A Quote by Mark Kac
... there are those who believe that mathematics can sustain itself and grow without any further contact with anything outside itself, and those who believe that nature is still and always will be one of the main (if not the main) sources of mathematical inspiration. The first group is identified as "pure mathematicians" (though "purist" would be more adequate) while the second is, with equal inadequacy, referred to as "applied".
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Outside observers often assume that the more complicted a piece of mathematics is, the more mathematicians admire it. Nothing could be further from the truth. Mathematicians admire elegance and simplicity above all else, and the ultimate goal in solving a problem is to find the method that does the job in the most efficient manner. Though the major accolades are given to the individual who solves a particular problem first, credit (and gratitude) always goes to those who subsequently find a simpler solution.
Relations between pure and applied mathematicians are based on trust and understanding.
Namely, pure mathematicians do not trust applied mathematicians, and applied mathematicians do not understand pure mathematicians.
In any bureaucratic organization there will be two kinds of people: those who work to further the actual goals of the organization, and those who work for the organization itself. Examples in education would be teachers who work and sacrifice to teach children, vs. union representative who work to protect any teacher including the most incompetent. The Iron Law states that in all cases, the second type of person will always gain control of the organization, and will always write the rules under which the organization functions.
The refusal to take sides on great moral issues is itself a decision. It is a silent acquiescence to evil. The Tragedy of our time is that those who still believe in honesty lack fire and conviction, while those who believe in dishonesty are full of passionate conviction.
Those who say they believe in God and yet neither love nor fear Him, do not in fact believe in Him but in those who have taught them that God exists. Those who believe that they believe in God, but without any passion in their heart, any anguish of mind, without uncertainty, without doubt, without an element of despair even in their consolation, believe only in the God-idea, not in God.
A world without right or wrong was a world that did not want itself, anything other than itself, or anything not those two things, but that still wanted something. A world without right or wrong invited you over, complained about you, and gave you cookies. Don't leave, it said, and gave you a vegan cookie. It avoided eye contact, but touched your knee sometimes. It was the world without right or wrong. It didn't have any meaning. It just wanted a little meaning.
Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians.
Time doesn't exist. It doesn't exist in any way. It's more subjective than real. Time doesn't exist. I believe in memory. Memory is the real inspiration. Memory creates time. Memory is pure power. Pure power and pure strength, and pure utilization of space and time (if time is something we can really ever label). But I don't believe in time itself.
Pure mathematics is on the whole distinctly more useful than applied... For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.
Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.
Mathematical thinking is not the same as doing mathematics - at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box - a valuable ability in today's world.
There is no doubt that even the greatest musical geniuses have sometimes worked without inspiration. This guest (inspiration) does not always respond to the first invitation. We must always work, and a self-respecting artist must not fold his hands on the pretext that he is not in the mood. If we wait for the mood, without endeavouring to meet it half-way, we easily become indolent and apathetic. We must be patient, and believe that inspiration will come to those who can master their disinclination.
Two thoughts occur to just about any parent whose child is about to enter college. The first is, 'I can't believe how quickly the years have gone by.' The second: 'I can't believe how much it costs.' As one of those parents, I did my best to get past the disturbing first thought and tried to calm my churning stomach while dealing with the second.
The three main sources of scepticism are first, that not every people desires freedom; second, that democracy in certain parts of the world would be dangerous; and third, that there is little the world's democracies can do to advance freedom outside their countries.
The three main sources of scepticism are first, that not every people desires freedom second, that democracy in certain parts of the world would be dangerous and third, that there is little the world's democracies can do to advance freedom outside their countries.
Where taxes are concerned, there are two clear-cut points of view. There are those who think they're too high and those who think they should be even higher because, after all, politicians spend our money far more wisely than we do. The obvious solution I'd propose is that the people in the first group would pay less and those in the second group would pay more. Lots more.
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