Proof By Contradiction Quotes

We both wondered whether these contradictions that one can't avoid if one begins to think of time and space may not really be proofs that the whole of life is a dream, and the moon and stars bits of nightmare.

Lieutenant Chatrand: I don’t understand this omnipotent-benevolent thing. Camerlengo Carlo Ventresca: You are confused because the Bible describes God as an omnipotent and benevolent deity. Lieutenant Chatrand: Exactly. Camerlengo Carlo Ventresca: Omnipotent-benevolent simply means that God is all-powerful and well-meaning. Lieutenant Chatrand: I understand the concept. It’s just... there seems to be a contradiction. Camerlengo Carlo Ventresca: Yes. The contradiction is pain. Man’s starvation, war, sickness... Lieutenant Chatrand: Exactly! Terrible things happen in this world. Human tragedy seems like proof that God could not possibly be both all-powerful and well-meaning. If He loves us and has the power to change our situation, He would prevent our pain, wouldn’t he? Camerlengo Carlo Ventresca: Would He? Lieutenant Chatrand: Well... if God Loves us, and He can protect us, He would have to. It seems He is either omnipotent and uncaring, or benevolent and powerless to help. Camerlengo Carlo Ventresca: Do you have children? Lieutenant Chatrand: No, signore. Camerlengo Carlo Ventresca: Imagine you had an eight-year-old son... would you love him? Lieutenant Chatrand: Of course. Camerlengo Carlo Ventresca: Would you let him skateboard? Lieutenant Chatrand: Yeah, I guess. Sure I’d let him skateboard, but I’d tell him to be careful. Camerlengo Carlo Ventresca: So as this child’s father, you would give him some basic, good advice and then let him go off and make his own mistakes? Lieutenant Chatrand: I wouldn’t run behind him and mollycoddle him if that’s what you mean. Camerlengo Carlo Ventresca: But what if he fell and skinned his knee? Lieutenant Chatrand: He would learn to be more careful. Camerlengo Carlo Ventresca: So although you have the power to interfere and prevent your child’s pain, you would choose to show you love by letting him learn his own lessons? Lieutenant Chatrand: Of course. Pain is part of growing up. It’s how we learn. Camerlengo Carlo Ventresca: Exactly.

I always make sure that the world will prove me right. It gives me the freedom to contradict myself.

It (proof by contradiction) is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

BLACK AND WHITE I was born into A religion of Light, But with so many other Religions and Philosophies, How do I know which ONE Is right? Is it not My birthright To seek out the light? To find Truth After surveying all the proof, Am I supposed To love Or fight? And why do all those who Try to guide me, Always start by dividing And multiplying me – From what they consider Wrong or right? I thought, There were no walls For whoever beams truth and light. And how can one speak on Light's behalf, lf all they do Is act black, But talk WHITE?

Life cannot be known by the “mind,” its secrets cannot be learned through the “mind.” The proof is, the ceaseless strife and contradiction of opinion among those who trust in the mind. Much less can the “mind” know itself, the more so, because it is pervaded by the illusion that it truly knows, truly is.

The first and most important field of philosophy is the application of principles such as “Do not lie.” Next come the proofs, such as why we should not lie. The third field supports and articulates the proofs, by asking, for example, “How does this prove it? What exactly is a proof, what is logical inference, what is contradiction, what is truth, what is falsehood?” Thus, the third field is necessary because of the second, and the second because of the first. The most important, though, the one that should occupy most of our time, is the first. But we do just the opposite. We are preoccupied with the third field and give that all our attention, passing the first by altogether. The result is that we lie – but have no difficulty proving why we shouldn’t.

Every argument for God and every attribute ascribed to Him rests on a false metaphysical premise. None can survive for a moment on a correct metaphysics.... Existence exists, and only existence exists. Existence is a primary: it is uncreated, indestructible, eternal. So if you are to postulate something beyond existence—some supernatural realm—you must do it by openly denying reason, dispensing with definitions, proofs, arguments, and saying flatly, “To Hell with argument, I have faith.” That, of course, is a willful rejection of reason. Objectivism advocates reason as man’s sole means of knowledge, and therefore, for the reasons I have already given, it is atheist. It denies any supernatural dimension presented as a contradiction of nature, of existence. This applies not only to God, but also to every variant of the supernatural ever advocated or to be advocated. In other words, we accept reality, and that’s all.

Science—the science to which I’ve devoted so much of my life—doesn’t contradict what I learned up there. But far, far too many people believe it does, because certain members of the scientific community, who are pledged to the materialist worldview, have insisted again and again that science and spirituality cannot coexist.

I now believe truth lies not in logic but in hope, both past and future. I believe hope can surprise you. It can survive the odds against it, all sorts of contradictions, and certainly any skeptic's rationale of relying on proof through fact.

Omnipotent-benevolent simply means that God is all-powerful and well-meaning.' 'I understand the concept. It's just . . . there seems to be a contradiction.' 'Yes. The contradiction is pain. Man's starvation, war, sickness . . .' 'Exactly!' Chartrand knew the camerlengo would understand. 'Terrible things happen in this world. Human tragedy seems like proof that God could not possibly be both all-powerful and well-meaning. If He loves us and has the power to change our situation, He would prevent our pain, wouldn't He?' The camerlengo frowned. 'Would He?' Chartrand felt uneasy. Had he overstepped his bounds? Was this one of those religious questions you just didn't ask? 'Well . . . if God loves us, and He can protect us, He would have to. It seems He is either omnipotent and uncaring, or benevolent and powerless to help.' 'Do you have children, Lieutenant?' Chartrand flushed. 'No, signore.' 'Imagine you had an eight-year-old son . . . would you love him?' 'Of course.' 'Would you let him skateboard?' Chartrand did a double take. The camerlengo always seemed oddly "in touch" for a clergyman. 'Yeah, I guess,' Chartrand said. 'Sure, I'd let him skateboard, but I'd tell him to be careful.' 'So as this child's father, you would give him some basic, good advice and then let him go off and make his own mistakes?' 'I wouldn't run behind him and mollycoddle him if that's what you mean.' 'But what if he fell and skinned his knee?' 'He would learn to be more careful.' The camerlengo smiled. 'So although you have the power to interfere and prevent your child's pain, you would choose to show your love by letting him learn his own lessons?' 'Of course. Pain is part of growing up. It's how we learn.' The camerlengo nodded. 'Exactly.

1. That reason is a gift of God and that we should believe in its ability to comprehend the world. 2. That they have been wrong who undermined confidence in reason by enumerating the forces that want to usurp it: class struggle, libido, will to power. 3. That we should be aware that our being is enclosed within the circle of its perceptions, but not reduce reality to dreams and the phantoms of the mind. 4. That truth is a proof of freedom and that the sign of slavery is the lie. 5. That the proper attitude toward being is respect and that we must, therefore, avoid the company of people who debase being with their sarcasm, and praise nothingness. 6. That, even if we are accused of arrogance, it is the case that in the life of the mind a strict hierarchy is necessary. 7. That intellectuals in the twentieth century were afflicted with the habit of baratin, i.e., irresponsible jabber. 8. That in the hierarchy of human activities the arts stand higher than philosophy, and yet bad philosophy can spoil art. 9. That the objective truth exists; namely, out of two contrary assertions, one is true, one false, except in strictly defined cases when maintaining contradiction is legitimate. 10. That quite independently of the fate of religious denominations we should preserve a "philosophical faith," i.e., a belief in transcendence as a measure of humanity. 11. That time excludes and sentences to oblivion only those works of our hands and minds which prove worthless in raising up, century after century, the huge edifice of civilization. 12. That in our lives we should not succumb to despair because of our errors and our sins, for the past is never closed down and receives the meaning we give it by our subsequent acts.

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 deepseated prejudices were heaven-sent intuitions.

the Tet surprise was regarded by Westy as proof of Hanoi’s weakness. Nowhere in his understanding of the war was there room for the size and quality of the force that had taken Hue. So the MACV in Saigon and General LaHue in Phu Bai simply refused to believe it had happened. Reports that contradicted this high-level understanding were dismissed as unreliable, the cries of men facing real combat for the first time, and panicking. Against the certainties of the American command, the truth never stood a chance.

If I say that I need the government to protect my property, but that the government is by definition a group of people who can violate my property rights at will, then I am caught in an insurmountable contradiction. I am saying that my property rights must be defended – and then I create an agency to defend them that can violate them at any time. This

All sciences have their mysteries and at certain points the apparently most obvious theory will be found in contradiction with experience. Politics, for example, offers several proofs of this truth. In theory, is anything more absurd than hereditary monarchy? We judge it by experience, but if government had never been heard of and we had to choose one, whoever would deliberate between hereditary and elective monarchy would be taken for a fool. Yet we know by experience that the first is, all things considered, the best that can be imagined, while the second is the worst. What arguments could not be amassed to establish that sovereignty comes from the people? However they all amount to nothing. Sovereignty is always taken, never given, and a second more profound theory subsequently discovers why this must be so. Who would not say the best political constitution is that which has been debated and drafted by statesmen perfectly acquainted with the national character, and who have foreseen every circumstance? Nevertheless nothing is more false. The best constituted people is the one that has the fewest written constitutional laws, and every written constitution is WORTHLESS.

Instead of proving all possible theorems in an axiomatic system (which Kurt Gödel showed is not possible), professional mathematicians continue to use a formal presentation of mathematics to specify and prove many theorems that are amenable to the formalist paradigm. This has generated a vast corpus of formal theory. Controversies continue unresolved. Some mathematicians continue to insist on giving explicit constructions of mathematical entities, and do not allow proof by contradiction. This is a valid approach in its own right with much to recommend it. In the end, however, the choice that is likely to lead to the greater conquests is the one that offers the greater power and at the moment, it is David Hilbert's formalism that continues to predominate, while steadily being expanded as mathematics expands." -David Tall (2013, p. 246) thinks though Formalism (mathematics) may have Lost the Battle it Still may Win the War.

True faith rests upon the character of God and asks no further proof than the moral perfections of the One who cannot lie. It is enough that God said it, and if the statement should contradict every one of the five senses and all the conclusions of logic as well, still the believer continues to believe. "Let God be true, but every man a liar," is the language of true faith. Heaven approves such faith because it rises above mere proofs and rests in the bosom of God.

The great advantage of the reductio method is that it allows us to tell if a statement is true, even if we do not know how to construct a proof for it. We can tell a statement is true by showing that its negation leads to a contradiction.

... I now believe truth lies not in logic but in hope, both past and future. I believe hope can surprise you. It can survive the odds against it, all sorts of contradictions, and certainly any skeptic's rationale on relying on proof through fact.

We may observe that the teaching of Our Lord Himself, in which there is no imperfection, is not given us in that cut-and-dried, fool-proof, systematic fashion we might have expected or desired. He wrote no book. We have only reported sayings, most of them uttered in answer to questions, shaped in some degree by their context. And when we have collected them all we cannot reduce them to a system. He preaches but He does not lecture. He uses paradox, proverb, exaggeration, parable, irony; even (I mean no irreverence) the 'wisecrack'. He utters maxims which, like popular proverbs, if rigorously taken, may seem to contradict one another. His teaching therefore cannot be grasped by the intellect alone, cannot be 'got up' as if it were a 'subject'. If we try to do that with it, we shall find Him the most elusive of teachers. He hardly ever gave a straight answer to a straight question. He will not be, in the way we want, 'pinned down'. The attempt is (again, I mean no irreverence) like trying to bottle a sunbeam.

What is a perceptual “proof”? You can observe the world forever and it will not explain itself to you. What is physical evidence? All physical evidence is interpreted according to some paradigm or other which is created by conception, not perception, hence is unperceivable and contradicts perceptualism.

Descartes’s argument turns out to be a reworking of Anselm’s Ontological Proof. When we doubt, the limitations and finite nature of the ego are revealed. Yet we could not arrive at the idea of “imperfection” if we did not have a prior conception of “perfection.” Like Anselm, Descartes concluded that a perfection that did not exist would be a contradiction in terms.

in churchmen, luxury is wrong, except in connection with representations and ceremonies. It seems to reveal habits which have very little that is charitable about them. An opulent priest is a contradiction. The priest must keep close to the poor. Now, can one come in contact incessantly night and day with all this distress, all these misfortunes, and this poverty, without having about one's own person a little of that misery, like the dust of labor? Is it possible to imagine a man near a brazier who is not warm? Can one imagine a workman who is working near a furnace, and who has neither a singed hair, nor blackened nails, nor a drop of sweat, nor a speck of ashes on his face? The first proof of charity in the priest, in the bishop especially, is poverty.

As we have so often seen, the task of ego consciousness in the second half of life is to step out of the way and embrace a larger spiritual agenda. Contrary to the fantasy of the youthful ego, this larger life will quite often be found in the savannahs of suffering—not on the lofty peaks of New Age transcendence, or in fundamentalism’s fearful flight from complexity, but down in what Yeats called “the fury and mire of human veins.” Only in this way do we grow, and do we find, amid suffering and defeat, the possibility of meaning so rich we can scarcely bear it. For this embrace of suffering, this acceptance of paradox, we deserve to be valued. As Jung put it so aptly, “This apparently unendurable conflict is proof of the rightness of your life. A life without inner contradiction is only half a life, or else a life in the Beyond which is destined only for angels. But God loves human beings more than the angels.

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.

We may observe that the teaching of Our Lord Himself, in which there is no imperfection, is not given us in that cut-and-dried, fool-proof, systematic fashion we might have expected or desired. He wrote no book. We have only reported sayings, most of them uttered in answer to questions, shaped in some degree by their context. And when we have collected them all we cannot reduce them to a system. He preaches but He does not lecture. He uses paradox, proverb, exaggeration, parable, irony; even (I mean no irreverence) the 'wisecrack'. He utters maxims which, like popular proverbs, if rigorously taken, may seem to contradict one another. His teaching therefore cannot be grasped by the intellect alone, cannot be 'got up' as if it were a 'subject'. If we try to do that with it, we shall find Him the most elusive of teachers. He hardly ever gave a straight answer to a straight question. He will not be, in the way we want, 'pinned down'. The attempt is (again, I mean no irreverence) like trying to bottle a sunbeam.

Contradiction. In the rational realm, the word was a blistering condemnation. Proof of flawed logic. To expose it in an adversary’s position was akin to delivering a deathblow, and she well recalled the triumphant gleam in his eyes in the instant he struck. But, she wondered now, where was the crime in that most human of capacities: to carry in one’s heart a contradiction, to leave it unchallenged, immune to reconciliation; indeed, to be two people at once, each true to herself, and neither denying the presence of the other? What vast laws of cosmology were broken by this human talent? Did the universe split asunder? Did reality lose its way?

Contradiction. In the rational realm, the word was a blistering condemnation. Proof of flawed logic. To expose it in an adversary’s position was akin to delivering a deathblow, and she well recalled the triumphant gleam in his eyes in the instant he struck. But, she wondered now, where was the crime in that most human of capacities: to carry in one’s heart a contradiction, to leave it unchallenged, immune to reconciliation; indeed, to be two people at once, each true to herself, and neither denying the presence of the other? What vast laws of cosmology were broken by this human talent? Did the universe split asunder? Did reality lose its way?

The little which she could understand, however, appeared to contradict the very few notions she had entertained on the matter before. It seemed as if a good view were no longer to be taken from the top of an high hill, and that a clear blue sky was no longer a proof of a fine day. She was heartily ashamed of her ignorance. A misplaced shame. Where people wish to attach, they should always be ignorant. To come with a well-informed mind is to come with an inability of administering to the vanity of others, which a sensible person would always wish to avoid. A woman especially, if she have the misfortune of knowing anything, should conceal it as well as she can.

are not to have, in any object of use or ornament, what would be a contradiction in fact.  You don’t walk upon flowers in fact; you cannot be allowed to walk upon flowers in carpets.  You don’t find that foreign birds and butterflies come and perch upon your crockery; you cannot be permitted to paint foreign birds and butterflies upon your crockery.  You never meet with quadrupeds going up and down walls; you must not have quadrupeds represented upon walls.  You must use,’ said the gentleman, ‘for all these purposes, combinations and modifications (in primary colours) of mathematical figures which are susceptible of proof and demonstration.  This is the new discovery.  This is fact.  This is taste.’   The

You haven't seen the proof of that either? Alright, alright. Let's say you have an infinite set A. We'll show how to produce another infinite set, B, which is even bigger than A. This B will simply be the set of all subsets of A, which is guaranteed to exist by the power set axiom. How do we know B is bigger than A? Well, suppose we could pair off every element a A with an element f(a) B, in such a way that no elements of B were left over. Then, we could define a new subset S A, consisting of every a that's not contained in f(a). Then S is also an element of B. But notice that S can't have been paired off with any a A – since otherwise, a would be contained in f(a) if and only if it wasn't contained in f(a), contradiction. Therefore, B is larger than A, and we've ended up with a bigger infinity than the one we started with.

What truth so obvious, so certain, as the being of a God, which the most ignorant ages have acknowledged, for which the most refined geniuses have ambitiously striven to produce new proofs and arguments? What truth so important as this, which is the ground of all our hopes, the surest foundation of morality, the firmest support of society, and the only principle which ought never to be a moment absent from our thoughts and meditations? But, in treating of this obvious and important truth, what obscure questions occur concerning the nature of that Divine Being; his attributes, his decrees, his plan of providence? These have been always subjected to the disputations of men: Concerning these, human reason has not reached any certain determination. But these are topics so interesting that we cannot restrain our restless inquiry with regard to them; though nothing but doubt, uncertainty, and contradiction have as yet been the result of our most accurate researches.

Fusionism as a political philosophy falls short (as do its modern analogues, such as “conservatarianism”) because, at the end of the day, liberty and order or freedom and virtue cannot be permanently reconciled. They are at once mutually dependent and at war, a bickering couple that cannot live without each other. At any given moment, one may have the better argument than the other, but tomorrow is another day. Life is full of contradictions and conflicts, and the story of Western civilization—the only true fundament of modern conservatism—is the story of these contradictions and conflicts being worked out over millennia. Fusionism is a failure if one looks to it as a source for what to think. But it is a shining success if one sees it as a guide for how to think. It tells us that we must always try to balance these conflicting principles—albeit with a thumb on the scales of liberty. That’s fine, because in the classical liberal tradition, the benefit of the doubt should always go to liberty, while the forces of coercion should meet an extra burden of proof.

Thanks to the popular policies of our Party and the Government of our Republic, all our people are provided by the state and society with all the practical conditions they need for adequate food, clothing and housing and enjoy an equally happy life. They are supplied by the state with provisions virtually free of charge and receive the benefits of free education, free medical care and all the conditions they need for adequate food, clothing and housing. Moreover, as a result of the abolition of taxation, this word has disappeared from their vocabulary. In our country the state takes responsible care of the old and disabled people and children who have no means of support. In our country preferential, social treatment is accorded to merited people, including veterans who have been disabled in the fight for the noble cause of the fatherland and the people, and the Party and the state take warm care of them. Our people receive many benefits from the Party and the state. The popular policies of our Party and the Government of the Republic are eloquent proof of the advantages of our socialist system which is centred upon the popular masses. The “welfare policies” pursued in capitalist countries are fundamentally different from the popular policies of a socialist society. They are aimed at disguising the class contradictions in that society and at pacifying the resistance of the working masses. Even if the “welfare policies” are enforced, this is done only in name and cannot improve the life of the working people.

What Cantor's Diagonal Proof does is generate just such a number, which let's call R. The proof is both ingenious and beautiful-a total confirmation of art's compresence in pure math. First, have another look at the above table. We can let the integral value of R be whatever X we want; it doesn't matter. But now look at the table's very first row. We're going to make sure R's first post-decimal digit, a, is a different number from the table's a1. It's easy to do this even though we don't know what particular number a1 is: let's specify that a=(a1-1) unless a1 happens to be 0, in which case a=9. Now look at the table's second row, because we're going to do the same thing for R's second digit b: b=(b2-1), or b=9 if b2=0. This is how it works. We use the same procedure for R's third digit c and the table's c3, for d and d4, for e and e5, and so on, ad inf. Even though we can't really construct the whole R (just as we can't really finish the whole infinite table), we can still see that this real number R=X.abcdefhi... is going to be demonstrably different from every real number in the table. It will differ from the table's 1st Real in its first post-decimal digit, from the 2nd Real in its second digit, from the 3rd Real in its third digit,...and will, given the Diagonal Method here, differ from the table's Nth Real in its nth digit. Ergo R is not-cannot be-included in the above infinite table; ergo the infinite table is not exhaustive of all the real numbers; ergo (by the rules of reductio) the initial assumption is contradicted and the set of all real numbers is not denumerable, i.e. it's not 1-1 C-able with the set of integers. And since the set of all rational numbers is 1-1C-able with the integers, the set of all reals' cardinality has got to be greater than the set of all rationals' cardinality. Q.E.D.*

Many of the really great, famous proofs in the history of math have been reduction proofs. Here's an example. It is Euclid's proof of Proposition 20 in Book IX of the Elements. Prop. 20 concerns the primes, which-as you probably remember from school-are those integers that can't be divided into smaller integers w/o remainder. Prop. 20 basically states that there is no largest prime number. (What this means of course is that the number of prime numbers is really infinite, but Euclid dances all around this; he sure never says 'infinite'.) Here is the proof. Assume that there is in fact a largest prime number. Call this number Pn. This means that the sequence of primes (2,3,5,7,11,...,Pn) is exhaustive and finite: (2,3,5,7,11,...,Pn) is all the primes there are. Now think of the number R, which we're defining as the number you get when you multiply all the primes up to Pn together and then add 1. R is obviously bigger than Pn. But is R prime? If it is, we have an immediate contradiction, because we already assumed that Pn was the largest possible prime. But if R isn't prime, what can it be divided by? It obviously can't be divided by any of the primes in the sequence (2,3,5,...,Pn), because dividing R by any of these will leave the remainder 1. But this sequence is all the primes there are, and the primes are ultimately the only numbers that a non-prime can be divided by. So if R isn't prime, and if none of the primes (2,3,5,...,Pn) can divide it, there must be some other prime that divides R. But this contradicts the assumption that (2,3,5,...,Pn) is exhaustive of all the prime numbers. Either way, we have a clear contradiction. And since the assumption that there's a largest prime entails a contradiction, modus tollens dictates that the assumption is necessarily false, which by LEM means that the denial of the assumption is necessarily true, meaning there is no largest prime. Q.E.D.

This experiment succeeds as hoped and promises to metaphysics, in its first part, which deals with those *a priori* concepts to which the corresponding objects may be given in experience, the secure course of a science. For by thus changing our point of view, the possibility of *a priori* knowledge can well be explained, and, what is still more, the laws which *a priori* lie at the foundation of nature, as the sum total of the objects of experience, may be supplied with satisfactory proofs, neither of which was possible within the procedure hitherto adopted. But there arises from this deduction of our faculty of knowing *a priori*, as given in the first part of metaphysics, a somewhat startling result, apparently most detrimental to that purpose of metaphysics which has to be treated in its second part, namely the impossibly of using this faculty to transcend the limits of possible experience, which is precisely the most essential concern of the science of metaphysics. But here we have exactly the experiment which, by disproving the opposite, establishes the truth of the first estimate of our *a priori* rational knowledge, namely, that it is directed only at appearances and must leave the thing in itself as real for itself but unknown to us. For that which necessarily impels us to to go beyond the limits of experience and of all appearances is the *unconditioned*, which reason rightfully and necessarily demands, aside from everything conditioned, in all things in themselves, so that the series of conditions be completed. If, then, we find that, under the supposition that our empirical knowledge conforms to objects as things in themselves, the unconditioned *cannot be thought without contradiction*, while under the supposition that our representation of things as they are given to us does not conform to them as things in themselves, but, on the contrary, that these objects as appearance conform to our mode of representation, then *the contradiction vanishes*; and if we find, therefore, that the unconditioned cannot be encountered in things insofar as we are acquainted with them (insofar as they are given to us), but only in things insofar as we are not acquainted with them, that is, insofar as they are things in themselves; then it becomes apparent that what we at first assumed only for the sake of experiment is well founded. However, with speculative reason unable to make progress in the field of the supersensible, it is still open to us to investigate whether in reason's practical knowledge data may not be found which would enable us to determine that transcendent rational concept of the unconditioned, so as to allow us, in accordance with the wish of metaphysics, to get beyond the limits of all possible experience with our *a priori* knowledge, which is possible in practical matters only. Within such a procedure, speculative reason has always at least created a space for such an expansion, even if it has to leave it empty; none the less we are at liberty, indeed we are summoned, to fill it, if we are able to do so, with practical *data* of reason." ―from_Critique of Pure Reason_. Preface to the Second Edition. Translated, edited, and with an Introduction by Marcus Weigelt, based on the translation by Max Müller, pp. 19-21

What I was reading contradicted what I'd long ago decided: that my parents had never been happy with each other, never had hope. I couldn't believe that my father could have been so articulate and vulnerable, or that my mother had ever adored him so intensely. My brain was in revolt. What I was reading seemed to have been written by stranger, not the people I'd known. I read their letters again and again, arguing with what I found. But as I continued to find proof of their love, I realized that I was defending the story I'd arrived with against mounting counter-evidence, and losing.

Torricelli fully realized the advantages and disadvantages of the method of indivisibles; and he suspected that the ancients possessed some such method for discovering difficult theorems, the proofs of which they cast in another form either "to hide the secret of their method or to avoid giving occasion for contradiction to jealous detractors.

Proof of the beginning of time probably ranks as the most theologically significant theorem. This great significance arises from the theorem establishing that the universe must be caused by some Entity capable of creating the universe entirely independent of space and time. Such an entity matches the attributes of the God of the Bible but is contradicted by the gods of the eastern (and indeed all other) religions who create within space and time.

Conspiracy theories, by contrast, are frustrating precisely because they are so intricate. Each rejoinder or contradiction only produces a more complicated theory. Conspiracy theorists manipulate all tangible evidence to fit their explanation, but worse, they will also point to the absence of evidence as even stronger confirmation. After all, what better sign of a really effective conspiracy is there than a complete lack of any trace that the conspiracy exists? Facts, the absence of facts, contradictory facts: everything is proof. Nothing can ever challenge the underlying belief.

If, according to Aristotle, the law of contradiction is the most certain of all principles, if it is the ultimate and most basic, upon which every demonstrative proof rests, if the principle of every axiom lies in it; then one should consider all the more rigorously what presuppositions already lie at the bottom of it. Either it asserts something about. actuality, about being, as if one already knew this from another source; that is, as if opposite attributes could not be ascribed to it. Or the proposition means: opposite attributes should not be ascribed to it. In that case, logic would be an imperative, not to know the true, but to posit and arrange a world that shall be called true by us. In short, the question remains open: are the axioms of logic adequate to reality or are they a means and measure for us to create reality, the concept "reality," for ourselves.?--To affirm the former one would, as already said, have to have a previous knowledge of being--which is certainly not the case. The proposition therefore contains no criterion of truth, but an imperative concerning that which should count as true.

A philosophic system is an integrated view of existence. As a human being you have no choice about the fact that you need a philosophy. Your only choice is whether you define your philosophy by a conscious, rational, disciplined process of thought and scrupulously logical deliberation - or let your subconscious accumulate a junk heap of unwarranted conclusions, false generalizations, undefined contradictions, undigested slogans, unidentified wishes, doubts and fears, thrown together by chance, but integrated by your subconscious into a kind of mongrel philosophy and fused into a single solid weight: self-doubt, like a ball and chain in the place where your mind's wings should have grown.

The method of arguing through a conclusion also contradicts the idea of God. Every argument derives conclusions from something that is given about something that is sought … Thomas Aquinas is correct when he rejects such an interpretation and asserts that what is first in itself may be last for our knowledge … The arguments for the existence of God neither are arguments nor are they proof of the existence of God. They are expressions of the question of God which is implied in human finitude. The question is their truth; every answer they give is untrue.

The world is complete without us. Intolerable fact. To which the poet responds by rebelling, wanting to prove otherwise. Out of wounded vanity or stubborn pride or desolate need, the poet lives in chronic dispute with fact, and an astonishment occurs: another fact is created, like a new element, in partial contradiction of the intolerable. Indelible voice, though it has no impact on the non-human universe,

But here is, as we have said, the same madness, in their allowing indeed that the apostles were ignorant of nothing, and preached not any (doctrines) which contradicted one another, but at the same time insisting that they did not reveal all to all men, for that they proclaimed some openly and to all the world, whilst they disclosed others (only) in secret and to a few, because Paul addressed even this expression to Timothy: 'O Timothy, guard that which is entrusted to thee; 'and again: 'That good thing which was committed unto thee keep.'What is this deposit? Is it so secret as to be supposed to characterize a new doctrine? or is it a part of that charge of which he says, 'This charge I commit unto thee, son Timothy? 'and also of that precept of which he says, 'I charge thee in the sight of God, who quickeneth all things, and before Jesus Christ who witnessed a good confession under Pontius Pilate, that thou keep this commandment?' Now, what is (this) commandment and what is (this) charge? From the preceding and the succeeding contexts, it will be manifest that there is no mysterious hint darkly suggested in this expression about (some) far-fetched doctrine, but that a warning is rather given against receiving any other (doctrine) than that which Timothy had heard from himself, as I take it publicly: 'Before many witnesses' is his phrase. Now, if they refuse to allow that the church is meant by these 'many witnesses,' it matters nothing, since nothing could have been secret which was produced 'before many witnesses.' Nor, again, must the circumstance of his having wished him to 'commit these things to faithful men, who should be able to teach others also,' be construed into a proof of there being some occult gospel. For, when he says 'these things,' he refers to the things of which he is writing at the moment. In reference, however, to occult subjects, he would have called them, as being absent, those things, not these things, to one who had a joint knowledge of them with himself.

The beliefs in individual competition and reason we have been discussing are the ones which in actuality have guided modern western development, and are not necessarily the ideal values. To be sure, the values accepted as ideal by most people have been those of the Hebrew-Christian tradition allied with ethical humanism, consisting of such precepts as love thy neighbor, serve the community, and so on. On the whole, these ideal values have been taught in schools and churches hand in hand with the emphasis on competition and individual reason. (We can see the watered-down influence of the values of “service” and “love” coming out in roundabout fashion in the “service clubs” and the great emphasis on being “well liked.”) Indeed, the two sets of values—the one running back many centuries to the sources of our ethical and religious traditions in ancient Palestine and Greece and the other born in the Renaissance—were to a considerable extent wedded. For example, Protestantism, which was the religious side of the cultural revolution beginning in the Renaissance, expressed the new individualism by emphasizing each person’s right and ability to find religious truth for himself. The marriage had a good deal to be said for it, and for several centuries the squabbles between the marriage partners were ironed out fairly well. For the ideal of the brotherhood of man was to a considerable extent furthered by economic competition—the tremendous scientific gains, the new factories and the more rapid moving of the wheels of industry increased man’s material weal and physical health immensely, and for the first time in history our factories and our science can now produce so much that it is possible to wipe starvation and material want from the face of the earth. One could well have argued that science and competitive industry were bringing mankind ever closer to its ethical ideals of universal brotherhood. But in the last few decades it has become clear that this marriage is full of conflict, and is headed for drastic overhauling or for divorce. For now the great emphasis on one person getting ahead of the other, whether it be getting higher grades in school, or more stars after one’s name in Sunday school, or gaining proof of salvation by being economically successful, greatly blocks the possibilities of loving one’s neighbor. And, as we shall see later, it even blocks the love between brother and sister and husband and wife in the same family. Furthermore, since our world is now made literally “one world” by scientific and industrial advances, our inherited emphasis on individual competitiveness is as obsolete as though each man were to deliver his own letters by his own pony express. The final eruption which showed the underlying contradictions in our society was fascist totalitarianism, in which the humanist and Hebrew-Christian values, particularly the value of the person, were flouted in a mammoth upsurgence of barbarism.

Here it is claimed is the fatal defect of all monistic schemes: they disclose hopeless contradiction of our necessary laws of thought and truths of experience, as their inevitable corollaries. Thus Spinoza, having assumed that all real existence must be absolute, existence and therefore one is obliged to teach that modes of extension and modes of thought can both qualify and at the same time be the law; and thus, that phenomenal beings as real and true to our experience as any a priori cognition, or as this very law itself, are both modes of the One, although a part of them are qualified by size, figure, ponderosity, impenetrability, color; and the other part universally and utterly lack every one of these qualities, and are qualified by thought, sensibility, desire, spontaneity, and self-action. But this is not a mystery, it is a self-contradiction. The qualities of matter and extension cannot be relevant to spirit, nor those of thought, feeling, and volition to matter. They utterly exclude each other. Descartes was right: the common sense of mankind is right in thus judging. The proof is that just so soon as we attempt to ascribe intelligence and will to matter, or qualities of extension to spirit, utterly absurd and impossible fancies are asserted. Spinoza teaches us that the Absolute Being must inevitably have an immutable sameness and necessity of being so strict, as to necessitate its absolute unity. Yet he has to teach, in order to carry out this monism, that this monad exists, at the same instant of time, not only in numberless diversities of mode, but in utterly opposite modes, as for instance, as solid, liquid, and gaseous at the same instant...Or, worse yet, that this One so necessary, eternal and absolute in its unity, may at the same moment of time, hate a Frenchman and love a Frenchman in the two modal manifestations of German and Gaul, and may hate sin and love the same sin in the two manifestations, at the same moment, of good souls and bad souls! Yet this Spinoza could not admit that infinite, eternal power and wisdom can make a beginning of real being objective to itself. Truly, this is 'straining out the gnat and swallowing the camel.

The mystics of both schools, who preach the creed of sacrifice, are germs that attack you through a single sore: your fear of relying on your mind. They tell you that they possess a means of knowledge higher than the mind, a mode of consciousness superior to reason—like a special pull with some bureaucrat of the universe who gives them secret tips withheld from others. The mystics of spirit declare that they possess an extra sense you lack: this special sixth sense consists of contradicting the whole of the knowledge of your five. The mystics of muscle do not bother to assert any claim to extrasensory perception: they merely declare that your senses are not valid, and that their wisdom consists of perceiving your blindness by some manner of unspecified means. Both kinds demand that you invalidate your own consciousness and surrender yourself into their power. They offer you, as proof of their superior knowledge, the fact that they assert the opposite of everything you know, and as proof of their superior ability to deal with existence, the fact that they lead you to misery, self-sacrifice, starvation, destruction.

The purpose of philosophy is not a sterile questioning of everything and game-playing with everything by those who cannot deal with real subjects but fancy themselves in the pursuit of endless games. The only merit of these games is that nobody can prove some phenomena, at least not yet. Such a state of affairs gives them sufficient reason to additionally question, scrutinize, disbelieve, and suspect because they think there is no proof to contradict them. But they forget that the only merit of such pursuits is a lack of merit. And since there is no merit, the only way to establish merit is by artificially producing questions and supposed philosophical problems, which would, in fact, only serve the proponents of such artificial issues.

MIRACULOUS!” . . . “Revolutionary!” . . . “Greatest ever!” We are inundated by a flood of extravagant claims as we channel surf the television or flip magazine pages. The messages leap out at us. The products assure that they are new, improved, fantastic, and capable of changing our lives. For only a few dollars, we can have “cleaner clothes,” “whiter teeth,” “glamorous hair,” and “tastier food.” Automobiles, perfume, diet drinks, and mouthwash are guaranteed to bring happiness, friends, and the good life. And just before an election, no one can match the politicians’ promises. But talk is cheap, and too often we soon realize that the boasts were hollow, quite far from the truth. “Jesus is the answer!” . . . “Believe in God!” . . . “Follow me to church!” Christians also make great claims but are often guilty of belying them with their actions. Professing to trust God and to be his people, they cling tightly to the world and its values. Possessing all the right answers, they contradict the gospel with their lives. With energetic style and crisp, well-chosen words, James confronts this conflict head-on. It is not enough to talk the Christian faith, he says; we must live it. “What good is it, my brothers and sisters, if someone claims to have faith but has no deeds? Can such faith save them?” (2:14). The proof of the reality of our faith is a changed life. Genuine faith will inevitably produce good deeds. This is the central theme of James’ letter, around which he supplies practical advice on living the Christian life. James begins his letter by outlining some general characteristics of the Christian life (1:1–27). Next, he exhorts Christians to act justly in society (2:1–13). He follows this practical advice with a theological discourse on the relationship between faith and action (2:14–26). Then James shows the importance of controlling one’s speech (3:1–12). In 3:13–18, James distinguishes two kinds of wisdom—earthly and heavenly. Then he encourages his readers to turn from evil desires and obey God (4:1–12). James reproves those who trust in their own plans and possessions (4:13—5:6). Finally, he exhorts his readers to be patient with each other (5:7–11), to be straightforward in their promises (5:12), to pray for each other (5:13–18), and to help each other remain faithful to God (5:19, 20). This letter could be considered a how-to book on Christian living. Confrontation, challenges, and a call to commitment await you in its pages. Read James and become a doer of the Word (1:22–25).

The strongest arguments for nominalism are questions like the following: Who set up these Platonic ideals? Why are they there? For the past several hundred years, scientists have made steady progress by eliminating metaphysical presuppositions. Why keep any such metaphysics in mathematics and in set theory? A nominalist would counter a Platonist's proof by saying that the mathematicans are not all isolated from each other. Before they entered their lonely writer's garret, they were all aware of the rules for being a good mathematician. They knew that if they were to write anything that would cause a contradiction, they would lose their status as a mathematician. They were not isolated because they knew the language beforehand.

You are to be in all things regulated and governed,’ said the gentleman, ‘by fact. We hope to have, before long, a board of fact, composed of commissioners of fact, who will force the people to be a people of fact, and of nothing but fact. You must discard the word Fancy altogether. You have nothing to do with it. You are not to have, in any object of use or ornament, what would be a contradiction in fact. You don’t walk upon flowers in fact; you cannot be allowed to walk upon flowers in carpets. You don’t find that foreign birds and butterflies come and perch upon your crockery; you cannot be permitted to paint foreign birds and butterflies upon your crockery. You never meet with quadrupeds going up and down walls; you must not have quadrupeds represented upon walls. You must use,’ said the gentleman, ‘for all these purposes, combinations and modifications (in primary colours) of mathematical figures which are susceptible of proof and demonstration. This is the new discovery. This is fact. This is taste.

But the fact that the New Testament winters used this version on many occasions supplies a new proof in opposition to the idea of its authority, for in not a few places they do not follow it, but they suppl)'• a version of their own which rightly represents the Hebrew text, although contradicting the Septuagint. The use, however, which the writers of the New Testament have made of the Septuagint version must always invest it with a peculiar interest; we thus see Avhat honour God may be pleased to put on an honestly-made version, since we find that inspired Avriters often used such a version, when it was sufficiently near the original to suit the purpose for which it was cited, instead of rendering the Hebrew text de novo on every occasion.

There appears to be a fifth way, that of eminence. According to this I argue that it is incompatible with the idea of a most perfect being that anything should excel it in perfection (from the corollary to the fourth conclusion of the third chapter) . Now there is nothing incompatible about a finite thing being excelled in perfection; therefore, etc. The minor is proved from this, that to be infinite is not incompatible with being; but the infinite is greater than any finite being. Another formulation of the same is this. That to which intensive infinity is not repugnant is not all perfect unless it be infinite, for if it is finite, it can be surpassed, since infinity is not repugnant to it. But infinity is not repugnant to being, therefore the most perfect being is infinite. The minor of this proof, which was used in the previous argument, [1] cannot, it seems, be proven *a priori*. For, just as contradictories by their very nature contradict each other and their opposition cannot be made manifest by anything more evident, so also these terms [viz. "being" and "infinite"] by their very nature are not repugnant to each other. Neither does there seem to be any way of proving this except by explaining the meaning of the notions themselves. "Being" cannot be explained by anything better known than itself. "Infinite" we understand by means of finite. I explain "infinite" in a popular definition as follows: The infinite is that which exceeds the finite, not exactly by reason of any finite measure, but in excess of any measure that could be assigned.—[2] The following persuasive argument can be given for what we intend to prove. Just as everything is assumed to be possible if its impossibility is not apparent, so also all things are assumed to be compatible if their incompatibility is not manifest. Now there is no incompatibility apparent here, for it is not of the nature of being to be finite; nor does finite appear to be an attribute coextensive with being. But if they were mutually repugnant, it would be for one or the other of these reasons. The coextensive attributes which being possesses seem to be sufficiently evident.—[3] A third persuasive argument is this. Infinite in its own way is not opposed to quantity (that is, where parts are taken successively); therefore, neither is infinity, in its own way, opposed to entity (that is, where perfection exists simultaneously) .—[4] If the quantity characteristic of power is simply more perfect than that characteristic of mass, why is it possible to have an infinity [of parts] in mass and not an infinite power? And if an infinite power is possible, then it actually exists (from the fourth conclusion of the third chapter).—[5] The intellect, whose object is being, finds nothing repugnant about the notion of something infinite. Indeed, the infinite seems to be the most perfect thing we can know. Now if tonal discord so easily displeases the ear, it would be strange if some intellect did not clearly perceive the contradiction between infinite and its first object [viz. being] if such existed. For if the disagreeable becomes offensive as soon as it is perceived, why is it that no intellect naturally shrinks from infinite being as it would from something out of harmony with, and even destructive of, its first object?" —from_A Treatise on God as First Principle_, 4.63-4.64

These days it takes courage to be religious. Atheism, is proof of lack of courage. It's nothing, neither fish nor fowl." I contradicted him. He let me speak but didn't listen. It was water off a duck's back.

Same premise holds for the liberal critique of “American hegemony” (a term from the 1990s and early 2000s that is very rarely used anymore, but the premise still holds); from the position that America will hold a permanent hegemonic position. This cynical disbelief in the *end* of the Empire’s hegemony is what sustains the decadence that accelerates its precipitous decline. Same thing with global warming or any other exigencies. The denial of the “end” is what accelerates the end. The admission of the crisis is what accelerates the action necessary to negate the crisis and, therefore, turn the cynic into someone who can justify non-belief in the crisis because in the evaporation of the emergency, the proof that the emergency existed is removed, and, therefore, the cynic becomes even more assured that their cynicism is justified. If you do not believe in global warming and policies change to the point where carbon emissions are reduced and global warming is averted, the first thing a cynic will say is, “See! I told you global warming was nothing to worry about!” Same thing with the COVID quarantine and vaccine measures. If they did not work to evade the virus the cynic would cry that the measures were not effective and therefore nobody should trust the government. If the measure did work and the virus was evaded, the cynic would cling to the cynicism and say, “See! The crisis of the virus would have been averted! What is all the fuss? Why did everyone worry?” An ideology succeeds when the facts that at first appearance should serve to contradict it start to serve as arguments in its favor. There is a sense that “enjoyment itself, which we experience as ‘transgression’, is in its innermost status something imposed, ordered—when we enjoy… this obscene call ‘Enjoy!’, is the superego.” Cultural downfall is pressed into action—by the decadent attitudes of those who suspend disbelief even as the demise of hegemonic power is unfolding before their eyes. The same holds true for those who disbelieve that the Other can actually overtake them, and the downplaying of cunning tendencies in the Neighbors. There is an infantilizing process that misunderestimated the possibility that the Other can mobilize itself into a force that can become a “master”-signifier where you are the “subject” to the Other, the condescension that “their” power will remain virtual and never become fully realized as an actual force enacting violence upon you, this is the condescension of liberal compassion towards the “Neighbors.

From an intuitionistic standpoint, the platonistic conception is the result of blatantly transferring, from the finite case to the infinite one, a picture appropriate only to the former. In making this transference, the platonist destroys the whole essence of infinity, which lies in the conception of a structure which is always in growth, precisely because the process of construction is never completed. The platonistic conception of an infinite structure as something which may be regarded both extensionally, that is, as the outcome of the process, and as a whole, that is, as if the process were completed, thus rests on a straightforward contradiction: an infinite process is spoken of as if it were merely a particularly long finite one. On an intuitionistic view, neither the truth-value of a statement nor any other mathematical entity can be given as the final result of an infinite process, since an infinite process is precisely one that does not have a final re- sult: that is why, when the domain of quantification is infinite, an existentially quantified statement cannot be regarded in advance as determinately either true or false, and a universally quantified one cannot be thought of as being true accidentally, that is independently of there being a proof of it, a proof which must depend intrinsically upon our grasp of the process whereby the domain is generated.

Hilbert justifies vicious circles by defining a ’proof’ only by postulates, thus turning it into a new dead mathematical element. But should not an existence proof or the absence of possible contradictions be given for this new symbol? And is this not just moving the difficulty? (...) From our point of view Hilbert’s ‘replâtrage’ is superfluous.

In the direction of formalization, there are two major successes in modern logic. First, the fairly well established conclusion that all of mathematics is reducible to axiomatic set theory and that, if one takes enough trouble, mathematical proofs can be reproduced in this system completely formally in the sense of mechanical checkability. Second, the results of Skolem and Herbrand according to which we can, by construing mathematical theorems as conditional theorems (viz. that the axioms imply the theorem) in the predicate calculus, search for each mathematical proof in a mechanical (in principle) way to determine whether a related Herbrand expansion contains a contradiction. Impressive as these results are, and encouraging as they are for the project of mechanizing mathematical arguments, they are only theoretical results which do not establish the strong conclusion that mathematical reasoning (or even a major part of it) is mechanical in nature.

I walked around those places, pious child that i was, thinking that my goodness was proof negative. "Look at me!" I wanted to shout. I wanted to be a living theorem, a Logos. Science and math had already taught me that if there was many exceptions to a rule, then the rule was not a rule. Look at me. This was all so wrongheaded, so backward, but I didn't know how to think any differently. The rule was never a rule, but I had mistaken it for one. I took me years of questioning and seeking to see more than my little piece, and even now I don't always see it.

This second metaphysical premise dates back to the time of Parmenides. It may be summed up as follows: “From nothing, only nothing comes.” If we don’t put any content into “nothing” (such as continuity, dimensionality, or orientability, as might be found in a spatial continuum), then “nothing” will not mistakenly refer to “something”; it will not mistakenly be thought to have characteristics, do something, or be somewhere, etc. Nothing is nothing. There is no such thing as “nothing.” To say anything else argues the most fundamental of contradictions. We may now proceed to our conclusion—combining a first premise from physics and a second premise from metaphysics. (1) If there is a reasonable likelihood of a beginning of the universe (prior to which there was no physical reality whatsoever), and (2) if it is apriori true that “from nothing, only nothing comes,” then it is reasonably likely that the universe came from something which is not physical reality. This is commonly referred to as a “transcendent cause of the universe (physical reality)” or “a creator of the universe.

Believers hold that every word in the Bible has not only been inspired but also literally dictated by God. Thus we are to believe every verse and every story as spoken directly by God, and this creates some serious problems, including: Intellectual difficulty with overgeneralizations, conflicts with science, and contradictions. Moral difficulties where God is portrayed at times as partial, vengeful, and deceptive, while in other parts of the Bible universal love is taught; the history of the Hebrews in the Bible shows progress in moral concern rather than a static code; injustice in the Bible including the slaughter of innocent people and minor transgressors. Moral difficulty with concept of endless torture in hell. Problem with occasions of Jesus expressing vindictiveness, discourtesy, narrow-mindedness, and ethnic and religious intolerance. Intellectual difficulties with the human decision-making process for deciding the books of the Bible and questions of the value of other writings not included. Non-uniqueness of Judeo-Christian teachings and practices. Other religions have similar rituals and beliefs, including sacrifice and vicarious atonement through the death of a god, union of a god and a virgin, trinities, the mother Mary (Myrrha, Maya, Maia, and Maritala), a place for good people who die and a hell of fire, an apocalypse, the first man falling from the god’s favor by doing something forbidden or having been tempted by some evil animal, catastrophic floods in which the whole race is exterminated (with details analogous to the story of the flood), a man being swallowed by a fish and then spat out alive, miracles as proof of power and divine messengers. Moral difficulties with intolerance and oppression in today’s society, which are based on the Bible. Intellectual difficulties with New Testament authors’ interpretation of events as fulfillment of Old Testament prophecy. There are a number of references to “scriptures” that simply don’t exist.

I’m a historian. I’m opposed to the destruction of documents, and I would love to see religious scholars have more information to ponder the exceptional life of Jesus Christ.” “You’re arguing both sides of my question.” “Am I? The Bible represents a fundamental guidepost for millions of people on the planet, in much the same way the Koran, Torah, and Pali Canon offer guidance to people of other religions. If you and I could dig up documentation that contradicted the holy stories of Islamic belief, Judaic belief, Buddhist belief, pagan belief, should we do that? Should we wave a flag and tell the Buddhists that we have proof the Buddha did not come from a lotus blossom? Or that Jesus was not born of a literal virgin birth? Those who truly understand their faiths understand the stories are metaphorical.” Sophie looked skeptical. “My friends who are devout Christians definitely believe that Christ literally walked on water, literally turned water into wine, and was born of a literal virgin birth.” “My point exactly,” Langdon said. “Religious allegory has become a part of the fabric of reality. And living in that reality helps millions of people cope and be better people.” “But it appears their reality is false.” Langdon chuckled. “No more false than that of a mathematical cryptographer who believes in the imaginary number ‘i’ because it helps her break codes.

We need by no means despair of reaching this goal [of a consistency proof for formalized mathematics], but nothing of mathematical value will thus be gained: an incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is none the less incorrect, just as a criminal policy is none the less criminal even if it cannot be inhibited by any court that would curb it.

In classical mathematics there occur non-constructive or indirect existence proofs, which intuitionists do not accept. For example, to prove there exists an n such that P(n), the classical mathematician may deduce a contradiction from the assumption for all n, not P(n). Under both the classical and the intuitionistic logic, by reductio ad absurdum this gives not for all n, not P(n). The classical logic allows this result to be transformed into there exists an n such that P(n), but not in general the intuitionistic … the classical meaning, that somewhere in the completed infinite totality of the natural numbers there occurs an n such that P(n), is not available to him, since he does not conceive the natural numbers as a completed totality.

This treatise on the church would seem to make the "true church" an article of faith that could be neither proved nor disproved, safe from refutation, in a realm beyond the range of reason's dreadful artillery. It represents a tour de force not only against the Catholic Church but also against the notion that history, tradition, or custom could be proof of the validity of any institution. In this direction lay perhaps Luther's most truly revolutionary impulse. The medieval mentality had been to say that anything long sanctioned by tradition was valid because tradition or custom represented the working out of the will of an almighty and invisible God. The Middle Ages much more than the age of Enlightenment might sing in almost unanimous chorus, "Whatever is, is right." This willingness to dismiss the authority of custom in favor of a scriptural text that might radically contradict custom-or tradition-could lead to the notion that all the institutions of society might be made over anew. These might include positive law-law as announced in statutes and royal decrees. They might include even the political order itself. Yet this thundering treatise against Catharinus with its hidden church known only to God ran against a stronger impulse of Luther's that was soon to assert itself once it became clear that he would have no reconciliation with the Catholic Church. He was by temperament conservative in his political views, and like other educated people of his time he feared the common people. Moreover, despite the apocalyptic fervor of this tractate, Luther was far from being one of those souls who look skyward every day expecting to hear the last trumpet and to see the heavens split with the return of Christ in judgment. The end times had begun; but he did not make predictions about how long they would last, nor did he claim that Christians might help things along by revolting against Catholic princes.

The USSR—and particularly Stalin, for it was he who almost single-handedly made key Soviet foreign policy decisions at that point—interpreted the contradictions in U.S. behavior as proof of the Americans’ bad faith.

Nennius tells us, what Gildas omits, the name of the British soldier who won the crowning mercy of Mount Badon, and that name takes us out of the mist of dimly remembered history into the daylight of romance. There looms, large, uncertain, dim but glittering, the legend of King Arthur and the Knights of the Round Table. Somewhere in the Island a great captain gathered the forces of Roman Britain and fought the barbarian invaders to the death. Around him, around his name and his deeds, shine all that romance and poetry can bestow. Twelve battles, all located in scenes untraceable, with foes unknown, except that they were heathen, are punctiliously set forth in the Latin of Nennius. Other authorities say, “No Arthur; at least, no proof of any Arthur.” It was only when Geoffrey of Monmouth six hundred years later was praising the splendours of feudalism and martial aristocracy that chivalry, honour, the Christian faith, knights in steel and ladies bewitching, are enshrined in a glorious circle lit by victory. Later these tales would be retold and embellished by the genius of Mallory, Spenser, and Tennyson. True or false, they have gained an immortal hold upon the thoughts of men. It is difficult to believe it was all an invention of a Welsh writer. If it was he must have been a marvellous inventor. Modern research has not accepted the annihilation of Arthur. Timidly but resolutely the latest and best-informed writers unite to proclaim his reality. They cannot tell when in this dark period he lived, or where he held sway and fought his battles. They are ready to believe however that there was a great British warrior, who kept the light of civilisation burning against all the storms that beat, and that behind his sword there sheltered a faithful following of which the memory did not fail. All four groups of the Celtic tribes which dwelt in the tilted uplands of Britain cheered themselves with the Arthurian legend, and each claimed their own region as the scene of his exploits. From Cornwall to Cumberland a search for Arthur’s realm or sphere has been pursued.The reserve of modern assertions is sometimes pushed to extremes, in which the fear of being contradicted leads the writer to strip himself of almost all sense and meaning. One specimen of this method will suffice: "It is reasonably certain that a petty chieftain named Arthur did exist, probably in South Wales. It is possible that he may have held some military command uniting the tribal forces of the Celtic or highland zone or part of it against raiders and invaders (not all of them necessarily Teutonic). It is also possible that he may have engaged in all or some of the battles attributed to him; on the other hand, this attribution may belong to a later date." This is not much to show after so much toil and learning. Nonetheless, to have established a basis of fact for the story of Arthur is a service which should be respected. In this account we prefer to believe that the story with which Geoffrey delighted the fiction-loving Europe of the twelfth century is not all fancy. If we could see exactly what happened we should find ourselves in the presence of a theme as well founded, as inspired, and as inalienable from the inheritance of mankind as the Odyssey or the Old Testament. It is all true, or it ought to be; and more and better besides. And wherever men are fighting against barbarism, tyranny, and massacre, for freedom, law, and honour, let them remember that the fame of their deeds, even though they themselves be exterminated, may perhaps be celebrated as long as the world rolls round. Let us then declare that King Arthur and his noble knights, guarding the Sacred Flame of Christianity and the theme of a world order, sustained by valour, physical strength, and good horses and armour, slaughtered innumerable hosts of foul barbarians and set decent folk an example for all time.

In an age like the present, it simply adds to one’s perils to be left to one’s instincts. The instincts contradict, disturb, and destroy each other; I have already defined modernism as physiological self-contradiction. A reasonable system of education would insist upon at least one of these instinct-systems being paralysed beneath an iron pressure, in order to[Pg 100] allow others to assert their power, to grow strong, and to dominate. At present, the only conceivable way of making the individual possible would be to prune him:—of making him possible—that is to say, whole. The very reverse occurs. Independence, free development, and laisser aller are clamoured for most violently precisely by those for whom no restraint could be too severe—this is true in politics, it is true in Art. But this is a symptom of decadence: our modern notion of “freedom” is one proof the more of the degeneration of instinct.

So how did an entire generation of billions of people today come to universally accept planet Earth as being 4.6 billion years old? The answer is simple: It is because most people just believe what they are told, and errant scientists have crammed their guesswork down their throats. Today, we are inundated daily with newspapers, magazines, newscasters, and schoolteachers regurgitating scientist’s theories. But, friend, let us call a spade a spade; a theory is a guess! Therefore, the theory of macroevolution and the big bang theory are nothing more than speculation. And these two theories demand a very old Earth and universe to be true, so scientist — without definitive evidence — falsely assume they are! And, unfortunately, their guesses are being taught as truth to our children in schools. Thus, we have the perfect storm for festering evil: A generation of Biblically illiterate kids are growing up believing whatever their teachers teach them in school, which contradicts the Bible trueness!

The Times story caused a sensation, not just because the emails contradicted Don junior’s explanations but because they offered the clearest proof yet of coordination—collusion—between the Trump campaign and Russia.

The legacy power players long associated with the Republican side of the aisle, such as the U.S. Chamber of Commerce and the Business Roundtable, argue without proof that further slashing taxes on themselves will create more jobs because they will invest their savings in job creation. The data wildly contradicts their assertions

Charity stems from compassion and generosity. The proof of this is that it is freely given. Contributions to welfare, on the other hand, are coerced. Everyone understands the difference between what we are made to do and what we choose voluntarily to do. Welfare is a form of obligatory charity to which all taxpayers are forced to contribute. It is fundamentally different from taxes that pay for the common good: roads, soldiers, diplomats, policemen, etc. In effect, our government says to its citizens, “You will give part of your money to poor people or you will go to jail for tax evasion.” Obligatory charity is thus a contradiction—not charity at all. If a beggar forces a citizen, under threat of violence, to hand over his money, this is robbery. If government forces the same citizen, under threat of jail, to give money to beggars, the same transfer is called welfare. As far as the citizen is concerned, the outcome is the same: He has been parted from his property against his will. In real life, obstreperous beggars are less of a threat than government, for they can be avoided or fought. Government cannot be avoided, and it is much more difficult to fight.

Here my friend the arts man observed that, although my ideas were not without interest, they had not been proved at all. At first, I was extremely surprised, but when I managed to understand the sense of what he was saying I saw that here, too, he was strictly consistent. He called proof only a text containing precisely formulated information, but not considerations about the subject raised. Of course, I did not agree with him. In that case, of course, I would have been obliged to assert that Prester John ruled in the "Three Indias"! Instead, I proposed putting my, obviously conventional, scheme down in terms of time and space and he would be convinced that the facts speak for themselves. For clarity the whole essential information has been reduced to a synchronic table and four historical maps with annotations, so that a broad historical panorama is obtained. We have not taken primary information as a standard, but first order generalisation obtained earlier as a result of precise analysis of details. Thus, the principle of the hierarchical organisation of information and a scale ensuring a review of the subject as a whole have been observed. In the accounting system offered, a "proved statement" will not be one which has a footnote to an authentic source, but one which does not contradict strictly established facts and logic, however paradoxical the conclusion based on such principles. Incidentally, this is how all natural scientists work.

With a little freedom in the use of words, one might actually call Jesus a “free spirit”[9]—he cares nothing for what is established: the word killeth,[10] whatever is established killeth. The idea of “life” as an experience, as he alone conceives it, stands opposed to his mind to every sort of word, formula, law, belief and dogma. He speaks only of inner things: “life” or “truth” or “light” is his word for the innermost—in his sight everything else, the whole of reality, all nature, even language, has significance only as sign, as allegory.—Here it is of paramount importance to be led into no error by the temptations lying in Christian, or rather ecclesiastical prejudices: such a symbolism par excellence stands outside all religion, all notions of worship, all history, all natural science, all worldly experience, all knowledge, all politics, all psychology, all books, all art—his “wisdom” is precisely a pure ignorance[11] of all such things. He has never heard of culture; he doesn’t have to make war on it—he doesn’t even deny it.... The same thing may be said of the state, of the whole bourgeoise social order, of labour, of war—he has no ground for denying “the world,” for he knows nothing of the ecclesiastical concept of “the world”.... Denial is precisely the thing that is impossible to him.—In the same way he lacks argumentative capacity, and has no belief that an article of faith, a “truth,” may be established by proofs (—his proofs are inner “lights,” subjective sensations of happiness and self-approval, simple “proofs of power”—). Such a doctrine cannot contradict: it doesn’t know that other doctrines exist, or can exist, and is wholly incapable of imagining anything opposed to it.... If anything of the sort is ever encountered, it laments the “blindness” with sincere sympathy—for it alone has “light”— but it does not offer objections....

So how did an entire generation of billions of people today come to universally accept planet Earth as being 4.6 billion years old? The answer is simple: It is because most people just believe what they are told, and errant scientists have crammed their guesswork down their throats. Today, we are inundated daily with newspapers, magazines, newscasters, and schoolteachers regurgitating scientist’s theories. But, friend, let us call a spade a spade; a theory is a guess! Therefore, the theory of macroevolution and the big bang theory are nothing more than speculation. And these two theories demand a very old Earth and universe to be true, so scientist — without definitive evidence — falsely assume they are! And, unfortunately, their guesses are being taught as truth to our children in schools. Thus, we have the perfect storm for festering evil: A generation of Biblically illiterate kids are growing up believing whatever their teachers teach them in school, which contradicts the Bible trueness! Friend, the great “end time” deception is in full force! Paul wrote about it: “evil men and seducers shall wax worse and worse, deceiving, and being deceived. But continue thou in the things which thou hast learned and hast been assured of … And that from a child thou hast known the Holy Scriptures” (II Timothy 3:13-15). As a 3 year Chemistry/Biochemistry majored collegiate student with a 3.84 GPA, let me assure you there is not a single test a scientist can perform to absolutely, positively verify the age of something! The tests all involve assumptions — including radiometric dating — and therein lay the error. Are you really going to throw away hard, cold, ancient, Biblical “end times” prophetic evidence, including the 7 day Creation story’s amazing prophetic verbiage, all of which support the truth of a 6,000 year old Earth, to believe in mankind’s scientific guesses? Allow me to tell you first HOW the world was created, for the answer is in Scripture! Then we will investigate how the great “end-time” deceptive theories like macroevolution & the big bang arrived, claiming a very old universe. Friend, the method God used to create the world is blatantly flaunted in a miracle Jesus performed twice. I want you to seriously consider the miracle, for it appears God wanted the miracle to be remembered above ALL other miracles, because it is the ONLY one contained in ALL 4 Gospels. And rightly so, for it should be contemplated by all: The

From the very first days of the occupation, the U.S. practiced what appeared from the outside to be a duplicitous policy toward denazification and decartelization of Germany. This was not surprising, considering that the policy was a product of an unresolved factional conflict within the U.S. government that went back a decade or more. The USSR—and particularly Stalin, for it was he who almost single-handedly made key Soviet foreign policy decisions at that point—interpreted the contradictions in U.S. behavior as proof of the Americans’ bad faith.

Father," Chartrand said, "may I ask you a strange question?" The camerlegno smiled. "Only if I may give you a strange answer." Chartrand laughed. "I have asked every priest I know, and I still don't understand." "What troubles you?" The camerlegno led the way in short, quick strides, his frock kicking out in front of him as he walked. His black, crepe-sole shoes seemed befitting, Chartrand thought, like reflections of the man's essence... modern but humble, and showing signs of wear. Chartrand took a deep breath. "I don't understand this omnipotent-benevolent thing." The camerlegno smiled. "You've been reading Scripture." "I try." "You are confused because the Bible describes God as an omnipotent and benevolent deity." "Exactly." "Omnipotent-benevolent simply means that God is all-powerful and well-meaning." "I understand the concept. It's just... there seems to be a contradiction." "Yes. The contradiction is pain. Man's starvation, war, sickness..." "Exactly!" Chartrand knew the camerlegno would understand. "Terrible things happen in this world. Human tragedy seems like proof that God could not possibly be both all-powerful and well-meaning. If He loves us and has the power to change our situation, He would prevent our pain, wouldn't He?" The camerlegno frowned. "Would He?" Chartrand felt uneasy. Had he overstepped his bounds? Was this one of those religious questions you just didn't ask? "Well... if God loves us, and He can protect us, He would have to. It seems He is either omnipotent and uncaring, or benevolent and powerless to help." "Do you have children, Lieutenant?" Chartrand flushed. "No, signore." "Imagine you had an eight-year-old son... would you love him?" "Of course." "Would you do everything in your power to prevent pain in his life?" "Of course." "Would you let him skateboard?" Chartrand did a double take. The camerlegno always seemed oddly "in touch" for a clergyman. "Yeah, I guess," Chartrand said. "Sure, I'd let him skateboard, but I'd tell him to be careful." "So as this child's father, you would give him some basic, good advice and then let him go off and make his own mistakes?" "I wouldn't run behind him and mollycoddle him if that's what you mean." "But what if he fell and skinned his knee?" "He would learn to be more careful." The camerlegno smiled. "So although you have the power to interfere and prevent your child's pain, you would choose to show your love by letting him learn his own lessons?" "Of course. Pain is part of growing up. It's how we learn." The camerlegno nodded. "Exactly.

The concept of “confirmation bias”—the tendency of people to favor information, true or not, that confirms their preexisting beliefs—was introduced in the 1960s by Peter Wason, a British psychologist. Wason did a famous series of experiments that explored how people give lesser weight to data that contradicts what they think is true. (As if we needed more proof that what’s hidden can make us draw the wrong conclusions.)

It would take salvation out of the realm of the volitional and place it in the mental, where according to the Scriptures, it surely does not belong. True faith rests upon the character of God and asks no further proof than the moral perfections of the One who cannot lie. It is enough that God said it, and if the statement should contradict every one of the five senses and all the conclusions of logic as well, still the believer continues to believe. “Let God be true, but every man a liar” (Romans 3:4), is the language of true faith. Heaven approves such faith because it rises above mere proofs and rests in the bosom of God.

True faith rests upon the character of God and asks no further proof than the moral perfections of the One who cannot lie. It is enough that God said it, and if the statement should contradict every one of the five senses and all the conclusions of logic as well, still the believer continues to believe. “Let God be true, but every man a liar” (Romans 3:4), is the language of true faith. Heaven approves such faith because it rises above mere proofs and rests in the bosom of God.