“Multiversism and Concepts of Set” revisited

A month ago I posted here a link to an interesting paper here by Neil Barton. There’s now a discussion exchange, which it would be a pity to leave buried unread in comments on an old posting: so here it is.

From Rowsety Moid. It’s an interesting paper, but to me it seems there are many questionable steps in its arguments, and I would like to know what people who know more than I do about set-theoretic multiverses would say.

The “algebraic” interpretation strikes me as incoherent, or else slight of hand. When he explains it on page 10, he seems to be saying it is not involved with existence and reference, but he then talks of “a group G”, of “elements” of G, and of “constructing new groups from old”, which all involve existence and reference. His way out of this seem to be to say the algebraic view is not “concerned with” such things (which is largely a matter of attitude, focus and interest), and that we can understand operations on groups “not as making any claims about existence and reference” (as if that were the only way issues of existence or reference could come in).

Or, page 11: “We do not make any claims as to what exists within the Multiverse, rather it is seen as an intuitive picture to facilitate algebraic reasoning concerning sets.” Even the sets don’t exist? “Given a structure” The structure doesn’t exist?

Also, it’s not clear whether he is addressing Hamkins’s actual view or a maximally “radical” alternative. For instance, on page 8 “we are interested in Multiversism in its most radical form” is given as a reason for assuming that every level of metalanguage is indeterminate.

Even when the aim doesn’t seem to be the maximalisation of radicalism, there are a number of questionable interpretations or restatements. On page 9: “One way to understand Hamkins’ suggestion is to hold that we refer to several universes at once via description”. By page 11, the “one way to understand” has dropped out. Hamkins saying “in this article I shall simply identify a set concept with the model of set theory to which it gives rise”, quoted on page 7, becomes “it was noted that the Multiversist thought that every model of set theory constituted a set concept” on page 13. The idea that the concept gives rise to the model has been lost, and the idea that every model constitutes a concept has been added. Between pages 7 and 8, Hamkins’s “Often the clearest way to refer to a set concept is to describe the universe of sets in which it is instantiated” becomes “each model is correlated with a set concept, and we refer (to?) this concept through a description.” Such examples can be multiplied.

On page 14, it turns out to be important for Neil Barton’s argument that every model (or, perhaps, every “cloud” of models) corresponds to a concept and so to a description. I don’t see how that could work. Aren’t there more models than descriptions? (Hamkins seemed to have it that there was a model for every concept, not that there was a concept for every model.)

Another important step in Barton’s argument is the idea that we (or at least “Hamkinsians”) can use only first-order descriptions and so “lack the conceptual resources to pin down a single universe precisely”. From that, via the “One way to understand Hamkins” mentioned above, we reach the idea that we end up referring to “clouds”. For some purposes, that restriction seems correct. But when we’re trying to ground reference and so avoid a vicious infinite regress? That’s not so clear. Natural language (English, for example) isn’t restricted to FOL, for a start.

In the background, there seems to be an ideological element to the argument. It’s difficult to pin it down, but I think it may become visible on page 15 when arguing that “the Hamkinsian can give no particular reason to focus on one stopping point rather than another” and then saying “the response that we simply stop somewhere (without being able to give any reason for a particular stopping point) seems, like Go ̈del, to ascribe unexplained powers to the human mind.”

The power to stop somewhere? Is that supposed to be mysterious? I supposed that, in a sense, it is unexplained; but only because pretty much everything about the mind is currently unexplained, if you push hard enough.

From Neil Barton First, let me say a big “Thank You!” to Peter for publicising my paper, and to Rowsety Moid for some excellent comments. Indeed, your remarks were very timely, as they highlighted a mistake that I corrected in the proofs (shameless self-promotion: the paper will come out in this volume.

You said, RM: “The “algebraic” interpretation strikes me as incoherent, or else slight of hand. When he explains it on page 10, he seems to be saying it is not involved with existence and reference, but he then talks of “a group G”, of “elements” of G, and of “constructing new groups from old”, which all involve existence and reference. His way out of this seem to be to say the algebraic view is not “concerned with” such things (which is largely a matter of attitude, focus and interest), and that we can understand operations on groups “not as making any claims about existence and reference” (as if that were the only way issues of existence or reference could come in).”

Sure: we can have the algebraic interpretation relate to issues of existence and reference. The point is just that one requires additional assumptions first. For example, if we assume that the relevant objects exist, then there is a class of structures which instantiate the relevant algebraic properties about which the algebraist talks. Indeed, then various claims she makes might be a good way of proving the existence of certain objects, and better than trying to construct these things `absolutely’ (I think non-standard models are possibility a vivid case here). But those additional assumptions are needed before her view gets off the ground.

To bring this out, imagine that nominalism about sets were true. The ontological interpretation would then be null and void, there simply is no multiverse. It seems that the algebraic interpretation might still live on, despite the fact that the relevant algebraic properties are uninstantiated. For, we could still say that IF we were given some objects satisfying the relevant properties, we would be able to do such and such operations (and similarly with other algebraic theories like group theory).

[N.B. It’s an interesting question how the algebraic interpretation relates to if-then-ism. It’s not clear that these are wholly the same because of the algebraists acceptance of indeterminacy in metalogical notions.]

RM: “Even the sets don’t exist? “Given a structure” The structure doesn’t exist?”

We say things like this all the time though. “If space-time is discrete, then such and such will hold, and it is theoretically possible to construct such and such kind of object.” That seems like a perfectly valid claim to make, even if space-time is not discrete. Similarly with sets. Give me a structure, and I will be able to do these sorts of operations. I make no claim on whether the structure exists.

[N.B. As someone who is generally of a realist persuasion, I tend to think that this sort of response depends on the existence of the structures anyway. But this is just a refusal to engage with the position, not a dialectically convincing response.]

RM: “Also, it’s not clear whether he is addressing Hamkins’ actual view or a maximally “radical” alternative. For instance, on page 8 “we are interested in Multiversism in its most radical form” is given as a reason for assuming that every level of metalanguage is indeterminate.”

You raise a good point here, that I think is a general feature of the debate: the exact positions on the table are unclear. The Hamkins paper, though both highly interesting and ingenious, is notoriously slippery when it comes to being fully precise about the commitments of his view. So: am I addressing Hamkins or a radical alternative? I don’t know: you’d have to ask Joel how well the view put forward coheres with his (who, it has to be mentioned, was very helpful in discussing the paper and I’ve found very approachable). However, the views I present in the paper are ones that can be extracted from some of the things that he says, and he’s often keen to embrace the radical consequences of his view. He’s very clear that he thinks that indeterminacy infects the metalanguage, and there is no definite concept of natural number.

RM: “there are a number of questionable interpretations or restatements.”

I think there are going to have to be with the literature as it stands. Nowhere is Hamkins explicit about the kinds of epistemology he envisages, or the full metaphysical character of his view. My paper is intended to be just as much filling out possible ways of taking Hamkins’ view as a criticism of some of the things he says. I would welcome it if there are alternative interpretations out there, or I have got something wrong in exegesis—that way we can be more precise about what views are available and tenable. But I need to see these additional interpretations before I can weigh them up against the ones I have put forward.

RM: “The idea that the concept gives rise to the model has been lost, and the idea that every model constitutes a concept has been added. Between pages 7 and 8, Hamkins’s “Often the clearest way to refer to a set concept is to describe the universe of sets in which it is instantiated” becomes “each model is correlated with a set concept, and we refer (to?) this concept through a description.” Such examples can be multiplied.

On page 14, it turns out to be important for Neil Barton’s argument that every model (or, perhaps, every “cloud” of models) corresponds to a concept and so to a description. I don’t see how that could work… (Hamkins seemed to have it that there was a model for every concept, not that there was a concept for every model.)”

This was sloppy on my part, and I made some alterations in the proofs as a result. You are right, we should say that we refer to the concept through the model. However, this is done through *description*: we refer to the concept by *describing* the model. However, since we can only use first-order descriptions, we can’t pin down a single model, so our reference must be indeterminate, but this requires fixing some other model (in which some concept is instantiated; I take it that every model instantiates a concept) and so on.

I think there’s a lot more to be said here (in fact it doesn’t seem impossible to me that we get a loop of concepts or models), but the challenge at least represents an invitation for the Hamkinsian Multiversist to be clear on their commitments, and explain why there’s no such problem. As it stands, the Ontological Interpretation is not developed in sufficient detail to explain how these problems are avoided.

RM: “Aren’t there more models than descriptions?”

That depends on where you live for the Hamkinsian. Since every universe’s multiverse is countable from the perspective of some other universe, the universes of a multiverse are bijective with the descriptions from a suitable perspective (though not through the natural mapping of a universe with a description of it, and not within any particular multiverse). In any case, to press the challenge I only require infinitely many concepts being used, so countable is enough. There’s also the question of whether or not Hamkins is allowing parameters, which would in turn make the issue a whole lot more complicated (given the emphasis on ultrapowers, I’m guessing he is allowing parameters for the ultrafilters), as then we could have proper-class-many descriptions. Again, this would be another area I would like clarification from the Hamkinsian.

RM: “But when we’re trying to ground reference and so avoid a vicious infinite regress? That’s not so clear. Natural language (English, for example) isn’t restricted to FOL, for a start.”

I’m in full agreement here! In fact I find the restriction to FOL excessive. Again though, this is a case where the dialectic with Hamkinsian is important. If he/she wants to admit non-FOL, s/he has to explain what is acceptable and what isn’t. Why is it okay for him/her to use non-FOL resources in giving an account of reference, yet an account of the semantic referents of the natural numbers or set theory in terms of properties/plurals/the ancestral relation/sets is not allowed? I want to know what the rules of the game are by which the Hamkinsian has a stronger position compared to the Universist.

RM:“but only because pretty much everything about the mind is currently unexplained, if you push hard enough.”

Sure! But here we’re seeing if the Hamkinsian has a decent response to Benacerraf’s challenge through description. So, given that this is the background for the paper, we can demand an explanation.

I think in general the paper shouldn’t be viewed as an all out attack on the Hamkinsian, but rather a request for him/her to clarify a number of points.

Thank you ever so much for the comments! I found them very helpful. Best Wishes, Neil

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Intro to Formal Logic 2?

I’m mulling over a proposal that I write a second edition of my Introduction to Formal Logic, first published thirteen years ago by CUP.

I’m tempted. I’m sure I could make a very much better job of it. Though, of course, I’m only too aware of how time consuming it would be to undertake, and I’d also like to crack on e.g. with the Gentle Introduction to category theory (which has been going slowly of late).

If I do write a second edition, I’d like to include chapters on natural deduction (Fitch-style, I think, for user-friendliness). But I wouldn’t want the book to get too unwieldy in length — and CUP wouldn’t let it! — so some stuff would also have to go. But I guess I could put online some of the current content that goes beyond a usual first logic course (e.g. the completeness proofs for trees for PL and QL!). Would that be a win-win solution? — a tighter book, with the stuff on natural deduction some people wanted, but with the “extras” still available for the small number of real enthusiasts?

I haven’t decided yet whether to take up the proposal, let alone how to reshape and add to the text if I do go with it. But any advice and suggestions (either here, or in emails, address at the end of the About page) — especially from those who have used the first edition or indeed, perhaps even better, from those who decided not to use it — will be very gratefully received!

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The difference a bit of history makes

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We keep car and bikes in the garage at the bottom of the garden. The lane there is very rough and ready. It is unclear who owns what and whose responsibility it all is. The potholes make cycling a hazard, and you have to inch a car along rather gingerly. A bit of a pain, we’ve always thought (though actually having a usable garage in central Cambridge is very unusual).

But now we find that the lane is medieval — a broad track used by people going along the river to cross over to Stourbridge Fair, once the largest fair in Europe (and in the sixteenth century, lasting a month).

Which puts the mud and rubble in a whole new light.

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A Note on Core Logic

Here’s a very short Note on Core Logic — that’s Core Logic in Neil Tennant’s sense — prompted by reading a forthcoming short paper by Joseph Vidal-Rosset which offers what are supposed to be three fatal objections (link to his paper in my Note, which I’m posting as a PDF as it involves proof diagrams). I’m not sure quite where I stand on Tennant’s project: but I really don’t think that Vidal-Rosset gets the measure of it at all. No doubt Tennant himself will reply in due course; but some might find this quick counterblast of interest.

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Quick book note: Prime Numbers and the Riemann Hypothesis

9781107499430Here’s a newly published book from CUP by Barry Mazur and William Stein. It’s delightfully short — just 128 pages before the endnotes, and in fact a fair proportion of those relatively few pages is taken up with graphs and diagrams, and photographs, and there is a lot of white space. The aim is to explain the nature of the Riemann hypothesis and how issues about the distribution of primes relate can be illuminated e.g. by devices from Fourier analysis: there is real mathematics going on here, not just popularising gossip or arm-waving history. The intended readership, at least for the first parts of the book, is beginning mathematics students (or curious amateurs who do have a little maths).

How well does the book succeed? The headline news: this may at the beginning be a bit too noddy and towards the end be a somewhat bumpier ride than the authors intend — but I indeed found the book very engaging and illuminating. I’d recommend it warmly. (OK, I know I’m not quite the audience the authors had in mind, but this isn’t maths I’m at all familiar with, so in this area I do count as a pretty naive reader.)

Before giving just a bit more detail, however, I’ll get out of the way a double presentational reservation, namely about the use of photographs (which might put off some potential readers). In Chapter 1, for example, we are given a story from Don Quixote that turns on the primeness of 17: cue illustration from a translation of that book. And we are also told the primeness of 17 has something to do with the fact that there are periodic cicadas which emerge every 17 years: cue snap  of a cicada. Later when talking about (mathematical) spectra, we get a dull stock photo of a rainbow, and — heavens above! — when talking about the kind of smoothing of data that produces read-outs of average vehicle speed we get a  big photo of a car dashboard. Oh dear. These silly illustrations are quite pointless and in fact spoil the aesthetics of an otherwise rather  attractively produced book and send quite the wrong message  (let the book be judged by the zestfulness and clarity of its mathematical expositions, not by false gestures to user-friendliness of this kind). But there’s another reservation, about the other pictures which are portraits. Unlike the popularising books of a familiar kind, we don’t get stories about the lives and eccentricities of past mathematicians: there’s almost no biography here. Yet we do still get unnecessary mug-shots of various mathematicians from Descartes to our contemporaries. Which emphasises all too vividly that those mentioned are — unsurprisingly, given the history — one and all male. Without wanting to come over all politically correct, I wonder whether — in a book trying to enthuse the young —  it is a bright idea in this way to unintentionally reinforce once again the impression that maths is a man’s game?

Back to the real content. Prime Numbers and the Riemann Hypothesis is divided into four parts. In Part I, we get an initial statement of the Riemann Hypothesis in the form

For any real number X, the number of prime numbers less than X is approximately Li(X) and this approximation is essentially square root accurate.

Informally, Li(X) is defined as the area of the region under the graph of 1/log(X) from 2 to X, and square root accuracy can be thought of as getting right at least the first half of the digits of  the number of prime numbers less than X. Thus explained, this version of the Riemann Hypothesis will therefore be available to any high school student who has met natural logarithms. Lots of computer-generated diagrams are used to make this Hypothesis look a decent bet (though the authors also give warnings about reading too much into such diagrams). And then, if you’ve met the cosine function, you’ll understand the rest of Part One too, with its hints of Fourier analysis. This is all, surely, very well done  though a little slow to start.

In Part II, we meet “generalized functions” like the Dirac delta function and members of the same family, and get to think rather informally about Fourier transformations of delta functions and the like. So we arrive at the idea of a trigonometric series having an infinite “spike” at various values, and develop the idea of a spectrum of spike values, and learn how to get powers of primes into the story. This requires a bit more more background, and a bit more sophistication, but is still very accessible. Computer-generated diagrams of the behaviour of various truncated  trigonometric series are again very effectively deployed in order to underpin plausible hypotheses about what happens in the limit.

Part III is called “The Riemann Spectrum of the Prime Numbers”. I’m not quite sure why the rather short Parts II and III are separated, as it doesn’t seem to require more mathematical sophistication to follow the further exposition here — just a bit more effort, perhaps. With few lapses, this is also clearly and engagingly done. But note, we are still working with those ideas of trig series and Fourier transformations from real analysis, and haven’t yet met any complex analysis, and hence not yet met the Riemann zeta function. That happens in the very short …

Part IV, “Back to Riemann”, where we begin to connect up the approach of previous chapters with Riemann’s own work, relating the “Riemann spectrum” of spiking points of a certain series with the non-trivial zeros of the zeta function. And here the level does indeed ratchet up. It’s not just that you need to at least dimly recall some notions from complex analysis, but the speed of exposition goes up. In fact, I rather wish the space wasted on pointless pictures had been used instead to take things more gently/expansively here and to illuminate more the connection between various versions, announced equivalent, of the Riemann Hypothesis. We are also, along the way, told that RH has lots of interesting implications: again, it would have been good to learn more.

It would also have been good to finish up with a couple of pages or so of an annotated reading list for those enthused enough to want to follow up the discussions (with more pointers, at any rate, than we get in passing on pp. 119-120). After all, Prime Numbers and the Riemann Hypothesis  is certainly accessible and intriguing enough to leave many readers wanting to know more.

Overall, then, at least modified rapture! And forgetting the bum notes hit by the photos, we could do with more maths books in this general style.

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CD choice #6

51ERLMsJmeLTime goes so very fast: it is Martha Argerich’s 75th birthday today. Here is perhaps still my favourite recording of hers, from 1980. Completely stunning Bach playing. I can’t put it better than Jessica Duchen has:  “Martha Argerich presents intense, minor-key Bach: fierce but pure, finely argued and rhetorical yet never losing the rhythm of dance; expressive but contained and articulated with beautiful clarity. Her intensity is completely compelling from the very first note of the C minor Toccata onwards: she demands, seizes and holds attention at every moment. Her personal vision displays the emotional darkness at the heart of these pieces without any suggestion of mannerism or gimmick.”   If you don’t know this disc, you have missed what is surely one of the greatest ever recordings (and it’s now immediately available for streaming if you have an Apple Music, or similar, subscription.)

But oh how we wish that Argerich had recorded more Bach over the years …

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The English art

IMG_1706We drive to and from Cornwall slowly each year, staying over for a night somewhere, visiting National Trust gardens en route — this year Coughton Court and Killerton as we drove down, Knightshayes and Hidcote on the way back to Cambridge. If the weather is bad we’ll take the tour of a historic house — but they are only shells of the homes they once were, while their grounds are bursting with life and colour in May.

The making of gardens is surely one of the art forms the English do best, and care about the most.

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Postcard from Cornwall

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In Cornwall again, back in St Mawes. As lovely as before. The photo was taken walking along another little bit of the South West Coastal Path this morning, this time along the cliff-tops between Portloe and Pendower. On the whole, kind weather. Too kind, at any rate, to be sitting indoors, reading much serious stuff: that, and logical blogposts, can wait until I return to Cambridge, far too soon.

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TYL 2017?

The Teach Yourself Logic 2016 Study Guide is linked here at Logic Matters but also at my (decidedly sparse) academia.edu page. Rather startlingly, the latter link has now been followed up over 50K times. Who knows how much impact the Guide really has. Still, I occasionally get appreciative emails (and equally cheeringly, I don’t get protests from colleagues complaining bitterly that I am leading the youth astray). So, hopefully, TYL 2016 is doing some good in spreading the logical word.

That’s the plus side. The downside is that, given it indeed seems to be used quite a bit, I suppose I should keep on updating the Guide. The year is already rattling by, and I guess I should soon start turning my mind to the time-consuming business of (re)reading around logic books old and new, familiar and less familiar, as background homework for producing TYL 2017. So if you do have suggestions for improvement, and in particular suggestions of recent books I should really take a look at over the next few months, do let me know sooner rather than later.

So many logic books, so little time …

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Kurt Gödel, Philosopher-Scientist #5

The next paper in Kurt Gödel, Philosopher-Scientist is by Paola Cantù, on “Peano and Gödel”. The headline claim is that Gödel’s philosophical notebooks indicate that he had read Peano (and in particular Peano’s contributions to the Formulaire des mathématiques /Formulario Mathematico) rather carefully — and that Gödel was alert to the important differences too between Peano and Russell.

Unfortunately, when it comes to trying to spell out carefully what Peano’s views were (e.g. about the nature of functions), how Russell’s differed, and exactly what Gödel’s response comes to, I found Cantù less than ideally clear.

Here’s just one example where Cantù seems muddled. Peano defines the “classe nullo” \Lambda to be the class of objects which are common to every class (“classe de objecto commune ad omne classe” Formulario, §6). Peano also defines the iota function (Formulario, §7) such that \iota x is the singleton class of x (the class of y such y = x). And then he adds a [partially defined] inverse iota function which undoes the effect of the iota function. I can’t typographically invert an iota here, so I’ll follow the later Peano and write \overline{\iota} instead: and then the idea is that, if a is a class other than \Lambda such that any two members of it are equal (i.e. if it is a singleton class), then {\overline{\iota}a} = {x} iff {a = \iota x}. Note, by the way, Peano’s {\overline{\iota}}, i.e. his inverted iota, is thus significantly different from Russell’s inverted iota!

Peano then shows, inter alia, something of this shape: \Lambda = \overline{\iota}K, where ‘K’ is in fact just one obvious way of characterizing the singleton of \Lambda. The finer details of ‘K‘  don’t matter.

So far so good. But Cantù comments on the latter rather trivial result as follows:

The definition of  \Lambda by means of the inverse iota operator allows us to define it as an individual object rather than as a set: nullo is the element associated to a set that contains all the x such that (a) x = a \land \neg a, i.e. the elements that, for any property, satisfy that property and its contradictory.

This is surely muddled (even forgetting about Cantù’s symbolic foul-up, though to be honest that doesn’t inspire confidence). For {{\Lambda} = {\overline{\iota}K}} no more defines \Lambda as an individual object in any sense that contrasts with its being a class than would do the close equivalent  {\Lambda} = {\overline{\iota}}{\iota}{\Lambda}\Lambda is still a class (though of course, it also an element — an element of its own singleton). Peano’s result which Cantù quotes doesn’t make nullo an element associated to a set containing just those things satisfying some contradictory condition (associated how?); rather it still is, as originally defined, a class

To move on, there is a quite separate issue for Peano when he later makes use of the inverted iota operation applied to classes defined by conditions not guaranteed to determine singletons. How are we to understand ‘{\overline{\iota}a}’ if {a} is not a singleton? Good question. Cantù gives an entry from Max Phil where Gödel (a) holds that Peano commits himself to the idea that there is a contradictory null object (Unding), and “\overline{\iota}a when {a} has no elements [or] several elements, is this null object”, but (b) Peano can avoid this obscure doctrine. It is not clear, however, that Peano is countenancing a null object in this sense  [which is not to be confused with the perfectly good class \Lambda!]. Gödel’s alternative isn’t clear either, and Cantù’s discussion of it here is not easy to follow. But I am not minded right now to sit down with more of the Formulario to try to work out what’s going on — fun though it is to decode Peano’s simplified Latin!  So for the moment, I’ll have to leave things in this unsatisfactory state.

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