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Carl Friedrich Gauss

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The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. ... Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.
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Problema, numeros primos a compositis dignoscendi, hosque in factores suos primos resolvendi, ad gravissima ac utilissima totius arithmeticae pertinere, et geometrarum tum veterum tum recentiorum industriam ac sagacitatem occupavisse, tam notum est, ut de hac re copiose loqui superfluum foret. ... [P]raetereaque scientiae dignitas requirere videtur, ut omnia subsidia ad solutionem problematis tam elegantis ac celebris sedulo excolantur.
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Disquisitiones Arithmeticae (1801) Article 329

 
Carl Friedrich Gauss

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Armed with his prime number tables, Gauss began his quest. As he looked at the proportion of numbers that were prime, he found that when he counted higher and higher a pattern started to emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out of the mist.

 
Carl Friedrich Gauss
 

It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning.

 
David Hilbert
 

The problem of induction is, roughly speaking, the problem of finding a way to prove that certain empirical generalizations which are derived from past experience will hold good also in the future. There are only two ways of approaching this problem on the assumption that it is a genuine problem, and it is easy to see that neither of them can lead to its solution.

 
Alfred Jules Ayer
 

It seems to me that the real problem is the mind itself, and not the problem which the mind has created and tries to solve. If the mind is petty, small, narrow, limited, however great and complex the problem may be, the mind approaches that problem in terms of its own pettiness. If I have a little mind and I think of God, the God of my thinking will be a little God, though I may clothe him with grandeur, beauty, wisdom, and all the rest of it. It is the same with the problem of existence, the problem of bread, the problem of love, the problem of sex, the problem of relationship, the problem of death. These are all enormous problems, and we approach them with a small mind; we try to resolve them with a mind that is very limited. Though it has extraordinary capacities and is capable of invention, of subtle, cunning thought, the mind is still petty. It may be able to quote Marx, or the Gita, or some other religious book, but it is still a small mind, and a small mind confronted with a complex problem can only translate that problem in terms of itself, and therefore the problem, the misery increases. So the question is: Can the mind that is small, petty, be transformed into something which is not bound by its own limitations?

 
Jiddu Krishnamurti
 

Pythagoras, as everyone knows, said that "all things are numbers." This statement, interpreted in a modern way, is logical nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music and the connection which he established between music and arithmetic survives in the mathematical terms "harmonic mean" and "harmonic progression." He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares or cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles (or as we would more naturally say, shot) required to make the shapes in question.

 
Pythagoras
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