We can invent as many theories we like, and any one of them can be made to fit the facts. But that theory is always preferred which makes the fewest number of assumptions.
We may affirm of Mr. Buffon, that which has been said of the chemists of old; though he may have failed in attaining his principal aim, of establishing a theory, yet he has brought together such a multitude of facts relative to the history of the earth, and the nature of its fossil productions, that curiosity finds ample compensation, even while it feels the want of conviction.
We must alter theory to adapt it to nature, but not nature to adapt it to theory.
What I remember most clearly was that when I put down a suggestion that seemed to me cogent and reasonable, Einstein did not in the least contest this, but he only said, 'Oh, how ugly.' As soon as an equation seemed to him to be ugly, he really rather lost interest in it and could not understand why somebody else was willing to spend much time on it. He was quite convinced that beauty was a guiding principle in the search for important results in theoretical physics.
When an investigator has developed a formula which gives a complete representation of the phenomena within a certain range, he may be prone to satisfaction. Would it not be wiser if he should say 'Foiled again! I can find out no more about Nature along this line.'
When someone admits one and rejects another which is equally in accordance with the appearances, it is clear that he has quitted all physical explanation and descended into myth.
Without theory, practice is but routine born of habit. Theory alone can bring forth and develop the spirit of invention. ... [Do not] share the opinion of those narrow minds who disdain everything in science which has not an immediate application. ... A theoretical discovery has but the merit of its existence: it awakens hope, and that is all. But let it be cultivated, let it grow, and you will see what it will become.
[Gauss calculated the elements of the planet Ceres] and his analysis proved him to be the first of theoretical astronomers no less than the greatest of 'arithmeticians.'
[In mathematics] There are two kinds of mistakes. There are fatal mistakes that destroy a theory, but there are also contingent ones, which are useful in testing the stability of a theory.
[Referring to Fourier's mathematical theory of the conduction of heat] ... Fourier's great mathematical poem.
I worked on true Baconian principles, and without any theory collected facts.
It took Einstein ten years of groping through the fog to get the theory of special relativity, and he was a bright guy.
As regards authority I so proceed. Boetius says in the second prologue to his Arithmetic, 'If an inquirer lacks the four parts of mathematics, he has very little ability to discover truth.' And again, 'Without this theory no one can have a correct insight into truth.' And he says also, 'I warn the man who spurns these paths of knowledge that he cannot philosophize correctly.' And Again, 'It is clear that whosoever passes these by, has lost the knowledge of all learning.'
Dissections daily convince us of our ignorance of the seats of diseases, and cause us to blush at our prescriptions. How often are we disappointed in our expectation from the most certain and powerful of our remedies, by the negligence or obstinacy of our patients! What mischief have we done under the belief of false facts and false theories! We have assisted in multiplying diseases. We have done more — we have increased their mortality.
If any human being earnestly desire to push on to new discoveries instead of just retaining and using the old; to win victories over Nature as a worker rather than over hostile critics as a disputant; to attain, in fact, clear and demonstrative knowlegde instead of attractive and probable theory; we invite him as a true son of Science to join our ranks.
Indeed, nothing more beautifully simplifying has ever happened in the history of science than the whole series of discoveries culminating about 1914 which finally brought practically universal acceptance to the theory that the material world contains but two fundamental entities, namely, positive and negative electrons, exactly alike in charge, but differing widely in mass, the positive electron-now usually called a proton-being 1850 times heavier than the negative, now usually called simply the electron.
The great revelation of the quantum theory was that features of discreteness were discovered in the Book of Nature, in a context in which anything other than continuity seemed to be absurd according to the views held until then.
What good does the theory [of economics] do if it is not working for people?
A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.
It is always noteworthy that all those who seriously study this science [the theory of numbers] conceive a sort of passion for it.
Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.
No one shall expel us from the paradise which Cantor has created for us. Expressing the importance of Cantor's set theory in the development of mathematics.
Poincaré was a vigorous opponent of the theory that all mathematics can be rewritten in terms of the most elementary notions of classical logic; something more than logic, he believed, makes mathematics what it is.
The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.
It is clear, then, that the idea of a fixed method, or of a fixed theory of rationality, rests on too naive a view of man and his social surroundings. To those who look at the rich material provided by history, and who are not intent on impoverishing it in order to please their lower instincts, their craving for intellectual security in the form of clarity, precision, "objectivity," "truth," it will become clear that there is only one principle that can be defended under all circumstances and in all stages of human development. It is the principle: anything goes.
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