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I toyed with the idea of playing Ravel's 'Pavane pour une infante defunte' but I couldn't remember if it's a tune or Latin prescription for piles.
The idea is that it will tell you where the importance lies, because it will be able to show you what impacts you in your everyday life.
Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying 'No, that's wrong.' Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.
The clearer the teacher makes it, the worse it is for you. You must work things out for yourself and make the ideas your own.
Notable enough, however, are the controversies over the series 1 - 1 + 1 - 1 + 1 - ... whose sum was given by Leibniz as 1/2, although others disagree. ... Understanding of this question is to be sought in the word "sum"; this idea, if thus conceived - namely, the sum of a series is said to be that quantity to which it is brought closer as more terms of the series are taken - has relevance only for convergent series, and we should in general give up the idea of sum for divergent series.
Gel'fand amazed me by talking of mathematics as though it were poetry. He once said about a long paper bristling with formulas that it contained the vague beginnings of an idea which could only hint at and which he had never managed to bring out more clearly. I had always thought of mathematics as being much more straightforward: a formula is a formula, and an algebra is an algebra, but Gel'fand found hedgehogs lurking in the rows of his spectral sequences!
The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.
One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That's so unlike the true nature of mathematics.
Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.
I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts... When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through the process of 'seeing'.
"The most powerful single idea in mathematics is the notion of a variable."
The revolutionary spirit is mighty convenient in this, that it frees one from all scruples as regards ideas. Its hard absolute optimism is repulsive to my mind by the menace of fanaticism and intolerance it contains. No doubt one should smile at these things; but, imperfect Esthete, I am no better Philosopher. All claim to special righteousness awakens in me that scorn and anger from which a philosophical mind should be free.
People ... have no idea what a hard job it is for two writers to be friends. Sooner or later you have to talk about each other's work.
Books work as an art form (and an economic one) because they are primarily the work of an individual.
Writing a book is a tremendous experience. It pays off intellectually. It clarifies your thinking. It builds credibility. It is a living engine of marketing and idea spreading, working every day to deliver your message with authority. You should write one.
But perhaps the rest of us could have separate classes in science appreciation, the wonder of science, scientific ways of thinking, and the history of scientific ideas, rather than laboratory experience.
...That is one of the main causes of this arrogance: the idea of power. Then you lose your true power which is to be part of all, and the only way you can be part of all is to understand it. And when you don't understand, you have to go humbly to it. You don't go to school and say, 'I know what you're going to teach me'.
The best society, the best human existence, arrives when humans most closely determine the truth, and act on the truth, and separate it from superstition, falsity, or misinformation. And there is no better system for determining the truth than free speech: Testing the validity of an idea in the waters of public discussion and debate.
Angels are able to know and understand better than the human intellect can, precisely because such knowledge and understanding comes to them by way of ideas infused in them by God.
I am very comfortable with the idea that we can override biology with free will.
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