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.

— As quoted in "A Paper of Omar Khayyam" by A.R. Amir-Moez in

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.

— A reply to Olbers' 1816 attempt to entice him to work on Fermat's Theorem. As quoted in

Science
[is] knowledge of the truth of **Propositions** and how things are
called.

— 1650 Human Nature, ch.6.

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.

— Morris Kline,

Literature is based not on life but on **Propositions** about life, of which this
is one.

— 1957 Opus Posthumous, Aphorisms,'Adagia'.

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.

— p. 20 (Language, Truth, and Logic (1936))

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.

— p. 77 (Language, Truth, and Logic (1936))

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 cabinet has no **Propositions** to make, but orders to give.

—

— in

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.

— p. 146 ("The Latest Attack on Metaphysics" (1937))

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.

— p. 149 ("The Latest Attack on Metaphysics" (1937))

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.

—

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.

— Preface, The Sacredness Of Criticism

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.

— "Men Made Out of Words"

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.

— "Systems of Logic Based on Ordinals," section 11: The purpose of ordinal logics (1938), published in

In a footnote to the first sentence, Turing added: "We are leaving out of account that most important faculty which distinguishes topics of interest from others; in fact, we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions."

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.

— p. 259

Variant : It is more important that a proposition be interesting than that it be true. This statement is almost a tautology. For the energy of operation of a proposition in an occassion of experience is its interest, and its importance. But of course a true proposition is more apt to be interesting than a false one.As extended upon in

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.

— Language as Conspiracy, p. 277

There are numerous theorems in economics that rely upon mathematically fallacious **Propositions**.

— Chapter 12, Don't Shoot Me, I'm Only The Piano, p. 259

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 On higher arithmetic.

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.

— T. L. Heath,

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.

— Florian Cajori,