Though our own hearts break, we cannot flinch; these are new times, sir. There is a misty plot afoot so subtle we should be criminal to cling to old respect and ancient friendships. I have seen too many frightful **proofs** in court the Devil is alive in Salem, and we dare not quail to follow wherever the accusing finger points!

—

Would you convey my compliments to the purist who reads your **proofs**
and tell him or her that I write in a sort of
broken-downpatoiswhichissomething likethewaya Swiss waiter talks,
and that when I split an infinitive,God damn it, I split it
so it will stay split.

— 1947 Letter to Edward Weeks at Atlantic Monthly,18 Jan.

As Eisenstein shows, his method for constructing elliptic functions applies beautifully to the simpler case of trigonometric functions. Moreover, this case provides not merely an illuminating introduction to his theory, but also the simplest **proofs** for a series of results, originally discussed by Euler.

— André Weil in

The establishment of formal standards for **proofs** about programs [...] and the proposal that the semantics of a programming language may be defined independently of all processors for that language, by establishing standards of rigor for **proofs** about programs in the language, appears to be novel.

— pp. 19–20 (Assigning Meanings to Programs (1967))

I HAVE no patience with the hypothesis occasionally expressed, and often implied, especially in tales written to teach children to be good, that babies are born pretty much alike, and that the sole agencies in creating differences between boy and boy, and man and man, are steady application and moral effort. It is in the most unqualified manner that I object to pretensions of natural equality. The experiences of the nursery, the school, the University, and of professional careers, are a chain of **proofs** to the contrary.

—

I mean the word proof not in the sense of the lawyers, who set two half **proofs** equal to a whole one, but in the sense of a mathematician, where ½ proof = 0, and it is demanded for proof that every doubt becomes impossible.

— In a letter to Heinrich Wilhelm Matthias Olbers (14 May 1826), defending Chevalier d'Angos against presumption of guilt (by Johann Franz Encke and others), of having falsely claimed to have discovered a comet in 1784; as quoted in

It is not particularly satisfactory to see equations set forth as direct results of observation and experiment, where we used to get long mathematical deductions as apparent **proofs** of them. Nevertheless, I believe that we cannot, without deceiving ourselves, extract much more from known facts than is asserted in the papers referred to. If we wish to lend more color to the theory, there is nothing to prevent us from supplementing all this and aiding our powers of imagination by concrete representations of the various conceptions as to the nature of electric polarisation, the electric current, etc.

—

For we may remark generally of our mathematical researches, that these auxiliary quantities, these long and difficult calculations into which we are often drawn, are almost always **proofs** that we have not in the beginning considered the objects themselves so thoroughly and directly as their nature requires, since all is abridged and simplified, as soon as we place ourselves in a right point of view.

— Louis Poinsot, translated by Charles Thomas Whitley (1834).

It has been observed before that images, however beautiful, though faithfully copied from nature, and as accurately represented in words, do not of themselves characterize the poet. They become **proofs** of original genius only as far as they are modified by a predominant passion; or by associated thoughts or images awakened by that passion; or when they have the effect of reducing multitude to unity, or succession to an instant; or lastly, when a human and intellectual life is transferred to them from the poet's spirit.

— Ch. XV (Biographia Literaria (1817))

No body wishes more than I do to see such **proofs** as you exhibit, that nature has given to our black brethren, talents equal to those of the other colors of men, and that the appearance of a want of them is owing merely to the degraded condition of their existence, both in Africa & America.

— Letter to Benjamin Banneker (August 30, 1791).

I see too many **proofs** of the imperfection of human reason, to entertain wonder or intolerance at any difference of opinion on any subject; and acquiesce in that difference as easily as on a difference of feature or form; experience having long taught me the reasonableness of mutual sacrifices of opinion among those who are to act together for any common object, and the expediency of doing what good we can, when we cannot do all we would wish.

— Letter to John Randolph, (1 December 1803), published in

With respect to the words "general welfare," I have always regarded them as qualified by the detail of powers connected with them. To take them in a literal and unlimited sense would be a metamorphosis of the Constitution into a character which there is a host of **proofs** was not contemplated by its creators.

— Letter to James Robertson (April 20, 1831)

The cookbook gives a detailed description of ingredients and procedures but no **proofs** for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. ... Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information.

— p. 219 (How to Solve It (1945))

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,

Algorithms are the computational content of **proofs**.

— Robert Harper, Benjamin C. Pierce et al.

If I shall be condemn'd Upon surmises, all **proofs** sleeping else But what your jealousies awake, I tell you, 'Tis rigour, and not law.

— William Shakespeare,

The belief in a certain idea gives to the researcher the support for his work. Without this belief he would be lost in a sea of doubts and insufficiently verified **proofs**.

— Konrad Zuse (2008) in Konrad Zuse on "Die Erfindergalerie", dpma.de, 2008

Perhaps there is no literature in Europe that mirrors so clearly as the Portuguese, the painful conflict in the minds of people who, on the one hand, by their humanistic education, not only knew better but also more uncritically admired, ancient learning than their medieval predecessors, and, who, on the other hand, in the same epoch, were confronted with abundant **proofs** of the insufficiency and fallibility of that same Antiquity.

— Reijer Hooykaas, "The Portuguese Discoveries and the Rise of Modern Science" in

It is worthy of notice that Apollonius nowhere introduces the notion of directrix for a conic, and that, though he incidentally discovered the focus of an ellipse and hyperbola, he did not discover the focus of a parabola. Conspicuous in his geometry is also the absence of technical terms and symbols, which renders the **proofs** long and cumbrous.

—

... that what is proved, by impossibility **proofs**, is lack of imagination.

— John Stewart Bell, On the imposible pilot wave, Ref.TH.3315-CERN, 1982, p. 15

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,

To those who have an active belief, reasoned **proofs** are needless and probably useless.

— Book IV, Chapter 17 (From St. Athanasius' Life of St. Antony)

A mathematician is a person who can find analogies between theorems ; a better mathematician is one who can see analogies between **proofs** and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.

— Stefan Banach, in Banach Spaces and Their Applications in Analysis: Proceedings of the ... (2007), p.5

Ever since Plato most philosophers have considered it part of their business to produce ‘**proofs**’ of immortality and the existence of God . They have found fault with the **proofs** of their predecessors Saint Thomas rejected Saint Anselm's **proofs**, and Kant rejected Descartes ' but they have supplied new ones of their own. In order to make their **proofs** seem valid, they have had to falsify logic , to make mathematics mystical , and to pretend that deep seated prejudices were heaven-sent intuitions .

— Bertrand Russell, in

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