Thanks for this! I didn’t know about the Single Truth Convention, and I will respect it from now on. Somehow I thought that, for example, constructivists would claim that there are at least 2 distinct truth notions, but I guess I was wrong about that (maybe you would say that they are working with a different set-concept?).
Of course I knew that the official version of Thin Realism takes both Type 1 (good set theory) and Type 2 (set theory as a foundation) evidence into account; but the fact is that in the present forum with only one exception we’ve been talking exclusively about axioms that are good for set theory, like PD and large cardinals, but not good for other areas of mathematics (the exception is when I brought in forcing axioms). More on this point below.
So there were “terminological” errors in what I said. But correcting those errors leaves the argument unchanged and in fact I will make the argument even stronger in this mail.
There are 3 kinds of evidence for Truth (not 3 kinds of Truth!), emanating from the 3 roles of Set Theory that I indicated: (1) A branch of mathematics, (2) A foundation for mathematics and (3) A study of the set-concept.
Now you will object immediately and say that there is no Type 3 evidence; Type 3 is just an engine (heuristic) for generating new examples of Types 1 and 2. Fine, but it still generates evidence, albeit indirectly! You would simply cleanse that evidence of its source (the maximality of the set-concept) and just regard it as plain good set theory or mathematics. It is hard to imagine (although not to be ruled out) that Type 3 will ever generate anything useful for mathematics outside of set theory, so let’s say that Type 3 provides an indirect way of generating evidence of Type 1. Clearly the way that this evidence is generated is not the usual direct way. In any case even your Thin Realist will take Type 3 generated evidence into account (as long as it entails good set theory).
So up to this point the only difference we have is that I regard Type 3 considerations as more than just an indirect way of generating new Type 1 evidence; I would like to preserve the source of that evidence and say that Type 3 considerations enhance our understanding of the set-concept through the MIC, i.e., they are good for the development of the philosophy of the set-concept. Of course the radical skeptic regards this as pure nonsense, I understand that. But I continue to think that there is more at play here than just good set theory or good math, there is also something valuable in better understanding the concept of set. For now we can just leave that debate aside and regard it just as a polite, collegial disagreement which can safely be ignored.
OK, so we have 3 sources for truth. But there is an important difference between Type 1 vs. Types 2, 3 and this regards the issue of “grounding” for the evidence.
Type 3 evidence (i.e. evidence evoked through Type 3 considerations, as you may prefer to say) is grounded in the maximal iterative conception. The HP is limited to squeezing out consequences of that. Of course there is some wiggle room here, but it is fairly robust to say that some mathematical criterion reflects the Maximality of the universe or synthesises two other such criteria.
Type 2 evidence is grounded in the practice of mathematics outside of set theory. Functional analysts, number theorists, toplogists, group theorists, … are not thinking about set theory directly, but axioms of set theory are of course useful for them. You cite AC, which is a perfect example, but in the contemporary setting we can look at the value for example of forcing axioms for the combinatorial power they provide for resolving problems in mathematics outside of set theory. Now just as Type 3 evidence is limited to what grounds it, namely considerations regarding the maximality of the set-concept, so is Type 2 evidence limited by what is valuable for the work of mathematicians who don’t have set theory in their heads, who are not thinking about actualism vs. potentialism, reflection principles, HOD, … To see that this is a nontrivial limitation, just note that two of the most discussed axioms of set theory in this forum, PD and large cardinals, appear to have nothing relevant to say about areas of mathematics outside of set theory. In this sense the Type 2 evidence for forcing axioms is overwhelming in comparison to Type 2 evidence for anything else we have been discussing here, including Ultimate-L or what is generated by the HP.
So there is a big gap in what we know about about evidence for set-theoretic truth. We have barely scratched the surface with the issue of what new axioms of set theory are good for the development of mathematics outside of set theory. Your great example, the Axiom of Choice, won the day for its value in the development of mathematics outside of set theory, yet for some reason this important point has been forgotten and the focus has been on what is good for the development of just set theory. (This may be the only point on which Angus MacIntyre and I may agree: To understand the fondations of mathematics one has to take a close look at mathematics and see what it needs, and not just play around with set theory all the time.)
In my view, Type 1 evidence is very poorly grounded, I would even say not grounded at all. It is evidence that says that some axiom of set theory is good for set theory. That could mean 100 different things. One person says that V = L is true because it is such a strong theory, another that forcing axioms are true because they have great combinatorial strength, Ultimate-L is true for reasons I don’t yet understand, …, not to forget Aczel’s AFA, or constructive set theory, … the list is almost endless. With just Type 1 evidence, we allow set-theorists to run rampant, declaring their own brand of set theory to be particularly “good set theory”. I think this is what I meant earlier when I griped about set-theorists promoting their latest discoveries to the level of evidence for truth.
Pen, can you give Type 1 evidence a better grounding? I’m not sure that I grasped the point of Hugh’s latest mail, but maybe he is hinting at a way of doing this:
“Assuming V = Ultimate L one can have inner models containing the reals of say MM. But assuming MM one cannot have an inner model containing the reals which satisfies V = Ultimate L.”
Perhaps (not to put words in Hugh’s mouth) he is saying that Axiom B is better than Axiom A if models of Axiom B produce inner models of Axiom A but not conversely. Is this a start on how to impose a justifiable preference for one kind of good set theory over another? But maybe I missed the point here, because it seems that one could have “MM together with an inner model of Ultimate-L not containing all of the reals”, in which case that would be an even better synthesis of the 2 axioms! (Here we go again, with maximality and synthesis, this time with first-order axioms, rather than with the set-concept and Hyperuniverse-criteria.)
Anyway, in the absence of a better grounding for Type 1 evidence I am strongly inclined to favour what is well-grounded, namely evidence of Types 2 and 3.
You raised the issue of conflict. It is clear that there can Type 1 conflicts, i.e. conflicts between different forms of Type 1 evidence, and that’s why I’m asking for a better grounding for Type 1. We don’t know yet if there are Type 2 conflicts, because we don’t know much about Type 2 evidence at all. And the hardest part of the HP is dealing with Type 3 conflicts; surely they arise but my “synthesis method” is meant to resolve them.
But what about conflicts between evidence of different Types (1, 2 or 3)? The Single Truth Convention makes this tough, I can’t weasel out anymore by simply saying that there are just different forms of Truth (too bad). Nor can I accept simply rejecting evidence of a particular Type (as you “almost” seemed to suggest when you hinted that Type 3 should defer to Types 1 and 2). This is a dilemma. To present a wild scenario, suppose we have:
(Type 1) The axioms that give the “best set theory” imply CH.
(Type 2) The axioms that give the best foundation for mathematics outside of set theory imply not-CH.
(Type 3) The axioms that follow from the maximality of the set-concept imply not-CH.
What do we tell Sol at that point? Does the majority win, 2 out of 3 for not-CH? I don’t think that Sol will be very impressed by that.
Sorry that this mail got so long. As always, I look forward to your reply.
All the best,