Year: 2007

Absolute Generality 10: Expanding background domains

Having tried twice and miserably failed to explain to a sullenly sceptical Hungarian girl that you don’t make an espresso macchiato by filling up the cup to the top with hot milk, I was perhaps not in the optimal mood to sit in a cafe and wrestle with the rest of Glanzberg’s paper again. Still, I tried my best.

What I was expecting, after the first three sections, was a story about how “background domains” get contextually set, a story drawing on the thoughts about what goes on in more common-or-garden contextual settings of restricted domains. Though I confess I was sceptical about how this could be pulled off — given that the business of restricting quantifications to some subdomain of everything available to be quantified over in a context and the business of (so to speak) reaching out to everything would seem to be intuitively quite different. But in fact, what Glanzberg gives us is not the whole story about background-domain-fixing (“very little has been said here about how an initial background domain is set” p. 71), but rather a story about how we might go about domain-expansion when we are (supposedly) brought to acknowledge new objects like the Russell class which cannot (on pain of paradox) already be in the domain of objects that we are currently countenancing as all that there are.

Now, Glanzberg says (p. 62) that although domain-expansion isn’t a case of setting a restricted domain, “it is still the setting of a domain of quantification … [so] it should be governed by the principles” discussed in Sec. 3 of the paper. But in fact, as we’ll see, at most one of the principles to do with common-or-garden domain fixing arguably features in the discussion here about domain-expansion (and I’d say not even that one).

What Glanzberg does discuss is the following line of thought (I’ll use a more familiar example, though he prefers to work with Williamson’s variant Russell paradox about interpretations). Suppose I’m cheerfully quantifying over everything, including sets. You then — how? invoking an All in One principle? — get me to countenance that domain as itself a something, an object, and then show me that it is one which on pain of paradox can’t be in the domain I started off with. Ok, so now this new object is a “topic of discourse”, and what is now covered by “(absolutely) everything” should now include that new object — and I suppose we could see this as an application of the same principle about domains including current topics which we mentioned as governing ordinary domain setting. (But equally, as I said before, it in fact isn’t at all clear how we should handle that supposed general principle in the case of ordinary domains. And the thought in the present case just needn’t invoke any wobbly notion of ‘topic’ but comes down to the following more basic point: if we are brought explicitly to acknowledge the existence of an object outside the previous domain of what we counted as “(absolutely) everything”, then that forces an expansion of what we must now — in our new situation — include in “(absolutely) everything”.)

So, to repeat, suppose you bring me to acknowledge e.g. a set-like object beyond those currently covered by “all sets”. I expand my domain of quantification to contain that too. But not just that too. I’ll also need to add … well, what? At least, presumably, all the other objects that I can define in terms of it, using notions that I already have. And so then what? Thinking of all these objects together with the old ones, I can — by the same move as before — take all those together as a domain, and now we have a new object again. And off we go, iterating the procedure. It is, of course, not for nothing that Dummett called this sort of expansion indefinitely extensible!

We are now in very familiar territory, but territory unconnected with the early sections of the paper. We can now ask: just how far along the ordinals should we iterate? Maybe — Glanzberg seems to be saying — it isn’t a matter of indefinite extensibility, but there is a natural limit. But I found the discussion here to be not very clear.

Where does all this leave us then? As I say, the early sections about common-or-garden domain setting in fact drop out as pretty irrelevant. If the paper has does have anything interesting to say, it is in the later sections, particularly in Sec. 6.2 about — so to speak — how indefinitely extensible indefinitely extensible concepts are. So, OK, I’ll return to have another bash at that. (Though I will grumpily add that I think the editors could have taken a firmer line in getting Glanzberg to make his arguments more accessible.)

Hartley Slater goes off-piste

B. Hartley Slater has a book out, The De-Mathematisation of Logic, collecting together some of his papers. You can in fact download it free by filling in the form here. As you’ll see just from reading the first few pages, he certainly thinks that most of us are skiing too fast down the conventional well-worn slopes, ending up in tangles of confusion. Methinks he protests too much; but still, you might find some thought provoking episodes here.

L’Eclisse

It is bad luck to return from blue skies in Milan to a miserably wet and cold Cambridge (it is one of those times when those notices in the Botanical Gardens classifying us as falling into a ‘semi-arid’ region seem a mockery). And term has ended so the faculty is almost deserted, so that’s not very cheering either. It all gives added attraction to the idea of spending a lot more of the year in Italy when I retire.

In an Italian mood, we’ve just watched L’Eclisse for the first time in very many years. It does remain quite astonishing. And what is remarkable is just how many of the images seem so very familiar, having been burnt into the memory by perhaps three viewings in the cinema decades ago. For a tiny example, there is a moment when Monica Vitti in longshot is unhappily walking home after a bad night alongside a grassy bank, carrying her handbag and perhaps a silk wrap or scarf, and she suddenly — as one might — swishes the scarf though some plants by the road. Why on earth should one remember that? Yet both of us watching did.

Absolute Generality 9: Restricting quantifiers

Section 3 of Glanzberg’s paper gives an overview of the ways in explicit and common-or-garden-contextual restrictions on quantifiers work (as background to a discussion in later sections about how “background domains” are fixed). This section isn’t intended to be more than just a quick set of reminders about familiar stuff, so we can be equally speedy here.

Going along with Glanzberg for the moment, suppose the “background domain” in a context is M (the absolutist says there is ultimately one fixed background domain containing everything: the contextualist says that M can vary from context to context). More or less explicit restrictions on quantifiers plus common-or-garden-contextual restrictions carve out from this background a subdomain D (so that Every A is B is interpreted as true just when the As in D are B, and so on). How?

Explicit restrictions are relatively unproblematic. But how is the contextual carving done? There are cases and cases. For example, there is carving by ‘anaphora on predicates from the context’, as in

1. Susan found most books which Bill needs, but few were important,

where ‘few’ is naturally heard as restricted to the books that Susan found and Bill needs. Then second, there is ‘accommodation’, where we rely perhaps on some Gricean mechanisms to read quantifiers so that claims made are sensible contributions to the conversational context. For example, as we are about to leave for the airport, I reassuringly say that, yes, I’m sure,

2. Everything is packed

when maybe some salient things (the passports, say) are in plain view in my hand and my keys are jingling in my pocket. Here, my claim is heard, and is intended to be heard, as generalizing over those things that it was appropriate to pack, or some such.

There’s a third, rather different way, in which context can constrain domain selection, that isn’t a matter of domain restriction but rather a matter of how, when an object which is already featuring prominently enough as a focal topic of discourse at a particular point, “we will expect contextually set quantifier domains to include it”. (Though I guess that this point has to be handled with a bit of care. The taxi for the airport arrives very early: we comment on it. “But,” I say, “we might as well leave in the taxi now. Everything is packed” The quantifier of course doesn’t now include the taxi, even though it is the current topic of discourse.)

Well, so far, let’s suppose, so good. But how are these reminders about common-or-garden contextual settings of domains going to help us with understanding what is going on in fixing ‘background’ domains? The story continues …

Twaddle about religion and science

The Guardian‘s Review of books on a Saturday is always worth reading, and the lead articles can be magnificent. This week, for example, it prints Doris Lessing’s Nobel prize acceptance speech. Still, the reviews do occasionally get me spluttering into my coffee.

For a shaming display of sheer intellectual incompetence, how about one Colin Tudge, reviewing today a book on the conflict or otherwise between science and religion. Here’s what he says about science:

Scientists study only those aspects of the universe that it is withintheir gift to study: what is observable; what is measurable and amenable to statistical analysis; and, indeed, what they can afford to study within the means and time available. Science thus emerges as a giant tautology, a “closed system”.

That is simply fatuous. Firstly, science isn’t and can’t be a tautology — science makes contentful checkable predictions, and no mere tautology (even in a stretched sense of “tautology”) can do that. Second, the fact the scientists are limited by their endowments (both cognitive and financial) in no way implies that science is a “closed system” on any sensible understanding of that phrase — at least in our early stage in the game, novel conjectures and new experimental techniques to test them are always possible.

But it gets worse:

Religion, by contrast, accepts the limitations of our senses and brains and posits at least the possibility that there is more going on than meets the eye – a meta-dimension that might be called transcendental.

First, the “by contrast” is utterly inept. Anyone of a naturalistic disposition accepts the limitations of our senses and brains; science indeed already tells us that there is a lot more going on than meets the eye; and it is a pretty good bet, on scientific grounds, that there is a more going than we will ever be able to get our brains around. So what? Accepting such limitations has nothing whatever to do with “meta-dimensions” (ye gods, what on earth could that even mean?); and it would be just a horribly bad pun to slide from the thought that there might be aspects of the natural world that transcend our ability to get our heads around them to the thought that a properly modest view of our own cognitive limitations means countenancing murky religious claims about the “transcendental”. Yet, we are told,

[A]theism – when you boil it down – is little more than dogma: simple denial, a refusal to take seriously the proposition that there could be more to the universe than meets the eye.

So, according to Tudge, no atheist takes a sensibly modest view about our cognitive limitations. What a daft thing to say. It is plainly entirely consistent — I don’t here say correct, but at least consistent — to hold that (a) our cognitive capacities are limited, but (b) among the things we do have very good reason to believe are that Zeus, Thor, Baal and the like are fictions, and that the Gods of contemporary believers are in the same boat. And that entirely consistent position is, for a start, the one held by the atheist philosophers I know (in fact, by most of the philosophers I know, full stop).

There’s more that is as bad in Tudge’s piece, as you can read for yourself. If this kind of utterly shabby thinking is the best that can be done on behalf of religion, then it does indeed deserve everything that Dawkins and co. throw at it.

Added later, in response to comments elsewhere If I wrote with a little heat, I was just echoing Colin Tudge, who says of Dennett and Dawkins:

On matters of theology their arguments are a disgrace: assertive without substance; demanding evidence while offering none; staggeringly unscholarly.

I think it is appropriate to point out, in similar words, that — whatever the merits or demerits of Dennett and Dawkins — Tudge’s arguments are a disgrace, assertive without substance, and staggeringly unscholarly.

Absolute Generality 8: Glanzberg on contextualism

According to Michael Glanzberg’s “Context and Unrestricted Quantification”, quantifiers always have to be understood as ranging over some contextually given domain; and paradoxes like Russell’s show that, ‘for any given context, there is a distinct context which provides a wider domain of quantification’. So he is defending ‘a contextualist version of the view that there is no absolutely unrestricted quantification’.

The aim of this paper, however, isn’t to directly defend the contextualist thesis as the best response to the paradoxes (Glanzberg has argued the case elsewhere), so much as to explore more closely how best to articulate the thesis, and in particular to explore how the idea that quantifiers always have to understood in terms of a background domain which is set contextually relates to more common-or-garden cases of quantifier domain restriction.

Consider, for example,

1. Every graduate student turned up to the party, and some undergraduates did too.
2. Everyone left before midnight.

In the first case there is are explicitly restricted quantifiers. But we of course don’t mean every graduate student in the world turned up: there is also a contextually supplied restriction to e.g. students in the Cambridge philosophy department. In the second case, context does all the restricting — e.g. to the people at the party.

So far, so familiar. But what about

3. Absolutely everything that exists, without exception, is self-identical?

Here there is no explicit restriction to a subclass of what exists; nor need there be any common-or-garden-contextual restriction of the ordinary kind. Still, Glanzberg wants to say, in any given context there is a background domain (‘the widest domain of quantification available’ in that context). This is the domain over which quantifications as in an occurrence of (3) range, when there is no explicit restriction and no common-or-garden-contextual restriction. And, the argument goes, there is a kind of contextual relativity in fixing this background domain (so, in a sense, the likes of (3) involve contextually relative although unrestricted quantifiers):

Whereas the absolutist holds there is one fixed background domain, which is simply ‘absolutely everything’, the contextualist holds that different contexts can have different background domains.

But of course the contextualist needs to say more than that: it isn’t just that different contexts might give different extensions to ‘absolutely everything’, it is also the case that there is no way of setting up a ‘maximal’ context in which our quantifiers do succeed in being maximally, unexpandably, all-embracing. For example, the contextualist must say that, even given a context of sophisticated philosophical reflection, when — in fall awareness of the issues — we essay a claim like

4. Absolutely everything that might fall under our quantifiers in any context whatsoever is self-identical,

we still somehow must fall short of our aim, because the context can be changed in a way that will expand what counts as everything. But how plausible is this? Well, we’ll have to see how the explanations develop over the rest of the paper.

Absolute Generality again

A while ago I made a start here on blogviewing Absolute Generality, edited by Augustín Rayo and Gabriel Uzquiano (OUP, 2006): but I only had a chance to comment on two papers before the chaos of term and other commitments brought things to a halt. I’m reviewing the book for the Bulletin of Symbolic Logic, so I really need to get back down to business, so I can write the ‘proper’ review over the next few weeks. Next up here, then, will be comments on the contribution by Michael Glanzberg: I’m planning to go through the pieces in roughly the order they are printed. If you want to know what I thought about the contributions by Kit Fine and Geoffrey Hellman, then — rather than trawl back through the blog — you can find what I wrote all in one place here.

Forcing myself back to logic!

Tim Chow has posted a draft “beginner’s guide to forcing“. I very much like these opening remarks:

All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves. The proofs should be ‘natural’ in Donald Newman’s sense: “This term … is introduced to mean not having any ad hoc constructions or brilliancies. A ‘natural’ proof, then, is one which proves itself, one available to the ‘common mathematician in the streets’.” I believe that it is an open exposition problem to explain forcing. Current treatments allow readers to verify the truth of the basic theorems, and to progress fairly rapidly to the point where they can use forcing to prove their own independence results …. However, in all treatments that I know of, one is left feeling that only a genius with fantastic intuition or technical virtuosity could have found the road to the final result.

Leaving aside the question of how well Tim Chow brings off his expository task — though it looks a very interesting attempt to my inexpert eyes, and I’m off to read it more carefully — I absolutely agree with him about the importance of such expository projects, giving “natural” proofs of key results in various levels of detail: these things are really difficult to do well yet are hugely worth attempting for the illumination that they bring.

Also, Philosophy of Mathematics: 5 Questions (which I’ve mentioned before as forthcoming) is now out. This is a rather different kind of exercise in standing back and trying to give an overview, with 28 philosophers and logicians giving their takes on the current state of play in philosophy of mathematics (the authors range from Jeremy Avigad, Steve Awodey and John L. Bell through to Philip Welch, Crispin Wright and Edward N. Zalta). The five questions are

  1. Why were you initially drawn to the foundations of mathematics and/or the philosophy of mathematics?
  2. What example(s) from your work (or the work of others) illustrates the use of mathematics for philosophy?
  3. What is the proper role of philosophy of mathematics in relation to logic, foundations of mathematics, the traditional core areas of mathematics, and science?
  4. What do you consider the most neglected topics and/or contributions in late 20th century philosophy of mathematics?
  5. What are the most important open problems in the philosophy of mathematics and what are the prospects for progress?

The answers should be very illuminating.

Postcard from Milan #3

Continuing to wax lyrical about the food here would quickly get very boring, so I won’t — I’ll just say that if you get a chance to eat at the Trattoria dei Cacciatori, something of a Milanese institution, in old farm buildings on the outskirts of the city, then do take it!

But enough already. It has been very good being here again, in all sorts of ways (not just gastronomic!). And in the gaps between other things, I’m having the welcome chance to idly turn over in my mind what my next work project(s) should be. For the first time in far too many years, I find myself a completely free agent. No journal to edit, no necessity to write anything to get brownie points for this or that purpose, no new courses I need to work up. The feeling of freedom is quite a novelty. Slightly disconverting, but I’m rather enjoying it. And some possible ideas are already beginning to take shape …

Postcard from Milan #2

Two of life’s mysteries. Why is it more or less impossible to get a decent cappuccino in England when any Autogrill stop on an Italian motorway can do a brilliant one? And just why is tagliatelle in butter with white  truffle shaved over it to die for?

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