Peter Smith's Blog, page 96

CD choice #2

I have come very late to appreciate Janáček’s piano music — something I largely owe to two discs by Ivana Gavrić. Her fine first CD, titled after that composer’s In the Mists, was a serendipitous find in an Oxfam shop (it also includes, among other things, an excellent performance of Schubert’s A minor Sonata, D. 784). But her second recording from 2011, From the Street, is even better. If you don’t know Janacek’s sequence of ten pieces On an Overgrown Path, then you have a delight awaiting you. Gavrić’s playing seems exceptional here: the Gramophone reviewer rightly wrote of “the intimacy, finely honed nuance, conversational flow and subtle underlining of the composer’s harmonic surprises that Gavrić brings to each of the short pieces”, and other reviewers were equally enthusiastic. There are, I have since discovered, some other terrific recordings available, including one by Marc-André Hamelin. But this still strikes me as outstanding.

Also on the CD are Janáček’s Sonata 1.X.1905 From the Street, Ravel’s Valses Nobles Et Sentimentalise, and not least a wonderful performance of Prokofiev’s Sonata no. 2. (I’m not usually a great one for mixed recital discs, and I usually listen to these performances separately; but actually the programming works very well). So indeed, all very warmly recommended, especially if the Janáček or Prokofiev isn’t familiar.

Footnote Ivana Gavrić won a BBC Music Magazine Award in 2011 for her first CD. It is now time to vote for this year’s Awards. The Pavel Haas Quartet are shortlisted in the Chamber Music section. So you know what you have to do …

Book Note: Tony Roy, Symbolic Logic, #3

After a longer than intended gap, I return to consider Parts III and IV of Tony Roy’s freely available  Symbolic Logic: An Accessible Introduction to Serious Mathematical Logic. The previous two , rather lukewarm, instalments discussing Parts I and II are here and here (but do please note Roy’s long comment in response).

Part of my beef against this very long text is indeed its length. In teaching maths, often the key task is to engender the kind of understanding that enables the student to see the wood for the trees, to see what are the Big Ideas and what is merely hack-work joining up the dots. The longer you bang on filling in every last detail of a proof, the greater the danger that you will obscure the overall contours of what’s going on (even if you scatter around an amount of signposting). We ask our students in exams to do “bookwork” questions outlining a proof of some major result, and here the name of the game is indeed to highlight the Big Ideas, the key moves, and to confidently know what can be gestured at, or when we can say “rather similarly, we can show …” etc. Where I take issue with Roy’s pedagogic style, then, is in thinking that writing at his length won’t really help foster these skills.

I mention this again because Part III on Classical Metalogic consists mostly of two very dense chapters, one of forty pages, one of fifty pages, going into rather over-the-top detail (by my lights) on relatively few results. So again I wouldn’t recommend these as primary reading for students encountering some metalogic for the first time.

However, on the positive side, the main content of Chapter 9 is unusual in one interesting respect. Roy has earlier introduced both an axiomatic and a natural deduction system for first-order logic. We can of course prove they are equivalent by going via the respective soundness/completeness proofs for the two systems. (That doesn’t really require two lots of proofs as we can point out in particular that what it takes for a Henkin  completeness proof to go through is available in both cases.) But we can also give a syntactic equivalence proof for the two systems by showing how to systematically manipulate in both directions a proof of the one sort into a proof of the other. Now, this tends to be the sort of thing one armwaves about in class, perhaps sketching in a few obvious moves. But I can’t offhand recall any textbook which spells out in any detail, for particular given axiomatic and ND systems, clear routines for moving between proofs in the two systems (any offers here?). Roy however does this at (slightly numbing) length. Good: we can now usefully point any students unsatisfied by classroom armwaving who want to know how such proofs go to Roy’s very careful working-out of detailed routines.

Chapter 10 then tackles soundness and completeness. The completeness proof for the sentential fragment takes eleven pages: another thirty pages labour over the completeness of a first-order calculus with identity. This is, for example, over twice the number of pages needed by the extraordinarily lucid, gently paced, Chiswell and Hodges. I won’t quote chunks, as you can make your own judgement, since the chapter (as with the rest of the Accessible Introduction) is freely available. But I honestly can’t imagine many students finding the extra length going with a doubling in clarity and understanding. Indeed, if a student were stuck on a Henkin completeness proof in one standard text, I’d first suggest looking at another snappy presentation in a different text (before mentioning Roy’s much longer efforts): for the problem would very likely be in seeing the overall strategy, the Big Ideas — and brisker presentations are likely to make those stand out better.

Part III also contains a fragmentary Chapter 11 which belatedly talks about expressive completeness for the sentential connectives, and then says something briefly about compactness and the L-S theorems. But I won’t say anything about this.

To be concluded.

Easing gently into 2016

There is a new version of the Gentle Introduction to Category Theory. There are no new chapters this time, but there are some significant additions (I now prove a result about Cartesian closed categories with natural numbers objects, which previously was only announced, and prove that free monoids can be thought of as initial objects in a certain comma category). And there are many improvements, both in content and presentation. Note in particular, I correct a mistake about the relation between different notions of diagrams, and clear up what was an unnecessarily messy chapter on the existence of limits. I am very grateful indeed to comments/corrections from Paolo G. Giarrusso and Yufei Cai for prompting some of these improvements.

Although now 178pp., this version is still very incomplete: you can find some rough-and-ready follow-up chapters at the categories page here where there is also an alternative link to the Gentle Intro for those without an academia.edu login.

A cheering start to the logical New Year?

Just before Christmas, I put a copy of Teach Yourself Logic 2016 on my rather sparse academia.edu page. It has since been browsed there over 45,000 times, and then the whole thing downloaded 2,500 times. I’m not sure how many times TYL has also been downloaded from Logic Matters, as the stats counter here is flakey (though the page it is linked from has been visited over 11,000 times in the last few weeks): but it is hundreds more.

The particular numbers don’t matter. But the trend is good.  At a personal level, this makes the effort I put into TYL continue to seem worthwhile. And more impersonally, this is serious logic we are talking about here: and caring about the future of the subject, it is really  good to find that there is enough interest out there for thousands of people to go to the bother of downloading the Guide, with at least some sense of what they are letting themselves in for. So this is all rather cheering, and puts a spring in my logical step, inspiring me to keep on tinkering with TYL and with related stuff.

But not quite yet! Back to category theory first …

A cheering logical start to 2016?

Just before Christmas, I put a copy of Teach Yourself Logic 2016 on my rather sparse academia.edu page. It has since been browsed there over 45,000 times, and then the whole thing downloaded 2,500 times. I’m not sure how many times TYL has also been downloaded from Logic Matters, as the stats counter here is flakey (though the page it is linked from has been visited over 11,000 times in the last few weeks): but it is hundreds more.

At a personal level, this makes the effort I put into TYL continue to seem worthwhile. And more impersonally, this is serious logic we are talking about here: and caring about the future of the subject, it is really  good to find that there is enough interest out there for thousands of people to go to the bother of downloading the Guide, with at least some sense of what they are letting themselves in for. So this is all rather cheering, and puts a spring in my logical step, inspiring me to keep on tinkering with TYL and with related stuff.

But not quite yet! Back to category theory first …

CD choice #1

Those end-of-year lists of recommended books are really rather depressing, aren’t they? Even setting aside the pretentious, the uninviting, the distinctly esoteric, there remain all those novels, all those biographies, all those histories, and much more, books that do sound so enticing, yet which you know — despite your best resolutions to read more, and idle less on the internet — you are never going to have the time to read.

Lists of the best CDs of the year, however, I find much more cheering. And with a subscription to Apple Music or the like, you can quickly sample a fair selection of the recommendations that you’d earlier missed, and then listen to a goodly virtual pile of the discs that grab you the most, all in the time it would take to get through that six hundred page history book you aren’t going to read ….

Well, I’ve missed the appropriate time to give my own recommendations from the new classical CDs released in 2015 — and anyway, to be honest, it wouldn’t have been that exciting, but mostly just a rather predictable subset of the monthly recommendations in the Gramophone (predictable, at any rate, given the sort of CDs and concerts mentioned here over the years). So let me begin the year by starting something hopefully a bit more interesting, namely a fairly regular series of  ‘CD choice’ posts, mentioning a disc that I’ve been listening to with enjoyment over the previous few days, perhaps emphasizing discs not as well known as they might be. I’ll try, by the way, mostly to mention recordings available on Apple Music (and presumably on other subscription services). It could be a new release, or an old disc that I forgotten that I had, a recent charity-shop find, something caught by chance on internet radio … Who knows? We’ll just see how it goes! [I was thinking of posting weekly, hence the initial title ‘CD of the week’, but I quickly thought better of it — not because I couldn’t recommend a  CD every seven days, but because that many posts on music would unbalance what is still supposed to be mainly a logic-related blog!]

First up, then, a delightful disc first released in 2014, the oboist Albrecht Mayer’s “Lost and Found”. This is subtitled “Oboenkonzerte des 18. Jahrhunderts von Hoffmeister, Lebrun, Fiala und Kozeluh”, which sounds potentially worthy but dull; but in fact, this is simply very, very enjoyable.

So these are concertos for oboe and cor anglais from around the 1780s, contemporary with Mozart and Haydn, from four other composers well known in the day (and not entirely “lost” since!). The music is immediately engaging yet certainly stands up to repeated listenings. The playing on the CD is terrific (as the Gramophone agrees). Try it!

CD of the week #1

Those end-of-year lists of recommended books are really rather depressing, aren’t they? Even setting aside the pretentious, the uninviting, the distinctly esoteric, there remain all those novels, all those biographies, all those histories, and much more, books that do sound so enticing, yet which you know — despite your best resolutions to read more, and idle less on the internet — you are never going to have the time to read.

Lists of the best CDs of the year, however, I find much more cheering. And with a subscription to Apple Music or the like, you can quickly sample a fair selection of the recommendations that you’d earlier missed, and then listen to a goodly virtual pile of the discs that grab you the most, all in the time it would take to get through that six hundred page history book you aren’t going to read ….

Well, I’ve missed the appropriate time to give my own recommendations from the new classical CDs released in 2015 — and anyway, to be honest, it wouldn’t have been that exciting, but mostly just a rather predictable subset of the monthly recommendations in the Gramophone (predictable, at any rate, given the sort of CDs and concerts mentioned here over the years). So let me begin the year by starting something hopefully a bit more interesting, namely a weekly  ‘CD of the week’ post, mentioning a disc that I’ve been listening to with enjoyment over the last few days, perhaps something a bit out of the ordinary. I’ll try to do this every Friday. I’ll also try, by the way, mostly to mention recordings available on Apple Music (and presumably on other subscription services). It could be a new release, or an old disc that I forgotten that I had, a recent charity-shop find, something caught by chance on internet radio … Who knows? We’ll see how it goes!

First up, a delightful disc first released in 2014, the oboist Albrecht Mayer’s “Lost and Found”. This is subtitled “Oboenkonzerte des 18. Jahrhunderts von Hoffmeister, Lebrun, Fiala und Kozeluh”, which sounds potentially worthy but dull; but in fact, this is simply very, very enjoyable.

So these are concertos for oboe and cor anglais from around the 1780s, contemporary with Mozart and Haydn, from four other composers well known in the day (and not entirely “lost” since!). The music is immediately engaging yet certainly stands up to repeated listenings. The playing on the CD is terrific (as the Gramophone agrees). Try it!

Sic transit …

The flood of freely available downloads of pre-2005 mathematics and philosophy books from Springer — including many logical classics, for which I posted a couple of very partial “taster menus” here — didn’t last long! Two days on, the free downloads are no longer available. I believe that there may have been issues about Springer making available books for which they didn’t have the full ownership of the copyright, without consulting authors.

It would be cruel to those who missed the party to leave up the previous posts detailing what they’ve missed; so those posts are for the moment deleted. We can only guess at the background story. We will just have to see whether, in due course, Springer do start making older books freely available when they can (I can see why it might be in their overall interests to do so).

Springer flood: more logic

In that Springer flood of not-so-old mathematics and philosophy books made available to download, there is a vast range of interesting finds — note, for example, there are all the pre-2005 volumes of the Synthese library. But logicians and their students might like to note in particular that all the first edition, and most of the second edition, of the often extremely useful Handbook of Philosophical Logic is freely available (but not, of course, the equally useful Handbook of the History of Logic, which has a different publisher!).

Volumes in the Handbook’s second edition in particular have previously been prohibitively expensive, and I can imagine that many less well-funded university libraries haven’t bought them all. The long survey essays are a bit patchy, but often excellent.  So let me give  links to contents lists and downloadable files. Here’s the first edition:

Vol I: Elements of Classical Logic
Vol II: Extensions of Classical Logic
Vol III: Alternatives to Classical Logic
Vol IV: Topics in the Philosophy of Language

And here are the freely available volumes of the wider-ranging second edition (the volumes are untitled so I give some partial indications of contents):

Vol 1: [First order, higher order logic: basic recursion theory]
Vol 2: [Types of deductive system; algebraic and many-valued logics]
Vol 3: [Modal Logic]
Vol 4: [Conditional logic, dynamic logic, etc.]
Vol 5: [Intuitionistic logic, free logic]
Vol 6: [Relevance logic, paraconsistent logic]
Vol 7: [Tense logic]
Vol 8: [Sequent systems for modal logics, deontic logic]
Vol 9: [Proof theory and meaning, negation, logical frameworks]
Vol 10: [Names and descriptions, indexicals, property theories]

I’ll need, when the mood takes me, to update the TYL Guide, both to give links to now freely available Springer books, but also in one or two places to make more use of some Handbook articles. And I will need to update the categories page too. But, hey, it is still the festive season, so one thing at a time!

Springer make many logic books freely downloadable

Springer have made very many mathematics and philosophy books more than ten years old freely downloadable. Logicians at various levels may in interested, for example, in the following:

Van Dalen, Logic and Structure (4th edn)
Monk, Mathematical Logic
Manin, A Course in Mathematical Logic
Andrews, An Introduction to Mathematical Logic and Type Theory
Fitting & Mendelsohn, First-Order Modal Logic
Poizat, Model Theory
Marker, Model Theory
Marcia & Toffalori, A Guide to Classical and Modern Model Theory
Schütte, Proof Theory
Hendricks et al, eds., Proof Theory: History and Philosophical Significance
Halmos, Naive Set Theory
Moschovakis, Notes on Set Theory (1st edn)
Devlin, The Joy of Sets
Jech, Set Theory
Kechris, Classical Descriptive Set Theory
Kanamori, The Higher Infinite
Hermes, Enumerability, Decidability, Computability
Bridges, Computability
Mac Lane, Categories for the Working Mathematician
Mac Lane & Moerdijk, Sheaves in Geometry and Logic

Well, that slightly random 20 is enough to start with — though for fun, let me also mention Aigner & Ziegler, Proofs from THE BOOK. Try searching Springer Link for many more. (Of course, depending on your university’s library policy, you may already have had access via Springer Link to these and newer books: but the free access to older books now, or at least for the moment, appears to be universally available.)

If anyone knows whether this is a new long-term policy at Springer, or is a more short-lived Christmas treat, do please let us know in the comments! For a start, I don’t want to spend time updating the TYL Guide with some of these links if the access is temporary.