It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word.

If I compare arithmetic with a tree that unfolds upward into a multitude of techniques and theorems while its root drives into the depths, then it seems to me that the impetus of the root.

Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician.

A judgment, for me is not the mere grasping of a thought, but the admission of its truth.

Without some affinity in human ideas art would certainly be impossible; but it can never be exactly determined how far the intentions of the poet are realized.

Being true is different from being taken as true, whether by one or by many or everybody, and in no case is it to be reduced to it. There is no contradiction in something's being true which everybody takes to be false. I understand by 'laws of logic' not psychological laws of takings-to-be-true, but laws of truth. ...If being true is thus independent of being acknowledged by somebody or other, then the laws of truth are not psychological laws: they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace. It is because of this that they have authority for our thought if it would attain truth. They do not bear the relation to thought that the laws of grammar bear to language; they do not make explicit the nature of our human thinking and change as it changes.

The ideal of strictly scientific method in mathematics which I have tried to realise here, and which perhaps might be named after Euclid I should like to describe in the following way... The novelty of this book does not lie in the content of the theorems but in the development of the proofs and the foundations on which they are based... With this book I accomplish an object which I had in view in my Begriffsschrift of 1879 and which I announced in my Grundlagen der Arithmetik. I am here trying to prove the opinion on the concept of number that I expressed in the book last mentioned.

A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.

Is it always permissible to speak of the extension of a concept, of a class? And if not, how do we recognize the exceptional cases? Can we always infer from the extension of one concept's coinciding with that of a second, that every object which falls under the first concept also falls under the second?

The historical approach, with its aim of detecting how things began and arriving from these origins at a knowledge of their nature, is certainly perfectly legitimate; but it also has its limitations. If everything were in continual flux, and nothing maintained itself fixed for all time, there would no longer be any possibility of getting to know about the world, and everything would be plunged into confusion.

We suppose, it would seem, that concepts grow in the individual mind like leaves on a tree, and we think to discover their nature by studying their growth; we seek to define them psychologically, in terms of the human mind. But this account makes everything subjective, and if we follow it through to the end, does away with truth. What is known as the history of concepts is really a history either of our knowledge of concepts or of the meanings of words.

Often it is only after immense intellectual effort, which may have continued over centuries, that humanity at last succeeds in achieving knowledge of a concept in its pure form, by stripping off the irrelevant accretions which veil it from the eye of the mind.

Gottlob Frege created modern logic including "for all," "there exists," and rules of proof. Leibniz and Boole had dealt only with what we now call "propositional logic" (that is, no "for all" or "there exists"). They also did not concern themselves with rules of proof, since their aim was to reach truth by pure calculation with symbols for the propositions. Frege took the opposite track: instead of trying to reduce logic to calculation, he tried to reduce mathematics to logic, including the concept of number.

O world, thou choosest not the better part! It is not wisdom to be only wise, And on the inward vision close the eyes, But it is wisdom to believe the heart. Columbus found a world, and had no chart, Save one that faith deciphered in the skies; To trust the soul's invincible surmise Was all his science and his only art.

In the Gospels, for instance, we sometimes find the kingdom of heaven illustrated by principles drawn from observation of this world rather than from an ideal conception of justice; ... They remind us that the God we are seeking is present and active, that he is the living God; they are doubtless necessary if we are to keep religion from passing into a mere idealism and God into the vanishing point of our thought and endeavour.

Although a poem be not made by counting of syllables upon the fingers, yet "numbers" is the most poetical synonym we have for verse, and "measure" the most significant equivalent for beauty, for goodness, and perhaps even for truth. Those early and profound philosophers, the followers of Pythagoras, saw the essence of all things in number, and it was by weight, measure, and number, as we read in the Bible, that the Creator first brought Nature out of the void.

There is nothing impossible in the existence of the supernatural: its existence seems to me decidedly probable.

They [the wise spirits of antiquity in the first circle of Dante's Inferno] are condemned, Dante tells us, to no other penalty than to live in desire without hope, a fate appropriate to noble souls with a clear vision of life.

Skepticism, like chastity, should not be relinquished too readily.

I leave you but the sound of many a word In mocking echoes haply overheard, I sang to heaven. My exile made me free,from world to world, from all worlds carried me.

The idea of Christ is much older than Christianity.

A child educated only at school is an uneducated child.

Religions are not true or false, but better or worse.

On fact, the whole machinery of our intelligence, our general ideas and laws, fixed and external objects, principles, persons, and gods, are so many symbolic, algebraic expressions. They stand for experience; experience which we are incapable of retaining and surveying in its multitudinous immediacy. We should flounder hopelessly, like the animals, did we not keep ourselves afloat and direct our course by these intellectual devices.

Beauty as we feel it is something indescribable: what it is or what it means can never be said.

Beauty is a pledge of the possible conformity between the soul and nature, and consequently a ground of faith in the supremacy of the good.

Everything ideal has a natural basis and everything natural an ideal development.

Even the most inspired verse, which boasts not without a relative justification to be immortal, becomes in the course of ages a scarcely legible hieroglyphic; the language it was written in dies, a learned education and an imaginative effort are requisite to catch even a vestige of its original force. Nothing is so irrevocable as mind.

Happiness is the only sanction of life; where happiness fails, existence remains a mad and lamentable experiment.

That life is worth living is the most necessary of assumptions and, were it not assumed, the most impossible of conclusions.

Fanaticism consists in redoubling your efforts when you have forgotten your aim.

Progress, far from consisting in change, depends on retentiveness. When change is absolute there remains no being to improve and no direction is set for possible improvement: and when experience is not retained, as among savages, infancy is perpetual. Those who cannot remember the past are condemned to repeat it. This famous statement has produced many paraphrases and variants: Those who cannot learn from history are doomed to repeat it. Those who do not remember their past are condemned to repeat their mistakes. Those who do not read history are doomed to repeat it. Those who fail to learn from the mistakes of their predecessors are destined to repeat them. Those who do not know history's mistakes are doomed to repeat them.

The highest form of vanity is love of fame.

The human race, in its intellectual life, is organized like the bees: the masculine soul is a worker, sexually atrophied, and essentially dedicated to impersonal and universal arts; the feminine is a queen, infinitely fertile, omnipresent in its brooding industry, but passive and abounding in intuitions without method and passions without justice.

To call war the soil of courage and virtue is like calling debauchery the soil of love.

It is not society's fault that most men seem to miss their vocation. Most men have no vocation.

Injustice in this world is not something comparative; the wrong is deep, clear, and absolute in each private fate.

Culture is on the horns of this dilemma: if profound and noble, it must remain rare, if common, it must become mean.

What renders man an imaginative and moral being is that in society he gives new aims to his life which could not have existed in solitude: the aims of friendship, religion, science, and art.

When men and women agree, it is only in their conclusions; their reasons are always different.

In proportion as a man's interests become humane and his efforts rational, he appropriates and expands a common life, which reappears in all individuals who reach the same impersonal level of ideas.

Most men's conscience, habits, and opinions are borrowed from convention and gather continual comforting assurances from the same social consensus that originally suggested them.

Experience has repeatedly confirmed that well-known maxim of Bacon's that "a little philosophy inclineth a man's mind to atheism, but depth in philosophy bringeth men's minds about to religion." At the same time, when Bacon penned that sage epigram... he forgot to add that the God to whom depth in philosophy brings back men's minds is far from being the same from whom a little philosophy estranges them.

Matters of religion should never be matters of controversy. We neither argue with a lover about his taste, nor condemn him, if we are just, for knowing so human a passion.

Fashion is something barbarous, for it produces innovation without reason and imitation without benefit.

Every moment celebrates obsequies over the virtues of its predecessor.

To know how just a cause we have for grieving is already a consolation.

The mind celebrates a little triumph whenever it can formulate a truth, however unwelcome to the flesh, or discover an actual force, however unfavourable to given interests.

Art like life should be free, since both are experimental.

History is nothing but assisted and recorded memory. It might almost be said to be no science at all, if memory and faith in memory were not what science necessarily rest on. In order to sift evidence we must rely on some witness, and we must trust experience before we proceed to expand it. The line between what is known scientifically and what has to be assumed in order to support knowledge is impossible to draw. Memory itself is an internal rumour; and when to this hearsay within the mind we add the falsified echoes that reach us from others, we have but a shifting and unseizable basis to build upon. The picture we frame of the past changes continually and grows every day less similar to the original experience which it purports to describe.