Peter Smith's Blog, page 44

Distractions

A number of distractions (good distractions!) from logical writing projects recently. For a start, we spent some days near the Suffolk coast, a few miles inland from Aldeburgh. The first time we’ve stayed away from Cambridge in these days of Covid. Initially, we were quite irrationally tentative about doing that — but this seems a quite common reaction after more than a year of restrictions. But we had a great time. The weather was good enough for lots of breezy walks. And being by the sea always very much lifts the spirits.

Back home to hours of nerdy amusement for me, playing around with a new bit of serious software, Affinity Publisher. Extraordinarily inexpensive but powerful alternative to Adobe InDesign. Lots of terrific reviews, and certainly seems quite excellent to me. Have used it to design good-enough covers — still big and red but slightly snappier — for hardback versions of the Big Red Logic Books, and I’ll use the same redesign for the republished paperbacks. More about this when the first hardbacks become available, shortly I hope.

Not least among the distractions, my new 24″ iMac has arrived. Entry-level in blue, since you asked.

Took a day to clear my study into a fit state to have somewhere to put it, and half another day to set it up, transferring files across, installing software and so on. But I’m bowled over by the result.

I’ve not had a desktop machine for well over a decade. So the difference between this and my trusty (but soon-to-be-traded-in) six year old MacBook Pro is wonderful. The big screen in particular is amazing. For LaTeX, being able to work with source code and PDF output side-by-side in big windows is a delight. And LaTeX compiles a large diagram-heavy file about 2.5 times faster too, which is a nice bonus.

The iMac itself is a thing of beauty (significantly better in reality, I’d say, than in the adverts). Even Mrs Logic Matters is impressed. The front-side colour is pleasingly muted. And those “white bezels” which Mac forums were complaining about a lot when the machine was launched in fact strike me as something of a design triumph. They aren’t glaring white — but a sort of slighty-grey off-white and so, in use, the edges of the screen shade off in your peripheral vision into your background wall (at least, if your background wall is fairly neutral like my study wall), in a way which works very well. I could go on. But if you were wavering about whether to get one, I’d say just do it!

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Mitsuko Uchida plays the Schubert Impromptus

The wonderful Mitsuko Uchida played the two sets of Schubert Impromptus at Wigmore Hall a few nights ago. You can catch the Radio 3 broadcast here. There is also a video recording on Wigmore Hall’s site, and on Youtube. Such extraordinary playing of such great music.

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Going back and forth

You might have missed Jacob Plotkin’s comment a week ago about about a reference (by me!) to “Cantor’s back-and-forth proof”:

Cantor did not invent/discover or use this method of proof. That honor belongs independently to Felix Hausdorff and E.V. Huntington.

Jacob gave a reference to the short paper where he spells out the evidence. I’ve now had a chance to read it, and it is interesting and instructive (and after all, it is always better to get our history right rather than wrong). So let me add a link to where you can find the paper!

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Now’s the time to buy …

For reasons I really won’t bore you with, I need to be republishing my three print-on-demand paperback books, in order to make them more widely available (e.g. in Australia). I’ll also take the opportunity of arranging for — still inexpensive — hardbacks to be published, which will become available through bookstores and library suppliers (with proper ISBNs, etc., to gladden your librarian’s heart). I hope this is all done by mid June.

At the moment you have to use Am*z*n: Intro to Formal Logic is just £8.99/$11; Intro to Gödel’s Theorems is £7.99/$9.99; and Gödel Without (Too Many) Tears is £3.99/$4.99 — with comparable prices in euros. Amazing bargains, of course :) But for reasons I also won’t bore you with, the paperback prices might have to increase a little on re-publishing. So now could be just the time to buy one of the Big Red Logic Books, if you’ve been dithering. Just saying ….

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PHQ and Boris Giltburg, radio recording

Boris Giltburg and the Pavel Haas Quartet were playing last night to a live audience at the Louisiana Museum of Modern Art (that’s in Denmark, near Elsinore, on the Øresund coast).

The concert — in which they performed the Brahms and Dvorak Piano Quintets — was broadcast on Danish radio, and the concert is available for listening online on DR P2. (Player at the foot of the page; their concert starts at 43:04). Great stuff!

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Roman Kossak’s Model Theory for Beginners

Following on from his Mathematical Logic (2018), Roman Kossak has now published Model Theory for Beginners: 15 Lectures (College Publications, 2021). As the title indicates, the fifteen chapters of this short book — just 138 pages — have their origin in introductory lectures given to graduate students in CUNY. Roughly speaking, the topics of the first half of this new book overlap quite closely with the second half of his previous book. And after grumbling a bit about Part I of that earlier book, I did warm considerably to the model-theoretic Part II, which I think makes for a very approachable elementary introduction to a cluster of issues about definability.

The new treatment is aimed at a rather more sophisticated reader, the writing is a bit less relaxed, and indeed becomes increasingly terse as the book progresses (in later chapters, I could often have done with a sentence or two more motivational chat). But overall, this strikes me as a welcome book. Though I’m at all not sure it is all suitable for beginners.

In a little more detail, after initial chapters on structures and (first-order) languages, Chapters 3 and 4 are on definability and on simple results such as that ordering is not definable in (Z, +). Chapter 5 introduces the notion of types, and e.g. gives Cantor’s back-and-forth proof that countable dense linearly ordered sets without endpoints are isomorphic to (Q,

So there is a somewhat different arrangement of initial topics here, compared with books whose first steps in model theory are applications of  compactness. But the early chapters are indeed nicely done. However, I don’t think that Kossak’s Chapter 8 will be found an outstandingly clear and helpful first introduction to applications of compactness — it will probably be best read after e.g. Goldrei’s nice final chapter in his logic text.

Chapter 9 is on categoricity — in particular,  countable categoricity. (Very sensibly, Kossak wants to keep his use of set theory in this book to a minimum; but he does have a section here looking at κ-categoricity for larger cardinals κ.) And now the book starts requiring rather more of its reader. Chapter 10 is on indiscernibility and the Ehrenfeucht-Mostowski Theorem: but it is difficult to get a sense from this chapter of quite why this matters.

Up to this point, the structures we’ve been looking at are all officially relational. Chapter 11 adds functions, and discusses Skolem functions and Skolemization (this could have been more relaxed and helpful). We return to arithmetic in Chapter 12; there’s a compressed  discussion leading up to a version of Robinson’s model-theoretic proof of Tarski’s theorem of the arithmetic undefinability of arithmetic truth. But I rather doubt that this will be readily accessible to someone who hasn’t already read e.g. some of Kaye’s book on non-standard models of PA and met ideas like overspill.

The last three chapters are more advanced still, on saturation, automorphisms of recursively saturated structures, and (very briefly) stability. Are these topics for those just starting out on model theory? That’s a judgement call. But I suspect that the mode of presentation could be found quite challenging by many beginners — for me, more classroom asides in later chapters would have been welcome.

So as with Kossak’s earlier Mathematical Logic, then, I have rather different reactions to the two halves of Beginning Model Theory. But I’d say that the first eight or nine chapters do work very well under the advertised title (and I’ll be recommending them in the Study Guide). Later chapters are probably to be read in parallel with familiar moderately advanced texts like Marker’s classic.

Finally, a bonus point for publishing very inexpensively with College Publications, and with tidy LaTeX layout too (however, they still can’t design a nice title page and verso!). But dock a point for the number of minor typos …

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Mathematical logic from a compsci angle?

In the Study Guide entry on First-Order Logic, after the list of main reading suggestions, there is a further list of suggested parallel/additional reading which ends by warmly recommending Melvin Fitting’s First-Order Logic and Automated Theorem Proving (Springer, 2nd edn 2012). Although published in a series aimed at compsci students, this should certainly appeal to logicians who are primarily philosophers or mathematicians but who want to know a little more about some themes (e.g. resolution proof systems) of special interest to computer science, and who want to re-encounter some familiar ideas approached from a slightly different angle. You can, if you want, skip some of the bits on Prolog and still get a particularly elegantly and clearly written account.

I’ve been wondering whether there are other books also written for computer scientists which could similarly appeal to the Guide’s intended readers. I know, of course, Michael Huth and Mark Ryan’s Logic in Computer Science (CUP, 2nd edn 2004). I haven’t reread this recently, but I recall it as being attractively and clearly written — the long first two chapters on propositional and predicate logic are well done, with a few interesting extras for the philosophical or mathematical reader (e.g. on SAT solvers). But then the later chapters go off in directions no doubt of key concern to computer scientist, but less interesting for the rest of us (for me, anyway!).

I’ve had Mordechai Ben-Ari’s Mathematical Logic for Computer Science (Springer 3rd edn 2012) recommended to me. But I thought this pretty second-rate.   The level of exposition is poor, and indeed at points seemingly outright confused (e.g. about the status of the Deduction Theorem for a Hilbert system). Someone who already has a grip on the standard math logic approaches could, I guess, get something out of the book by diving straight into the chapters on propositional resolution, SAT solvers, and first-order resolution, for example. But I didn’t find this material well explained: it is surely treated more pleasingly elsewhere.

To repeat, then, I’m interested in locating logic books coming from a compsci angle which will however also appeal to someone whose main interest remains in philosophy or mathematics. Luis Augusto’s book is advertised as aimed at such readers, but you know what I think of that. So are there other options?  I’d be very interested to hear!

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Not so unlocked

Cows sheltering under a tree, in the meadow below Ely Cathedral. Not the most clement spring day, but still a lovely sight. We are still erring on the side of caution in our strayings from home (because too many others are evidently not). But Ely makes a welcome change of scene from our small corner of Cambridge. And there’s the wonderful Topping’s book shop to visit. Small steps out of lockdown. There’s a longish way for us to go on to whatever is the new normal; but I think it is the same story for many of our generation.

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Luis Augusto’s Formal Logic

[image error]“Of making many logic books there is no end; and much study is a weariness of the flesh.” The author of logical Ecclesiastes had probably just been reading the likes of Luis M. Augusto’s unnecessary Formal Logic: Classical Problems and Proofs (College Publications, 2019). I’ve noted before that this publisher’s quality control is lousy. Fortunately, because its books are relatively inexpensive, you can take the chance and order one which has a tempting-seeming  blurb, without cursing too much if the punt doesn’t come off.

This particular book aims to highlight problems which, though they “feature in introductory logic textbooks aimed at computer science students, … are largely or wholly absent from textbooks targeting a mathematical or philosophical studentship.” Looking for books with a compsci angle for the Logic Guide, I was intrigued.  But, apart from being written in poor almost-English, the technical exposition here is unappealingly hard going, and the level of motivational explanation third-rate. It could be so much better. So this is just a warning note: if you are similarly tempted by the blurb for this book, simply resist. And if that sounds a bit tetchy, it could be because my flesh is more than a bit weary after trying to study it for a day.

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Luis Augusto, Formal Logic

[image error]“Of making many logic books there is no end; and much study is a weariness of the flesh.” The author of logical Ecclesiastes probably had in mind the likes of Luis M. Augusto’s unnecessary Formal Logic: Classical Problems and Proofs (College Publications, 2019). I’ve noted before that this publisher’s quality control is lousy. Fortunately, because its books are relatively inexpensive, you can take the chance and order one which has a tempting-seeming  blurb, without cursing too much if the punt doesn’t come off.

This particular book aims to highlight problems which, though they “feature in introductory logic textbooks aimed at computer science students, … are largely or wholly absent from textbooks targeting a mathematical or philosophical studentship.” Looking for books with a compsci angle for the Logic Guide, I was intrigued.  But, apart from being written in poor almost-English, the technical exposition here is unappealingly hard going, and the level of motivational explanation third-rate. It could be so much better. So this is just a warning note: if you are similarly tempted by the blurb for this book, simply resist. And if that sounds a bit tetchy, it could be because my flesh is more than a bit weary after trying to study it for a day.

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