Euclid... gave his famous definition of a point: "A point is that which has no parts, or which has no magnitude." ...A point has no existence by itself. It exists only as a part of the pattern of relationships which constitute the geometry of Euclid. This is what one means when one says that a point is a mathematical abstraction. The question, What is a point? has no satisfactory answer. Euclid's definition certainly does not answer it. The right way to ask the question is: How does the concept of a point fit into the logical structure of Euclid's geometry? ...It cannot be answered by a definition.
Freeman DysonRelated topics
abstraction answer ask concept definition euclid fit geometry magnitude point question right satisfactory way means partsRelated quotes
The student who would build his knowledge on solid foundations, and proceed by just degrees to the pinnacles of truth, is directed by the great philosopher of France to begin by doubting of his own existence. In like manner, whoever would complete any arduous and intricate enterprise, should, as soon as his imagination can cool after the first blaze of hope, place before his own eyes every possible embarrassment that may retard or defeat him. He should first question the probability of success, and then endeavour to remove the objections that he has raised.
Samuel Johnson
Prayer, fasting, vigils, and all other Christian practices, however good they may be in themselves, certainly do not constitute the aim of our Christian life: they are but the indispensable means of attaining that aim. For the true aim of the Christian life is the acquisition of the Holy Spirit of God. As for fasts, vigils, prayer and almsgiving, and other good works done in the name of Christ, they are only the means of acquiring the Holy Spirit of God. Note well that it is only good works done in the name of Christ that bring us the fruits of the Spirit.
Seraphim of Sarov
Powerful groups set constraints on and may even determine the actions of state authority. They constitute the society that is protected by the state; their particular concept of justice reigns. The definition and functioning of property rights tend to advance their interest and welfare. ...the basic domestic function of the state is to define and protect property rights of individuals and groups. ...The primary external function of the state is to protect the property rights and personal security of its members vis-à-vis the citizens and actions of other states.
Robert Gilpin
In mathematics, as in any scientific research, we find two tendencies... [T]he tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects... a live rapport with them... which stresses the concrete meaning of their relations. ...[I]ntuitive understanding plays a major role in geometry. ...[S]uch concrete intuition is of great value not only for the research worker, but... for anyone who wishes to study and appreciate the results of research in geometry.
David Hilbert
Philosophers might care to ask themselves ... how often they are accomplices in increasing the audience for a second-rate article simply because their introductory course needs a simple-minded version of a bad idea that even freshmen can refute. Some of the most frequently reprinted articles in twentieth-century philosophy are famous precisely because nobody believes them; everybody can see what's wrong with them. ... The confirmation of this claim is left as an exercise for the reader. Among the memes that structure the infosphere and hence affect the transmission of other memes are the laws of libel.
Daniel Dennett