[A]ll the important things we can say about an act of knowledge consist of pointing out the third-world objects of the act—a theory or proposition—and its relation to other third-world objects, such as the arguments bearing on the problem as well as the objects known. 
I have given here some reasons for the autonomous existence of an objective [World 3] because I hope to make a contribution to the theory of understanding (‘hermeneutics’), which has been much discussed by students of the humanities (‘Geisteswissenschaften’, ‘moral and mental sciences’). Here I will start from the assumption that it is the understanding of objects belonging to [World 3] which constitutes the central problem of the humanities. This, it appears, is a radical departure from the fundamental dogma accepted by almost all students of the humanities (as the term indicates), and especially by those who are interested in the problem of understanding. I mean of course the dogma that the objects of our understanding belong mainly to [World 2], or that they are at any rate to be explained in psychological terms. 
What I suggest is that we can grasp a theory only by trying to reinvent it or to reconstruct it, and by trying out, with the help of our imagination, all the consequences of the theory which seem to us to be interesting and important.
Understanding is an active process, not just a process of merely staring at a thing and waiting for enlightenment. One could say that the process of understanding and the process of the actual production or discovery of World 3 objects are very much alike. 
Or take as an example Bohr’s theory (1913) of the hydrogen atom. This theory was describing a model, and was therefore intuitive and visualizable. Yet it was also very perplexing. Not because of any intuitive difficulty, but because it assumed, contrary to Maxwell’s and Lorentz’s theory and to well-known experimental effects, that a periodically moving electron, a moving electric charge, need not always create a disturbance of the eletromagnetic field, and so need not always send out electromagnetic waves. This difficulty is a logical one – a clash with other theories. And no one can be said to understand Bohr’s theory who does not understand this difficulty and the reasons why Bohr boldly accepted it, thus departing in a revolutionary way from earlier and well-established theories.
But the only way to understand Bohr’s reasons is to understand his problem – the problem of combining Rutherford’s atom model with a theory of emission and absorption of light, and thus with Einstein’s photon theory, and with the discreteness of atomic spectra. The understanding of Bohr’s theory does not lie in visualizing it intuitively but in gaining familiarity with the problems it tries to solve, and in the appreciation of both the explanatory power of the solution and the fact, that the new difficulty that it creates constitutes an entirely new problem of great fertility.
The question whether or not a theory or a conjecture is more or less satisfactory or, if you like prima facie acceptable as a solution of the problem which it sets out to solve is largely a question of purely deductive logic. It is a matter of getting acquainted with the logical conclusions which may be drawn from the theory, and of judging whether or not these conclusions (a) yield the desired solution and (b) yield undesirable by-products – for example some insoluble paradox, some absurdity. 
He who knows only his own side of the case, knows little of that. His reasons may be good, and no one may have been able to refute them. But if he is equally unable to refute the reasons on the opposite side; if he does not so much as know what they are, he has no ground for preferring either opinion. The rational position for him would be suspension of judgment, and unless he contents himself with that, he is either led by authority, or adopts, like the generality of the world, the side to which he feels most inclination. Nor is it enough that he should hear the arguments of adversaries from his own teachers, presented as they state them, and accompanied by what they offer as refutations. That is not the way to do justice to the arguments, or bring them into real contact with his own mind. He must be able to hear them from persons who actually believe them; who defend them in earnest, and do their very utmost for them. He must know them in their most plausible and persuasive form; he must feel the whole force of the difficulty which the true view of the subject has to encounter and dispose of; else he will never really possess himself of the portion of truth which meets and removes that difficulty. Ninety-nine in a hundred of what are called educated men are in this condition; even of those who can argue fluently for their opinions. Their conclusion may be true, but it might be false for anything they know: they have never thrown themselves into the mental position of those who think differently from them, and considered what such persons may have to say; and consequently they do not, in any proper sense of the word, know the doctrine which they themselves profess. They do not know those parts of it which explain and justify the remainder; the considerations which show that a fact which seemingly conflicts with another is reconcilable with it, or that, of two apparently strong reasons, one and not the other ought to be preferred. All that part of the truth which turns the scale, and decides the judgment of a completely informed mind, they are strangers to; nor is it ever really known, but to those who have attended equally and impartially to both sides, and endeavoured to see the reasons of both in the strongest light. So essential is this discipline to a real understanding of moral and human subjects, that if opponents of all important truths do not exist, it is indispensable to imagine them, and supply them with the strongest arguments which the most skilful devil’s advocate can conjure up. [ch. II, 47-8]
But on every subject on which difference of opinion is possible, the truth depends on a balance to be struck between two sets of conflicting reasons. Even in natural philosophy, there is always some other explanation possible of the same facts; some geocentric theory instead of heliocentric, some phlogiston instead of oxygen; and it has to be shown why that other theory cannot be the true one: and until this is shown, and until we know how it is shown, we do not understand the grounds of our opinion. But when we turn to subjects infinitely more complicated, to morals, religion, politics, social relations, and the business of life, three-fourths of the arguments for every disputed opinion consist in dispelling the appearances which favour some opinion different from it. [ch. II, 46-7]
So the growth of objective scientific knowledge cannot be explained in the Kuhnian picture. It is no good trying to pretend that successive explanations are better only in terms of their own paradigm. There are objective differences. We can fly, whereas for most of human history people could only dream of this. The ancients would not have been blind to the efficacy of our flying machines just because, within their paradigm, they could not conceive of how they work. The reason why we can fly is that we understand ‘what is really out there’ well enough to build flying machines. The reason why the ancients could not is that their understanding was objectively inferior to ours. 
Do abstract, non-physical entities exist? Are they part of the fabric of reality? I am not interested here in issues of mere word usage. It is obvious that numbers, the laws of physics, and so on do ‘exist’ in some senses and not in others. The substantive question is this: how are we to understand such entities? Which of them are merely convenient forms of words, referring ultimately only to ordinary, physical reality? Which are merely ephemeral features of our culture? Which are arbitrary, like the rules of a trivial game that we need only look up? And which, if any, can be explained only in a way that attributes an independent existence to them? Things of this last type must be part of the fabric of reality as defined in this book, because one would have to understand them in order to understand everything that is understood. [222-3]