I think we’ve reached the crux, but let me try one more time to summarize your position accurately:
We reject any ‘external’ truth to which we must be faithful, either in the form of a platonist ontology or some form of truth-value realism, but we also deny that the resulting ‘true-in-V’ arises strictly out of the practice (as my Arealist would have it). One key is that ‘true-in-V’ is answerable to various intrinsic considerations. The other key is that it’s also answerable to some set-theoretic claims, namely ZFC and the consistency of LCs.
The intrinsic constraints aren’t limited to items that are implicit in the concept of set. One of the items present in this concept is a notion of maximality. The new intrinsic considerations arise when we begin to consider, in addition, the concept of a universe of sets. We investigate this new concept with the help of a mathematical construct, the hyperuniverse. This analysis reveals a new notion of maximality that’s implicit in the concept of a universe of sets and that generates the schema of Logical Maximality and its various instances (and more, set aside for now).
At this point, we have ZFC+the consistency of LCs and various maximality principles. If the consequences of the maximality principles conflict with ZFC+the consistency of LCs, they’re subject to serious question. They’re further tested by their ability to settle independent questions. Once we’ve settled on a principle, we use it to define ‘preferred universe’ and count as ‘true-in-V’ anything that’s true in all preferred universes.
Now two remarks (not ‘attacks’ for goodness sake!):
1. About ‘external’ and the ‘concept':
We reject any ‘external’ truth to which we must be faithful, but we also deny that the
resulting ‘true-in-V’ arises strictly out of the practice (as my Arealist would have it).
I think that I agree but am not entirely clear about your use of the term “external truth”. For example, I remain faithful to the extrinsically-confirmed fact that large cardinal axioms are consistent. Is that part of what you mean by “external truth”? With this one exception, my concept of truth is entirely based on intrinsic evidence.
Actually, I adopted the word ‘external’ from you, Sy: ‘No “external” constraint is imposed … such as an already existing reality to which one must be faithful’ (BSL, p. 80). Later, in these exchanges, you also distanced yourself from truth-value realism:
In my reply to Sol I only made reference to truth-value realism for the purpose of illustrating that one can ascribe meaning to set-theoretic truth without being a platonist. Indeed my view of truth is very far from the truth-value realist, it is entirely epistemic in nature.
So my concern was just that you’d need an account of what your ‘concepts’ are that doesn’t end you up with something you find uncomfortably close to these things you reject. In response, you write:
The concept of set is clear enough in the discussion, I have not proposed any change to its usual meaning.
Maybe so, but what is a concept? An abstract item (a property? a universal? a meaning?) Something mental? Just a fancy way of talking about our shared practices in using a particular word? If we’re to understand what your intrinsic justifications come to, we have to know what grounds them, and for that we have to know what a concept is. My guess is that your aversion to abstract ontology and truth-value realism would lead you to rule out the first. Would you be happy with a view of concepts as some kind of mental construction, of an individual or of a group (however that would work)? Would it be possible for these mental concepts to have features that we don’t now know about but can somehow discover? (Can we discover how long Sherlock Holmes’s nose is?) Wanting there to be a fact-of-the-matter we’re out to discover is what pushes many people in the direction of the first option (some kind of abstract item). I don’t have any horse in this race — I’m a bit of a concept&meaning-phobe myself — I’m just pointing out that you need a notion of concept that does all the things you want it to do and doesn’t land you in a place you don’t like.
2. About truth.
Given the general tenor of your position, this sounds to me like the right move for you to make:
I think that I just fell over the edge and am ready to revoke my generous offer of “veto power” to the working set-theorist. Doing so takes the thrust out of intrinsically based discoveries about truth. You are absolutely right, “veto power” would constrain the necessary freedom one needs in the investigation of intrinsically-based criteria for the choice of preferred universes.
Then I come back to my original concern about intrinsic justifications in general:
The challenge we friends of extrinsic justifications like to put to defenders of intrinsic justifications is this: suppose some candidate principle generates a lot of deep-looking mathematics, but conflicts with intrinsically generated principles; would you really want to say ‘gee, that’s too bad, but we have to jettison that deep-looking mathematics’?
You seem perfectly prepared to say just that:
The only way to avoid that would be to hoard together a group of brilliant young set-theorists whose minds have not yet been influenced (polluted?) by set-theoretic practice, deny them access to the latest results in set theory and set them to work on the HP in isolation. From time to time somebody would have to drop by and provide them with the mathematical methods they need to create preferred universes. Then after a good amount of time we could see what conclusions they reach! LC? PD? CH? What?
Whatever they come up with wins, even if it means jettisoning what looks like deep and important mathematics.
So that’s the crux. To me this sounds like a reductio. To you it sounds revolutionary. Here’s your defense:
I would like to have a notion of truth in set theory that is immune to the influence of fads, forceful personalities, available grant money, … I really am not confident that what we now consider to be important in the subject will be important in the future; I am more confident about the “stability” of Sy truth. Second, and this may appeal to you more, it is already clear that the new approach taken by the HP has generated new mathematical ideas that never would have been generated through the usual practice. Taking a practice-independent look at set-theoretic truth generates new forms of set-theoretic practice. And I do like the practice of set theory, even though I don’t want it to dictate investigations of truth! It is valuable to develop set theory in new directions.
I sympathize with your desire to know, right now, what the good math is and what the lousy math is, but the history of the subject seems to me to show that we can’t know this for sure right now, that it can take decades, or longer, for matters to sort themselves out. Of course there are nihilists who think that there’s really no difference between good and lousy, that it’s all just fads, personalities, politics, etc., but despite the undeniable fact that factors like these are always in play, again the history of the subject makes me hopeful that we do, eventually, attain a fair view of the terrain.
As for your second point, yes, I do like it! I have no more desire to curtail the HP program than to curtail any other promising mathematical avenue. (My take is that your so-called ‘intrinsic justifications’ are actually functioning as heuristics that are helping you get to some interesting ideas, but that the justification will come from the extrinsic success of those ideas.) My gripe only comes in when you lay claim to an intrinsic way of limiting everyone else. In the end, of course, I hope that whatever good comes out of your program and out of Hugh’s program and out of other programs, can be combined into one overarching subject it seems natural to continue to call ‘set theory’, but if not, well, we’ll face that when it comes. But here’s another of my wagers: however we do it, we won’t decide to throw out any good mathematics.
PS: You asked about my notion of truth. I haven’t been out to expound my own position here; I only threw in the Arealist because Sol suggested that something of mine might help clarify your views. For what it’s worth, my Arealist doesn’t think that truth is what we’re after in doing set theory; rather, we’re doing our best to devise an effective theory to do certain mathematical jobs. But while he’s doing set theory, the Arealist is happy to use the word ‘true’ in conventional ways: for example, if he accepts, works in, a theory with lots of LCs, he’s happy to say things like ” ‘MCs exist’ is true in V, but not in L’. My own position is close to the Arealist’s but not identical. I realize it’s tedious of me to have written a book, let alone books, but if you’re ever curious, the one to read is Defending the Axioms. It’s very short!