mathematics

More often than not, at the end of the day (or a month, or a year), you realize that your initial idea was wrong, and you have to try something else. These are the moments of frustration and despair. You feel that you have wasted an enormous amount of time, with nothing to show for it. This is hard to stomach. But you can never give up. You go back to the drawing board, you analyze more data, you learn from your previous mistakes, you try to come up with a better idea. And every once in a while, suddenly, your idea starts to work. It's as if you had spent a fruitless day surfing, when you finally catch a wave: you try to hold on to it and ride it for as long as possible. At moments like this, you have to free your imagination and let the wave take you as far as it can. Even if the idea sounds totally crazy at first.

The __eriousness_ of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is __ignificant_ if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences.

See appendix A for a proof that Winston Churchill was a carrot.

I don__ deny that it was more than a coincidence which made things turn out as they did, it was a whole train of coincidences. But what has providence to do with it? I don__ need any mystical explanation for the occurrence of the improbable; mathematics explains it adequately, as far as I__ concerned.Mathematically speaking, the probable (that in 6,000,000,000 throws with a regular six-sided die the one will come up approximately 1,000,000,000 times) and the improbable (that in six throws with the same die the one will come up six times) are not different in kind, but only in frequency, whereby the more frequent appears a priori more probable. But the occasional occurrence of the improbable does not imply the intervention of a higher power, something in the nature of a miracle, as the layman is so ready to assume. The term probability includes improbability at the extreme limits of probability, and when the improbable does occur this is no cause for surprise, bewilderment or mystification.

Woodrow Wilson, like most other educated Americans of his time, despised mathematics, complaining that "the natural man inevitably rebels against mathematics, a mild form of torture that could only be leaned by painful process of drill.

And, believe me, if I were again beginning my studies, I should follow the advice of Plato and start with mathematics.

In a way, mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite. That's why mathematics is really my only interest.

...in pure mathematics the mind deal only with its own creations and imaginations. The concepts of number and form have not been derived from any source other than the world of reality. The ten fingers on which men learned to count, that is, to carry out the first arithmetical operation, may be anything else, but they are certainly not only objects that can be counted, but also the ability to exclude all properties of the objects considered other than their number-and this ability is the product of a long historical evolution based on experience. Like the idea of number, so the idea of form is derived exclusively from the external world, and does not arise in the mind as a product of pure thought.

Overstimulation has been the real drawback. I need to find ways to stop thinking about analysis of algorithms, in order to do various other things that human beings ought to do.

With me, everything turns into mathematics.

The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to no

When he said someone had 'died", Erd_s meant that that the person had stopped doing mathematics. When he said someone had "left", the person had died.

Nothing comforted Sabine like long division. That was how she had passed time waiting for Phan and then Parsifal to come back from their tests. She figured the square root of the date while other people knit and read. Sabine blamed much of the world's unhappiness on the advent of calculators.

All mathematicians live in two different worlds. They live in a crystalline world of perfect platonic forms. An ice palace. But they also live in the common world where things are transient, ambiguous, subject to vicissitudes. Mathematicians go backward and forward from one world to another. They__e adults in the crystalline world, infants in the real one.

Really, there was only one problem with Mr. Davis, as far as Gregory was concerned; He taught math.

Nothing takes place in the world whose meaning is not that of some maximum or minimum.

What is it, in fact, that we are supposed to abstract from, in order to get, for example, from the moon to the number 1? By abstraction we do indeed get certain concepts, viz. satellite of the Earth, satellite of a planet, non-self-luminous heavenly body, heavenly body, body, object. But in this series 1 is not to be met with; for it is no concept that the moon could fall under. In the case of 0, we have simply no object at all from which to start our process of abstracting. It is no good objecting that 0 and 1 are not numbers in the same sense as 2 and 3. What answers the question How many? is number, and if we ask, for example, "How many moons has this planet?", we are quite as much prepared for the answer 0 or 1 as for 2 or 3, and that without having to understand the question differently. No doubt there is something unique about 0, and about 1000; but the same is true in principle of every whole number, only the bigger the number the less obvious it is. To make out of this a difference in kind is utterly arbitrary. What will not work with 0 and 1 cannot be essential to the concept of number.

Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler__ equation reaches down into the very depths of existence.