The science of pure mathematics ... may claim to be the most original creation of the human spirit.
The study of mathematics is apt to commence in disappointment....We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this greatest science eludes the efforts of our mental weapons to grasp it.
Mathematics as a science, commenced when first someone, probably a Greek, proved propositions about "any" things or about "some" things, without specifications of definite particular things.
So far as the mere imparting of information is concerned, no university has had any justification for existence since the popularization of printing in the fifteenth century.
No Roman ever died in contemplation over a geometrical diagram.
[A reference to the death of Archimedes.]
Life is an offensive, directed against the repetitious mechanism of the Universe.
There is no nature at an instant.
Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings.
There is a tradition of opposition between adherents of induction and of deduction. In my view it would be just as sensible for the two ends of a worm to quarrel.
It is a profoundly erroneous truism, repeated by all copy books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.
Our minds are finite, and yet even in these circumstances of finitude we are surrounded by possibilities that are infinite, and the purpose of life is to grasp as much as we can out of that infinitude.
In modern times the belief that the ultimate explanation of all things was to be found in Newtonian mechanics was an adumbration of the truth that all science, as it grows towards perfection, becomes mathematical in its ideas.
Algebra reverses the relative importance of the factors in ordinary language. It is essentially a written language, and it endeavors to exemplify in its written structures the patterns which it is its purpose to convey. The pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghostlike character of the real variable.
By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race.
Everything of importance has been said before by somebody who did not discover it.
Seek simplicity, and distrust it.
Fundamental progress has to do with the reinterpretation of basic ideas.
We think in generalities, but we live in details.
Apart from blunt truth, our lives sink decadently amid the perfume of hints and suggestions.
"Necessity is the mother of invention" is a silly proverb. "Necessity is the mother of futile dodges" is much nearer the truth.
It is more important that a proposition be interesting than that it be true. This statement is almost a tautology. For the energy of operation of a proposition in an occasion of experience is its interest and is its importance. But of course a true proposition is more apt to be interesting than a false one.
War can protect; it cannot create.
The progress of Science consists in observing interconnections and in showing with a patient ingenuity that the events of this ever-shifting world are but examples of a few general relations, called laws. To see what is general in what is particular, and what is permanent in what is transitory, is the aim of scientific thought.
Through and through the world is infested with quantity: To talk sense is to talk quantities. It is not use saying the nation is large .. How large? It is no use saying the radium is scarce ... How scarce? You cannot evade quantity. You may fly to poetry and music, and quantity and number will face you in your rhythms and your octaves.
"One and one make two" assumes that the changes in the shift of circumstance are unimportant. But it is impossible for us to analyze this notion of unimportant change.
I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much. But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming ... and a little mad.
In the study of ideas, it is necessary to remember that insistence on hard-headed clarity issues from sentimental feeling, as it were a mist, cloaking the perplexities of fact. Insistence on clarity at all costs is based on sheer superstition as to the mode in which human intelligence functions. Our reasonings grasp at straws for premises and float on gossamers for deductions.
Familiar things happen, and mankind does not bother about them. It requires a very unusual mind to undertake the analysis of the obvious.
It is a safe rule to apply that, when a mathematical or philosophical author writes with a misty profundity, he is talking nonsense.
The race that does not value trained intelligence is doomed.
Let us grant that the pusuit of mathematics is a divine madness of the human spirit.
A Unitarian is a person who believes in at most one God.
... to be an abstraction does not mean that an entity is nothing. It merely means that its existence is only a factor of a more concrete element of nature.
... algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world.
The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. ... Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.
But in the prevalent discussion of classes, there are illegitimate transitions to the notions of a 'nexus' and of a 'proposition'. The appeal to a class to perform the services of a proper entity is exactly analogous to an appeal to an imaginary terrier to kill a real rat.
Now in creative thought common sense is a bad master. Its sole criterion for judgement is that the new ideas shall look like the old ones. In other words it can only work by suppressing originality.
I regret that it has been necessary for me in this lecture to administer a large dose of four-dimensional geometry. I do not apologise, because I am realyy not responsible for the fact that nature in its most fundamental aspect is four-dimensional. Things are what they are ...
Mathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning.
The utmost abstractions are the true weapons with which to control our thought of concrete fact.
Order is not sufficient. What is required, is something much more complex. It is order entering upon novelty; so that the massiveness of order does not degenerate into mere repetition; and so that the novelty is always reflected upon a background of system.
Education is the acquisition of the art of the utilisation of knowledge.
There are no whole truths; all truths are half truths. It is trying to treat them as whole truths that plays the devil.