Quotes are from the book Karl Popper, by Anthony O'Hear (AOH). It's in "The Arguments of the Philosophers" series edited by Ted Honderich. (Be careful, AOH has two other books with titles beginning with "Karl Popper".)
Popper's attempt to dispense with induction is unsuccessful. [ch. 4, p. 57]
AOH says his reason, which he'll attempt to show, is:
any coherent conceptualization of the experience requires the assumption of a stable order in the world. [ch. 4, p. 58, emphasis added]
Previously, AOH wrote:
But, argues Popper, we can see on logical grounds that there is no such thing as a perfect repetition of any event. Similarity in all respects would mean that the two events were really identical, and so there would actually be only one event. So the repetitions we experience are only approximate. But this means that some features of repetition B of event A will be different from some features of A. Thus B is to be seen as a repetition of A only to the extent that we discount those features in which B differs from A. [ch. 2, p. 13]
So AOH ought to address the question: "Stable in which respects?" He ought to know that the world is stable in some respects and not others, just as the future resembles the past in some ways and not others, and any two observations are similar to each other in some ways and not others.
Saying the world is "stable" means just as little as saying two observations are "similar". Claiming a stable world means claiming some things stay the same over time (or at least only change a small amount, according to some suitable measure). Of course not all things stay the same over time.
So AOH needs to say what type of stability he's talking about for his claim to mean anything.
One of the standard problems with inductivists is their routine failure to understand this general problem (that when we compare non-identical things they're always both similar and different, and you have to specify what sort of comparison you're doing). What does AOH do about this issue? Nothing. After the "stable" claim I quoted, he immediately changes the subject to solipsism. He's apparently unaware of this issue, even though he discussed it earlier in the book.
AOH proceeds (p. 59) to talk about regularities and patterns of experience without talking about which ones. Of course there are some regularities and some non-regularities in the world. AOH's approach to epistemology is basically "We live in a stable world, so recognize regularities and project them into the future." This is standard inductivist, and misses the point in the standard ways, such as the issue of which regularities to project into the future and how to find them (how does thinking work? AOH just takes for granted that we find these regularities somehow – that is, his epistemology presupposes intelligent thought and fails to explain how thinking actually works. He starts in the middle.) Then:
Our notion of an objective world, then, is reflected by the degree of continuing order and regularity that is to be found within our perceptions. [ch. 4, p. 59]
But Popper already explained the problem with this, and AOH already included that in this book. There is no such thing as "order" or "regularity" out of context. You have to first say which things you want to be the same which you'll count as being orderly or regular. Different aspects of the world are always similar (orderly, regular) in some ways and different in other ways. AOH doesn't address this.
I also found this bizarre statement:
That a belief in induction is not something which can be dropped without substantial alterations elsewhere in our conceptual scheme is why the failure of Popper to develop a truly non-inductive science is not a chance result, but one with deep roots. [ch 4, p. 60]
But Popper was aware of this issue, and wrote about it, and did develop substantial alterations in our conceptual scheme. I would understand if someone thought Popper's substantial alterations were mistaken, or if someone was unfamiliar with Popper's writing. But AOH has studied Popper a lot, and then is apparently unaware this substantial alterations even exist. AOH even quotes and discusses some of them, but apparently(?) doesn't recognize their meaning and importance. This is just like the similar in which respects issue, where AOH quoted Popper about it and discussed it – but then later on he writes as if he was unaware of it (which I take to mean he doesn't fully understand it).
the assumption that the world is not going to [suddenly become chaotic] [ch 4. p. 61]
The world is already chaotic in some ways and not others. So what does this mean? AOH doesn't say.
Does it mean the world won't suddenly become chaotic in all respects? But what would a world that is chaotic in all respects even mean? AOH doesn't address the issue and it's highly problematic.
One fairly technical way to approach the matter is via the theory of computation: consider whether there exist long bitstrings which can't be compressed by any compression algorithm (or, equivalently, can't be the output of any computer program, in any language, which is much shorter than the bitstring). Such a bitstring would be chaotic in all respects. But the answer is no, such a bitstring doesn't exist.
AOH might imagine that, all of a sudden, all the ways the world is stable stop working, and some new ones take their place. But that doesn't make sense, because no matter what happens, you can always retrospectively find regularities in the bigger picture including both before and after the so-called descent into chaos. All that's happened is this: from the infinitely many regularities compatible with the data you have, you favored some (why those? how were they chosen?), and found out those favored regularities were mistaken. (Meanwhile this so-called descent into chaos is fully compatible with some of the other data-compatible claims about regularities you could have made before it happened.)
So the assumption of the world's stability really means assuming your favored theories are correct. Why did you favor them over other theories, compatible with the same data, which make different predictions about the future? From the perspective of those rival theories, the future you predict is a descent into chaos. So when you say the world won't descend into chaos, you just mean the future will happen as you expect and not as your rivals expect – you mean the world will descend into chaos for the people who disagree with you, just not for yourself.
Thus, I am not simply saying that our ability to distinguish between true experience and illusions depends on our once having experienced an orderly world, but that it depends on the continuance of whatever order we had previously recognized. But to assume this is just what, according to Popper, is deeply irrational, and which should be eliminated from our conceptual scheme. [ch 4. p. 61]
Yes, it is irrational. Because it consists of assuming you're right.
What does "whatever order we had previously recognized" refer to? There are infinitely many theories compatible with the data you've observed previously. To recognize some order means to choose some of those of those theories (why those? why not others?) to provide order to your thinking. Then to assume the continuance of that order means to assume that your choice of which theories to prefer won't turn out to be mistaken in the future.
The solution to all this is what Popper said: critical and explanatory thinking (which is literally evolution). We can only conjecture which of the infinite regularities (or, preferably, explanatory theories) compatible with our data are correct. And we can correct errors with criticism, which is how progress is made. (Part of this is explained by AOH, pp. 171-177)
AOH also objects to Popper's corroboration, and I agree that corroboration is a mistake. I have fixed that aspect of Critical Rationalism. You can find my solution here. For a quick overview, I also offer a free short argument.