failed to perceive that what is really confirmed or corroborated by observation is not the validity of our logical deductions (which we suppose to be correct in the ordinary sense), but the validity of the assumptions from which our calculations started. If some day an astronomer should not find a planet at the place which he had calculated for its position, he would not think that the mistake lay in his using ordinary logic in his deductions, but he would know, that something was wrong with the hypotheses from which he deduced the position of the planet. (These hypotheses would concern the laws of motion, the initial position of the planets, the absence of disturbing influences etc.) A sceptic might object that in principle the failure of the astronomer could be explained in two ways: (1) by inadequate hypotheses, (2) by inadequate logic. But the second explanation is impossible. It is based on the fundamental error that the calculation, as it were, adds something to the hypothesis, and that the result of the calculation is the product of two factors: initial assumptions and logical deduction. But this is not so. On the contrary, it is clear that the initial hypotheses alone determine the position of the planet, the deduction or computation cannot be regarded as introducing as a new hypothesis the validity of ordinary logic, which may or may not be fulfilled. No, the assumption that the motion of the planet follows certain laws, etc. is the assumption that the planet will have a certain position at a given time, and (of course) certain other positions at other times; the law of motion is nothing but a short way of saying that the planet will occupy a certain series of positions at definite times — it must not be misunderstood as a kind of imperative order given by nature that the planet must and shall move in a prescribed orbit. A natural "law" is a formula which describes, it does not prescribe. The mathematical calculation by which the present position of the planet is "deduced" from the general law does not do anything but show that the proposition about the particular place of the planet is already contained in the law; that is to say: that proposition is not a result of the Law plus Logic, but the Law is an abbreviated way of asserting an indefinite number of propositions. One of these is picked out — that is all. Thus, if such a proposition is found to be false by observation, this proves that the law is false, it has nothing to do with logic.
It must be clear by this time that the validity of logic (and mathematics)
