Year: 2022

Beginning Category Theory: Chs 1 to 11 (and more)

To avoid readers having to juggle two PDFs, and to keep at least some cross-references between new and old material functioning, I have decided to put the newly revised chapters together with the old unrevised chapters from the Gentle Intro into one long document. So here is Beginning Category Theory which starts with eleven revised chapters, followed by all the remaining old chapters [with prominent headline warnings about their unrevised status].

The two newly revised chapters are

  1. Pairs and products, pre-categorially [Motivational background]
  2. Categorial products introduced [Definitions, examples, and coproducts too]

Note: these early revised chapters are not final versions. Revised chapters get  incorporated when I think that they are at least better than what they replace, not when I think they are as good as they could be. So, needless to say, all comments and corrections will be very gratefully received. Onwards!

Greg Restall, Proofs and Models in Philosophical Logic

I notice that Juliette Kennedy’s book on Gödel’s incompleteness theorems in the Cambridge Elements series has now also been published. I’ll no doubt get round to commenting on that in due course, along with John Bell’s short book on type theory. But first, let me say something about Greg Restall’s contribution to the series: as I said, for the coming few days you can freely download a PDF here.

There does seem little consistency in the level/intended audience of the various books in this series. As we will see, Bell’s book is pretty hard-core graduate level, and mathematical in style and approach. Burgess’s book I found to be a bit of a mixed bag: the earlier sections are nicely approachable at an introductory level; but the later overview of topics in higher set theory — though indeed interesting and well done — seems written for a different, significantly more mathematically sophisticated, audience. It is good to report, then, that Greg Restall — as his title promises — does keep philosophers and philosophical issues firmly in mind; he writes with great clarity at a level that should be pretty consistently accessible to someone who has done a first formal logic course.

After a short scene-setting introduction to the context, there are three main sections, titled ‘Proofs’, ‘Models’ and ‘Connections’. So, the first section is predictably on proof-styles — Frege-Hilbert proofs, Gentzen natural deduction, single-conclusion sequent calculi, multi-conclusion sequent calculi — with, along the way, discussions of ‘tonk’, of the role of contraction in deriving certain paradoxes, and more. I enjoyed reading this, and it strikes me as extremely well done (a definite recommendation for motivational reading in the proof-theory chapter of the Beginning Math Logic guide).

I can’t myself muster quite the same enthusiasm for the ‘Models’ section — though it is written with the same enviable clarity and zest. For what we get here is a discussion of variant models (at the level of propositional logic) with three values, with truth-value  gaps, and truth-value gluts, and with (re)-definitions of logical consequence to match, discussed with an eye on the treatment of various paradoxes (the Liar, the Curry paradox, the Sorites). I know there are many philosophers who get really excited by this sort of thing. Not me. However, if you are one, then you’ll find Restall’s discussion a very nicely organized introductory overview.

The shorter ‘Connections’ section, as you’d expect, says something technical about soundness and completeness proofs; but it also makes interesting remarks about the philosophical significance of such proofs, depending on whether you take a truth-first or inferentialist approach to semantics. (And then this is related back to the discussion of the paradoxes.)

If you aren’t a paradox-monger and think that truth-value gluts and the like are the work of the devil, you can skim some bits and still get a lot out of reading Restall’s book. For it is always good to stand back and see an area — even one you know quite well — being organised by an insightful and eminently clear logician. Overall, then, an excellent and very welcome Element.

John Burgess, Set Theory

Of making many logic books there is no end. So a project like the Beginning Mathematical Logic study guide has no terminus. I hit the “pause” button at an arbitrary moment, and published the first book edition a couple of months ago, because I wanted to concentrate for a while on other things. But needless to say, there are already a number of new publications which are possible candidates for being mentioned in the next edition. In particular there are the four first books in the Cambridge Elements series on Philosophy and Logic. Three have already been published — in fact, the first of them just as I was finishing the guide, namely John Burgess’s Set Theory. Then we have John Bell on Higher-Order Logic and Type Theory, and Greg Restall on Proofs and Models in Philosophical Logic. Juliette Kennedy’s Gödel’s Incompleteness Theorems is due any day. The first three should already be accessible, then, via Cambridge Core if your library has a subscription; and even better, Proofs and Models is free to download here for another week. I’ll try to say something brief about each of these books over the coming days.

First then Burgess on sets. Like other Elements, this little book is about seventy, not-very-packed, pages (perhaps 30K words?). More than an encyclopedia article, or a handbook chapter, but half the length of a short book like my Gödel Without Tears. Books in the series are aimed at providing “a dynamic reference resource for graduate students [and] researchers”. And that’s already a tall order for a book on this topic, at any rate: for most graduate students in philosophy (even if logic-minded) are likely to be pretty much beginners when it comes to tackling some set theory — and a book accessible to such beginners isn’t likely to also to be of much interest to researchers.

OK, forget the impossible prospectus, and let me try to assess the book in its own terms. First, I certainly enjoyed a quick read. It is engagingly written. And at various points in the later pages Burgess very helpfully put some order into my fragmentary knowledge, or offered genuinely illuminating remarks. So I endorse Rowsety Moid’s comment below, when he writes “I especially liked the second half — on ‘higher set theory’ — and the picture it gives of the various subject areas (descriptive set theory, continuum questions, combinatorial set theory) and techniques (large cardinals, forcing, inner models, infinite games, …) and of how they’re interrelated. I can’t recall anything else that gives as good an overview so briefly.” However, although set theory isn’t my special thing, I didn’t exactly come to this innocent of prior knowledge. And I do have to doubt whether later pages of the book will really be accessible to many of the intended student audience. OK, if may be that all the materials have officially been given to understand e.g. the Levy Reflection Principle on p. 55: but I suspect that a significant amount of mathematical maturity, as they say, would be needed to really appreciate what’s going on.

In headline terms, then, I don’t think that the book as a whole would work as advertised for many students. Still, the first half does make a nice motivating introduction, but one to be followed by (or as RM again comments, perhaps better read in conjunction with) a standard accessible introduction to set theory like Goldrei or Enderton. And then the enthusiast can return to try reading from §8 “Topics in Higher Set Theory” onwards to get a first overview of a few further more advanced topics, with a hope of getting a first inkling of what some of the interesting issues might be, before tackling a second-level set theory text.

Updated to use a comment from RM!

Like it or not ….

Drat, I missed the blog’s birthday. Now sweet sixteen!

In an experimental way, I have added a “like” button, just to appear on blog posts. So please “like” what you do like enough. It will be interesting to know what finds the most favour, and that might even affect a bit what I choose to blog about. Or not, as the case might be …

For a world gone awry

“The St Matthew Passion is one of the most moving experiences of our common humanity that it is possible to share. The story of the Passion of Christ remains today a living drama and moral dilemma of universal relevance, in which — whatever our spirituality or culture — all of us are confronted with our own mortality, our own search for answers. We all share its humanity. Bach’s immense genius is to step completely outside the liturgical framework by placing us at the very heart of the drama: we become the actors, we take part in the action, we feel it, in our sensibility and even physically. We traverse a drama that is above all human: injustice, betrayal, love, sacrifice, forgiveness, remorse, compassion, pity … In quite unprecedented fashion, Bach conveys and makes us feel the fragility and failings of humanity and describes a world gone awry, where love and faith are the only answers. In seeking to challenge and console the human conscience, he offers us genuine ‘balm for the soul’, universal and timeless.”

Thus Raphaël Pichon, in the booklet for his stunning new recording of the Matthew Passion on Harmonia Mundi with his Ensemble Pygmalion and a stella cast of singers. The short video linked above is of Sabine Devieilhe singing the aria ‘Aus Liebe will mein Heiland sterben’ from an earlier performance. And here is a wonderful video of a complete performance from last year.

Beginning Category Theory: Chs 1 to 9

Slow progress again but, as I said before, any progress is better than none. So here are Chapters 1 to 9 of Beginning Category Theory. [As always you may need to force a reload to get the latest version.]

And no, there isn’t really a new chapter. I’ve split what was becoming a baggy chapter about kinds of arrows into two, and I hope to have made some of it a fair bit clearer and better organised. The chapters are

  1. Introduction [The categorial imperative!]
  2. One structured family of structures. [Revision about groups, and categories of groups introduced]
  3. Groups and sets [Why I don’t want to assume straight off the bat that structures are sets]
  4. Categories defined [General definition, and lots of standard examples]
  5. Diagrams [Reading commutative diagrams]
  6. Categories beget categories [Duals of categories, subcategories, products, slice categories, etc.]
  7. Kinds of arrows [Monos, epics, inverses]
  8. Isomorphisms [why they get defined as they do]
  9. Initial and terminal objects

Ch. 3 has been mildly revised again, and as I said Ch.7 has been significantly improved. Various minor typos have been corrected. And there have been quite a few small stylistic improvements (including, I’m embarrassed to say, deleting over 50 occurrences of the word “indeed” …).

The Logic of Number

Having been so very struck by Russell and Frege as a student —  a long time ago in a galaxy far, far away — I have always wanted some form of logicism to be true. And the deviant form canvassed by Tennant back in his rich 1987 book Anti-Realism and Logic (I mean, deviating from neo-Fregean version we all know) is surely worth more attention than it has widely received. So I’m looking forward to reading Tennant’s latest defence of his form of logicism, in his new book The Logic of Number, published a few weeks ago at a typically extortionate OUP price.

On a quick browse, the book goes in a somewhat different direction to what I was expecting, having read Tennant’s 2008 piece ‘Natural Logicism via the Logic of Orderly Pairing’. The additional new apparatus for developing arithmetic added in that paper seems not to be in play again here. Rather, the book doesn’t (I think) develop arithmetic as far, but instead goes on to discuss logicist accounts of rationals and reals.

I’ll perhaps try to say something more about the book in due course, when some other reading commitments are done and dusted. Meanwhile you might well in fact be able to take a look if interested. For The Logic of Number, I’m glad to say, has just been also made available at Oxford Scholarship Online. So if your university library has a suitable OUP subscription, it should be free to read.

Beginning Category Theory: Chs 1–8

Here now are Chapters 1 to 8 of Beginning Category Theory

The new chapter is on initial and terminal objects; there have only been minor changes to other chapters from Chapter 4 onwards. These new chapters 1 to 8 are I think a significant improvement to the corresponding Chapters 1 to 6 of the old Gentle Introduction. Or at least, they are a significant improvement in clarity of content. But I don’t think I have yet quite hit the mark as far as tone/reader-friendliness is concerned. So I need to let these pages marinade for a few days, and then return to them (particularly to the last couple of action-packed chapters) to make them a little more relaxed. Onwards!

Beginning Category Theory: Chs 1 to 7

Here now are Chapters 1 to 7 of Beginning Category Theory

The new chapter is on kinds of arrows. I have also revised Chapter 3 (now preferring to talk about implementing structures in set theory, rather than to talk of set-theoretic surrogates or proxies — the change of rhetoric isn’t really a change of view, but will I hope slightly mollify some readers!).

I should say that Beginning Category Theory is very much a work in progress, and I can imagine these early chapters getting significantly revised to better fit the later ones in content and tone. But, for all that, I’m putting them online as I go along, when I get to the point of thinking that a new chapter is at least better than the corresponding old one in the Gentle Intro!

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