Conjectures and Refutations refers, naturally. But what is so charming about Popper is the way that an idea, that in our debased times would amount to an entire book, is merely a single chapter. Although the material I'm discussing here, I am obliged to admit, is not quite up to the first two sections; ah well.The Nature of Philosophical Problems and their Roots in Science (66-96)
Popper returns to the theme that philosophy is about solving problems not talking about philosophy; and points out that most problems in philosophy arise outside it. But that may not be obvious from the works themselves; they may have so sublimed the initial problem that is is not mentioned.
This is I think true and quite telling; it makes me pause and wonder about my reaction to certain works. Popper's examples here are Plato's theory of Forms, and Kant's Critique.
Before that, since Popper briefly discusses pseudo-problems, and since it fits my view of much of philosophy, I quote his My first thesis is that every philosophy... is liable to degenerate in such a way that its problems become practically indistinguishable from pseudo-problems, and its cant, accordingly, practically indistinguishable from meaningless babble. His second thesis is that what appears to be the prima facie method of teaching philosophy is liable to produce a philosophy which answers Wittgenstein's description [as babble]... that of giving the beginner... the works of the great philosophers to read... A new world of astonishingly subtle and vast abstractions opens itself before the reader; abstractions on an extremely high and difficult level. Thoughts and arguments are put before his mind which sometimes are not only hard to understand, but which seem to him irrelevant because he cannot find out what they may be relevant to; and so we return to the problems that provoked the philosophy.
Let's take his second example first, because I think it is simpler. Popper's contention is that Newtonian physics was so utterly successful and completely accepted by Kant's time that people had mistaken it for an absolute truth; and Kant therefore felt he needed to provide an explanation of how we could know such a thing a priori. I believe that (a) this is likely correct; (b) would not be universally accepted, because Kant is so obscure that people disagree over what he actually means, and are moreover desperate not to admit that he was wrong. My own reaction is yes: this is a nice explanation, it helps to understand Kant, but it also helps by pointing out that I really don't need to bother reading him: he is wrong.
The question of Plato's theory of forms is more complex, and also I think more speculative. Popper says that the discover of irrational numbers gets in the way of the atomisation and arithmetisation of nature, by which he means the programme - apparently that of those times - of associating numbers with things and deducing properties by counting; which assumes a smallest unit length scale, by which all other things in principle could be measured by counting (this programme is obscure and mystical, but that doesn't mean it wasn't their programme). Popper associates this with a switch to geometry; and then somehow connects this to the theory of forms. My own suspicion is that Popper is jumping backwards through hoops to rescue Plato's theory, probably from people like me who think it is silly, but I have to admit that it does perhaps help illuminate the writings.
Three Views of Human Knowledge (97-119)
This one isn't so good. It starts with a rather poor summary of "the Galileo affair" that totally misses the main point - that big G was trying to tell the Church how to interpret the Bible; and then tells us that the Gregorian calendrical reform made full use of big C, which I doubt. But leaving that aside, he is mostly discussing the status of scientific theory in the lights of what he calls Instrumentalism (theories are but tools, and don't describe an underlying reality, which most people using epicycles subscribed to) and Essentialism (theories describe "the realities behind appearances"). And his own preferred view: rejecting Instrumentalism, theories attempt to describe the real world, so not going all the way to Essentialism. All this seems to be not very interesting, and to come rather close to arguing over labels, a thing he usually disdains.
Towards a Rational Theory of Tradition (120-135)
Also not his finest; note that it was given to the Third Annual Conference of the Rationalist Press Association at Magdalen College; how delightful that such things once existed. But the ideas here are largely those of previous chapters.
Back to the Pre-Socratics (136-153)
A great deal of discussion of Pre-Socratic philosophy, doubtless very fine if you're interested in that history, but not of obvious modern relevance; he does though re-emphasise the importance of the emergence of a tradition of critical rational discussion. He does say two very strange things in the introduction to this section: (1) all science is cosmology; and (2) philosophy must return to cosmology. Admittedly he was talking in 1958 and wiki tells me that the CMB wasn't discovered until '64. Now, I would say, cosmology has almost entirely left the realm of philosophy and is part of science; that trend must have been obvious in '58. And (1) is only true if at all in the most general and useless sense.
Every now and again one reads a paper saying something along the lines of "ZOMG we're all going to die of starvation". Perhaps not literally that, but it will say something like crop yields decline under global warming. And totally fail to mention that other factors - better farming, fertilisers, and so on - are increasing yields faster. So this is just a page for me to tag with the obvious rebuttals, and link to some of my old posts. My pic is from 
