But to be content with this would be to forget too far the origin of these symbols; it remains to explain how we have been led to attribute to them a sort of concrete existence, and, besides, does not the difficulty begin even for the fractional numbers themselves? Should we have the notion of these numbers if we had not previously known a matter that we conceive as infinitely divisible, that is to say, a continuum? THE PHYSICAL CONTINUUM.--We ask ourselves then if the notion of the mathematical continuum is not simply drawn from experience. If it were, the raw data of experience, which are our sensations, would be susceptible of measurement. We might be tempted to believe they really are so, since in these latter days the attempt has been made to measure them and a law has even been formulated, known as Fechner's law, according to which sensation is proportional to the logarithm of the stimulus. But if we examine more closely the experiments by which it has been sought to establish this law, we shall be led to a diametrically opposite conclusion. It has been observed, for example, that a weight _A_ of 10 grams and a weight _B_ of 11 grams produce identical sensations, that the weight _B_ is just as indistinguishable from a weight _C_ of 12 grams, but that the weight _A_ is easily distinguished from the weight _C_. Thus the raw results of experience may be expressed by the following relations: _A_ =_B_, _B_ = _C_, _A_ < _C_, which may be regarded as the formula of the physical continuum. But here is an intolerable discord with the principle of contradiction, and the need of stopping this has compelled us to invent the mathematical continuum. We are, therefore, forced to conclude that this notion has been created entirely by the mind, but that experience has given the occasion. We can not believe that two quantities equal to a third are not equal to one another, and so we are led to suppose that _A_ is different from _B_ and _B_ from _C_, but that the imperfection of our senses has not permitted of our distinguishing them. CREATION OF THE MATHEMATICAL CONTINUUM.--_First Stage._ So far it would suffice, in accounting for the facts, to intercalate between _A_ and _B_ a few terms, which would remain discrete. What happens now if we have recourse to some instrument to supplement the feebleness of our senses, if, for example, we make use of a microscope? Terms such as _A_ and _B_, before indistinguishable, appear now distinct; but between _A_ and _B_, now become distinct, will be intercalated a new term, _D_, that we can distinguish neither from _A_ nor from _B_. Despite the employment of the most highly perfected methods, the raw results of our experience will always present the characteristics of the physical continuum with the contradiction which is inherent in it.