A Quote by Carl Friedrich Gauss
I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.
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A theorem is a proposition which is a strict logical consequence of certain definitions and other propositions.
It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with so much success, did not leave us with the proofs of the theorems he discovered. In truth, Messrs Euler and Lagrange, who have not disdained this kind of research, have proved most of these theorems, and have even substituted extensive theories for the isolated propositions of Fermat. But there are several proofs which have resisted their efforts.
If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.
I realized that anything to do with Fermat's Last Theorem generates too much interest.
About Thomas Hobbes: He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition "By God", sayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry.
To make our position clearer, we may formulate it in another way. Let us call a proposition which records an actual or possible observation an experiential proposition. Then we may say that it is the mark of a genuine factual proposition, not that it should be equivalent to an experiential proposition, or any finite number of experiential propositions, but simply that some experiential propositions can be deduced from it in conjunction with certain other premises without being deducible from those other premises alone.
Before beginning [to try to prove Fermat's Last Theorem] I should have to put in three years of intensive study, and I haven't that much time to squander on a probable failure.
Mathematics is a logical method. . . . Mathematical propositions express no thoughts. In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics.
What can I do my friends, if I do not know? I am neither Christian nor Jew, nor Muslim nor Hindu. What can I do? What can I do? Not of the East, nor of the West, Nor of the land, nor of the sea, Not of nature's essence, nor of circling heavens. What could I be?
In the first place a philosophical proposition must be general. It must not deal specially with things on the surface of the earth, or within the solar system, or with any other portion of space and time. . . . This brings us to a second characteristic of philosophical propositions, namely that they must be a priori. A philosophical proposition must be such as can neither be proved nor disproved by empirical evidence. . . . Philosophy, if what has been said is correct, becomes indistinguishable from logic as that word has now come to be used.
...the great movement of apostasy being organized in every country for the establishment of a One-World Church which shall have neither dogmas, nor hierarchy, neither discipline for the mind, nor curb for the passions, and which, under the pretext of freedom and human dignity, would bring back to the world (if such a Church could overcome) the reign of legalized cunning and force, and the oppression of the weak, and of all those who toil and suffer. [...] Indeed, the true friends of the people are neither revolutionaries, nor innovators: they are traditionalists.
The chief difference between me and others is that I have plenty of time not only because I am without a multitude of responsibilities and without daily tasks, which demand attention: But also because I am basically without ambition. Neither the present nor the future has claims on me.
People have murdered each other, in massive wars and guerilla actions, for many centuries, and still murder each other in the present, over Ideologies and Religions which, stated as propositions, appear neither true nor false to modern logicians- meaningless propositions that look meaningful to the linguistically naive.
Neither fear nor self-interest can convert the soul. They may change the appearance, perhaps even the conduct, but never the object of supreme desire... Fear is the motive which constrains the slave; greed binds the selfish man, by which he is tempted when he is drawn away by his own lust and enticed (James 1:14). But neither fear nor self-interest is undefiled, nor can they convert the soul. Only charity can convert the soul, freeing it from unworthy motives.
All that was neither a city, nor a church, nor a river, nor color, nor light, nor shadow: it was reverie. For a long time, I remained motionless, letting myself be penetrated gently by this unspeakable ensemble, by the serenity of the sky and the melancholy of the moment. I do not know what was going on in my mind, and I could not express it; it was one of those ineffable moments when one feels something in himself which is going to sleep and something which is awakening.
Thought must never submit, neither to a dogma, nor to a party, nor to a passion, nor to an interest, nor to a preconceived idea, nor to whatever it may be, save to the facts themselves, because, for thought, submission would mean ceasing to be.
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