Peter Smith's Blog, page 96

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.

A Christmas card

Gentile da Fabriano, Adoration of the Magi (detail), 1423

With all good wishes for a very happy and peaceful Christmas.

Serendipity

We have far too many books for a house this small, despite our best efforts to make room on shelves by passing on the unloved or the never-to-be-reread to Oxfam. So these days, we try not to buy a newly published book if neither of us is going to read it more or less immediately (of course, a different rule applies to happy second-hand finds). That’s why, when Kate Atkinson’s companion piece to her truly brilliant Life after Life came out, despite our both very much looking forward to reading it, we didn’t buy it immediately, both already having tall piles of books waiting for us. And then it got to the point when a paperback was announced for the end of this month, and we decided to hang on and get it to read over festive season.

But ah,  a couple of weeks ago, in  the National Trust shop at Wimpole, there it was — an almost pristine copy of A God in Ruins. The beautifully produced hardback for less than half the cost of a paperback. And it’s quite unreasonably cheering — isn’t it? — this kind of serendipitous find. Well ok, not  that serendipitous, when you think about it: Kate Atkinson is a best-selling author, we often drop in to take a quick look at the book shelves of charity shops large and small, so I suppose it was pretty likely that we’d stumble on a copy over the months since it came out. But even that thought doesn’t make the find less cheering. It isn’t the matter of saving a few pounds (or of giving the money to a charity rather than a chain bookstore); that’s nice, to be sure, but it doesn’t account for the pleasure, the happy feeling engendered by the little smidgin of good fortune. And it’s the sort of little thing that sticks in the mind,  “Do you remember finding that by chance when we were on holiday in …?”, it becomes part of your history of your encounter with the book in a way that just marching into Blackwells and picking the volume off a pile never does.

It is, or rather was, the same with CDs, the pleasure of the happy find in charity shops of something well-known that you’ve been wanting for a while, or of something quite obscure but intriguing. I had much more enjoyment over some years acquiring the complete Hyperion Schubert Edition — all 37 volumes of Graham Johnson’s astonishing exploration of all Schubert’s songs with various singers — mostly second-hand than I ever would have done just buying the lot new. I have had more fun again discovering baroque composers you’ve never heard of, or later obscure Bohemians, and the rest. (Of course there are plenty of misses as well as the hits, but the few pounds have gone to charity, so the misses don’t matter at all.)  But now I mostly use Apple Music to stream music, for there is precious little space for more CDs; and I miss that kind of serendipity. Yes, of course, there is an unending source of music to explore there for the small subscription. Yes, of course, you make happy finds. But, dinosaur that I am, it just doesn’t feel the same.

No way, though, are we swapping real books for e-books. Putting a few on the iPads when travelling is fine: but a well-made real book remains a continuing delight that we are not giving up.  So we’ll carry on the pleasurable truffling through the charity shops: it’s the splendid Oxfam bookshop in Saffron Walden tomorrow, I think …

And how was A God in Ruins? As good as they say. Just wonderful.

Teach Yourself Logic 2016

You need something good to read over the holidays; so here is the Teach Yourself Logic 2016: A Study Guide!

For anyone who doesn’t know TYL, it is aimed at philosophers (who have already done a baby logic course) and mathematicians who are trying to teach themselves some mathematical logic and need a guide through the very large literature.

For those who are familiar with the 2015 version, there are some presentational changes aimed at making the Guide look a bit less daunting and more manageable, a few new recommendations or changes of view, and a lot of minor tinkering.

As always, constructive comments are immensely welcome. And many thanks to those who have made suggestions over the last year. I fear I may have overlooked/forgotten about some good ones: in which case, nag me! But most logic teachers are busy people, and haven’t time to wade through an 89 page Guide, however zestfully written it is. So, in the new year, I might start putting some cut-down excerpts here on the blog, to ask for feedback on specific sections. I’m sure all kinds of improvements could be made; but having spent quite a bit of time on the new version already, and to prevent myself tinkering away more over the holidays when I should be doing more fun things, here it is. Enjoy!

Publicize your new book here?

It is difficult to know which statistics package to believe. Google Analytics says that Logic Matters steadily gets something in the order of 12,000 unique visitors a month, AWStats puts it around 18,000. And both say that the number of visits a month is at least three times as high. The WordPress stats are significantly higher again. It’s really nice to know that I’m not talking to myself!

The costs of hosting, backup, the domain names, etc. are not trivial. But they are only of the order of £200 a year, and since I do get a lot out of keeping Logic Matters going, that’s fine by me. So I’m not inclined to go to the bother of trying to emulate some Well-Known Blogs and seek paid advertising (I doubt that I’m missing out on much, given that the traffic on this site, though substantial, is not in the same league).

However … If you have a new book in mathematical logic or philosophy of maths/philosophy of logic that you would like to publicize at the top of the side bar that appears on most pages — i.e. where the cover image of my Gödel book currently appears –then get in touch (email at the foot of the “About” page). If I like the sound of it, then the deal will be that you send me a free copy of the book, and then you get at least four weeks in the sidebar (cover image linked to publisher’s site or Amazon). I save a bit on book bills, and you — for the cost of postage and one of your free author’s copies — get exposure to a large but very select audience of logic fans. Fair swap?

Conference: Philosophy of mathematics — truth, existence and explanation.

Philosophy of maths AND Italy — what’s not to like? So let me note that the second conference of the Italian Network for the Philosophy of Mathematics has been announced for 26-28 May 2016, University of Chieti-Pescara, Chieti, Italy.  

The invited speakers are Volker Halbach (University of Oxford), Enrico Moriconi (University of Pisa), Achille Varzi (Columbia University) together with ‘early career’ speakers Marianna Antonutti Marfori (IHPST, Paris) and Luca Incurvati (University of Amsterdam).

This is an English language conference, and there is a call for abstracts for contributed talks “in any area of philosophy of mathematics connected with the issues of truth, existence, and explanation”.  All the details can be found at the FilMat website here.

Book Note: Leary and Kristiansen, A Friendly Introduction …

I’ll return to say more about Tony Roy’s text in a day or two, but here is a new version of my earlier Book Note on the ‘Friendly Introduction to Mathematical Logic’, revised to take into account the expanded second edition. As you will see, very warmly recommended.

Christopher C. Leary and Lars Kristiansen’s A Friendly Introduction to Mathematical Logic (Milne Library 2015: pp. 364 — ISBN 978-1-942341-07-9) is the second, significantly expanded, edition of a fine book originally just authored by Leary (Prentice Hall, 2000: pp. 218).  The book is now available at a very attractive price; the  main differences between the editions are a long new chapter on computability theory, and some 75 pages of solutions to exercises.

So how friendly is  A Friendly Introduction? – meaning, of course, ‘friendly’ by the standard of logic books!

I do like the tone a great deal (without being the least patronizing, it is indeed relaxed and inviting), and the level of exposition seems to me to be very well-judged for an introductory course. The book is officially aimed mostly at mathematics undergraduates without assuming any particular background knowledge. But as the Preface notes, it should also be accessible to logic-minded philosophers who are happy to work at following rather abstract arguments (and, I would add, who are also happy to skip over just a few inessential elementary mathematical illustrations).

What does the book cover? Basic first-order logic (up to the L-S theorems), the incompleteness theorems, and some computability theory. But by being so tightly focused, this book rarely seems to rush at what it does cover: the pace is pretty even. The authors do opt for a Hilbertian axiomatic system of logic, with fairly brisk explanations. (If you’d never seen before a serious formal system for first-order logic this could initially make for a somewhat dense read: if on the other hand you have been introduced to logic by trees or seen a natural deduction presentation, you would perhaps welcome a paragraph or two explaining the advantages for present purposes of the choice of an axiomatic approach here.) But the clarity is indeed exemplary.

Some details Ch. 1, ‘Structures and languages’, starts by talking of first-order languages (The authors make the good choice of not starting over again with propositional logic, but assume that most readers will know their truth-tables so just give quick revision). The chapter  then moves on to explaining the idea of first order structures, and truth-in-a-structure. There is a good amount of motivational chat as we go through, and the exercises – as elsewhere in the book – seem particularly well-designed to aid understanding. (The solutions to exercises added to the new edition makes the book even more suitable for self-study.)

Ch. 2, ‘Deductions’, introduces an essentially Hilbertian logical system and proves its soundness: it also considers systems with additional non-logical axioms. The logical primitives are ‘ ∨ ’, ‘¬’, ‘∀’ and ‘ = ’. Logical axioms are just the identity axioms, an axiom-version of ∀-elimination (and its dual, ∃-introduction): the inference rules are ∀-introduction (and its dual) and a rule which allows us to infer φ from a finite set of premisses Γ if it is an instance of a tautological entailment. I don’t think this is the friendliest ever logical system (and no doubt for reasons of brevity, the authors don’t pause to consider alternative options); but it certainly is not horrible either. If you take it slowly, the exposition here should be quite manageable even for the not-very-mathematical.

Ch. 3, ‘Completeness and compactness’, gives a nice version of a Henkin-style completeness theorem for the described deductive system, then proves compactness and the upward and downward Löwenheim-Skolem theorems (the latter in the version ‘if L is a countable language and is an L-structure, then has a countable elementary substructure’ [the proof might be found just a bit tricky though]). So there is a little model theory here as well as the completeness proof: and you could well read this chapter without reading the previous ones if you are already reasonably up to speed on structures, languages, and deductive systems. And so, in a hundred pages, we wrap up what is indeed a pretty friendly introduction to FOL.

Ch. 4,  ‘Incompleteness, from two points of view’ is a helpful bridge chapter, outlining the route ahead, and then defining   and wffs (no subscripts in their usage, and exponentials are atomic  — maybe a footnote would have been wise, to help students when they encounter other uses). Then in Ch. 5, ‘Syntactic Incompleteness—Groundwork’, the authors (re)introduce the theory they call N, a version of Robinson Arithmetic with exponentiation built in. They then show that (given a scheme of Gödel coding) that the usual numerical properties and relations involved in the arithmetization of syntax – such as, ultimately, Prf(m, n), i.e. m codes for an N-proof of the formula numbered n – can be represented in N. They do this by the direct method. That is to say, instead of [like my IGT] showing that those properties/relations are (primitive) recursive, and that N can represent all (primitive) recursive relations, they directly write down wffs which represent them. This is inevitably gets more than a bit messy: but they have a very good stab at motivating every step working up to showing that N can express Prf(m, n) by a  wff. If you want a full-dress demonstration of this result, then this is one of the most user-friendly available.

Ch. 6, ‘The Incompleteness Theorems’, is then pretty short: but all the groundwork has been done to enable the authors now to give a brisk but very clear presentation, at least after they have proved the Diagonalization Lemma. I did complain that, in the first edition, the proof of the Lemma was slightly too rabbit-out-of-a-hat for my liking. This edition I think notably softens the blow (one of many such small but significant improvements, as well as the major additions). And with the Lemma in place, the rest of the chapter goes very nicely and accessibly. We get the first incompleteness theorem in its semantic version, the undecidability of arithmetic, Tarksi’s theorem, the syntactic version of incompleteness and then Rosser’s improvement. Then there is nice section giving Boolos’s proof of incompleteness echoing the Berry paradox. Finally, the second theorem is proved by assuming (though not proving) the derivability conditions.

The newly added Ch. 7, ‘Computability theory’ starts with a very brief section on historical origins, mentioning Turing machines etc.: but we then settle to exploring the $\latex \mu$-recursive functions. We get some way, including the S-m-n theorem and a full-dress proof of Kleene’s Normal Form Theorem (with due apologies for the necessary hacking though details) and meet the standard definition of the set  where is the domain of the computable function with index . The uncomputability of is then used, in the usual sort of way, to prove  the undecidability of the Entscheidungsproblem, to re-prove the incompleteness theorem, and in tackling Hilbert’s 10th problem. This is all nicely done in the same spirit and with the same level of accessibility as the previous chapters.

Summary verdict If you have already briefly met a formally presented deductive system for first-order logic, and some account of its semantics, then you’ll find the opening two chapters of this book very manageable (if you haven’t they’ll be a bit more work). The treatment of completeness etc. in Ch. 3 would make for a nice stand-alone treatment even if you don’t read the first two chapters. Or you could just start the book by reading §2.8 (where N is first mentioned), and then read the excellent ensuing chapters on incompleteness and computability with a lot of profit. A Friendly Introduction is indeed in many ways a very unusually likeable introduction to the material it covers, and has a great deal to recommend it. Very warmly recommended (so ensure that your library gets a copy!)

Postcard from Florence

Bronzino meets the Ponte Vecchio

To Florence for the better part of six days. The city is as usual wonderful in December, far from the summer heat and the summer crowds of foreign tourists (though we forgot about the Immacolata Concezione holiday, so there are Italian crowds for a couple of days). Blue skies and bright sun too.

The Uffizi is at long last being renovated and pictures rehung (the parts so far done are a huge improvement): the Leonardo Annunciation particularly beautifully displayed — On a sunny winter’s day, the view from San Miniato is breathtaking — Buy soap in that most beautiful shop, the Officina Profumo-Farmaceutica di Santa Maria Novella, complete with its own frescos — The Magi Chapel in the Palazzo Medici Riccardi as stunning as ever, and almost empty, so we can stay as long as we want — Night time walks round Florence, with added light shows (as above) — Rigoletto, conducted by Zubin Mehta, wonderfully sung with a fine cast and only slightly daft staging, then walking back along the Arno late on a fine night — A fresco day (Santa Croce, the Brancacci Chapel, and not least Ghirlandaio at Santa Maria Novella) — Fiesole — and more …

We eat and drink very well too (notably at Olio e Convivium and Il Santo Bevitore). So we have an exceptionally good time. We know we are lucky to be able to do this kind of thing.