A Quote by Leopold Kronecker
Definitions must contain the means of reaching a decision in a finite number off steps, and existence proofs must be conducted so that the quantity in question can be calculated with any degree of accuracy.
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If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.
The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great.
It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you may meet from time to time are merely the products of a deranged imagination.
To me, there's no question that using a metronome develops your speed and accuracy. If you're learning scales or you're jamming on parts that you can't quite pull off, it's a must.
Mathematicians are proud of the fact that, generally, they do their work with a piece of chalk and a blackboard. They value hand-done proofs above all else. A big question in mathematics today is whether or not computational proofs are legitimate. Some mathematicians won't accept computational proofs and insist that a real proof must be done by the human hand and mind, using equations.
Anything is one of a million paths. Therefore, a warrior must always keep in mind that a path is only a path; if he feels that he should not follow it, he must not stay with it under any conditions. His decision to keep on that path or to leave it must be free of fear or ambition. He must look at every path closely and deliberately. There is a question that a warrior has to ask, mandatorily: 'Does this path have a heart?'
In any finite region of space, matter can only arrange itself in a finite number of configurations, just as a deck of cards can be arranged in only finitely many different orders. If you shuffle the deck infinitely many times, the card orderings must necessarily repeat.
...art must must carry man's craving for the ideal, must be an expression of his reaching out towards it; that art must give man hope and faith. And the more hopeless the world in the artist's version, the more clearly perhaps must we see the ideal that stands in opposition - otherwise life becomes impossible! Art symbolises the meaning of our existence.
In a world where God does not exist, any reasonable person must see that the propensity toward selfishness should be the most venerated of all human traits. From this it stands to reason that any man, knowing the total expanse of his existence to be finite, who does not devote every moment of his life to acts of pure selfishness, must be seen as nothing more than a fool.
In all pointed sentences, some degree of accuracy must be sacrificed to conciseness.
... keep practicing. After a great deal of practice, we no longer think about all the necessary movements we must make; they become part of our existence. Before reaching that stage, however, you must practice and repeat. And if that's not enough, you must practice and repeat some more.
The war against Russia is an important chapter in the German nation's struggle for existence. [...] The objective of this battle must be the demolition of present-day Russia and must therefore be conducted with unprecedented severity. Every military action must be guided in planning and execution by an iron resolution to exterminate the enemy remorselessly and totally. In particular, no adherents of the contemporary Russian Bolshevik system are to be spared.
In the mass of mankind, I fear, there is too great a majority of fools and knaves; who, singly from their number, must to a certain degree be respected, though they are by no means respectable.
Each thing that exists remains forever, and that very existence of existence is proof of its eternity. But without that realization, which is the knowledge of perfect being, man would never know whether there was existence or non-existence. If eternal existence is altered, then it must become more beautiful; and if it disappears, it must return with more sublime image; and if it sleeps, it must dream of a better awakening, for it is ever greater upon its rebirth.
In order to climb properly on big peak one must free oneself of fear. This means you must write yourself off before any big climb. You must say to yourself, I may die here.
I must destroy everybody that comes in front of me. I must execute them and I must do it in grand style. I must slay Robert Allen like I must slay De La Hoya or anybody else that steps in that ring.
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