1 do not believe there is anything useful which men can know with exactitude that they cannot know by arithmetic and algebra.

During the eighteenth and nineteenth centuries we can see the emergence of a tension that has yet to be resolved, concerning the attitude of scientists towards the usefulness of science. During this time, scientists were careful not to stress too much their relationships with industry or the military. They were seeking autonomy for their activities. On the other hand, to get social support there had to be some perception that the fruits of scientific activity could have useful results. One resolution of this dilemma was to assert that science only contributed at the discovery stage; others, industrialists for example, could apply the results. ... Few noted the ... obvious paradox of this position; that, if scientists were to be distanced from the 'evil' effects of the applications of scientific ideas, so too should they receive no credit for the 'good' or socially beneficial, effects of their activities.
Co-author with Philip Gummett (1947- ), -British social scientist

Every common mechanic has something to say in his craft about good and evil, useful and useless, but these practical considerations never enter into the purview of the mathematician.

I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world... Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value.

I looked for it [heavy hydrogen, deuterium] because I thought it should exist. I didn't know it would have industrial applications or be the basic for the most powerful weapon ever known [the nuclear bomb] ... I thought maybe my discovery might have the practical value of, say, neon in neon signs.
[He was awarded the 1931 Nobel Prize in Chemistry for discovering deuterium.]

It is not enough to discover and prove a useful truth previously unknown, but that it is necessary also to be able to propagate it and get it recognized.

Next came the patent laws. These began in England in 1624, and in this country with the adoption of our Constitution. Before then any man [might] instantly use what another man had invented, so that the inventor had no special advantage from his own invention. The patent system changed this, secured to the inventor for a limited time exclusive use of his inventions, and thereby added the fuel of interest to the fire of genius in the discovery and production of new and useful things.

Of what use are the great number of petrifactions, of different species, shape and form which are dug up by naturalists? Perhaps the collection of such specimens is sheer vanity and inquisitiveness. I do not presume to say; but we find in our mountains the rarest animals, shells, mussels, and corals embalmed in stone, as it were, living specimens of which are now being sought in vain throughout Europe. These stones alone whisper in the midst of general silence.

Science … has no consideration for ultimate purposes, any more than Nature has, but just as the latter occasionally achieves things of the greatest suitableness without intending to do so, so also true science, as the imitator of nature in ideas, will occasionally and in many ways further the usefulness and welfare of man,—but also without intending to do so.

The mathematician may be compared to a designer of garments who is utterly oblivious of the creature whom his garments may fit.

The more experiences and experiments accumulate in the exploration of nature, the more precarious the theories become. But it is not always good to discard them immediately on this account. For every hypothesis which once was sound was useful for thinking of previous phenomena in the proper interrelations and for keeping them in context. We ought to set down contradictory experiences separately, until enough have accumulated to make building a new structure worthwhile.

The process of natural selection has been summed up in the phrase 'survival of the fittest'. This, however, tells only part of the story. 'Survival of the existing' in many cases covers more of the truth. For in hosts of cases the survival of characters rests not on any special usefulness or fitness, but on the fact that individuals possessing these characters have inhabited or invaded a certain area. The principle of utility explains survivals among competing structures. It rarely accounts for qualities associated with geographic distribution.
The nature of animals which first colonize a district must determine what the future fauna will be. From their specific characters, which are neither useful nor harmful, will be derived for the most part the specific characters of their successors.
It is not essential to the meadow lark that he should have a black blotch on the breast or the outer tail-feather white. Yet all meadow larks have these characters just as all shore larks have the tiny plume behind the ear. Those characters of the parent stock, which may be harmful in the new relations, will be eliminated by natural selection. Those especially helpful will be intensified and modified, but the great body of characters, the marks by which we know the species, will be neither helpful nor hurtful. These will be meaningless streaks and spots, variations in size of parts, peculiar relations of scales or hair or feathers, little matters which can neither help nor hurt, but which have all the persistence heredity can give.

The scientist discovers a new type of material or energy and the engineer discovers a new use for it.

Those who eat most, and who take the most exercise, are not in better health than they who eat just as much as is good for them; and in the same way it is not those who know a great many things, but they who know what is useful who are valuable men.

[Science] dissipates errors born of ignorance about our true relations with nature, errors the more damaging in that the social order should rest only on those relations. TRUTH! JUSTICE! Those are the immutable laws. Let us banish the dangerous maxim that it is sometimes useful to depart from them and to deceive or enslave mankind to assure its happiness.

…all models are approximations. Essentially, all models are wrong, but some are useful. However, the approximate nature of the model must always be borne in mind…