Knowing God without knowing our own wretchedness makes for pride. Knowing our own wretchedness without knowing God makes for despair. Knowing Jesus Christ strikes the balance because he shows us both God and our own wretchedness.

Nothing is so important to man as his own state; nothing is so formidable to him as eternity. And thus it is unnatural that thereshould be men indifferent to the loss of their existence and to the perils of everlasting suffering.

The incredulous are the more credulous. They believe the miracles of Vespasian that they may not believe those of Moses. [Fr., Incredules les plus credules. Ils croient les miracle de Vespasien, pour ne pas croire ceux de Moise.]

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's Equation reaches down into the very depths of existence.

The art of progress is to preserve order amid change, and to preserve change amid order. Life refuses to be embalmed alive. The more prolonged the halt in some unrelieved system of order, the greater the crash of the dead society.

In a living civilization there is always an element of unrest, for sensitiveness to ideas means curiosity, adventure, change. Civilized order survives on its merits and is transformed by its power of recognizing its imperfections.

I am also greatly indebted to Bergson, William James, and John Dewey. One of my preoccupations has been to rescue their type of thought from the charge of anti-intellectualism, which rightly or wrongly has been associated with it.

The only thing which consoles for our miseries is diversion, and yet this is the greatest of our miseries. For it is this which principally hinders us from reflecting upon ourselves and which makes us imperceptibly ruin ourselves.

A mathematician's work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.

I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author.

In recent years we have become much more preoccupied with streamlining and organizing our subject than with maintaining its overall vitality. If we are not careful, a great adventure of the mind will become yet another profession.

Instead of trying to produce a programme to simulate the adult mind, why not rather try to produce one which simulates the child's? If this were then subjected to an appropriate course of education one would obtain the adult brain.

Reflect on death as in Jesus Christ, not as without Jesus Christ. Without Jesus Christ it is dreadful, it is alarming, it is the terror of nature. In Jesus Christ it is fair and lovely, it is good and holy, it is the joy of saints.

Man finds nothing so intolerable as to be in a state of complete rest, without passions, without occupation, without diversion, without effort. Then he feels his nullity, loneliness, inadequacy, dependence, helplessness, emptiness.

The exterior must be joined to the interior to obtain anything from God, that is to say, we must kneel, pray with the lips, and soon, in order that proud man, who would not submit himself to God, may be now subject to the creature.

A town, a landscape are when seen from afar a town and a landscape; but as one gets nearer, there are houses, trees, tiles leaves, grasses, ants, legs of ants and so on to infinity. All this is subsumed under the name of landscape.

The real work of us mathematicians, from now until, roughly, fifty years from now, when computers won't need us anymore, is to make the transition from human-centric math to machine-centric math as smooth and efficient as possible.

To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples.

Finally we shall place the Sun himself at the center of the Universe. All this is suggested by the system of procession of events and the harmony of the whole Universe, if only we face the facts, as they say, "with eyes wide open."

...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly generality is, in essence, the same as a small and concrete special case.

The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which of times they are unable to account.

The frequencies of the notes in a scale—do, re, mi, fa, sol, la, ti, do—sound to us like they’re rising in equal steps. But objectively their vibrational frequencies are rising by equal multiples. We perceive pitch logarithmically.

To find recreation in amusements is not happiness; for this joy springs from alien and extrinsic sources, and is therefore dependent upon and subject to interruption by a thousand accidents, which may minister inevitable affliction.

It is not from space that I must seek my dignity, but from the government of my thought. I shall have no more if I possess worlds. By space the universe encompasses and swallows me up like an atom; by thought I comprehend the world.

Caesar was too old, it seems to me, to go off and amuse himself conquering the world. Such a pastime was all right for Augustus and Alexander; they were young men, not easily held in check, but Caesar ought to have been more mature.

Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate.

When a problem seems intractable, it is often a good idea to try to study "toy" versions of it in the hope that as the toys become increasingly larger and more sophisticated, they would metamorphose, in the limit, to the real thing.

The teacher can seldom afford to miss the questions: What is the unknown? What are the data? What is the condition? The student should consider the principal parts of the problem attentively, repeatedly, and from from various sides.

If the prior distribution, at which I am frankly guessing, has little or no effect on the result, then why bother; and if it has a large effect, then since I do not know what I am doing how would I dare act on the conclusions drawn?

As I say, there was this movement to try to bring philosophers and mathematicians together into an organization where they would talk to each other. An organization wasn't effective unless you had a journal. That's about all I know.

If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).

If God exists, not seeking God must be the gravest error imaginable. If one decides to sincerely seek for God and doesn't find God, the lost effort is negligible in comparison to what is at risk in not seeking God in the first place.

We do not weary of eating and sleeping every day, for hunger and sleepiness recur. Without that we should weary of them. So, without the hunger for spiritual things, we weary of them. Hunger after righteousness--the eighth beatitude.

Wherever Mathematics is mixed up with anything, which is outside its field, you will find attempts to demonstrate these merely conventional propositions a priori, and it will be your task to find out the false deduction in each case.

Quite often, when an idea that could be helpful presents itself, we do not appreciate it, for it is so inconspicuous. The expert has, perhaps, no more ideas than the inexperienced, but appreciates more what he has and uses it better.

It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.

Since geometry is co-eternal with the divine mind before the birth of things, God himself served as his own model in creating the world (for what is there in God which is not God?), and he with his own image reached down to humanity.

As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection.

To exist (in mathematics), said Henri Poincaré, is to be free from contradiction. But mere existence does not guarantee survival. To survive in mathematics requires a kind of vitality that cannot be described in purely logical terms.

The manner in which Epictetus, Montaigne, and Salomon de Tultie wrote, is the most usual, the most suggestive, the most remembered, and the oftener quoted; because it is entirely composed of thoughts born from the common talk of life.

There was once in man a true happiness of which there now remain to him only the mark and empty trace, which he in vain tries to fill from all his surroundings, seeking from things absent the help he does not obtain in things present.

On two occasions I have been asked, 'Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?' I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

The larger the mass of collected things, the less will be their usefulness. Therefore, one should not only strive to assemble new goods from everywhere, but one must endeavor to put in the right order those that one already possesses.

Indeed every monad must be different from every other. For there are never in nature two beings, which are precisely alike, and in which it is not possible to find some difference which is internal, or based on some intrinsic quality.

Mathematicians do not deal in objects, but in relations between objects; thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.

That in affairs of very considerable importance men should deal with one another with satisfaction of mind, and mutual confidence, they must receive competent assurances concerning the integrity, fidelity, and constancy each of other.

A popular cliche in philosophy says that science is pure analysis or reductionism, like taking the rainbow to pieces; and art is pure synthesis, putting the rainbow together. This is not so. All imagination begins by analyzing nature.

The men who made the Industrial Revolution are usually pictured as hardfaced businessmen with no other motive than self-interest. That is certainly wrong. For one thing, many of them were inventors who had come into business that way.

Aspiring to these wide generalizations, the analysis of quadratic functions soars to a pitch from whence it may look proudly down on the feeble and vain attempts of geometry proper to rise to its level or to emulate it in its flights.

A mind is accustomed to mathematical deduction, when confronted with the faulty foundations of astrology, resists a long, long time, like an obstinate mule, until compelled by beating and curses to put its foot into that dirty puddle.

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