Peter Smith's Blog, page 59

Luca Incurvati’s Conceptions of Set, 3

Well, I’m half-way through the task of writing up answers to the Exercises for Chapter 41 of ILF2 (and since I have the space for a few additional exercises, I’ll be trying to think up some more). But there is only so much excitement I can take! So let me return for a bit to reading Conceptions of Set. And by the way, do note that Luca has now commented on my first tranche of comments.

Chapter 2 is called ‘The Iterative Conception’, and really divides into two parts. The first part outlines this conception (and explains its relation to [some of] the axioms of set theory). The second critically considers whether the conception can be grounded (as some have supposed) in the thought that there is a fundamental relation of metaphysical dependence between collections and their members. More on this very interesting second part in my next posting. For now, let’s just think a bit about the iterative conception itself, mention some issues about the height and width of the cumulative hierarchy, and then say something about some set theories which tally with this conception.

Luca’s discussion starts like this:

On the iterative conception, sets are formed in stages. In the beginning we have some previously given objects, the individuals. At any finite stage, we form all possible collections of individuals and sets formed at earlier stages, and collect up the sets formed so far. After the finite stages, there is a stage, stage ω. The sets formed at stage ω are all possible collections of items formed at stages earlier than ω – that is, the items formed at stages 0, 1, 2, 3, etc. After stage ω, there are stages ω + 1, ω + 2, ω + 3, etc., each of which is obtained by forming all possible collections of items formed at the preceding stage and collecting up what came before. …

Of course, that’s exactly the usual story! But perhaps we should discern two thoughts here. There’s the core iterative idea that sets are built up in stages, and that after each stage there is another one where we can form new sets from individuals and/or the sets we have formed before. This captures an idea of indefinite extensibility, while rejecting the idea that at any stage we have formed all the sets (so we develop this thought, it looks as if we are going to avoid entangling ourselves with the familiar paradoxes). Then we have the further idea that we can iterate the set-building transfinitely; there are set-building stages indexed by limit ordinals, where we can collect together everything formed so far.

Luca of course stresses that the iterative conception itself leaves it open how far the cumulative hierarchy goes (what the ‘height’ of the universe is). But I think he is more concerned with how far into the transfinite we should go, while I would have liked him to pause longer here at the start, over the question of why we need to go into the transfinite at all. After all, it might be said, if we are allowing individuals, then a set universe where we have the natural numbers as individuals and then the finite levels of the hierarchy gives us a capacious setting in which arguably most mathematics can be carried out. So someone might ask: why commit ourselves to more, why go transfinite? But we’ll no doubt be coming back to issues of ‘height’

The iterative conception also leaves it open what exactly we are to make of forming ‘all possible collections of items’ from earlier stages. How ‘wide’ or ‘fat’ is each stage? ‘All possible’ certainly seems intended to be more generous than e.g. ‘all describable’; which is why we think the axiom of constructibility V = L gives us a cumulative hierarchy of rather anorexic stages, less than we intended, and why the axiom of choice can seem so natural. We are tempted to say: if all (banging the table, yes ALL!) sets are formed at each stage, then surely the needed choice sets are formed in particular. But as Luca nicely points out, following Boolos, that tempting thought is on second thoughts not so convincing, unless we build in another thought which is not itself part of the core iterative conception. The extra we seem to need is the combinatorial conception’s thought that “the existence of a set does not depend on the existence of a condition satisfied by all the members or of a rule for selecting them, [so] nothing seems to stand in the way of the choice sets being formed”. But again, we’ll need to come back to issues of ‘width’.

And what about the individuals at the ground level of the hierarchy? Do we need to consider set theories with urelements? Luca makes a familiar point:

From the mathematician’s perspective, starting with no individuals makes a lot of sense: mathematicians tend to be interested in structures up to isomorphism, and it is usually assumed that — no matter how complex or big a putative set of individuals might be — there will always be a corresponding set in the hierarchy of the same [size].

(Actually, Luca writes ‘order type’ rather than ‘size’; I’m not sure why.) So for many mathematical purposes we can do without individuals, and Luca proposes to typically focus his attention on pure set theories.

OK, so far so good: now turn to the question of what set theories the iterative conception might give its blessing to.

There are familiar worries about replacement and choice, so Luca shelves those for later consideration. And set aside extensionality as already underwritten by our very concept of set. Then Luca argues — in a familiar way — that the iterative conception sanctions the other axioms of Zermelo set theory Z. But he discusses other theories too: the stage theory ST of Shoenfield and Boolos; the theory Z+ which you get by replacing the Axiom of Foundation with an axiom which asserts that every set is the subset of some level of the hierarchy; and SP (a version of) Scott-Potter set theory. Luca argues, plausibly enough, that the iterative conception underwrites not only ST (which implies the axioms of Z leaving aside extensionality), but also Z+ and SP (those two theories in fact being equivalent).

Those claims are all persuasive. If I have a comment, then, it is about presentation rather than content. Luca’s Chapter One finishes with a couple of Appendices, two pages on cardinals and ordinals, Cantor/Frege/Russell vs the standard ZFC treatment, and one page on Cantor’s Theorem. Fine. But if a reader needs those explanations of some absolute basics, then I suspect they are going to need significantly more explanation here. For many a reader will only have encountered standard Zermelo Fraenkel set theory, and would surely have welcomed a less rushed treatment (or another chapter Appendix) elaborating on those neighbouring alternatives — especially given that some of these embody the iterative conception in a particularly direct and appealing way.

To be continued, with a discussion of Luca on grounding (or not grounding) the iterative conception in some idea of collections ‘depending’ on their members.

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Luca Incurvati’s Conceptions of Set, 2

We are still on Chapter 1 of Luca’s book. Sorry about taking longer than I had intended to get back to this. But I’d promised myself to get the answers to the Exercises for Chs 32 and 33 of IFL2 (on natural deduction for quantifier arguments) done and dusted. Thirty eight pages(!) of work later, they are online!

Let’s take it that the concept of set is (at least in part) characterized by Luca’s three conditions — Unity (a set is in some sense a unity, distinct from its members), Unique Decomposition (a set decomposes into its members in just one way), Extensionality.

Which leaves more to be said, no doubt. But then there are various possible views of the role of the further story we need.

Suppose, for example, that you hold that the concept of set, as pre-theoretically grasped, is governed by the following assumption: that for any coherent predicate there is a set of objects which satisfy it. Then, rapidly, we get to a classically inconsistent naive set theory. Put on hold for now the option of revising your logic as a palliative. Then you’ll want to work with a classically consistent replacement concept of set*. And the further story we need is an elaboration of this replacement concept.

Suppose alternatively that, as far as it goes, the concept of set is consistent enough. Then that leaves open a spectrum of possible views (at least I take it there is a spectrum here, though Luca highlights the endpoints). At one end, the idea will be that there is not much more to be said about the basic concept of set. We can go on, though, to sharpen the notion in a number of distinct ways, coming up with different, more refined, concepts — though it may turn out that one sharpening is particularly fruitful, mathematically speaking. [Possible model: we have a rough-and-ready concept of a computable function. This can be refined in various ways, though one direction — giving us the notion of an effectively computable function, where we abstract from considerations of computation length or storage costs, etc. — turns out to be particularly fruitful.]

At the other end of the spectrum, the idea will be that our pre-theoretical dealings with the notion of set reveal our partial grasp of a single, sharply definite, concept. So now what we need is not a sharpening/refinement/filling-in-of-the-conceptual gaps, but an analysis of this concept, a concept which we perhaps initially ‘perceive’ only through a glass darkly. [Perhaps Gödel had some such view of ‘perceiving’ mathematical concepts.]

Now, whether we want something on the sharpening/analysis spectrum or want replacement, Luca talks of this being provided (or at least a start being made) by elaborating a conception of sets — which he characterizes as a (possibly partial) answer to the question what is it to be a set, an answer which “someone could agree or disagree with … without being reasonably deemed not to possess the concept” set.

I’m happy with the spirit of all this, and with Luca’s view that to make progress on the interesting questions, we don’t really need to worry too much whether we are sharpening, analysing or replacing! But I suppose we could niggle about the letter of his discussion. A self-conscious sharpener (we might reasonably argue) isn’t saying what it is to be a set, tout court, but what it is e.g. to be a set in the iterative hierarchy (compare, a sharpener talking about computable functions isn’t saying what it is to be computable, in the one true sense, but e.g. what it is to be effectively computable). Likewise, a replacer isn’t saying what it is to be a set — nothing falls under that inconsistent concept, says he — but rather is saying what it is to be a set*, where this a concept which will actually do much of the work we want in a coherent way. Maybe Luca’s framework gets a bit procrustean here.

But as I say, I’m happy to grant the basic point: there’s a difference between outlining what anyone who counts as having the pre-theoretical concept of set needs to grasp, and going on to articulate a conception of sets in the sense of some guiding thoughts about what sets might be that can shape fully-fledged theory-construction.

In this initial chapter, Luca has something to say about three such guiding thoughts. One we have already touched on, the thought that every contentful predicate has a set as extension, which lands us with naive set theory. Luca then gives a familiar diagnosis of what goes wrong. Say a concept is (i) indefinitely extensible if, taking any set of things which fall under C, there is an operation which produces a further thing which falls under C. Say a concept C is (ii) collectivizing [Potter] or universal [Luca] if there is a set of everything that falls under the concept C. Then naive theory takes the concept of set to be both (i) indefinitely extensible and (ii) universal. And that way contradiction lies. A similar diagnosis can be given, as Luca nicely explains, for Cantor’s Paradox, the Burali-Forti Paradox and Mirimanoff’s Paradox. So we want our developed consistent set theory to allow only one of extensibility and universality. And Luca promises to discuss theories of both kinds. And now we see that one role for a conception of sets in the sense of some guiding thoughts can be (i) to indeed guide us in one direction or the other, and (ii) also give us some initial confidence that we are going to avoid falling into paradox.

Luca at the end of his chapter discusses two more guiding thoughts, what he calls the logical and combinatorial conception of sets. A logical conception treats sets as essentially associated with some predicate, concept, or property (the naive theory involves a naive version of this thought). A combinatorial conception arguably goes back to Cantor himself — and can be associated with images such as a sequence of random choices of what goes into the set. Thus Bernays writes that on this conception, “one views a set of integers as the result of infinitely many independent acts deciding for each number whether it should be included or excluded.” Which might well raise some philosophical eyebrows. Indeed there’s a long tradition that worries that standard set theory is conceived in sin, an unholy alliance between a logical conception (which gives us infinite sets but not arbitrary collections) and a combinatorial conception (which gives us finite arbitrary collections but not infinite ones, unless we are going to countenance Bernays-style supertasks — and why, a Weyl might ask, suppose that such a fairy-story even makes sense?). Luca doesn’t take the occasion to comment, though: maybe he will return to question.

To be continued: Chapter 2 on the Iterative Conception next. But not until I’ve got answers to Exercises 41 online!

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From a small corner of Cambridge, 3

Some quite beautiful spring weather in recent days, and it’s rather frustrating to be confined to home. But while our whatever-it-has-been bug still hangs around, we should do what we can not to spread it. At least we are no longer feeling dog-tired much of the day, and want to get out. Tomorrow maybe. If we survive the excitement of a Waitrose delivery later.

Decorating plans are on hold. Guess who left it too late to order a delivery of what we needed. Which is a very small example of the irrationality that seems to hit people: at some level I knew perfectly well it was foolish to delay making the order when I first thought about it, and that  spending a day or two wavering over colour choices was likely to be silly. But it is difficult to really adjust your thinking to the strange times, isn’t it? — so waver we did, and now we’ll have to wait, until perhaps I find another source for at least some of the kit we need. Still, we seem to have plenty to keep us busy. Friends and family are keeping safe. The house is very comfortable even if the paintwork isn’t perfect, and the larder is well stocked. We are extremely lucky when so very many are not. The real world beyond this very small corner seems pretty grim in places. And April could be the cruellest month.

Distractions are needed. There’s a lot of High Culture being provided free online as various opera houses open up their back catalogue of recorded performances for streaming. For instance, we  caught the Swan Lake from Paris Opera which is available until April 5.  But mostly we are in the mood for unchallenging comfort viewing. So we watched the 2009 BBC adaption of Emma again. We enjoyed it enormously (thinking it even better on a second viewing). Of course you can cavil. You can argue that no filmed version can do even a quarter justice to Austen’s knowing narration as it slips in and out of her character’s minds. You can complain that, in this Emma, Hartfield is too grand, Mr Knightly a  bit too young, Mr Elton too caricatured. But as Austen adaptations go, this is surely as true to its original as any. And Romola Garai is terrific in capturing Emma’s vivacity and youth and fundamental good-heartedness (and shortcomings!). So that is this week’s recommendation for something to distract you!

The Amazonian ontological argument: if it is listed, it does/will exist.  So I better crack on with getting more answers to exercises online …

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Illusionists of logic

My friend Hugh Mellor sent this on to me, one of the pieces in J.B. Priestly’s Delights, a book of short essays from 1949, another difficult time, mostly about simple pleasures. Or in this case, something a bit more philosophical:

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Luca Incurvati’s Conceptions of Set, 1

I’m really pleased to see that Luca Incurvati’s long-awaited Conceptions of Set and the Foundations of Mathematics has now been published by CUP. It’s currently jolly expensive. So let’s hope for an early inexpensive paperback. Happily, though, you will be able now to read an e-version of the book for free if your library has appropriate access to the Cambridge Core platform. So I’m going to assume I’m not the only one with access to the book! — and will dive in and comment slowly, chapter by chapter, over the next few weeks. I’ll be very interested (of course) to hear other readers’ reactions.

The first chapter is titled ‘Concepts and conceptions’. Not that Luca wants to suggest a sharp distinction here between concept and conception. But roughly, to characterize the concept of set is to characterize what someone has to grasp if they are to count as understanding ‘set’ (in the right way). But that characterization will leave open a lot of fundmental questions about the nature of sets and about what sorts of sets there are, about how sets relate to their members, and so on. And our answers to such questions will typically be guided by a conception of sets, which tells us something about what it is to be a set (a story which “someone could agree or disagree with though without being reasonably deemed not to possess the concept” of set). Take for example the iterative conception of sets: you don’t have to have grasped that surprisingly late arrival on the scene to count as understanding ‘set’. (I suppose that we might wonder about the understanding of someone who couldn’t see that the iterative conception, once presented, was at least a candidate for an appropriate conception of the world of sets: but reasoned rejection of the now standard conception would surely not debar you from counting as talking about sets.)

OK, so what does belong to the core concept of set as opposed to a more elaborated conception? Luca suggests three key elements:

Unity: “A set is … a single object, over and above its members.”
Unique decomposition: “A set has a unique decomposition” into its members.
Extensionality: The familiar criterion of identity for sets — sets are identical if and only if they have the same members.

By the way, in talking about members here, it isn’t (I take it) being assumed that we we can call on a prior, fully articulated, notion of membership. The notions ‘set (of)’ and ‘member (of)’ have to be elucidated in tandem — just as e.g.  ‘fusion (of)’ and ‘part (of)’ have to be elucidated in tandem (and similarly for some other pairs).

These three aspects of the concept of set distinguish it from neighbouring ideas. (1) is needed to distinguish sets from mere pluralities — it distinguishes the set of Tom, Dick and Harry from those men. (2) is needed to distinguish sets from mereological fusions which can be carved into parts in arbitrarily many ways. (3) is needed to distinguish the relation between a set and its members, and the relation between an intensionally individuated property and the objects which have the property (different properties can have the same extension).

Let’s pause though over (1). We have two sets of Trollope’s Barchester Chronicles in the house. We can distinguish the two sets of six books, and count the sets — two sets, twelve individual books. One set is particularly beautifully produced, the other was a lucky find in an Oxfam bookshop. In a thin logical sense (if we can refer to Xs, count Xs, predicate properties of Xs, then Xs are objects in this thin sense) the sets can therefore be thought of as objects. But are the sets of books objects “over and above” the books themselves? Trying that thought out on Mrs Logic Matters, she firmly thought that talking of the set (singular) is just talking of the books (plural), and balked at the thought that the set was something over and above the matching books. Does that mean she doesn’t understand talk of a ‘set of books’?

The distinction we need here is the one made by Paul Finsler as early as 1926 in a lovely quote Luca gives:

It would surely be inconvenient if one always had to speak of many things in the plural; it is much more convenient to use the singular and speak of them as a class. […] A class of things is understood as being the things themselves, while the set which contains them as its elements is a single thing, in general distinct from the things comprising it. […] Thus a set is a genuine, individual entity. By contrast, a class is singular only by virtue of linguistic usage; in actuality, it almost always signifies a plurality.

In this sense, I’d say that (as with Mrs Logic Matters and the Trollopian sets) much ordinary set talk is surely class talk, is singular talk of pluralities. Luca cheerfully claims “if I say that the set of books on my table has two elements, you [as an English speaker] understand what I am saying”, I rather suspect that the non-mathematician, non-philosopher (i) is going to find the talk of ‘elements’ really rather peculiar, and (ii) is in any case not going to be thinking of the set as something over and above the two books, there on the table.

There’s little to be gained, however, in spending more time wondering how much set talk “as it occurs in everyday parlance” (as Luca puts it) really is set talk in Finsler’s sense, as characterized by Luca’s (1). I think it is probably less than Luca thinks. But be that as it may. Let’s move on to ask: what is the cash value of the claim that a set (the real thing, not a mere class) is a “genuine [bangs the table!] individual entity” [Finsler], is “a single object, over and above its members” [Luca]?

One key thought is surely that sets of objects are themselves objects in the sense that they too, the sets, can be collected together to form more sets. Suppose someone just hasn’t grasped that sets are the sorts of thing that themselves can straightforwardly be members of sets, would we say that they have fully cottoned on to the idea of sets (in the sense we want that contrasts with Finsler’s classes)?

Let’s take that thought more slowly. Suppose we for the moment take the idea of an object in the most colourless, all-embracing, way — just to mean a single item of some type or another. Then e.g. Fregean concepts are indeed items distinct from the objects that fall under them; fixing the world, there’s a unique answer to what falls under them; and they are individuated extensionally — same extension, same Fregean Begriff. This isn’t the place to assess Frege’s theory of concepts! The point, though, is that (1) talk of a single item distinct from the plurality it subsumes, plus (2) and (3), doesn’t distinguish sets from Fregean concepts. And similarly, I think, if we are to distinguish sets from (extensionally individuated) types in the sense of type theory.

But why should we distinguish Fregean concepts or types from sets? What, apart from some rhetoric and motivational chat is the real difference? Surely, one key difference is that Fregean concepts or types are, well, typed — only certain kinds of items are even candidates for falling under a given Fregean concept, or for inhabiting a given type. Sets are, by contrast, promiscuously formed. Take any assortment of objects, as different in type as you like — the number three, the set of complex fifth roots of one, the Eiffel Tower, Beethoven’s op.131 Quartet [whatever exactly that is!] — and then there is a set of just those things. At least, so the usual story goes.

Maybe that example is a step too whacky, and you could deny that there is such a set without being deemed not to know what sets are. But still, you’ll standardly want to countenance at least e.g. a set whose members are a basic set [either empty or with some urelement], a set with that set as it member, a set with those two sets as its members, etc. The set-forming operation does not discriminate the types of such things, but cheerfully bundles them together.

Isn’t the usual idea, in short, that a set of objects (objects apt for being collected into a set) is itself an object in the sense that it is, inter alia, apt to be collected into a set — indeed, collected alongside those very objects we started from? Whereas e.g. a Fregean concept has objects falling under it, but can’t be regarded as itself another item that could fall under a concept with those same objects — that offends against Frege’s type discipline.

I suppose — well, we’ll see when we come to his discussion of the iterative conception — that Luca could treat the idea of sets being (in a sense) type promiscuous as part of a certain conception of sets, something that elaborates rather than is part of our core concept. Given neither of us think there is a sharp concept/conception distinction to be drawn anyway, it certainly wouldn’t be worth getting into a fight about this. But my feeling remains that if we don’t say something more about how (1)’s understanding of sets as objects allows them to be themselves members of sets alongside other objects, then we won’t have done enough to distinguish the concept of set from the concept of more intrinsically typed items.

To be continued (with some comments on Chapter 1’s conception of ‘conceptions‘)

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Luca Incurvati’s Conceptions of Set — 1

I’m really pleased to see that Luca Incurvati’s long-awaited Conceptions of Set and the Foundations of Mathematics has now been published by CUP. It’s currently jolly expensive. So let’s hope for an early inexpensive paperback. Happily, though, you will be able now to read an e-version of the book for free if your library has appropriate access to the Cambridge Core platform. So I’m going to assume I’m not the only one with access to the book! — and will dive in and comment slowly, chapter by chapter, over the next few weeks. I’ll be very interested (of course) to hear other readers’ reactions.

The first chapter is titled ‘Concepts and conceptions’. Not that Luca wants to suggest a sharp distinction here between concept and conception. But roughly, to characterize the concept of set is to characterize what someone has to grasp if they are to count as understanding ‘set’ (in the right way). But that characterization will leave open a lot of fundmental questions about the nature of sets and about what sorts of sets there are, about how sets relate to their members, and so on. And our answers to such questions will typically be guided by a conception of sets, which tells us something about what it is to be a set (a story which “someone could agree or disagree with though without being reasonably deemed not to possess the concept” of set). Take for example the iterative conception of sets: you don’t have to have grasped that surprisingly late arrival on the scene to count as understanding ‘set’. (I suppose that we might wonder about the understanding of someone who couldn’t see that the iterative conception, once presented, was at least a candidate for an appropriate conception of the world of sets: but reasoned rejection of the now standard conception would surely not debar you from counting as talking about sets.)

OK, so what does belong to the core concept of set as opposed to a more elaborated conception? Luca suggests three key elements:

Unity: “A set is … a single object, over and above its members.”
Unique decomposition: “A set has a unique decomposition” into its members.
Extensionality: The familiar criterion of identity for sets — sets are identical if and only if they have the same members.

By the way, in talking about members here, it isn’t (I take it) being assumed that we we can call on a prior, fully articulated, notion of membership. The notions ‘set (of)’ and ‘member (of)’ have to be elucidated in tandem — just as e.g.  ‘fusion (of)’ and ‘part (of)’ have to be elucidated in tandem (and similarly for some other pairs).

These three aspects of the concept of set distinguish it from neighbouring ideas. (1) is needed to distinguish sets from mere pluralities — it distinguishes the set of Tom, Dick and Harry from those men. (2) is needed to distinguish sets from mereological fusions which can be carved into parts in arbitrarily many ways. (3) is needed to distinguish the relation between a set and its members, and the relation between an intensionally individuated property and the objects which have the property (different properties can have the same extension).

Let’s pause though over (1). We have two sets of Trollope’s Barchester Chronicles in the house. We can distinguish the two sets of six books, and count the sets — two sets, twelve individual books. One set is particularly beautifully produced, the other was a lucky find in an Oxfam bookshop. In a thin logical sense (if we can refer to Xs, count Xs, predicate properties of Xs, then Xs are objects in this thin sense) the sets can therefore be thought of as objects. But are the sets of books objects “over and above” the books themselves? Trying that thought out on Mrs Logic Matters, she firmly thought that talking of the set (singular) is just talking of the books (plural), and balked at the thought that the set was something over and above the matching books. Does that mean she doesn’t understand talk of a ‘set of books’?

The distinction we need here is the one made by Paul Finsler as early as 1926 in a lovely quote Luca gives:

It would surely be inconvenient if one always had to speak of many things in the plural; it is much more convenient to use the singular and speak of them as a class. […] A class of things is understood as being the things themselves, while the set which contains them as its elements is a single thing, in general distinct from the things comprising it. […] Thus a set is a genuine, individual entity. By contrast, a class is singular only by virtue of linguistic usage; in actuality, it almost always signifies a plurality.

In this sense, I’d say that (as with Mrs Logic Matters and the Trollopian sets) much ordinary set talk is surely class talk, is singular talk of pluralities. Luca cheerfully claims “if I say that the set of books on my table has two elements, you [as an English speaker] understand what I am saying”, I rather suspect that the non-mathematician, non-philosopher (i) is going to find the talk of ‘elements’ really rather peculiar, and (ii) is in any case not going to be thinking of the set as something over and above the two books, there on the table.

There’s little to be gained, however, in spending more time wondering how much set talk “as it occurs in everyday parlance” (as Luca puts it) really is set talk in Finsler’s sense, as characterized by Luca’s (1). I think it is probably less than Luca thinks. But be that as it may. Let’s move on to ask: what is the cash value of the claim that a set (the real thing, not a mere class) is a “genuine [bangs the table!] individual entity” [Finsler], is “a single object, over and above its members” [Luca]?

One key thought is surely that sets of objects are themselves objects in the sense that they too, the sets, can be collected together to form more sets. Suppose someone just hasn’t grasped that sets are the sorts of thing that themselves can straightforwardly be members of sets, would we say that they have fully cottoned on to the idea of sets (in the sense we want that contrasts with Finsler’s classes)?

Let’s take that thought more slowly. Suppose we for the moment take the idea of an object in the most colourless, all-embracing, way — just to mean a single item of some type or another. Then e.g. Fregean concepts are indeed items distinct from the objects that fall under them; fixing the world, there’s a unique answer to what falls under them; and they are individuated extensionally — same extension, same Fregean Begriff. This isn’t the place to assess Frege’s theory of concepts! The point, though, is that (1) talk of a single item distinct from the plurality it subsumes, plus (2) and (3), doesn’t distinguish sets from Fregean concepts. And similarly, I think, if we are to distinguish sets from (extensionally individuated) types in the sense of type theory.

But why should we distinguish Fregean concepts or types from sets? What, apart from some rhetoric and motivational chat is the real difference? Surely, one key difference is that Fregean concepts or types are, well, typed — only certain kinds of items are even candidates for falling under a given Fregean concept, or for inhabiting a given type. Sets are, by contrast, promiscuously formed. Take any assortment of objects, as different in type as you like — the number three, the set of complex fifth roots of one, the Eiffel Tower, Beethoven’s op.131 Quartet [whatever exactly that is!] — and then there is a set of just those things. At least, so the usual story goes.

Maybe that example is a step too whacky, and you could deny that there is such a set without being deemed not to know what sets are. But still, you’ll standardly want to countenance at least e.g. a set whose members are a basic set [either empty or with some urelement], a set with that set as it member, a set with those two sets as its members, etc. The set-forming operation does not discriminate the types of such things, but cheerfully bundles them together.

Isn’t the usual idea, in short, that a set of objects (objects apt for being collected into a set) is itself an object in the sense that it is, inter alia, apt to be collected into a set — indeed, collected alongside those very objects we started from? Whereas e.g. a Fregean concept has objects falling under it, but can’t be regarded as itself another item that could fall under a concept with those same objects — that offends against Frege’s type discipline.

I suppose — well, we’ll see when we come to his discussion of the iterative conception — that Luca could treat the idea of sets being (in a sense) type promiscuous as part of a certain conception of sets, something that elaborates rather than is part of our core concept. Given neither of us think there is a sharp concept/conception distinction to be drawn anyway, it certainly wouldn’t be worth getting into a fight about this. But my feeling remains that if we don’t say something more about how (1)’s understanding of sets as objects allows them to be themselves members of sets alongside other objects, then we won’t have done enough to distinguish the concept of set from the concept of more intrinsically typed items.

To be continued (with some comments on Chapter 1’s conception of ‘conceptions‘)

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Covered in blue …

So here’s the cover for IFL2 (click for a larger version).  The painting, my choice, is Kandinsky’s Blue Painting (Blaues Bild) from January 1924. It has been slightly cropped by CUP’s designers but I do think it still makes for a good-looking cover. I’m really very pleased with the result.

I’m working away putting answers to end-of-chapter exercises online (currently I’m having fun with quantifier natural deduction).  This is actually rather a good task to have on the go right now. It is distracting enough to keep my mind off other things during a long afternoon stuck at home; but it hardly demands prolonged concentration trying to get my head round Difficult Stuff.  I’m making some very slight improvements to the exercises as I go along which can be incorporated into the final final book version when CUP call for it in a week or two. And so  on we go.

I’d like, though, to get back to thinking about category theory. Here’s one question: category theorists, or some influential ones among them at any rate, seemingly work with a non-standard conception of sets: but what is it, exactly (when you try to cash out the ‘bag of dots’ metaphor that gets trotted out)? As a warm up exercise (although I don’t think he tackles this question) I’m going to be sitting down to read carefully Luca Incurvati’s recent Conceptions of Set and the Foundations of Mathematics. If your library subscribes to the Cambridge Core system, you should be able to get it online. I’ll start posting about this book, chapter by chapter, in the next few days.

 

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From a small corner of Cambridge, 2

In the London Review of Books a couple of issues ago, there was a strangely timely review of a book on the plague in Florence in the early seventeenth century. The plague approached the city, only temporarily halted by the natural barrier of the Appenines

On the other side of the mountains, Florence braced itself. The officials of the Sanità, the city’s health board, wrote anxiously to their colleagues in Milan, Verona, Venice, in the hope that studying the patterns of contagion would help them protect their city. Reports came from Parma that its ‘inhabitants are reduced to such a state that they are jealous of those who are dead’. The Sanità learned that, in Bologna, officials had forbidden people to discuss the peste, as if they feared you could summon death with a word. Plague was thought to spread through corrupt air, on the breath of the sick or trapped in soft materials like cloth or wood, so in June 1630 the Sanità stopped the flow of commerce and implemented a cordon sanitaire across the mountain passes of the Apennines. But they soon discovered that the boundary was distressingly permeable. Peasants slipped past bored guards as they played cards. In the dog days of the summer, a chicken-seller fell ill and died in Trespiano, a village in the hills above Florence. The city teetered on the brink of calamity.

By August, Florentines were dying. The archbishop ordered the bells of all the churches in the city to be rung while men and women fell to their knees and prayed for divine intercession. In September, six hundred people were buried in pits outside the city walls. As panic mounted, rumours spread: about malicious ‘anointers’, swirling infection through holy water stoups, about a Sicilian doctor who poisoned his patients with rotten chickens. In October, the number of plague burials rose to more than a thousand. The Sanità opened lazaretti, quarantine centres for the sick and dying, commandeering dozens of monasteries and villas across the Florentine hills. In November, 2100 plague dead were buried. A general quarantine seemed the only answer. In January 1631, the Sanità ordered the majority of citizens to be locked in their homes for forty days under threat of fines and imprisonment.

And so it went (the review is gripping). And so, with some marked similarities though a thankfully much less virulent infection, it goes. The coronavirus lockdown has now reached Cambridge. Probably some days after it should have done, and probably it is still less stringent that it ought to be. The English, or far too many of them it seems, are still not conducting themselves particularly well. Some national myths are being unmade as we watch. Grim times.

The now official lockdown will, I suppose, make relatively little difference to us, at least for the moment. We’ve been ultra-cautious about going out and about in our small corner of the town for the past two weeks. We rather suspect that, like very many, we might in fact already have had the virus in a pretty mild way. Who knows? We’ll have to wait for when an antibody test becomes available, though it would be very good to know. When we feel a bit more lively, DV, time for some serious gardening (Mrs Logic Matters) and decorating.

We were much looking forward to a Wigmore Hall concert a couple of days ago with the countertenor Iestyn Davies and the lutenist Thomas Dunford. But that of course was cancelled like all London concerts. To see what we missed, there are some really nice videos of Iestyn Davies singing on his site here. And looking ahead we’ve had to cancel our usual spring stay in Cornwall, which is a sad (apart from other reasons, they are now rightly strongly asking visitors to stay away). With that missed trip I mind, I picked up again, and read from end to end, a book (not Cornish but at least West Country in theme) that we bought on one trip in the delightful indie bookshop in Falmouth — namely Alice Oswald’s Dart, that follows the river and its people from source to sea. This is a wonderfully ambitious  many-layered, many-voiced, poetic journey full of allusion and mythic echoes and observation of nature and more: no wonder it won the T.S. Eliot Prize. This week’s warm recommendation!

Just when, after lean years, independent booksellers like the Falmouth shop I mentioned seem to have been having doing rather better, they will be hit hard by the lockdown. So do support your favourite ones by giving them orders. There’s a website here listing indie shops offering an online service: book love in the time of coronavirus.

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Groups and sets

Standard algebra texts define a group to be a set of objects together with a binary operation on those objects (obeying familiar conditions) — or indeed, they define a group to be an ordered pair of a set and a suitable binary function (where ordered pairs and functions are to be treated as sets too, in familiar ways).

But what work is the conventional set talk here actually doing? — is it really doing much more than the set talk back in the days of ‘new math’? Can these sets be treated as Quinean virtual classes? Can we in fact perfectly well think of a group as some objects (plural — or at least one) equipped with a group operation, and drop the set talk in favour of common-or-garden plural locutions?

There are a number of general philosophical issues here about the commitments of various grades of set talk, and about the commitments of plural talk too. But there’s an interesting technical question: how far can we get in group theory before we have to call on substantial ideas from set theory?

Let’s sharpen that question: what issues are there that can readily be understood by a relative beginner in group theory that depend for their answers on set-theoretic matters (e.g. whose answers can depending on the ambient set theoretical principles we countenance?). It is well known, for example, that Shelah showed in 1974 that the Whitehead problem is independent of ZFC. But the beginning student is surely not going to understand the significance of that (indeed picking up a couple of heavyweight standard texts, Dummit and Foote’s Abstract Algebra and Aluffi’s Algebra: Chapter 0, neither even mention the Whitehead problem). So put aside problems at that level. To repeat, then, what student-accessible problems have answers dependent on set theoretic ideas?

One suggestion might be

“Given objects X, is there always a group (X, e, *) with those objects, and some identity e and some group operation *?”

The answer is positive if and only if the Axiom of Choice is true. Though I suppose here we might now have a serious debate about whether the Axiom of Choice is at bottom an essentially set-theoretic principle at all (after all, some plural logics have versions of choice, and then there are type-theoretic versions).

Here’s a perhaps better example. The following

“Take a group G, its automorphism group Aut(G), the automorphism group of that, i.e. Aut(Aut(G)), the automorphism group of that, i.e. Aut(Aut(Aut(G))), etc. Does this automorphism tower terminate (count it as terminating when successive groups are isomorphic)?”

Joel Hamkins has shown that the answer is yes, but the very same group can lead to towers with wildly different heights in different set theoretic universes. (Why should this be, on reflection, no great surprise? — which is not to diminish Hamkins theorem one jot! Because, as he points out, there’s a sense in which going from a group to its automorphism group is ‘going up a level’; and we can play forcing tricks as we go up the levels.)

OK that’s a lovely example. But what others are there? I asked this on math.stackexchange here and the question got a surprising number of up-votes (suggesting I’m not alone in  finding the issue interesting).

And I didn’t get any more problems which are as immediately student-accessible as the automorphism towers problem, rather re-inforcing my guess that you can get a pretty substantial way into group theory without really tangling with set theory (especially if choice principles are assigned to the logic side of the ledger rather than the intrinsically set-theoretic side). But I did get two nice suggestions a notch or two up in sophistication, which you might be interested to see (if you know some algebra).

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Dale Jacquette’s biography of Frege, again

In a posting here almost a year ago, I was rather scathing about Dale Jacquette’s book  Frege: A Philosophical Biography (surprisingly, to my mind, published by CUP in 2019). But I frankly admitted to having read only a hundred pages or so before giving up on the book. Was I being unfair, then? Did I miss much by not reading on? Or did the book continue in the same hopelessly amateurish and confused way?

It seems the latter. Here’s a review by Wolfgang Kienzler (a shorter version appeared in History and Philosophy of Logic in January this year). Kienzler was evidently no more impressed than I was.

(A nice example of Gricean implicature, from this review: “The author seems to enjoy using as many German words and expressions as possible across his text, and it must be admitted that almost all of them are spelled correctly.” Ouch.)

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