Procedure Quotes - page 5
No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical. Rather than behaving anti-didactically, one should recognise that the learner is entitled to recapitulate in a fashion of mankind. Not in the trivial matter of an abridged version, but equally we cannot require the new generation to start at the point where their predecessors left off.
Hans Freudenthal
Unfortunately, truth is neither a listable nor a decidable property; nor is the truth of a statement of arithmetic. The American logician John Myhill has used the term 'prospective' to characterize those attributes of the world that are neither listable nor decidable. They are properties that cannot be recognized by the application of some formula, made to conform to a rule, or generated by some computer program. They are characterized by incessant novelty that cannot be encompassed by any finite set of rules. 'Beauty', 'ugliness', 'truth', 'harmony', simplicity', and 'poetry' are names we give to some of the attributes of this sort. There is no way of listing all examples of beauty or ugliness, nor any procedure for saying whether or not something possesses either of those attributes, without redefining them in some more restrictive fashion that kills their prospective character.
John D. Barrow
In Plato's Republic, when the interlocutor of Socrates appears to bring certain plausible reasons to bear upon the mathematical sciences, to show that they are useful to human life, arithmetic for reckoning, distributions, contributions, exchanges, and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation and other undertakings, revealing beforehand the proper procedure and suitable season, Socrates, reproaching him says: "You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld."
Nicomachus