Re: Paper and slides on indefiniteness of CH

Dear Harvey,

I realize you’re talking about ‘good mathematics’ rather than ‘deep mathematics’, but I hope you won’t mind if I take the liberty of transposing your remarks to the context of the workshop.

I have never heard mathematicians talk directly about what good mathematics is in any kind of generality.

Actually, for example, Urquhart presents some interesting quotations from people like Timothy Gowers and Terence Tao.

In talking separately to different kinds of mathematicians over the years, it is obvious that there is a huge amount of disagreement about how to evaluate mathematics, what it’s purpose it, what it means, and so forth.

Sure.  And it may turn out that ‘depth’ is just a term people use to mean ‘I like it’.  But that isn’t entirely clear yet.  In fact, many of the ideas you touch on were discussed at the workshop as potential symptoms or features of depth.  For example …

If the problem has resisted solution for a very long time, and it is known that some mathematicians with very strong reputations worked on it and failed to solve it, then mathematicians will generally regard the solution as extremely good mathematics.

This looks more like a symptom than a feature.

the solution uses considerable machinery … that promises further solutions to further problems.

This looks like ‘fruitfulness’, which may or may not be connected to depth.

There is a major premium paid for interactions between areas of mathematics – particularly if the interaction is unexpected.

This last is perhaps the suggestion most uniformly embraced.  The group pondered the problem of what counts as an ‘area’ of mathematics — is this something inherent in the math, or is it just a reflection of our human ways of dividing things up?  There’s also the question of ‘unexpected’ — is this just a matter of what we humans happen to be surprised by or is it a sign of something more fundamental in the math itself?

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Harvey,

Because ‘mathematical depth’ is a notion that hasn’t received much philosophical attention, this workshop was largely exploratory.  Maybe it would help to say a bit more about how it was organized:

This workshop will bring together mathematicians and historians and philosophers of mathematics to try to get a preliminary sense of whether or not the notion of ‘mathematical depth’ can play a useful role in our understanding of the nature and practice of mathematics. Speakers have been encouraged to present an example or examples of concepts, theorems, subject areas that they think qualify as deep, or as not deep, and to lay out the particular mathematical features of those examples that lead them to make those judgments. Then, in discussion, we aim to do several things:

  1. See if there is agreement on which examples are deep and not deep.
  2. See if there are commonalities in the kinds of features cited in defense of depth and non-depth assessments in the various examples.
  3. Ask whether depth is or isn’t the same as fruitfulness, surprisingness, importance, elegance, difficulty, fundamentalness, explanatoriness, etc.
  4. Ask whether depth seems likely to be an objective feature or something essentially tied to our interests, abilities, and so on. (Even natural science is tied to our interests and abilities, in that we might be drawn to certain areas of inquiry by our interests, hampered or helped by certain of our abilities, etc. The question is whether depth is tied to our interests and abilities in some more fundamental way.)

One possible outcome would be: this is a non-starter. Another would be: this is worthy of further study.

So, yes, Stillwell largely gave examples, which were then most helpful in the discussions.  Arana’s talk had more analytical content in pursuit of just one example.  Gray talked about Gauss’s notion of depth.  Lange discussed some very interesting ‘Martin Gardner level’ examples of comparative depth.  Urquhart considered a range of things mathematicians have said, in print or online, on the topic.  And so on.  But if you want a sense of the real consolidating work that was done, you might dip into the discussions, which gathered steam over the two days.

On the other hand, if you’re interested in something more polished, you might wait for the special issue.  It’s projected to include papers by Arana, Gray, Lange, Stillwell and Urquhart based on their talks, and a substantial editorial Foreword and Afterword that summarizes and explores the high points of the discussions.

Again, this is only a preliminary investigation, beginning from cases and asking what, if anything, can be gleaned from them.  There’s no final answer on offer here — at best it’s a start.

All best,
Pen

PS:  An aside to Sy —  I believe my concerns about your notions of ‘truth’ and ‘the concepts of set and set-theoretic universe’ are independent of my own ‘radical’ views on these matters.

Re: Paper and slides on indefiniteness of CH

Dear Harvey,

  1. People seem to be blackboxing the idea of “good mathematics” or “good set theory.” These notions are desperately in need of some sort of elucidation particularly by people with foundational sensibilities.

In a recent workshop, some mathematicians, historians, and philosophers undertook a preliminary, collaborative exploration of the notion of ‘mathematical depth’.

A special issue of Philosophia Mathematica is in the works.

With all best wishes,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

I didn’t imagine I’d been keeping a secret of my views about the irrelevance of ‘truth’ to understanding what’s going on in set theory!  (This isn’t because I’m otherwise allergic to ‘truth’ — exploration of that notion takes up a fair chunk of another of my too-many books.)  But I’m not so sure of my beliefs that I don’t take an interest in what other people are thinking about truth and intrinsic justification, and I’m not claiming that what you’ve said is meaningless.

You ask some questions about my position, but maybe this is a topic for another time.  I’ve done my best to lay it out succinctly in Defending the Axioms.

One last remark.  You write:

Fine, but as I understand it in your view there is no possibility whatsoever that Hugh or I will “solve the continuum problem” during our lifetimes because for you the truth of CH is either just meaningless or could only be resolved by developing mathematics of time-tested importance that proves or refutes it, and given that Hugh and I are about 60 years old there just ain’t enough time.

I wouldn’t say the CH is meaningless.  It does seem to me at least possible that the mathematics will progress quickly enough for you and Hugh (and me, too, though I’m a few years older than the both of you) to have a pretty good idea of how the CH will ultimately be settled.  But you’re probably right that any final resolution, whatever it might look like, will come too late for us.  The citation for Hugh’s recent Hausdorff prize read:

This work is innovative, courageous and deep, and no matter what the final solution of this major problem in set theory will look like, it is clear that it will have to depend on the contributions Woodin made in these papers.

The immediate topic was inner model theory, but the CH is surely in the wings.

Thanks for your patience with my questioning, Sy.  The opportunity to delve into your views was most welcome.

With all best wishes,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

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:

  1. 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’.
  2. 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’.
  3. 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!

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

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.

All best,
Pen

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!

Re: Paper and slides on indefiniteness of CH

Dear Sy,

Thanks for these clarifications and amendments!  (The only changes of mind that strike me as objectionable are the ones where the person pretends it hasn’t happened.)  I’m still keen to formulate a nice tight summary of your approach, and then to raise a couple of questions, so let me take another shot at it. Here’s a revised version of the previous summary:

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).  One key is that ‘true-in-V’ is answerable, not to a realist ontology or some sort of ‘truth value realism’, but to various intrinsic considerations.  The other key is that it’s also answerable to a certain restricted portion of the practice, the de facto set-theoretic claims.  These are the ones that need to be be taken seriously as we evaluate any candidate for a new set theoretic axiom or principle. They include 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 the hyperuniverse. One of the items present in this concept is a new notion of maximality, building on the old, that generates the schema of Logical Maximality and its various instances (and more, set aside for now).

At this point, we have the de facto part of practice and various maximality principles.  If the principles conflict with the de facto part, they’re subject to serious question (see below).  They’re further tested by their ability to settle independent questions.  Once we’re 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.

I hope this has inched a bit closer!  Assuming so, here are the two questions I wanted to pose to you:

  • What is the status of ‘the concept of set’ and ‘the concept of set-theoretic universe’?

This might sound like a nicety of interest only to philosophers, but there’s a real danger of falling into something ‘external’, something too close for comfort to an ontology of abstracta or a form of truth-value realism.

  • 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’?  (I’d argue that this isn’t entirely hypothetical.  Choice was initially controversial largely because it conflicted with one strong theme in the contemporary concept of set, namely, the idea that a set is determined by a property.  The mathematics generated by Choice was so irresistible that (much of the) mathematical community switched to the iterative conception. Trying to shut down attractive mathematical avenues has been a fool’s errand in the history of mathematics.)

You’ve had some pretty interesting things to say about this!  This remark to Hugh, which you repeat, was what made me realize I’d so badly misunderstood you the first time around:

The basic problem with what you are saying is that you are letting set-theoretic practice dictate the investigation of set-theoretic truth!

And these remarks to Sol also jumped out:

Another very interesting question concerns the relationship between truth and practice. It is perfectly possible to develop the mathematics of set theory without consideration of set-theoretic truth. Indeed Saharon has suggested that ZFC exhausts what we can say regarding truth but of course that does not force him to work just in ZFC. Conversely, the HP makes it clear that one can investigate truth in set theory quite independently from set-theoretic practice; indeed the IMH arose from such an investigation and some would argue that it conflicts with set-theoretic practice (as it denies the existence of inaccessibles). 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?

And today, to me, you add:

I see that the HP is the correct source for axiom *candidates* which must then be tested against current set-theoretic practice. There is no naturalist leaning here, as I am in no way allowing set-theoretic practice to influence the choice of axiom-candidates; I am only allowing a certain veto power by the mathematical community. The ideal situation is if an (intrinsically-based) axiom candidate is also evidenced by set-theoretic practice; then a strong case can be made for its truth.

But I am very close to dropping this last “veto power” idea in favour of the following (which I already mentioned to Sol in an earlier mail): Perhaps we should accept the fact that set-theoretic truth and set-theoretic practice are quite independent of each other and not worry when we see conflicts between them. Maybe the existence of measurable cardinals is not “true” but set theory can proceed perfectly well without taking this into consideration.

Let me just make two remarks on all this.  First, if you allow the practice to have ‘veto power’, I don’t see how you aren’t allowing it to influence the choice of principles.  Second, if you don’t allow the practice to have ‘veto power’, but you also don’t demand that the practice conform to truth (as I was imagining in my generic challenge to intrinsic justification given above), then — to put it bluntly — who cares about truth?  I thought the whole project was to gain useful guidance for the future development of set theory.

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

Thanks so much for your patient responses to my elementary questions!  I now see that I was viewing those passages in your BSL paper through the wrong lens, but rather than detailing the sources of my previous errors, I hope you’ll forgive me in advance for making some new ones.  As I now (mis?)understand your picture, it goes roughly like this…

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).  One key is that ‘true-in-V’ is answerable, not to a realist ontology or some sort of ‘truth value realism’, but to various intrinsic considerations.  The other key is that it’s also answerable to a certain restricted portion of the practice, the de facto set-theoretic claims.  These are the ones that ‘due to the role that they play in the practice of set theory and, more generally, of mathematics, should not be contradicted by any further candidate for a set-theoretic statement that may be regarded as ultimate and unrevisable’ (p. 80).  (Is it really essential that these statements be ‘ultimate and unrevisable’?  Isn’t it enough that they’re the ones we accept for now, reserving the right to adjust our thinking as we learn more?)  These include ZFC and the consistency of LCs.

The intrinsic constraints aren’t limited to items that are ‘implicit in the concept of set’.  They also include items ‘implicit in the concept of a set-theoretic universe’.  (This sounds reminiscent of Tony’s reading in ‘Gödel’s conceptual realism’.  Do you find this congenial?)  One of the items present in the latter concept is a notion of maximality.  The new intrinsic considerations arise at this point, when we begin to consider, not just V, but a range of different ‘pictures of V’ and their interrelations in the hyperuniverse.  When we do this, we come to see that the vague principle of maximality derived from the concept of a set-theoretic universe can be made more precise — hence the schema of Logical Maximality and its various instances.

At this point, we have the de facto part of practice and various maximality principles (and more, but let’s stick with this example for now).  If the principles conflict with the de facto part, they’re rejected.  Of the survivors, they’re further tested by their ability to settle independent questions.

Is this at least a bit closer to the story you want to tell?

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Thank you for the plug, Sol.  Sy says some interesting things in his BSL paper about ‘true in V':  it doesn’t ‘reflect an ontological state of affairs concerning the universe of all sets as a reality to which existence can be ascribed independently of set-theoretic practice’, but rather ‘a façon de parler that only conveys information about set-theorists’ epistemic attitudes, as a description of the status that certain statements have or are expected to have in set-theorist’s eyes’ (p. 80). There is ‘no “external” constraint … to which one must be faithful’, only ‘justifiable procedures’ (p. 80); V is ‘a product of our own, progressively developing along with the advances of set theory’ (p. 93).  This sounds more or less congenial to my Arealist (a non-platonist):   in the course of doing set theory, when we adopt an axiom or prove a theorem from axioms we accept, we say it’s ‘true in V’, and the Arealist will say this along with the realist; the philosophical debate is about what we say when we’re describing set-theoretic activity itself, and here the Arealist denies (and the realist asserts) that it’s out to discover the truth about some objectively existing abstracta.  (By the way, I don’t think ‘truth-value realism’ is the way to go here.  In its usual form, it avoids abstract entities, but there remains an external fact-of-the-matter quite independent of the practice to which we’re supposed to be faithful.)  Unfortunately the rest of my story of the Arealist as it stands won’t be much help because the non-platonistic grounds given there in favor of embracing various set-theoretic methods or principles are fundamentally extrinsic and Sy is out to find a new kind of intrinsic support.

I’m probably insufficiently attentive, or just plain dim, but I confess to being confused about how this new intrinsic evidence is intended to work.   It isn’t a matter of being part of the concept of set, nor is it given by the clear light of mathematical intuition.  It does involve, quoting from Gödel, ‘a more profound understanding of basic concepts underlying logic and mathematics’, and in particular, in Sy’s words, ‘a logical-mathematical analysis of the hyperuniverse’ (p. 79).  Is it just a matter of switching from the concept of set to the concept of the hyperuniverse?  (My guess is no.)  Our examination of the hyperuniverse is supposed to ‘evoke’ (p. 79) certain general principles (the principles are ‘based on’ general features of the hyperuniverse (p. 87)), which will in turn ‘suggest’ (pp. 79, 87) criteria for singling out the preferred universes — and the items ultimately supported by these considerations are the first-order statements true in all preferred universes.

One such general principle is maximality, but I’d like to understand better how it arises intrinsically out of our contemplation of the hyperuniverse (at the top of p. 88).  On p. 93, the principle (or its more specific versions) is said to be ‘the rigorous expression of what it means for an element of the hyperuniverse, i.e., a countable transitive model of ZFC, to display “maximal properties”‘.  Does this mean that maximality for the hyperuniverse derives from a prior principle of maximality inherent in the concept of set?

With all best wishes,
Pen