Tag Archives: Good mathematics

Re: Paper and slides on indefiniteness of CH

Dear Pen,

Wallpaper? Interior decoration? Pen Maddy, you have outdone yourself! But you are ignoring the obvious; I explain. As you wrote:

OK!  Now we’re getting somewhere.  Your position begins like this:

  1. The relevant concept is the familiar iterative conception, which includes a rough idea of maximality in ‘height’ and ‘width’.
  2. To give an intrinsic justification or intrinsic evidence for a set-theoretic principle is to show that it is implicit in the concept in (1).
  3. The HP is a method for extracting more of the implicit content of the concept in (1) than has heretofore been possible.

The next step is to look into the mechanisms by which (3) is accomplished.”

Or as you usefully paraphrased it several times, this is a question of how the HP succeeds in “squeezing more out of Maximality”, a feature implicit in the concept of set.

So this discussion is not about the philosophy of mathematics! It is about the concept of set, i.e. part of the philosophy of set theory! The obvious point you ignore is that there is a philosophy of set theory, independent of what is done in mathematics, which is not interior decoration! An important task of the kind of philosophy of set theory under discussion is to understand the consequences of the Maximality feature of the set concept.

So getting back to where I thought we were before wallpaper entered the picture: I am claiming that the HP is a legitimate contribution to the philosophy of the set-concept, part of the philosophy of set theory. Of course there is more to the philosophy of set theory than just the philosophy of the set-concept, namely the philosophy of what is done in set theory as a branch of mathematics, but that is a different thing. Probably the latter is the only part of the philosophy of set theory that interests you, as it is indeed a part of the philosophy of mathematics. But to say that what I am doing is not part of the philosophy of set theory at all is just plain wrong.

I thought that we were getting somehwere. First you rejected the HP since I didn’t care whether it generated “good math” (clarification: this means “good set theory”), then when I offered to care about that, you now claim that the HP has nothing to do with the philosophy of set theory! Can I at least get you to agree that this is indeed a philosophical programme concerned with the concept of set whose specific aim is to understand the consequences of the Maximality of the set concept? If we get that far then we can focus the debate on whether my approach to “squeezing more out of Maximality”, as you would say, is legitimate, interesting, justified, worth exploring, whatever … ?

(By the way, was it important that I care whether the HP generates “good set theory”; wasn’t it enough that it does that, whether or not I care that it does that?)

Re: Paper and slides on indefiniteness of CH

Dear Pen,

Pen, it’s nice to have you back! I thought I’d lost you due to my pig-headed ban on allowing “good math” to play a role in how the HP is judged. Please don’t be too bored if I repeat my reasons for having been so pig-headed: I don’t want to be yet another set-theorist who gets all excited about his latest mathematical discoveries and then chooses to declare them as revealing for the concept of “truth”. As soon as I hint that maybe the HP could be valued for its mathematical output I risk being thrown into the same barn with your Thin Realists; no thanks! This would be a gross injustice to the programme! Just consider the following simple fact: The ingredients that I use in the HP go back to the “basics” of set theory (elementary aspects of inner and outer models, end-extensions, a bit of definability, …); they are about fundamentals and consequently “practice-independent”. Maybe you are not impressed by that but you have to at least admit that the style of my approach is radically different than what you find in, for example, what John Steel and Hugh Woodin have to say about truth.

Moreover, I calculated that it would be better to take a radical position (you are no stranger to that!) and then later soften it; moving in the other direction is much harder.

So yes, I admit it, I have softened my position! Thanks for rubbing my nose in it with your long quotes. (Just kidding, I’m not offended.) I softened it in two ways: First, I now allow the HP to be appreciated not only for its value as a theory of truth but also for the “good math” that comes out of it. I made this move mostly to try to win back your interest (it worked!) but also because, although slightly risky, this doesn’t necessarily force any change in the way the programme runs (one can still squeeze things out of Maximality whilst ignoring their mathematical consequences). So this softening was more or less painless. Second, I now confess that it may take time for you and others to assess whether my way of squeezing new consequences of Maximality through the comparison of universe-pictures is compelling. I’m convinced by it but I don’t presume that everybody else will be, at least not in short time.

But haven’t you softened your position as well? You went from:

I don’t see why anyone has reason to sign onto this project, or to care about it one way or the other, unless it reveals some mathematical interest despite you.


my strongest protest — why should mathematicians care about a program that doesn’t even aim to produce any good mathematics? — is no longer valid. And my interest revives …

You changed the word “anyone” to “mathematicians”! Of course I agree that mathematicians shouldn’t care unless there is “good math” coming out! My disappointment was with your earlier version, which implied that not even philosophers should care unless there is “good math” coming out! That I took as a denunciation of the HP on purely philosophical grounds.

OK, we can forget the protests now and I’ll hope you’ll agree that there is indeed a solid HP approach to deriving new consequences from Maximality based on radical potentialism.

So here I was intrigued by your exchange with Peter. But I blush to admit that I still haven’t grasped your answer to the flat-footed question: if there is no actual V, in width or height, what are we asking about when we ask about CH?

Finally you ask an easy question! (Your other questions were all very challenging.)

Answer to this question: We have many pictures of V. Through a process of comparison we isolate those pictures which best exhibit the feature of Maximality, the “optimal” pictures. Then we have 3 possibilities:

a. Does CH hold in all of the optimal pictures?
b. Does CH fail in all of the optimal pictures?
c. Otherwise

In Case a, we have inferred CH from Maximality, in Case b we have inferrred -CH from Maximality and in Case c we come to no definitive conclusion about CH on the basis of Maximality.


Re: Paper and slides on indefiniteness of CH

Dear Hugh,

From your message to Pen on 19 Aug:

Goal: The goal is to arrive at a single optimal criterion which best synthesises the different intrinsically-based criteria, or less ambitiously a small set of such optimal criteria. Elements of the Hyperuniverse which obey one of these optimal criteria are called “preferred universes” and first-order properties shared by all preferred universes are regarded as intrinsically-based set-theoretic truths. Although the process is dynamic and therefore the set of such truths can change the expectation is that intrinsically-based truth will stabilise over time. (I expect that more than a handful will consider this to be a legitimate notion of intrinsically-based truth.)

I will try one more time. At some point HP must identify and validate a new axiom. Otherwise HP is not a program to find new “axioms”. It is simply part of the study of the structure of countable wellfounded models no matter what the motivation of HP is.

It seems that to date HP has not done this. Suppose though that HP does eventually isolate and declare as true some new axiom. I would like to see clarified how one envisions this happens and what the force of that declaration is. For example, is the declaration simply conditioned on a better axiom not subsequently being identified which refutes it? This seems to me what you indicate in your message to Pen.

Out of LC comes the declaration “PD is true”. The force of this declaration is extreme, within LC only the inconsistency of PD can reverse it.

You ignore the dynamic aspects of set-theoretic truth and of set-theoretic practice.

As was unveiled in my discussion with Pen, there is a notion of set-theoretic truth which operates quite independently of practice, what I call “intrinsic truth”. These are properties derivable from intrinsic features of the universe of sets. Then of course there is set-theoretic practice, whose aim is to develop “good mathematics”. (Following Pen I refrain from invoking a notion of “practice-based truth”, but simply deal with practice directly.) Certain principles, like large cardinal axioms and forcing axioms, play a special role in set-theoretic practice because they are especially valuable for the development of good mathematics. “Intrinsic truth” is made more precise in my 19.August e-mail to Pen; “good mathematics” is of course left undefined, but I don’t see that as a problem.

As I see it, you and I are talking about 2 different things. In the HP my aim is to develop an understanding of intrinsic truth; your work for example with Ultimate L is aimed at uncovering ideas which are important for the development of good mathematics. It could be that you would not consider what I do in the HP to be good mathematics, and conversely I may regard some of the principles you are developing as being irrelevant to a further understanding of intrinsic truth. Furthermore, there is no a priori reason to think that what the HP regards as true will be compatible with the principles you consider of the greatest importance for the benefit of good mathematics. This is the “truce” that I declared in my 8.August e-mail to Pen.

Now as I have tried to emphasize in my current discussion of intrinsic truth, there are no definitive, ultimate declarations of the truth of new axioms; sorry to disappoint you. (Originally there were, but I changed my mind; see my 7.August e-mail to Pen.) Instead it is a dynamic exploratory process as described in the “Goal” you quoted above. In other words, I consider the search for intrinsic truth to be so subtle that at best we can only hope to “converge” towards the correct understanding, with corrections being made along the way (the process is “\Delta_2” and not “\Sigma_1“). Moreover, and this is key to the HP, I feel that the understanding of first-order truth demands non-first-order considerations, such as comparing universes within the Hyperuniverse.

You appear to regard this as a defect in the programme. But I maintain that you are wrong if you think that set-theoretic practice is very different. I quote Pen:

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.

And myself:

But what are ‘the goals of set theory’? Who chooses those? I can imagine set-theorists looking back and assessing the most important achievements of the past, but surely they will be wrong when they guess about the most important achievements of the future.

In other words I claim that understanding “good mathematics” is also a process of convergence, as for intrinsic truth. In particular claims like “large large cardinals exist” or “PD is true” are also subject to revision in the future development of the subject.

On the other hand, principles that are supported by both programmes, i.e. both intrinsically and in terms of their benefit for good mathematics, have in my view a better chance of retaining their significance for set theory. For this reason I plan to look more closely at some of what you (and John) have written to see if what you say can be made intrinsic and thereby incorporated into the HP. A special problem is the repeated use of the genericity concept which I have yet to find an intrinsic basis for.


Re: Paper and slides on indefiniteness of CH

Dear Pen,

Thanks for the info. I’m hoping to spend some more time on those videos, but in any case I will be quite interested to look at the resulting papers.

Recall that I was pointing to the blackboxing of the notion of “good mathematics”. Of course, as you clearly know, this is not the same as depth, though for most mathematicians, depth is a substantial component.

Here are my thoughts.

  1. I have never heard mathematicians talk directly about what good mathematics is in any kind of generality. Normally, they talk about the relative merits of some particular developments in some particular areas, without delving into any general considerations or criteria.
  2. 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.
  3. Just about the only thing that there is widespread agreement among mathematicians is the following.
    1. 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. There are exceptions to this, and the main exception, which will sometimes split the opinion, is whether or not the solution uses considerable machinery. This is considered an extreme plus over it being solved by extremely clever special methods. There is rationale for this, mainly that if big machines are used, then that promises further solutions to further problems more than an extremely clever special method. However, this kind of attitude is somewhat bothersome to me because it illustrates how far the mathematicians generally are from evaluated the mathematics on the basis of information content. To bring in an obvious example, I couldn’t care less in my evaluation of Goedel’s Second Incompleteness Theorem whether the proof was very easy, easy, fairly hard, hard, or extremely hard. Or for that matter whether the proof was very deep, deep, fairly deep, deep, or extremely deep. Information content is all I personally care about, but making that coherent across mathematics seems to require something like 4 below.
    2. There is a major premium paid for interactions between areas of mathematics – particularly if the interaction is unexpected.
    3. There is increasingly a premium paid to problems of a concrete nature, particularly if there is some finite computerizable component. There has long been a premium paid for stuff of clear geometric meaning.
  4. Here is what I think mathematics needs most, in terms of a practical project. It needs a thorough systematic foundational exposition. I think I know how to go about this, generally speaking, and would love to find the time I don’t have to do a chunk of this. But I haven’t seen, nor have I done, a foundational exposition of even mathematical logic – which has largely veered away from its foundational roots.
  5. For me, I take mathematics to be both a tool and an object of study. I am a foundationalist who uses and studies mathematics. But I am still interested in the dynamics of the mathematics community, even if I find many features of it rather unattractive intellectually.


Re: Paper and slides on indefiniteness of CH

Obviously Pen and Sy are sort of talking past each other, which is understandable given the abstract nature of a discussion on general strategy of research in set theory with almost no mathematical specifics presented. In order to help the discussion, I have a few suggestions.

  1. Woodin and other close associates like John Steel have definite specific proposals for “settling” CH through explicit or reasonably definite conjectures. The statements are very complex, even for most set theorists, let alone logicians generally, or philosophers.
  2. Sy has raised various kinds of objections to their proposals for “settling” CH. Sy has been offering an alternative plan for “settling” CH. Sy is claiming some substantial advantages of his plan over Woodin/Steel plans. In particular, either explicitly or by inference, Sy is claiming that his plans are comparatively straightforward.
  3. I was intrigued by the offering up of a comparatively straightforward plan to “settle” CH. So I asked for Sy to provide some account of these comparatively straightforward plans here in this forum, so people can comment on them. I was hoping that this would not only help the present discussion, but also bring in some other people to comment on the advantages and disadvantages of Woodin/Steel versus Sy et al. I don’t know why I have been ignored. Sy?

Now here are some other matters related to the discussion.

  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. My working definition during my entire career has been “mathematics with a clear foundational purpose”. I do recognize that there can be “good mathematics” that does not – at least not obviously – fit that criterion. This is a crucial issue – what is good math or good set theory – as on at least one kind of reading, almost none of it has any clear foundational purpose.
  2. Sol’s use of the phrase “mental picture” in his earlier email reminded me of some ideas that I had left up in the air for some time. I buckled down and wrote the following posting to the FOM email list. I close with a copy of the substantive part.

Harvey Friedman

I have an extended abstract on some mental pictures, which can be used to arguably justify the consistency of certain formal systems.

#84  August 11, 2014

by Harvey M. Friedman*
August 11, 2014

*This research was partially supported by the John Templeton Foundation grant ID #36297. The opinions expressed here are those of the author and do not necessarily reflect the views of the John Templeton Foundation.

Abstract. All mathematicians rely on mental pictures of structures. Some can be used to offer justifications for certain axiom systems. Here we use them to make arguable justifications ranging from Zermelo set theory to ZFC to various large cardinal hypotheses.

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,

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!