… 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.

[When nature appears complicated:] The moment we contemplate it as it is, and attain a position from which we can take a commanding view, though but of a small part of its plan, we never fail to recognize that sublime simplicity on which the mind rests satisfied that it has attained the truth.

A man does not attain the status of Galileo merely because he is persecuted; he must also be right.

A scientist is happy, not in resting on his attainments but in the steady acquisition of fresh knowledge.

An inventor is an opportunist, one who takes occasion by the hand; who, having seen where some want exists, successfully applies the right means to attain the desired end. The means may be largely, or even wholly, something already known, or there may be a certain originality or discovery in the means employed. But in every case the inventor uses the work of others. If I may use a metaphor, I should liken him to the man who essays the conquest of some virgin alp. At the outset he uses the beaten track, and, as he progresses in the ascent, he uses the steps made by those who have preceded him, whenever they lead in the right direction; and it is only after the last footprints have died out that he takes ice-axe in hand and cuts the remaining steps, few or many, that lift him to the crowning height which is his goal.

Architects who have aimed at acquiring manual skill without scholarship have never been able to reach a position of authority to correspond to their pains, while those who relied only upon theories and scholarship were obviously hunting the shadow, not the substance. But those who have a thorough knowledge of both, like men armed at all points, have the sooner attained their object and carried authority with them.

Ask a follower of Bacon what [science] the new philosophy, as it was called in the time of Charles the Second, has effected for mankind, and his answer is ready; “It has lengthened life; it has mitigated pain; it has extinguished diseases; it has increased the fertility of the soil; it has given new securities to the mariner; it has furnished new arms to the warrior; it has spanned great rivers and estuaries with bridges of form unknown to our fathers; it has guided the thunderbolt innocuously from heaven to earth; it has lighted up the night with the splendour of the day; it has extended the range of the human vision; it has multiplied the power of the human muscles; it has accelerated motion; it has annihilated distance; it has facilitated intercourse, correspondence, all friendly offices, all dispatch of business; it has enabled man to descend to the depths of the sea, to soar into the air, to penetrate securely into the noxious recesses of the earth, to traverse the land in cars which whirl along without horses, to cross the ocean in ships which run ten knots an hour against the wind. These are but a part of its fruits, and of its first-fruits; for it is a philosophy which never rests, which has never attained, which is never perfect. Its law is progress. A point which yesterday was invisible is its goal to-day, and will be its starting-point to-morrow.”

Attainment and science, retainment and art—the two couples keep to themselves, but when they do meet, nothing else in the world matters.

Attainment is a poor measure of capacity, and ignorance no proof of defect.

Doubtless the reasoning faculty, the mind, is the leading and characteristic attribute of the human race. By the exercise of this, man arrives at the properties of the natural bodies. This is science, properly and emphatically so called. It is the science of pure mathematics; and in the high branches of this science lies the truly sublime of human acquisition. If any attainment deserves that epithet, it is the knowledge, which, from the mensuration of the minutest dust of the balance, proceeds on the rising scale of material bodies, everywhere weighing, everywhere measuring, everywhere detecting and explaining the laws of force and motion, penetrating into the secret principles which hold the universe of God together, and balancing worlds against worlds, and system against system. When we seek to accompany those who pursue studies at once so high, so vast, and so exact; when we arrive at the discoveries of Newton, which pour in day on the works of God, as if a second fiat had gone forth from his own mouth; when, further, we attempt to follow those who set out where Newton paused, making his goal their starting-place, and, proceeding with demonstration upon demonstration, and discovery upon discovery, bring new worlds and new systems of worlds within the limits of the known universe, failing to learn all only because all is infinite; however we may say of man, in admiration of his physical structure, that “in form and moving he is express and admirable,” it is here, and here without irreverence, we may exclaim, “In apprehension how like a god!” The study of the pure mathematics will of course not be extensively pursued in an institution, which, like this [Boston Mechanics’ Institute], has a direct practical tendency and aim. But it is still to be remembered, that pure mathematics lie at the foundation of mechanical philosophy, and that it is ignorance only which can speak or think of that sublime science as useless research or barren speculation.

During the century after Newton, it was still possible for a man of unusual attainments to master all fields of scientific knowledge. But by 1800, this had become entirely impracticable.

Even as a coin attains its full value when it is spent, so life attains its supreme value when one knows how to forfeit it with grace when the time comes.

Even though the realms of religion and science in themselves are clearly marked off from each other, nevertheless there exist between the two strong reciprocal relationships and dependencies. Though religion may be that which determines the goal, it has, nevertheless, learned from science, in the broadest sense, what means will contribute to the attainment of the goals it has set up. But science can only be created by those who are thoroughly imbued with the aspiration toward truth and understanding. This source of feeling, however, springs from the sphere of religion. To this there also belongs the faith in the possibility that the regulations valid for the world of existence are rational, that is, comprehensible to reason. I cannot conceive of a genuine scientist without that profound faith. The situation may be expressed by an image: science without religion is lame, religion without science is blind.

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.

I conclude that, while it is true that science cannot decide questions of value, that is because they cannot be intellectually decided at all, and lie outside the realm of truth and falsehood. Whatever knowledge is attainable, must be attained by scientific methods; and what science cannot discover, mankind cannot know.

I should like to draw attention to the inexhaustible variety of the problems and exercises which it [mathematics] furnishes; these may be graduated to precisely the amount of attainment which may be possessed, while yet retaining an interest and value. It seems to me that no other branch of study at all compares with mathematics in this. When we propose a deduction to a beginner we give him an exercise in many cases that would have been admired in the vigorous days of Greek geometry. Although grammatical exercises are well suited to insure the great benefits connected with the study of languages, yet these exercises seem to me stiff and artificial in comparison with the problems of mathematics. It is not absurd to maintain that Euclid and Apollonius would have regarded with interest many of the elegant deductions which are invented for the use of our students in geometry; but it seems scarcely conceivable that the great masters in any other line of study could condescend to give a moment’s attention to the elementary books of the beginner.

I would be the last to deny that the greatest scientific pioneers belonged to an aristocracy of the spirit and were exceptionally intelligent, something that we as modest investigators will never attain, no matter how much we exert ourselves. Nevertheless … I continue to believe that there is always room for anyone with average intelligence … to utilize his energy and … any man could, if he were so inclined, be the sculptor of his own brain, and that even the least gifted may, like the poorest land that has been well-cultivated and fertilized, produce an abundant harvest..

I would... establish the conviction that Chemistry, as an independent science, offers one of the most powerful means towards the attainment of a higher mental cultivation; that the study of Chemistry is profitable, not only inasmuch as it promotes the material interests of mankind, but also because it furnishes us with insight into those wonders of creation which immediately surround us, and with which our existence, life, and development, are most closely connected.

If our intention had been merely to bring back a handful of soil and rocks from the lunar gravel pit and then forget the whole thing, we would certainly be history's biggest fools. But that is not our intention now—it never will be. What we are seeking in tomorrow's [Apollo 11] trip is indeed that key to our future on earth. We are expanding the mind of man. We are extending this God-given brain and these God-given hands to their outermost limits and in so doing all mankind will benefit. All mankind will reap the harvest…. What we will have attained when Neil Armstrong steps down upon the moon is a completely new step in the evolution of man.

If we would indicate an idea … striving to remove the barriers which prejudice and limited views of every kind have erected among men, and to treat all mankind, without reference to religion, nation, or color, as one fraternity, one great community, fitted for the attainment of one object, the unrestrained development of the physical powers. This is the ultimate and highest aim of society.

In Man the brain presents an ascensive step in development, higher and more strongly marked than that by which the preceding subclass was distinguished from the one below it. Not only do the cerebral hemispheres overlap the olfactory lobes and cerebellum, but they extend in advance of the one, and further back than the other. Their posterior development is so marked, that anatomists have assigned to that part the character of a third lobe; it is peculiar to the genus Homo, and equally peculiar is the 'posterior horn of the lateral ventricle,' and the 'hippocampus minor,' which characterize the hind lobe of each hemisphere. The superficial grey matter of the cerebrum, through the number and depth of the convolutions, attains its maximum of extent in Man. Peculiar mental powers are associated with this highest form of brain, and their consequences wonderfully illustrate the value of the cerebral character; according to my estimate of which, I am led to regard the genus Homo, as not merely a representative of a distinct order, but of a distinct subclass of the Mammalia, for which I propose a name of 'ARCHENCEPHALA.'

It is true that when pride releases energies and serves as a spur to achievement, it can lead to a reconciliation with the self and the attainment of genuine self-esteem.

Man is not a machine, ... although man most certainly processes information, he does not necessarily process it in the way computers do. Computers and men are not species of the same genus. .... No other organism, and certainly no computer, can be made to confront genuine human problems in human terms. ... However much intelligence computers may attain, now or in the future, theirs must always be an intelligence alien to genuine human problems and concerns.

Mathematics is the life supreme. The life of the gods is mathematics. All divine messengers are mathematicians. Pure mathematics is religion. Its attainment requires a theophany.

Nonconformity is the highest evolutionary attainment of social animals.

Not fact-finding, but attainment to philosophy is the aim of science.

Only by following out the injunction of our great predecessor [William Harvey] to search out and study the secrets of Nature by way of experiment, can we hope to attain to a comprehension of 'the wisdom of the body and the understanding of the heart,' and thereby to the mastery of disease and pain, which will enable us to relieve the burden of mankind.

Science has hitherto been proceeding without the guidance of any rational theory of logic, and has certainly made good progress. It is like a computer who is pursuing some method of arithmetical approximation. Even if he occasionally makes mistakes in his ciphering, yet if the process is a good one they will rectify themselves. But then he would approximate much more rapidly if he did not commit these errors; and in my opinion, the time has come when science ought to be provided with a logic. My theory satisfies me; I can see no flaw in it. According to that theory universality, necessity, exactitude, in the absolute sense of these words, are unattainable by us, and do not exist in nature. There is an ideal law to which nature approximates; but to express it would require an endless series of modifications, like the decimals expressing surd. Only when you have asked a question in so crude a shape that continuity is not involved, is a perfectly true answer attainable.

Science, regarded as the pursuit of truth, which can only be attained by patient and unprejudiced investigation, wherein nothing is to be attempted, nothing so minute as to be justly disregarded, must ever afford occupation of consummate interest, and subject of elevated meditation.

That ability to impart knowledge … what does it consist of? … a deep belief in the interest and importance of the thing taught, a concern about it amounting to a sort of passion. A man who knows a subject thoroughly, a man so soaked in it that he eats it, sleeps it and dreams it—this man can always teach it with success, no matter how little he knows of technical pedagogy. That is because there is enthusiasm in him, and because enthusiasm is almost as contagious as fear or the barber’s itch. An enthusiast is willing to go to any trouble to impart the glad news bubbling within him. He thinks that it is important and valuable for to know; given the slightest glow of interest in a pupil to start with, he will fan that glow to a flame. No hollow formalism cripples him and slows him down. He drags his best pupils along as fast as they can go, and he is so full of the thing that he never tires of expounding its elements to the dullest.
This passion, so unordered and yet so potent, explains the capacity for teaching that one frequently observes in scientific men of high attainments in their specialties—for example, Huxley, Ostwald, Karl Ludwig, Virchow, Billroth, Jowett, William G. Sumner, Halsted and Osler—men who knew nothing whatever about the so-called science of pedagogy, and would have derided its alleged principles if they had heard them stated.

The advancement of science is slow; it is effected only by virtue of hard work and perseverance. And when a result is attained, should we not in recognition connect it with the efforts of those who have preceded us, who have struggled and suffered in advance? Is it not truly a duty to recall the difficulties which they vanquished, the thoughts which guided them; and how men of different nations, ideas, positions, and characters, moved solely by the love of science, have bequeathed to us the unsolved problem? Should not the last comer recall the researches of his predecessors while adding in his turn his contribution of intelligence and of labor? Here is an intellectual collaboration consecrated entirely to the search for truth, and which continues from century to century.
[Respecting how the work of prior researchers had enabled his isolation of fluorine.]

The attainment of knowledge is the high and exclusive attribute of man, among the numberless myriads of animated beings, inhabitants of the terrestrial globe. On him alone is bestowed, by the bounty of the Creator of the universe, the power and the capacity of acquiring knowledge. Knowledge is the attribute of his nature which at once enables him to improve his condition upon earth, and to prepare him for the enjoyment of a happier existence hereafter.

The fascination of any search after truth lies not in the attainment, which at best is found to be very relative, but in the pursuit, where all the powers of the mind and character are brought into play and are absorbed by the task. One feels oneself in contact with something that is infinite and one finds joy that is beyond expression in sounding the abyss of science and the secrets of the infinite mind.

The first [quality] to be named must always be the power of attention, of giving one's whole mind to the patient without the interposition of anything of oneself. It sounds simple but only the very greatest doctors ever fully attain it. … The second thing to be striven for is intuition. This sounds an impossibility, for who can control that small quiet monitor? But intuition is only interference from experience stored and not actively recalled. … The last aptitude I shall mention that must be attained by the good physician is that of handling the sick man's mind.

The great difference between science and technology is a difference of initial attitude. The scientific man follows his method whithersoever it may take him. He seeks acquaintance with his subject­matter, and he does not at all care about what he shall find, what shall be the content of his knowledge when acquaintance-with is transformed into knowledge-about. The technologist moves in another universe; he seeks the attainment of some determinate end, which is his sole and obsessing care; and he therefore takes no heed of anything that he cannot put to use as means toward that end.

The great testimony of history shows how often in fact the development of science has emerged in response to technological and even economic needs, and how in the economy of social effort, science, even of the most abstract and recondite kind, pays for itself again and again in providing the basis for radically new technological developments. In fact, most people—when they think of science as a good thing, when they think of it as worthy of encouragement, when they are willing to see their governments spend substance upon it, when they greatly do honor to men who in science have attained some eminence—have in mind that the conditions of their life have been altered just by such technology, of which they may be reluctant to be deprived.

The reason it is so hard to attain to something good in any of the arts and sciences is that it involves attaining to a certain stipulated point; to do something badly according to a predetermined rule would be just as hard, if indeed it would then still deserve to be called bad.

The sense for style … is an aesthetic sense, based on admiration for the direct attainment of a foreseen end, simply and without waste. Style in art, style in literature, style in science, style in logic, style in practical execution have fundamentally the same aesthetic qualities, namely, attainment and restraint. The love of a subject in itself and for itself, where it is not the sleepy pleasure of pacing a mental quarter-deck, is the love of style as manifested in that study. Here we are brought back to the position from which we started, the utility of education. Style, in its finest sense, is the last acquirement of the educated mind; it is also the most useful. It pervades the whole being. The administrator with a sense for style hates waste; the engineer with a sense for style economises his material; the artisan with a sense for style prefers good work. Style is the ultimate morality of the mind.

The study of … simple cases would, I think, often be of advantage even to students whose mathematical attainments are sufficient to enable them to follow the solution of the more general cases. For in these simple cases the absence of analytical difficulties allows attention to be more easily concentrated on the physical aspects of the question, and thus gives the student a more vivid idea and a more manageable grasp of the subject than he would be likely to attain if he merely regarded electrical phenomena through a cloud of analytical symbols.

The useless search of philosophers for a cause of the universe is a regressus in infinitum (a stepping backwards into the infinite) and resembles climbing up an endless ladder, the recurring question as to the cause of the cause rendering the attainment of a final goal impossible.

There is no short cut, nor “royal road” to the attainment of medical knowledge. The path which we have to pursue is long, difficult, and unsafe. In our progress, we must frequently take up our abode with death and corruption, we must adopt loathsome diseases for our familiar associates, or we shall never be acquainted with their nature and dispositions ; we must risk, nay, even injure our own health, in order to be able to preserve, or restore that of others.

Unless you expect the unexpected you will never find it, for it is hard to discover and hard to attain.

We had a clear, unmistakable, specific objective. Although at first there was considerable doubt whether we could attain this objective, there was never any doubt about what it was. Consequently the people in responsible positions were able to tailor their every action to its accomplishment.

We may, I think, draw a yet higher and deeper teaching from the phenomena of degeneration. We seem to learn from it the absolute necessity of labour and effort, of struggle and difficulty, of discomfort and pain, as the condition of all progress, whether physical or mental, and that the lower the organism the more need there is of these ever-present stimuli, not only to effect progress, but to avoid retrogression. And if so, does not this afford us the nearest attainable solution of the great problem of the origin of evil? What we call evil is the essential condition of progress in the lower stages of the development of conscious organisms, and will only cease when the mind has become so thoroughly healthy, so well balanced, and so highly organised, that the happiness derived from mental activity, moral harmony, and the social affections, will itself be a sufficient stimulus to higher progress and to the attainment of a more perfect life.

We must somehow keep the dreams of space exploration alive, for in the long run they will prove to be of far more importance to the human race than the attainment of material benefits. Like Darwin, we have set sail upon an ocean: the cosmic sea of the Universe. There can be no turning back. To do so could well prove to be a guarantee of extinction. When a nation, or a race or a planet turns its back on the future, to concentrate on the present, it cannot see what lies ahead. It can neither plan nor prepare for the future, and thus discards the vital opportunity for determining its evolutionary heritage and perhaps its survival.

We should have positive expectations of what is in the universe, not fears and dreads. We are made with the realization that we’re not Earthbound, and that our acceptance of the universe offers us room to explore and extend outward. It’s like being in a dark room and imagining all sorts of terrors. But when we turn on the light – technology - suddenly it’s just a room where we can stretch out and explore. If the resources here on Earth are limited, they are not limited in the universe. We are not constrained by the limitations of our planet. As children have to leave the security of family and home life to insure growth into mature adults, so also must humankind leave the security and familiarity of Earth to reach maturity and obtain the highest attainment possible for the human race.

Wonder [admiratio astonishment, marvel] is a kind of desire for knowledge. The situation arises when one sees an effect and does not know its cause, or when the cause of the particular effect is one that exceeds his power of understanding. Hence, wonder is a cause of pleasure insofar as there is annexed the hope of attaining understanding of that which one wants to know. ... For desire is especially aroused by the awareness of ignorance, and consequently a man takes the greatest pleasure in those things which he discovers for himself or learns from the ground up.

You cannot do without one specialty. You must have some base-line to measure the work and attainments of others. For a general view of the subject, study the history of the sciences. Broad knowledge of all Nature has been the possession of no naturalist except Humboldt, and general relations constituted his specialty.