Mathematical Logic Quotes

Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.

I am convinced that the act of thinking logically cannot possibly be natural to the human mind. If it were, then mathematics would be everybody's easiest course at school and our species would not have taken several millennia to figure out the scientific method.

I know that two and two make four - and should be glad to prove it too if I could - though I must say if by any sort of process I could convert 2 and 2 into five it would give me much greater pleasure.

Pure mathematics is in its way the poetry of logical ideas.

Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning.

Histories make men wise; poets, witty; the mathematics, subtle; natural philosophy, deep; moral, grave; logic and rhetoric, able to contend.

Satan, on the contrary, is thin, ascetic and a fanatical devotee of logic. He reads Machiavelli, Ignatius of Loyola, Marx and Hegel; he is cold and unmerciful to mankind, out of a kind of mathematical mercifulness. He is damned always to do that which is most repugnant to him: to become a slaughterer, in order to abolish slaughtering, to sacrifice lambs so that no more lambs may be slaughtered, to whip people with knouts so that they may learn not to let themselves be whipped, to strip himself of every scruple in the name of a higher scrupulousness, and to challenge the hatred of mankind because of his love for it--an abstract and geometric love.

[The Old Astronomer to His Pupil] Reach me down my Tycho Brahe, I would know him when we meet, When I share my later science, sitting humbly at his feet; He may know the law of all things, yet be ignorant of how We are working to completion, working on from then to now. Pray remember that I leave you all my theory complete, Lacking only certain data for your adding, as is meet, And remember men will scorn it, 'tis original and true, And the obloquy of newness may fall bitterly on you. But, my pupil, as my pupil you have learned the worth of scorn, You have laughed with me at pity, we have joyed to be forlorn, What for us are all distractions of men's fellowship and smiles; What for us the Goddess Pleasure with her meretricious smiles. You may tell that German College that their honor comes too late, But they must not waste repentance on the grizzly savant's fate. Though my soul may set in darkness, it will rise in perfect light; I have loved the stars too fondly to be fearful of the night. What, my boy, you are not weeping? You should save your eyes for sight; You will need them, mine observer, yet for many another night. I leave none but you, my pupil, unto whom my plans are known. You 'have none but me,' you murmur, and I 'leave you quite alone'? Well then, kiss me, -- since my mother left her blessing on my brow, There has been a something wanting in my nature until now; I can dimly comprehend it, -- that I might have been more kind, Might have cherished you more wisely, as the one I leave behind. I 'have never failed in kindness'? No, we lived too high for strife,-- Calmest coldness was the error which has crept into our life; But your spirit is untainted, I can dedicate you still To the service of our science: you will further it? you will! There are certain calculations I should like to make with you, To be sure that your deductions will be logical and true; And remember, 'Patience, Patience,' is the watchword of a sage, Not to-day nor yet to-morrow can complete a perfect age. I have sown, like Tycho Brahe, that a greater man may reap; But if none should do my reaping, 'twill disturb me in my sleep So be careful and be faithful, though, like me, you leave no name; See, my boy, that nothing turn you to the mere pursuit of fame. I must say Good-bye, my pupil, for I cannot longer speak; Draw the curtain back for Venus, ere my vision grows too weak: It is strange the pearly planet should look red as fiery Mars,-- God will mercifully guide me on my way amongst the stars.

The law of gravity and gravity itself did not exist before Isaac Newton." ...and what that means is that that law of gravity exists nowhere except in people's heads! It 's a ghost!" Mind has no matter or energy but they can't escape its predominance over everything they do. Logic exists in the mind. numbers exist only in the mind. I don't get upset when scientists say that ghosts exist in the mind. it's that only that gets me. science is only in your mind too, it's just that that doesn't make it bad. or ghosts either." Laws of nature are human inventions, like ghosts. Law of logic, of mathematics are also human inventions, like ghosts." ...we see what we see because these ghosts show it to us, ghosts of Moses and Christ and the Buddha, and Plato, and Descartes, and Rousseau and Jefferson and Lincoln, on and on and on. Isaac Newton is a very good ghost. One of the best. Your common sense is nothing more than the voices of thousands and thousands of these ghosts from the past.

How are you going to teach logic in a world where everybody talks about the sun setting, when it’s really the horizon rising? [Cal Craig]

The person who wishes to attain human perfection should study logic first, next mathematics, then physics, and, lastly, metaphysics.

Sapiens don’t behave according to a cold mathematical logic, but rather according to a warm social logic. We are ruled by emotions.

Logic, n. The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. The basic of logic is the syllogism, consisting of a major and a minor premise and a conclusion - thus: Major Premise: Sixty men can do a piece of work sixty times as quickly as one man. Minor Premise: One man can dig a post-hole in sixty seconds; Therefore- Conclusion: Sixty men can dig a post-hole in one second. This may be called syllogism arithmetical, in which, by combining logic and mathematics, we obtain a double certainty and are twice blessed.

It is all about numbers. It is all about sequence. It's the mathematical logic of being alive. If everything kept to its normal progression, we would live with the sadness--cry and then walk--but what really breaks us cleanest are the losses that happen out of order.

Logic issues in tautologies, mathematics in identities, philosophy in definitions; all trivial, but all part of the vital work of clarifying and organising our thought.

In the Principia Mathematica, Bertrand Russell and Alfred Whitehead attempted to give a rigorous foundation to mathematics using formal logic as their basis. They began with what they considered to be axioms, and used those to derive theorems of increasing complexity. By page 362, they had established enough to prove "1 + 1 = 2.

Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [...] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.

The theories of the social sciences do not consist of “laws” in the sense of empirical rules about the behavior of objects definable in physical terms. All that the theory of the social sciences attempts is to provide a technique of reasoning which assists us in connecting individual facts, but which, like logic or mathematics, is not about the facts. It can, therefore, and this is the second point, never be verified or falsified by reference to facts.

Both for practical reasons and for mathematically verifiable moral reasons, authority and responsibility must be equal - else a balancing takes place as surely as current flows between points of unequal potential. To permit irresponsible authority is to sow disaster; to hold a man responsible for anything he does not control is to behave with blind idiocy. The unlimited democracies were unstable because their citizens were not responsible for the fashion in which they exerted their sovereign authority... other than through the tragic logic of history... No attempt was made to determine whether a voter was socially responsible to the extent of his literally unlimited authority. If he voted the impossible, the disastrous possible happened instead - and responsibility was then forced on him willy-nilly and destroyed both him and his foundationless temple.

Russell's books should be bound in two colours, those dealing with mathematical logic in red — and all students of philosophy should read them; those dealing with ethics and politics in blue — and no one should be allowed to read them.

Mathematics, rightly viewed, possesses not only truth, but supreme beauty— a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.

Mathematicians can be persnickety about logical niceties. We're the kind of people who think it's funny, when asked, "Do you want soup or salad with that?" to reply, "Yes.

In my short stay I realized that without a deep understanding of human psychology, without the acceptance that we are all crazy, irrational, impulsive, emotionally driven animals, all the raw intelligence and mathematical logic in the world is little help in the fraught, shifting interplay of two people negotiating.

I lose faith in mathematics, logical and rigid. What with those that even zero doesn’t accept?

Mathematics is the logical study of how logical things work.

We especially need imagination in science. It is not all logic, nor all mathematics, but is somewhat beauty and poetry.

When the twins asked what cuff-links were for—“To link cuffs together,” Ammu told them—they were thrilled by this morsel of logic in what had so far seemed an illogical language. Cuff+link = cuff-link. This, to them, rivaled the precision of logic and mathematics. Cuff-links gave them an inordinate (if exaggerated) satisfaction, and a real affection for the English language.

Praxeology is a theoretical and systematic, not a historical, science. Its scope is human action as such, irrespective of all environmental, accidental, and individual circumstances of the concrete acts. Its cognition is purely formal and general without reference to the material content and the particular features of the actual case. It aims at knowledge valid for all instances in which the conditions exactly correspond to those implied in its assumptions and inferences. Its statements and propositions are not derived from experience. They are, like those of logic and mathematics, a priori. They are not subject to verification or falsification on the ground of experience and facts.

And I think that if God did exist, he had many children. I think Jesus proves this. Jesus must be the seventh son of God. A-sus, B-sus, C-sus, D-sus, E-sus, F-sus, G-sus. That's just logic. That's just mathematical. And T-sus would always be fucking about. And P-sus does deliveries. C-sus started the Roman Empire. Cae-sus. F-sus, City in Turkey. B-sus was covered in something. Some people applauding there; other people going, "What?" ... B-sus was covered in bees.

Every civilisation has had its irrational but reassuring myth. Previous civilisations have used their culture to sing about it and tell stories about it. Ours has used its mathematics to prove it.

Mathematics was actually a logical puzzle with endless variations—riddles that could be solved. The trick was not to solve arithmetical problems. Five times five would always be twenty-five. The trick was to understand combinations of the various rules that made it possible to solve any mathematical problem whatsoever.

At night I would return home, set out a lamp before me, and devote myself to reading and writing. Whenever sleep overcame me or I became conscious of weakening, I would turn aside to drink a cup of wine, so that my strength would return to me. Then I would return to reading. And whenever sleep seized me I would see those very problems in my dream; and many questions became clear to me in my sleep. I continued in this until all of the sciences were deeply rooted within me and I understood them as is humanly possible. Everything which I knew at the time is just as I know it now; I have not added anything to it to this day. Thus I mastered the logical, natural, and mathematical sciences, and I had now reached the science.

Most people would have probably lost count around seven. This was, Harry knew from his extensive reading on logic and arithmetic, the largest number that most people could visually appreciate. Put seven dots on a page, and most people can take a quick glance and declare, “Seven.” Switch to eight, and the majority of humanity was lost.

Женщина с ребенком во чреве не может быть контрапримером для тезиса, что люди имеют одну голову.

Someone. Everyone. Anyone. No-one. One. One can't be everyone, but there isn't more than one everyone, at the same time. And at the same time no-one can't be someone, but anyone can be one, and also anyone can be a no-one. To sum up - everyone is someone, and any-one becomes a no-one if you divide the one part long enough by every part of every-one, so in conclusion, I have no idea what I’m talking about, basically.

Mathematics has always shown a curious ability to be applicable to nature, and this may express a deep link between our minds and nature. We are the Universe speaking out, a part of nature. So it is not so surprising that our systems of logic and mathematics sing in tune with nature.

گروهى بدون آن كه به فن منطق مجهز باشند سراغ علوم عقلى آمده، نتوانسته اند حق را از باطل جدا كنند و پنداشته اند علوم، نسبى اند. چنين كسانى را بايد با آموزش قوانين منطق و علوم رياضى معالجه كرد.

Take this neat little equation here. It tells me all the ways an electron can make itself comfortable in or around an atom. That's the logic of it. The poetry of it is that the equation tells me how shiny gold is, how come rocks are hard, what makes grass green, and why you can't see the wind. And a million other things besides, about the way nature works.

(1) disengaging their minds; sabotaging their mental activities; providing a low-quality program of public education in mathematics, logic, systems design and economics; and discouraging technical creativity.

Mathematics is not a deductive science - that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

To mention Boston we use 'Boston' or a synonym, and to mention 'Boston' we use ' 'Boston' ' or a synonym. ' 'Boston' ' contains six letters and just one pair of quotation marks; 'Boston' contains six letters and no quotation marks; and Boston contains some 800,000 people.

Language as putative science. - The significance of language for the evolution of culture lies in this, that mankind set up in language a separate world beside the other world, a place it took to be so firmly set that, standing upon it, it could lift the rest of the world off its hinges and make itself master of it. To the extent that man has for long ages believed in the concepts and names of things as in aeternae veritates he has appropriated to himself that pride by which he raised himself above the animal: he really thought that in language he possessed knowledge of the world. The sculptor of language was not so modest as to believe that he was only giving things designations, he conceived rather that with words he was expressing supreame knowledge of things; language is, in fact, the first stage of occupation with science. Here, too, it is the belief that the truth has been found out of which the mightiest sources of energy have flowed. A great deal later - only now - it dawns on men that in their belief in language they have propagated a tremendous error. Happily, it is too late for the evolution of reason, which depends on this belief, to be put back. - Logic too depends on presuppositions with which nothing in the real world corresponds, for example on the presupposition that there are identical things, that the same thing is identical at different points of time: but this science came into existence through the opposite belief (that such conditions do obtain in the real world). It is the same with mathematics, which would certainly not have come into existence if one had known from the beginning that there was in nature no exactly straight line, no real circle, no absolute magnitude.

So how does one go about proving something like this? It's not like being a lawyer, where the goal is to persuade other people; nor is it like a scientist testing a theory. This is a unique art form within the world of rational science. We are trying to craft a "poem of reason" that explains fully and clearly and satisfies the pickiest demands of logic, while at the same time giving us goosebumps.

I believe neither in luck nor in destiny,” he declared. “I trust only the science of probabilities. I have studied mathematical statistics, combinatorial analysis, mass function, and random variables, and they have never held any surprises for me. You don’t seem fully to grasp the destabilizing effect that someone like you can have on someone like me.

Science attempts to find logic and simplicity in nature. Mathematics attempts to establish order and simplicity in human thought.

Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

Although the method is simple, it shows how, mathematically, random brute force can overcome precise logic. It's a numerical approach that uses quantity to derive quality.

Mathematics is the poetry of logic and the music of reason.

… understanding consists in reducing one type of reality to another.” Claude Levi-Strauss

ethics is not about platitudes, let alone tautologies, logic or mathematics, but about difficult choices - dilemmas.

Certainly one of the most important things I learned is that numbers can be deceiving. There is a logic to mathematics, but there is also the underlying human element that must be considered. Numbers can't lie, but the people who create those numbers can and do. As so many people have learned, forgetting to include human nature in an equation can be devastating.

If a mathematician wishes to disparage the work of one of his colleagues, say, A, the most effective method he finds for doing this is to ask where the results can be applied. The hard pressed man, with his back against the wall, finally unearths the researches of another mathematician B as the locus of the application of his own results. If next B is plagued with a similar question, he will refer to another mathematician C. After a few steps of this kind we find ourselves referred back to the researches of A, and in this way the chain closes.

And of course in the long run, if there is a constant fight, the graceful is bound to be defeated and the efficient mind will win, because the world understands the language of mathematics, not of love.

There is no great religion without a great schism. All of them have it. And that's because you're dealing with something called faith. And faith is not something you can prove; faith is personal opinion. Uh, when you're dealing with something with certainty, like, y'know, science or logic, you don't have the--there's no wiggle room; that's why history is not filled with warring math cults, y'know, because you can settle the issue; you can prove something to be right or wrong, and that's the end of the argument: next case. Whereas, when you're dealing with faith, you can forever argue your point, or another point, because you're dealing with intangibles. Personally, I think, faith is what you ask of somebody when you don't have the goods to prove your point.

There was a basic logical paradox that I called the 'fraudulence paradox' that I had discovered more or less on my own while taking a mathematical logic course in school...The fraudulence paradox was that the more time and effort you put into trying to appear impressive or attractive to other people, the less impressive or attractive you felt inside - you were a fraud. And the more of a fraud you felt like, the harder you tried to convey an impressive or likable image of yourself so that other people wouldn't find out what a hollow, fraudulent person you really were.

The philosophers make still another objection: "What you gain in rigour," they say, "you lose in objectivity. You can rise toward your logical ideal only by cutting the bonds which attach you to reality. Your science is infallible, but it can only remain so by imprisoning itself in an ivory tower and renouncing all relation with the external world. From this seclusion it must go out when it would attempt the slightest application.

Like Molière’s M. Jourdain, who spoke prose all his life without knowing it, mathematicians have been reasoning for at least two millennia without being aware of all the principles underlying what they were doing. The real nature of the tools of their craft has become evident only within recent times A renaissance of logical studies in modern times begins with the publication in 1847 of George Boole’s 'The Mathematical Analysis of Logic'.

Don't be scared by the word authority. Believing things on authority only means believing them because you've been told them by someone you think trustworthy. Ninety-nine per cent of the things you believe are believed on authority. I believe there is such a place as New York. I haven't seen it myself. I couldn't prove by abstract reasoning that there must be such a place. I believe it because reliable people have told me so. The ordinary man believes in the Solar System, atoms, evolution, and the circulation of the blood on authority -because the scientists say so. Every historical statement in the world is believed on authority. None of us has seen the Norman Conquest or the defeat of the Armada. None of us could prove them by pure logic as you prove a thing in mathematics. We believe them simply because people who did see them have left writings that tell us about them: in fact, on authority. A man who jibbed at authority in other things as some people do in religion would have to be content to know nothing all his life.

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.

If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.

Brains operate...not by logic but by pattern recognition. This process is not precise, as is logic and mathematics. Instead, it trades off specificity and precision, if necessary, to increase its range. It is likely, for example, that early human thought proceeded by metaphor, which, even with the late acquisition of precise means such as logic and mathematical thought, continues to be a major source of imagination and creativity in adult life.

As a teacher, Tengo pounded into his students' heads how voraciously mathematics demanded logic. Here things that could not be proven had no meaning, but once you had succeeded in proving something, the world's riddles settled into the palm of your hand like a tender oyster.

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.

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.

So a)To what extent might human relationships be expressed in a mathematical or logical formula? And b) If so, what signs might be placed between the integers? Plus and minus, self- evidently; sometimes multiplication, and yes. division. But these signs are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total zero; whereas an entirely successful one can be represented by both addition and multiplication. But what of most relationships? Do they not require to be expressed in notations which are logically insoluble?

The assumption that numbers and mathematical or logical laws are mental is due to the even more widespread notion that only particular sensible entities exist in nature, and that relations abstractions, or universals cannot have any such objective existence - hence they are given a shadowy existence in the mind.

Moreover, we look in vain to philosophy for the answer to the great riddle. Despite its noble purpose and history, pure philosophy long ago abandoned the foundational questions about human existence. The question itself is a reputation killer. It has become a Gorgon for philosophers, upon whose visage even the best thinkers fear to gaze. They have good reason for their aversion. Most of the history of philosophy consists of failed models of the mind. The field of discourse is strewn with the wreckage of theories of consciousness. After the decline of logical positivism in the middle of the twentieth century, and the attempt of this movement to blend science and logic into a closed system, professional philosophers dispersed in an intellectual diaspora. They emigrated into the more tractable disciplines not yet colonized by science – intellectual history, semantics, logic, foundational mathematics, ethics, theology, and, most lucratively, problems of personal life adjustment. Philosophers flourish in these various endeavors, but for the time being, at least, and by a process of elimination, the solution of the riddle has been left to science. What science promises, and has already supplied in part, is the following. There is a real creation story of humanity, and one only, and it is not a myth. It is being worked out and tested, and enriched and strengthened, step by step. (9-10)

When we talk mathematics, we may be discussing a secondary language, built on the primary language truly used by the central nervous system. Thus the outward forms of our mathematics are not absolutely relevant from the point of view of evaluating what the mathematical or logical language truly used by the central nervous system is. However, the above remarks about reliability and logical and arithmetical depth prove that whatever the system is, it cannot fail to differ considerably from what we consciously and explicitly consider as mathematics.

No muscles without strength, friendship without trust, opinion without consequence, change without aesthetics, age without values, life without effort, water without thirst, food without nourishment, love without sacrifice, power without fairness, facts without rigor, statistics without logic, mathematics without proof, teaching without experience, politeness without warmth, values without embodiment, degrees without erudition, militarism without fortitude, progress without civilization, friendship without investment, virtue without risk, probability without ergodicity, wealth without exposure, complication without depth, fluency without content, decision without asymmetry, science without skepticism, religion without tolerance, and, most of all: nothing without skin in the game.

Another explanation for the failure of logic and observation alone to advance medicine is that unlike, say, physics, which uses a form of logic - mathematics - as its natural language, biology does not lend itself to logic. Leo Szilard, a prominent physicist, made this point when he complained that after switching from physics to biology he never had a peaceful bath again. As a physicist he would soak in the warmth of a bathtub and contemplate a problem, turn it in his mind, reason his way through it. But once he became a biologist, he constantly had to climb out of the bathtub to look up a fact.

Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems—general and specific statements—can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.

Faith in God, then, is not at all the same as the kind of logical certainty that we attain in Euclidean geometry. God is not the conclusion to a process of reasoning, the solution to a mathematical problem. To believe in God is not to accept the possibility of his existence because it has been “proved” to us by some theoretical argument, but it is to put our trust in One whom we know and love. Faith is not the supposition that something might be true, but the assurance that someone is there.

Therefore, mathematically...there was no chance at all that [she] could have existed. A zero in ten-to-the-power-of-forever chance. And yet there she was, in front of me, and I was quite taken aback by it all; I really was. Suddenly it made me realise why religion was such a big thing around here. Because, yes, sure, God could not exist. But then neither could humans. So, if they believed in themselves - the logic must go - why not believe in something that was only a fraction more unlikely?

Russell is reputed at a dinner party once to have said, ‘Oh, it is useless talking about inconsistent things, from an inconsistent proposition you can prove anything you like.’ Well, it is very easy to show this by mathematical means. But, as usual, Russell was much cleverer than this. Somebody at the dinner table said, 'Oh, come on!’ He said, 'Well, name an inconsistent proposition,’ and the man said, 'Well, what shall we say, 2 = 1.’ 'All right,’ said Russell, 'what do you want me to prove?’ The man said, 'I want you to prove that you’re the pope.’ 'Why,’ said Russell, 'the pope and I are two, but two equals one, therefore the pope and I are one.

Dear friends & fellow characters, you all know the importance we attach to the power of collective prayer in this our desperate struggle for survival. Some of us have more existence than others, at various times according to fashion. But even this is becoming extremely shadowy & precarious, for we are not read, & when read , we are read badly, we are not lived as we used to be, we are not identified with & fantasized, we are rapidly forgotten. Those of us who have the good fortune to be read by teachers, scholars, & students are not read as we used to be read, but analyzed as schemata, structures, functions within structures, logical & mathematical formulae, aporia, psychic movements, social significances & so forth.

The world of being is unchangeable, rigid, exact, delightful to the mathematician, the logician, the builder of metaphysical systems, and all who love perfection more than life. The world of existence is fleeting, vague, without sharp boundaries, without any clear plan or arrangement, but it contains all thoughts and feelings, all the data of sense, and all physical objects, everything that can do either good or harm, everything that makes any difference to the value of life and the world. According to our temperaments, we shall prefer the contemplation of the one or of the other.

I don't know which is worse—to have a bad teacher or no teacher at all. In any case, I believe the teacher's work should be largely negative. He can't put the gift into you, but if he finds it there, he can try to keep it from going in an obviously wrong direction. We can learn how not to write, but this is a discipline that does not simply concern writing itself but concerns the whole intellectual life. A mind cleared of false emotion and false sentiment and egocentricity is going to have at least those roadblocks removed from its path. If you don't think cheaply, then there at least won't be the quality of cheapness in your writing, even though you may not be able to write well. The teacher can try to weed out what is positively bad, and this should be the aim of the whole college. Any discipline can help your writing: logic, mathematics, theology, and of course and particularly drawing. Anything that helps you to see, anything that makes you look. The writer should never be ashamed of staring. There is nothing that doesn't require his attention.

The semanticists maintained that everything depends on how you interpret the words “potato,” “is” and “moving.” Since the key here is the operational copula “is,” one must examine “is” rigorously. Whereupon they set to work on an Encyclopedia of Cosmic Semasiology, devoting the first four volumes to a discussion of the operational referents of “is.” The neopositivists maintained that it is not clusters of potatoes one directly perceives, but clusters of sensory impressions. Then, employing symbolic logic, they created terms for “cluster of impressions” and “cluster of potatoes,” devised a special calculus of propositions all in algebraic signs and after using up several seas of ink reached the mathematically precise and absolutely undeniable conclusion that 0=0.

A distinguished writer [Siméon Denis Poisson] has thus stated the fundamental definitions of the science: 'The probability of an event is the reason we have to believe that it has taken place, or that it will take place.' 'The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible' (equally like to happen). From these definitions it follows that the word probability, in its mathematical acceptation, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. With the degree of information which we possess concerning the circumstances of an event, the reason we have to think that it will occur, or, to use a single term, our expectation of it, will vary. Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities.

Turing attended Wittgenstein's lectures on the philosophy of mathematics in Cambridge in 1939 and disagreed strongly with a line of argument that Wittgenstein was pursuing which wanted to allow contradictions to exist in mathematical systems. Wittgenstein argues that he can see why people don't like contradictions outside of mathematics but cannot see what harm they do inside mathematics. Turing is exasperated and points out that such contradictions inside mathematics will lead to disasters outside mathematics: bridges will fall down. Only if there are no applications will the consequences of contradictions be innocuous. Turing eventually gave up attending these lectures. His despair is understandable. The inclusion of just one contradiction (like 0 = 1) in an axiomatic system allows any statement about the objects in the system to be proved true (and also proved false). When Bertrand Russel pointed this out in a lecture he was once challenged by a heckler demanding that he show how the questioner could be proved to be the Pope if 2 + 2 = 5. Russel replied immediately that 'if twice 2 is 5, then 4 is 5, subtract 3; then 1 = 2. But you and the Pope are 2; therefore you and the Pope are 1'! A contradictory statement is the ultimate Trojan horse.

Ultimately Russell himself admitted that he made his greatest efforts in the field of traditional philosophy – in epistemology, the search for the ultimate grounds of our knowledge about the world. How can we be certain that what we claim to know is true? Where lies the certainty in our experience of the world? Can even the most precise knowledge – such as mathematics – be said to rest on any sure logical foundation? These were the questions that Russell sought to answer during the periods of his most profound philosophical thinking. They have remained the perennial questions of philosophy from Plato and Aristotle through Descartes, Hume, and Kant, to Russell and Wittgenstein.

The “Empirical Fallacy” is that experience is knowledge when in fact it is just experience. A person could have infinite experiences and literally know nothing about what reality is. A person could perform a trillion observations and have no more clue about what reality is than someone performing divination in the ancient world, or a cockroach. It is not perceptualism that has led to humanity’s body of knowledge, it is conceptualism. Humanity doesn’t perceive better today, it conceives better, and that is purely thanks to mathematics, reason and logic.

From the moment I bought my ticket, I had a premonition I wasn’t returning to New York anytime soon. You Know, this happens a lot to Russians. The Soviet Union is gone, and the borders are as free and passable as they’ve ever been. And yet, when a Russian moves between the two universes, this feeling of finality persists, the logical impossibility of a place like Russia existing alongside the civilized world, of Ann Arbor, Michigan, sharing the same atmosphere with, say, Vladivostok. It was like those mathematical concepts I could never understand in high school: if, then. If Russia exists, then the West is a mirage; conversely, if Russia does not exist, then and only then is the West real and tangible. No wonder young people talk about “going beyond the cordon” when they talk of emigrating, as if Russia were ringed by a vast cordon sanitaire. Either you stay in the leper colony or you get out into the wider world and maybe try to spread your disease to others.

Pure mathematics consists entirely of asseverations to the extent that, if such and such a proposition is true of anything, then such and such another proposition is true of anything. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true ... If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

Says the Cardinal: "Freethought leads to Atheism, to the destruction of social and civil order, and to the overthrow of government." I accept the gentleman's statement; I credit him with much intellectual acumen for perceiving that which many freethinkers have failed to perceive: accepting it, I shall do my best to prove it, and then endeavor to show that this very iconoclastic principle is the salvation of the economic slave and the destruction of the economic tyrant. ... Hence the freethinker who recognizes the science of astronomy, the science of mathematics, and the equally positive and exact science of justice, is logically forced to the denial of supreme authority. For no human being who observes and reflects can admit a supreme tyrant and preserve his self-respect. No human mind can accept the dogma of divine despotism and the doctrine of eternal justice at the same time; they contradict each other, and it takes two brains to hold them. The cardinal is right: freethought does logically lead to atheism, if by atheism he means the denial of supreme authority.

Consider a cognitive scientist concerned with the empirical study of the mind, especially the cognitive unconscious, and ultimately committed to understanding the mind in terms of the brain and its neural structure. To such a scientist of the mind, Anglo-American approaches to the philosophy of mind and language of the sort discussed above seem odd indeed. The brain uses neurons, not languagelike symbols. Neural computation works by real-time spreading activation, which is neither akin to prooflike deductions in a mathematical logic, nor like disembodied algorithms in classical artificial intelligence, nor like derivations in a transformational grammar.

Whether we like it or not, if we are to pursue a career in science, eventually we have to learn the “language of nature”: mathematics. Without mathematics, we can only be passive observers to the dance of nature rather than active participants. As Einstein once said, “Pure mathematics is, in its way, the poetry of logical ideas.” Let me offer an analogy. One may love French civilization and literature, but to truly understand the French mind, one must learn the French language and how to conjugate French verbs. The same is true of science and mathematics. Galileo once wrote, “[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to understand a single word.

The artistically inclined delight in the Game because it provides opportunities for improvisation and fantasy. The strict scholars and scientists despise it – and so do some musicians also – because, they say, it lacks that degree of strictness which their specialties can achieve. Well and good, you will encounter these antinomies, and in time you will discover that they are subjective, not objective – that, for example, a fancy-free artist avoids pure mathematics or logic not because he understands them and could say something about them if he wished, but because he instinctively inclines toward other things. Such instinctive and violent inclinations and disinclinations are signs by which you can recognize the pettier souls. In great souls and superior minds, these passions are not found. Each of us is merely one human being, merely an experiment, a way station. But each of us should be on the way toward perfection, should be striving to reach the center, not the periphery. Remember this: one can be a strict logician or grammarian, and at the same time full of imagination and music. One can be a musician or Glass Bead Game player and at the same time wholly devoted to rule and order. The kind of person we want to develop, the kind of person we aim to become, would at any time be able to exchange his discipline or art for any other. He would infuse the Glass Bead Game with crystalline logic, and grammar with creative imagination. That is how we ought to be. We should be so constituted that we can at any time be placed in a different position without offering resistance or losing our heads.

Aside from the encounter with the Sphinx, there is little in Oedipus to connect him to the common run of Greek heroic figures. He strikes us today as a modern tragic hero and political animal; it is hard to picture him shaking hands with Heracles or joining the crew of the Argo. many scholars and thinkers, most notably Friedrich Nietzsche in his book The Birth of Tragedy, have seen in Oedipus a character who works out on stage the tension in Athenians (and all of us) between the reasoning, mathematically literate citizen and the transgressive blood criminal; between the thinking and the instinctual being; between the superego and the id; between the Apollonian and the Dionysian impulses that contend within us. Oedipus is a detective who employs all the fields of enquiry of which the Athenians were so proud -- logic, numbers, rhetoric, order and discovery -- only to reveal a truth that is disordered, shameful, transgressive and bestial.

Why do I know I exist if I also know I will not? Why was I given access to logical space and the mathematical structure of the world? Just to lose them when my body is destroyed? Why do I wake up in the night with the thought that I will die, why do I sit up, drenched in sweat, and scream and slap myself and try to suppress the thought that I will disappear for all eternity, that I will never be again, to the end of time? Why will the world end with me? We age: we stand quietly in line with those condemned to death. We are executed one after the other in a sinister extermination camp. We are first stripped of our beauty, youth, and hope. We are next wrapped in the penitential robe of illness, weariness, and decay. Our grandparents die, our parents are executed in front of us, and suddenly time gets short, you suddenly see your reflection in the axeblade. And only then do you realize you are living in a slaughterhouse, that generations are butchered and swallowed by the earth, that billions are pushed down the throat of hell, that no one, absolutely no one escapes. That not one person that you see coming out of the factory gates in a Mélies film is still alive. That absolutely everyone in an eighty-year-old sepia photograph is dead. That we all come into this world from a frightening abyss without our memories, that we suffer unimaginably on a speck of dust, and that we then perish, all in a nanosecond, as though we had never lived, as though we had never been.

Human nature is not taken into account, it is excluded, it's not supposed to exist! They don't recognise that humanity, developing by a historical living process, will become at last a normal society, but they believe that a social system that has come out of some mathematical brain is going to organise all humanity at once and make it just and sinless in an instant, quicker than any living process! That's why they instinctively dislike history, 'nothing but ugliness and stupidity in it,' and they explain it all as stupidity! That's why they so dislike the living process of life; they don't want a living soul! The living soul demands life, the soul won't obey the rules of mechanics, the soul is an object of suspicion, the soul is retrograde! But what they want though it smells of death and can be made of india-rubber, at least is not alive, has no will, is servile and won't revolt! And it comes in the end to their reducing everything to the building of walls and the planning of rooms and passages in a phalanstery! The phalanstery is ready, indeed, but your human nature is not ready for the phalanstery--it wants life, it hasn't completed its vital process, it's too soon for the graveyard! You can't skip over nature by logic.

Thinking in art and morals and even mathematics is neither the reflection in consciousness of a mechanical order in the brain nor the tracing with the mind’s eye of some empirical order in its object, but an endeavour to realize in thought an ideal order which would satisfy an inner demand. The nearer thought comes to its goal, the more it finds itself under constraint by that goal, and dominated in its creative effort by aesthetic or moral or logical relevance. These relations of relevance are not physical or psychological relations. They are normative relations that can enter into the mental current because that current is . . . teleological. Their operation marks the presence of a different type of law, which supervenes upon physical and psychological laws when purpose takes control.

Sienna gave him a solemn shrug. “Robert, speaking from a purely scientific standpoint—all logic, no heart—I can tell you without a doubt that without some kind of drastic change, the end of our species is coming. And it’s coming fast. It won’t be fire, brimstone, apocalypse, or nuclear war … it will be total collapse due to the number of people on the planet. The mathematics is indisputable.” Langdon stiffened. “I’ve studied a fair amount of biology,” she said, “and it’s quite normal for a species to go extinct simply as a result of overpopulating its environment. Picture a colony of surface algae living in a tiny pond in the forest, enjoying the pond’s perfect balance of nutrients. Unchecked, they reproduce so wildly that they quickly cover the pond’s entire surface, blotting out the sun and thereby preventing the growth of the nutrients in the pond. Having sapped everything possible from their environment, the algae quickly die and disappear without a trace.” She gave a heavy sigh. “A similar fate could easily await mankind. Far sooner and faster than any of us imagine.

Some could say it is the external world which has molded our thinking-that is, the operation of the human brain-into what is now called logic. Others-philosophers and scientists alike-say that our logical thought (thinking process?) is a creation of the internal workings of the mind as they developed through evolution "independently" of the action of the outside world. Obviously, mathematics is some of both. It seems to be a language both for the description of the external world, and possibly even more so for the analysis of ourselves. In its evolution from a more primitive nervous system, the brain, as an organ with ten or more billion neurons and many more connections between them must have changed and grown as a result of many accidents. The very existence of mathematics is due to the fact that there exist statements or theorems, which are very simple to state but whose proofs demand pages of explanations. Nobody knows why this should be so. The simplicity of many of these statements has both aesthetic value and philosophical interest.

5.4 The question of accumulation. If life is a wager, what form does it take? At the racetrack, an accumulator is a bet which rolls on profits from the success of one of the horse to engross the stake on the next one. 5.5 So a) To what extent might human relationships be expressed in a mathematical or logical formula? And b) If so, what signs might be placed between the integers?Plus and minus, self-evidently; sometimes multiplication, and yes, division. But these sings are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total of zero; whereas an entirely successful one can be represented by both addition and multiplication. But what of most relationships? Do they not require to be expressed in notations which are logically improbable and mathematically insoluble? 5.6 Thus how might you express an accumulation containing the integers b, b, a (to the first), a (to the second), s, v? B = s - v (*/+) a (to the first) Or a (to the second) + v + a (to the first) x s = b 5.7 Or is that the wrong way to put the question and express the accumulation? Is the application of logic to the human condition in and of itself self-defeating? What becomes of a chain of argument when the links are made of different metals, each with a separate frangibility? 5.8 Or is "link" a false metaphor? 5.9 But allowing that is not, if a link breaks, wherein lies the responsibility for such breaking? On the links immediately on the other side, or on the whole chain? But what do you mean by "the whole chain"? How far do the limits of responsibility extend? 6.0 Or we might try to draw the responsibility more narrowly and apportion it more exactly. And not use equations and integers but instead express matters in the traditional narrative terminology. So, for instance, if...." - Adrian Finn

All this attempt to control... We are talking about Western attitudes that are five hundred years old... The basic idea of science - that there was a new way to look at reality, that it was objective, that it did not depend on your beliefs or your nationality, that it was rational - that idea was fresh and exciting back then. It offered promise and hope for the future, and it swept away the old medieval system, which was hundreds of years old. The medieval world of feudal politics and religious dogma and hateful superstitions fell before science. But, in truth, this was because the medieval world didn't really work any more. It didn't work economically, it didn't work intellectually, and it didn't fit the new world that was emerging... But now... science is the belief system that is hundreds of years old. And, like the medieval system before it, science is starting to not fit the world any more. Science has attained so much power that its practical limits begin to be apparent. Largely through science, billions of us live in one small world, densely packed and intercommunicating. But science cannot help us decide what to do with that world, or how to live. Science can make a nuclear reactor, but it can not tell us not to build it. Science can make pesticide, but cannot tell us not to use it. And our world starts to seem polluted in fundamental ways - air, and water, and land - because of ungovernable science... At the same time, the great intellectual justification of science has vanished. Ever since Newton and Descartes, science has explicitly offered us the vision of total control. Science has claimed the power to eventually control everything, through its understanding of natural laws. But in the twentieth century, that claim has been shattered beyond repair. First, Heisenberg's uncertainty principle set limits on what we could know about the subatomic world. Oh well, we say. None of us lives in a subatomic world. It doesn't make any practical difference as we go through our lives. Then Godel's theorem set similar limits to mathematics, the formal language of science. Mathematicians used to think that their language had some inherent trueness that derived from the laws of logic. Now we know what we call 'reason' is just an arbitrary game. It's not special, in the way we thought it was. And now chaos theory proves that unpredictability is built into our daily lives. It is as mundane as the rain storms we cannot predict. And so the grand vision of science, hundreds of years old - the dream of total control - has died, in our century. And with it much of the justification, the rationale for science to do what it does. And for us to listen to it. Science has always said that it may not know everything now but it will know, eventually. But now we see that isn't true. It is an idle boast. As foolish, and misguided, as the child who jumps off a building because he believes he can fly... We are witnessing the end of the scientific era. Science, like other outmoded systems, is destroying itself. As it gains in power, it proves itself incapable of handling the power. Because things are going very fast now... it will be in everyone's hands. It will be in kits for backyard gardeners. Experiments for schoolchildren. Cheap labs for terrorists and dictators. And that will force everyone to ask the same question - What should I do with my power? - which is the very question science says it cannot answer.

Wars and chaoses and paradoxes ago, two mathematicians between them ended an age d began another for our hosts, our ghosts called Man. One was Einstein, who with his Theory of Relativity defined the limits of man's perception by expressing mathematically just how far the condition of the observer influences the thing he perceives. ... The other was Goedel, a contemporary of Eintstein, who was the first to bring back a mathematically precise statement about the vaster realm beyond the limits Einstein had defined: In any closed mathematical system--you may read 'the real world with its immutable laws of logic'--there are an infinite number of true theorems--you may read 'perceivable, measurable phenomena'--which, though contained in the original system, can not be deduced from it--read 'proven with ordinary or extraordinary logic.' Which is to say, there are more things in heaven and Earth than are dreamed of in your philosophy, Horatio. There are an infinite number of true things in the world with no way of ascertaining their truth. Einstein defined the extent of the rational. Goedel stuck a pin into the irrational and fixed it to the wall of the universe so that it held still long enough for people to know it was there. ... The visible effects of Einstein's theory leaped up on a convex curve, its production huge in the first century after its discovery, then leveling off. The production of Goedel's law crept up on a concave curve, microscopic at first, then leaping to equal the Einsteinian curve, cross it, outstrip it. At the point of intersection, humanity was able to reach the limits of the known universe... ... And when the line of Goedel's law eagled over Einstein's, its shadow fell on a dewerted Earth. The humans had gone somewhere else, to no world in this continuum. We came, took their bodies, their souls--both husks abandoned here for any wanderer's taking. The Cities, once bustling centers of interstellar commerce, were crumbled to the sands you see today.

The essence of Hilbert's program was to find a decision process that would operate on symbols in a purely mechanical fashion, without requiring any understanding of their meaning. Since mathematics was reduced to a collection of marks on paper, the decision process should concern itself only with the marks and not with the fallible human intuitions out of which the marks were reduced. In spite of the prolonged efforts of Hilbert and his disciples, the Entscheidungsproblem was never solved. Success was achieved only in highly restricted domains of mathematics, excluding all the deeper and more interesting concepts. Hilbert never gave up hope, but as the years went by his program became an exercise in formal logic having little connection with real mathematics. Finally, when Hilbert was seventy years old, Kurt Godel proved by a brilliant analysis that the Entscheindungsproblem as Hilbert formulated it cannot be solved. Godel proved that in any formulation of mathematics, including the rules of ordinary arithmetic, a formal process for separating statements into true and false cannot exist. He proved the stronger result which is now known as Godel's theorem, that in any formalization of mathematics including the rules of ordinary arithmetic there are meaningful arithmetical statements that cannot be proved true or false. Godel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.

Some books are to be tasted, others to be swallowed, and some few to be chewed and digested; that is, some books are to be read only in parts; others to be read, but not curiously; and some few to be read wholly, and with diligence and attention. Some books also may be read by deputy, and extracts made of them by others; but that would be only in the less important arguments, and the meaner sort of books, else distilled books are like common distilled waters, flashy things. Reading maketh a full man; conference a ready man; and writing an exact man. And therefore, if a man write little, he had need have a great memory; if he confer little, he had need have a present wit: and if he read little, he had need have much cunning, to seem to know, that he doth not. Histories make men wise; poets witty; the mathematics subtile; natural philosophy deep; moral grave; logic and rhetoric able to contend. ---- Alcuni libri devono essere gustati, altri masticati e digeriti, vale a dire che alcuni libri vanno letti solo in parte, altri senza curiosità, e altri per intero, con diligenza ed attenzione. Alcuni libri possono essere letti da altri e se ne possono fare degli estratti, ma ciò riguarderebbe solo argomenti di scarsa importanza o di libri secondari perché altrimenti i libri sintetizzati sono come l’acqua distillata, evanescente. La lettura completa la formazione di un uomo; il parlare lo fa abile, e la scrittura lo trasforma in un uomo preciso. E, pertanto, se un uomo scrive poco, deve avere una grande memoria, se parla poco ha bisogno di uno spirito arguto; se legge poco deve avere bisogno di molta astuzia in modo da far sembrare di sapere quello che non sa. Le storie fanno gli uomini saggi; i poeti arguti; la matematica sottile; la filosofia naturale profondi; la logica e la retorica abili nella discussione.

Everything with them is ‘the influence of environment,’ and nothing else. Their favourite phrase! From which it follows that, if society is normally organised, all crime will cease at once, since there will be nothing to protest against and all men will become righteous in one instant. Human nature is not taken into account, it is excluded, it’s not supposed to exist! They don’t recognise that humanity, developing by a historical living process, will become at last a normal society, but they believe that a social system that has come out of some mathematical brain is going to organise all humanity at once and make it just and sinless in an instant, quicker than any living process! That’s why they instinctively dislike history, ‘nothing but ugliness and stupidity in it,’ and they explain it all as stupidity! That’s why they so dislike the living process of life; they don’t want a living soul! The living soul demands life, the soul won’t obey the rules of mechanics, the soul is an object of suspicion, the soul is retrograde! But what they want though it smells of death and can be made of India-rubber, at least is not alive, has no will, is servile and won’t revolt! And it comes in the end to their reducing everything to the building of walls and the planning of rooms and passages in a phalanstery! The phalanstery is ready, indeed, but your human nature is not ready for the phalanstery—it wants life, it hasn’t completed its vital process, it’s too soon for the graveyard! You can’t skip over nature by logic. Logic presupposes three possibilities, but there are millions! Cut away a million, and reduce it all to the question of comfort! That’s the easiest solution of the problem! It’s seductively clear and you musn’t think about it. That’s the great thing, you mustn’t think! The whole secret of life in two pages of print!