“Ostensive demonstration is that which concludes directly to that which is in question. Reduction ad impossibile is that which, when something the opposite of that which is in question has been assumed, concludes with some other proposition directly to a known and manifest impossibility, from the opposite of which the investigator is led back to the original proposition in question. But there is a difference between ostensive demonstration and reduction ad impossibile, because the former proves from things prior in the order of nature but the latter from things posterior in the order of nature. When things prior in nature are better known in the intellect of the person making the demonstration the process is carried out ostensively; but when posterior things are better known to his intellect then the demonstration is carried out per impossibile… in demonstration carried out per impossibile the showing of the original thing in question is carried out by means of things posterior to it in the order of nature… And there is in the contrary, falsely supposed in predicate of subject, a connecting term by which something is implied to be which impossible in the nature of things.”

— Robert Grosseteste

i. 17, f. 18<sup>r</sup>
Commentarius in Posteriorum Analyticorum Libros (c. 1217-1220)

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Robert Grosseteste 22

English bishop and philosopher 1175–1253

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“Rules for Demonstrations. I. Not to undertake to demonstrate any thing that is so evident of itself that nothing can be given that is clearer to prove it. II. To prove all propositions at all obscure, and to employ in their proof only very evident maxims or propositions already admitted or demonstrated. III. To always mentally substitute definitions in the place of things defined, in order not to be misled by the ambiguity of terms which have been restricted by definitions.”

Blaise Pascal (1623–1662) French mathematician, physicist, inventor, writer, and Christian philosopher

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“In every country it always is and must be the interest of the great body of the people to buy whatever they want of those who sell it cheapest. The proposition is so very manifest that it seems ridiculous to take any pains to prove it; nor could it ever have been called in question had not the interested sophistry of merchants and manufacturers confounded the common sense of mankind. Their interest is, in this respect, directly opposite to that of the great body of the people.”

Adam Smith (1723–1790) Scottish moral philosopher and political economist

Source: (1776), Book IV, Chapter III, Part II, p. 531.

“Understand the problem. What kind of problem is it? There are three main types of problems:
‘Show that …’ or ‘Evaluate …’ questions, in which a certain statement has to be proved true, or a certain expression has to be worked out;
‘Find a…’ or ‘Find all…’ questions, which requires one to find something (or everything) that satisfies certain requirements;
‘Is there a …’ questions, which either require you to prove a statement or provide a counterexample”

Terence Tao (1975) Australian mathematician

and thus is one of the previous two types of problem
Source: Solving Mathematical Problems (2nd ed., 2006), Ch. 1 : Strategies in problem solving

“In regard, then, to the sacred tradition of humanity, we learn that it consists, not in propositions or statements which are to be accepted and believed on the authority of the tradition, but in questions rightly asked, in conceptions which enable us to ask further questions, and in methods of answering questions. The value of all these things depends on their being tested day by day.”

William Kingdon Clifford (1845–1879) English mathematician and philosopher

The Ethics of Belief (1877), The Weight Of Authority
Context: In regard, then, to the sacred tradition of humanity, we learn that it consists, not in propositions or statements which are to be accepted and believed on the authority of the tradition, but in questions rightly asked, in conceptions which enable us to ask further questions, and in methods of answering questions. The value of all these things depends on their being tested day by day. The very sacredness of the precious deposit imposes upon us the duty and the responsibility of testing it, of purifying and enlarging it to the utmost of our power. He who makes use of its results to stifle his own doubts, or to hamper the inquiry of others, is guilty of a sacrilege which centuries shall never be able to blot out. When the labours and questionings of honest and brave men shall have built up the fabric of known truth to a glory which we in this generation can neither hope for nor imagine, in that pure and holy temple he shall have no part nor lot, but his name and his works shall be cast out into the darkness of oblivion for ever.

“Mr. Darwin and Mr. Wallace say that the opposite hypothesis, which assumes that every species is capable of varying indefinitely from its original type, is not a whit more arbitrary, and has this manifest claim to be preferred, that it will account for a multitude of phenomena which the ordinary theory is incapable of explaining.”

Charles Lyell (1797–1875) British lawyer and geologist

Source: The Geological Evidences of the Antiquity of Man (1863), Ch.21, p. 411
Context: Every naturalist admits that there is a general tendency in animals and plants to vary; but it is usually taken for granted, though we have no means of proving the assumption to be true, that there are certain limits beyond which each species cannot pass under any circumstances, or in any number of generations. Mr. Darwin and Mr. Wallace say that the opposite hypothesis, which assumes that every species is capable of varying indefinitely from its original type, is not a whit more arbitrary, and has this manifest claim to be preferred, that it will account for a multitude of phenomena which the ordinary theory is incapable of explaining.

“One finds in this subject a kind of demonstration which does not carry with it so high a degree of certainty as that employed in geometry; and which differs distinctly from the method employed by geometers in that they prove their propositions by well-established and incontrovertible principles, while here principles are tested by inferences which are derivable from them. The nature of the subject permits of no other treatment. It is possible, however, in this way to establish a probability which is little short of certainty. This is the case when the consequences of the assumed principles are in perfect accord with the observed phenomena, and especially when these verifications are numerous; but above all when one employs the hypothesis to predict new phenomena and finds his expectations realized.”

Christiaan Huygens book Treatise on Light

Treatise on Light (1690) - preface, Translated by Michael R. Matthews, Scientific Background to Modern Philosophy. 1989. p. 126

“These sublime and profound themes admit of no proof whatever… In all questions that cannot be demonstrated, we must adopt the method which we have adopted in this question about God's Omniscience. Note it.”

Maimónides book The Guide for the Perplexed

Source: Guide for the Perplexed (c. 1190), Part III, Ch.21
Context: He fully knows His unchangeable essence, and has thus a knowledge of all that results from any of His acts. If we were to try to understand in what manner this is done, it would be the same as if we tried to be the same as God, and to make our knowledge identical with His knowledge. Those who seek the truth, and admit what is true, must believe that nothing is hidden from God; that everything is revealed to His knowledge, which is identical with His essence; that this kind of knowledge cannot be comprehended by us; for if we knew its method, we would possess that intellect by which such knowledge could be acquired.... Note this well, for I think that this is an excellent idea, and leads to correct views; no error will be found in it; no dialectical argument; it does not lead to any absurd conclusion, nor to ascribing any defect to God. These sublime and profound themes admit of no proof whatever... In all questions that cannot be demonstrated, we must adopt the method which we have adopted in this question about God's Omniscience. Note it.

“In one case as in the other, the question which gives away the sacrificial mentality underlying group belonging is the same: are you for us, or are you one of them? It is the question which reveals the impossibility of a cracking of heart, and thus the impossibility of Eucharist.”

James Alison (1959) Christian theologian, priest

Source: Faith Beyond Resentment: Fragments Catholic and Gay (2001), " Theology amidst the stones and dust http://girardianlectionary.net/res/alison_elijah.htm", p. 35.

“A horse in movement is not a stationary horse that moves but a horse in a movement, which is to say something other, that should be conceived and expressed as something completely different. It is a question of conceiving objects in movement over and above the motion they carry within themselves. That is, a question of finding a form which is the expression of this new absolute... A question of studying the aspects that life has taken on in haste and in consequent simultaneity.”

Umberto Boccioni (1882–1916) Italian painter and sculptor

In Futurism, ed. Didier Ottinger; Centre Pompidou / 5 Continents Editions, Milan, 2008.
1914 - 1916, Pittura e scultura futuriste' Milan, 1914

“…In that same period there was Newton, there was Harvey studying the circulation of the blood, there were people with methods of analysis by which progress was being made! You can take every one of Spinoza's propositions and take the contrary propositions and look at the world--and you can't tell which is right. Sure, people were awed because he had the courage to take on these great questions, but it doesn't do any good to have the courage if you can't get anywhere with the question…”

Richard Feynman book The Pleasure of Finding Things Out

From Omni interview, "The Smartest Man in the World" (1979) or from the book p. 195.
The Pleasure of Finding Things Out (1999)

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Robert Grosseteste 22

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