… the three positive characteristics that distinguish mathematical knowledge from other knowledge … may be briefly expressed as follows: first, mathematical knowledge bears more distinctly the imprint of truth on all its results than any other kind of knowledge; secondly, it is always a sure preliminary step to the attainment of other correct knowledge; thirdly, it has no need of other knowledge.

“If there are two theories, one simpler man the other, the simpler one is to be preferred.” At first sight this does not seem quite so bad, but a little thought shows that our tendency to prefer the simpler possibility is psychological rather than scientific. It is less trouble to think that way. Experience invariably shows that the more correct a theory becomes, the more complex does it seem. … So this … interpretation of [Ockham’s Razor] is … worthless.

[E.H.] Moore was presenting a paper on a highly technical topic to a large gathering of faculty and graduate students from all parts of the country. When half way through he discovered what seemed to be an error (though probably no one else in the room observed it). He stopped and re-examined the doubtful step for several minutes and then, convinced of the error, he abruptly dismissed the meeting—to the astonishment of most of the audience. It was an evidence of intellectual courage as well as honesty and doubtless won for him the supreme admiration of every person in the group—an admiration which was in no wise diminished, but rather increased, when at a later meeting he announced that after all he had been able to prove the step to be correct.

Dilbert: Maybe I’m unlucky in love because I’m so knowledgeable about science that I intimidate people. Their intimidation becomes low self-esteem, then they reject me to protect their egos.
Dogbert: Occam’s Razor.
Dilbert: What is “Occam's Razor”?
Dogbert: A guy named Occam had a rule about the world. Basically he said that when there are multiple explanations for something the simplest explanation is usually correct. The simplest explanation for your poor love life is that you’re immensely unattractive.
Dilbert: Maybe Occam had another rule that specifically exempted this situation, but his house burned down with all his notes. Then he forgot.
Dogbert: Occam’s Razor.
Dilbert: I’m an idiot.
Dogbert: I don’t think we can rule it out at this point.

~~[Attributed]~~ Prudens quaestio dimidium scientiae.
Half of science is asking the right questions.

A book should have either intelligibility or correctness; to combine the two is impossible, but to lack both is to be unworthy of a place as Euclid has occupied in education.

A disease which new and obscure to you, Doctor, will be known only after death; and even then not without an autopsy will you examine it with exacting pains. But rare are those among the extremely busy clinicians who are willing or capable of doing this correctly.

A man is flying in a hot air balloon and realizes he is lost. He reduces height, spots a man down below and asks,“Excuse me, can you help me? I promised to return the balloon to its owner, but I don’t know where I am.”
The man below says: “You are in a hot air balloon, hovering approximately 350 feet above mean sea level and 30 feet above this field. You are between 40 and 42 degrees north latitude, and between 58 and 60 degrees west longitude.”
“You must be an engineer,” says the balloonist.
“I am,” replies the man.“How did you know?”
“Well,” says the balloonist, “everything you have told me is technically correct, but I have no idea what to make of your information, and the fact is I am still lost.”
The man below says, “You must be a manager.”
“I am,” replies the balloonist,“but how did you know?”
“Well,” says the engineer,“you don’t know where you are, or where you are going. You have made a promise which you have no idea how to keep, and you expect me to solve your problem.The fact is you are in the exact same position you were in before we met, but now it is somehow my fault.”

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 dealing 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 human wills, is continuous. To understand the laws of this continuous movement is the aim of history. … Only by taking an infinitesimally small unit for observation (the differential of history, that is, the individual tendencies of man) 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.

A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe.

A Vulgar Mechanick can practice what he has been taught or seen done, but if he is in an error he knows not how to find it out and correct it, and if you put him out of his road, he is at a stand; Whereas he that is able to reason nimbly and judiciously about figure, force and motion, is never at rest till he gets over every rub.

Access to more information isn’t enough—the information needs to be correct, timely, and presented in a manner that enables the reader to learn from it. The current network is full of inaccurate, misleading, and biased information that often crowds out the valid information. People have not learned that “popular” or “available” information is not necessarily valid.

Alexander Langmuir was quoted in the early 1960s instructing incoming Epidemic Intelligence Service (EIS) officers that the only need for the laboratory in an outbreak investigation was to “prove their conclusions were right.” (2011)

Biot, who assisted Laplace in revising it [The Mécanique Céleste] for the press, says that Laplace himself was frequently unable to recover the details in the chain of reasoning, and if satisfied that the conclusions were correct, he was content to insert the constantly recurring formula, “Il est àisé a voir” [it is easy to see].

But … the working scientist … is not consciously following any prescribed course of action, but feels complete freedom to utilize any method or device whatever which in the particular situation before him seems likely to yield the correct answer. … No one standing on the outside can predict what the individual scientist will do or what method he will follow.

But does Man have any “right” to spread through the universe? Man is what he is, a wild animal with the will to survive, and (so far) the ability, against all competition. Unless one accepts that, anything one says about morals, war, politics, you name it, is nonsense. Correct morals arise from knowing what man is, not what do-gooders and well-meaning old Aunt Nellies would like him to be. The Universe will let us know—later—whether or not Man has any “right” to expand through it.

But, you might say, “none of this shakes my belief that 2 and 2 are 4.” You are quite right, except in marginal cases—and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a meter. Two must be two of something, and the proposition “2 and 2 are 4” is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you are doubtful whether two of them are dogs. “Well, at any rate there are four animals,” you may say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. “Well, then living organisms,” you say. But there are things of which it is doubtful whether they are living organisms or not. You will be driven into saying: “Two entities and two entities are four entities.” When you have told me what you mean by “entity,” we will resume the argument.

By the early 1960s Pauling had earned a reputation for being audacious, intuitive, charming, irreverent, self-promoting, self-reliant, self-involved to the point of arrogance and correct about almost everything.

Careful and correct use of language is a powerful aid to straight thinking, for putting into words precisely what we mean necessitates getting our own minds quite clear on what we mean.

Chief Seattle, of the Indians that inhabited the Seattle area, wrote a wonderful paper that has to do with putting oneself in tune with the universe. He said, “Why should I lament the disappearance of my people! All things end, and the white man will find this out also.” And this goes for the universe. One can be at peace with that. This doesn’t mean that one shouldn’t participate in efforts to correct the situation. But underlying the effort to change must be an “at peace.” To win a dog sled race is great. To lose is okay too.

Don’t confuse hypothesis and theory. The former is a possible explanation; the latter, the correct one. The establishment of theory is the very purpose of science.

Evolution is an obstacle course not a freeway; the correct analogue for long-term success is a distant punt receiver evading legions of would-be tacklers in an oddly zigzagged path toward a goal, not a horse thundering down the flat.

Exercise in the most rigorous thinking that is possible will of its own accord strengthen the sense of truth and right, for each advance in the ability to distinguish between correct and false thoughts, each habit making for rigour in thought development will increase in the sound pupil the ability and the wish to ascertain what is right in life and to defend it.

First, as concerns the success of teaching mathematics. No instruction in the high schools is as difficult as that of mathematics, since the large majority of students are at first decidedly disinclined to be harnessed into the rigid framework of logical conclusions. The interest of young people is won much more easily, if sense-objects are made the starting point and the transition to abstract formulation is brought about gradually. For this reason it is psychologically quite correct to follow this course.
Not less to be recommended is this course if we inquire into the essential purpose of mathematical instruction. Formerly it was too exclusively held that this purpose is to sharpen the understanding. Surely another important end is to implant in the student the conviction that correct thinking based on true premises secures mastery over the outer world. To accomplish this the outer world must receive its share of attention from the very beginning.
Doubtless this is true but there is a danger which needs pointing out. It is as in the case of language teaching where the modern tendency is to secure in addition to grammar also an understanding of the authors. The danger lies in grammar being completely set aside leaving the subject without its indispensable solid basis. Just so in Teaching of Mathematics it is possible to accumulate interesting applications to such an extent as to stunt the essential logical development. This should in no wise be permitted, for thus the kernel of the whole matter is lost. Therefore: We do want throughout a quickening of mathematical instruction by the introduction of applications, but we do not want that the pendulum, which in former decades may have inclined too much toward the abstract side, should now swing to the other extreme; we would rather pursue the proper middle course.

For it is the duty of an astronomer to compose the history of the celestial motions or hypotheses about them. Since he cannot in any certain way attain to the true causes, he will adopt whatever suppositions enable the motions to be computed correctly from the principles of geometry for the future as well as for the past.

Formal thought, consciously recognized as such, is the means of all exact knowledge; and a correct understanding of the main formal sciences, Logic and Mathematics, is the proper and only safe foundation for a scientific education.

Go, wondrous creature! mount where Science guides,
Go, measure earth, weigh air, and state the tides;
Instruct the planets in what orbs to run,
Correct old Time, and regulate the Sun.

Good lawyers know that in many cases where the decisions are correct, the reasons that are given to sustain them may be entirely wrong. This is a thousand times more likely to be true in the practice of medicine than in that of the law, and hence the impropriety, not to say the folly, in spending your time in the discussion of medical belief and theories of cure that are more ingenious and seductive than they are profitable.

He attends constantly the Meetings both of ye Society and the Council; noteth the Observables said and done there; digesteth ym in private; takes care to have ym entered in the Journal- and Register-Books; reads over and corrects all entrys; sollicites the performances of taskes recommended and undertaken; writes all Letters abroad and answers the returns made to ym, entertaining a correspondence with at least 30. persons; employs a great deal of time, and takes much pain in inquiring after and satisfying foorain demands about philosophical matters, dispenseth farr and near store of directions and inquiries for the society’s purpose, and sees them well recommended etc.

How indispensable to a correct study of Nature is a perception of her true meaning. The fact will one day flower out into a truth. The season will mature and fructify what the understanding had cultivated. Mere accumulators of facts—collectors of materials for the master-workmen—are like those plants growing in dark forests, which “put forth only leaves instead of blossoms.”

I know that to personalize the Earth System as Gaia, as I have often done and continue to do in this book, irritates the scientifically correct, but I am unrepentant because metaphors are more than ever needed for a widespread comprehension of the true nature of the Earth and an understanding of the lethal dangers that lie ahead.

I would like it if everyone could make the prejudice vanish as I have that there is really a problem whether ants are machines, whether my brother is a machine, whether we are in the world, or the world is in us, if perhaps behind the word there is matter, power pushes or not, or if Locke is right that the intellect is between us and things. Or whether we are free or not free…

I would teach the world that science is the best way to understand the world, and that for any set of observations, there is only one correct explanation. Also, science is value-free, as it explains the world as it is. Ethical issues arise only when science is applied to technology – from medicine to industry.

I’m very intense in my work. At any given moment, I think I know the answer to some problem, and that I’m right. Since science is the only self-correcting human institution I know of, you should not be frightened to take an extreme stand, if that causes the stand to be examined more thoroughly than it might be if you are circumspect. I’ve always been positive about the value of the Hubble constant, knowing full well that it probably isn’t solved.

If Einstein’s theory [of relativity] should prove to be correct, as I expect it will, he will be considered the Copernicus of the twentieth century.

If the resident zoologist of Galaxy X had visited the earth 5 million years ago while making his inventory of inhabited planets in the universe, he would surely have corrected his earlier report that apes showed more promise than Old World monkeys and noted that monkeys had overcome an original disadvantage to gain domination among primates. (He will confirm this statement after his visit next year–but also add a footnote that one species from the ape bush has enjoyed an unusual and unexpected flowering, thus demanding closer monitoring.)

If there is a lesson in our story it is that the manipulation, according to strictly self-consistent rules, of a set of symbols representing one single aspect of the phenomena may produce correct, verifiable predictions, and yet completely ignore all other aspects whose ensemble constitutes reality.

Imagination comes first in both artistic and scientific creations, but in science there is only one answer and that has to be correct.

In [1950 in] South Africa all the geologists were disciples of Alfred Wegener and A. L. du Toit, and were anxious to correct my failure to accept continental drift, but I remained inflexible for another nine years.

In the secondary schools mathematics should be a part of general culture and not contributory to technical training of any kind; it should cultivate space intuition, logical thinking, the power to rephrase in clear language thoughts recognized as correct, and ethical and esthetic effects; so treated, mathematics is a quite indispensable factor of general education in so far as the latter shows its traces in the comprehension of the development of civilization and the ability to participate in the further tasks of civilization.

Intelligence is an extremely subtle concept. It’s a kind of understanding that flourishes if it’s combined with a good memory, but exists anyway even in the absence of good memory. It’s the ability to draw consequences from causes, to make correct inferences, to foresee what might be the result, to work out logical problems, to be reasonable, rational, to have the ability to understand the solution from perhaps insufficient information. You know when a person is intelligent, but you can be easily fooled if you are not yourself intelligent.

It appears that anything you say about the way that theory and experiment may interact is likely to be correct, and anything you say about the way that theory and experiment must interact is likely to be wrong.

It does appear that on the whole a physicist… tries to reduce his theory at all times to as few parameters as possible and is inclined to feel that a theory is a “respectable” one, though by no means necessarily correct, if in principle it does offer reasonably specific means for its possible refutation. Moreover the physicist will generally arouse the irritation amongst fellow physicists if he is not prepared to abandon his theory when it clashes with subsequent experiments. On the other hand it would appear that the chemist regards theories—or perhaps better his theories (!) —as far less sacrosanct, and perhaps in extreme cases is prepared to modify them continually as each bit of new experimental evidence comes in.

It is … a sign of the times—though our brothers of physics and chemistry may smile to hear me say so—that biology is now a science in which theories can be devised: theories which lead to predictions and predictions which sometimes turn out to be correct. These facts confirm me in a belief I hold most passionately—that biology is the heir of all the sciences.

It is now necessary to indicate more definitely the reason why mathematics not only carries conviction in itself, but also transmits conviction to the objects to which it is applied. The reason is found, first of all, in the perfect precision with which the elementary mathematical concepts are determined; in this respect each science must look to its own salvation .... But this is not all. As soon as human thought attempts long chains of conclusions, or difficult matters generally, there arises not only the danger of error but also the suspicion of error, because since all details cannot be surveyed with clearness at the same instant one must in the end be satisfied with a belief that nothing has been overlooked from the beginning. Every one knows how much this is the case even in arithmetic, the most elementary use of mathematics. No one would imagine that the higher parts of mathematics fare better in this respect; on the contrary, in more complicated conclusions the uncertainty and suspicion of hidden errors increases in rapid progression. How does mathematics manage to rid itself of this inconvenience which attaches to it in the highest degree? By making proofs more rigorous? By giving new rules according to which the old rules shall be applied? Not in the least. A very great uncertainty continues to attach to the result of each single computation. But there are checks. In the realm of mathematics each point may be reached by a hundred different ways; and if each of a hundred ways leads to the same point, one may be sure that the right point has been reached. A calculation without a check is as good as none. Just so it is with every isolated proof in any speculative science whatever; the proof may be ever so ingenious, and ever so perfectly true and correct, it will still fail to convince permanently. He will therefore be much deceived, who, in metaphysics, or in psychology which depends on metaphysics, hopes to see his greatest care in the precise determination of the concepts and in the logical conclusions rewarded by conviction, much less by success in transmitting conviction to others. Not only must the conclusions support each other, without coercion or suspicion of subreption, but in all matters originating in experience, or judging concerning experience, the results of speculation must be verified by experience, not only superficially, but in countless special cases.

It is of interest to note that while some dolphins are reported to have learned English—up to fifty words used in correct context—no human being has been reported to have learned delphinese.

It is the facts that matter, not the proofs. Physics can progress without the proofs, but we can’t go on without the facts … if the facts are right, then the proofs are a matter of playing around with the algebra correctly.

It would be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said, “Yes, it is true, but you won’t be able to understand the proof.”

It’s only through honesty and courage that science can work at all. The Ptolemaic understanding of the solar system was undermined and corrected by the constant pressure of more and more honest reporting.

Law springs from experiment, but not immediately. Experiment is individual, the law deduced from it is general; experiment is only approximate, the law is precise, or at least pretends to be. Experiment is made under conditions always complex, the enunciation of the law eliminates these complications. This is what is called ‘correcting the systematic errors’.

Living is like working out a long addition sum, and if you make a mistake in the first two totals you will never find the right answer. It means involving oneself in a complicated chain of circumstances.

Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated to do so. Our system is quite good at producing reliable theorems that can be backed up. It’s just that the reliability does not primarily come from mathematicians checking formal arguments; it come from mathematicians thinking carefully and critically about mathematical ideas.

No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful.

No society has ever yet been able to handle the temptations of technology, to mastery, to waste, to exuberance, to exploration and exploitation. We have to create something new, something that has never existed in the world before. We have to learn to cherish this Earth and cherish it as something that is fragile, that’s only one, that’s all we have, and we have to set up a system that is sufficiently complex to continue to monitor the whole. We have to use our scientific knowledge to correct the dangers that have come from science and technology.

Now do you not see that the eye embraces the beauty of the whole world? It counsels and corrects all the arts of mankind... it is the prince of mathematics, and the sciences founded on it are absolutely certain. It has measured the distances and sizes of the stars it has discovered the elements and their location... it has given birth to architecture and to perspective and to the divine art of painting.

One of my complaints is that you’ve got far more scientists than ever before but the pace of discovery has not increased. Why? Because they’re all busy just filling in the details of what they think is the standard story. And the youngsters, the people with different ideas have just as big a fight as ever and normally it takes decades for science to correct itself. But science does correct itself and that’s the reason why science is such a glorious thing for our species.

People are usually not very good in checking formal correctness of proofs, but they are quite good at detecting potential weaknesses or flaws in proofs.

People may believe correct things for the damnedest and weirdest of wrong reasons.

Physical Science and Industrialism may be conceived as a pair of dancers, both of whom know their steps and have an ear for the rhythm of the music. If the partner who has been leading chooses to change parts and to follow instead, there is perhaps no reason to expect that he will dance less correctly than before.

Psychiatry enables us to correct our faults by confessing to our parents’ shortcomings.

Psychology appeared to be a jungle of confusing, conflicting, and arbitrary concepts. These pre-scientific theories doubtless contained insights which still surpass in refinement those depended upon by psychiatrists or psychologists today. But who knows, among the many brilliant ideas offered, which are the true ones? Some will claim that the statements of one theorist are correct, but others will favour the views of another. Then there is no objective way of sorting out the truth except through scientific research.

Read, and found correct.
Written, with Einstein’s signature, below this statement written by an admirer: “A Short Definition of Relativity: There is no hitching post in the Universe—as far as we know.”

Since the world is what it is, it is clear that valid reasoning from sound principles cannot lead to error; but a principle may be so nearly true as to deserve theoretical respect, and yet may lead to practical consequences which we feel to be absurd. There is therefore a justification for common sense in philosophy, but only as showing that our theoretical principles cannot be quite correct so long as their consequences are condemned by an appeal to common sense which we feel to be irresistible.

Symbolism is useful because it makes things difficult. Now in the beginning everything is self-evident, and it is hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we must invent a new and difficult symbolism in which nothing is obvious. … Thus the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions.

That which is perfect in science, is most commonly the elaborate result of successive improvements, and of various judgments exercised in the rejection of what was wrong, no less than in the adoption of what was right.

The [Pentium] co-processor is designed to give you 19 digits correct.… For it to give you only 10 is just utterly atrocious To get a result that poor from that co-processor is like having the transmission fall out of your Ford.

The advantage is that mathematics is a field in which one’s blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician’s reputation.

The argument of the ‘long view’ may be correct in some meaninglessly abstract sense, but it represents a fundamental mistake in categories and time scales. Our only legitimate long view extends to our children and our children’s children’s children–hundreds or a few thousands of years down the road. If we let the slaughter continue, they will share a bleak world with rats, dogs, cockroaches, pigeons, and mosquitoes. A potential recovery millions of years later has no meaning at our appropriate scale.

The conundrum that baffled Medieval philosophers, How many angels can dance on the head of a pin?, can have only one correct answer: All of them.

The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each such set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to discover such systems. Any such system is a branch of mathematics.

The establishment of the periodic law may truly be said to mark a line in chemical science, and we anticipate that its application and and extension will be fraught With the most important consequences. It reminds us how important above all things is the correct determination of the fundamental constants of our science—the atomic weights of the elements, about which in many cases great uncertainty prevails; it is much to be desired that this may not long remain the case. It also affords the strongest encouragement to the chemist to persevere in the search for new elements.

The fact that your patient gets well does not prove that your diagnosis was correct.

The great horde of physicians are always servile imitators, who can neither perceive nor correct the faults of their system, and are always ready to growl at and even to worry the ingenious person that could attempt it. Thus was the system of Galen secured in the possession of the schools of physic.

The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible.

The idea that time may vary from place to place is a difficult one, but it is the idea Einstein used, and it is correct—believe it or not.

The laboratory work was the province of Dr Searle, an explosive, bearded Nemesis who struck terror into my heart. If one made a blunder one was sent to ‘stand in the corner’ like a naughty child. He had no patience with the women students. He said they disturbed the magnetic equipment, and more than once I heard him shout ‘Go and take off your corsets!’ for most girls wore these garments then, and steel was beginning to replace whalebone as a stiffening agent. For all his eccentricities, he gave us excellent training in all types of precise measurement and in the correct handling of data.

The mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.

The mind comprehends a thing the more correctly the closer the thing approaches toward pure quantity as its origin.

The opinion of Bacon on this subject [geometry] was diametrically opposed to that of the ancient philosophers. He valued geometry chiefly, if not solely, on account of those uses, which to Plato appeared so base. And it is remarkable that the longer Bacon lived the stronger this feeling became. When in 1605 he wrote the two books on the Advancement of Learning, he dwelt on the advantages which mankind derived from mixed mathematics; but he at the same time admitted that the beneficial effect produced by mathematical study on the intellect, though a collateral advantage, was “no less worthy than that which was principal and intended.” But it is evident that his views underwent a change. When near twenty years later, he published the De Augmentis, which is the Treatise on the Advancement of Learning, greatly expanded and carefully corrected, he made important alterations in the part which related to mathematics. He condemned with severity the pretensions of the mathematicians, “delidas et faslum mathematicorum.” Assuming the well-being of the human race to be the end of knowledge, he pronounced that mathematical science could claim no higher rank than that of an appendage or an auxiliary to other sciences. Mathematical science, he says, is the handmaid of natural philosophy; she ought to demean herself as such; and he declares that he cannot conceive by what ill chance it has happened that she presumes to claim precedence over her mistress.

The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth.

The question of the origin of the hypothesis belongs to a domain in which no very general rules can be given; experiment, analogy and constructive intuition play their part here. But once the correct hypothesis is formulated, the principle of mathematical induction is often sufficient to provide the proof.

The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face a blackboard and to turn his back on the class. He writes a, he says b, he means c, but it should be d. Some of his sayings are handed down from generation to generation:
“In order to solve this differential equation you look at it till a solution occurs to you.”
“This principle is so perfectly general that no particular application of it is possible.”
“Geometry is the science of correct reasoning on incorrect figures.”
“My method to overcome a difficulty is to go round it.”
“What is the difference between method and device? A method is a device which you used twice.”

The trick is to recognize the flaw in yourself and work to correct it.

The work of a pioneer in science of technique often consists of finding a correct solution, or creating a working mechanism, based on laws that are not yet discovered.

Theories rarely arise as patient inferences forced by accumulated facts. Theories are mental constructs potentiated by complex external prods (including, in idealized cases, a commanding push from empirical reality) . But the prods often in clude dreams, quirks, and errors–just as we may obtain crucial bursts of energy from foodstuffs or pharmaceuticals of no objective or enduring value. Great truth can emerge from small error. Evolution is thrilling, liberating, and correct. And Macrauchenia is a litoptern.

There are reported to be six species of metals, namely, gold, silver, iron, copper, tin, and lead. Actually there are more. Mercury is a metal although we differ on this point with the chemists. Plumbum cinereum (gray lead) which we call bisemutum was unknown to the older Greek writers. On the other hand, Ammonius writes correctly many metals are unknown to us, as well as many plants and animals.

There was once an Editor of the Chemical Society, given to dogmatic expressions of opinion, who once duly said firmly that 'isomer' was wrong usage and 'isomeride' was correct, because the ending 'er' always meant a 'do-er'. 'As in water?' snapped Sidgwick.

There’s a new science out called orthomolecular medicine. You correct the chemical imbalance with amino acids and vitamins and minerals that are naturally in the body.

Two of his [Euler’s] pupils having computed to the 17th term, a complicated converging series, their results differed one unit in the fiftieth cipher; and an appeal being made to Euler, he went over the calculation in his mind, and his decision was found correct.

We have also here an acting cause to account for that balance so often observed in nature,—a deficiency in one set of organs always being compensated by an increased development of some others—powerful wings accompanying weak feet, or great velocity making up for the absence of defensive weapons; for it has been shown that all varieties in which an unbalanced deficiency occurred could not long continue their existen The action of this principle is exactly like that of the centrifugal governor of the steam engine, which checks and corrects any irregularities almost before they become evident; and in like manner no unbalanced deficiency in the animal kingdom can ever reach any conspicuous magnitude, because it would make itself felt at the very first step, by rendering existence difficult and extinction almost sure soon to follow.

What else can the human mind hold besides numbers and magnitudes? These alone we apprehend correctly, and if piety permits to say so, our comprehension is in this case of the same kind as God’s, at least insofar as we are able to understand it in this mortal life.

When one considers how hard it is to write a computer program even approaching the intellectual scope of a good paper, and how much greater time and effort have to be put in to make it “almost” formally correct, it is preposterous to claim that mathematics as we practice it is anywhere near formally correct.

When the logician has resolved each demonstration into a host of elementary operations, all of them correct, he will not yet be in possession of the whole reality, that indefinable something that constitutes the unity ... Now pure logic cannot give us this view of the whole; it is to intuition that we must look for it.

Where do correct ideas come from? Do they drop from the skies? No. They come from social practice, and from it alone; they come from three kinds of social practice, the struggle for production, the class struggle and scientific experiment.

While seeing any number of black crows does not prove all the crows are black, seeing one white crow disproves it. Thus science proceeds not by proving models correct but by discarding false ones or improving incomplete ones.