On truth –
3. The HP and the current practice chug along side-by-side; they’re just ‘investigating different notions of truth’.
Yes, but in (3) I wouldn’t say “the current practice” but rather “the practice-based investigation of truth”. I think that doing set theory and investigating set-theoretic truth based on practice are different things.
Now you’ve got me confused. Here’s the original question you raised to Sol:
So what is the relationship between truth and practice? If there are compelling arguments that the continuum is large and measurable cardinals exist only in inner models but not in V will this or should this have an effect on the development of set theory? Conversely, should the very same compelling arguments be rejected because their consequences appear to be in conflict with current set-theoretic practice?
This is a question about the relations between the HP and current practice, isn’t it? And here again are the three options I thought we’d settled on:
- The current practice has ‘veto power’. That is, if the HP followers come up with a principle that contradicts the existence of MCs, they say, ‘oops, back to the drawing board’.
- The HP has veto power. That is, if the HP followers come up with a principle that contradicts the existence of MCs, they say to the community, ‘terribly sorry, but you’ll have to give that up’.
- The HP and the current practice chug along side-by-side; they’re just ‘investigating different notions of truth’.
How did ‘the current practice’ drop out and ‘a practice-based investigation of truth’ slip in? (If you want to insist on the later, I’m going to have to ask what it is and who’s doing it. What I’ve been advocating on my own behalf is a move away from ‘truth’ as the relevant notion in any of this.)
Hugh’s project is a trickier issue as it raises the following question: When is mathematics relevant to the investigation of truth and when is it just good mathematics? You may feel that this question doesn’t need answering, and we should welcome any investigation which a mathematician reassures us is relevant to the investigation of truth.
What I think is that doing good mathematics is the goal of mathematical practice, in set theory and elsewhere. A person can call this the search for truth if he likes (as Hugh and my other figure, the Thin Realist, both do), but if so (I say) then the grounding of this truth is in the goodness of the mathematics. (So I guess you might say that my other figure, the Thin Realist, is in pursuit of ‘practice-based truth’, but if so, she conducts this pursuit just by doing set theory.)
But surely if the conclusions of such an investigation are interesting, such as a solution to CH, we would want to verify that the arguments which led there were well-grounded philosophically and that there were not mathematical choices made along the way just to make things work.
I don’t see anything at all wrong with ‘mathematical choices made along the way just to make things work’ — or as I might phrase it more generously, ‘mathematical choices made along the way in order to uncover good mathematics’. This is how we form the various central concepts of mathematics (e.g., group) and I would say it’s how we chose (or ought to choose) new set-theoretic axioms.
Specifically with regard to Hugh’s projects, it is worrisome that huge mathematical prerequisites are required to understand even the statements of, let alone the motivation for, what Hugh presents as his key conjectures. As a mathematician I find this difficult, it must be even more difficult for the philosopher.
This is an entirely different matter. Hugh’s mathematics is very difficult, largely inaccessible to many of us. This makes it hard for the community to come to informed judgments about its ‘goodness’ or ‘depth’. But there’s no reason at this point not to applaud his efforts, and to wait for the inevitable progress of mathematics to better digest what he’s doing and for the inevitable judgment of history to determine its value.
On concepts –
First question: Is this your personal picture or one you share with others?
I don’t know, but maybe I have persuaded some subset of Carolin Antos, Tatiana Arrigoni, Radek Honzik and Claudio Ternullo (HP collaborators) to have the same picture; we could ask them.
Why do you ask? Unless someone can refute my picture then I’m willing to be the only “weirdo” who has it.
Now here you surprise me, Sy! Most people who go in for conceptualism of some brand or other take the relevant concepts to be shared by the community — ‘we’re all out to investigate the concept of set (or set-theoretic universe)’, or something like that. I thought you might hesitate to claim that set-theoretic truth is determined by a picture in Sy Friedman’s head (though others are welcome to be instructed by him on its contours). No?
Here you seem to say the same thing:
(Does the phrase ‘refinement of what we take as true’ trouble you at all? Don’t ‘true’ and ‘what we take as true’ at least potentially diverge?)
I have no concept of “true” other than “what I take as true based on my picture of V”, which is constantly being refined.
What’s true in set theory is what Sy Friedman takes to be true based on his picture of V, which he constantly refines as he sees fit?
On my second question –
In the second kind of case, what grounds those refinements? It can’t be that they’re faithful to the concepts, so what is it?
They are faithful to the motivating intrinsic philosophical principles such as maximality.
But those motivating intrinsic principles are supposed to be implicit in the concept, aren’t they? If not, where do they come from?
In general –
At this point, it sounds as if the HP works like this. SF has a picture, he refines the picture. He eschews any extrinsic standard (now removing even the clause about principles being tested ‘by their ability to settle independent questions’). He’s willing to follow this notion of set-theoretic truth even if the mathematics generated is trivial and boring (‘a risk I have to take’). He assures us that ‘the mathematics is secondary’.
Now the question of why we should care becomes acute. Why should someone want to learn your concept and help develop it if it doesn’t produce good mathematics? If the goal isn’t to produce good mathematics, if it’s not to be judged by shared mathematical standards, in what sense is it even mathematics?
For me the practical point is this: even if you don’t give a hoot about extrinsic success, it doesn’t follow that you aren’t, in fact, generating some good mathematics, despite yourself so to speak. It doesn’t matter if Newton thought he was writing down the thoughts in the mind of God, what he actually did was science of the highest order. This is what I meant a while back by saying I thought that your analysis of your concepts was actually functioning as a sort of heuristic for generating ideas that would then be judged extrinsically. But of course this means I don’t see how you can lay special claim to some privileged notion of set-theoretic truth.
I have no objection to other investigations of set-theoretic truth, but I do think that we need philosophers to play a role in deciding what qualifies as an investigation of truth and what is just good mathematics.I can tell you as a mathematician that it is not hard to deceive oneself into thinking that one’s exciting new results have important implications for truth in set theory. That is why we need philosophers to police the situation. Tatiana, Claudio and other philosophers have helped to keep me honest. And aren’t I being currently subjected to a valuable “grilling” by an expert in the philosophy of mathematics (you)? I think that any mathematician who claims to investigate truth should be subjected to such a “grilling”. Philosophers of mathematics: We need you!
By now it should be clear that this philosopher has no interest whatsoever in distinguishing between ‘an investigation of set-theoretic truth’ and ‘the pursuit of good mathematics’ — or for that matter in ‘policing’ anyone. (If extrinsic considerations are the proper measure, as I claim, then these matters are to be judged on mathematical, not philosophical grounds. While you’re right that it’s often hard for mathematicians to tell immediately what’s good and what’s not, this is no reason at all to defer to philosophers, who are much more poorly placed to make that call.) I also haven’t intended to ‘grill’ you, and apologize that it came across that way. I have been trying to figure out precisely what your position is, and I have then pointed to some areas where the answers seem to me to be problematic. But as I’ve said before, one person’s reductio is another’s revolution!