The word 'proof' should strictly only be used when we are dealing with deductive inferences... Popper claimed that scientists only need to use deductive inferences... So if a scientist is only interested in demonstrating that a given theory is false, she may be able to accomplish her goal without the use of inductive inferences... When a scientist collects experimental data, her aim might be to show that a particular theory... is false. She will have to resort to inductive reasoning... So Popper's attempt to show that science can get by without induction does not succeed.
The aim was to regulate the value of money by increasing or diminishing the quantity of it. The effects of these measures appeared to provide an inductive proof of the correctness of this superficial version of the Quantity Theory, and incidentally concealed the weaknesses of its logic.
Ludwig von Mises
It is difficult to distinguish deduction from what in other circumstances is called problem-solving. And concept learning, inference, and reasoning by analogy are all instances of inductive reasoning. (Detectives typically induce, rather than deduce.) None of these things can be done separately from each other, or from anything else. They are pseudo-categories.
This is the greatest mystery of the human mind-the inductive leap. Everything falls into place, irrelevancies relate, dissonance becomes harmony, and nonsense wears a crown of meaning. But the clarifying leap springs from the rich soil of confusion, and the leaper is not unfamiliar with pain.
The seers of ancient India had, in their experiments and efforts at spiritual training and the conquest of the body, perfected a discovery which in its importance to the future of human knowledge dwarfs the divinations of Newton and Galileo , even the discovery of the inductive and experimental method in Science was not more momentous...
The intellectual equipment needed for the job of the future is an ability to define problems, quickly assimilate relevant data, conceptualize and reorganize the information, make deductive and inductive leaps with it, ask hard questions about it, discuss findings with colleagues, work collaboratively to find solutions and then convince others.
The increasing technicality of the terminology employed is also a serious difficulty. It has become necessary to learn an extensive vocabulary before a book in even a limited department of science can be consulted with much profit. This change, of course, has its advantages for the initiated, in securing precision and concisement of statement; but it tends to narrow the field in which an investigator can labour, and it cannot fail to become, in the future, a serious impediment to wide inductive generalisations.
Thomas George Bonney
In the "commentatio" (note presented to the Russian Academy) in which his theorem on polyhedra (on the number of faces, edges and vertices) was first published Euler gives no proof. In place of a proof, he offers an inductive argument: he verifies the relation in a variety of special cases. There is little doubt that he also discovered the theorem, as many of his other results, inductively.
A woman anticipates danger by instinct, rather than inductive reasoning. Due to this, when faced with danger due to passionate feelings related to their basic needs, women are impelled by reasoning, conditioned by instincts acquired from family traditions and the conventions of her social stratum, much more than men are.
I have said that science is impossible without faith. ... Inductive logic, the logic of Bacon, is rather something on which we can act than something which we can prove, and to act on it is a supreme assertion of faith ... Science is a way of life which can only fluorish when men are free to have faith.
The very hope of experimental philosophy, its expectation of constructing the sciences into a true philosophy of nature, is basedon induction, or, if you please, the a priori presumption, that physical causation is universal; that the constitution of nature is written in its actual manifestations, and needs only to be deciphered by experimental and inductive research; that it is not a latent invisible writing, to be brought out by the magic of mental anticipation or metaphysical mediation.
Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
[Vestiges begins] from principles which are at variance with all sober inductive truth. The sober facts of geology shuffled, so as to play a rogue's game; phrenology (that sinkhole of human folly and prating coxcombry); spontaneous generation; transmutation of species; and I know not what; all to be swallowed, without tasting and trying, like so much horse-physic!! Gross credulity and rank infidelity joined in unlawful marriage, and breeding a deformed progeny of unnatural conclusions!
It is possible to express the laws of thermodynamics in the form of independent principles , deduced by induction from the facts of observation and experiment, without reference to any hypothesis as to the occult molecular operations with which the sensible phenomena may be conceived to be connected; and that course will be followed in the body of the present treatise. But, in giving a brief historical sketch of the progress of thermodynamics, the progress of the hypothesis of thermic molecular motions cannot be wholly separated from that of the purely inductive theory.
William John Macquorn Rankine
[Patricia Greenfield] concluded that 'every medium develops some cognitive skills at the expense of others.' Our growing use of the Net and other screen-based technologies has led to the 'widespread and sophisticated development of visual-spatial skills.' We can, for example, rotate objects in our minds better than we used to be able to. But our 'new strengths in visual-spatial intelligence' go hand in hand with a weakening of our capacities for the kind of 'deep processing' that underpins 'mindful knowledge acquisition, inductive analysis, critical thinking, imagination, and reflection.
Georgie, I've got it, " she said. "I've guessed what it means." Now though Georgie was devoted to his Lucia, he was just as devoted to inductive reasoning, and Daisy Quantock was, with the exception of himself, far the most powerful logician in the place. "What is it, then?" he asked. "Stupid of me not to have thought of it at once, " said Daisy. "Why, don't you see? Pepino is Auntie's heir, for she was unmarried, and he's the only nephew, and probably he has been left piles and piles. So naturally they say it's a terrible blow. Wouldn't do to be exultant. They must say it's a terrible blow, to show they don't care about the money. The more they're left, the sadder it is. So natural. I blame myself for not having thought of it at once...
What is important is to understand the true boundaries of reality, not the probable boundaries of possible future events. Although boundary conditions operate on the future, they are probabilistic constraints, not absolutely determined fact. We assume that ten minutes hence, the room we are in will still exist. It is a boundary condition that will define the next ten minutes in our space/time coordinate. But we cannot know who will be in the room ten minutes hence; that is free to be determined. One may ask if we can really know that the room will exist at any future moment. This is where induction enters the picture, since in truth we cannot know with certainty. There is no absolutely rigorous way of establishing that. But we can make the inductive leap of faith that has to do with accumulated experience. We project that the existence of the room will remain a boundary condition, but in principle in the next ten minutes there could be an earthquake and this building might not be left standing. However, for that to happen, the boundary condition will have to be radically disrupted in some unexpected and improbable manner. What is so curious is that such a thing could occur.
But why did she come back and take her card away?" asked Miss Mackintosh. "I told Florence that Miss Mapp had heard something dreadful about her. And how did she know that Lady Deal was coming here at all? The house was taken in my name." "That's just what we all long to find out, " said Diva eagerly. "She said that somebody in London told her." "But who?" asked Miss Mackintosh. "Florence only settled to come at lunch time that day, and she told her butler to ring up Susie and say she would be arriving." Diva's eyes grew round and bright with inductive reasoning. "I believe we're on the right tack, " she said. "Could she have received Lady Deal's butler's message, do you think? What's your number?" "Tilling 76, " said Miss Mackintosh. Evie gave three ecstatic little squeaks. "Oh, that's it, that's it!" she said. "Elizabeth Mapp is Tilling 67. So careless of them, but all quite plain. And she did hear it from somebody in London. Quite true, and so dreadfully false and misleading, and so like her. Isn't it, Diva? Well, it does serve her right to be found out." Miss Mackintosh was evidently a true Tillingite. "How marvellous!" she said.