognized, of which one inheres in the other, or one is spatially distant from the other, or exists in some relation to the other, immediately in virtue of that non-propositional cognition of those things, it is known if the thing inheres or does not inhere, if it is spatially distant or not, and the same for other true contingent Propositions, unless that cognition is flawed or there is some impediment.
It is as easy to count atomies as to resolve the Propositions of a lover.
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.