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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.
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On higher arithmetic. Mathematical Circles Adieu (1977) by Howard W. Eves

 

Carl Friedrich Gauss

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It has been observed here this morning that we are called fanatics. Bless me! That is nothing. Who has not been called a fanatic who has discovered anything new in philosophy or science? We have all read of Galileo the astronomer who, contrary to the system of astronomy that had been received for ages before his day, taught that the sun, and not the earth, was the centre of our planetary system? For this the learned astronomer was called "fanatic," and subjected to persecution and imprisonment of the most rigorous character. So it has been with others who have discovered and explained new truths in science and philosophy which have been in opposition to long-established theories; and the opposition they have encountered has endured until the truth of their discoveries has been demonstrated by time. The term "fanatic" is not applied to professors of religion only...I will tell you who the real fanatics are: they are they who adopt false principles and ideas as facts, and try to establish a superstructure upon a false foundation. They are the fanatics; and however ardent and zealous they may be, they may reason or argue on false premises till doomsday, and the result will be false. If our religion is of this character we want to know it; we would like to find a philosopher who can prove it to us. We are called ignorant; so we are: but what of it? Are not all ignorant? I rather think so. Who can tell us of the inhabitants of this little planet that shines of an evening, called the moon? When we view its face we may see what is termed "the man in the moon," and what some philosophers declare are the shadows of mountains. But these sayings are very vague, and amount to nothing; and when you inquire about the inhabitants of that sphere you find that the most learned are as ignorant in regard to them as the most ignorant of their fellows. So it is with regard to the inhabitants of the sun. Do you think it is inhabited? I rather think it is. Do you think there is any life there? No question of it; it was not made in vain. It was made to give light to those who dwell upon it, and to other planets; and so will this earth when it is celestialized. Every planet in its first rude, organic state receives not the glory of God upon it, but is opaque; but when celestialized, every planet that God brings into existence is a body of light, but not till then. Christ is the light of this planet. God gives light to our eyes.

 

Brigham Young

 

A principle of induction would be a statement with the help of which we could put inductive inferences into a logically acceptable form. In the eyes of the upholders of inductive logic, a principle of induction is of supreme importance for scientific method: "... this principle", says Reichenbach, "determines the truth of scientific theories. To eliminate it from science would mean nothing less than to deprive science of the power to decide the truth or falsity of its theories. Without it, clearly, science would no longer have the right to distinguish its theories from the fanciful and arbitrary creations of the poet's mind."
Now this principle of induction cannot be a purely logical truth like a tautology or an analytic statement. Indeed, if there were such a thing as a purely logical principle of induction, there would be no problem of induction; for in this case, all inductive inferences would have to be regarded as purely logical or tautological transformations, just like inferences in inductive logic. Thus the principle of induction must be a synthetic statement; that is, a statement whose negation is not self-contradictory but logically possible. So the question arises why such a principle should be accepted at all, and how we can justify its acceptance on rational grounds.

 

Karl Popper

 

Ere many generations pass, our machinery will be driven by a power obtainable at any point of the universe. This idea is not novel. Men have been led to it long ago by instinct or reason; it has been expressed in many ways, and in many places, in the history of old and new. We find it in the delightful myth of Antheus, who derives power from the earth; we find it among the subtle speculations of one of your splendid mathematicians and in many hints and statements of thinkers of the present time. Throughout space there is energy. Is this energy static or kinetic! If static our hopes are in vain; if kinetic — and this we know it is, for certain — then it is a mere question of time when men will succeed in attaching their machinery to the very wheelwork of nature.

 

Nikola Tesla

 

I mean to lead a simple life, to choose a simple shell I can carry easily — like a hermit crab. But I do not. I find that my frame of life does not foster simplicity. My husband and five children must make their way in the world. The life I have chosen as a wife and mother entrains a whole caravan of complications.

 

Anne Morrow Lindbergh

 

Wherever Mathematics is mixed up with anything, which is outside its field, you will find attempts to demonstrate these merely conventional propositions a priori, and it will be your task to find out the false deduction in each case.

 

Carl Gustav Jacob Jacobi