[FOM] Pluralistic Foundational Crisis?/set theory/completed

Fri Apr 22 17:50:53 EDT 2016

Dear Toby,

Thanks for your very rich and interesting mail. I'm really looking forward
to the symposium!

There's lots to talk about, but I have one quibble:

My anecdotally derived impression, is that the diehard (one-universe) view
> is the mainstream one.

You're clear that this is anecdotally derived, but I just don't get this
sense at all. In my experience, for many (philosophically interested) set
theorists I've spoken to, philosophical view and set-theoretic practice are
inextricably bound together. It seems to me that there's a lot of people
working in different frameworks, with no-one view having a strong majority.

Since that's also anecdotal, I made a (one question!) survey to
(non-rigorously) test this. You can answer it here:

https://www.surveymonkey.com/r/7K6H52M

It comprises one question:

1. Is there a unique, maximal, proper class universe of set theory?

Available answers are:

A1 *Yes.*

A2 *No.* There are maximal universes, but they are incompatible (our
set-concept bifurcates).

A3 *No.* There are many universes extending each other in height. The
powerset operation is determinate, and there is an unbounded sequence of
universes, each of which is a rank initial segment of the next.

A4 *No.* There are many universes extending each other in width. Privileged
universes contain all the ordinals, but the powerset operation in not
determinate.

A5 *No.* There are many universes, and any universe can be extended in both
height and width. The (set-theoretic representatives of) natural numbers
are determinate though.

A6 *No.* Any first-order model of some set theory is a universe as
legitimate as any other.

A7 I am *strongly agnostic* about this question.

A8 I think this is a *bad question* (say because one thinks that set theory
is just a bad foundation, or the issue is intractable).

A9 Other (please specify)

I think it would be interesting, at least from a sociological perspective,
to see the spread of responses from within the FOM community to this
question. After a week, I'll record the results and try a more `open'
version on social media and soforth.

All Best,

Neil

On 22 April 2016 at 06:16, Toby Meadows <toby.meadows at gmail.com> wrote:

> Dear FOMers,
>
> I wanted to respond to some of Professor Friedman's insightful comments.
> Sorry about the delay. I've just moved to Australia from the UK and I'm
> still catching up with a lot of things.
>
> I'm going to attempt to use gmail's quote-embedding to respond. Moreover,
> I'll delete those parts of Friedman's comments that I'm not responding to
> (in this email). This will be the third layer, but if there are any clarity
> issues, I'll happily give it a tidy later.
>
>
>> > Set theory is in the throes of a foundational crisis, the results of
>> > which may radically alter our understanding of the infinite and
>> > mathematics as a whole.
>>
>> It is doubtful if the specific "pluralistic crisis" being discussed
>> with alter our understanding of the kind of mathematics that dominates
>> mathematical practice for decades, and is expected to dominate
>> mathematical practice for the foreseeable future. Working
>> mathematicians have a very limited connection to set theory, using
>> just enough set theory to provide a convenient underpinning for their
>> real mathematical interests. When the set theory takes a life of its
>> own and ceases to be convenient for their intended mathematical
>> purposes, they seek to avoid it. We obviously cannot talk about
>> everybody, but rather overwhelming dominating modus operandi.
>>
>> Nevertheless, if one adopts the viewpoint that the most, or at least
>> very general concepts of set are of intrinsic importance and worth
>> investigating in their own right, then one can come to be concerned
>> with this "pluralistic crisis" even if it is intrinsically completely
>> divorced from and totally independent from mathematical practice.
>>
>> In fact, there is a different kind of "foundational conundrum" in set
>> theory which does promise to have a substantial and potentially
>> enormous impact on mathematics as a whole. I don't think that those
>> who talk about Set Theoretic Pluralism here and elsewhere have this
>> different kind of "foundational conundrum" in mind, although one could
>> merge the two under some wider banner. I will say a bit more about
>> this later on here.
>>
>>
> I think Friedman's response here is quite charitable. My opening line is
> a little over the top and I think Friedman is right to point out that
> whatever crisis is brought upon set theory via multiverse interpretations,
> that crisis is unlikely to have too much effect on everyday mathematical
> activity.
>
> That said, if one takes set theory seriously in a foundational sense, then
> questions about how its language should be interpreted (e.g., multiverse
> versus universe) seem obviously important. Moreover and depending on how
> one understands the role of a mathematical foundation, the answers to such
> questions could impact our understanding of mathematics as a whole, albeit
> from a more philosophical (or foundational) perspective than that usually
> taken up by practising mathematicians.
>
>
>>
>> > On
>> > the second, however, significant problems have emerged. The most
>> > dramatic example of this is the continuum hypothesis (CH). While the
>> > large cardinal programme initially appeared to promise a means of
>> > solving these kinds of problems, it is now well-known that CH is
>> > independent of anything we could foreseeably think of as a large
>> > cardinal assumption.
>>
>> It would be interesting to see a suitably general transparent notion
>> of "large cardinal axiom" for which we can establish that no such can
>> prove or refute CH over ZFC.
>>
>>
> I suppose that the closest we get here is that large, large cardinals
> tend to be nicely represented by objects like ultrafilters and extenders.
> Using these representations, it can then be shown that their distinctive
> properties are preserved under small forcings.
>
> This is hardly mathematical precision, although one could sharpen this
> kind of thing to obtain precise definitions, which are incomplete in the
> sense that they leave some large cardinals out.
>
> I suppose the interesting question here is with regard to what kind of
> problem this is. Are we seeing a failure to come up with the correct
> analysis of large cardinal when there is a correct analysis to be had? Or
> is the lack of a definitive analysis a kind of symptom for any programme
> trying to demarcate ever higher reaches of the indefinitely extending
> hierarchy of consistency strengths?
>
>
>
>> But CH is in the family of statements whose abstract set theoretic
>> intensity is way way way higher than what the working mathematician is
>> focused on or feels that they really need.
>
>
> That sounds right to me in one sense and not in another.
>
> It is surely correct that CH, as a Sigma^2_1 statement is
> stratospherically out of the field of concern of working mathematicians.
>
> But from a more "conceptual" (for want of a better word) perspective, CH
> seems like a perfectly reasonable mathematical question. One can explain
> Cantor's theorem to any mathematician within a few minutes. The next - and
> very natural (it seems to me) - question to ask is: are there any
> cardinalities strictly between that of the natural numbers and that of the
> reals. It seems like a combinatorial problem.
>
> Perhaps this "conceptual" perspective is illusory. Perhaps CH is
> essentially tangled in metamathematics and set theoretic frivolity.
>
> I also note that my talk of a "conceptual" perspective is somewhat
> tangential to Friedman's remark and as such, it's not really a
> counterpoint. It just seemed apropos.
>
>
>
>> See
>> http://www.cs.nyu.edu/pipermail/fom/2016-March/019584.html So even if
>> the large cardinal hypotheses were to settle CH, this would not have
>> been a compelling reason for the mathematical community to enlarge the
>> usual ZFC foundations with large cardinal hypotheses.
>>
>>
> Yes, that sounds right to me. That said, it would have been - at the
> least - an intriguing piece of evidence in its favour.
>
>
>> > In the last few years and in response to these epistemic challenges a
>> > number of new perspectives on set theory have emerged which attempt to
>> > engage with these problems by avoiding the fixed ontology of the
>> > cumulative hierarchy and replacing it with a plurality of universes.
>> > For multiverse approaches, a problem like CH is treated as misleading
>> > way of asking which universe we happen to be working in.
>>
>> However, it should be noted that this does not seem to apply to
>> problems unlike CH in that they lie within the realm of concrete
>> mathematics, of the kind that mathematicians are focused on. In
>> particular, not for sentences of arithmetic or sentences of low levels
>> of the analytic hierarchy.
>>
>>
> I think one might see Steel's version of the generic multiverse playing
> an interesting part here. By considering a multiverse closed (loosely
> speaking) under generic extension and refinement, we are able to see CH as
> a non-concrete question. Depending on how much set theory we use to govern
> each universe, we can fix more questions as being concrete. For example, if
> we demand that each universe contain unboundedly many measurable cardinals,
> then the Sigma^1_3 questions will be concrete in the sense that their truth
> value is constant across all universes.
>
> This leaves open the question of how much set theory one should take up,
> but from a foundational perspective, I think it is interesting to develop a
> system that discards as meaningless many of those questions which are
> beyond the reach of ordinary mathematics.
>
> (I'd like to stress that this is an extremely brief and crude overview of
> Steel's view but hopefully the broad strokes paint a little of the picture.)
>
> Perhaps there are other means of achieving this effect. That would be very
> interesting to me.
>
>
>
>> > For example,
>> > Joel Hamkins has proposed that set theory should be construed in
>> > better faith with its practice.
>>
>> Perhaps foundations of set theory should be construed in better faith
>> with mathematical practice?
>>
>
> Sure.
>
>
>>
>> > In accord with contemporary set
>> > theory's fascination with models, Hamkins suggests that the models
>> > themselves should be added its ontology (Hamkins, 2012).
>>
>> This reaction to different models satisfying different major set
>> theoretic statements is to accept and study different models, while
>> generally limiting any value judgments as to the appropriateness of
>> these models.
>>
>> But there is another reaction to different models satisfying different
>> major set theoretic statements. This is to focus on one particular
>> model. A variant would be to focus on a limited group of models.
>>
>> We can go around in circles and say that we should pick (V,epsilon) as
>> the one particular model. However, the current view based on
>> experience is that this model is underdetermined. Of course, the
>> diehards will in fact take the position that there is exactly one
>> (V,epsilon) by definition. And therefore there is only one truth value
>> of CH in that model. We just haven't yet figured out what it is.
>>
>> Of course, the pluralistic view is that (V,epsilon) is in fact
>> underdetermined, and is not really a single model. So let's take this
>> view.
>>
>
> My anecdotally derived impression, is that the diehard (one-universe)
> view is the mainstream one.
>
>
>>
>> An obvious move is to focus on one particular model, and if we are
>> going to focus on one particular model, the most obvious focus would
>> be on (L,epsilon).
>> 
>>
>> Of course (L,epsilon) has been a particularly unpopular model
>> especially from those inclined to think that there is only one
>> (V,epsilon). Nevertheless, it has some tremendous advantages.
>>
>> First and foremost, almost every single one of the known natural set
>> theoretically intense statements about V(omega + omega) and many
>> others, have a known truth value in (L,epsilon).
>>
>> So on purely mathematical set theory grounds, (L,epsilon) solves all
>> of the issues, period. So why is (L,epsilon) such an unpopular choice
>> of model to focus on?
>>
>> The usual reason given is that in (L,epsilon), there are no measurable
>> cardinals (and somewhat weaker). Taking this way down in abstraction
>> level, down to the relatively concrete, but still way way higher than
>> the overwhelming focus of mathematical practice, (L,epsilon) satisfies
>> the negation of some "good" assertions. Most notably, perhaps, is "any
>> two analytic non Borel sets of reals are Borel isomorphic". This
>> pleasing assertion is known to be false in (L,epsilon), but is
>> provable using the existence of a measurable cardinal.
>>
>> I'm not convinced of the strength of this argument rejecting a focus
>> on (L,epsilon). It has too much the flavor of "gee, I'll lose all this
>> beautiful set theory that I've grown up with".
>>
>
> In "Does Mathematics need New Axioms?" (
> http://projecteuclid.org/euclid.bsl/1182353727), Steel proposes a more
> nuanced argument against V=L. Essentially (and again, very briefly and
> crudely), he proposes a principle of maximising interpretability. This
> isn't quite a mathematically precise definition in that for a theory T to
> be able to interpret (in this sense) another theory U, the interpretation
> must preserve "meaning". So we might think that ZFC+\exists meas can
> provide an interpretation of ZFC + V=L which preserves meaning, but we
> might discard interpretations which we concoct via diagonal arguments and
> Gödelian trickery as failing to preserve meaning. The definition is not
> sharp.
>
> Now Steel then argues that theories with greater interpretability (in this
> restricted sense) strength are better since when we use a greater theory,
> we can obtain all the results of the weaker theory by applying the
> meaning-preserving interpretation. In the case of large cardinals, we might
> use this principle to vaunt having a measurable cardinal above that of V=L.
>
> There are certainly things to worry about here, but it seems to be more
> principled than a mere desire to retain beautiful set theory.
>
>
>>
>> Another argument in favor of focusing on (L,epsilon) is that
>> (L,epsilon) is the only "tangible" model containing all
>> ordinals. In fact, it is the minimum model containing all ordinals, in
>> the appropriate sense.
>>
>> It is pretty clear that no forcing extension of (L,epsilon) is going
>> to be in any reasonable sense "tangible". That is, even if there are
>> any. If we are in (L,epsilon) then there won't be any. Some
>> interesting technical issues arise here in making clear sense of this
>> (I think mostly resolved), but the main point is solid - there are no
>> tangible forcing extensions of (L,epsilon). One can only look to
>> studying families, abandoning the very idea of focusing on one model.
>>
>> ADDED 4/9/16: However, there are tangible Boolean valued models.
>> Boolean valued models are not "official" models. I am still not clear
>> how or whether Boolean valued models enters the philosophical picture.
>> One view is that they are perfectly legitimate models as much as
>> official models are. Another view is that they are simply a very
>> convenient way of talking about a big family of non tangible models.
>>
>> Granted there is a level of tangibility around in models like
>> (L(mu),epsilon), where mu is a kappa additive measure on a measurable
>> cardinal kappa. But still there is the question of where that kappa
>> and where that measure comes from. How do we determine whether a
>> subset of kappa is to have measure 0 or have measure 1? It is my
>> impression that the attempts to deal with this issue are not
>> satisfactory. You seem to have to say that some involved process
>> actually miraculously works to get this.
>>
>
> I'd like to get a better idea of the notion of "tangibility" at play
> here.
>
> Rather than consider L(\mu), what about L[0#]. Would that count as
> tangible? What about the hierarchy of models that then lead us from L[0#]
> upward to core models?
>
>
>>
>> We an also go further and consider the notion of ordinal as
>> underdetermined just as we have considered the notion of V as
>> underdetermined. Then we arrive at the so called minimum transitive
>> model, which is a countable (L(lambda),epsilon). Enough of this
>> discussion...
>>
>> > John Steel
>> > takes the impressive impact of the large cardinal programme on
>> > descriptive set theory and turns our ordinary understanding of sets on
>> > its head. Rather than thinking of set theory as describing some
>> > pre-existing structure in which mathematics can be seen to take place,
>> > we should rather see it as a congenial scaffolding through which
>> > further concrete mathematics can be interpreted (Steel, 2012).
>>
>> This use of the word "concrete" must be distinguished from normal
>> mathematical usage. I would say that the Borel measurable world (in
>> Polish spaces) is at the outer limits of anything that would even
>> remotely be regarded as concrete from the point of view of
>> mathematical practice. But I need to take a careful look at Steel,
>> 2012 to further address Steel's viewpoint.
>>
>>
> Yes, I think the use of the word "concrete" is somewhat vexed, but I'm
> not sure this is merely a definition debate. I think what is - at least in
> part - at stake in something like Steel's programme is an effort to expand
> the ordinary mathematical usage of the term. 
>
>
>> > Finally, Friedman’s hyperuniverse programme attempts to combine
>> > features of both the universe and multiverse perspectives. By tracking
>> > first order properties of universes in multiverses constrained by
>> > natural principles, Friedman aims to discover new axiom candidates to
>> > characterise the universe of sets V.
>>
>> There was a very extensive discussion of S. Friedman's hyperuniverse
>> program on a blog some time ago, and I think that this blog has been
>> retained for retrieval by Koellner.
>>
>> The program was at least originally billed as generated by a new
>> analysis of the idea of V being "maximal". However, S. Friedman's
>> approach to this in terms of inner models is very brittle in the sense
>> that everywhere you turn, a slight change bites you in the face with
>> an inconsistency.
>>
>> To address this brittleness, I proposed that this idea of "maximality
>> of V" be revisited carefully with philosophical coherence already with
>> PA, Z2, Z, to ZFC, looking for new ideas beyond the existing
>> literature on maximality in set theory.
>>
>> At the time the ideas in this HP program looked much more like a
>> systematic study of countable models of ZFC, a perfectly respectable
>> technical program in set theory, like Hamkin's. At least that was the
>> then assessment of both Woodin and me.
>>
>> Also the jump from features of countable models of ZFC to what is or
>> should be true in V had a lot of difficulties at the time.
>>
>>
> I don't have anything to add here. The extension discussion from last
> year is well worth reading.
>
>
>
>> > Väänänen uses his dependence
>> > logic, in particular the concept of team semantics, to make sense of
>> > the multiverse idea. His starting point is general first order logic
>> > with multiverse structures and he applies this to set theory.
>>
>> I am not familiar with this approach, and would like to take a look.
>>
>
> Here is a link:
> http://www.math.helsinki.fi/logic/people/jouko.vaananen/multiverse5.pdf.
>
>
>> >
>> > Each of these pictures admits a kind of pluralistic ontology and
>> > indeterminacy into foundations. The move is controversial. Hugh Woodin
>> > has argued that the kind of generic multiverse offered by Steel
>> > reduces set theory to a species of formalism that betrays its
>> > Cantorian roots (Woodin, 2012).
>>
>> Yes, this conforms to my impression that Woodin, along with his thesis
>> advisor Solovay, are among the really prominent diehards arguing for
>> absolute reality and matters of fact for (V,epsilon).
>>
>> > Moreover, Tony Martin has offered a
>> > naïve re-working of Zermelo's categoricity argument to claim that the
>> > indeterminacy revealed by CH is of a merely epistemic nature and thus,
>> > that the metaphysical re-imaginings of Hamkins and Steel are
>> > unwarranted (Martin, 2001; Zermelo, 1976). In a related vein, a
>> > criticism of the pluralist account of foundations is given by Väänänen
>> > in his comparison of the second order logic and set theory approaches
>> > (Väänänen, 2012).
>>
>> Martin has said a long time ago that the longer we go without any
>> convincing assignment of a truth value to CH, the weaker the case for
>> naive realism, let alone Platonic reality. But Martin thinks that the
>> advances in set theory since he said this makes him more rather than
>> less optimistic. I don't share that assessment of what has transpired.
>> >
>> > Beyond the mathematical challenges involved in addressing these
>> > programmes, there are significant overlaps with recent work in
>> > mainstream analytic philosophy, particularly in metaphysics and
>> > philosophical logic.
>>
>> My general impression has been that work in metaphysics and
>> philosophical logic would not be able to come to grips with the rather
>> focused issues in foundations of set theory.
>>
>> Actually I hope that my general impression is wrong! This would open
>> up a rather exciting and possibly productive adventure.
>>
>
> I'd like to be more optimistic here. I think there are a number of
> younger philosophers who possess the requisite skills and interest for this
> adventure. Time will tell!
>
>
>>
>> However, there is one thing that I strongly believe. In order to get
>> the philosophical side of things to really engage with the foundations
>> of set theory, there has to be a very much upgraded line of
>> communication between people proving deep theorems and people doing
>> intricate philosophy. In order for this to happen, both sides will
>> have to rethink their entire approaches from first principles and
>> engage with each other.
>>
>> I cannot tell from the ensuing paragraphs to what extent this
>> interactive soul searching is taking place.
>>
>>
> I think this is right. Personally, this why I wanted to set up this
> network. I'd like to see more philosophers and mathematicians just talking
> to each other. Indeed, I should note that this network was largely inspired
> by a conference in London set up by Sean Walsh in 2011. It was
> well-subscribed by mathematicians and philosophers and, I think, a lot of
> productive relationships came out of it.
>
>
>
>> > A key problem in metaontology is Putnam’s
>> > paradox, which is a generalisation of Skolem’s paradox to language and
>> > semantics at large.
>>
>> Already right here, there is probably a kind of clash of cultures.
>> Math logicians have been trained to think that there is no paradox of
>> any kind in Skolem.
>>
>>
> I tend to the no-paradox view myself, however, I cited this topic more as
> a location where philosophers can benefit from the insights of mathematical
> logicians. For example, the starting point for this argument occurs in
> Putnam's "Models and Reality" (
> https://www.princeton.edu/~hhalvors/teaching/phi520_f2012/putnam1980.pdf),
> which contains a relatively technical argument (by philosophical standards)
> about L and omega-models. I should also stress that Putnam's paradox, while
> related to Skolem's paradox, is a little more subtle and difficult to
> escape.
>
>
>
>> > Using model theoretic techniques, it is argued
>> > that we are caught in a regress of theory augmentation whenever we
>> > seek to give a full account of the meaning of our expressions.
>>
>> Meaning that if we have a theory, and we want to reflect on it, and
>> treat the meaning of the expressions used, we must add to the theory,
>> a la Tarski undefinability of truth, etc.
>>
>>
> It's not so much a Tarskian hierarchy in the sense that the
> metalinguistic jumps are forced by a diagonal argument. Rather, we try to
> provide an extensional theory for the meaning of the terms of our language.
> Then using what is known as the permutation argument, we find that those
> terms could actually denote different things. So we expand our theory in
> attempt to clarify this. Then the permutation argument can be applied again
> ... and so on.
>
> To be honest, I'm not doing this any justice. A good source for this is
> David Lewis's "Putna's Paradox" (
> http://web.mit.edu/rvm/www/metaphysics%20reading%20group/Lewis%20-%20Putnam's%20Paradox.pdf
> ).
>
>
>
>> > Without
>> > such an account, we lose the ability to anchor our ontology to our
>> > language.
>>
>> I half understand this sentence, but would be interested in having it
>> explained.
>>
>>
> I think I'll demur and direct you back to the Lewis paper. 
>
>
>
>> > A response emerges with Lewis and has been developed by
>> > Sider, Schaffer and Williams. They argue that there is a privileged
>> > language which carves nature at it joins and that this is the goal of
>> > our best theories.
>>
>> I would like to see this in action, so I get a feel for what "carves
>> nature at it joins" means. And examples of "privileged languages".
>>
>>
> I sit on other (more Carnapian, less platonism) side of the fence to
> these views, but I can point out a couple of good places to start.
>
>    - David Lewis's "Putnam's Paradox" again! but toward the end. (
>    http://web.mit.edu/rvm/www/metaphysics%20reading%20group/Lewis%20-%20Putnam's%20Paradox.pdf
>    )
>    - Ted Sider's "Writing the Book of the World" (sample -
>    http://tedsider.org/books/wbotw_sample.pdf)
>
> I think it's fair to say that these kinds of views are big business in
> mainstream analytic philosophy's metaphysical quarters. 
>
>
>
>> > For multiverse debates, these approaches are
>> > particularly useful for the one-universe adherent.
>>
>> I'm curious what the one-universe adherents like Woodin and Solovay
>> and Koellner do with this.
>>
>>
> Perhaps I should have given this remark more thought. My inclination is
> to think of "joint carving" theories as being - more or less (much
> discussion required) - platonist in close to the Gödelian sense of that
> term.
>
> I've also been tempted to think that belief in a one-universe requires
> some kind of intense realism somewhere in the vicinity of platonism. So one
> might argue that there is one universe on the basis of the axioms of ZFC
> carving at the joins of nature in the sense of articulating truths about
> "the" membership relation.
>
> That said, there is certainly some wiggle room here. I don't see why the
> multiverse adherant couldn't be a platonist (Hamkins claims to be one); and
> I don't see why the universe adherent needs to be a platonist: perhaps the
> one-universe stance is ultimately more congenial. Nonetheless, I still get
> the feeling that realism sits more comfortably with the one-universe view;
> perhaps this is just a prejudice.
>
>
>
>> > Related work by
>> > Kennedy (2013) suggests a pluralistic approach involving generalised
>> > constructibility and more widely the concept of "formalism freeness",
>> > and its dual, the concept of the entanglement of a semantically given
>> > object with its underlying formalism.
>>
>> I am a little bit familiar with generalized constructibility, and I
>> assume that "formalism freeness" means a deliberate non commitment to
>> any fixed formalism? I would like an elaboration of what
>> "entanglement" means here, e.g., by examples.
>>
>>
> I might invite Professor Kennedy to talk about this.
>
>
>
>> > On the other hand, there has
>> > also been recent work into the identification of substantive debates.
>> > Stemming from Carnap (1956) and Ryle (1954) – and emerging more
>> > recently with Thomason (2009), Chalmers (2011) and Sider (2011), it is
>> > argued that some metaphysical debates are merely verbal. Such debates
>> > are pointless as although the parties to the debate are in conflict
>> > nothing substantive hangs on the result. With multiverse debates,
>> > these approaches provide a means of arguing that some questions are
>> > meaningless.
>>
>> Yes, there really is the crucial question of whether phrases like
>> "matters of fact", "objective reality", "one or multi universes", are
>> meaningful.
>>
>
> This is an area where I believe philosophers can add a lot to the debate.
> This is a very general issue affecting a lot of disciplines and it has
> received a lot of philosophical attention in recent years. The sources
> cited above are very good. Moreover, they attempt to provide criteria for
> assessing the substantive-ness of questions.
>
> I also believe that the sharpness of mathematical examples could help
> further understanding of these problems in philosophy. 
>
>
>> More broadly, we can demand that philosophical debates be
>> "productive". But then, what kind of "production" are we looking for,
>> or should be looking for?
>>
>> Speaking for myself, I like to make the distinction between
>> foundations and philosophy. E.g., foundations of mathematics is NOT
>> the same as philosophy of mathematics, although there are a lot of
>> interactions and common interests.
>>
>>
> I think this is important, but I'd propose looking at it from a slightly
> different perspective.
>
> I'd like to suggest that foundations of mathematics fits straddles both
> philosophy and mathematics (mostly mathematical logic).
>
> However, on the philosophical side, I see there are some reasons for
> reservation. Much activity in philosophy of mathematics has focussed on
> questions of metaphysics and epistemology. So philosophers have asked
> questions like: what is a number? and how can we know about the numbers?
> Such questions are certainly of core philosophical concern, however, I
> don't think they are of much interest to working mathematicians, nor are
> they particularly relevant to the foundations of mathematics.
>
> I think this could give one reason to think foundations aren't part of
> philosophy of mathematics. But I think this is misleading. It seems to me
> that beyond the technical "internal" questions (involving the solving of
> mathematical problems etc), the kinds of "external" questions about which
> framework is good, right, etc are clearly philosophical ... at least they
> don't look like the kinds of questions working mathematicians tend to ask.
>  
> Perhaps this will degenerate into a definition debate.
>
>
>
>> For me, "productive" means whether it furthers foundational research,
>> and I have some sort of working "definition" for what foundational
>> research is, and what ideally comes out of it.
>>
>> In general, I have not found that philosophers generally make the
>> moves that further foundational research. However, I do find that when
>> I talk to philosophers one on one, and try to explain foundational
>> research to them, things come up that I haven't thought of before, and
>> this often opens up new lines of foundational research.
>>
>>
> Perhaps you could give an example of a move that furthers foundational
> research. I suppose I suspect that I'll agree that philosophers aren't
> making those moves, but perhaps there are other concerns that philosophers
> can (are?) contributing to which have foundational value. I don't know;
> perhaps this will be an epicycle on the definition of foundations. 
>
>
>> > With regard to philosophical logic, a significant amount of recent
>> > activity has been devoted to problems of indeterminacy; in particular,
>> > problems caused by vagueness and the liar paradox.
>>
>> Of course, the whole thrust of mathematics is to avoid vagueness and
>> paradoxes. And of course, vagueness and paradoxes are firmly embedded
>> in ordinary language.
>>
>>
> Agreed, although the techniques for dealing with vagueness are frequently
> applicable to other paradoxes and situations involving indeterminacy.
>
>
>> > A prominent
>> > response to these problems is known as supervaluation. Observing that
>> > indeterminacy results where there are different possibilities none of
>> > which is determined as correct, supervaluation tells us that the
>> > determinate propositions are those which are true regardless of which
>> > possibility we select.
>>
>> Of course, the great classic thingie like this is Goedel's
>> Completeness Theorem. I recall that this has been tried with languages
>> that support directly and indirectly, self reference as an attempt to
>> deal with Liar Paradox and related paradoxes. But I never found what
>> came out of this very attractive. There is absolutely no comparison
>> between the fixes of the Russell Paradox and the fixes of the Liar
>> Paradox. Actually, I have had it on my wish list to fix the Liar
>> Paradox with the same level of clarity and robustness as has been done
>> with the Russell Paradox. I think this can be done, but seems
>> difficult.
>>
>>
> I'd like to hear more about this!
>
>
>> > In the context of the multiverse, a proposition
>> > is meaningful if it is true in every universe.
>>
>> Do you mean to say "a proposition is meaningful if it has the same
>> truth value in every universe", or "a proposition is meaningful if and
>> only if it has the same truth value in every universe"?
>>
>>
> That was sloppy of me. I should have written the latter of your
> formulations. 
>
>
>> > one of many different approaches to indeterminacy which include
>> > epistemicism, fuzzy logic, non-standard consequence relations and
>> > paraconsistency (Williamson 2008).
>>
>> I have not seen any of these things interact decently with foundations
>> of set theory - at least not yet.
>>
>>
> I'm hoping that something along at least one of these lines will emerge
> from our network.
>
>
>> > It has been observed that any
>> > approach to indeterminacy developed in one area can be generalised
>> > into an analogous response in another. This raises interesting
>> > questions about the applicability of a wider variety of techniques in
>> > philosophical logic to the multiverse.
>>
>> I would like to see what "analogous response" means here.
>>
>>
> Here's what I had in mind. When considering the Sorites paradox, one can
> consider a number of "solutions" including: strong Kleene semantics;
> supervalution; epistemicism and the logic of paradox (dialeithism).
>
> Each of these approaches has a relatively natural counterpart "solution"
> for the liar paradox. One may question just how natural the counterpart
> relation is here.
>
> My (wildly speculative) thought here was that multiverse solutions tend
> provide a supervaluation approach to indeterminacy. They discard CH since
> it is true in some valuations/worlds and false in others. One might wonder
> if other approaches to indeterminacy (from philosophical logic) could also
> be used to deal with set theoretic indeterminacy. One might ask: to what
> end? Well, perhaps a different perspective could make the indeterminacy
> easier to explain an understand (a predominantly philosophical goal).
> Regardless, the idea was a bit of a lob into open territory. Perhaps
> someone out there will have the requisite skills and background to see how
> to do this easily or, indeed, why it is a non-starter.
>
>
>
>> Harvey Friedman
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>>
>
>
>
> --
> Toby Meadows
> Lecturer, Department of Philosophy
> School of Divinity, History and Philosophy
> University of Aberdeen · Old Brewery · High Street · Aberdeen · AB24 3UB
> https://sites.google.com/site/tobymeadows/
> https://www.facebook.com/PhilUniAb
>
> Set Theoretic Pluralism: https://sites.google.com/site/pluralset/home
>
> *I am a physical object sitting in a physical world. Some of the forces of
> this physical world impinge upon my surface. Light rays strike my retinas;
> molecules bombard my eardrums and fingertips. I strike back, emanating
> concentric air waves. These take the form of a torrent of discourse about
> tables, people, molecules, light rays, retinas, air waves, prime numbers,
> infinite classes, joy and sorrow, good and evil. [Quine]*
>
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