Demonstrate Quotes (53 quotes)

*Nature and nurture*are an inseparable blend of influences that work together to produce our behavior. A growing band of researchers are demonstrating that the bedrock of behaviors that make up the concerns of everyday life, such as sex, language, cooperation, and violence have been carved out by evolution over the eons, and this Stone Age legacy continues to influence modern life today.

According to the theory of aerodynamics, as may be readily demonstrated through wind tunnel experiments, the bumblebee is unable to fly. This is because the size, weight and shape of his body in relation to the total wingspread make flying impossible.

*But the bumblebee, being ignorant of these scientific truths, goes ahead and flies anyway—and makes a little honey every day.*
As soon as somebody demonstrates the art of flying, settlers from our species of man will not be lacking [on the moon and Jupiter]… Given ships or sails adapted to the breezes of heaven, there will be those who will not shrink from even that vast expanse.

Each species has evolved a special set of solutions to the general problems that all organisms must face. By the fact of its existence, a species demonstrates that its members are able to carry out adequately a series of general functions. … These general functions offer a framework within which one can integrate one’s view of biology and focus one’s research. Such a view helps one to avoid becoming lost in a morass of unstructured detail—even though the ways in which different species perform these functions may differ widely. A few obvious examples will suffice. Organisms must remain functionally integrated. They must obtain materials from their environments, and process and release energy from these materials. … They must differentiate and grow, and they must reproduce. By focusing one’s questions on one or another of these obligatory and universal capacities, one can ensure that one’s research will not be trivial and that it will have some chance of achieving broad general applicability.

Euclid and Archimedes are allowed to be knowing, and to have demonstrated what they say: and yet whosoever shall read over their writings without perceiving the connection of their proofs, and seeing what they show, though he may understand all their words, yet he is not the more knowing. He may believe, indeed, but does not know what they say, and so is not advanced one jot in mathematical knowledge by all his reading of those approved mathematicians.

Exobiology—a curious development in view of the fact that this “science” has yet to demonstrate that its subject matter exists!

Geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring.

Gravitation is demonstrable by leaving a body unsupported.

He made an instrument to know If the moon shine at full or no;

That would, as soon as e’er she shone straight,

Whether ‘twere day or night demonstrate;

Tell what her d’ameter to an inch is,

And prove that she’s not made of green cheese.

That would, as soon as e’er she shone straight,

Whether ‘twere day or night demonstrate;

Tell what her d’ameter to an inch is,

And prove that she’s not made of green cheese.

Here lies Sir Isaac Newton, Knight, who by a vigour of mind almost supernatural, first demonstrated, the motions and Figures of the Planets, the Paths of the comets, and the Tides of the Oceans ... Let Mortals rejoice that there has existed such and so great an ornament of Nature.

— Epitaph

I do not claim that intelligence, however defined, has no genetic basis–I regard it as trivially true, uninteresting, and unimportant that it does. The expression of any trait represents a complex interaction of heredity and environment ... a specific claim purporting to demonstrate a mean genetic deficiency in the intelligence of American blacks rests upon no new facts whatever and can cite no valid data in its support. It is just as likely that blacks have a genetic advantage over whites. And, either way, it doesn’t matter a damn. An individual can’t be judged by his group mean.

I never really paused for a moment to question the idea that the progressive Spiritualization of Matter—so clearly demonstrated to me by Paleontology—could be anything other, or anything less, than an

*irreversible*process. By its gravitational nature, the Universe, I saw, was falling—falling forwards—in the direction of spirit as upon its stable form. In other words, Matter was not ultra-materialized as I would at first have believed, but was instead metamorphosed in Psyche.
I was inspired by the remarks in those books; not by the parts in which everything was proved and demonstrated [but by] the remarks about the fact that this doesn’t make any sense. … So I had this as a challenge and an inspiration.

I would think I knew nothing in physics if I could say only how things could be but, without demonstrating that they can’t be otherwise.

Iamblichus in his treatise On the Arithmetic of Nicomachus observes p. 47- “that certain numbers were called amicable by those who assimilated the virtues and elegant habits to numbers.” He adds, “that 284 and 220 are numbers of this kind; for the parts of each are generative of each other according to the nature of friendship, as was shown by Pythagoras. For some one asking him what a friend was, he answered,

*another*I (ετεϑος εγω) which is demonstrated to take place in these numbers.” [“Friendly” thus: Each number is equal to the sum of the factors of the other.]
If it were always necessary to reduce everything to intuitive knowledge, demonstration would often be insufferably prolix. This is why mathematicians have had the cleverness to divide the difficulties and to demonstrate separately the intervening propositions. And there is art also in this; for as the mediate truths (which are called

*lemmas*, since they appear to be a digression) may be assigned in many ways, it is well, in order to aid the understanding and memory, to choose of them those which greatly shorten the process, and appear memorable and worthy in themselves of being demonstrated. But there is another obstacle, viz.: that it is not easy to demonstrate all the axioms, and to reduce demonstrations wholly to intuitive knowledge. And if we had chosen to wait for that, perhaps we should not yet have the science of geometry.
In a sense, of course, probability theory in the form of the simple laws of chance is the key to the analysis of warfare;… My own experience of actual operational research work, has however, shown that its is generally possible to avoid using anything more sophisticated. … In fact the wise operational research worker attempts to concentrate his efforts in finding results which are so obvious as not to need elaborate statistical methods to demonstrate their truth. In this sense advanced probability theory is something one has to know about in order to avoid having to use it.

In Pure Mathematics, where all the various truths are necessarily connected with each other, (being all necessarily connected with those hypotheses which are the principles of the science), an arrangement is beautiful in proportion as the principles are few; and what we admire perhaps chiefly in the science, is the astonishing variety of consequences which may be demonstrably deduced from so small a number of premises.

In the discovery of hidden things and the investigation of hidden causes, stronger reasons are obtained from sure experiments and demonstrated arguments than from probable conjectures and the opinions of philosophical speculators of the common sort...

In the mathematical investigations I have usually employed such methods as present themselves naturally to a physicist. The pure mathematician will complain, and (it must be confessed) sometimes with justice, of deficient rigour. But to this question there are two sides. For, however important it may be to maintain a uniformly high standard in pure mathematics, the physicist may occasionally do well to rest content with arguments which are fairly satisfactory and conclusive from his point of view. To his mind, exercised in a different order of ideas, the more severe procedure of the pure mathematician may appear not more but less demonstrative. And further, in many cases of difficulty to insist upon the highest standard would mean the exclusion of the subject altogether in view of the space that would be required.

In this manner the whole substance of our geometry is reduced to the definitions and axioms which we employ in our elementary reasonings; and in like manner we reduce the demonstrative truths of any other science to the definitions and axioms which we there employ.

Little could Plato have imagined, when, indulging his instinctive love of the true and beautiful for their own sakes, he entered upon these refined speculations and revelled in a world of his own creation, that he was writing the grammar of the language in which it would be demonstrated in after ages that the pages of the universe are written.

Mathematic is either Pure or Mixed: To Pure Mathematic belong those sciences which handle Quantity entirely severed from matter and from axioms of natural philosophy. These are two, Geometry and Arithmetic; the one handling quantity continued, the other dissevered. … Mixed Mathematic has for its subject some axioms and parts of natural philosophy, and considers quantity in so far as it assists to explain, demonstrate and actuate these.

Mathematics, from the earliest times to which the history of human reason can reach, has followed, among that wonderful people of the Greeks, the safe way of science. But it must not be supposed that it was as easy for mathematics as for logic, in which reason is concerned with itself alone, to find, or rather to make for itself that royal road. I believe, on the contrary, that there was a long period of tentative work (chiefly still among the Egyptians), and that the change is to be ascribed to a revolution, produced by the happy thought of a single man, whose experiments pointed unmistakably to the path that had to be followed, and opened and traced out for the most distant times the safe way of a science. The history of that intellectual revolution, which was far more important than the passage round the celebrated Cape of Good Hope, and the name of its fortunate author, have not been preserved to us. … A new light flashed on the first man who demonstrated the properties of the isosceles triangle (whether his name was Thales or any other name), for he found that he had not to investigate what he saw in the figure, or the mere concepts of that figure, and thus to learn its properties; but that he had to produce (by construction) what he had himself, according to concepts

*a priori*, placed into that figure and represented in it, so that, in order to know anything with certainty*a priori*, he must not attribute to that figure anything beyond what necessarily follows from what he has himself placed into it, in accordance with the concept.
Nature never “fails.” Nature complies with its own laws.

*Nature is the law*. When Man lacks understanding of Nature’s laws and a Man-contrived structure buckles unexpectedly, it does not fail. It only demonstrates that Man did not understand Nature’s laws and behaviors. Nothing failed. Man’s knowledge or estimating was inadequate.
Our failure to discern a universal good does not record any lack of insight or ingenuity, but merely demonstrates that nature contains no moral messages framed in human terms. Morality is a subject for philosophers, theologians, students of the humanities, indeed for all thinking people. The answers will not be read passively from nature; they do not, and cannot, arise from the data of science. The factual state of the world does not teach us how we, with our powers for good and evil, should alter or preserve it in the most ethical manner.

Our present work sets forth mathematical principles of philosophy. For the basic problem of philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces. It is to these ends that the general propositions in books 1 and 2 are directed, while in book 3 our explanation of the system of the world illustrates these propositions.

Perhaps a species that has accumulated ... tons of explosive per capita has already demonstrated its biological unfitness beyond any further question.

Politics is a science. You can demonstrate that you are right and that others are wrong.

Science has sometimes been said to be opposed to faith and inconsistent with it. But all science in fact rests on a basis of faith for it assumes the permanence and uniformity of natural laws—a thing which can never be demonstrated.

So the astronomer is on common ground with the physicist both in the subject and in the predicate of the conclusion, but the physicist demonstrates the predicate to belong to the subject by nature, whereas the astronomer does not care whether it belongs by nature or not. What, therefore, is the predicate for the physicist, is abstracted as the subject for the pure mathematician.

Surely the claim of mathematics to take a place among the liberal arts must now be admitted as fully made good. Whether we look at the advances made in modern geometry, in modern integral calculus, or in modern algebra, in each of these three a free handling of the material employed is now possible, and an almost unlimited scope is left to the regulated play of fancy. It seems to me that the whole of aesthetic (so far as at present revealed) may be regarded as a scheme having four centres, which may be treated as the four apices of a tetrahedron, namely Epic, Music, Plastic, and Mathematic. There will be found a

*common*plane to every three of these,*outside*of which lies the fourth; and through every two may be drawn a common axis*opposite*to the axis passing through the other two. So far is certain and demonstrable. I think it also possible that there is a centre of gravity to each set of three, and that the line joining each such centre with the outside apex will intersect in a common point the centre of gravity of the whole body of aesthetic; but what that centre is or must be I have not had time to think out.
The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.

The discovery in 1846 of the planet Neptune was a dramatic and spectacular achievement of mathematical astronomy. The very existence of this new member of the solar system, and its exact location, were demonstrated with pencil and paper; there was left to observers only the routine task of pointing their telescopes at the spot the mathematicians had marked.

The Excellence of Modern Geometry is in nothing more evident, than in those full and adequate Solutions it gives to Problems; representing all possible Cases in one view, and in one general Theorem many times comprehending whole Sciences; which deduced at length into Propositions, and demonstrated after the manner of the Ancients, might well become the subjects of large Treatises: For whatsoever Theorem solves the most complicated Problem of the kind, does with a due Reduction reach all the subordinate Cases.

The Greeks in the first vigour of their pursuit of mathematical truth, at the time of Plato and soon after, had by no means confined themselves to those propositions which had a visible bearing on the phenomena of nature; but had followed out many beautiful trains of research concerning various kinds of figures, for the sake of their beauty alone; as for instance in their doctrine of Conic Sections, of which curves they had discovered all the principal properties. But it is curious to remark, that these investigations, thus pursued at first as mere matters of curiosity and intellectual gratification, were destined, two thousand years later, to play a very important part in establishing that system of celestial motions which succeeded the Platonic scheme of cycles and epicycles. If the properties of conic sections had not been demonstrated by the Greeks and thus rendered familiar to the mathematicians of succeeding ages, Kepler would probably not have been able to discover those laws respecting the orbits and motions of planets which were the occasion of the greatest revolution that ever happened in the history of science.

The mediocre teacher tells. The good teacher explains. The superior teacher demonstrates. The great teacher inspires.

The more a science advances, the more will it be possible to understand immediately results which formerly could be demonstrated only by means of lengthy intermediate considerations: a mathematical subject cannot be considered as finally completed until this end has been attained.

There are three leading objects in the study of truth:—one, to discover it; another, to demonstrate it when discovered; the last, to separate it from all admixture of falsehood.

There is no art so difficult as the art of observation: it requires a skillful, sober spirit and a well-trained experience, which can only be acquired by practice; for he is not an observer

*who only sees the thing before him with his eyes, but he who sees of what parts the thing consists, and in what connexion the parts stand to the whole.*One person overlooks half from inattention; another relates more than he sees while he confounds it with that which he figures to himself; another sees the parts of the whole, but he throws things together that ought to be separated. ... When the observer has ascertained the foundation of a phenomenon, and he is able to associate its conditions, he then proves while he endeavours to produce the phenomena at his will, the correctness of his observations by*experiment*. To make a series of experiments is often to decompose an opinion into its individual parts, and to prove it by a sensible phenomenon. The naturalist makes experiments in order to exhibit a phenomenon in all its different parts. When he is able to show of a series of phenomena, that they are all operations of the same cause, he arrives at a simple expression of their significance, which, in this case, is called a Law of Nature. We speak of a simple property as a Law of Nature when it serves for the explanation of one or more natural phenomena.
This political movement has patently demonstrated that it will not defend the integrity of science in any case in which science runs afoul of its core political constituencies. In so doing, it has ceded any right to govern a technologically advanced and sophisticated nation.

Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the

*logic piano*imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.
We are accustomed to say that every human being displays both male and female instinctual impulses, needs, and attributes, but the characteristics of what is male and female can only be demonstrated in anatomy, and not in psychology.

We may have three principal objects in the study of truth: one to discover it when it is sought; another to demonstrate it when it is possessed; and a third, to discriminate it from the false when it is examined.

When you try to explain the behavior of water, remember to demonstrate the experiment first and the cause next.

Whenever … a controversy arises in mathematics, the issue is not whether a thing is true or not, but whether the proof might not be conducted more simply in some other way, or whether the proposition demonstrated is sufficiently important for the advancement of the science as to deserve especial enunciation and emphasis, or finally, whether the proposition is not a special case of some other and more general truth which is as easily discovered.

Who, by vigor of mind almost divine, the motions and figures of the planets, the paths of comets, and the tides of the seas, his mathematics first demonstrated.

Whoever … proves his point and demonstrates the prime truth geometrically should be believed by all the world, for there we are captured.

You know the formula

*m*over naught equals infinity,*m*being any positive number? [*m*/0 = ∞]. Well, why not reduce the equation to a simpler form by multiplying both sides by naught? In which case you have*m*equals infinity times naught [*m*= ∞ × 0]. That is to say, a positive number is the product of zero and infinity. Doesn't that demonstrate the creation of the Universe by an infinite power out of nothing? Doesn't it?
[1158] There is no certainty in sciences where one of the mathematical sciences cannot be applied, or which are not in relation with these mathematics.

[There was] in some of the intellectual leaders a great aspiration to demonstrate that the universe ran like a piece of clock-work, but this was was itself initially a religious
aspiration. It was felt that there would be something defective in Creation itself—something not quite worthy of God—unless the whole system of the universe could be shown to be interlocking, so that it carried the pattern of reasonableness and orderliness. Kepler, inaugurating the scientist’s quest for a mechanistic universe in the seventeenth century, is significant here—his mysticism, his music of the spheres, his rational deity demand a system which has the beauty of a piece of mathematics.

~~[Dubious]~~ I demonstrate by means of philosophy that the earth is round, and is inhabited on all sides; that it is insignificantly small, and is borne through the stars.

…from the same principles, I now demonstrate the frame of the System of the World.