Tag Archives: Thin realist

Re: Paper and slides on indefiniteness of CH

Dear Sy,

Peter is right, Sy. There’s no difference of opinion here between Peter and me about what counts as evidence, whether we call it ‘good set theory’ or ‘P and Vs’.

There is another point. Wouldn’t you want a discussion of truth in set theory to be receptive to what is going on in the rest of mathematics?

I don’t mean to be cranky about this, Sy, but I’ve lost track of how many times I’ve repeated that my Thin Realist recognizes evidence of both your Type 1 (from set theory) and Type 2 (from mathematics). I think I’ve mentioned that the foundational goal of set theory in particular plays a central role (especially in Naturalism in Mathematics).

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

Let’s first note that in the wake of independence, it’s going to be a pretty hard-line Universist (read “nutty Universist”) who asserts that we shouldn’t be studying truth across models in order to understand V better.

Pen Maddy is not a nut! You can simply ground truth in what is good set theory and mathematics, as would a Thin Realist (right, Pen?) and not bother with all of this talk about models.

This doesn’t really bear on what you and Neil are discussing, but for the record, the Thin Realist will study models of set theory as readily as the next set theorist.  Such a study might well turn out some good set theory and/or good mathematics more generally (which is how she comes ‘to understand V better’).  And, as remarked earlier, she doesn’t regard herself as infallible:  her commitment to a single universe could change in light of further developments.

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Pen, Sol and others,

It occurs to me that some of my disagreements with Pen and Sol could be resolved just by being clear about how the term “Set Theory” is used.

As I see it, Set Theory is three things:

  1. It is a branch of mathematics.
  2. It is a foundation for mathematics.
  3. It is the study of the concept of set.

Regarding 3: It is plain as pie that there is indeed a “concept of set”, familiar to schoolchildren who are victims of the “new math” (Venn diagrams, essentially). Even kids understand basic set-theoretic operations; probably once they are out of short pants they understand what we mean by powerset.

Now take a look at the “standard” axioms of ZFC. Why are they “standard”? It’s because we all seem to feel that they are “essential to Set Theory”. But there are two distinct sources for believing that:

As Boolos clarified in his paper on the iterative conception (IC), the axioms of Zermelo set theory are derivable from the concept of set as expressed by that conception. Replacement is not derivable from the IC, but it easily follows once we invoke Maximality, i.e. we strengthen the IC to the MIC (maximal iterative conception), also part of the concept of set.

As Pen has clearly expressed, the Axiom of Choice is a different matter: It does not follow from the MIC, but it does follow from the role of Set Theory as a foundation for mathematics. She can say this better than I, but the idea is that mathematics did much better once the old restrictive idea of set given by a rule was liberated through AC.

Now here we come to an important distinction that is ignored in discussions of Thin Realism: The Axiom of Choice didn’t get elected to the club because it is beneficial to the development of Set Theory! It got elected only because of its broader value for the development of mathematics outside of Set Theory, for the way it strengthens Set Theory as a foundation of mathematics. It is much more impressive for a statement of Set Theory to be valuable for the foundations of mathematics than it is for it to be valuable for the foundations of just Set Theory itself!

In other words when a Thin Realist talks about some statement being true as a result of its role for producing “good mathematics” she almost surely means just “good Set Theory” and nothing more than that. In the case of AC it was much more than that.

This has a corresponding effect on discussions of set-theoretic truth. Corresponding to the above 3 roles of Set Theory we have three notions of truth:

  1. True in the sense of Pen’s Thin Realist, i.e. a statement is true because of its importance for producing “good Set Theory”.
  2. True in the sense assigned to AC, i.e., a statement is true based on Set Theory’s role as a foundation of mathematics, i.e. because it is important for the development of areas of mathematics outside of Set Theory.
  3. True in the intrinsic sense, i.e., derivable from the maximal iterative conception of set.

Examples:

  1. Pen’s model Thin Realist John Steel will go for Hugh’s Ultimate-L axiom, assuming certain hard math gets taken care of. Will he then regard it as “true” based on its importance for producing “good Set Theory”? I assume so. If not, then maybe Pen will have to look for a new Thin Realist.
  2. Examples here are much harder to find! What have axioms beyond ZFC done for areas of math outside of Set Theory? Surely forcing axioms have had some dramatic combinatorial consequences, but large cardinals haven’t yet had a similar impact. Descriptive Set Theory has had recent and major implications for functional analysis, but the DST being used is just part of good old ZFC. To understand this situation better I think it’s time for set-theorists to stop being so self-centered and to take a close look at independence outside of set theory, with the aim of seeing which axioms beyond ZFC are the most fruitful for resolving those cases of independence (I’m happy to lead the charge!).
  3. Small large cardinals come easily out of the MIC. Precisely what I am doing with the HP is to derive further consequences. Maybe the negation of CH! Work in progress.

Now I see absolutely no argument for rejecting any of these three notions of Truth in Set Theory. Nor do I see an argument that they should reach common conclusions! Maybe you’ll find this to be excessively diplomatic, taking the heat and excitement out of the Great Set Theory Truth Debate, but I’m sure that even if we agree to this proposed Grand Truce, we’ll still find interesting things to argue about.

As I understand it (I am happy to be corrected), Pen is no fan of Type 3 truth and Sol is no fan of Type 1 truth. OK, I have nothing against aesthetic preferences. But to say that an answer to the Continuum Problem based on one of these three takes on Truth is “illegitimate” is going too far. If someone is going to say that CH is true (or false) then she has to say what notion of Truth is being referenced. Indeed, maybe CH is Type 2 true but Type 3 false!

In any case, it is clearly very hard (but in my view possible) to come to conclusions about what is true in any of these senses. As I have emphasized in the HP (Type 3 truth), for me to make a verdict about CH I will have to first produce “optimal” maximality criteria and show that CH is decided in the same way by those criteria. That is very hard work. For Type 2 truth one would similarly have to show that the statements of Set Theory which are most fruitful for the further development of Set Theory as a foundation for mathematics converge on a theory which settles CH. We have barely begun an investigation of the class of such statements!

I am most pessimistic about Type 1 truth (Thin Realism). To get any useful conclusions here one would not only have to talk about “good Set Theory” but about “the Best Set Theory”, or at least show that all forms of “good Set Theory” reach the same conclusion about something like CH. Can we really expect to ever do that? To be specific: We’ve got an axiom proposed by Hugh which, if things work out nicely, implies CH. But then at the same time we have all of the “very good Set Theory” that comes out of forcing axioms, which have enormous combinatorial power, many applications and imply not CH. So it seems that if Type 1 truth will ever have a chance of resolving CH one would have to either shoot down Ultimate L, shoot down forcing axioms or argue that one of these is not “good Set Theory”. Pen, how do you propose to do that? Forcing axioms are here to stay as “good Set Theory”, they can’t be “shot down”. And even if Ultimate L dies, there will very likely be something to replace it. Why should we expect this replacement for Ultimate L to come to the same conclusion about CH that forcing axioms reach (i.e. that CH is false)?

Nevertheless, as a stubborn optimist I do still expect that at least one of these forms of truth will generate some useful conclusions. But I have given up on the idea that there is a unique, supreme notion of truth in Set Theory that overrides all others; there are at least three distinct and legitimate forms to be taken seriously (despite my pessimism about Thin Realism). And maybe there is even yet a another form of set-theoretic truth that I have overlooked.

Best to all,
Sy

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