Christianity claims that the supernatural is as reasonable as the natural, that man himself is supernatural as truly as he is natural, and that the Bible is so clearly the word of God by proofs that are unanswerable, that it is unreasonable to disbelieve its divine truths.
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!Arthur Miller
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.Raymond Chandler
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.ferdinand eisenstein
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.robert w floyd
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.francis galton
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.carl friedrich gauss
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.heinrich hertz
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.
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.Samuel Taylor Coleridge
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.thomas jefferson
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.thomas jefferson
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.james madison
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.george pólya
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.
Algorithms are the computational content of proofs.
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.
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.
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.
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.Florian Cajori
... that what is proved, by impossibility proofs, is lack of imagination.
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.
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.
To those who have an active belief, reasoned proofs are needless and probably useless.
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.
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 .
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