The Abnormal Distribution

Michael. Everyone knew him as Michael.

I was a freshman at Oxford in mathematics, interested in logic. I had been reading Chomsky in my first quarter because I had been told Chomsky had mathematised language. My tutor in algebra, Ian Macdonald (same jacket as in the picture!), an algebraic geometer, suggested I could look at a logic textbook he recommended (which I read with some difficulty over the Christmas break). Derek Goldrei, a graduate student tutoring in logic at my college Magdalen, suggested I listen to Michael’s lectures in set theory.

Michael didn’t lecture. Michael thought out loud. He distributed notes telling his listeners what he was going to be thinking about during that appointment. I learnt, by watching and listening, how to think. About set theory. About inference rules. About non-classical logic (Michael was drawn to intuitionist thinking about mathematics, because he thought it was right to base your assertions on the concrete evidence you had).

I had been attending freshman mathematics lectures, which went “Theorem” “Proof” “Let x be…” and had despaired of ever being the kind of person who thought like that. Then I attended Michael’s thinking-out-loud sessions and understood what really went on in people’s minds; how the symbols were shorthand for notating thoughts. And, in my second year, I could do it! Just like Michael! Actually, not just like Michael. Not anywhere near “just like Michael”. For, as John Mackie is reported to have said in The Times’s obituary, Michael was a genius. Michael was ineffable.

Michael was different. A mass of wavy white hair, he would array himself longitudinally on a bench in the lecture hall and clean his cigarette holder while leaning on an elbow, with his head just above the seat backs, and crack jokes about his friends and colleagues while waiting for the lecture to begin. At which point the jokes would reduce in number as he concentrated on what was being said. If there is anything any undergraduate wished to be in the course of study he had in large part created, Maths and Philosophy, it was to be “just like Michael”.

Simply put, Michael taught me how to think, in logic; by extrapolation, in mathematics. About the deep philosophical questions concerning truth, mathematics, the use of language. Differently put, I learned how to think by watching and listening to him.

When I graduated in 1973, I attended a ceremony in the Sheldonian Theatre, in Latin, much foreshortened from the original, during which my degree was conferred. A ceremony designed over centuries to give its recipients the indelible impression: you have done it! I had done it! I felt it and they’d said it in Latin! After the ceremony, I went straight across the road in my academic dress to purchase a copy of Michael’s new book, on Frege’s philosophy of language. Michael had shown how to think about these matters in pellucid English prose.

I went right afterwards to the other side of the Northern Hemisphere, to Berkeley in California. Michael had helped me get there, for he had written me a recommendation for graduate school. I have no idea what he said, but I it can’t have been all disastrous. (I can imagine: “Ladkin is mortal and does OK for one. But I’m afraid I don’t really know much about mortals.”)

I was required at the end of my first year by Bill Craig, my advisor in Berkeley, he of Craig’s Interpolation Theorem, to take the qualifying exam in philosophy. I protested and threw tantrums and all that, but you know you can’t really rebel. Bill said “you will do it” so I did it. I read Michael’s book, and its seemingly impenetrable prose. And I read it again. And understood more. And again. And more. And again. It wasn’t that Michael’s prose was impenetrable. Michael wrote exactly what he was thinking and his thinking was non-trivial and exact. It took me a while to absorb his train of thought. His prose was, indeed, pellucid. When I had done so, I went into the exam room (actually the philosophy library) for six hours and wrote exactly what I thought about the matters about which I had learned from reading Michael’s book so carefully. Non-trivially and exactly. I think Ernie Adams graded the exam. I passed. Turns out I was the first student in the history of Tarski’s program to pass the philosophy exam in my first year. Thank you, Michael!

(You have to understand – I was rotten at written exams. I got so nervous I couldn’t even read the questions straight. It’s a miracle I ever got into and out of Oxford, at which assessment is based on a student’s brilliance at written exams.)

I saw Michael in Berkeley once. He gave an evening lecture which I attended. I did get to exchange a brief word, amongst all the others earnest to talk with him.

I saw him again in 2009, at the 40th anniversary reunion of Maths and Philosophy graduates in Oxford, of the course which he had done so much to establish, and to which I owe my subsequent career. Derek Goldrei was the First Graduate (he switched in his final year; graduating in 1969 when the course was established). I in 1973. I was one of only two or three from that era at the reunion and felt quite The Establishment. Michael was there, and Dana Scott. Michael was old and frail. Gave an endearing and well-constructed speech. When I approached him after the dinner, he didn’t remember who I was, but then so many had passed through the gate since I had. I simply thanked him. He accepted graciously.

Michael is gone, on 27 December 2011. For me, he was Philosophy. When he was with us, Philosophy was alive. Now he is gone, Philosophy is gone. Maybe not, but it sure feels like it. It turns out I seem to have assumed he was immortal. Apparently not. It is -let me say- hard for me to adjust.

Here is The Guardian’s take. The Times has a fine obituary, forwarded to me by Chris Miller, but it lies behind a paywall, just as now Michael does, though with a currency which I only wish I had. As an atheist without this currency, I can only say: God be with you, as you wished.

Some Coincidences.

Racism. Two of the killers of Stephen Lawrence were convicted in early January 2012. Here is a poem about it by Poet Laureate Carol Ann Duffy. Michael and his wife Ann devoted themselves to race relations in 1960′s and early 1970′s Britain, efforts well documented in the obituaries. He only returned to philosophical work after he felt the efforts to turn Britain away from racist habits had failed. But they haven’t failed, Michael, and neither had you.

Brains. Apparently some people claim now that our brains start to go downhill at age 45 It is not clear this is news: The Guardian had something about it 12 years ago. Michael published his first book at 48, and there followed many more, all of them worth reading very carefully indeed.

Note Added 11.01.2012

It’s not just philosophy. Thinking it over, there are three fundamental developments in technical elementary logic which I have kept coming back to throughout my career. Things which are simple, clear, brilliant, which increase one’s understanding almost instantly, and continually prove to be useful. One is Dana Scott’s Consequence Relations, a formulation of logics which, to me, turns out to be the most efficient way to perform formal deductions, the raw material of logic. I keep meaning to translate into LaTeX the mimeographed notes which Dana handed out almost 40 years ago now. Another is Saul Kripke’s possible-worlds semantics for normal modal logics, and his similar epistemic-worlds semantics for logics of belief and evidence, such as inference in intuitionistic mathematics, and the inferences of Pen Maddy’s “Second Philosopher”. I learnt these partly from Michael. The third is Michael’s and John Lemmon’s formal correspondence between the modal logics from S4 to S5 and the propositional logics between intuitionist and classical.

Second Note Added 11.01.2012

Timothy Williamson, Michael’s successor in the Wykeham chair of Logic (David Wiggins came between Michael and Tim), pointed me to a series of tributes in the New York Times Opinionator blog last week.

John McCarthy has died. The great John McCarthy. Brilliant and entertaining, fun to be around, accessible unlike many of his stature, who carried an aura about him which blessed you with the feeling, if you came within it, that you were doing the Thinking That Really Mattered. Even if you were just flapping around at a loss for ideas.

The German Wikipedia describes him as a logician and computer scientist. The English version as a computer scientist and cognitive scientist. The German has it right.

John used to be quite happy to get in discussions with everyone about anything and became well known for it as Internet news groups really got started in the mid-1980′s. He had a knack for posing simple questions that turned out to be hard to answer.

And not just in AI. For example, check out his proposal for a new civil right on what counts for him as his personal page:

Remark: Ideally one would put all the information that one considers public about oneself on a page like this. When asked to fill out a form, one would simply put down the URL in place of any information that is on the page and tell the recipient of the form to just look it up.

One step beyond that is that any program needing this public information would just take it from the somewhat standardized web page.

More precisely, here’s a proposed new civil right. No Government agency, educational institution or business should ever be able to require anyone to supply anew information that the institution already has or is publicly available.

Typically for John, it is simple, doable, but somehow not done, and has significant social consequences. Let’s consider it a little further.

There are inadvertent violations. I tried to hand in a technical review of a paper submitted to an IEEE Transactions to the IEEE ScholarOne “system” (I use the word loosely) and found it wouldn’t let me do it without requiring me to fill in a lot of personal information. (I sent the review by email, and someone else has now tweaked the system enough to let me file, apparently.)

But the phenonemon is also used – and this, I suspect, is an insight of John – for political purposes. I had been asked on five or six occasions in the last year by a grant-supported institution with which I am associated to deliver information about activities (publications and talks and so on). Always the same stuff, but somehow not in quite the right format, or not quite the right selection. I began to suspect that someone is looking for a “reason” to ease me out, and so it turned out. Bureaucracy-overload as a political instrument; and of course always deniable.

The focus of this institution is, well, the successor “discipline” (I use the word loosely) to AI. John would have loved it!

Jon Hind informed a mailing list on Tuesday 25th October of the Guardian obituary that had just come out. There is a joint obituary with Dennis Ritchie in The Economist which appeared in the Novermber 5th print edition.

The Economist suggests that John did not suffer fools gladly. That is not quite how it was, as I recall. He engaged with all sorts of people – students mouthing off on Internet bulletin boards, for example, which nobody else did at the time. But he didn’t condescend. Anyone could talk to and argue with John, but he didn’t adjust his intellect to your capabilities – you had to adjust yours to his; for almost everyone an impossibly tall order. As well as being exactly what bright Stanford students need.

The Guardian article seems to me to miss most of what John was about during the 1980′s and 1990′s (the Economist, unusually, even more). Of course, after the invention of LISP, still the longest living programming language with over half a century of use (C, eleven years its junior, still has to catch up), one could regard anything else as a coda. But it was just a start. I’ll talk about the decade I know about, from the mid-1980′s to the mid-1990′s.

John had discovered, or invented, the Frame Problem, with Pat Hayes, and then came up with the cleanest purely technical proposal for solving it, Circumscription. Unfortunately, Circumscription didn’t turn out to solve the Frame Problem sui generis, but it did start a little industry all of its own. This little industry frustrated people such as Danny Bobrow, then-editor in the 1980′s of the premier journal in the field, Artificial Intelligence. Danny is a programmer through and through, who feels that to do AI you have to build stuff, that is, to program. The Circumscription industry consisted of a largish collection of mostly ex-Eastern European logicians, many of them eminent and all of them both capable and productive, who wrote great technical papers in mathematical logic and sent them to the Artifical Intelligence journal- where of course they had to be sent off to be refereed by other members of the group, and they took over about the third of the journal with all that ***** Math!! All good stuff no doubt, but it didn’t seem to some as if much was getting built………

It mirrored a significant split in AI, indeed in all computer science, which continues to this day. There are people who incline to solve problems intellectually before they solve them practically, and there are people who attempt practical solutions and solve, or resolve, issues as they come up to them. In AI in the 1980′s, they were known respectively as “neats” and “scruffies”. The neats have it right in that you cannot program solutions you do not have. The scruffies have it right in that a computational solution to a problem consists in an implementation. You might imagine that they could agree on a division of labor, but research is a little messier than that. The neats have it wrong in that abstraction is also a fine way of subtly changing the problem to fit the solution you happen to have, and the scruffies have it wrong in that a clever programmer can build wonderful programs which fail to solve the problem they set themselves, but “almost” do – the permanent, ineffable “almost”, which turns out to mean “never”.

John’s view on progress was that you knew a field was technically mature when you couldn’t understand the work of someone working on a different problem from you. Let’s turn that on itself. In some sense the division of AI research into neats and scruffies, say Danny’s frustrated view of all that math, could thereby have constituted a proof of some sort of maturity, although the way the squabbles were conducted left many wondering if that was the word for it! Maybe that was John’s point?

And John was the living contradiction to this view on progress. Of course. He could explain to anyone with a modicum of understanding of propositional calculus exactly what he was interested in and what problems he thought were worth solving. Check it out at his group WWW site. They were all so simple! Until you realised that, John being John, if they were really as simple as they looked, he would have solved them already. I recall one evening after dinner at the IJCAI conference in Detroit in 1989 when a bunch of us formal people were chatting away after late dinner. Along comes John. Says, “you know, I was thinking about this…… do you know how to do it?” and posed, as usual, what appeared to be a simple problem in propositional logic. Well, after a few minutes, everyone else made their excuses and left. I couldn’t solve it. Then came another problem. And another. All simple, all propositional logic, all needing to be solved if machines were going to mimic human decision capabilities. And, of course, AI meant that machines should be able to do this, somehow, so even if you programmed them with genetic algorithms or neural-network problem solvers, they would still just have been solving John’s “easy” problems in propositional logic. Surely a problem posed in logic can be solved in logic? Well, sometimes.

This went on for an hour and longer. At such sessions one could choose to feel stupid and frustrated at not being able to solve anything, or to revel in the creativity exhibited before your very eyes. For anyone can solve problems, but very, very few people know how to ask exactly the right questions. John was one. Astonishing performances, puzzles rolling off his tongue as if he were discussing the bus timetable. Anyone – and there were many – who claimed that “symbolic AI” was dead just hadn’t been listening properly. Symbolic AI wasn’t and isn’t “dead”. John’s simple problems need to be solved one way or another. But no longer by him, unfortunately.

Circumscription? Let me have a go. Circumscription is a syntactic (and therefore computationally feasible) way of doing the following. Say you have a description of part of the world in front of you, and what is going on in it. Say your description is in some language which allows deductive inference. Circumscription is a way of drawing rich inferences about features (“predicates”) of that scenario under the supposition that the world doesn’t have anything else in it but those objects expressly described plus whatever else needs to be there for the description to be accurate. Not just rich inferences, more than you could obtain with deduction alone, but rich, correct inferences. To logicians, it is a set of inference rules about what is true in certain minimal models of the set of sentences.

That is logically very important. Modern logic arose with Frege considering the logic of arithmetic, of counting and adding and so on. But in Frege’s logic, it turns out that you can’t just restrict your talk to the positive whole numbers. Circumscription was a way of trying to do just that for “worlds” which had a finite number of objects in them. It resolves many of the issues in the Frame Problem (maybe more appropriately called the Framing Problem), by implicitly defining what you are framing. However, it doesn’t neatly resolve the conundrum posed in 1986 by Steve Hanks and Drew McDermott and known as the Yale Shooting Problem. That was first resolved by using other principles. The conundrum it posed has now dropped out of fashion, as far as I know.

To see the rich the tradition around Circumscription, one may look at the Stanford Encyclopedia of Philosophy entries on Non-Monotonic Logic, on Defeasible Reasoning, on Logic and Artificial Intelligence, on the Frame Problem, on Ceteris Paribus laws (that is, rules based on “all other things being equal”).

John was not concerned merely with mimicking intelligence with machines, but with the more elusive reasoning phenomenon of common sense, which occurred in the title to a collection of his papers in 1990. There was a whole branch of AI research devoted to “common-sense” reasoning about the world; which in turn spawned a branch of reasoning called Qualitative Physics: how the world works ; check out for example this book chapter by Ken Forbus. (John, though, would have distinguished common-sense physics from qualitative physics.) If you put a round ball on a slanting table, it will roll down the slope and drop off the edge and hit the floor just beyond the edge, and how far beyond depends largely on its speed relative to the table when it drops. This phenomenon is known to every two-year-old a couple of decades before they can understand the Newtonian version, but we adults have far more trouble getting a handle on the qualitative reasoning than we do on Mr. Newton’s mathematics. Yet another delicious irony.

And one could go on. Maybe without end. Qualitative physics will not end; it’s a phenomenally hard problem. It may go out of fashion, but it’ll come back. And somebody will have to solve all those common-sense physics problems as well, and maybe differently. Circumscription didn’t solve the Yale Shooting Problem, but it did open up the study of rigorous forms of defeasible reasoning.

And always there was an irony, a delightful little joke in the tail. Somehow, you never quite knew whether you were thinking about a new subject or an elaborate joke. Look closely at the picture in the Guardian. Can you, also, maybe, see the slight smirk that I always thought I saw? Maybe, just maybe, AI was his very biggest joke of all…..

Those of us interested in commercial aviation accidents have to deal with a lot of what I shall call screwy reasoning.

Last week, I read a September 2 article in The Times on the crash of AF447 and its aftermath which I felt was somewhat screwy. It suggested that Air France’s attempt to introduce specialised training for the scenario of loss of airspeed data in cruise on A330/340 machines had “provoked anger”. The angered party was unspecified, but the initial paragraph had a “pilot’s union” “accusing” AF of trying to cover up the cause of the AF447 crash.

At the end of the article, the author, Charles Bremner, says that simulator training for “speed problems” at cruising altitude had not previously been offered by Air France, but this is now being done “at the request of all airlines’ unions.”

Assuming the pilots’ union was the party to have been angered, there is an obvious contradiction between a union requesting training, and being provoked to anger by (apparently) the same. So, if not the pilots’ union, it would be good to know who it was who is being angered by Air France’s specialised training.

Not that Air France has much choice in the matter. European Aviation Safety Agency (EASA) Safety Information Bulletin 2009-17, issued on 9 June,2009, which may be found by searching EASA’s Safety Information Bulletin list, requires inter alia that “familiarisation of flight crews with unreliable airspeed indication procedures should be provided through adequate training.” It is hard to imagine how anger can be provoked through airlines following a regulatory safety directive.

The Times article began by suggesting the the Air France pilots’ union had accused “accident investigators”, by which I take it is meant the BEA-convened AF 447 investigation committee, of “covering up the cause” of the crash of AF447. I doubt this is correct, whatever the union is annoyed about. One can only cover up a cause if one knows what the cause is, and I doubt that any professional thinks the BEA or AF or anyone else knows.

Bremner has written quite a lot on AF447. He is a pilot (a private pilot I take it) and he does have useful information on the negotiations between stakeholders (pilots’ unions, airline, government, manufacturer, investigating agency, maybe even passengers….) to share. However, he seems to do so in a selective way which I often find unhelpful, as in this case, and his phraseology is sometimes unfortunate, also as here.

I wrote some comments on the article and the discussion, which appear in the Commentary section under the article. Colleagues who look at the aviation discussion forums have been amazed at how often the 1988 Habsheim accident comes up whenever anything Airbusy surfaces, and so it does in Commentary to this article. But, in these contributions, it is no longer Habsheim. It is the “Paris Air Show”. One wonders what, if anything, is going on in these people’s heads.

Well, there are partial answers. Sociologists have been studying people’s actual reasoning behavior in detail, but I was always missing a link into the literature. Now I have one, courtesy of the New York Times’s TierneyLab blog, which discusses the issue of who was first to the North Pole. One possible answer: Roald Amundsen in an airship in 1926. On the ground – well, ice – it might well have been Ralph Plaisted in 1968 on snowmobile, and Wally Herbert a year later by dogsled.

Tierney points to research on belief formation and retention, by Steve G. Hoffmann and colleagues, who used the Saddam-9/11 non-link to conduct a survey and then interview some responders, and wrote an article about it in the journal Sociological Inquiry, published in 2009. A number of belief formation and retention mechanisms have been proposed in the literature, and Hoffmann et al consider them. I find the paper very readable and it contains a number of references to related work. So this is a way in to studies of reasoning phenomena, for those of us who might be curious about it.

As I pointed out yesterday when considering a critique of Swinburne, a lot of popular science blogging is concerned with the arguments for intelligent-design creationism: the argument that the nature and complexity of the world yields empirical evidence for the existence of a god, an “intelligent designer” who designed the world.

Such arguments have a long provenance. Thomas Aquinas published his five proofs of God’s existence near the beginning of his Meisterwerk, the Summa Theologiae, written somewhere between 1265 and 1268 when he was in Rome (see Anthony Kenny, Medieval Philosophy, vol. 2 of A New History of Western Philosophy, Oxford: Clarendon Press, 2005, pp 67-9).

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I recently looked through Nature’s List of Top 50 Science Blogs and ended up reading quite a lot of the Good Math, Bad Math Weblog of Mark Chu-Carroll, a computer scientist at a major industry research center who is enamoured of mathematics.

The Science Top 50 Weblogs seems to be heavy on biology and the U.S. Theme du Jour – maybe Theme du Siecle – which seems to be evolution versus creationism. Chu-Carroll came across an argument for the likelihood of the resurrection of the incarnation of God, proposed by Richard Swinburne, Emeritus Nolloth Professor of the Philosophy of the Christian Religion at the University of Oxford, and Fellow of the British Academy. Swinburne proposed his argument in a book, The Resurrection of God Incarnate, published by the Clarendon Press of Oxford University Press in 2003.

Now, I am no connoisseur of arguments about a god – any god – nor do I usually have much interest in such. However, Chu-Carroll’s comment awoke my interest, not only because of its less than deferential nature, but also because I recall Swinburne writing a book on Bayesian confirmation theory in the early 1970′s, which I tried to use – and failed – as a student trying to distinguish good arguments from bad arguments when the premisses were not certain (if the premisses are certain, deductive logic is good enough – I leave aside the question of which deductive logic……). And so it seemed to me that assigning Swinburne’s argument to a category of “bad math”, indeed, according to Chu-Carroll, “mind-numbingly stupid math”, was probably mistaken.

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