Wilfrid Hodges' logic page
Personal
Dr Wilfrid Hodges
Herons Brook
Sticklepath
Okehampton
Devon EX20 2PY
Phone 01837 840154
Email my first and last names with a dot between them, at btinternet.com.
Married since 1965 to Helen M. Hodges, Emeritus Professor, Institute of Psychiatry, Kings College, University of London.
Children (all with Facebook pages):
 Sally Watson = Dr Sally Hodges, Tavistock and Portman NHS Foundation Trust;
 Gale Hodges;
 Edwin (Eddy) Hodges.
Directions for reaching me by car are here, and pictures of the house and property are here. For the South Zeal Open Gardens of June 2012 we put together a guide to the mining remains at Herons Brook.
Bibliography and CV
A list of my publications is here.
My CV is here.
Corrigenda to publications
 Corrigenda to 'Model Theory'
 Corrigenda (18 September 2012) to 'A Shorter Model Theory'
 Corrigenda to 'Logic'
 Corrigenda to 'Building Models by Games' are incorporated in the Dover edition.
 Corrigenda to 'Mathematical Logic' (with Ian Chiswell)
 Corrigenda to 'Elementary Predicate Logic' (in 'Handbook of Philosophical Logic' Vol. 1; to be added)
Several of these are due for updating.
The lectures below come with two health warnings. First, when I lecture I add explanations and comments; you will have to imagine these. Second, there are some mistakes, some of which are corrected in the published versions indicated.
Arabic logic and semantics
Translations of Ibn Sina
 Madkhal i.6. A translation of section i.6 of Ibn Sina's commentary on Porphyry's Eisagoge. This passage contains his reasons for rejecting Porphyry's notion of an inseparable accident and replacing it by his own notion of a necessary accident. Briefly, the issue is that Porphyry took logic to be about what we can or can't imagine to be the case, whereas Ibn Sina switches attention to what we can infer by logical rules. At first sight Ibn Sina defines logical rules in terms of the operation of the intellect, which he places in the rear cerebral ventricles. But this is unfair; Ibn Sina is certainly not under the impression that logical rules could depend on brain function. The rules are those followed when the inference engine (the 'working mind') in the intellect operates with 'correctness'. Ibn Sina points out towards the end of this section that 'correctness' can be understood in two ways. First, the inference engine operates correctly when it follows correct logical inference rules. Second, the inference engine operates correctly when it doesn't tolerate things that couldn't be true in the world. These notions of correctness are distinct because the inference engine can only handle single applications of inference rules (local formalising!) and is unable to reach conclusions that need two or more inference steps. (The translation has not yet been checked by a native Arabic speaker.) (29 September 2013)
 Ibara ii.1. 'Affirmative and negative in Ibn Sina', a preprint of a paper submitted for a Festschrift. It contains a translation of section ii.1 of Ibn Sina's commentary on De Interpretatione, in which he discusses which sentences count as negative, and claims that all true affirmative predicative sentences have a nonempty subject. I comment on his discussion and argue that parts of it should count today as linguistics rather than logic. (27 June 2012.)
 Ibara ii.4. 'Ibn Sina on modes', a translation of
section ii.4 of his commentary on De Interpretatione. Warning: the translation is complete but it
has not yet been checked by a native Arabic speaker, and the
accompanying notes are incomplete. In this section Ibn Sina doesn't add much to what was in the Alexandrian commentators. But one thing seems to be original in him, namely the distinction between 'Possibly every A is a B' and 'Every A is possibly a B'. Sadly his account of the logical differences between the two is a mess, which will leave some doubt about what he thinks is going on in proofs that something doesn't follow from something else. The two sections Qiyas iii.4, 5 (which are now on my list to translate) develop the material here in more interesting ways; see the opening notes to this file. Modal logic was never my thing, but I thank Zia Movahed for urging me to take an interest in this material. (9 February 2010.)
 Ibara ii.5. 'Ibn Sina and conflict in logic', for a Festschrift. (Revised draft 18 August 2009.) This contains a commented translation of Ibn Sina's commentary on the final section of Aristotle, Peri Hermeneias. Ibn Sina tackles some particularly incomprehensible remarks in Aristotle with the aid of some thoughts about lengths of proofs. There is also a polemical section which seems to be intended to show that metatheorems of logic  in particular those of the kind illustrated by the more recent laws of distribution and interpolation theorems  are just as useful for philosophical argument as the laws of logic itself. If Ibn alTayyib is Ibn Sina's target in this passage, it may help to explain why Ibn alTayyib came to dislike Ibn Sina so much.
 Qiyas i.3. 'Ibn Sina on understood time and aspect', a translation of section i.3 of Qiyas. This is a very rich section
containing some of his most original work. It can serve as an introduction to his teaching that what we mean, even in careful scientific discourse, generally goes a long way beyond what we say. I put it up partly so that I can refer people to it. But like the translation two above, the translation has not yet been checked by a native Arabic speaker and the notes are incomplete. (5 March 2010.)
 Qiyas i.5. 'Ibn Sina on contradictories of absolutes, a commented translation of section i.5 of Qiyas. It picks up from the section i.3 translated above, and draws consequences about possible sentence structures. The text is important for establishing Ibn Sina's links with the earlier tradition, in particular Theophrastus and Alexander. But my present interest in it is its rather abortive discussion of quantifier scopes in terms of function representations rather than syntax. The paper is severely unfinished, but I put it up in order to be able to cite it for a talk on scope at the September 2010 Fest for Jouko Väänänen. (10 September 2010)
 Qiyas ii.2. 'Qiyas ii.2: Conversion of absolutes.
This and the next six items are text and translation of seven of the fifteen sections covering the formal logic of modalities (and more besides) in the book Qiyas. It will be obvious that the translations are work in progress and very much at draft stage. Also with this kind of material, raw translations without commentary have limited value. But in view of the increasing interest in Ibn Sina's modal logic it seems sensible to make them available. More will follow as soon as time allows. (13 January 2013)
 Qiyas ii.3. 'Qiyas ii.3: Conversion of necessaries and possibles. (4 November 2012)
 Qiyas ii.4. 'Qiyas ii.4: Recombinant syllogisms. (13 January 2013)
 Qiyas iii.1. 'Qiyas iii.1: On mixed syllogisms with absolute and necessary.
(4 November 2012)
 Qiyas iv.1. 'Qiyas iv.1: On possibility syllogisms in the first figure.
(4 November 2012)
 Qiyas iv.2. 'Qiyas iv.2: On syllogisms that are mixtures of possible and absolute in the first figure.
(4 November 2012)
 Qiyas iv.3. 'Qiyas iv.3: On syllogisms that are mixtures of possible and necessary in the first figure.
(13 January 2013)
 Qiyas v.1. 'Ibn Sina on conditionals', work in progress, so please don't quote without checking with me. I put it up because of the centrality of the topic. (18 August 2010.)
 Qiyas vi.4. A translation of Qiyas vi.4. This is a very boring section on propositional logic, but hidden inside it is the idea of a novel and surprisingly strong proof rule, which Ibn Sina invokes in his treatment of reductio ad absurdum. (2 Oct 2013)
 Qiyas viii.3. 'Ibn Sina on reductio ad absurdum, a draft paper. 'Ibn Sina's explanation of reductio ad absurdum' was an expository lecture on this for a workshop in Brussels. (8 October 2013.)
 Qiyas ix.3. 'Ibn Sina on patterns of proofs' (Revised draft 6 October 2009.) In Qiyas ix.3, Ibn Sina defines two different notions of compound syllogism and studies the relations between them.
 Qiyas ix.6. 'Ibn Sina on analysis: 1. Proof search. Or: Abstract State Machines as a tool for history of logic', for a Festschrift for Yuri Gurevich. (Revised draft 6 September 2009.) This contains a commented translation of a section of Ibn Sina's commentary on a couple of paragraphs of Aristotle's Prior Analytics. Ibn Sina seems to be giving his reader a string of 64 exercises to train the reader in a proof search algorithm for syllogisms. I confirm that he is doing this, by extracting from his text all the essential ingredients of an Abstract State Machine for the algorithm. Here is my accompanying lecture at the YuriFest in Brno where the volume was presented to Yuri. Here is a short talk on the same subject for the symposium Arabic Foundations of Science for the International Congress of History of Science and Technology, Manchester, July 2013. (16 July 2013.)
 Raw materials for a book with Amirouche Moktefi on what skills Ibn Sina thinks he is teaching by teaching logic, under the title 'Ibn Sina on Logical Analysis'. This includes preliminary translations of Qiyas ii.4, ix.34, 69, and it will rework the paper above on proof search. (25 January 2013.)
 Other commented translations of Ibn Sina's logic will appear here as they become available.
Other items on Arabic logic
The earlier items below contain things that I no longer believe. They will gradually be replaced by things that I do believe, and the process might even converge.
 Lecture on Definitions in Ibn Sina's Jadal, from conference 'The Topics in the Arabic and Latin Traditions', CRASSH, Cambridge 2006.
 Lecture on Ibn Sina's semantics, Seminar, Department of Philosophy, St Andrews June 07.
 Lecture on Ibn Sina's syllogistic,
Oxford, November 07. This lecture had the strictly limited aim of describing in modern terms a formal system adequate to support what Ibn Sina says about the semantics of predicate syllogisms. But the truth turns out to be rather more complicated than I realised. Ibn Sina identifies a number of sentence forms that are important in scientific discourse and are missing from the aristotelian tradition before him; they include ∀∃ sentences (see the section above on time and aspect, and the next lecture below) and sentences where a modal operator has a quantifier within its scope (see the section above on modes). But he shows little interest in developing a proof theory to cover sentences of these forms. I have the strong impression that he thinks the gap can be made up by 'analysis', reducing more complicated sentences to the standard categorical forms.
 Lecture on Ibn Sina's cyclotron, for a conference on formalising ancient logics, chiefly Indian, Hamburg, June 2010. The lecture discusses some of the sentence forms mentioned in the comments above on the previous lecture. (Note, 8 October 2013. Michael Carter has just revealed to me that Richard Lorch, a leading authority on medieval Arabic thought in the area that Ibn Sina's 'cyclotron' sentence comes from, is the same Richard Lorch that I knew at the Cathedral Choir School in Oxford for four years in the early 1950s. I'm hoping that Richard can throw some light on where Ibn Sina might have taken this sentence from.)

Draft text of my talk on `Ibn Sina's view of
the practice of logic' for World Philosophy Day, Tehran, November 2010. The talk describes Ibn Sina's procedures for checking the validity of
natural language arguments, and contrasts them with those of Aristotle,
Fakhr alDin Razi and modern undergraduate logic classes. (Revised 18 November 2010.) The slides of my talk to SIHSPAI, December 2010, cover similar ground.

A draft paper (with accompanying talk) which attempts to collect together Ibn Sina's assumptions about sentence structure from his logical writings. This sort of exercise seems to be essential if we are to get a realistic picture of what he understood, what he didn't understand and what he believed in semantics. For example it emerges that he had no notion of scope. (18 November 2010)

A note on what Ibn Sina took the 'subject of logic' to be. I don't regard the question of the subject of logic as particularly important, and there is evidence that Ibn Sina himself didn't either. But it does worry me that published accounts of Ibn Sina's view on the question are generally very unhistorical. The note is mostly about the steps that would need to be taken in order to pin down Ibn Sina's view of the matter. It would be nice to find time to write out the conclusion with full supporting arguments. (17 June 2011)
 For a workshop on later Arabic logic and philosophy of language, a talk on the role of modality in Ibn Sina's logic. There is an accompanying handout. (17 November 2011)
 For a seminar in Munich, a talk on whether and in what sense Ibn Sina had first order logic. There is an accompanying handout. (18 March 2012)
 For the CNRS workshop 'Ancient and Arabic Logic', 29 March 2012, a talk on Ibn Sina's assumptions about the grammatical structure
of compound meanings, and some applications in his logic. (26 March 2012)
 For the workshop 'Modal Logic in the Middle Ages' (St Andrews, November 2012), a talk and accompanying handout on Ibn Sina's adaptation of the notions of permanence and necessity which he took from his Aristotelian predecessors. (24 September 2012)
 For the workshop 'Medieval Philosophy Network, Warburg Institute London 7 June 2013', a talk on Syntax and meaning in alSirafi and Ibn Sina based on joint work with Manuela E. B. Giolfo. There is a handout. (6 June 2013)
 Notes for a paper The grammar of meanings in Ibn Sina and Frege. I have to apologise for this one. A journal invited me to submit a paper comparing Ibn Sina and Frege. Since Ibn Sina and Frege are the two main figures in the Aristotelian tradition who developed a syntax of meanings, this was a stimulating challenge. The paper has been submitted, but it contains no references and few quotations; instead I gave a reference to the present paper for a fuller account. The paper is not yet written, though it's a good deal further down the line than these notes might suggest. I will get the whole thing put up here as soon as I can. (1 January 2013)
 The slides of the 2013 Lindström Lectures at Göteborg. The first discusses Ibn Sina as a logician generally, and in particular his view of the 'subjectterm' of logic; his view of this broadly puts him in the mainstream along with Pascal, Bolzano and early Tarski, but he has radical ideas about the types of entity needed for handling compound meanings. (None of this is reflected in the standard encyclopedia accounts.) The second discusses Ibn Sina's treatment of the making and discharging of assumptions; see above. (11 November 2013)
 These 'Notes on a remark of Street' are a preliminary
announcement on Ibn Sina's system of 'modal syllogistics'. The system describes the logical relationships between some sentences with two or three independent quantifiers, that Ibn Sina discusses in a number of places. In effect it creates a new branch of formal logic, quite unlike anything else before modern times. But Ibn Sina never gave a good proof theory for it, probably because he wanted to do that on the pattern of Aristotle's Prior Analytics, which are not a suitable basis for a logic this advanced. Also the system is not really modal at all, though it's an open question whether Ibn Sina expected it to have modal applications. (6 January 2014)
 Sprenger's translation of Risala alShamsiyya by alQazwini alKatibi. Tony Street has a new translation in preparation.
General history of logic

The history of British logic, British Logic Colloquium, Manchester, September 2001. It would be good to have the history of the origins of the BLC written up before too many memories die. I'm not offering to write it, but if I get a chance I will put here some of my old BLC archives. Meanwhile some corrections on this talk: (Page 3) The distinction was certainly noticed before Lukasiewicz, but it came slowly. Already Ibn Sina studies conversions between forms like A and forms like B. The rule of arrowintroduction is explicit in the PortRoyal Logic, though only for onestep inferences. Frege discusses the notion of making and resolving assumptions. (Page 16) As Tim Williamson pointed out, Lewis Carroll should be mentioned here. (Page 17) The word 'quantifier' appears first in De Morgan, as an abbreviation for 'quantifying phrase'  which comes from Hamilton as De Morgan acknowledges.

Indirect proofs and proofs from assumptions. I wrote this in 2004 to clarify in my own mind some issues raised by Paolo Mancosu, without any plans to publish it as it is. I still think it is correct in what it covers, but the absence of medieval logicians is unbalanced. In particular Ibn Sina should be discussed alongside the PortRoyal Logic and Frege. Mancosu has seen the paper but not responded.

What happened to fallacies?, Indian Logic Circle, Kolkata October 2005.

Logic versus theory of language in the late 19th century, Augustus De Morgan Workshop, Kings London November 2000.

Tutorial on Tarski and decidable theories, Mumbai 10 January 2005. A fuller version is in the Proceedings.

Tarski on Padoa's method, International Conference on Logic,
NavyaNyaya and Applications, Homage to Bimal Krishna Matilal, Kolkata
January 2007. A fuller version is in the Proceedings.

The history of model theory. This was written for a handbook on the history of mathematical logic (not the more recent GabbayWoods manyvolume handbook). Before the paper was finished, the terms of reference were changed to ones that I didn't have the time to meet. So after some discussion I withdrew the chapter. Later the handbook itself folded. Anybody wanting to use any of this material in writing a history of early model theory is welcome to do that; if they contact me I might be able to supply some corrections and background.

Bill Marsh's unpublished 1966 DPhil thesis on uncountably but not countably categorical theories (included with his permission).

My paper 'Detecting the logical content: Burley's "Purity of Logic"' refers to a checklist of consequentiae in Burley's book. That checklist is here.

Where Frege is coming from, seminar, Amsterdam March 2009. The talk compares Ibn Sina, Leibniz and Frege on some key issues in logic and semantics, and aims to illuminate Frege by putting him into this context.

Why modern logic took so long to arrive, three lectures for Cameleon, March 2009. I trace the development, over the last two thousand years, of (1) relational reasoning, (2) proof rules that discharge assumptions, (3) typetheoretic semantics. There is a collection of texts to accompany the lectures, including eight pages of translation of Ibn Sina.

Traditional logic, modern logic and natural language, a paper that looks for the fundamental dividing lines between traditional logic and modern logic. Comparisons between traditional and modern logic tend to be based on an assumption that traditional logicians were aiming to do what modern logicians do, but weren't so good at it. Wouldn't it be better history, and fairer to all concerned, to try to establish what the traditionals themselves thought they were aiming to do? This was a draft (with personal references removed) of a paper which appeared in an issue of the Journal of Philosophical Logic in honour of Johan van Benthem. Here is a more recent lecture in the same general area but with more emphasis on deathtraps in history of logic (for the Edinburgh University Philosophy Society, November 2010).
 How Boole broke through the top syntactic level. For the meeting History of Modern Algebra: 19th Century and Later, in memory of Maria Panteki, Thessaloniki October 2009. The talk notes that Boole broke with traditional logic by allowing substitutions at any depth in a formula. I discuss why this breakthrough marks a fundamental difference between traditional and modern logic, and how far Boole understood what he was doing in making the advance. Here is a draft of the writeup for the Proceedings.
 A talk for the 60th birthday of Oleg Belegradek, Istanbul, December 2009. The talk contains a mixture of mathematical and personal reminiscences about how the Kemerovo model theorists (Belegradek and Zilber) came into contact with the British model theorists across the iron curtain.
 Including history in a mathematics module  practice and theory. For the meeting History of Science in Practice, Athens, May 2010. The talk discusses the pros and cons of including history in an undergraduate mathematics module, with some examples from my own experience.
 Basing logic on semantics  some historical themes, a discussion of the use of dependency grammars by various traditional linguists and logicians, with particular emphasis on Ibn Sina's use of them for describing compound thoughts. For a meeting in Cambridge, March 2011.
 Reconciling Greek mathematics and Greek logic  Galen's question and Ibn Sina's answer, for a workshop of Catarina Dutilh's project in Groningen. Some progress on a problem I mentioned in earlier lectures: why did Aristotelian logicians regard Euclid as the summit of logical reasoning if they couldn't represent his reasoning within their own logic? Galen at least indicated the problem, and Alexander of Aphrodisias applied some sticking plaster to the gap pointed out by Galen. But Ibn Sina seems to have the question much better under control. I give a formal calculus consisting of items available to Ibn Sina, which is complete for firstorder logic. This shows in principle that Ibn Sina was right to suppose there was nothing in Euclid that he couldn't justify within his own logic. But for a historian of logic an equally important question is what Ibn Sina would have counted as validating Euclid's arguments. It's clear that he would not have used any formal calculus for the purpose. Further historical and logical details are in the handout. (24 November 2011)
 A lecture on the history of the notion of logical scope, including some remarks on Ibn Sina's attempt to do without it, for a seminar in Amsterdam. (26 November 2011)
 A lecture 'Tarski through a wideangle lens' on Tarski's place in the broad history of logic, with particular reference to semantics. (3 June 2012)
Semantics in natural language, mathematics, engineering

From sentence meanings to full semantics, Mumbai
10 January 2005. A fuller version is in the Proceedings.

The interplay of fact and theory in separating
syntax from meaning from Esslli 05 Edinburgh.

Architectural questions about theories of sentence and word meanings, Bristol October 2006.

The mathematical core of Tarski's truth definition, Unilog 2, Xi'an August 2007.

Composition of meanings for a workshop at Duesseldorf in 2003.

Necessity in mathematics was written for the Proceedings of a conference on Necessity and Contingency. But I can't get any confirmation that the Proceedings will appear. The paper was based on a review of the modal terminology in the first hundred pages of Birkhoff and Mac Lane, A Survey of Modern Algebra. The review contains more material, which I hope to report on soon.

Necessity in mathematics II. Slides for a talk at the 'Practicebased Philosophy of Logic and Mathematics' Workshop, Amsterdam, AugustSeptember 2009. The talk picks up some ideas from the preceding piece as evidence for the processes involved in understanding a mathematical textbook. A draft of the writeup 'Modality in mathematics' is here.

Models in science and technology, for a workshop in Delft, April 2004. This was an early stage in the writing of Functional modelling and mathematical models, to appear in a pioneering volume on the philosophy of technology.

The model theory of specification, Conference in
memory of Sauro Tulipani, Camerino April 2006.

Model theory as Peacock's revenge. This is the unpublished original of a paper that appeared in Spanish translation. If retrospective dedications are allowed, I would like to dedicate this original to the memory of Maria Panteki, a learned and generous authority on George Peacock; she died in 2008, long before her time.

Compositionality: Its history and formalism. A tutorial for the workshop Dependence Logic, Dagstuhl, February 2013.

The choice of semantics as a methodological question. A talk at Second Conference on ProofTheoretic Semantics, Tübingen, March 2013.
Mathematical logic

Short model theory course, Johannesburg December 1999.

Three tutorials in logic, Tbilisi 610 October 2003.

Draft of lecture on nonstructure, Hattingen July 1999. Chris Laskowski and I gave lectures on nonstructure; I reviewed the 'classical' material and Chris introduced more recent work. There was a plan for a joint paper, but it never materialised.

Classification over a predicate, Istanbul Bilgi University March 2001. The published version was devoted to the special case of theories of linear orderings. But see also the paper of Rami Grossberg in the same volume. Istanbul is an earthquake zone, and apparently there was an earthquake during my talk, but I never noticed.

Model theory of pairs of abelian groups, for St Petersburg July 2005. Here is an archived preprint of the published version, which includes some nontrivial corrections. Anatoly Yakovlev was in the audience at the talk, and the paper includes his answer to the final Question on relatively categorical pairs of finite abelian groups.

Relatively categorical abelian groups III. This lecture applies the analysis of the preceding item to prove that Gaifman's conjecture holds for theories of relatively categorical abelian group pairs: Any model B of the Ptheory can be extended to a model A of the whole theory so that the Ppart of A is B. The occasion for the lecture was the International Conference on Fundamental Structures of Algebra, held at Constanța on the Black Sea in April 2010 in honour of Șerban Basarab. A draft of the writeup of the proof for the Conference Proceedings is here.

Fully abstract valuations for subgames, De Morgan Workshop on Interactive Logic, London November 2005. A fuller version is in the Proceedings.

Maze games, proof games and some others, for
'Games in logic, language and computation', Amsterdam September 2005.
Here is the PowerPoint. The talk contains some personal opinions about the motivation of gametheoretic modelling.

Four paradigms for logical games, for the meeting 'Modelling interaction, dialog, social choice, and vagueness', Amsterdam, March 2010. Benedikt Löwe asked me to review samenesses and differences between Obligationes, Lorenzen dialogue games, EF backandforth games and Hintikka gametheoretic semantics. I did this with special reference to the modelling involved. The handout contains bibliography and some mathematical underpinning.

Mathematics of imperfect information, Kings London February 2000.

Definability versus definabilityuptoisomorphism, in groups and fields (revised 29 May 2010). One of several reports I've given on the proof (with Saharon Shelah) that there is no formula of set theory which provably in ZFC defines an algebraic closure for each field. This report was to the Antalya Algebra Day XII in May 2010. The proof is in two papers; one of them has appeared, and the other is in preparation (the end is in sight).
Cognitive aspects of logic
 Some notes I put up for my students at Queen Mary: Psychologists' tips on how and how not to learn. Though I take responsibility for the contents, I checked them with my wife who is a professional psychologist.
 Some things a logician would like to know about
human reasoning, CogSci 2001, Edinburgh August 2001.
 Efforts to make mathematics infallible, Workshop on
semantic processing, logic and cognition, Tuebingen November 2005.

How reasoning depends on representations,
two lectures to Centre for Cognitive Science, Jadavpur University,
Kolkata, October 2005. The work of Keith Stenning that I discuss here has moved on: see Keith Stenning and Michiel van Lambalgen, Human Reasoning and Cognitive Science, MIT Press, Cambridge Mass. 2008.

Logical rules at deep syntactic levels, a talk to the London Reasoning Workshop, Birkbeck College London, July 2010. The talk was an appeal for help in measuring the relative difficulty or 'naturalness' of some types of inference rule. The type that particularly interest me are those which apply monotonicity to make substitutions at arbitrary syntactic depth. My belief is that most people find these rules impossible to follow intuitively when the depth is more than 3 or so, and have to resort to explicit calculation. John of Salisbury claimed just this in the 12th century, but without solid evidence. Several logicians have suggested in recent years that these rules are particularly 'natural', which seems to me a paradoxical description.
Mathematics and music

Talk on Pythagoras, QMW Oct 98. A revised version of this talk was given as the Coulter McDowell Annual Lecture at Royal Holloway in 2001. Pythagoras must have been an extraordinary person, but modern scholarship has stripped him of most of his supposed scientific advances. One that is still sometimes attributed to him is the correlation between subdivisions of a vibrating string and points of the musical scale. I argued, relying on the shapes of kitharas and the known techniques for playing them, that this correlation was probably clear to professionals at least a generation before Pythagoras. It would be consistent with what we do know about Pythagoras if he took this known correlation and converted it into some kind of music therapy. (Sad footnote: I tried unsuccessfully to discuss this with the late and much missed David Fowler, who was an expert both on Greek mathematics and on the mathematics of the musical scale. Later I learned that at the time he was already severely ill with the brain tumour that took him not long after.)

Some raw material on mathematical and musical beauty, for a meeting of mathematicians and artists organised by Juliette Kennedy in Utrecht, November 2007. For copyright and technical reasons, most of the music itself is missing here. The talk draws on an archive of geometrical patterns in music. Material from the same archive went into my Third Annual Venn Lecture in Hull in 2003, and into two published papers (first), (second), one joint with Robin Wilson. For some years I have hoped to turn the archive into a book, but it is not high priority.

Time, Music and Mathematics. Slides for a talk at the Greenwich Observatory, November 2009.

The geometry of music, one of a group of lectures on mathematics and music arranged by Robin Wilson at Gresham College to mark his retirement from the post of Professor of Geometry at Gresham. My handout is here. The lecturers were Robin himself and me (November 2009) and Jonathan Cross (December 2009).
Dartmoor

Copper and arsenic processing at Ramsley mine, a short talk for the Industrial Archaeology group of the Devonshire Association (February 2013).
IUHPS DLMPS
The website of the South Tawton and District Local History Group is at http://www.southtawtonhistory.org.uk. The website of the IUHPS Division of Logic, Methodology and Philosophy of Science is at http://www.dlmpst.org.
Author : Wilfrid Hodges
Last updated 10 March 2014 