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 FreudenthalRelated topics
beauty complex fashion generation hot idea inversion learner left matter off point problem procedure should start teaching turn version wayRelated quotes
With the collapse of the old welfare states, all this has come to seem decidedly quaint. As the language of antibureaucratic individualism has been adopted, with increasing ferocity, by the Right, which insists on "market solutions” to every social problem, the mainstream Left has increasingly reduced itself to fighting a kind of pathetic rearguard action, trying to salvage remnants of the old welfare state: it has acquiesced with–often spearheaded–attempts to make government efforts more "efficient” though the partial privatization of services and the incorporation of ever-more "market principle,” "market incentives,” and market-based "accountability processes” into the structure of the bureaucracy itself.
David Graeber
People may wonder why I am content to prescribe such a general and apparently vague formula as "Every day, in every way, I am getting better and better" for all and every ailment. The reason is, strange as it may seem, that our subconscious mind does not need the details. The general suggestion that everything "in every way" is going well is quite sufficient to set up the procedure of persuasion which will carry its effects to the different organs and improve every function. I have had remarkable demonstration of this in the course of my long teaching and experiments. Time and again I have seen patients cured, not only of the particular disease for which they sought relief, but also of minor disabilities which they had almost forgotten.
Émile Coué
We are indeed living at the end of the Golden Age of Mankind, or at least the end of this particular human civilization. That almost everything we take for granted today may, only a couple of decades farther along, seem entirely remarkable, that our most mundane artefacts and toys will stand as incredible examples of luxury and excess. That all of this will pass away, and the "simplest" bits of our day-to-day lives will become miracles of a half-remembered past. A past which will be responsible for that future-present misery. It is difficult to force myself through the trivial routine of my days when these thoughts are front and center. It is difficult to see beyond the veil they draw up about me and difficult to push it all aside long enough to write my silly little stories.
Caitlín R. Kiernan
To the biologist the problem of socialism appears largely as a problem of size. The extreme socialists desire to run every nation as a single business concern. I do not suppose that Henry Ford would find much difficulty in running Andorra or Luxembourg on a socialistic basis. He has already more men on his pay-roll than their population. It is conceivable that a syndicate of Fords, if we could find them, would make Belgium Ltd. or Denmark Inc. pay their way. But while nationalization of certain industries is an obvious possibility in the largest of states, I find it no easier to picture a completely socialized British Empire or United States than an elephant turning somersaults or a hippopotamus jumping a hedge.
J. B. S. Haldane
If you have weapons, take them home; if you do not have them, please do not seek to get them. We cannot solve this problem through retaliatory violence. We must meet violence with nonviolence. Remember the words of Jesus: «He who lives by the sword will perish by the sword.» We must love our white brothers, no matter what they do to us. We must make them know that we love them. Jesus still cries out in words that echo across the centuries: «Love your enemies; bless them that curse you; pray for them that despitefully use you.» This is what we must live by. We must meet hate with love. Remember, if I am stopped, this movement will not stop, because God is with the movement. Go home with this glowing faith and this radiant assurance.
Martin Luther King Jr.