Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted.

Roemer used eclipses of the satellites of Jupiter, and sought how much the event fell behind its prediction. But... this prediction is made... by... astronomic laws; for instance Newton's... The velocity of light... is adopted, such that the astronomic laws compatible with this value may be as simple as possible.

He has begun by supposing that light has a constant velocity... the same in all directions. This... could never be verified directly by experiment... The postulate... resembling the principle of sufficient reason... furnishes us with a new rule for the investigation of simultaneity.

When an astronomer tells me that some stellar phenomenon, which his telescope reveals to him at this moment, happened... fifty years ago... I... ask... how he has measured the velocity of light.

If all the parts of the universe are interchained in a certain measure, any one phenomenon will not be the effect of a single cause, but the resultant of causes infinitely numerous; it is, one often says, the consequence of the state of the universe the moment before.

Have we really the right to speak of the cause of a phenomenon?

Behold... the only... rule we can follow: when a phenomenon appears... as the cause of another, we regard it as anterior. ...Therefore by cause... we define time; but...how do we recognize which is the cause and which the effect? We assume... the anterior fact, the antecedent, is the cause of the... consequent. It is then by time that we define cause. ...Shall we escape from this vicious circle?

Is not my present nearer my past of yesterday than the present of Sirius?

This shows, perhaps, why we have tried to put all physical phenomena into the same frame. But that can not pass for a definition of simultaneity, since this hypothetical intelligence, even if it existed, would be for us impenetrable. It is... necessary to seek something else.

Yet when we speak of time... do we not unconsciously adopt this hypothesis... and put ourselves in the place of this imperfect god... Do not even the atheists put themselves in the place where god would be..?

We should like to represent... the... universe, and... feel... we understood it. We... never can attain this representation: our weakness is too great. But... we desire... to conceive an infinite intelligence... which should see all, and... classify all in its time, as we classify, in our time, the little we see. ...This supreme intelligence would be only a demigod; infinite in one sense... limited in another, since it would have... imperfect recollection of the past... otherwise all recollections would be equally present... and for it there would be no time.

If... it be supposed that another way of measuring time is adopted... enunciation of the law would be... translated into another language... much less simple. So that the definition implicitly adopted by the astronomers may be summed up..: Time should be so defined that the equations of mechanics may be as simple as possible... [i.e.,] there is not one way of measuring time more true... only more convenient.

Astronomers... define duration in the following way: time... so defined that Newton's law and that of vis viva [or of the conservation of energy] may be verified. Newton's law is an experimental truth... only approximate... We still have only a definition by approximation.

We should say: 'Causes almost identical take almost the same time to produce almost the same effects.'

When navigators... determine a longitude... they must... calculate Paris time…with a chronometer set for Paris. The qualitative problem of simultaneity is made to depend upon the quantitative problem of the measurement of time.

Or... they observe an astronomic phenomenon... an eclipse of the moon, and... suppose... this... is perceived simultaneously from all points of the earth. That is not altogether true, since the propagation of light is not instantaneous; if absolute exactitude were desired, there would be a correction... according to a complicated rule.

We have not a direct intuition of simultaneity, nor of the equality of two durations. If we think we have this intuition, this is an illusion. We replace it by the aid of certain rules which we apply almost always without taking count of them....We ...choose these rules, not because they are true, but because they are the most convenient, and we may recapitulate them as follows: "The simultaneity of two events, or the order of their succession, the equality of two durations, are to be so defined that the enunciation of the natural laws may be as simple as possible. In other words, all these rules, all these definitions, are only the fruit of an unconscious opportunism."

Logic teaches us that on such and such a road we are sure of not meeting an obstacle; it does not tell us which is the road that leads to the desired end. For this, it is necessary to see the end from afar, and the faculty which teaches us to see is intuition. Without it, the geometrician would be like a writer well up in grammar but destitute of ideas.

It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.

The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious. It is through it that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality.

I think I have already said somewhere that mathematics is the art of giving the same name to different things.

It is because simplicity and vastness are both beautiful that we seek by preference simple facts and vast facts; that we take delight, now in following the giant courses of the stars, now in scrutinizing the microscope that prodigious smallness which is also a vastness, and now in seeking in geological ages the traces of a past that attracts us because of its remoteness.

Sociology is the science which has the most methods and the least results.

All that is not thought is pure nothingness; since we can think only thought and all the words we use to speak of things can express only thoughts, to say there is something other than thought, is therefore an affirmation which can have no meaning.And yet-strange contradiction for those who believe in time-geologic history shows us that life is only a short episode between two eternities of death, and that, even in this episode, conscious thought has lasted and will last only a moment. Thought is only a gleam in the midst of a long night. But it is this gleam which is everything.

It is only through science and art that civilization is of value. Some have wondered at the formula: science for its own sake; an yet it is as good as life for its own sake, if life is only misery; and even as happiness for its own sake, if we do not believe that all pleasures are of the same quality...Every act should have an aim. We must suffer, we must work, we must pay for our place at the game, but this is for seeing's sake; or at the very least that others may one day see.

Now what is science? ...it is before all a classification, a manner of bringing together facts which appearances separate, though they are bound together by some natural and hidden kinship. Science, in other words, is a system of relations. ...it is in relations alone that objectivity must be sought. ...it is relations alone which can be regarded as objective.External objects... are really objects and not fleeting and fugitive appearances, because they are not only groups of sensations, but groups cemented by a constant bond. It is this bond, and this bond alone, which is the object in itself, and this bond is a relation.

What is objective must be common to many minds and consequently transmissible from one to the other, and as this transmission can only come about by... discourse... we are even forced to conclude: no discourse no objectivity.

As we can not give a general definition of energy, the principle of the conservation of energy signifies simply that there is something which remains constant.

All laws are... deduced from experiment; but to enunciate them, a special language is needful... ordinary language is too poor...This... is one reason why the physicist can not do without mathematics; it furnishes him the only language he can speak. And a well-made language is no indifferent thing;...the analyst, who pursues a purely esthetic aim, helps create, just by that, a language more fit to satisfy the physicist....law springs from experiment, but not immediately. Experiment is individual, the law deduced from it is general; experiment is only approximate, the law is precise...In a word, to get the law from experiment, it is necessary to generalize... But how generalize? ...in this choice what shall guide us?It can only be analogy. ...What has taught us to know the true profound analogies, those the eyes do not see but reason divines?It is the mathematical spirit, which disdains matter to cling only to pure form.

Mathematics have a triple aim. They must furnish an instrument for the study of nature. But that is not all: they have a philosophic aim and, I dare maintain, an esthetic aim. They must aid the philosopher to fathom the notions of number, of space, of time. And above all, their adepts find therein delights analogous to those given by painting and music. They admire the delicate harmony of numbers and forms; they marvel when a new discovery opens to them an unexpected perspective; and has not the joy they thus feel the esthetic character, even though the senses take no part therein? Only a privileged few are called to enjoy it fully, it is true, but is not this the case for all the noblest arts?This is why I do not hesitate to say that mathematics deserve to be cultivated for their own sake, and the theories inapplicable to physics as well as the others. Even if the physical aim and the esthetic aim were not united, we ought not to sacrifice either.

In physical reality one cause does not produce a given effect, but a multitude of distinct causes contribute to produce it, without our having any means of discriminating the part of each of them. ...Causes which have produced a certain effect will never be reproduced except approximately.

What postulate do we implicitly admit? It is that the duration of two identical phenomena is the same; or... that the same causes take the same time to produce the same effects. ...Is it impossible that experiment may some day contradict our postulate?

It is the sidereal day, that is, the duration of the rotation of the earth, which is the constant unit of time. ...However ...many astronomers ...think that the tides act as a check on our globe, and that the rotation of the earth is becoming slower and slower. Thus would be explained the apparent acceleration of the motion of the moon...

Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone.

But, one will say, if raw experience can not legitimatize reasoning by recurrence, is it so of experiment aided by induction? We see successively that a theorem is true of the number 1, of the number 2, of the number 3 and so on; the law is evident, we say, and it has the same warranty as every physical law based on observations, whose number is very great but limited. But there is an essential difference. Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us. Mathematical induction, that is, demonstration by recurrence, on the contrary, imposes itself necessarily, because it is only the affirmation of a property of the mind itself.

We can not... escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction. ...Neither can this rule come to us from experience... This rule, inaccessible to analytic demonstration and to experience, is the veritable type of the synthetic a priori judgment. On the other hand, we can not think of seeing in it a convention, as in some of the postulates of geometry. ...it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it.

There is no science apart from the general. It may even be said that the very object of the exact sciences is to spare us these direct verifications.

To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection.

Thought is only a flash between two long nights, but this flash is everything.

Point set topology is a disease from which the human race will soon recover.

The ones who are preoccupied by logic are above all; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The others are guided by intuition and, at the first stroke, make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.

Thought must never submit, neither to a dogma, nor to a party, nor to a passion, nor to an interest, nor to a preconceived idea, nor to whatever it may be, save to the facts themselves, because, for thought, submission would mean ceasing to be.

Everyone is sure of this [that errors are normally distributed], Mr. Lippman told me one day, since the experimentalists believe that it is a mathematical theorem, and the mathematicians that it is an experimentally determined fact.

We do not have and cannot have any means of discovering whether or not we are carried along in a uniform motion of translation.

The task of the educator is to make the child's spirit pass again where its forefathers have gone, moving rapidly through certain stages but suppressing none of them. In this regard, the history of science must be our guide.

It is the simple hypotheses of which one must be most wary; because these are the ones that have the most chances of passing unnoticed.

We see that experience plays an indispensable role in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science. If it were experimental it would be only approximative and provisional. And what rough approximation!...The object of geometry is the study of a particular 'group'; but the general group concept pre-exists... in our minds. It is imposed on us, not as form of our sense, but as form of our understanding. Only, from among all the possible groups, that must be chosen... will be... the standard to which we shall refer natural phenomena.Experience guides us in this choice without forcing it upon us; it tells us not which is the truest geometry, but which is the most convenient.Notice that I have been able to describe the fantastic worlds... imagined without ceasing to employ the language of ordinary geometry.

Is the position tenable, that certain phenomena, possible in Euclidean space, would be impossible in non-Euclidean space, so that experience, in establishing these phenomena, would directly contradict the non-Euclidean hypothesis? For my part I think no such question can be put. To my mind it is precisely equivalent to the following, whose absurdity is patent to all eyes: are there lengths expressible in meters and centimeters, but which can not be measured in fathoms, feet, and inches, so that experience, in ascertaining the existence of these lengths, would directly contradict the hypothesis that there are fathoms divided into six feet?

The Scientist must set in order. Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.

We have not a direct intuition of the equality of two intervals of time.