In the first place, the discoverer is not in possession of the knowledge he covets. It is for him a desire, an aspiration, an aim to be attained. Proof, on the other hand, presupposes knowledge. Not only must the demonstrator _know_ the assured truths he uses as premisses, not only must he have a supply of absolutely certain truths if his proof is not to remain hypothetical (§ 9), but he must already _know_ the conclusion he exhibits. He cannot be ignorant, like the discoverer, of the result he is to arrive at. He is not engaged in _discovering_ new truth, he is only showing how it _follows from_ old truths. His retrospective contemplation has merely to retrace the history of its attainment, or rather to rearrange it in the more pleasing order which he calls ‘logical’. This order is not that in which it _was_ discovered, nor even that in which it _could be_ discovered. For there are such things as necessary errors, indispensable artifices, and indefensible fictions, and the way to a truth often lies through them. Thus from time immemorial mathematicians have represented the continuous by the discrete, quantities by numbers, knowing full well what fictions their practice involved. Again, mathematical calculation of shapes, areas, and motions necessarily presupposes the fictions that bodies have the ideal and regular forms to which they ‘approximate’, and that their ‘mass’ is concentrated at their (ideal) ‘centre of gravity’. It is more than doubtful whether the notion of an ‘evolution’ of species could ever have been reached, except by starting from the false notion of the fixity of species, or whether the true nature of the mobility and development of meanings could have been understood except by correcting the Platonic theory of immutable and eternal ‘universals’. To ‘proof’ all these incidents and accidents of the history of discovery are irrelevant; all that has to be done is to show that the new truth can be deduced from the old, and that a ‘logical connexion’ exists between them.