Tuesday, 9 January 2018

Robin Hanson Responds

I recently posted criticisms of Robin Hanson and Kevin Simler's excellent new social science book The Elephant in the Brain. Hanson responds here. The response is short so I will reproduce it here:

The fourth blog review was 1500 words, and is the one on a 4-rank blog, by philosopher Tristan Haze. He starts with praise:

A fantastic synthesis of subversive social scientific insight into hidden (or less apparent) motives of human behaviour, and hidden (or less apparent) functions of institutions. Just understanding these matters is an intellectual thrill, and helpful in thinking about how the world works. Furthermore – and I didn’t sufficiently appreciate this point until reading the book, … better understanding the real function of our institutions can help us improve them and prevent us from screwing them up. Lots of reform efforts, I have been convinced (especially for the case of schooling), are likely to make a hash of things due to taking orthodox views of institutions’ functions too seriously.
But as you might expect from a philosopher, he has two nits to pick regarding our exact use of words.
I want to point out what I think are two conceptual shortcomings in the book. … The authors seem to conflate the concept of common knowledge with the idea of being “out in the open” or “on the record”. … This seems wrong to me. Something may satisfy the conditions for being common knowledge, but people may still not be OK talking about it openly. … They write: ‘Common knowledge is the difference between (…) a lesbian who’s still in the closet (though everyone suspects her of being a lesbian), and one who’s open about her sexuality; between an awkward moment that everyone tries to pretend didn’t happen and one that everyone acknowledges’ (p. 55). If we stick to the proper recursive explanation of ‘common knowledge’, these claims just seem wrong.
We agree that the two concepts are in principle distinct. In practice the official definition of common knowledge almost never applies, though a related concept of common belief does often apply. But we claim that in practice a lack of common belief is the main reason for widely known things not being treated as “out in the open”. While the two concepts are not co-extensive, one is the main cause of the other. Tristan’s other nit:
Classical decision theory has it right: there’s no value in sabotaging yourself per se. The value lies in convincing other players that you’ve sabotaged yourself. (p. 67).
This fits the game of chicken example pretty well. But it doesn’t really fit the turning-your-phone-off example: what matters there is that your phone is off – it doesn’t matter if the person wanting the favour thinks that your phone malfunctioned and turned itself off, rather than you turning it off. … It doesn’t really matter how the kidnapper thinks it came about that you failed to see them – they don’t need to believe you brought the failure on yourself for the strategy to be good.
Yes, yes, in the quote above we were sloppy, and should have instead said “The value lies in convincing other players that you’ve been sabotaged.” It matters less who exactly caused you to be sabotaged.
So Hanson paints me as a nitpicky philosopher, but nevertheless takes the points. He didn't mention the second point under the second heading, about theory of mind, which I think is maybe the most important. This omission better lets him get away with painting me as a nitpicky philosopher. But I am happy to see the response, and will not be daunted in making conceptual points that in fast-and-loose mode may seem like mere nitpicks.

What may seem like mere nitpicks at the stage of airing these ideas and getting them a hearing can turn into important substantive points in the context of actually trying to develop them further and make them more robust. 

Wednesday, 3 January 2018

Two Critical Remarks on The Elephant in the Brain

UPDATE: See my response to Robin Hanson's response.

The Elephant in the Brain, the new book by Robin Hanson and Kevin Simler, is a fantastic synthesis of subversive social scientific insight into hidden (or less apparent) motives of human behaviour, and hidden (or less apparent) functions of institutions. Just understanding these matters is an intellectual thrill, and helpful in thinking about how the world works. Furthermore - and I didn't sufficiently appreciate this point until reading the book, despite being exposed to some of the ideas on Hanson's blog and elsewhere - better understanding the real function of our institutions can help us improve them and prevent us from screwing them up. Lots of reform efforts, I have been convinced (especially for the case of schooling), are likely to make a hash of things due to taking orthodox views of institutions' functions too seriously.

Without trying to summarise the book here, I want to point out what I think are two conceptual shortcomings in the book. This is friendly criticism. Straightening these confusions out will, I think, help us make the most of the insights contained in this book. Also, avoiding these errors, which may cause some to be unduly hostile, in future or revised presentations of these insights may aid in their dissemination.

I'm not sure how important the first shortcoming is. It may be fairly trifling, so I'll be quick. The second one I suspect might be more important.

1. Being Common Knowledge Confused With Being Out in the Open

One conceptual issue came up for me in Chapter 4, 'Cheating'. Here, around p. 55 - 57, the authors seem to conflate the concept of common knowledge with the idea of being "out in the open" or "on the record".

A group of people have common knowledge of P if everyone in the group knows that P, and knows that everyone in the group knows that P, and knows that everyone in the group knows that everyone in the group knows that P, and so on.

On the other hand, a bit of knowledge is on the record or out in the open if it is 'available for everyone to see and discuss openly' (p. 55). 

The authors conflate these ideas, asserting that 'Common knowledge is information that's fully "on the record," available for everyone to see and discuss openly' (p. 55). (This comes shortly after the proper recursive explanation of 'common knowledge'.)

This seems wrong to me. Something may satisfy the conditions for being common knowledge, but people may still not be OK talking about it openly. The popular notion of an open secret gets at this point (somewhat confusingly for present purposes, since here the word 'open' gets used on the other side of the distinction). Something may be widely known, indeed even commonly known in the special recursive sense, while being taboo or otherwise unavailable for free discussion.

In addition to muddying the proper recursive explanation by asserting that common knowledge is that which is on the record and out in the open, the authors give supplementary example-based explanations of 'common knowledge' which seem to pull this expression further towards being unhelpfully synonymous with 'out in the open' and 'on the record'. For instance when they write: 'Common knowledge is the difference between (...) a lesbian who's still in the closet (though everyone suspects her of being a lesbian), and one who's open about her sexuality; between an awkward moment that everyone tries to pretend didn't happen and one that everyone acknowledges' (p, 55). If we stick to the proper recursive explanation of 'common knowledge', these claims just seem wrong. There could be cases where a lesbian is not open about being a lesbian, yet the hierarchy of conditions for common knowledge is fulfilled. Likewise for the awkward moment that everyone wants swept under the rug.

2. Excessive Preconditions Posited for Adaptive 'Self-Sabotage'

The authors give fascinating, instructive explanations of how what they call 'self-sabotage' can be adaptive in some situations (pp. 66 - 67). One example they give is visibly removing and throwing out your steering wheel in a game of chicken (provided you do it first, this is a good strategy, since your opponent then knows that their only hope of avoiding collision is to turn away themselves, losing the game of chicken). Another is closing or degrading a line of communication, e.g. turning your phone off when you think you might be asked a favour you don't want to grant. Another is avoiding seeing your kidnapper's face so that they don't kill you in order to prevent you identifying them to authorities. Another example is a general believing, despite contrary evidence, that they are in a good position to win a battle - while epistemically bad, this may cause the general (and in turn the troops) to be more confident and intimidating, and could even change the outcome in the general's favour.

But some of the things they then say about this sort of thing seem confused or wrong to me. The underlying problem, I think, is hasty generalisation. For instance:
Classical decision theory has it right: there's no value in sabotaging yourself per se. The value lies in convincing other players that you've sabotaged yourself. (p. 67).
This fits the game of chicken example pretty well.

But it doesn't really fit the turning-your-phone-off example: what matters there is that your phone is off - it doesn't matter if the person wanting the favour thinks that your phone malfunctioned and turned itself off, rather than you turning it off. Indeed having them think the former thing may be even better. But still, it might be right in this case that it's important that the person calling believes that you were uncontactable. If you have your phone off but they somehow nevertheless believe they succeeded in speaking to you and asking the favour, you may not have gained anything by turning it off.

It similarly doesn't fit the example of the kidnapper. It doesn't really matter how the kidnapper thinks it came about that you failed to see them - they don't need to believe you brought the failure on yourself for the strategy to be good. But still, it seems right in this case that it's important that they believe you didn't see their face.

Now it really doesn't fit the example of the general, and here the failure of fit is worse than in the previous two cases. If the point is that the epistemically dodgy belief of the general makes them more confident and intimidating, potentially causing them to win, then it doesn't matter how the general got the belief. The "sabotage" could just as well be due to an elaborate ruse carried out by a small cadre of the general's subordinates. And here there's not even a 'but still' of the sort in the two previous cases. The general's epistemically dodgy belief does not have to be known to be epistemically dodgy by the enemy in order for it to intimidate them and cause them to lose. Indeed, that would undermine the effectiveness of the strategy!

So, things are not as simple as the above quote suggests. Realising this and appreciating the nuances here could pay dividends.

Another claim made about this sort of thing which may at first seem striking and insightful, but which I think does not hold up, is this:
Sabotaging yourself works only when you're playing against an opponent with a theory-of-mind (p. 68).
(Theory-of-mind is the ability to attribute mental states to oneself and others.)

This doesn't really fit the game of chicken example, or at least it doesn't fit possible cases with a similar structure. It may be that to truly have a game of chicken, you need theory-of-mind on both sides, but you could have a situation where you're up against a robotic car with no theory-of-mind, and it may still be best to throw out your steering wheel. (As to why you wouldn't just forfeit the "game of chicken": there may be (theory-of-mind-less) systems monitoring you which will bring about your death if you swerve.)

I don't think it really fits the kidnapper case in a deep way. It may be a contingent fact that this sort of thing only works in our world with kidnappers with theory-of-mind, but one can easily imagine theory-of-mind-less animals who have evolved, rather than worked out by thinking, the behaviour of killing captives when seen by them.

I think it quite clearly doesn't fit the general example. Imagine the general and their army were fighting beasts with no theory-of-mind. All that matters is that the beasts can be intimidated by the confident behaviour caused by the general's dodgy belief. No theory-of-mind in the opponent required.

This seems like more than a quibble, for going along with this mistaken overgeneralization may stop us from seeing this kind of mechanism at work in lots of situations where there is no theory-of-mind at work on the other end of the adaptive sabotage.

Monday, 11 December 2017

Contingent Examples of Term-Relative Intrinsicality?

Zylstra's work shows that, if we are going to try to analyze essence in terms of necessity and intrinsicality and deliver the goods on Fine's celebrated Socrates/{Socrates} example (Socrates does not belong essentially to {Socrates}, but {Socrates} essentially contains Socrates), we had better understand intrinsicality as term-relative, at least in the case of relations. That is, we can't just say that some relations are intrinsic and others are extrinsic and that's it - rather, some two-place relations are, so to speak, intrinsic on one side but extrinsic on the other.

But can we really explicate such a concept of intrinsicality? Or is this really just going to be the concept of essence which we end up explicating? If we can do the job, then we should get something that, when supplemented with necessity, yields the notion of essence. This suggests that we should be able to find contingent cases of such asymmetric intrinsicality. And so that now seems to be the big question, if we're wondering whether essence should be accounted for in terms of necessity and something else, or the other way around. (Or at least whether intrinsicality should be involved if we pursue the first strategy.)

Thinking about parts of things, where those things could nevertheless have had different parts, may be one way of looking. For instance, perhaps 'My laptop contains the chip C' provides such an example. If the chip is intrinsic to the laptop, then we can say that the laptop intrinsically contains the chip, but that the chip is not intrinsically inside the laptop. But the laptop could have had another chip or perhaps no chip in that place, so it does not contain the chip necessarily.

I wonder how solid and convincing this sort of example is, though, and I wonder if there are other sorts available.

Saturday, 9 December 2017

Sticking Up for 'Essence = Necessity + Intrinsicality' in the Face of Zylstra's Argument

Followup: Contingent Examples of Term-Relative Intrinsicality?

UPDATE 11/12/2017: The more I think about Zylstra's argument, the more I think I've been overly critical, and not sufficiently open to changing my views. I have moderated some of the worst excesses by editing the below a little bit. I continue to think about the lessons which we should draw from Zylstra's argument, and may come back to the matter in a future post. One thing which has just begun to bother me is that, if we try to take the lesson to show that we'd better make intrinsicality term-relative when it comes to relations, is that the stuff which comes to mind when trying to explicate the resulting notion of "intrinsicality" - I found myself thinking things like 'x bears R to y intrinsically if part of what it is to be x is to be R-related to y' - just ends up sounding like a characterisation of essence; the necessity-ish bit seems to come of its own accord. So maybe there are grounds here for serious doubt about the overall E = N + I approach to essence.

An interesting new paper by Justin Zylstra attempts to cast doubt on the project of analyzing essence in terms of necessity plus something else. As Fine famously pointed out, it is plausible that the set {Soctrates} essentially contains Socrates but that Socrates does not essentially belong to {Socrates}. Being a member of that set does not have enough to do with Socrates as he is in himself, we might say, to count as an essential property of Socrates. Nevertheless, Socrates necessarily belongs to {Socrates}; in no possible world do we find Socrates but not the set containing him.

So essential properties aren't just the necessarily-possessed properties, or so it seems. Fine makes the further proposal that we give up trying to analyze essence in terms of necessity and instead go the other way around. But others have accepted that the essential properties aren't just the necessarily-possessed ones, but sought to supplement the analysis of essence in terms of necessity. I am sympathetic to this approach, and particularly to the idea - prominently defended by Denby - that essence = necessity + intrinsicality. Let's call this the E = N + I approach.

(Denby, it is important to note, favours an account of intrinsicality on which the property of containing Soctrates is not intrinsic, but extrinsic, to {Socrates}. This leads him to push back against the prima facie plausible Finean thesis that containing Socrates is essential to {Socrates}. In my view, this was a mistake on Denby's part, and we should instead try to understand 'intrinsic' in such a way that it does come out true that the property of containing Socrates is intrinsic to {Socrates}.)

You can imagine my interest in Zylstra's paper, which is supposed to cast serious doubt on this approach. Here I want to explain why I think it does no such thing. I won't reconstruct Zylstra's detailed and technically sophisticated argument in full. To fully assess what I'm saying, in particular to verify that I speak the truth about what Zylstra does in his paper, you'd have to look at the paper.

To understand why Zylstra's argument goes as wrong as I think it does, it helps to note that he aims his criticisms more generally at any attempt to supplement a necessity-based analysis of essence so that it delivers the goods on Fine's celebrated examples, provided it is of a certain general form. He intends this form to cover the E = N + I approach. The trouble is, it is very easy to formulate a version of that approach which does not take general form in question.

The central problem with Zylstra's handling of the E = N + I approach is that he considers only Denby's version, which proceeds as if the relevant notion of intrinsicality can be treated as a sentential operator. It is intrinsic that p. But no friend of the E = N + I approach should want to do that.

The whole point of bringing in intrinsicality, I would have thought, is that it is plausibly intrinsic to {Socrates} that it contains Socrates, but not intrinsic to Socrates that he is contained by {Socrates}. But if we represent our idea of intrinsicality as a sentential operator, all we can say is:

It is intrinsic that Socrates is a member of {Socrates}.


It is intrinsic that {Socrates} contains Socrates.

or whatever.

Now, this doesn't really even make sense without explanation, but putting that aside, and assuming that such claims will either be true or be false, Zylstra is able to show that an analysis of essence in terms of necessity and this weird intrinsicality sentential operator can't deliver the goods.

But so what? This just shows that the relevant notion of intrinsicality can't be captured as a sentential operator! Indeed, in his last section, entitled 'A glimmer of hope', Zylstra suggests that instead of supplementing a necessity-based analysis of essence with a notion that can be expressed as a sentential operator, we might be able to use an operator that takes a sentence and a noun phrase and produces a sentence:
Recall that the Supplemented Necessity Analysis involved an existentially bound variable O that functions syntactically as a monadic sentential operator. But nothing prohibits us from introducing a further type of variable Xt that functions syntactically as a binary term-sentence operator. (Zylstra (forthcoming), Section 5.)
Considering as he is all analyses of the relevant, sentential-operator form, rather than just the weird instrinsicality-as-a-sentential-operator instance, he never comes back to consider that maybe the E = N + I approach should be pursued with a binary term-sentence operator. (Another reason for Zylstra's neglecting to do this, perhaps, is that it is Denby's version of the approach that Zylstra considers, and that version - ill-advisedly, as I suggested in a parenthesis near the beginning of this post - fails to deliver the intuitive Finean verdict that containing Socrates is essential to {Socrates}.) But really, that's just the natural view when you think about this. The weird sentential-operator form is just an especially bad version of the E = N + I approach which no one sympathetic to that approach should allow.

I conclude that Zylstra's new paper poses no real threat at all to the E = N + I approach to understanding essence. Rather, the lesson that the friend of the E = N + I approach should draw is that intrinsicality is not to be expressed using a monadic sentential operator. Nor will it do to think of it, in general, as something which relations possess or fail to possess tout court. A relation like the set-membership relation is, so to speak, extrinsic on Socrates’s end but intrinsic on {Socrates}’s end.

In a way, this is really just a criticism about emphasis. Rather than presenting his argument as if it were a serious threat to the E = N + I approach, and then offering a 'glimmer of hope', Zylstra should, in my view, have just presented his argument as showing something instructive about how a friend of the E = N + I should, and should not, try to formulate it.


Denby, David A. (2014). Essence and Intrinsicality. In Robert Francescotti (ed.), Companion to Intrinsic Properties. De Gruyter. pp. 87-109.Author-archived version currently available open-access at http://philpapers.org/rec/DENIAE-3.

Fine, Kit (1994). Essence and modality. Philosophical Perspectives 8:1-16.

Zylstra, Justin (forthcoming). Essence, necessity, and definition. Philosophical Studies:1-12. Currently available open-access at the author's Academia.edu page, the URL of which is currently http://vermont.academia.edu/JustinZylstra.

Thursday, 9 November 2017

Two-Dimensional Semantics and Counterfactual Invariance Deciders

For a long time I have wondered, with an uneasy feeling that there was something I couldn't see, about the relationship between two-dimensional semantics and my approach to analysing subjunctive necessity de dicto. As I flagged in the previous post, this has become even more urgent in light of my new, relational account involving the notion of a counterfactual invariance (CI) decider.

I think I've finally made a breakthrough here, and found a clear connection. There is more to say, but here it is briefly.

Recall that my account states that a proposition is necessary (i.e. necessarily true or necessarily false) iff it has a true positive counterfactual invariance (CI) decider.

(P is a positive counterfactual invariance decider for Q iff Q does not vary across genuine counterfactual scenario descriptions for which P is held true.)

A close analogue of this account can be stated in terms of two-dimensional semantics: a proposition Q is necessary iff there is a true proposition P such that for every scenario S in which P is true, Q's two-dimensional intension maps, for all W, <S, W> to the same truth-value.

And I think I can maintain, as CI deciderhood is plausibly a priori tractable and arguably a semantic matter, so too is the question whether, given some propositions P and Q, P is such that for every scenario S in which P is true, Q's two-dimensional intension maps, for all W, <S, W> to the same truth-value.

This makes clear one major way in which my analysis goes beyond the normal two-dimensional account of subjunctive necessity in terms of secondary (or C) intension - and this way can then be translated into two-dimensional terms. And looking at necessity this way, as opposed to with just the usual two-dimensional account of subjunctive necessity, gives us a finer grained picture of the role played by what Kripke called 'a priori philosophical analysis' in our knowledge of necessity. You don't have to know which scenario is actual to know that a proposition is necessary - you just need to know that you're in one of some range of scenarios such that, if they were actual, the proposition would be necessary. And such a range can be characterized by a proposition which you can know a priori to be a CI decider for the necessary proposition in question.

Monday, 6 November 2017

Old Account May Not Be False After All, But New One Still Better (and New Frontier: Relation to Two-Dimensionalism)

Last Thursday I gave a talk at Sydney University's philosophy department about Kipper's bombshell, my old account of necessity, and my new account involving counterfactual invariance deciders. I was asked many good questions and got a lot out of it.

In preparing the talk, I came to realise that I may have been too quick to assume that 'Air is airy' disproves my old account, according to which a proposition is necessarily true iff it is in the deductive closure of the set of propositions which are both true and inherently counterfactually invariant. Because 'There is nothing more to being air than being airy' is plausibly true and ICI, and it does - at least on a rich enough notion of impication - imply 'Air is airy'.

Now, if that's right, what follows? Are my new ideas about abandoning, in the analysis of necessity, the property of ICI for a relation of deciderhood, to be thrown out? I don't think so. Even if I was pushed towards them by the possibly wrong idea that my old account can't be defended from 'Air is airy', they still seem to give us an account which seems better. The old account now seems clumsy, so to speak. Maybe it can be understood in a way - with a rich notion of implication - so that it doesn't go wrong on 'Air is airy'. But this still seems like a kind of lucky break, and it's not clear to me that there aren't more threatening examples in the offing. The new account, on which a proposition is necessary iff it has a true positive counterfactual invariance decider, seems to reveal the notion's workings more faithfully, and seems less hostage to as-yet-unconsidered examples.

(Also note that, with the new account, you can use 'There's nothing more to being air than being airy' as your decider, but it seems like you can also use something like 'Air has no underlying nature' or 'Air is not a natural kind', and these do not seem to imply 'Air is airy' - they do not seem to contain that information. And since it seems you can plug these into the new account and conclude that 'Air is airy' is necessary, but cannot conclude the same on the same basis with the old account, that the new account is superior here, in enabling us to conclude necessity on a sometimes slenderer basis than we can using the old account.)

In the talk I gave, there were a number of questions and examples suggested which could look like they may disprove my account, but I was able to respond to all of them straightforwardly and to my account's credit. (With some elements of the new account, it's hard to see immediately why they're there and are as they are, but working through some examples clarifies things.) I also fielded a question (thanks to N.J.J. Smith) about how my account goes beyond what we already find in Kripke. There too I was able to give what I think is a satisfactory answer: the account isolates a plausibly a priori tractable, maybe broadly semantic, aspect to necessity. Kripke's work doesn't do this. He says a proposition is necessary if it holds in all the ways things could have been, and one of his main points is that we don't in general know a priori what these ways are. True, he also allows that we know by 'a priori philosophical analysis' (this occurs in 'Identity and Necessity') that 'Hesperus is Phosphorus' is necessarily true if true at all, but that isn't true of all examples. You might thus wonder, with respect to examples that don't work that way, what part 'a priori philosophical analysis' might play in our knowledge of their modal status. My account gives us an answer to this.

But another sort of question arose in the talk was how my account relates to two-dimensional semantics, and I was less satisfied with what I had to say on that. The true CI deciding proposition(s) in my account seem to play a role close to the role played by what world is actual in two-dimensional semantics.  I worry that some in the audience were beginning to suspect that I've just laboriously re-arrived at two-dimensionalism along a somewhat different path. (And I'm getting a bit suspicious myself.)

So, I think that now, the most pressing task is to clarify the relationship of my new account to two-dimensional semantics, rather than to defend it further from counterexample. (This has always been a background concern, even with my old account, but now it has become urgent.) The notions in my account come up in a different way, and most formulations of two-dimensionalism seem to bring up difficulties which I may be able to avoid. My account seems more minimal and focused on its topic, and thus potentially more instructive.

Such anyway is my hunch, but it remains to make this clear.

Thursday, 21 September 2017

A Dialogue on Mathematical Propositions

I wrote the following dialogue as an antidote to the dogmatism I felt myself falling into when trying to write a paper about a priori propositions. The characters A and B are present-day analytic philosophers. Roughly, A represents the part of me which wanted to write the paper I was working on, and B represents the part which made trouble for the project.

A: I've got a view about a priori propositions I'd like to discuss with you. I don't think you're going to like it.

B: Intriguing! I'll try to put up a good fight.

A: Good. Still, you won't just defend the opposite view no matter what, will you? I'm certainly going into this ready to modify my view, if not to completely relinquish it.

B: Sure. No, I won't just set myself up as an opponent debater. Let's try to give each other as much ground as our philosophical consciences allow, and see if we can agree on some things.

A: OK, great. So, here's the view: what is special about a priori propositions, which enables them to be known independently of experience, is that they have their truth values essentially. They do not reach outside themselves to get their truth values, but carry them within as part of their nature.

B: OK. Interesting use of the notion of essence. I'm used to associating views which tie a priori propositions' truth or falsity closely to meaning with more deflationary attitudes, not with philosophers who make positive use of metaphysical notions like that of essence.

A: Exactly. That's one of the exciting things about my view, I think. It brings out the fact that that sort of tight connection between meaning and truth value can be posited without embracing any problematic conventionalist or deflationary attitudes about essence or meaning.

B: I think you have a point there. A meaning-based view of a priori truth doesn't need to be deflationary or conventionalist. Still, I think it's wrong. Your view overlooks the fact that a priori propositions, or many of them at least, are about something, and we often have to inquire into that something to know them. When mathematicians discover new truths, they don't sit and try to get insight into the essences of the propositions they are wondering about. They try to get insight into the things that the propositions are about, like numbers, or sets, or graphs.

A: That is true, but does not affect what I am saying. Look, the a priori truths of mathematics either have their truth essentially, or accidentally. And if they really had to reach outside themselves for their truth, then they would only be true accidentally. And in that case it should be possible to depict those very propositions reaching out but getting the opposite truth value. But you can't even begin to imagine a situation where someone has expressed what is actually an a priori truth, but which in that situation is a false proposition. And it's not like the case of propositions whose instantiation vouchsafes their truth, like 'Language exists'. Instead, their truth is of their very essence. Now, we all agree that an a priori truth can have its actual truth value, but what would it look like for it to have the other one? The onus is on you to flesh out an answer here, and it seems to me that nothing you could say on this point would satisfy.

B: I do not dispute that I couldn't really flesh out a description of a situation where the same a priori proposition gets the opposite truth value, but I don't think I have to be able to. I can still maintain that these a priori truths do not have their truth off their own bat, due to meaning alone. The source of their truth lies in what they are about. However, unlike with empirical truths, what they are about is rigid and unmoving - necessarily the way it is. So it is no real objection that I cannot depict a situation in which their source of truth or falsity yields them a different truth value, since that is just because their source is necessarily the way it is. That doesn't make their source any less of a source.

A: So you are saying that the meanings of these a priori propositions are out there in a rigid, unmoving space of possible meanings, and that they get their truth or falsity from an equally rigid, unmoving space of mathematical objects. But since all this stuff is rigid, unmoving, and necessarily the way it is, it seems to me that your talk of sourcing is just empty talk. The very idea of sourcing seems dubious here. Granted, you may seem to have an advantage in the fact that our knowledge of these truths must have some source. But the sourcing you are talking about is all going on in Plato's Heaven. It does nothing to explain how we get the knowledge. So you might as well not posit it.

B: You are trying to cast aspersions on my talk of sourcing, but I want to suggest that what you are saying is, on examination, more dubious than what I am saying. You are no nominalist, no denier of the independent existence of mathematical objects. Right?

A: Sure. I mean, I think when people object to claims like 'Mathematical objects exist independently', they are perhaps bothered by something that really should bother them. But I do think that understood properly, such claims do make a sound and correct point.

B: OK, fine. And so, it seems to me that if you are saying that a priori truths about these objects have their truth essentially and off their own bat, you are positing a kind of harmony between the meanings and what they carry inside them on the one hand, and the mathematical objects on the other. But this harmony seems dubious. It cries out for explanation. Why should it exist? Coming around to the proper view, that the propositions are about the mathematical objects, and therefore the mathematical objects' being the way they are is the source of these propositions' truth values, the difficulty disappears.

A: I don't see how the harmony you complain about is particularly strange or objectionable. Don't parts of mathematics mirror and reflect each other in weird and wonderful ways? Since we accept that, it seems that it's not particularly costly to acknowledge that the meanings of mathematical truths are also part of this crystalline structure. Crucially, it seems less dubious than your sourcing talk - more of a piece with things we already acknowledge. And it seems to me that your view overdoes the analogy between mathematical and empirical truths, leading to confusion.

B: Do you see any positive value in your view? Or is it all about stopping that over-assimilation?

A: Well, perhaps my view helps with the problem of how we get mathematical knowledge. It seems to me an easier problem to say how we get in touch with meanings, than to say how we get in touch with things like numbers and sets. Our talk and thought instantiates meanings, I want to say, even if the meanings themselves are abstract, like numbers and sets.

B: But there are also "instantiation relationships", arguably more straightforward, between, say, numbers and piles of apples.

A: Hmm. Well, I don't know, I'll have to think more about that - but perhaps stopping the over-assimilation is enough. What value do you see in your view, anyway?

B: When I think about what is fundamentally wrong with your view, apart from my complaints about it being mysterious and ill-motivated, it seems to me that, in your effort to block the over-assimilation of mathematical and empirical propositions, you bring about another over-assimilation. Namely, between mathematical propositions which can be hard to discover the truth about, and what you might call paradigmatically analytic propositions - propositions where it really does seem that the way to know the truth about them is just to have insight into their meanings. Those propositions may perhaps be said to have their truth values essentially, since they don't seem to say anything substantial about anything, whether their subject matter be empirical or mathematical. And your view wrongly depicts substantial mathematical propositions as being like them. My view has the virtue of avoiding that over-assimilation. It may be that the over-assimilation you worry about is also a problem, but it should be combated in a different way.

A: Well, I am - or at least have been, up to having this conversation - inclined to think the corresponding thing about the over-assimilation that you are worried about. Positing a mysterious sourcing relationship between mathematical propositions and mathematical objects seems like a crude expedient. But I must acknowledge that the over-assimilation that bothers you is also a problem.

B: OK. So, it seems we can both agree that our respective views may have some power to prevent a certain over-assimilation, a different one in each case. And perhaps we can also agree that each of our respective views, when adopted, may increase the danger of falling into the over-assimilation targeted by the opposite view.

A: Hmm. I suppose we can both agree about that.

B: Now, isn't this worrying? I mean, where does it leave us? We have a question: Do mathematical propositions have their truth values essentially, intrinsically, inherently, off their own bat - or do they not? And it seems like our opposing answers have opposing strengths and opposing weaknesses. I feel the weakness of your view much more acutely, but I can't deny that your feeling that my view might be a somewhat crude expedient makes some sense as well.

A: I'm glad you're staying true to your intention of not just defending your view tooth and nail. Now it's starting to look like both our views have some merit, but that these merits crowd each other out. I am beginning to think that perhaps both our views can be said to suffer from crudeness on that score. We are both inclined to use a certain picture to ward off the over-assimilation which has most bothered us. And the pictures conflict, or at least seem to. Now, could it be that if our views were made clearer, these pictures could be seen to apply in different ways, so that there is no inconsistency in using one in its way, and the other in its way? The task then would be to clarify the difference between these two ways of using what appear to be conflicting pictures.

B: That is sounding more and more reasonable to me as a diagnosis of what's going on in this case. How Wittgensteinian! And to be honest, the Wittgensteinian-ness of this view worries me a bit, since this sort of approach, to this sort of problem, seems like it will turn many people off right away. If we are to try to resolve our difficulties this way, and if we expect the resolution to be given a fair hearing, I suppose we will also have to be careful to defend our resolution from objections which lump it together with features of Wittgenstein's views which people don't like.

A: I agree that is a worry. And it may be even worse than you are suggesting. What if the things people don't like and have turned their back on include this very power to resolve our difficulties!

B: Well, I see what you're saying. People are invested in a certain way of doing things, and in defending views of a certain type. And those ways of doing things may come naturally, at least to people with a certain background (including us), so that one slides back into them. But I think we may just have to try to give the naysayers about this method plenty of credit, and allow that there are serious problems with the sort of resolution we're talking about now. After all, why wouldn't there be? It could be that it's very promising, and still ultimately our best hope, but that there are serious difficulties with it which, in our desire to resolve our present issue, we aren't currently alive to.

A: I suppose I'm on board with what you're saying. As exciting and powerful as this approach may seem now, we must beware of coming off as if we think there's a silver bullet, a simple solution we've already got here. And I think that comes out more clearly when we come back from talking about pictures and consider the question, framed in terms of 'essence' or 'intrinsic' or what have you. Something about the idea of pictures makes us quite willing to allow different applications. Ambiguities, if you like. But it seems as though people, ourselves included, may be inclined to take a certain attitude to words like 'essence' and 'intrinsic', such that the word analogue of the move where we say 'These pictures appear to conflict, but if you look at their application, you see it's only an apparent conflict' seems less appealing. There is a feeling that with such words that for each there is a big, important, single job that they should be doing.

B: I think you're right. But again, I think you may be overplaying people's resistance. Yes, there will be people who just get turned off at the suggestion that such words should be understood as having various quite important roles to play. But probably, with many of the sort of people you have in mind, you must admit that they are willing to countenance such things as long as you keep things relatively clear and definite. I mean, if you start banging on about how complex and multifaceted it all is with these words, then yes, that will turn people off, because it sounds defeatist. It sounds like shirking hard and maybe very interesting work. But these sorts of people - and let's face it we're among them a lot of the time when we aren't just talking but trying to write papers - are quite willing to distinguish certain senses of weighty-seeming words, using little subscripts for example. So we shouldn't be too discouraged.

A: Yes, I suppose that's right. So, we should be ready to float the idea that our different pictures each having a role to play, but that just giving the picture and saying 'That's how things are' is a bit crude until we clarify and distinguish the application of the picture in each case. And we should be ready to try to take exactly this approach when it comes to our difficulties as posed in philosophical jargon, but be on guard against defeatist or wishy-washy sounding attitudes. I confess I'm worried about the extent to which this is possible. I mean, maybe once we try, we will find that the distinctions we might want to make by putting little subscripts on words like 'essence' tend to fall apart in our hands, or that possibilities multiply very quickly. But on the other hand, I must admit we haven't seriously tried yet. And maybe there is some progress to be made in that way, even if it does give out and get confusing again in a way similar to our original disagreement. So we should keep working on this.

B: Agreed.

A: I think I'm pretty worn out for now, though. And I suspect there are further problems with your view that I haven't brought out.

B: Same here, on both counts.

A: I hope we can find what it takes to continue soon.

B: So do I.