These primitive propositions ... suffice to deduce all the properties of the numbers that we shall meet in the sequel. There is, however, an infinity of systems which satisfy the five primitive propositions. ... All systems which satisfy the five primitive propositions are in one-to-one correspondence with the natural numbers. The natural numbers are what one obtains by abstraction from all these systems; in other words, the natural numbers are the system which has all the properties and only those properties listed in the five primitive propositions.
Giuseppe PeanoRelated topics
abstraction correspondence five infinity meet natural primitive sequel words one-to-oneRelated quotes
It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.
Pierre-Simon Laplace
Suppose that the organism is given the problem of determining the analysis of a stimulus at a certain level of representation - e.g., the problem of determining which sequence of words a given utterance encodes. Since, in the general case, transducer outputs underdetermine perceptual analyses, we can think of the solution of such problems as involving processes of nondemonstrative inference. In particular, we can think of each input system as a computational mechanism which projects and confirms a certain class of hyputheses on the basis of a certain body of data.
Jerry Fodor
The face of "evil" is always the face of total need. A dope fiend is a man in total need of dope. Beyond a certain frequency need knows absolutely no limit or control. In the words of total need: "Wouldn't you?" Yes you would. You would lie, cheat, inform on your friends, steal, do anything to satisfy total need. Because you would be in a state of total sickness, total possession, and not in a position to act in any other way. Dope fiends are sick people who cannot act other than they do. A rabid dog cannot choose but bite.
William S. Burroughs
If every day a man takes orders in silence from an incompetent superior, if every day he solemnly performs ritual acts which he privately finds ridiculous, if he unhesitatingly gives answers to questionnaires which are contrary to his real opinions and is prepared to deny his own self in public, if he sees no difficulty in feigning sympathy or even affection where, in fact, he feels only indifference or aversion, it still does not mean that he has entirely lost the use of one of the basic human senses, namely, the sense of humiliation.
Václav Havel
The nude gains its enduring value from the fact that it reconciles several contrary states. It takes the most sensual and immediately interesting object, the human body, and puts it out of reach of time and desire; it takes the most purely rational concept of which mankind is capable, mathematical order, and makes it a delight to the senses; and it takes the vague fears of the unknown and sweetens them by showing that the gods are like men and may be worshiped for their life-giving beauty rather than their death-dealing powers.
Kenneth Clark