These old pieces are of very varying levels of sophistication, difficulty and interest: but I haven’t tried to impose any more order on the list other than a rough (reverse) chronological order.
- Notes on ‘the contemporary conception of logic’ gives evidence against a claim by Goldfarb (endorsed by some others) about the supposedly dominant conception of logic as dealing in schemata. (2013)
- Tennenbaum’s Theorem (a rewritten version of a reasonably accessible proof, with the old short discussion of its sometimes supposed conceptual significance which fed into the joint paper with Tim Button). 2013, revised 2014
- Critical Notice of Volker Halbach Axiomatic Theories of Truth and Leon Horsten The Tarskian Turn in Analysis 2013
- Review of Baaz et al. Kurt Gödel and the Foundations of Mathematics: Horizons of Truth in Philosophia Mathematica 2012.
- Review of Penelope Maddy Defending the Axioms (with Luca Incurvati) in Mind 2012
- Is ‘no’ a force-indicator? Sometimes, possibly (with Luca Incurvati) Analysis 2012
- The philosophical significance of Tennenbaum’s Theorem (with Tim Button) Philosophia Mathematica 201
- Review of Alan Weir Truth Through Proof in Mind 2011.
- Four lectures on the First Theorem given to maths students, Easter 2011.
- Squeezing arguments (on Kreisel’s argument — what it does and doesn’t show: expanding a bit on part of the Squeezing Church’s Thesis talk). Draft of paper in Analysis 2011.
- The MRDP Theorem (introduction to what it says and why it matters)
- Cuts, consistency and axiomatized theories (A short and naive introduction to the chapter in Negri and von Plato’s Structural Proof Theory on cut-elimination for certain kinds of theories — though you don’t have to have read the book to follow the story.)
- On chs 5 & 6 of Mary Leng’s Mathematics and Reality (concerning naturalism about mathematics)
- The Galois connection between syntax and semantics (explains Lawvere’s remark about ‘the familiar Galois connection between sets of axioms and classes of models, for a fixed [signature]’).
- Field on truth: how complex is too complex? (worries sceptically about the conceptual significance of Field’s intricate constructions).
- Curry’s paradox, Lukasiewicz and Field (some cheerfully naive notes introducing Ch. 4 of Field’s Saving Truth from Paradox, for a reading group).
- Kleene’s Normal Form Theorem entails Gödel’s Incomplete Theorem (explained in just two pages).
- Squeezing Church’s Thesis again (a talk given a few times in New Zealand, explaining what I was up to at the end of my Gödel book, but a little better!)
- Induction and predicativity (another talk given in New Zealand, for a non-expert audience)
- Entailment, with nods to Lewy and Smiley (an introductory talk to a seminar, something of an exercise in Cambridge piety: the promised next talk on Tennant wasn’t given though for reasons I can’t recall!)
- Back to basics: revisiting the Incompleteness Theorems (46 page handout for three lectures aimed at graduates)
- Rejection and valuations (with Luca Incurvati, draft of Analysis paper published in Jan 2010)
- Critical Notice of Charles Parsons’s Mathematical Thought and Its Objects (published in Analysis Reviews, 2009; excerpted from postings on the blog)
- Wittgenstein on mathematics and games (discussing §108 of the Big Typescript)
- Review of Rayo and Uzquiano (eds), Absolute Generality (short review in BSL, excerpted from postings on the blog)
- There are sea-serpents, Jim, but not as we know them (talk given to a metaphysics group in Cambridge, commenting on Zoltan Szabo’s “Believing in things”)
- Ancestral arithmetic and Isaacson’s thesis (stand-alone paper published in Analysis, reworking ideas in my Gödel book)
- Induction, more or less (expanded version of talk given at Dan Isaacson’s seminar in Oxford in 2007, mostly on ACA0)
- Review discussion of Adam Olszewski et al. Church’s Thesis after 70 Years (originally written, paper by paper, for my blog).
- Over 100 links to short notes giving advice/info answering student questions first posted on the forum math.stackexchange. Many of these are aimed at beginners or near beginners in logic.
General advice about writing
- Developing a writing style. (First written for beginning graduate students, but no doubt undergraduates could use the advice too!)