Mathematical Proof Quotes

If we leave that general descriptive talk where everything which looks like a poem can be called a poem and turn instead to normative talk, we will of course not recognize as a poem everything that looks like a poem. A real poem has to be a successful poem, a successful speech act. In approximately the same way that only a mathematical proof which really proves something can be called a mathematical proof. It is not enough that it looks like a proof. The proof has to prove. For the poem it is not enough to look like a poem. It has to achieve something. ~ Lars Gustafsson

Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof. ~ David Hilbert

A mathematical proof must be perspicuous. ~ Ludwig Wittgenstein

Poverty is a mathematical proof of the fact that mankind is a big failure! ~ Mehmet Murat Ildan

Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? Where does the proof use the hypothesis? ~ Paul Halmos

In a way, composing on the melodic level is an expression of a melodic truth, almost like a geometric truth. If it has clarity, other people will recognize it. There's no way of isolating it in a gallery on a white wall and saying, "This is a work of art. This is a mathematical proof." ~ Eyvind Kang

A journey is an achievement, Maria, just as much as a mathematical proof. ~ Sara Sheridan

Science proceeds by inference, rather than by the deduction of mathematical proof. A series of observations is accumulated, forcing the deeper question: What must be true if we are to explain what is observed? What "big picture" of reality offers the best fit to what is actually observed in our experience? American scientist and philosopher Charles S. Peirce used the term "abduction" to refer to the way in which scientists generate theories that might offer the best explanation of things. The method is now more often referred to as "inference to the best explanation." It is now widely agreed to be the philosophy of investigation of the world characteristic of the natural sciences. ~ Alister E. McGrath

I had made an empirical discovery and it carried all the weight of a mathematical proof. ~ Paul Auster

In the Christian religion, though perhaps not in any other, we frequently find a conception of god that is selfcontradictory and therefore corresponds to nothing. That is the conception formed by the following three propositions taken together:
1. God is all-powerful.
2. God is all-benevolent.
3. There is much misery in the world.
A god who was all-powerful but left much misery in the world would not be all-benevolent. An all-benevolent god in a world containing much misery would not be an all-powerful god. A world containing a god who was both all-powerful and all-benevolent would contain no misery.
Here, then, we have a mathematical proof bearing on a common religious doctrine. Anyone who is confident that he frequently comes across misery in the world may conclude with equal confidence that there is no such thing as an all-powerful and all-benevolent god. And this mathematically disposes of official Christianity, as has long been known. ~ Richard Robinson

What a mathematical proof actually does is show that certain conclusions, such as the irrationality of , follow from certain premises, such as the principle of mathematical induction. The validity of these premises is an entirely independent matter which can safely be left to philosophers. ~ Timothy Gowers

No human investigation can claim to be scientific if it doesn't pass the test of mathematical proof. ~ Leonardo Da Vinci

A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way. ~ G.H. Hardy

A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details. ~ Hermann Weyl

I read in the proof sheets of Hardy on Ramanujan: "As someone said, each of the positive integers was one of his personal friends." My reaction was, "I wonder who said that; I wish I had." In the next proof-sheets I read (what now stands), "It was Littlewood who said ... " ~ John Edensor Littlewood

I am sorry for your disappointment,' he continued, glancing into her face. Their eyes having met, became, as it were, mutually locked together, and the single instant only which good breeding allows as the length of such a look, became trebled: a clear penetrating ray of intelligence had shot from each into each, giving birth to one of those unaccountable sensations which carry home to the heart before the hand has been touched or the merest compliment passed, by something stronger than mathematical proof, the conviction, 'A tie has begun to unite us.' Both faces also unconsciously stated that their owners had been much in each other's thoughts of late. Owen had talked to the young architect of his sister as freely as to Cytherea of the young architect. ~ Thomas Hardy

Zero has had a long history. The Babylonians invented the concept of zero; the ancient Greeks debated it in lofty terms (how could something be nothing?); the ancient Indian scholar Pingala paired Zero with the numeral 1 to get double digits; and both the Mayans and the Romans made Zero a part of their numeral systems. But Zero finally found its place around AD 498, when the Indian astronomer Aryabhatta sat up in bed one morning and exclaimed, "Sthanam sthanam dasa gunam" - which translates, roughly as, "place to place in ten times in value". With that, the idea of decimal based place value notion was born. Now Zero was on a roll: It spread to the Arab world, where it flourished; crossed the Iberian Peninsula to Europe (thanks to the Spanish Moors); got some tweaking from the Italians; and eventually sailed the Atlantic to the New World, where zero ultimately found plenty of employment (together with the digit 1) in a place called Silicon Valley. ~ Dan Ariely

A mathematical proof is beautiful, but when you're finished, it's really only about one thing. A story can be about many things. ~ Bonnie Jo Campbell

In this passage we are taught how hateful a thing is calumny in all free States, as, indeed, in every society, and how we must neglect no means which may serve to check it. And there can be no more effectual means for checking calumny than by affording ample facilities for impeachment, which is as useful in a commonwealth as the other is pernicious. And between them there is this difference, that calumny needs neither witness, nor circumstantial proof to establish it, so that any man may be calumniated by any other; but not impeached; since impeachment demands that there be substantive charges made, and trustworthy evidence to support them. ~ Niccolo Machiavelli

Only small fish swim in schools. (p. 31)

"Don't ever see another human being, lock yourself up and live like a misanthrope." And another power tells me to accept people, talk to them. There is a struggle all the time within me. I also know that if I stay away from people, then I have to deal with only one human being-myself - and I 'd rather have other people than to all the time have only myself for an associate. When you are with yourself your egotism grows, your bitterness and suspicions grow. You become twice as meshuga as before. (p. 32)

An assimilated Jew is a man who is ashamed of his origin, who denies his roots. He wants to make believe that he's somebody else. (p. 60)

I believe in God but I have my doubts about revelation. I would say that I have no proof whatsoever
that God reveals Himself or tells us how to behave, what He wants. I believe that God is a silent God, and He must have a very good reason why He is silent. If He would begin to talk, He would have to speak in three thousand languages and in all kinds of dialects. God speaks in deeds, but the language of deeds is so large its vocabulary is as large as the universe perhaps. So we only understand a very small part of His language. Everything man says about God is pure guesswork. But since I believe in God's existence and since God created man and formed his brain, I believe also that there must be something of the divine in men's ideas about Him even if they are far fr ~ Isaac Bashevis Singer

Malicious tongues spread their poison abroad and nothing here below is proof against them. ~ Moliere

Roughly speaking what Fourier developed was a mathematical way of converting any pattern, no matter how complex, into a language of simple waves. He also showed how these wave forms could be converted back into the original pattern. In other words, just as a television camera converts an image into electromagnetic frequencies and a television set converts those frequencies back into the original image, Fourier showed how a similar process could be achieved mathematically. The equations he developed to convert images into wave forms and back again are known as Fourier transforms. ~ Michael Talbot

The essence of all religions is love, compassion and tolerance. Kindness is my true religion. No matter whether you are learned or not, whether you believe in the next life or not, whether you believe in God or Buddha or some other religion or not, in day-to-day life you must be a kind person. ~ Dalai Lama

The apex of mathematical achievement occurs when two or more fields which were thought to be entirely unrelated turn out to be closely intertwined. Mathematicians have never decided whether they should feel excited or upset by such events. ~ Gian-Carlo Rota

She only modelled for him once,' Max said stubbornly, leaning the canvases back against the wall and replacing the sheet.
'Once, twice or umpteen times, it's proof she knew Spataro ... how shall we put it? ... on terms a man who loved her might resent.'
'There are lots of artists in Montparnasse, Appelby, and lots of artists' models.'
'I wouldn't like it. And I bet Sir Henry didn't like it either.'
'There was nothing between Corinne and Spataro.'
'That's the problem, isn't it?' Appelby pointed with the stem of his pipe at the shrouded paintings. 'There may have been *literally* nothing between them. ~ Robert Goddard

You'll stay," he said firmly.
"But-"
He crossed his arms. "Do I look like a man in the mood to be argued with?"
She stared at him mutinously.
"If you run," he warned, "I will catch you."
Sophie eyed the distance between them, then tried to judge the distance back to My Cottage.If he stopped to pull on his clothing she might have a chance of escaping, but if he didn't...
"Sophie," he said, "I can practically see the steam coming out of your ears. Stop taxing your brain with useless mathematical computations and do as I asked."
One of her feet twitched. Whether it was itching to run home or merely turn around, she'd never know.
"Now," he ordered.
With a loud sigh and grumble, Sophie crossed her arms and turned around to stare at a knothole in the tree trunk in front of her as if her very life depended on it The inferal man wasn't being particularly quiet as he went about his business, and she couldn't seem to keep herself from listening to and trying to identify every sound that rustled and splashed behind her.Now he was emerging from the water, now he was reaching for his breeches, now he was...
It was no use.She had a dreadfully wicked imagination, and there was no getting around it.
He should have just let her return to the house. Instead she was forced to wait, utterly mortified, while he dressed. Her skin felt like it was on fire, and she was certain her cheeks must be eight different shades of red. A gentleman would ~ Julia Quinn

NIKKI: Yep! I'm trying to help her earn a cooking badge for Scouts. Any ideas for a super-EZ brat-proof snack? BRANDON: ~ Rachel Renee Russell

For Newton, the world's very existence and the mathematical regularity of the observed cosmos were evidence for God's presence. ~ Mario Livio

The rise of a hypersexual culture is not proof that we have reached full equality; rather, it has reflected and exaggerated the deeper imbalances of power in our society. Without thoroughgoing economic and political change, what we see when we look around us is not the equality we once sought; it is a stalled revolution. ~ Natasha Walter

We shall see that the mathematical treatment of the subject [of electricity] has been greatly developed by writers who express themselves in terms of the 'Two Fluids' theory. Their results, however, have been deduced entirely from data which can be proved by experiment, and which must therefore be true, whether we adopt the theory of two fluids or not. The experimental verification of the mathematical results therefore is no evidence for or against the peculiar doctrines of this theory. ~ James Clerk Maxwell

Our vision is so limited we can hardly imagine a love that does not show itself in protection from suffering. The love of God is of a different nature altogether. It does not hate tragedy. It never denies reality. It stands in the very teeth of suffering. The love of God did not protect His own Son. The cross was the proof of His love – that He gave that Son, that He let Him go to Calvary's cross, though "legions of angels" might have rescued Him. He will not necessarily protect us - not from anything it takes to make us like His Son. A lot of hammering and chiseling and purifying by fire will have to go into the process. ~ Elisabeth Elliot

You are an idealist, which means that you are destined to be disappointed, and perhaps even wounded. You seek a gospel of benevolence and miracle, which leaves no room for the sorrows of existence. You are like William Paley, arguing that the perfection of every design in the universe is proof of God's love for us. ~ Elizabeth Gilbert

You collect a hunderd pennies in one minute by Master's timepiece and you bought yourself free. Then they shakes a jar of coppers onto the belly of a shovel and holds it over the campfire long time. When they think it fun enough, they say, get yourself ready, and now you go, boy. The pennies go in the dirt around the fire and I got to pick them up and keep hold on them." "Oh," said Ada, with a sound like a kind of punch in the air, or out of it. "I gets forty-two hot cents before that minute up," he said, holding out his hands again. "Here's the proof." "Oh," she said. "And no freedom." "Nothing next, but that we left," he said. "One by one, and not on the same path, turns out." "And here you are." "And wherever this be, I don't know. Some mystery or t'other." She lifted her shoulders and dropped them. "Well. With me I suppose. ~ Gregory Maguire

If any proof were needed of the progress of the cause for which I have worked, it is here tonight. The presence on the stage of these college women, and in the audience of all those college girls who will some day be the nation's greatest strength, will tell their own story to the world. ~ Susan B. Anthony

The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking. ~ John Von Neumann

The Sun, the Moon, the Stars, the Birds, the Animals, the Flowers, are all proof that a Universal Power exists.-RVM ~ R.v.m.

What would you pack for Armageddon? Sunscreen and shades? Flame-proof underwear? Maybe a travel guide to the Underworld? ~ Jana Oliver

Accordingly, France Had Voltaire, and his school of negative thinkers, and England (or rather Scotland) had the profoundest negative thinker on record, David Hume: a man, the peculiarities of whose mind qualified him to detect failure of proof, and want of logical consistency, at a depth which French skeptics, with their comparatively feeble powers of analysis and abstractions stop far short of, and which German subtlety alone could thoroughly appreciate, or hope to rival. ~ John Stuart Mill

The six elements of her Fail Proof Broken-Heart Curing Treatment: Vitamin E, get much sleep, drink much water, travel to a place far away from the person you loved, meditate and teach your heart that this is destiny. ~ Elizabeth Gilbert

In struggling with misfortunes lies the true proof of virtue. ~ William Shakespeare

Simplification of modes of proof is not merely an indication of advance in our knowledge of a subject, but is also the surest guarantee of readiness for farther progress. ~ Lord Kelvin

The credit crunch was based on a climate (the post-Cold War victory party of free-market capitalism), a problem (the sub-prime mortgages), a mistake (the mathematical models of risk) and a failure, that of the regulators. ~ John Lanchester

Numbers written on restaurant checks within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the Universe. ~ Douglas Adams

As is true in many families where children create enormously upsetting diversions from core family secrets, Elana's family fixed on her behavior as further proof that the secret should remain buried. When behavior is truly a distraction for the unspeakable, therapist after therapist may be brought in, each attempting to intervene at the wrong level, mistakenly addressing such signals at the biological, behavioral, intrapsychic, or interpersonal level, while completely missing the unimaginable maze of secrecy in which such actions are embedded. ~ Evan Imber-Black

If there was a second alias under her smile, it was going to take a long time for me to dig it out – my best course of action at this point would be to play nice and accept her help. Then, when I had more proof, I'd slice layers off of her until the real woman appeared. ~ Aprille Legacy

I think a strong claim can be made that the process of scientific discovery may be regarded as a form of art. This is best seen in the theoretical aspects of Physical Science. The mathematical theorist builds up on certain assumptions and according to well understood logical rules, step by step, a stately edifice, while his imaginative power brings out clearly the hidden relations between its parts. A well constructed theory is in some respects undoubtedly an artistic production. A fine example is the famous Kinetic Theory of Maxwell ... The theory of relativity by Einstein, quite apart from any question of its validity, cannot but be regarded as a magnificent work of art. ~ Ernest Rutherford

The thing about having an audience right there laughing is that critics can write what they want, but the proof is right there in front of you. ~ Chris Rock

Nietzsche needed no flamboyance, no outward proof of genius, so long as he had his illness. It enabled him to live numberless lifetimes within one. She noticed how his life fell into a general pattern. A regular recurrent decline into sickness always demarcated one period of his life from another. Every illness was a death, a dip down into Hades. Every recuperation was a joyful rebirth, a regeneration. This mode of existence refreshed him. During each fleeting recuperation the world gleamed anew. And so each recuperation became not only his own rebirth, but also the birth of a whole new world, a new set of problems that demanded new answers. It was like the annual fertility cycle of the god being plowed into the ground. ~ Sue Prideaux

... Our sunsets have been reduced to wavelengths and frequencies. The complexities of the universe have been shredded into mathematical equations. Even our self-worth as human beings has been destroyed. ~ Dan Brown

Another proof of the conquest of a soul for Christ will be found in a real change of life. If the man does not live differently from what he did before, both at home and abroad, his repentance needs to be repented of and his conversion is a fiction ~ Charles Spurgeon

The perverse presumption that places the burden of proof on the challenger of spending must be inverted, back to the rule that applies elsewhere in life: 'Prove to me why we should.' ~ Mitch Daniels

One early terracotta statuette from Catal Huyuk in Anatolia depicts an enthroned female in the act of giving birth, supported by two cat-like animals that form her seat (Plate 1). This figure has been identified as a 'birth goddess' and it is this type of early image that has led a number of feminist scholars to posit a 'reign of the goddess' in ancient Near Eastern prehistory. Maria Gimbutas, for whom such images are proof of a perfect matriarchal society in 'Old Europe' , presents an ideal vision in which a socially egalitarian matriarchal culture was overthrown by a destructive patriarchy (Gimbutas 1991). Gerda Lerner has argued for a similar situation in the ancient Near East; however, she does not discuss nude figurines at any length (Lerner 1986a: 147). More recently, critiques of the matriarchal model of prehistory have pointed out the flaws in this methodology (e.g. Conkey and Tringham 1995; Meskell 1995; Goodison and Morris 1998). In all these critiques the identification of such figures as goddesses is rejected as a modern myth. There is no archaeological evidence that these ancient communities were in fact matriarchal, nor is there any evidence that female deities were worshipped exclusively. Male gods may have worshipped simultaneously with the 'mother goddesses' if such images are indeed representations of deities. Nor do such female figures glorify or show admiration for the female body; rather they essentialise it, reducing it to nothing more nor less than a re ~ Zainab Bahrani

I trust in due time to lay the perfidy, meanness, and wickedness of [Henry] Clay naked before the American people. I have lately got an intimation of some of his secret movements, which, if I can reach with positive and responsible proof, I will wield to his political, and perhaps his actual, destruction. ~ Andrew Jackson

No one thing shows the greatness and power of the human intellect or the loftiness and nobility of man more than his ability to know and to understand fully and feel strongly his own smallness. When, in considering the multiplicity of worlds, he feels himself to be an infinitesimal part of a globe which itself is a negligible part of one of the infinite number of systems that go to make up the world, and in considering this is astonished by his own smallness, and in feeling it deeply and regarding it intently, virtually blends into nothing, and it is as if he loses himself in the immensity of things, and finds himself as though lost in the incomprehensible vastness of existence, with this single act of thought he gives the greatest possible proof of the nobility and immense capability of his own mind, which, enclosed in such a small and negligible being, has nonetheless managed to know and understand things so superior to his own nature, and to embrace and contain this same intensity of existence and things in his thought. ~ Giacomo Leopardi

You grow up to become living proof of your parents' limitations. Their less-than masterpiece. ~ Chuck Palahniuk

Even today many educated people think that the victory of Christianity over Greek philosophy is a proof of the superior truth of the former - although in this case it was only the coarser and more violent that conquered the more spiritual and delicate. So far as superior truth is concerned, it is enough to observe that the awakening sciences have allied themselves point by point with the philosophy of Epicurus, but point by point rejected Christianity. ~ Friedrich Nietzsche