The chief characteristic of inference in the exact sciences, the characteristic of which Greek logicians tried to give a theoretical account when they formulated the rules of the syllogism, is a kind of logical compulsion whereby a person who makes certain assumptions is forced, simply by so doing, to make others. He has freedom of choice in two ways: he is not compelled to make the initial assumption (a fact technically expressed by saying that 'the starting-points of demonstrative reasoning are not themselves demonstrable'), and when once he has done so he is still at liberty, whenever he likes, to stop thinking. What he cannot do is to make the initial assumption, to go on thinking, and to arrive at a conclusion different from that which is scientifically correct.
In what is called 'inductive' thinking there is no such compulsion. The essence of the process, here, is that having put certain observations together, and having found that they make a pattern, we extrapolate this pattern indefinitely, just as a man who has plotted a few points on squared paper and says to himself 'the points I have plotted suggest a parabola', proceeds to draw as much of the parabola as he likes in either direction. This is technically described as 'proceeding from the known to the unknown', or 'from the particular to the universal'. It is essential to 'inductive' thinking, though the logicians who have tried to construct a theory of such thinking have not always realized this, that the step so described is never taken under any kind of logical compulsion. The thinker who takes it is logically free to take it or not to take it, just as he pleases. There is nothing in the pattern formed by the observations he or some-one else has actually made which can oblige him to extrapolate in that particular way, or indeed to extrapolate at all. The reason why this very obvious truth has been so often overlooked is that people have been hypnotized by the prestige of Aristotelian logic into thinking that they see a closer resemblance than actually exists between 'deductive' and 'inductive' thinking, that is, between exact science and the sciences of observation and experiment. In both cases there are, for any given piece of thinking, certain starting-points, traditionally called premisses, and a certain terminal point, traditionally called a conclusion ; and in both cases the premisses 'prove' the conclusion. But whereas in exact science this means that they enforce the conclusion, or make it logically obligatory, in the sciences of observation and experiment it means only that they justify it, that is, authorize anybody to think it who wishes to do so. What they provide, when they are said to 'prove' a certain conclusion, is not compulsion to embrace it, but only permission, a perfectly legitimate sense of the word 'prove' (approuver, probare), as there should be no need to show.
If in practice this permission, like so many permissions, amounts to virtual compulsion, that is only because the thinker who avails himself of it does not regard himself as free to extrapolate or not, just as he pleases. He regards himself as under an obligation to do so, and to do it in certain ways: obligations which, when we inquire into their history, we find to have their roots in certain religious beliefs about nature and its creator God. It would be out of place to develop this statement more fully here ; but not, perhaps, to add that if to-day it seems to some readers paradoxical, that is only because the facts have been obscured by a smoke-screen of propagandist literature, beginning with the 'illuminist' movement of the eighteenth century and prolonged by the 'conflict between religion and science' in the nineteenth, whose purpose was to attack Christian theology in the supposed interests of a 'scientific view of the world' which in fact is based upon it and could not for a moment survive its destruction. Take away Christian theology, and the scientist has no longer any motive for doing what inductive thought gives him permission to do. If he goes on doing it at all, that is only because he is blindly following the conventions of the professional society to which he belongs. -- The Idea of History
Popper was solving a problem that only exist once one tries to remove science from the Christian theology which grounds it.