All our surest statements about the nature of the world are mathematical statements, yet we do not know what mathematics “is”… and so we find that we have adapted a religion strikingly similar to many traditional faiths. Change “mathematics” to “God” and little else might seem to change. The problem of human contact with some spiritual realm, of timelessness, of our inability to capture all with language and symbol—all have their counterparts in the quest for the nature of Platonic mathematics.
Certainly he who can digest a second or third fluxion need not, methinks, be squeamish about any point in divinity.
And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?
I have often admired the mystical way of Pythagoras, and the secret magic of numbers.
The structures with which mathematics deals are more like lace, the leaves of trees and the play of the light and shadow on a human face than they are like buildings and machines, the least of their representatives.
From the time of Kepler to that of Newton, and from Newton to Hartley, not only all things in external nature, but the subtlest mysteries of life and organization, and even of the intellect and moral being, were conjured within the magic circle of mathematical formulae.
In mathematics we find the primitive source of rationality; and to mathematics must the biologists resort for means to carry out their researches.
Thus metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role.
The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight!
Every new body of discovery is mathematical in form, because there is no other guidance we can have.
Numbers are the free creation of the human mind.
Intuition is the conception of an attentive mind, so clear, so distinct, and so effortless that we cannot doubt what we have so conceived.
This therefore is Mathematics:
She reminds you of the invisible forms of the soul;
She gives life to her own discoveries;
She awakens the mind and purifies the intellect;
She brings light to our intrinsic ideas;
She abolishes oblivion and ignorance which are ours by birth.
The creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.
The laws of mathematics are not merely human inventions or creations. They simply ‘are;’ they exist quite independently of the human intellect. The most that any(one)…can do is to find that they are there and to take cognizance of them.
Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.
Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician.
I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated…geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.
We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds.
Either mathematics is too big for the human mind or the human mind is more than a machine.
I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.
Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.
Mathematics is the science of what is clear by itself.
[The] sole end of science is the honor of the human mind, and…under this title a question about numbers is worth as much as a question about the system of the world.
Every existence above a certain rank has its singular points; the higher the rank the more of them. At these points, influences whose physical magnitude is too small to be taken account of by a finite being may produce results of the greatest importance.
To create a good philosophy you should renounce metaphysics but be a good mathematician.