A Quote by Julian Coolidge
[Before the time of Benjamin Peirce it never occurred to anyone that mathematical research] was one of the things for which a mathematical department existed. Today it is a commonplace in all the leading universities. Peirce stood alone-a mountain peak whose absolute height might be hard to measure, but which towered above all the surrounding country.
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The mathematical is that evident aspect of things within which we are always already moving and according to which we experience them as things at all, and as such things. The mathematical is this fundamental position we take toward things by which we take up things as already given to us, and as they must and should be given. Therefore, the mathematical is the fundamental presupposition of the knowledge of things.
One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?
Mathematics is a logical method. . . . Mathematical propositions express no thoughts. In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics.
If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming.
The "seriousness" 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.
The name was supposed to be 'Googol,' which is the mathematical term for a 1 followed by 100 zeroes. It was before the Google spellchecker existed.
The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc'd to a Mathematical Reasoning, and when they cannot, it's a sign our Knowledge of them is very small and confus'd; and where a mathematical reasoning can be had, it's as great folly to make use of any other, as to grope for a thing in the dark when you have a Candle standing by you.
If I were asked to name, in one word, the pole star round which the mathematical firmament revolves, the central idea which pervades the whole corpus of mathematical doctrine, I should point to Continuity as contained in our notions of space, and say, it is this, it is this!
Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of
our science and at the secrets of its development during future centuries? What particular goals will there be toward
which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the
wide and rich field of mathematical thought will the new centuries disclose?
Whereas I think: I’m lying here in a haystack... The tiny space I occupy is so infinitesimal in comparison with the rest of space, which I don’t occupy and which has no relation to me. And the period of time in which I’m fated to live is so insignificant beside the eternity in which I haven’t existed and won’t exist... And yet in this atom, this mathematical point, blood is circulating, a brain is working, desiring something... What chaos! What a farce!
The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics.
What expressions we used - in part taken over and in part newly invented! above all, the famous 'wholly other' breaking in upon us 'perpendicularly from above,' the not less famous 'infinite qualitative distinction' between God and man, the vacuum, the mathematical point, and the tangent in which alone they must meet.
The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth.
Absolute, true, and mathematical time, in and of itself and of its own nature, without reference to anything external, flows uniformly and by another name is called duration. Relative, apparent, and common time is any sensible and external measure (precise or imprecise) of duration by means of motion; such as a measure-for example, an hour, a day, a month, a year-is commonly used instead of true time.
I meditate for the last time on this mountain that is bare, though others all around are white with snow. Like the bare peak of the koan, this one is not different from myself. I know this mountain because I am this mountain, I can feel it breathing at this moment, as its grass tops stray against the snows. If the snow leopard should leap from the rock above and manifest itself before me - S-A-A-O! - then in that moment of pure fright, out of my wits, I might truly perceive it, and be free.
Mathematical Mark all mathematical heads, which be only and wholly bent to those sciences, how solitary they be themselves, how unfit to live with others, and how unapt to serve in the world.
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