The Science Of Deduction Quotes

What a lovely thing a rose is!" He walked past the couch to the open window and held up the drooping stalk of a moss-rose, looking down at the dainty blend of crimson and green. It was a new phase of his character to me, for I had never before seen him show any keen interest in natural objects. "There is nothing in which deduction is so necessary as religion," said he, leaning with his back against the shutters. "It can be built up as an exact science by the reasoner. Our highest assurance of the goodness of Providence seems to me to rest in the flowers. All other things, our powers, our desires, our food, are all really necessary for our existence in the first instance. But this rose is an extra. Its smell and its color are an embellishment of life, not a condition of it. It is only goodness which gives extras, and so I say again that we have much to hope from the flowers.

[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.

There is nothing in which deduction is so necessary as in religion," said he, leaning with his back against the shutters. "It can be built up as an exact science by the reasoner. Our highest assurance of the goodness of Providence seems to me to rest in the flowers. All other things, our powers, our desires, our food, are all really necessary for our existence in the first instance. But this rose is an extra. Its smell and its colour are an embellishment of life, not a condition of it. It is only goodness which gives extras, and so I say again that we have much to hope from the flowers.

It cannot be denied that he has had many exceptional ideas, and that he is a highly intelligent man. For my part, however, I have always been taught to take a broad overview of things, in order to be able to deduce from them general rules, which might be applicable elsewhere.

She associated English accents with singing teapots, schools for witchcraft, and the science of deduction.

Mr Baley", said Quemot, "you can't treat human emotions as though they were built about a positronic brain". "I'm not saying you can. Robotics is a deductive science and sociology an inductive one. But mathematics can be made to apply in either case.

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.

What do you want to swear to?" she asks. "Science? Deductive reasoning? The ghost of Friedrich Nietzsche?

There can be no ultimate statements science: there can be no statements in science which can not be tested, and therefore none which cannot in principle be refuted, by falsifying some of the conclusions which can be deduced from them.

The sound of an English accent distracted her and lifted her spirits. She associated English accents with singing teapots, schools for witchcraft, and the science of deduction. This wasn't, she knew, terribly sophisticated of her, but she had no real guilt about it. She felt the English were themselves to blame for her feelings. They had spent a century relentlessly marketing their detectives and wizards and nannies, and they had to live with the results.

When we talk about the big bang or the fabric of space, what we are doing is not a continuation of the free and fantastic stories that humans have told nightly around campfires for hundreds of thousands of years. It is the continuation of something else: of the gaze of those same men in the first light of day looking at tracks left by antelope in the dust of the savannah - scrutinising and deducting from the details of reality in order to pursue something that we can't see directly but can follow the traces of. In the awareness that we can always be wrong, and therefore ready at any moment to change direction if a new track appears; but knowing also that if we are good enough we will get it right and will find what we are seeking. That is the nature of science.

Science of Deduction

The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms.

If the weight comes from bacon you can so deduct it off the scale total to get your true weight. #science

The sound of an English accent distracted her and lifted her spirits. She associated English accents with singing teapots, schools for witchcraft, and the science of deduction.

Like all other arts, the Science of Deduction and Analysis is one which can only be acquired by long and patient study nor is life long enough to allow any mortal to attain the highest possible perfection in it.

And some scientists and other intellectuals are convinced—too eagerly in my view—that the question of God's existence belongs in the forever inaccessible PAP category. From this, as we shall see, they often make the illogical deduction that the hypothesis of God's existence, and the hypothesis of his non-existence, have exactly equal probability of being right.

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

Even science is inductive, relying on observations and best explanations, not always deductive conclusions.

Geometry is a Deductive Science.

She associated English accents with singing teapots, schools for witchcraft, and the science of deduction. This wasn’t, she knew, terribly sophisticated of her, but she had no real guilt about it. She felt the English were themselves to blame for her feelings. They had spent a century relentlessly marketing their detectives and wizards and nannies, and they had to live with the results.

When we talk about the big bang or the fabric of space, what we are doing is not a continuation of the free and fantastic stories that humans have told nightly around campfires for hundreds of thousands of years. It is the continuation of something else: of the gaze of those same men in the first light of day looking at tracks left by antelope in the dust of the savannah—scrutinizing and deducting from the details of reality in order to pursue something that we can’t see directly but can follow the traces of. In the awareness that we can always be wrong, and therefore ready at any moment to change direction if a new track appears; but knowing also that if we are good enough we will get it right and will find what we are seeking. This is the nature of science. The confusion between these two diverse human activities—inventing stories and following traces in order to find something—is the origin of the incomprehension and distrust of science shown by a significant part of our contemporary culture. The separation is a subtle one: the antelope hunted at dawn is not far removed from the antelope deity in that night’s storytelling. The border is porous. Myths nourish science, and science nourishes myth. But the value of knowledge remains. If we find the antelope, we can eat.

If you can’t explain it simply you don’t understand it well enough. The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypothesis or axioms.

From a drop of water," said the writer, "a logician could infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other. So all life is a great chain, the nature of which is known whenever we are shown a single link of it. Like all other arts, the Science of Deduction and Analysis is one which can only be acquired by long and patient study nor is life long enough to allow any mortal to attain the highest possible perfection in it.

If perception is not absolute, no deduction from perception can be absolute. No matter how ingeniously one juggles with approximations, they do not magically turn into certainties; at best, they become the most accurate possible approximations.

If you suspect that you suck, you will probably suck. This is called the self-fulfilling prophecy. If you suspect you are great, you will probably suck. This is called the Dunning-Kruger effect. There’s an obvious deduction to be made here, but you probably missed it.

There is nothing distinctively scientific about the hypothetico-deductive process. It is not even distinctively intellectual. It is merely a scientific context for a much more general stratagem that underlies almost all regulative processes or processes of continuous control, namely feedback, the control of performance by the consequences of the act performed. In the hypothetico-deductive scheme the inferences we draw from a hypothesis are, in a sense, its logical output. If they are true, the hypothesis need not be altered, but correction is obligatory if they are false. The continuous feedback from inference to hypothesis is implicit in Whewell's account of scientific method; he would not have dissented from the view that scientific behaviour can be classified as appropriately under cybernetics as under logic.

The word 'proof' should strictly only be used when we are dealing with deductive inferences.... Popper claimed that scientists only need to use deductive inferences.... So if a scientist is only interested in demonstrating that a given theory is false, she may be able to accomplish her goal without the use of inductive inferences.... When a scientist collects experimental data, her aim might be to show that a particular theory...is false. She will have to resort to inductive reasoning.... So Popper's attempt to show that science can get by without induction does not succeed.

Generalizations in biology are almost invariably of a probabilistic nature. As one wit has formulated it, there is only one universal law in biology: 'All biological laws have exceptions.' This probabilistic conceptualization contrasts strikingly with the view during the early period of the scientific revolution that causation in nature is regulated by laws that can be stated in mathematical terms. Actually, this idea occurred apparently first to Pythagoras. It has remained a dominant idea, particularly in the physical sciences, up to the present day. Again and again it was made the basis of some comprehensive philosophy, but taking very different forms in the hands of various authors. With Plato it gave rise to essentialism, with Galileo to a mechanistic world picture, and with Descartes to the deductive method. All three philosophies had a fundamental impact on biology.

I'd left her in Sciences 442, after a long, trying day. The spectacularly bitchy text war she pitched with her brother wasn't even the worst of it. She didn't show me the original message she sent him, but I saw the ones he'd returned. No, you didn't find my spy, he insisted. He's obviously still at large. For instance, I can tell you right now that you're wearing all black, and that Jamie Watson is annoyed with you. I have eyes watching you right now. THAT IS NOT SPYING THAT IS SHODDY AMATEUR DEDUCTION AND IT IS INCORRECT, she replied furiously. She was, of course, wearing all black.

Cryptography is a science of deduction and controlled experiment; hypotheses are formed, tested and often discarded. But the residue which passes the test grows until finally there comes a point when the experimenter feels solid ground beneath his feet: his hypotheses cohere, and fragments of sense emerge from their camouflage. The code 'breaks'. Perhaps this is best defined as the point when the likely leads appear faster than they can be followed up. It is like the initiation of a chain-reaction in atomic physics; once the critical threshold is passed, the reaction propagates itself.

Imagine what would have happened had the logicist endeavor been entirely successful. This would have implied that mathematics stems fully from logic-literally from the laws of thought. But how could such a deductive science so marvelously fit natural phenomena? What is the relation between formal logic (maybe we should even say human formal logic) and the cosmos? The answer did not become any clearer after Hilbert and Godel. Now all that existed was an incomplete formal "game," expressed in mathematical language. How could models based on such an "unreliable" system produce deep insights about the universe and its workings?

Social psychology, the science of how people behave toward one another, is often a mishmash of interesting phenomena that are “explained” by giving them fancy names. Missing is the rich deductive structure of other sciences, in which a few deep principles can generate a wealth of subtle predictions—the kind of theory that scientists praise as “beautiful” or

Philosophy means and includes five fields of study and discourse: logic, aesthetics, ethics, politics, and metaphysics. Logic is the study of ideal method in thought and research: observation and introspection, deduction and induction, hypothesis and experiment, analysis and synthesis - such are the forms of human activity which logic tries to underhand the guide; it is a dull study for most of us, and yet the great events in the history of understand are the improvements men have made in their methods of thinking and research. Aesthetics is the study of ideal form, or beauty; it is the philosophy of art. Ethics is the study of ideal conduct; the highest knowledge, said Socrates, is the knowledge of good and evil, the knowledge of good and evil, the knowledge of the wisdom of life. Politics is the study of ideal social organization (it is not, as one might suppose, the art and science of capturing and keeping office); monarchy, aristocracy, democracy, socialism, anarchism, feminisim - these are the dramatis personae of political philosophy. And lastly, metaphysics (which gets into so much trouble because it is not, like the other forms of philosophy, an attempt to coordinate the real in the light of the ideal) is the study of the "ultimate reality" of all things: of the real and final nature of "matter" (ontology), of "mind" (philosophical psychology), and of the interrelation of "mind" and "matter" in the processes of perception and knowledge (epistemology).

Modern science has arrived at its conclusions through mathematical deduction. Religion has arrived at its conclusions through belief. But the yogi is a hard nut who will not settle for deduction or belief. He or she seeks to experience the ultimate through the enhancement of individual perception. Consequently, the yogic tradition does not mention God, or deny it either.

Free will is an idealization of human beings that makes the ethics game playable. Euclidean geometry requires idealizations like infinite straight lines and perfect circles, and its deductions are sound and useful even though the world does not really have infinite straight lines or perfect circles. The world is close enough to the idealization that the theorems can usefully be applied. Similarly, ethical theory requires idealizations like free, sentient, rational, equivalent agents whose behavior is uncaused, and its conclusions can be sound and useful even though the world, as seen by science, does not really have uncaused events. As long as there is no outright coercion or gross malfunction of reasoning, the world is close enough to the idealization of free will that moral theory can meaningfully be applied to it.

The Four Errors. Man has been reared by his errors: firstly, he saw himself always imperfect; secondly, he attributed to himself imaginary qualities; thirdly, he felt himself in a false position in relation to the animals and nature; fourthly, he always devised new tables of values, and accepted them for a time as eternal and unconditioned, so that at one time this, and at another time that human impulse or state stood first, and was ennobled in consequence. When one has deducted the effect of these four errors, one has also deducted humanity, humaneness, and "human dignity".

More important to me, however, was a comment by the Dalai Lama himself about the difference between Western science and Buddhism. Both disciplines, he argued, are based on observation and logical deduction, and both give experience precedence over belief in their quest for the truth. But, he added, Western scientists seem to seek understanding about how the world works in order to change and control the natural world. That is, the goal of most scientists is to gain mastery over our environment. Buddhists, on the other hand, seek understanding about how the world works in order to live more harmoniously with it.

All deductions having been made, democracy has done less harm, and more good, than any other form of government. It gave to human existence a zest and camaraderie that outweighed its pitfalls and defects. It gave to thought and science and enterprise the freedom essential to their operation and growth. It broke down the walls of privilege and class, and in each generation it raised up ability from every rank and place. Under its stimulus Athens and Rome became the most creative cities in history, and America in two centuries has provided abundance for an unprecedentedly large proportion of its population. Democracy has now dedicated itself resolutely to the spread and lengthening of education, and to the maintenance of public health. If equality of educational opportunity can be established, democracy will be real and justified. For this is the vital truth beneath its catchwords: that though men cannot be equal, their access to education and opportunity can be made more nearly equal. The rights of man are not rights to office and power, but the rights of entry into every avenue that may nourish and test a man’s fitness for office and power. A right is not a gift of God or nature but a privilege which it is good for the group that the individual should have.

The very possibility of mathematical science seems an insoluble contradiction. If this science is deductive in appearance only, from where does it get its perfect rigor that no one dares to doubt? If, on the contrary, all the propositions it sets forth can be derived from one another by the rules of formal logic, why is mathematics not reducible to an immense tautology? Syllogism can teach us nothing that is essentially new and, if everything originated in the principle of identity, it should also be possible to reduce everything to it. Are we then to concede that the statements of all those theorems filling so many volumes are merely roundabout ways of saying that A is A?

The power of the deductive network produced in physics has been illustrated in a delightful article by Victor F. Weisskopf. He begins by taking the magnitudes of six physical constants known by measurement: the mass of the proton, the mass and electric charge of the electron, the light velocity, Newton's gravitational constant, and the quantum of action of Planck. He adds three of four fundamental laws (e.g., de Broglie's relations connecting particle momentum and particle energy with the wavelength and frequency, and the Pauli exclusion principle), and shows that one can then derive a host of different, apparently quite unconnected, facts that happen to be known to us by observation separately ....

What a lovely thing a rose is!’ He walked past the couch to the open window, and held up the drooping stalk of a moss rose, looking down at the dainty blend of crimson and green. It was a new phase of his character to me, for I had never before seen him show any keen interest in natural objects. ‘There is nothing in which deduction is so necessary as in religion,’ said he, leaning with his back against the shutters. ‘It can be built up as an exact science by the reasoner. Our highest assurance of the goodness of Providence seems to me to rest in the flowers. All other things, our powers our desires, our food, are all really necessary for our existence in the first instance. But this rose is an extra. Its smell and its color are an embellishment of life, not a condition of it. It is only goodness which gives extras, and so I say again that we have much to hope from the flowers.

The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting? If, on the contrary, all the propositions it enunciates can be deduced one from another by the rules of formal logic, why is not mathematics reduced to an immense tautology? The syllogism can teach us nothing essentially new, and, if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill so many volumes are nothing but devious ways of saying A is A! ...Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive? ...If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism.

A Spinoza in poetry becomes a Machiavelli in philosophy. Mysticism is the scholastic of the heart, the dialectic of the feelings. So long as our scholastic education takes us back to antiquity and furthers the study of the Greek and Latin languages, we may congratulate ourselves that these studies, so necessary for the higher culture, will never disappear. If we set our gaze on antiquity and earnestly study it, in the desire to form ourselves thereon, we get the feeling as if it were only then that we really became men. The pedagogue, in trying to write and speak Latin, has a higher and grander idea of himself than would be permissible in ordinary life. If one has not read the newspapers for some months and then reads them all together, one sees, as one never saw before, how much time is wasted with this kind of literature. The classical is health; and the romantic, disease. When Nature begins to reveal her open secret to a man, he feels an irresistible longing for her worthiest interpreter, Art. For all other Arts we must make some allowance; but to Greek Art alone we are always debtors. The dignity of Art appears perhaps most conspicuously in Music; for in Music there is no material to be deducted. It is wholly form and intrinsic value, and it raises and ennobles all that it expresses. Art rests upon a kind of religious sense: it is deeply and ineradicably in earnest. Thus it is that Art so willingly goes hand in hand with Religion. Art is essentially noble; therefore the artist has nothing to fear from a low or common subject. Nay, by taking it up, he ennobles it; and so it is that we see the greatest artists boldly exercising their sovereign rights. Ignorant people raise questions which were answered by the wise thousands of years ago. To praise a man is to put oneself on his level. In science it is a service of the highest merit to seek out those fragmentary truths attained by the ancients, and to develop them further.

When Nature begins to reveal her open secret to a man, he feels an irresistible longing for her worthiest interpreter, Art. For all other Arts we must make some allowance; but to Greek Art alone we are always debtors. The dignity of Art appears perhaps most conspicuously in Music; for in Music there is no material to be deducted. It is wholly form and intrinsic value, and it raises and ennobles all that it expresses. Art rests upon a kind of religious sense: it is deeply and ineradicably in earnest. Thus it is that Art so willingly goes hand in hand with Religion. Art is essentially noble; therefore the artist has nothing to fear from a low or common subject. Nay, by taking it up, he ennobles it; and so it is that we see the greatest artists boldly exercising their sovereign rights. Ignorant people raise questions which were answered by the wise thousands of years ago. In science it is a service of the highest merit to seek out those fragmentary truths attained by the ancients, and to develop them further. The Classical is health, the Romantic disease.

Perhaps the most influential person ever associated with Samos was Pythagoras,* a contemporary of Polycrates in the sixth century B.C. According to local tradition, he lived for a time in a cave on the Samian Mount Kerkis, and was the first person in the history of the world to deduce that the Earth is a sphere. Perhaps he argued by analogy with the Moon and the Sun, or noticed the curved shadow of the Earth on the Moon during a lunar eclipse, or recognized that when ships leave Samos and recede over the horizon, their masts disappear last. He or his disciples discovered the Pythagorean theorem: the sum of the squares of the shorter sides of a right triangle equals the square of the longer side. Pythagoras did not simply enumerate examples of this theorem; he developed a method of mathematical deduction to prove the thing generally. The modern tradition of mathematical argument, essential to all of science, owes much to Pythagoras. It was he who first used the word Cosmos to denote a well-ordered and harmonious universe, a world amenable to human understanding.

Half a century ago Ostwald (1910) distinguished classicists and romanticists among the scientific investigators: the former being inclined to design schemes and to use consistently the deductions from working hypotheses; the latter being more fit for intuitive discoveries of functional relations between phenomena and therefore more able to open up new fields of study. Examples of both character types are Werner and Hutton. Werner was a real classicist. At the end of the eighteenth century he postulated the theory of “neptunism,” according to which all rocks including granites, were deposited in primeval seas. It was an artificial scheme, but, as a classification system, it worked quite satisfactorily at the time. Hutton, his contemporary and opponent, was more a romanticist. His concept of 'plutonism' supposed continually recurrent circuits of matter, which like gigantic paddle wheels raise material from various depths of the earth and carry it off again. This is a very flexible system which opens the mind to accept the possible occurrence in the course of time of a great variety of interrelated plutonic and tectonic processes.

Typically, proof of survival is held to standards that are rarely met in other areas of research, the hard sciences included. Much of what the hard sciences present as proven is more extrapolation from a set of effects than fact. If this and that are observed to happen, why they happen is deduced. From these deductions, a workable hypothesis is formed and then tested. We don’t know for sure, for instance, if there was ever a Big Bang, that stunning first moment in no-space, no-time, when something infinitely smaller than an atom exploded into what 13.7 billion years later would become the universe; nor do we know whether wormholes or even black holes actually exist. There has been no direct observation of these cosmic identities. The assumptions that they do exist derive from a set of discernible conditions that can best be explained — in the current state of our knowledge — by a bang or hole. The sorts of things astrophysicists and nuclear physicists now consider as probable conditions of reality also include equally fantastic notions such as the God particle (the Higgs boson), the many-worlds interpretation, string theory with its eleven dimensions — some of them “compactified” so we don’t see them — the zero-point field theory, and the hidden-worlds theory, which all read like the wildest science fiction and make any theory of postmortem survival look as dull as dishwater.

The part played by deduction in science is greater than Bacon supposed. Often, when a hypothesis has to be tested, there is a long deductive journey from the hypothesis to some consequence that can be tested by observation. Usually the deduction is mathematical, and in this respect Bacon underestimated the importance of mathematics in scientific investigation. The problem of induction by simple enumeration remains unsolved to this day. Bacon was quite right in rejecting simple enumeration where the details of scientific investigation are concerned, for in dealing with details we may assume general laws on the basis of which, so long as they are taken as valid, more or less cogent methods can be built up. John Stuart Mill framed four canons of inductive method, which can be usefully employed so long as the law of causality is assumed; but this law itself, he had to confess, is to be accepted solely on the basis of induction by simple enumeration. The thing that is achieved by the theoretical organization of science is the collection of all subordinate inductions into a few that are very comprehensive—perhaps only one. Such comprehensive inductions are confirmed by so many instances that it is thought legitimate to accept, as regards them, an induction by simple enumeration. This situation is profoundly unsatisfactory, but neither Bacon nor any of his successors have found a way out of it.

The main ones are the symbolists, connectionists, evolutionaries, Bayesians, and analogizers. Each tribe has a set of core beliefs, and a particular problem that it cares most about. It has found a solution to that problem, based on ideas from its allied fields of science, and it has a master algorithm that embodies it. For symbolists, all intelligence can be reduced to manipulating symbols, in the same way that a mathematician solves equations by replacing expressions by other expressions. Symbolists understand that you can’t learn from scratch: you need some initial knowledge to go with the data. They’ve figured out how to incorporate preexisting knowledge into learning, and how to combine different pieces of knowledge on the fly in order to solve new problems. Their master algorithm is inverse deduction, which figures out what knowledge is missing in order to make a deduction go through, and then makes it as general as possible. For connectionists, learning is what the brain does, and so what we need to do is reverse engineer it. The brain learns by adjusting the strengths of connections between neurons, and the crucial problem is figuring out which connections are to blame for which errors and changing them accordingly. The connectionists’ master algorithm is backpropagation, which compares a system’s output with the desired one and then successively changes the connections in layer after layer of neurons so as to bring the output closer to what it should be. Evolutionaries believe that the mother of all learning is natural selection. If it made us, it can make anything, and all we need to do is simulate it on the computer. The key problem that evolutionaries solve is learning structure: not just adjusting parameters, like backpropagation does, but creating the brain that those adjustments can then fine-tune. The evolutionaries’ master algorithm is genetic programming, which mates and evolves computer programs in the same way that nature mates and evolves organisms. Bayesians are concerned above all with uncertainty. All learned knowledge is uncertain, and learning itself is a form of uncertain inference. The problem then becomes how to deal with noisy, incomplete, and even contradictory information without falling apart. The solution is probabilistic inference, and the master algorithm is Bayes’ theorem and its derivates. Bayes’ theorem tells us how to incorporate new evidence into our beliefs, and probabilistic inference algorithms do that as efficiently as possible. For analogizers, the key to learning is recognizing similarities between situations and thereby inferring other similarities. If two patients have similar symptoms, perhaps they have the same disease. The key problem is judging how similar two things are. The analogizers’ master algorithm is the support vector machine, which figures out which experiences to remember and how to combine them to make new predictions.

Where has logic originated in men’s heads? Undoubtedly out of the illogical, the domain of which must originally have been immense. But numberless beings who reasoned otherwise than we do at present, perished; albeit that they may have come nearer to truth than we! Whoever, for example, could not discern the "like" often enough with regard to food, and with regard to animals dangerous to him, whoever, therefore, deduced too slowly, or was too circumspect in his deductions, had smaller probability of survival than he who in all similar cases immediately divined the equality. The dominant tendency however, to treat as equal that which is merely similar - an illogical tendency for there is nothing equal - is what first created the whole basis for logic. It was just so (in order that the conception of substance should originate, this being indispensable to logic, although in the strictest sense nothing actual corresponds to it) that for a long period the changing process in things had to be overlooked, and remain unperceived; the beings not seeing correctly had an advantage over those who saw everything "in flux" In itself every high degree of circumspection in conclusions, every skeptical inclination, is a great danger to life. No living being might have been preserved unless the contrary inclination to affirm rather than suspend judgment, to mistake and fabricate rather than wait, to assent rather than deny, to decide rather than be in the right had been cultivated with extra ordinary assiduity. The course of logical thought and reasoning in our modern brain corresponds to a process and struggle of impulses, which singly and in themselves are all very illogical and unjust; we experience usually only the result of the struggle, so rapidly and secretly does this primitive mechanism now operate in us.

The theory of relativity is a beautiful example of the basic character of the modern development of theory. That is to say, the hypotheses from which one starts become ever more abstract and more remote from experience. But in return one comes closer to the preeminent goal of science, that of encompassing a maximum of empirical contents through logical deduction with a minimum of hypotheses or axioms. The intellectual path from the axioms to the empirical contents or to the testable consequences becomes, thereby, ever longer and more subtle. The theoretician is forced, ever more, to allow himself to be directed by purely mathematical, formal points of view in the search for theories, because the physical experience of the experimenter is not capable of leading us up to the regions of the highest abstraction. Tentative deduction takes the place of the predominantly inductive methods appropriate to the youthful state of science. Such a theoretical structure must be quite thoroughly elaborated in order for it to lead to consequences that can be compared with experience. It is certainly the case that here, as well, the empirical fact is the all-powerful judge. But its judgment can be handed down only on the basis of great and difficult intellectual effort that first bridges the wide space between the axioms and the testable consequences. The theorist must accomplish this Herculean task with the clear understanding that this effort may only be destined to prepare the way for a death sentence for his theory. One should not reproach the theorist who undertakes such a task by calling him a fantast; instead, one must allow him his fantasizing, since for him there is no other way to his goal whatsoever. Indeed, it is no planless fantasizing, but rather a search for the logically simplest possibilities and their consequences.

Origin of the Logical. Where has logic originated in men’s heads? Undoubtedly out of the illogical, the domain of which must originally have been immense. But numberless beings who reasoned otherwise than we do at present, perished; albeit that they may have come nearer to truth than we! Whoever, for example, could not discern the "like" often enough with regard to food, and with regard to animals dangerous to him, whoever, therefore, deduced too slowly, or was too circumspect in his deductions, had smaller probability of survival than he who in all similar cases immediately divined the equality. The preponderating inclination, however, to deal with the similar as the equal - an illogical inclination, for there is no thing equal in itself - first created the whole basis of logic. It was just so (in order that the conception of substance should originate, this being indispensable to logic, although in the strictest sense nothing actual corresponds to it) that for a long period the changing process in things had to be overlooked, and remain unperceived; the beings not seeing correctly had an advantage over those who saw everything "in flux." In itself every high degree of circumspection in conclusions, every sceptical inclination, is a great danger to life. No living being might have been preserved unless the contrary inclination - to affirm rather than suspend judgment, to mistake and fabricate rather than wait, to assent rather than deny, to decide rather than be in the right - had been cultivated with extraordinary assiduity. - The course of logical thought and reasoning in our modern brain corresponds to a process and struggle of impulses, which singly and in themselves are all very illogical and unjust; we experience usually only the result of the struggle, so rapidly and secretly does this primitive mechanism now operate in us.

Origin of the Logical. — Where has logic originated in men's heads? Undoubtedly out of the illogical, the domain of which must originally have been immense. But numberless beings who reasoned otherwise than we do at present, perished; albeit that they may have come nearer to truth than we! Whoever, for example, could not discern the “like” often enough with regard to food, and with regard to animals dangerous to him, whoever, therefore, deduced too slowly, or was too circumspect in his deductions, had smaller probability of survival than he who in all similar cases immediately divined the equality. The preponderating inclination, however, to deal with the similar as the equal — an illogical inclination, for there is no thing equal in itself — first created the whole basis of logic. It was just so (in order that the conception of substance should originate, this being indispensable to logic, although in the strictest sense nothing actual corresponds to it) that for a long period the changing process in things had to be overlooked, and remain unperceived; the beings not seeing correctly had an advantage over those who saw everything “in flux.” In itself every high degree of circumspection in conclusions, every sceptical inclination, is a great danger to life. No living being might have been preserved unless the contrary inclination — to affirm rather than suspend judgment, to mistake and fabricate rather than wait, to assent rather than deny, to decide rather than be in the right — had been cultivated with extraordinary assiduity. — The course of logical thought and reasoning in our modern brain corresponds to a process and struggle of impulses, which singly and in themselves are all very illogical and unjust; we experience usually only the result of the struggle, so rapidly and secretly does this primitive mechanism now operate in us.

Absolute continuity of motion is not comprehensible to the human mind. Laws of motion of any kind become comprehensible to man only when he examines arbitrarily selected elements of that motion; but at the same time, a large proportion of human error comes from the arbitrary division of continuous motion into discontinuous elements. There is a well known, so-called sophism of the ancients consisting in this, that Achilles could never catch up with a tortoise he was following, in spite of the fact that he traveled ten times as fast as the tortoise. By the time Achilles has covered the distance that separated him from the tortoise, the tortoise has covered one tenth of that distance ahead of him: when Achilles has covered that tenth, the tortoise has covered another one hundredth, and so on forever. This problem seemed to the ancients insoluble. The absurd answer (that Achilles could never overtake the tortoise) resulted from this: that motion was arbitrarily divided into discontinuous elements, whereas the motion both of Achilles and of the tortoise was continuous. By adopting smaller and smaller elements of motion we only approach a solution of the problem, but never reach it. Only when we have admitted the conception of the infinitely small, and the resulting geometrical progression with a common ratio of one tenth, and have found the sum of this progression to infinity, do we reach a solution of the problem. A modern branch of mathematics having achieved the art of dealing with the infinitely small can now yield solutions in other more complex problems of motion which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when it deals with separate elements of motion instead of examining continuous motion. In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable arbitrary human wills, is continuous. To understand the laws of this continuous movement is the aim of history. But to arrive at these laws, resulting from the sum of all those human wills, man's mind postulates arbitrary and disconnected units. The first method of history is to take an arbitrarily selected series of continuous events and examine it apart from others, though there is and can be no beginning to any event, for one event always flows uninterruptedly from another. The second method is to consider the actions of some one man—a king or a commander—as equivalent to the sum of many individual wills; whereas the sum of individual wills is never expressed by the activity of a single historic personage. Historical science in its endeavor to draw nearer to truth continually takes smaller and smaller units for examination. But however small the units it takes, we feel that to take any unit disconnected from others, or to assume a beginning of any phenomenon, or to say that the will of many men is expressed by the actions of any one historic personage, is in itself false. It needs no critical exertion to reduce utterly to dust any deductions drawn from history. It is merely necessary to select some larger or smaller unit as the subject of observation—as criticism has every right to do, seeing that whatever unit history observes must always be arbitrarily selected. Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.

At this point, the cautious reader might wish to read over the whole argument again, as presented above, just to make sure that I have not indulged in any 'sleight of hand'! Admittedly there is an air of the conjuring trick about the argument, but it is perfectly legitimate, and it only gains in strength the more minutely it is examined. We have found a computation Ck(k) that we know does not stop; yet the given computational procedure A is not powerful enough to ascertain that facet. This is the Godel(-Turing) theorem in the form that I require. It applies to any computational procedure A whatever for ascertaining that computations do not stop, so long as we know it to be sound. We deduce that no knowably sound set of computational rules (such as A) can ever suffice for ascertaining that computations do not stop, since there are some non-stopping computations (such as Ck(k)) that must elude these rules. Moreover, since from the knowledge of A and of its soundness, we can actually construct a computation Ck(k) that we can see does not ever stop, we deduce that A cannot be a formalization of the procedures available to mathematicians for ascertaining that computations do not stop, no matter what A is. Hence: (G) Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth. It seems to me that this conclusion is inescapable. However, many people have tried to argue against it-bringing in objections like those summarized in the queries Q1-Q20 of 2.6 and 2.10 below-and certainly many would argue against the stronger deduction that there must be something fundamentally non-computational in our thought processes. The reader may indeed wonder what on earth mathematical reasoning like this, concerning the abstract nature of computations, can have to say about the workings of the human mind. What, after all, does any of this have to do with the issue of conscious awareness? The answer is that the argument indeed says something very significant about the mental quality of understanding-in relation to the general issue of computation-and, as was argued in 1.12, the quality of understanding is something dependent upon conscious awareness. It is true that, for the most part, the foregoing reasoning has been presented as just a piece of mathematics, but there is the essential point that the algorithm A enters the argument at two quite different levels. At the one level, it is being treated as just some algorithm that has certain properties, but at the other, we attempt to regard A as being actually 'the algorithm that we ourselves use' in coming to believe that a computation will not stop. The argument is not simply about computations. It is also about how we use our conscious understanding in order to infer the validity of some mathematical claim-here the non-stopping character of Ck(k). It is the interplay between the two different levels at which the algorithm A is being considered-as a putative instance of conscious activity and as a computation itself-that allows us to arrive at a conclusion expressing a fundamental conflict between such conscious activity and mere computation.

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

In light of Hipparchus and Ptolemy’s extraordinarily successful predictive theories, the goal of astronomy was to find the right combination of circles to describe the motion of the heavenly bodies around the Earth. Contrary to expectations, it turned out that Earth was itself one of the heavenly bodies. After Copernicus, the goal appeared to be to find the right combination of moving spheres that would reproduce the motion of the planets around the Sun. Contrary to expectations, it turned out that abstract elliptical trajectories were better than spheres. After Newton, it seemed clear that the aim of physics was to find the forces acting on bodies. Contrary to this, it turned out that the world could be better described by dynamical fields rather than bodies. After Faraday and Maxwell, it was clear that physics had to find laws of motion in space, as time passes. Contrary to assumptions, it turned out that space and time are themselves dynamical. After Einstein, it became clear that physics must only search for the deterministic laws of Nature. But it turned out that we can at best give probabilistic laws. And so on. Here are some sliding definitions for what scientists have thought science to be: deduction of general laws from observed phenomena, finding out the ultimate constituents of Nature, accounting for regularities in empirical observations, finding provisional conceptual schemes for making sense of the world. (The last one is the one I like.) Science is not a project with a methodology written in stone, or a fixed conceptual structure. It is our ever-evolving endeavor to better understand the world. In the course of its development, it has repeatedly violated its own rules and its own stated methodological assumptions.

Popper demonstrated that science can proceed in a truly deductive manner through a process of conjectures and refutations. As long as we concede that all our scientific theories are held tentatively, not as “truths” but as conjectures that may only be approximations to the truth, then Hume’s dilemma is resolved. They may be falsified by a deductive procedure but never verified in any logically valid manner. In other words, scientific theories are not verified by observations consistent with them; rather, they are corroborated by unsuccessful attempts at refutation.

When we talk about the Big Bang or the fabric of space, what we are doing is not a continuation of the free and fantastic stories which humans have told nightly around campfires for hundreds of thousands of years. It is the continuation of something else: of the gaze of those same men in the first light of day looking at tracks left by antelope in the dust of the savannah – scrutinizing and deducting from the details of reality in order to pursue something which we can’t see directly but can follow the traces of. In the awareness that we can always be wrong, and therefore ready at any moment to change direction if a new track appears; but knowing also that if we are good enough we will get it right and will find what we are seeking. This is the nature of science.

All science is full of statements where you put the best face on your ignorance, where you say: true enough, we know awfully little about this, but more or less irrespective of the stuff we don’t know about, we can make certain useful deductions.

The basic point of all the scientific ideas we threw at you is that there is a lot of disagreement about how the flow of time works and how or whether one thing causes another. If you take home one idea out of all of these, make it that the everyday feeling that the future has no effect on the present is not necessarily true. As a result of the current uncertainty about time and causality in philosophical and scientific circles, it is not at all unreasonable to talk in a serious way about the possibility of genuine precognition. We also hope that our brief mention of spirituality has opened your mind to the idea that there may be a spiritual perspective as well. Both Theresa and Julia treasure the spiritual aspects of precognition, because premonitions can act as reminders that there may be an eternal part of us that exists outside of time and space. There may well be a scientific explanation for this eternal part, and if one is found, science and spirituality will become happy partners. Much of Part 2 will be devoted to the spiritual and wellbeing components of becoming a Positive Precog, and we will continue to marry those elements with scientific research as we go. 1 Here, physics buffs might chime in with some concerns about the Second Law of Thermodynamics. Okay, physics rock stars! As you know, the Second Law states that in a closed system, disorder is very unlikely to decrease – and as such, you may believe this means that there is an “arrow of time” that is set by the Second Law, and this arrow goes in only the forward direction. As a result, you might also think that any talk of a future event influencing the past is bogus. We would ask you to consider four ideas. 2 Here we are not specifically talking about closed timelike curves, but causal loops in general. 3 For those concerned that the idea of messages from the future suggests such a message would be travelling faster than the speed of light, a few thoughts: 1) “message” here is used colloquially to mean “information” – essentially a correlation between present and future events that can’t be explained by deduction or induction but is not necessarily a signal; 2) recently it has been suggested that superluminal signalling is not actually prohibited by special relativity (Weinstein, S, “Superluminal signaling and relativity”, Synthese, 148(2), 2006: 381–99); and 3) the no-signalling theorem(s) may actually be logically circular (Kennedy, J B, “On the empirical foundations of the quantum no-signalling proofs”, Philosophy of Science, 62(4), 1995: 543–60.) 4 Note that in the movie Minority Report, the future was considered set in stone, which was part of the problem of the Pre-Crime Programme. However, at the end of the movie it becomes clear that the future envisioned did not occur, suggesting the idea that futures unfold probabilistically rather than definitely.

Science is a method that oscillates between induction and deduction—we observe patterns, propose explanations, and test them to see how well they predict things we do not yet know. We thus generate models of the world that, when we do the scientific work correctly, achieve three things: they predict more than what came before, assume less, and come to fit with one another, merging into a seamless whole.

The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle. Are we then to admit that the enunciations of all the theorems with which so many volumes are filled, are only indirect ways of saying that A is A?

Phenomenology calls us to return, as Husserl put it, “to the things themselves.” By “things” (Sachen) Husserl meant not real (concrete) objects, but the ideal (abstract) forms and contents of experience as we live them, not as we have learned to conceive and describe them according to the categories of science and received opinion. Phenomenology is thus a descriptive, not an explanatory or deductive enterprise, for it aims to reveal experience as such, rather than frame hypotheses or speculate beyond its bounds.

Every conclusion supposes premises; these premises themselves either are self-evident and need no demonstration, or can be established only by relying upon other propositions, and since we can not go back thus to infinity, every deductive science, and in particular geometry, must rest on a certain number of undemonstrable axioms.

We may still insist that space, although mainly emptiness, is completely “contaminated” by all the forces in the Universe—particles, waves, and so on. Still, based on this deduction or inference, we cannot prove that what is curved is space since nothing remains nothing and cannot be transformed into something based on the laws of physics. If the Nothing does not convert into something, the curvature of space is impossible. To prove the curvature of nothing (space), we must prove that nothing can transform itself into something based on the laws of physics. Suppose we further insist that this is only a problem of linguistics and philosophy and not physics since we stated that space is how we define it and not as it is. This reasoning would be insufficient because we must first prove that actual space is what it is and not what we say it is to fit our arguments. There can be no answers to the most critical questions of contemporary science, physics, philosophy, and even religions if we are not as precise as possible, linguistically, experimentally, or in any other way. These questions appear to be self-evident, self-explanatory truths and axioms, yet they are neither self-evident nor clear and precise to reflect the actual underlying reality. If we are not as precise as possible, we create theories and paradigms presented as facts, although the starting premises are undefined and unanswered. However, if we do not answer the starting causes and premises, we cannot adequately describe the laws of nature, nor can we understand them. Regardless of how sure we are about space and the curvature of space, we still cannot claim we are correct without describing the nature of nothing. Without the Nothing, there can be no space. There will be an immediate argument, though, that the Nothing has no nature and, therefore, there is nothing to describe. The answer is that its passivity and lack of properties are its most potent “property” because they enable creation and existence. Without the Nothing or void, there is no creation and no existence. We converted nothing into something by our thoughts and language, using deduction or inference, and concluded that space is the consequence of this thought process, not the actual process. We applied our definite language to our indefinite “understanding,” ideas, or reasoning to prove the “fact.” However, the fact is or is not, regardless of our ideas, language, or reasoning. We must use the opposite and apply language to the facts rather than our understanding. Our understanding is limited, and facts are impersonal and independent of our knowledge. We cannot falsify the language to fit and understand the facts better. We cannot change facts or affect them with our ideas, but facts are verifiable up to a point. There are facts beyond the verifiable point by humans since human beings are limited. Still, language is verifiable, and our theories are both falsifiable and verifiable to a large extent.

It’s a matter of balance between deduction and induction—between reason and empiricism—and in 1620 the English philosopher Francis Bacon published his Novum Organum, or “new instrument,” which described science as a blend of sensory data and reasoned theory. Ideally, Bacon argued, one should begin with observations, then formulate a general theory from which logical predictions can be made, then check the predictions against experiment.37 If you don’t give yourself a reality check you end up with half-baked (and often fully baked) ideas,

Infidels construct their [124] theories from the supposed deductions of sciences, and reject the revealed word of God. They presume to pass sentence upon God’s moral government; they despise his law and boast of the sufficiency of human reason. Then, “because sentence against an evil work is not executed speedily, therefore the heart of the sons of men is fully set in them to do evil.” Ecclesiastes 8:11.

I am a practitioner of the science of deduction, of using the known facts in a case to unveil the unknown.’ - Sherlock Holmes, Sherlock Holmes and the Whitechapel Murders

Agnostic philosopher Antony O’Hear agrees: ‘there are aspects of our experience and existence more fundamental than science, and on which science depends for its possibility. So science cannot be used, as it often is, to undermine those features of our natures.’35 The idea that the mind is ‘nothing but’ the brain is ultimately a deduction from the assumption of naturalism.

The Baconian view gave birth to a complementary approach. Here, the earlier deductive thinking made way for an inductive or empirical method in science. Instead of starting from classically derived principles and theories one now started with observation.

Chapter 1: What to Expect in Investing   Similar to other concepts that you may grasp, there is no “real way” to prepare for investing. Initially, you just have to jump in to start learning how you can manage different aspects of this activity as you go along. However, you still have to make the necessary preparations by learning the fundamental techniques and other intricacies included in this art and science. Also included in the things you should know are the expectations that you need to have when investing. "You need to expect tax deductions from the government.

When we talk about the big bang or the fabric of space, what we are doing is not a continuation of the free and fantastic stories that humans have told nightly around campfires for hundreds of thousands of years. It is the continuation of something else: of the gaze of those same men in the first light of day looking at tracks left by antelope in the dust of the savannah—scrutinizing and deducting from the details of reality in order to pursue something that we can’t see directly but can follow the traces of. In the awareness that we can always be wrong, and therefore ready at any moment to change direction if a new track appears; but knowing also that if we are good enough we will get it right and will find what we are seeking. This is the nature of science.

One of the first things I learned when I began teaching courses in critical thinking in 1974 was that those of us trained in philosophy needed to supplement our philosophical training with the study of various cognitive, perceptual, and affective biases or illusions. We had a lot to learn from the social scientists if we were to teach our students to think critically. Identifying fallacies, learning to test deductive arguments for validity, and the like would not be enough.

According to L. E. J . Brouwer, the founder of the intuitionist school, the simplest mathematical ideas are implied in the customary lines of thought of everyday life and all sciences make use of them; the mathematician is distinguished by the fact that he is conscious of these ideas, points them out clearly, and completes them. The only source of mathematical knowledge is, in Brouwer's opinion, the intuition that makes us recognize certain concepts and conclusions as absolutely evident, clear, and indubitable. However, he does not assume that it is possible to list in a precise and complete way all basic fundamental concepts and elementary methods of deduction, which in this sense are to serve as a basis of mathematical derivations. It should always be possible to supplement the once fixed set of assumptions by accepting new ones, if a further intuition leads that way.

It is obvious, to my eyes anyway, that if one eliminates the supernatural element from the universe, it is necessary to eliminate, at the same time, a number of fundamental notions—all of which were, in the past, deduced from supernatural tenets. All of morality and all of the ideas that serve as a base for law and for the political sciences, in their present-day conceptions, constitute a series of deductions of which the first term assumes the existence of a personal divinity… Remove all validity from this source, and there is nothing left.

The only religion that could ever be taken seriously is rationalism religion. The likes of Pythagoras, Plato, Aristotle, Descartes, Leibniz, Hegel, and Godel have done the hard work necessary to make such a religion viable. Wouldn't you want a religion that every logician, mathematician, philosopher, and scientist on earth could embrace? In fact, that religion already exists. It's called ontological mathematics, predicated on the principle of sufficient reason and Occam's razor. It constitutes a coherent, holistic, a priori, rationalist, analytic, deductive religion, metaphysics and physics. Ontological mathematics explains all.

As another brilliant logician, Bertrand Russell, pointed out, Abelard tended to overrate the value of Aristotelian logic and of deduction as the only path to truth. Abelard had no interest in the one sphere where inductive reasoning is most important to yielding new knowledge, namely science.35 Aristotle’s writings on that subject were still unavailable to him. Reason for Abelard and his followers was a tool for understanding what was already known, especially about God and the Bible, not opening new vistas for human investigation.

Inductive logic offers a source of new knowledge, based on empirical observation. Aristotle recognized the value of induction; his own sciences were founded on it.13 Still, his real focus was always the logic of deduction.

The cellularists, it is but fair to recall, regarding the cellule as the simplest anatomical element, believed it proceeded necessarily from a former cellule, omnis cellula e cellula, holding it to be the vital unit, living per se, and regarded an entire organism as the sum of these units. But we now know that that was a deduction from incomplete and superficial observations, for the cellule, a transitory anatomical element, has the microzyma for its anatomical element. It is this which alone possesses all the characters of an anatomical element, living per se, and which must be regarded as the unit of life. It is what I have already stated in the following terms: The microzyma is at the beginning and at the end of every living organization. It is the fundamental anatomical element whereby the cellules, the tissues, the organs, the whole, of an organism are constituted living.

Artists recontextualize reality and offer visions that were previously invisible. Creativity engages the brain’s daydreaming mode directly and stimulates the free flow and association of ideas, forging links between concepts and neural nodes that might not otherwise be made. In this way, engagement in art as either a creator or consumer helps us by hitting the reset button in our brains. Time stops. We contemplate. We reimagine our relationship to the world. Being creative means allowing the nonlinear to intrude on the linear, and to exercise some control over the output. The major achievements in science and art over the last several thousand years required induction, rather than deduction—required extrapolating from the known to the unknown and, to a large extent, blindly guessing what should come next and being right some of the time.

As the present civilization now stands, there are two education methods. One utilizes what is called Science, the other what is called Religion. As far as science is concerned, the deductions and speculations are based on observation and logic. The logic may be good enough; but the powers of observation, upon which the fundamentals of its logic rest, are restricted to the very imperfect faculties of sensual perception, and therefore your science is based entirely upon external illusions and is consequently a superficial and illusive science, knowing nothing about the inner life, which is far more important than external phenomena. Scientific doctrines regarding the fundamental laws of Nature are wrong, and therefore all the deductions are wrong as soon as it leaves the plane of illusions.

Different cultures have different systems for learning in part because of the philosophers who influenced the approach to intellectual life in general and science in particular. Although Aristotle, a Greek, is credited with articulating applications-first thinking (induction), it was British thinkers, including Roger Bacon in the thirteenth century and Francis Bacon in the sixteenth century, who popularized these methodologies among modern scholars and scientists. Later, Americans, with their pioneer mentality and disinclination toward theoretical learning, came to be even more applications-first than the British. By contrast, philosophy on the European continent has been largely driven by principles-first approaches. In the seventeenth century, Frenchman René Descartes spelled out a method of principles-first reasoning in which the scientist first formulates a hypothesis, then seeks evidence to prove or disprove it. Descartes was deeply skeptical of data based on mere observation and sought a deeper understanding of underlying principles. In the nineteenth century, the German Friedrich Hegel introduced the dialectic model of deduction, which reigns supreme in schools in Latin and Germanic countries. The Hegelian dialectic begins with a thesis, or foundational argument; this is opposed by an antithesis, or conflicting argument; and the two are then reconciled in a synthesis.

There is, however, a completely satisfactory way of avoiding the paradoxes without betraying our science. The desires and attitudes which help us find this way and show us what direction to take are these: 1. Wherever there is any hope of salvage, we will carefully investigate fruitful definitions and deductive methods. We will nurse them, strengthen them, and make them useful. No one shall drive us out of the paradise which Cantor has created for us. 2. We must establish throughout mathematics the same certitude for our deductions as exists in ordinary elementary number theory, which no one doubts and where contradictions and paradoxes arise only through our own carelessness.

The mathematical technique of modelling consists of ignoring this trouble and speaking about your deductive model in such a way as if it coincided with reality. The fact that this path, which is obviously incorrect from the point of view of natural science, often leads to useful results in physics is called "the inconceivable effectiveness of mathematics in natural sciences" (or "the Wigner principle").

There is one thing I enjoy about STEM: I love how words such as therefore, because, since, and thus can often be used to deeply comprehend a topic in math and science. These words all precede some form of logical deduction, and that is what makes STEM so beautiful: with math and science, you can always learn the logic behind everything. From quantum mechanics to biomedicine, science always finds a way to explain the universe.

Nerdy jokes are amazing, but the greatest form of nerdy jokes are STEM jokes. While STEM jokes, like any other form of nerdy jokes, consist of something witty and possibly a little awkward, they are a unique and noteworthy category of jokes, as they require technical knowledge to be immediately comprehended. For those who do not specialize in the scientific discipline that is being referenced in a joke, a little bit of logical deduction and inference would be required to understand the joke. They are like riddles.

In game theory, as in applications of other technologies that use RPT [Revealed Preference Theory], the purpose of the machinery is to tell us what happens when patterns of behavior instantiate some particular strategic vector, payoff matrix, and distribution of information—for example, a PD [Prisoner's Dilemma]—that we’re empirically motivated to regard as a correct model of a target situation. The motivational history that produced this vector in a given case is irrelevant to which game is instantiated, or to the location of its equilibrium or equilibria. As Binmore (1994, pp. 95–256) emphasizes at length, if, in the case of any putative PD, there is any available story that would rationalize cooperation by either player, then it follows as a matter of logic that the modeler has assigned at least one of them the wrong utility function (or has mistakenly assumed perfect information, or has failed to detect a commitment action) and so made a mistake in taking their game as an instance of the (one-shot) PD. Perhaps she has not observed enough of their behavior to have inferred an accurate model of the agents they instantiate. The game theorist’s solution algorithms, in themselves, are not empirical hypotheses about anything. Applications of them will be only as good, for purposes of either normative strategic advice or empirical explanation, as the empirical model of the players constructed from the intentional stance is accurate. It is a much-cited fact from the experimental economics literature that when people are brought into laboratories and set into situations contrived to induce PDs, substantial numbers cooperate. What follows from this, by proper use of RPT, not in discredit of it, is that the experimental setup has failed to induce a PD after all. The players’ behavior indicates that their preferences have been misrepresented in the specification of their game as a PD. A game is a mathematical representation of a situation, and the operation of solving a game is an exercise in deductive reasoning. Like any deductive argument, it adds no new empirical information not already contained in the premises. However, it can be of explanatory value in revealing structural relations among facts that we otherwise might not have noticed.

I often said I wouldn't have pursued programming as a career if I still did drugs. This is probably true, since weed was always immensely crippling for me. I would have weed hangovers for days, and while stoned, was unable to read or do much of anything besides clean and play video games. Whether or not this would have turned out to be true is academic, but it's definitely true that I wouldn't have become a programmer if I hadn't lost my mind, because the recovery process taught me my most valuable skill as a programmer: how to not think. Programming requires the acceptance that you are entering meaningless symbols into a machine that's going to spit out other meaningless symbols, and this can be hard to accept. It requires abandoning all hope for an answer for the existential "why?" in favour of shuffling boolean values ad infinitum. By no interpretation of the concept of understanding does a computer understand what you're telling it or what it's telling you. On top of that, programming as an act is more often hindered than helped by thinking. Despite zero years of training in computer science, I've found I have an edge in debugging because I never look or ask for an explanation. Ninety percent of the computer bugs in a program are tiny, one-line errors, and you just have to find that error. Holding the entire logical structure of a million lines of code in your mind in futile. The task is to find the references and connections and track them back until you hit the problem. If I get an error message, I copy it into Google, because someone somewhere has encountered and solved the problem, probably by tracking down the people who originally wrote the program. In seven years of programming, I've solved exactly two undocumented bugs via pure deductive reasoning.

The whole shtick of scientism revolves around sensory evidence. How many times must it be said that there is no such thing as self-explanatory sensory evidence? All evidence must be interpreted, and the interpretation is not a perceiving activity but a judging activity. The catastrophic error that worshipers of scientism (autistic sensing types) commit is that they privilege perceiving over judging. They believe that perception is the most important thing, and that judgment must be directed to the maximum degree possible at the perception, and minimize and indeed eliminate any reference to anything that has not been perceived. For worshipers of scientism, perception comes first, and judging is secondary, determined by perceiving. That is what empiricism is all about. It claims that all knowledge comes from experience, that there are no innate ideas, and it revolves around synthetic propositions and a posteriori knowledge. Rationalism, by total contrast, asserts that knowledge comes from logical, rational deduction and that innate ideas form the only secure basis for knowledge. It deals with analytic propositions and a priori knowledge.

Where does scientific knowledge come from? You know the process. A good scientist pushes to the edge of knowledge and then reaches beyond, forming a conjecture—a hypothesis—about how things work in that unknown territory. If the scientist avoids the edge, working with what is already well known and established, life will be comfortable, but there will be neither fame nor honor. In the same way, a good business strategy deals with the edge between the known and the unknown. Again, it is competition with others that pushes us to edges of knowledge. Only there are found the opportunities to keep ahead of rivals. There is no avoiding it. That uneasy sense of ambiguity you feel is real. It is the scent of opportunity. In science, you first test a new conjecture against known laws and experience. Is the new hypothesis contradicted by basic principles or by the results of past experiments? If the hypothesis survives that test, the scientist has to devise a real-world test—an experiment—to see how well the hypothesis stands up. Similarly, we test a new strategic insight against well-established principles and against our accumulated knowledge about the business. If it passes those hurdles, we are faced with trying it out and seeing what happens. Given that we are working on the edge, asking for a strategy that is guaranteed to work is like asking a scientist for a hypothesis that is guaranteed to be true—it is a dumb request. The problem of coming up with a good strategy has the same logical structure as the problem of coming up with a good scientific hypothesis. The key differences are that most scientific knowledge is broadly shared, whereas you are working with accumulated wisdom about your business and your industry that is unlike anyone else’s. A good strategy is, in the end, a hypothesis about what will work. Not a wild theory, but an educated judgment. And there isn’t anyone more educated about your businesses than the group in this room. This concept breaks the impasse. After some discussion, the group begins to work with the notion that a strategy is a hypothesis—an educated guess—about what will work. After a while, Barry starts to articulate his own judgments, saying “I think in my business we can …” When engineers use a nice clean deductive system to solve a problem, they call it winding the crank. By this they mean that it may be hard work, but that the nature and quality of the output depends on the machine (the chosen system of deduction), not on the skill of the crank winder. Later, looking back, I realize the group had expected strategy to be an exercise in crank winding. They had expected that I would give them a “logical machine” that they could use to deduce business plans—a system for generating forecasts and actions.

I deny your deduction. It may be because we think overmuch—in our science, our profession, our jurisprudence, our intellectual composition, our political career, or whatever be the pursuit which we follow—that we are disinclined to think in a place of mere amusement, after our dinners." "It would seem, then, that the decline of the drama resolves itself into a mere question of eating." "You are very perverse," said Denzil. "What I say is,—that the mind is always so highly strained at its work in our day, that it refuses to make any additional effort in its mere relaxations. When you have been thinking all day, with little pause or peace, you do not want to think in the evening, when your mental strain is relaxed: you want light, gayety, noise, pretty pictures—something that needs no thought whatever.

observation-model-prediction. That is, in science, you figure out what happened, then figure out why it happened, then, using that knowledge, cleverly deduce what may happen in the future. An example would be Isaac Newton observing an apple falling to Earth, deducing universal gravitation, and from his deductions realizing that the moon orbits the Earth because of Earth’s gravitational pull. He can then generalize this logic to predict the future: he may realize that an apple will fall on the surface of any heavenly body, like the moon or Mars, and not just the Earth. This is science, running from cause to effect, observation to prediction, and past to future. Accounting is materially the opposite. In accounting, you start out with what is going to happen in the future: ie: that you are not going to pay any taxes, then figure out why you’re not going to pay any taxes, and then, from this, decide what, for tax purposes, happened in the past.

Ontological mathematics: science with the scientific method relegated to the study of empirical, phenomenal mathematics only, and the mathematical method used to study noumenal, metaphysical reality: ultimate existence itself, i.e. the very thing that religion sought to understand. Ontological mathematics is religion, philosophy and science transmuted into a perfect, unified, closed, analytic, a priori, deductive, logical, complete and consistent rational system.

Chesterton's topic is nothing less than the fundamental contrast between deductive logic, true of all possible worlds, and inductive logic, capable only of telling us how we may reasonably expect this world to behave. Let us hasten to add that Chesterton's analysis is in full agreement with the views of modern logicians. Perhaps his "test of the imagination" is not strictly accurate--who can "imagine" the four-dimensional constructions of relativity?-but in essence his position is unassailable. Logical and mathematical statements are true by definition. They are "empty tautologies," to use a current phrase, like the impressive maxim that there are always six eggs in half a dozen. Nature, on the other hand, is under no similar constraints. Fortunately, her "weird repetitions," as GK calls them, often conform to surprisingly low-order equations. But as Hume and others before Hume made clear, there is no logical reason why she should behave so politely.

Think of facts and figures as bricks and cement, and science or scientific understanding as a building. Without the vision of the architect, it's impossible to construct the building no matter how much bricks and cement you have.

God cautions us in Isaiah 55:9 that his ways are not ours and his thoughts are higher than our thoughts (undoubtedly one of the grander understatements). God is warning us that he is not logical and that believing him to be logical will lead to all kinds of disappointment. Logic has been defined as 'the science or history of the human mind, as it traces the progress of our knowledge from our first conceptions through their different combinations, and the numerous deductions that result from comparing them with one another.' Doesn't sound much like God. Yet, we so often strain our relatively minuscule brains to conceive, combine, compare, and deduce. Then we fault God when his conclusions disagree. The repetition of this useless exercise leads to a form of insanity which ultimately manifests in denial of the existence of such an illogical God.

Romance is the deepest thing in life; romance is deeper even than reality. For even if reality could be proved to be misleading, it still could not be proved to be unimportant or unimpressive. Even if the facts are false, they are still very strange. And this strangeness of life, this unexpected and even perverse element of things as they fall out, remains incurably interesting. ... People wonder why the novel is the most popular form of literature; people wonder why it is read more than books of science or books of metaphysics. The reason is very simple; it is merely that the novel is more true than they are. Life may sometimes legitimately appear as a book of science. Life may 102 sometimes appear, and with a much greater legitimacy, as a book of metaphysics. But life is always a novel. Our existence may cease to be a song; it may cease even to be a beautiful lament. Our existence may not be an intelligible justice, or even a recognizable wrong. But our existence is still a story. In the fiery alphabet of every sunset is written, “to be continued in our next.” If we have sufficient intellect, we can finish a philosophical and exact deduction, and be certain that we are finishing it right. With the adequate brain-power we could finish any scientific discovery, and be certain that we were finishing it right. But not with the most gigantic intellect could we finish the simplest or silliest story, and be certain that we were finishing it right.

People wonder why the novel is the most popular form of literature; people wonder why it is read more than books of science or books of metaphysics. The reason is very simple; it is merely that the novel is more true than they are. Life may sometimes legitimately appear as a book of science. Life may sometimes appear, and with a much greater legitimacy, as a book of metaphysics. But life is always a novel. Our existence may cease to be a song; it may cease even to be a beautiful lament. Our existence may not be an intelligible justice, or even a recognizable wrong. But our existence is still a story. In the fiery alphabet of every sunset is written, “to be continued in our next.” If we have sufficient intellect, we can finish a philosophical and exact deduction, and be certain that we are finishing it right. With the adequate brain-power we could finish any scientific discovery, and be certain that we were finishing it right. But not with the most gigantic intellect could we finish the simplest or silliest story, and be certain that we were finishing it right.