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.

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))

Variant translation: A poet must leave traces of his passage, not **proofs**. Only traces bring about dreams.

— As quoted in

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.

—

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).

On the whole, however, the conclusions I have drawn from the **proofs** quoted may, I believe, safely be relied on. Assuredly they will not be disturbed either by the lays of a poet displaying the exaggeration of his craft, or by the compositions of the chroniclers that are attractive at truth's expense; the subjects they treat of being out of the reach of evidence, and time having robbed most of them of historical value by enthroning them in the region of legend.

— Book I, 21-[1]

Since fresh examples and **proofs** could always be found of the alleged relation between guilt and punishment: if you behave in such and such a way, it will go badly with you. Now, as it generally does go badly, the allegation was constantly confirmed; and thus popular morality, a pseudo- science on a level with popular medicine, continually gained ground.

— p. 26 (cf. Daybreak, § 11)

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))

I hate the world and almost all the people in it. I hate the Labour Congress and the journalists who send men to be slaughtered, and the fathers who feel a smug pride when their sons are killed, and even the pacifists who keep saying human nature is essentially good, in spite of all the daily **proofs** to the contrary. I hate the planet and the human race – I am ashamed to belong to such a species.

— Letter to Colette, December 28, 1916.It is preoccupation with possession, more than anything else, that prevents men from living freely and nobly.

So many really divine individuals humanity has not already produced:;! Heros in the moral sense, who never got tired to practice renouncement and charity; bright intelligences who opened to the mind new ways and horizons; poets and wonderful artists, who created for him the image of an ideal world, the reflect of perfection. These are as many **proofs** of the presence of the absolu in the midst of humanity, for him that does not discover the immediate proof of that in himself.

— p. 44 (Words of a Sage : Selected thoughts of African Spir (1937))

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.

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.

—

The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic **proofs** or solutions.

— Florian Cajori,

A poet should leave traces of his passage, not **proofs**. Traces alone engender dreams.

— As quoted in

... 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

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