Year: 2014

The consistency of NF

Randall Holmes on his website now says he

 … currently claim[s] to have a proof of the consistency of New Foundations. The current document describing the proof is here. Comments and questions from those who understand the technical issues are very welcome; I apologize in advance for the still difficult state of the text, and I am sure that I don’t yet have all the bugs out.

If you have an amateur philosophical interest in NF, I do not think it likely that you will get anything out of this very technical and not yet very polished document, and I am not likely to answer your questions about it. Be advised that in my opinion (which I know is not universal) the famed NF consistency problem has nothing at all to do with philosophical issues which Quine’s set theory might be taken to address: I think that NFU addresses these issues to exactly the same extent and its consistency and mathematical strength have been settled issues since 1969.

Teach Yourself Logic, new layout

The Teach Yourself Logic Guide has a new look.

Now, instead of being a standard A4 PDF, it should be an ideal size for reading on screen. Read it either (i) on an iPad (download in Safari, open e.g. in iBooks), or (ii) on a lap-top (e.g. read two pages side-by-side using Adobe Reader in full-screen mode). The changes in this new version are modest, then, but should I hope make the Guide significantly more readable.

If you do want to print it out, then using the two pages side-by-side format with Adobe Reader works very well. (Printing from other readers, you may need to change the page set-up of the printer to recognise the reduced paper size, approximately A5.)

Comments on the new layout, for or against, are welcome.

The Pavel Haas Quartet live, again

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One of the delights and frustrations of concert-going is how unpredictable the experience can be. Fine ensembles can have off-days. Less regarded players can capture the moment and transport you for a couple of hours.

On the frustrating side, then, the last two outings to hear the usually stellar Academy of Ancient Music were, for different reasons, pretty disappointing. Richard Tognetti’s playing style just didn’t gel with the band’s on the evening I heard him as a guest director in Cambridge. And Richard Egarr’s exaggerated fortepiano embellishments in Haydn at their next concert were positively camp rather than appropriately playful.

On the side of delight, there was the Pavel Haas Quartet again, at their most recent concert at the Wigmore Hall. We had high hopes. Though they hadn’t chosen the most audience-pleasing of programmes (which perhaps is why there was a surprising sprinkling of empty seats). But from the very first moments when they launched into Smetena’s second quartet, they were on fire. Even the composer acknowledged that the first movement “is quite unusual in style and difficult to follow, as if the whole movement were the product of whim….”. And by the time we get to the harmonically dissonant final movement, lesser performances can leave us lost. But the Pavel Haas made this their own, gave such shape and coherence to the piece, played with such attack and mutual understanding, that there was a storm of applause. By far the best performance of the Smetena I have ever heard, on disc or otherwise.

There followed a fine performance of Dvorak’s “Slavonic” quartet. And then after the interval, the Pavel Haas played the Brahms Op. 51, No. 2. Again, not the easiest work to communicate (I for one find don’t find myself drawn to listen and re-listen to Brahms); but by this time the quartet had the audience utterly in their spell, and played with mesmerising authority to produce another astonishing performance. Once more, this was better than any recording I know of the piece, by some way. This was playing of unsurpassed musical understanding and intensity.

If you ever have the chance to hear the Pavel Haas Quartet play live, grab it.>

Teach Yourself Logic, again

There’s an even bigger, even better, shiny new Version 10.0 of the TYL Study Guide now available at the usual URL https://logicmatters.net/tyl/  Form an orderly queue …

The  structure of the Guide has significantly changed again (hence the jump in version number). The key section on basic first-order logic at the beginning of the mathematical logic chapter was getting more and more sprawling: it has now been hived off into a separate chapter, divided into sections, and further expanded. My sense is that quite a few readers are particularly interested in getting advice on this first step after baby logic, so nearly all the effort in this particular revision of the Guide has been concentrated on making improvements here. So, inter alia, there are new comments on four outstanding relatively elementary books: Derek Goldrei’s Propositional and Predicate Calculus, Melvin Fitting’s  First-Order Logic and Automated Theorem Proving,  Raymond Smullyan’s Logical Labyrinths, and (not least) Jan von Plato’s very recent Elements of Logical Reasoning.

I still intend sometime to return to say more about the last of these when I’ve had a chance to re-read it: it is in many ways a very welcome addition to the literature. For the moment, I just remark that this book is based on the author’s introductory lectures. I rather suspect that without his lectures and classroom work to round things out, the fairly bare bones presented here in a relatively short compass would be quite tough as a first introduction, as von Plato talks about a number of variant natural deduction and sequent calculi. But suppose you have already met one system of natural deduction, and (still a beginner) want to know rather more about ‘proof-theoretic’ aspects of this and related systems. Suppose, for example, that you want to know about variant ways of setting up ND systems, about proof-search, about the relation with so-called sequent calculi, etc. Then this is a very clear, very approachable and interesting book. Experts will see that there are some novel twists, with deductive systems tweaked to have some very nice features: beginners will be put on the road towards understanding some of the initial concerns and issues in proof theory.

Logic matters, but not that much, apparently …

Looking at the Leiter blog … and no, this isn’t going to be about certain recent kerfuffles, where there has perhaps been rather too much rushing to judgement.

As I was saying, looking at the Leiter blog, I’m struck by the new list of “recent hires“. So far, there are [updated] eighty three “tenure-track and post-doc” appointments listed. Only four mention logic at all in any shape or form, and judging from publication lists and websites, the relevant people’s interests are in non-technical philosophy of logic overlapping with the philosophy of language and epistemology (topics like vagueness, theories of truth,  the epistemology of basic logical principles, the sense in which Russell was a logicist). Only one person on the whole list mentions philosophy of mathematics at all (and again, only with that same historical paper on Russell’s logicism to show for it). Fine topics to be interested in, but not perhaps at the core of logic or philosophy of maths.

It could be that Leiter’s list — or rather the list provided on his comment thread — is a bit unrepresentative. But it could be one more straw in the wind, showing how far the direction in most philosophy departments has turned from a central engagement with two of the founding disciplines of analytical philosophy.

A few quiet updates

Some recent changes/additions to the site:

  1. There was a quiet update of Godel Without (too many) Tears a couple of weeks ago adding a new section, and slightly tinkering with what I say about recursive-but-not-primitive-recursive functions to remove a possible suggestio falsi.
  2. There has also been an update to a handout on Tennenbaum’s Theorem which adds a section on how not to prove the theorem and tinkers elsewhere.
  3. The writing of exercises-and-solutions for the Gödel book proceeds at a snail’s pace, but there is a possibly interesting set of exercises on (informal) induction now added.

Next task of this kind: to get back to the Teach Yourself Logic Guide. For a start, I’ve three introductory books on my desk with different virtues, that I’d like to add notes on. In particular, Jan von Plato’s Elements of Logical Reasoning is very recently out with CUP and provides a not-so-familiar route for the logical beginner, and although intended as an introductory book for students  has elements that will certainly interest their teachers too. More in due course …

Logical snippets

For about eighteen months now, I’ve been a regular visitor to the useful question-and-answer site, math.stackexchange.com — this is a student-orientated forum, not to be confused with the truly wonderful mathoverflow.net which is its research-level counterpart. OK, you can think of these visits as (hopefully) constructive procrastination on my part …

Of course, many of the questions on the site, including many I’ve found myself answering, are very ephemeral or very localized or based on very specific confusions. But a small proportion of the exchanges to which I’ve contributed might, for one reason or another, be of some interest/use to other students — or at least, to beginners and near beginners in logic.

So I’ve put together a page of links to these logical scraps, morsels, excerpts, … snippets, shall we say. I’ve grouped the links by level and/or topic.

In praise of … Rachel Podger/ATOS Trio

We’ve been to two exceptional concerts in the last few days. First we went up to the Wigmore Hall for the Schubert birthday concert, where the ATOS trio played the two Schubert piano trios to deserved acclaim from a rapt audience. Wonderfully nuanced playing, deeply felt. About as good as it gets for performances of these stunning pieces  (it is time the ATOS recorded them). If you get the chance to see the trio, they really are quite outstanding.

Then a very different evening, listening to Rachel Podger and Brecon Baroque playing eight of the concertos from Vivaldi’s L’estro Armonico  in the antechapel at King’s College. I love their deservedly multi-award-winning recording of La Stragavanzaand the live concert was just terrific — played with verve and enjoyment, playfulness and charm, and a lot of light and shade. Technically brilliant too. The performances made the case wonderfully well for Rachel Podger’s description of these works, in her lovely talk to the audience after the first concerto, as intriguingly complex and rule-bending. The audience was sadly a bit thin, but again was bowled over. A recording of L’estro Armonico is the next project for these players together: you will want the CDs when they are out, and in the meantime get their earlier Vivaldi recording (their Bach is brilliant too …).

GWT2

A complete first draft of the revised version of Gödel Without Tears is now available for download here.

I hope in due course to improve the suggestions for further reading, particularly with pointers to resources on the web. But the notes are at least now online in a reasonable initial form, and are certainly better than what they replace. Please do let me know about typos/thinkos!

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