Relatively recently, we've started having an occasional in-house one-day faculty colloquium, where staff and grad students give papers on their current work. I've just been asked if I'd like to talk to the next one, so I offered to chat about induction in second-order arithmetics (introducing some of the themes from this paper). But the organizer wasn't sold on the suitability of the idea: a non-expert member of the audience might get to see what the issues are, but "could not him/herself hope to contribute". So I'm off the hook.

But that response got me thinking. Once upon a time -- in my philosophical lifetime, indeed -- you could "keep up" over quite a wide front, and so dive in and intelligently discuss issues across quite a range with colleagues and visiting speakers. But really, how possible is that nowadays? Editing a journal made me vividly aware that, with almost any narrow topic, there's a now serious, sophisticated, well-developed, very clever literature out there, where the moves, counter-moves, counter-counter-moves are analysed and explored many levels deep. So I wonder if there is any way in which the non-expert can seriously hope to "contribute" off the cuff in response to a talk (unless that just means asking intelligent questions for further elucidation). The problem is obvious with the technical philosophy of maths, for example: but isn't it now actually the same pretty much right across the board? Is the conception of a wide-ranging colloquium with discussions to which the audience generally might hope to "contribute" past it sell-by date? I rather suspect so. Or do younger and more energetic philosophers feel differently?

## Sunday, August 31, 2008

### Contributing off the cuff?

Posted by Peter Smith at 4:04 PM 2 comments

## Saturday, August 30, 2008

### Honest Toil

I'm just catching up with the (fairly) new and excellent blog Honest Toil from Chris Pinnock.

Posted by Peter Smith at 8:00 PM 0 comments

### Naturalism in the Philosophy of Mathematics

Alex Paseau has a new entry in the Stanford Encyclopedia on 'Naturalism in the Philosophy of Mathematics'. If I was being picky, I'd perhaps say that student readers will find that it dives in the deep end a bit quickly. But I found Alex's useful distinction-making and rather sceptical reflections very helpful.

Posted by Peter Smith at 7:22 PM 0 comments

Labels: Logic

## Monday, August 25, 2008

### Mediocrity and bullshit

"I have never pretended to political correctness, so I can happily abhor the sanctimonious politically correct bullshit that made the British contribution to the Olympic closing ceremony so appalling. The most dull dancing imaginable, completely unsuited in scale to the ceremony, and mismatching the Royal Ballet with (wait for it) a South London Hip-Hop ensemble and a dance group featuring able and disabled dancers. The quality produced was risible - it would not have graced a county fair, let alone the Olympics.

The PR bullshit said we were "honouring diversity". No, we were honouring mediocrity, and then apparently honouring Hello magazine by introducing Leona Lewis and David Beckham. I think I should run in the 100 metres in 2012, thus honouring diversity by vastly increasing our representation of overweight and unhealthy middle-aged men."

My thoughts exactly. Or rather, Craig Murray's thoughts, but I concur wholeheartedly.

Posted by Peter Smith at 10:34 PM 2 comments

### Back to Sammartini

An old friend has just given me, burnt on to a CD, some symphonies of Sammartini, taken from an old Saga LP from the 1960s. Instantly takes me back to my student rooms in Trinity. And quite wonderful to hear again.

Saga was one of the cheap record labels of the time (another was the terrific Supraphon, from which I got to know a lot of East European and Russian music). It wasn't that cheap though, by modern standards: buying a record was still quite an event. If I'm remembering right, Saga records were about 12/6 -- that's twelve shillings and sixpence to you! -- when full-price records were about 32/6. To put things into perspective, that was then about the cost of three Penguin books, five pints of beer, or twenty five Mars bars. Compare now when Naxos CDs are cheaper than Penguins, cost not much more than two pints, or a dozen Mars bars. And of course there's about twice as much music on a classical CD compared with an LP, and they aren't instantly damaged by dodgy student turntables!

Posted by Peter Smith at 10:05 PM 0 comments

## Sunday, August 24, 2008

### MacBook Air, one month on

Anyone out there who is wavering about getting a MacBook Air might be interested in some comments from a new owner. Everyone else can, of course, just cheerfully ignore this posting! (The headline summary is: get one! -- though perhaps not quite today as there is rumour of a chip upgrade soon.)

- The portability is fantastic. No question. Just to compare: I've had a 15" Titanium PowerBook, a 15" G4 PowerBook, and a 17" MacBook Pro before; and they've of course been portable in the sense I could heave them from home to my office and back. But all of them were just too heavy/bulky to make that particularly convenient. I very rarely bothered to take them elsewhere, e.g. to a coffee shop. (You might well ask why on earth, in that case, I had portables at all! Answer: Partly because our Cambridge house is very small, my "study" is the size of a large cupboard, and I very much like to be able to work in the kitchen for a change of scene, or answer emails with a computer on my knees in the living room in the evening. And partly I wanted to be able to drive data projectors when lecturing.) Anyway, by contrast with the earlier portables, I can and do cheerfully tote the MBA (in its snug protective sleeve) anywhere, without really thinking about it, whether or not I'm definitely planning to use it. It just is so light and convenient.
- Some reviews complained about the MBA's footprint, saying that it isn't a genuine ultraportable. Well, true, the footprint isn't in fact that much smaller that the 15" machine, and I can imagine e.g. that very frequent fliers would find it a pain to use in the cramped conditions of an airline seat. But that sort of issue doesn't arise for me. The thinness means that you can carry it so comfortably in a hand, and otherwise the footprint goes with the stunningly good, uncramped, screen and the generous keyboard.
- I don't use the MBA to watch movies, or do anything else very processor-intensive. So I've never had a temperature-induced core shut-down. And the battery life seems just fine: well over three hours for writing, text-browsing, reading. Recharging though is pretty slow: but if you need to take it with you, then -- unlike the small brick for the 17" -- the MBA's charger is very small and portable (though I've bought a second one for the office, and so don't find in practice I need to carry it around).
- One main reason I traded up a couple of years ago from the 15" G4 machine was that LaTeX ran pretty slowly: nearly 30 seconds to typeset my Gödel book on the G4, about 4 seconds on the new intel MacBook Pro. The MBA, despite its slower chip, seems almost as fast running LaTeX , and indeed in most other ways: occasionally, e.g. when opening an application, the MBA is noticeably slower -- but it has never been an irritating issue. So this is plenty fast enough.
- And the reason, when I previously traded up, I chose the 17" MBP model was to have enough "real estate" to have a TeXShop editing window and the PDF output side-by-side and comfortably readable. Obviously, I'm now looking at 1280 x 800 pixels, rather than 1680 x 1050 (so that's just 58% as much). But this is manageable, and the screen quality is really terrific. Of course it is nicer e.g. for extended on-screen reading to plug in an external monitor as well. But that's a luxury, not any sort of necessity.
- What about the paucity of ports, mentioned critically by all the reviewers, or the absence of an onboard CD drive? With one caveat, I've found those features no problem at all. Just not been an issue for me since day one. (So the one caveat indeed concerns day one. Since there is no firewire port, you can't migrate files from your old computer to your new MBA using the usual firewire connection. And using a wireless connection to migrate is both painfully slow and seems flaky. Is that a problem? I didn't find really it so. I installed the necessary additional software, like the LaTeX installation, over the web, and then copied my documents folder and other bits and pieces from a SuperDuper! clone of the old hard disk on an external drive. Quick to do, and resulting in a clean and tidy MBA.)
- So that's all very, very positive. Are there any negatives? The flat keyboard is surprisingly nice to use (much better than I imagined it would be). But, unlike the almost silent similarly flat new iMac keyboards, this is a bit noisier (a bit more so than the MBP keyboard). But that's a very marginal disappointment.
- I thought, when I bought the MBA a month ago, I'd be using it very much as a second machine, carrying on using the 17" MPB (and external monitor) as a main, quasi-desktop, set-up. In fact I find myself increasingly heavily favouring the MBA. I've hardly used the MPB.
- So, assuming a three year life cycle (and it seems very well built so should last longer with a battery refresh after a while), the MBA after education discount costs much less than half a pint of beer a day. Or one modestly decent bottle of Chianti Classico a fortnight if you prefer. Put like that, how can you resist?
- And then, of course, there is the "Wow!"-factor ...

Posted by Peter Smith at 7:24 AM 2 comments

Labels: Geek stuff

## Saturday, August 23, 2008

###
Parsons's *Mathematical Thought*: Secs 24-26, Intuition

Chapter 5 of Parsons's book is called "Intuition". And I guess I should declare an interest (or rather, lack of interest!) here. I've never really understood talk about intuition: and I'm certainly not helped when Parsons writes "I shall be concerned to develop a conception of mathematical intuition that is in a general way Kantian", since Kant is pretty much a closed book to me. So perhaps I'm not the best reader for this chapter! But still, let's proceed ...

Sec. 24, "Intuition: Basic distinctions". Parsons distinguishes supposed intuition of objects from intuition that such-and-such is the case. And he stresses that in his usage, intuition that isn't factive. So is an intuition that such-and-such just a non-inferential belief? Well note, for example, that "knowledge without observation" of our own bodily movements is non-inferential, but is not normally counted as intuitive. So what differentiates intuition properly so-called? Parsons promises an answer by a "development of the concept ... in the Kantian tradition".

Sec. 25, "Intuition and perception". Now, the headline suggestion here is that "It is hard to see what could make a cognitive relation to objects [intuition of] count as intuition if not some analogy with perception" (cf. e.g. Gödel). Further, intuition that is intimately connected with intuition of, rather as perception that is grounded in perception of. Well, fair enough: but that, of course, already does make claims about intuitions of mathematical objects very puzzling. Which leads to ...

Sec. 26, "Objections to the very idea of mathematical intuition". Start with the following point. Ordinary perception is (so to speak) evident to the subject -- when I see an object, my computer screen say, "there is a phenomenological datum here". But "it is hard to maintain that the case is the same for mathematical objects ... [Are] there any experiences we can appeal to in the mathematical cases that are anywhere near as indisputed as my present experience of seeing the computer screen?" This seems to undermine any alleged analogy between "intuition of mathematical entities" and ordinary perception. So how are we to defend the analogy, given the different phenomenologies? Unfortunately, Parsons next remarks here are Kantian obscurities I can do nothing with. So I'm left stumped.

(Parsons also raises a question about the relation between structuralist thoughts and claims about intuition. The worry seems to be one about how a particular intuition can latch on to a particular object, if mathematical objects are indentified by their places in structures. The point, however, is rather rushed. But since I think Parsons is going to return to these matters, I won't say more at the moment.)

Posted by Peter Smith at 2:34 PM 0 comments

Labels: Parsons's new book

## Thursday, August 21, 2008

### Shoesmith and Smiley to be reprinted

It is announced on the CUP website that Shoesmith and Smiley's Multiple-Conclusion Logic is to be reprinted early next year. That's terrific news.

Posted by Peter Smith at 4:37 PM 0 comments

###
Parsons's *Mathematical Thought*: Secs 19-23, A problem about sets

These sections make up the short Chapter 4 of Parsons's book (they are a slightly expanded version of a 1995 paper in a festschrifft for Ruth Barcan Marcus). The issue is whether there are special problems giving a broadly structuralist account of set theory. Since the last section of Chapter 3 left me puzzled about what, exactly, Parsons counted as a structuralist view, I'm not entirely sure I have the problem in sharp focus. But I'll try to comment all the same.

It's perhaps clear enough what the problem is for the *eliminative* structuralist (whether or not he modalizes). His idea is that an ordinary mathematical claim *A* is to be read as disguising a quantified claim of the form *for all ...., if Ω(...) then A**(...), where Ω is an appropriate set of axioms for the relevant mathematical domain, *A** is a suitable formal rendering of *A*, and where the quantification is over kosher non-mathematical whatnots, and perhaps possible world indices too. This account escapes making *A* vacuously true only if Ω is satisfied somewhere (at some index). Now if Ω is suitably modest -- axioms for arithmetic say -- we might conceive of it being satisfied by some physical realization at this (or at least, at some not-too-remote) world. I'm not sure this is right because of issues about theories Ω with full second-order quantification (which Parsons himself touches on); but let that pass. For certainly, if Ω is a rich set theory, then it cetainly *doesn't* seem so plausible to say that the relevant structure is realized somewhere. Unless, that is, we allow into our possible worlds abstracta to do the job -- in which case the point of the eliminative structuralist manoeuvre is undermined. (The structuralist could just bite the bullet of course, as I remarked before, and say so much the worse for set theory. After all, what's so great about something like ZFC? -- we certainly don't need it anything as exotic to construct applicable mathematics.)

But suppose we *do* want to endorse ZFC, and remain broadly structuralist. Even if we eschew eliminativist ambitions, presumably the idea will be at least that there isn't a given unique universe of determinately identified objects, *the sets, *which set theory aims to describe. And on the face of it, this runs against the motivating stories told at the beginning of typical set theory texts, which do (it seems) purport to describe a unique universe of sets. For example, in the case of pure set theory without urelemente, take the empty set (isn't that determinately unique?); now form its singleton; now form the sets whose members are what we have already; now do that again at the next level; keep on going ... Thus iterative story is a familiar one, and seems (or so the authors of many texts apparently suppose) to fix a unique universe.

The main burden of Parsons's discussion is to argue that familiar story isn't in as good order as we might like to think. For a start, the metaphors of "forming" and "levels" don't bear the weight that is put on them: "when we come to [a set] of sufficiently high rank, it is difficult to take seriously the idea that all the intermediate sets that arise in the construction of this set ... can be formed by us". And then there are problems wrapped up in the temporal metaphor of "keeping on going", when the relevant ordinal structure we are supposed to grasp is much richer than that of time. Further, it is aguable that additional thoughts, over and above the basic conception of an iterative hierarchy, are needed to underpin all the axioms of ZFC -- that's arguably the case for replacement, and possibly even for the full powerset axiom.

I'm not going to try to assess Parsons's arguments here. The idea that the iterative story is problematic and doesn't get us everything we want is by now a familiar one; there are interesting and important discussions by George Boolos, Alex Paseau, Michael Potter and others, and I don't have anything to add. But let's suppose he is right. What then? Parsons writes that his "discussion of the arguments that are actually in the literature should make plausible that there is not a set of persuasive, direct and "intuitive" considerations in favour of the axioms of ZF that are incompatible with a structuralist conception of what talk of sets is." But that seems too sanguine. For it isn't that there are multiple lines of thought in the literature which, each taken separately, give us a conception of some structure that satisfies the ZF axioms (first or second order), indicating -- perhaps -- the kind of multiple realizability that is grist to the structuralist argument. No, the worry is that no familiar line of thought (e.g. the iterative conception, the idea of "limitation of size", not to mention the ideas shaping NF) warrants all the axioms. So it isn't, after all, clear we have an intuitive grasp of *any* structure that satisfies the axioms. Hence, the worry continues, for all we know maybe there *is* no structure that satisfies them. Which seems to take us back to vacuity worries for structuralism.

Posted by Peter Smith at 4:21 PM 0 comments

Labels: Parsons's new book

## Wednesday, August 20, 2008

###
Parsons's *Mathematical Thought*: Sec. 18, A noneliminative structuralism

The previous two sections critically discussed a modal version of eliminative structuralism (though to my mind, the objections raised weren't particularly telling). Parsons now moves on characterize his own preferred "noneliminative structuralism", and responds to some potential obections.

I wish I could give a sharp characterization of the position Parsons wants to occupy here in the longest section of his book. But I do have to confess bafflement. "We have emphasized the point going back to Bernays that reference to mathematical objects is relative to a background structure." Further, structures aren't themselves objects, and "[Parsons's] structuralist account of a particular kind of mathematical object does not view statements about that kind of object as about structures at all". But surely there's thus far nothing that e.g. the Fregean need dissent from. The Fregean can agree that numbers, for example, don't come (so to speak) independently, but come all together forming an intrinsically order structured: and in identifying the number 42 as such, we necessarily give its position in relation to other numbers. So what more is Parsons saying about (say) numbers that distinguishes his position? Well, I've read the section three times and I'm still rather lost, and won't ramble here. If any other reader of the book can offer some crisp clarifying comments, I for one would be very grateful!

Posted by Peter Smith at 3:44 PM 0 comments

Labels: Parsons's new book

###
Parsons's *Mathematical Thought*: Secs 16, 17, Modalism

In Sec. 16, "Modalism", Parsons considers the stratgegy of rescuing eliminativist structuralism from the vacuity problem by going modal.

To recap, we're considering the schematic idea that an ordinary arithmetical statement is elliptical for something along the lines of

For any N, 0, S, if Ω(N, 0, S) then A(N, 0, S),where Ω(N, 0, S) lays down the conditions for a set N (equipped with a distinguished element 0, and a mapping S: N -> N - {0}) to be "simply infinite", and A(N, 0, S) is appropriately correlated with the ordinary statement. And the idea is, of course, that the quantifications here are restricted to kosher, unproblematic, collections of physical objects N, and mappings on them (such as arrays of space-time points, and a 'go to the next point' map): in this way, problematic purely abstract entities drop out of the picture. And the vacuity problem is: what if, as a matter of physical fact, ours is a finite, granular, world and there is no infinite physical collection and physically realized mapping for which the condition Ω is true? In that case, the quantified conditional holds vacuously, and all arithmetical statements come out as indiscriminately true.

Now, the obvious modal gambit is to respond: Ah, we should require the quantified conditional to hold necessarily. For, even if in this world there are no physically realized simply infinite systems, there could be other possible worlds (maybe where the physical laws are very different) which do realize simply infinite systems. Why not? So even if the plain unmodalized conditional is vacuously true for any A, the modalized version won't be.

Note, though, that the natural way of construing the modality here is surely in that sense of necessity which is spelt out as truth in all possible worlds (or at last, truth in all worlds that are in some sense physical worlds lacking abstract denizens) -- as some kind of metaphysical necessity, in other words. So I'm a bit stumped as to why Parsons says "I will assume that the modal operators are understood either in the sense of mathematical modality or of metaphysical modality." I'm quite unclear what the notion of mathematical modality would do for us here.

Anyway, Parsons considers two objections to modal structuralism. The first, however, is not an objection to the modal version in particular: "It is falsifying the sense of discourse about natural numbers [to take] arithmetical statements to be really about every simply infinite system." And surely a structuralist of any sort will think this a pretty feeble objection: even if the structuralist account seems prima facie to do some violence to our intuitive model of what we mean, its defender will just say that that shows we are gripped by a bad philosophical picture of what we are really up to in doing mathematics.

Parsons considers another objection, by pressing "whether the modalist's apparatus really does offer an elimination of mathematical objects" in Sec. 17, "Difficulties of modalism". But in fact the way he develops the point seems not to tell against a modal structuralist account of arithmetic or indeed applicable analysis (given we can construct applicable analysis in such weak systems of second-order arithmetic). Rather the worry seems to be whether there could be structures realized in some sort of alternative physical world which have anything like the richness to be a model of higher set theory. Well, maybe not (though I'm not sure how one goes about settling the issue!). But then why shouldn't the structuralist just say, so much the worse for higher set theory -- it's just a fiction, or a jeux d'esprit?

Posted by Peter Smith at 9:19 AM 0 comments

Labels: Parsons's new book

## Tuesday, August 19, 2008

### The joys of Italian TV

This is fun. We've just got a small satellite dish installed and can now watch Italian TV (for free) while at home in Cambridge. The hope is that we pick up a little more of the language in a fairly painless way. "Italian TV? But that's just girls in bikinis in every programme isn't it?" Well, actually no. It's nowhere near as bad as its reputation. In fact, it can be a bit old-fashioned in a rather charming way. For example, there's a couple of quiz gameshows we've watched before in Italy (good for learners, because there are lots of pauses!) which seem much more gentle and warm-hearted occasions than the English equivalents. And the pride in la bella paese, the extolling of local food and wine and so on, that repeatedly comes across in the morning magazine programmes makes a nice change from our world-weary cynicism.

And I just love the sound of the language. Must be all those hours and hours spent once upon a time in Cambridge cinemas, at a very impressionable age, watching the likes of Monica Vitti (pictured!).

Posted by Peter Smith at 3:53 PM 1 comments

Labels: Italian matters

###
*Killing Time*

I've only just got round to reading Paul Feyerabend's short autobiography Killing Time, which he finished just before his death. (I was stunned when I opened it to find that it was published in 1995: I can't believe that I've been meaning to read this for 13 years!) I gulped it down in a couple of sittings. If you don't know the book, it is a terrific read, written with great verve about a world that already seems remote: and, in the end, it is touching too. Recommended.

Posted by Peter Smith at 3:28 PM 0 comments

###
Parsons's *Mathematical Thought*: Sec. 15, Mathematical modality

Chapter 3 of Mathematical Thought and Its Objects is called "Modality and structuralism". Before turning to discuss modal structuralism in Secs. 16 and 17, Parsons discusses what kind modality it might involve. Setting aside epistemic modalities as not to the present purpose, he considers (i) physical (or natural) necessity, (ii) metaphysical necessity (truth in all possible worlds), (iii) mathematical necessity, (iv) logical necessity (meant in a narrow sense that can be explicated model-theoretically).

Parsons argues that we don't want to spell out a modal structuralism in terms of (i) natural modalities: "it demands too much to ask that the structures considered in mathematics be physically possible; indeed, in the case of higher set theory, there is every reason to believe that they are not physically possible." I'll buy that.

Second, Parsons argues that logical possibility -- in the sense explicated via the idea of there being a suitable model -- reveals itself as itself a mathematical notion, given that models are (at least typically) mathematical entities. So(?), "It is very doubtful that a generous notion of logical possibility would be distinguishable in a principled way from ... mathematical possibility."

But there is surely something rather odd here. For the idea, to repeat, is that we explicate "it is logically possible that P" (in the generous sense of allowed-at-least-by-considerations-of-logical-form, that runs beyond metaphysical possibility) in terms of there being a mathematical model on which P can be interpreted as true. It seems we don't have a modality in the explanans here. Indeed, Parsons himself remarks on the common view that a mathematical truth (falsehood) is necessarily true (false): and on that view the very idea of a kind of "mathematical possibility" distinguished from plain truth evaporates.

So I'm left puzzled when Parsons concludes that the two runners for the kind of modality that might be involved in a modal structuralism are metaphysical modality and mathematical modality: for I just don't have a grip on the latter notion.

(Relatedly, Parsons reads Putnam as holding that "it is mathematically possible that there should be no sets of uncountable rank, although it is a theorem of ZF that there are such sets". Again, I really just don't know how to construe that "mathematically possible" if that is supposed to be neither epistemic nor equivalent to "true".)

Posted by Peter Smith at 2:17 PM 0 comments

Labels: Parsons's new book

## Monday, August 18, 2008

###
Parsons's *Mathematical Thought*: Sec 14, Structuralism and application

We're considering the schematic idea that an ordinary arithmetical statement is elliptical for something generalizing over structures, along the lines of

For any N, 0, S, if Ω(N, 0, S) then A(N, 0, S),where Ω(N, 0, S) lays down the conditions for a set N (equipped with a distinguished element 0, and a mapping S: N -> N - {0}) to be "simply infinite", and A(N, 0, S) is appropriately correlated with the ordinary statement. (Parsons, you'll recall, associates such a view with Dedekind. That doesn't seem historically correct. But let that pass.)

Does this "eliminative structuralist" view have a problem accounting for the application of numbers as cardinals? Recall Frege's remark: "It is applicability alone that raises arithmetic from the rank of a game to that of a science. Applicability therefore belongs to it of necessity." And Frege further takes it that an account of numbers should start from their use in counting (so a structuralist understanding that explains the nature of arithmetical truths prior to explaining their application is going wrong). But, Parsons argues, our structuralist in fact can resist that further thought.

I'm not sure I fully have the measure of Parsons thinking here. Part of the trouble is that he slips back into talking of numbers as objects (e.p. pp. 74--75), while I thought the attraction of the eliminative structuralism was to get rid of numbers as a special kind of object. But I take it the thought is something like this. Counting some objects involves putting them into one-one correspondence with an initial segment of some paradigm simply infinite system (of numerals, say). That involves setting up some external relations between some members of the relevant simply infinite system, over an above the internal relations which constitute their being a such a system. But now, via the Dedekind categoricity theorem, we see that these external relations will engender a one-one correspondence with an isomorphic initial segment of any simply infinite system. So, in counting, we automatically get an implicit generalization over simply infinite systems -- which is what, according to the eliminative structuralist, talk of numbers amounts to. Hence, as Frege wanted, even on the structuralist view, we do after all have an essential connection between numbers and their application in counting.

That, I think, does deal with the supposed general problem. Now, Dummett has raised a more specific problem -- roughly, defining a simply infinite system doesn't tell us whether its initial element is to be treated as 0 or 1 (or indeed, I suppose, 42). But Parsons (rightly in my view) doesn't find this worry a telling one for the structuralist. He can regard it as just a matter of pragmatic convention whether, in applications, we start counting at 0 or 1, depending on how much we care about having a number for empty collections.

One final comment on this section. Having quieted worries about the structuralist view, Parsons remarks that as well as the natural number 3, we have the integer 3, the rational 3, the real number 3 and the complex number 3 (not to mention more exotic constructions). And the structuralist can say that the use of "3" each time signifies not the same entity but the same structural role, a point congenial to his general account of the significance of number words. But, contra Parsons, I don't see that the multiple use of "3" counts at all against the Fregean view that numbers are specific objects. The Fregean can just say that there are here a number of different terms, ("natural number 3", "rational number 3", etc.) with different objects as reference -- with the common elements of the referring terms justified by the likeness of the role of the denoted objects in the respective families.

Posted by Peter Smith at 12:37 PM 0 comments

Labels: Parsons's new book

## Sunday, August 17, 2008

### One book done, another started ...

So, at long last, I've actually sent off to CUP the PDFs for the reprint of An Introduction to Formal Logic. And no, I've not been spending months non-stop on the revisions. But overall, I have now spent quite a bit of time on this; and I think it's been well worth the effort. It's still the same book, and I guess that there aren't really enough changes to make this count as a forthcoming new edition. But there are some new paragraphs here and there, and there's a large number of small changes in phraseology -- not to mention, of course, the correction of known typos/thinkos in the first printing. The cumulative effect is a book I'm markedly happier with -- even if it's not the book I'd be writing if were starting over from scratch. Still, I'm glad that's done.

Another matter I've been turning over in my mind more seriously in the last month or so is the question of which book to write next. There's a number of possibilities, including an intermediate logic text following on from the Introduction. I did at one point say I was going write a sequel to the Gödel book -- as it might be, Incompleteness After Gödel. But the more I've been thinking about that, the more dauntingly wide the field seems to become. So I'm now minded to be much less ambitious, and to write around and about just one theme, namely proofs of the consistency of arithmetic. In particular, what do we make of the significance of Gentzen's various proofs? Indeed, how do those proofs actually work? (Not that I want to write a primarily historical book -- I've not got the skills or the knowledge for that. I'm after neat rational reconstructions.) I've a ton of work to do to get more on top of this stuff. But it should be fun.

Meanwhile, for light relief, I must get back to Charles Parsons's book, as I've a review to write in the next few weeks. So over the coming days it will be back to blogging about that.

(Oh, I'm sure that other congenital procrastinators will recognize the self-motivating technique here! If you tell the world you are going to do X -- where X is something you really quite want to do anyway, but fear you might be distracted from -- then avoiding the sheer embarrassment consequent on publicly not doing X now becomes a big added reason to stop faffing around and get down to it. So ... to work!)

Posted by Peter Smith at 1:07 PM 3 comments

## Sunday, August 10, 2008

### Wittgenstein's trousers

Jeanette Winterson in yesterday's Books supplement to The Times.

I am not an absent-minded person, and certainly not like Wittgenstein who had to summon his friends to help him fasten his braces -- he couldn't match the loops to his trouser buttons, and if left to dress unaided, went forth into Cambridge in a tweedy mass of clown clothes.Complete bollocks of course about a notoriously fastidious and meticulous man. Which leaves me wondering first where on earth Winterson got that absurdity, and second why the supplement's editor couldn't be bothered to check.

Posted by Peter Smith at 7:01 AM 2 comments

## Sunday, August 03, 2008

###
*IFL* again

It turned out that it wasn't exactly urgent to get the revised version of my *Intro to Formal Logic* off to CUP when I got back from Italy. So I put it to one side to take another look at when I was less busy. I'm now working though it again. Which is just about proving worth doing, as I'm still finding little ways of clarifying a sentence here, sharpening a point there. It's all a bit time consuming though. So this must be the last revision, and I will then force myself to let it go.

Posted by Peter Smith at 7:55 PM 2 comments

## Saturday, August 02, 2008

### [P] Portishead live at La musicale

Maybe one of my many wise and insightful readers could explain why someone whose main musical passions are for keyboard and chamber music from Bach to Schubert (especially Schubert) should be so smitten with Portishead. Well, an odd conjunction perhaps, but there it is.

Anyway, for other enthusiasts, here's a really rather stunning series of live performances, recently recorded for French TV in front of a small (and, hooray, during songs, silent) audience. You can watch the videos in high quality too.

Posted by Peter Smith at 1:51 PM 2 comments