Since reading Logicomix (which, as I said, I really like), I’ve been wondering about the “logic and madness” theme that runs through the book. In the making-of movie (which I also recommend), Papadimitriou says at the beginning, “We were both interested in this very curious fact, that the majority of the protagonists of this intellectual adventure [the quest for mathematical foundations] ended up insane” and Doxiadis cites the well known line from Gian-Carlo Rota’s Indiscrete Thoughts:

It cannot be a complete coincidence that several outstanding logicians of the twentieth century found shelter in asylums at some point in their lives: Cantor, Zermelo, Gödel and Post are some. (p. 4)

And, if you’ve read the book, you’ll probably agree that the “logic and madness” theme does make for a great story. But is it true? Is there a link between logic and madness?

First, some facts, and corrections of claims of facts. It is well known that Georg Cantor underwent psychiatric treatment and “died in an asylum”. But as Grattan-Guiness and Dauben have documented, it was neither Kronecker’s attacks on Cantor’s set theory, nor Cantor’s failure to solve the continuum hypothesis that drove him mad. Cantor suffered from bipolar affective disorder, i.e., he was manic depressive, and stress such as that caused by having your work viciously attacked by a leading member of the profession, or that caused by expending every last effort and yet failing to prove a theorem, caused the onset of manic periods. He would have had such attacks also if he hadn’t invented set theory (see Dauben’s Georg Cantor, Ch. 12, esp. p. 285; Dauben is very critical of people like E. T. Bell here, and offers a very nuanced interpretation of the relationship between Cantor’s mental health and his mathematics). Remember, this all happened between 1884 and 1918, when no effective treatment for bipolar disorder was available; lithium wasn’t used until the 1950s and approved by the FDA for this use only in 1970.

Emil Post likewise was manic depressive, and died from a heart attack following electric shock therapy he was undergoing in 1954. And Gödel did die because he starved himself to death as he suffered from the paranoid fear that people were trying to poison him. But also Gödel’s mental health problems manifested themselves quite early, and not as a result of a lifetime of work on logic, or because he couldn’t prove the continuum hypothesis (see Dawson’s Logical Dilemmas). In addition to Cantor, Post, and Gödel, Moses Schönfinkel, the inventor of combinatory logic, is reported to have been mentally ill.

What about the others? Rota mentions Peano and Zermelo. I couldn’t find any evidence that either of them had mental health problems. Both spent time in medical institutions, but underwent treatment not for mental health problems but for lung disease.

That leaves Frege. In Logicomix, Frege is portrayed as a raving lunatic spewing paranoid anti-semitic nonsense. In the biographical section at the end of the book, one reads the following:

In the last decades of his life, Frege became increasingly paranoid, writing a series of rabid treatises attacking parliamentary democracy, labor unions, foreigners, and especially, the Jews, even suggesting a “final solution” to the “Jewish problem”. (p. 325-326)

The source of these claims is Frege’s infamous “political diary” (edited by Gottfried Gabriel and Wolfgang Kienzler, “Frege’s politisches Tagebuch”, Deutsche philosophische Zeitschrift 42:6 (1994) p.105–1098; translated by Richard Mendelsohn, “Diary: Written by professor Dr Gottlob Frege in the time from 10 March to 9 April 1924“, Inquiry 39 (1996) 303–342; you can get a taste for them with a bit of background in Stroll’s Twentieth-century Analytic Philosophy and in the chapter on Frege in Martin Davis’ The Universal Computer). As you can see for yourself, the diaries reveal the very dark side of Frege’s political views: reactionary, anti-semitic, anti-catholic, anti-socialist. But: Frege didn’t write “increasingly rabid treatises” over “the last decades of his life”—these are diary entries written over two months in the very last year before he died. As far as I can tell, he never advocated a “final solution” to the “Jewish problem” with anything like the meaning that these terms have taken on, and he didn’t use this Nazi terminology. There is no indication that he admired Hitler (he opposed the Munich Putsch of 1923), and there’s no indication that his anti-semitism was racially motivated or anywhere near the level of the Nazis. But most importantly: He wasn’t clinically paranoid. As objectionable as his views are, they were widespread in Germany at the time (Had they not been, Hitler would never have come to power). Moreover, if he had been paranoid, this would, I think, absolve Frege of moral responsibility. After all, we don’t hold people morally (or legally) responsible for their actions when they’re insane. So: Frege: reactionary anti-semite, but no Nazi, and not insane.

Of the “protagonists of this intellectual adventure”, four (Cantor, Schönfinkel, Gödel, Post) had mental health issues. We don’t know enough about Schönfinkel, Cantor and Post were manic depressives, Gödel more than the others, and probably paranoid schizophrenic. Is that “a majority”? Is it even a statistically significant increase from the norm?

The National Institutes of Mental Health puts the percentage of the US population with “serious mental illness” at 6%. What’s the percentage of pioneers of logic with a serious mental illness? We’ve found four, but what’s the sample? Let’s say Rota had in mind the authors of papers in van Heijenoort’s From Frege to Gödel. That’s 30, and doesn’t even include Tarski, Lukasiewicz, Church, Fraenkel, Gentzen, Turing (all not insane), or many of the less well known people working in foundations at around that time. So: 13% of the pioneers of logic had a serious mental illness. But with a sample of 30, the margin of error has to be huge. I’m no statistician, but using the standard formula, I get a margin of error of ±12% (ok, I know you probably shouldn’t use the standard formula for samples this small; if you know stats, help me out, please). This suggests that there’s good reason to think that Rota’s claim is just wrong: it may very well be pure coincidence.

All this of course doesn’t detract from the good story told in Logicomix, which, after all is mainly about Russell, about his personal life, and about his struggle with the foundations of mathematics; the “logic and madness” theme isn’t that pronounced. But that story does play into a myth that, if taken on its own, is not exactly the image any field of science wants to project (or have painted) of itself: that it’s the domain of lunatics. It’s not only detrimental to the field and hurtful to the people working in it, it also distorts and minimizes the actual personal struggles of the protagonists and the interesting historical context. All of these people lived through one world war, many of them through two and the toughest economic times of the last 100 years. Some were forced to flee their home countries, some faced persecution and prejudice, some personal tragedy, some professional misfortune. Most of them produced their groundbreaking results despite these obstacles. These are the important stories, not any myths about how doing logic drives people mad.

A revised version of Hubert Kennedy‘s 1980 biography of Giuseppe Peano, is available as a free download and a cheap print-on-demand paperback through lulu.com: Peano: Life and Works of Giuseppe Peano.

Sydney Padua has produced a number of amazing and funny comics on Ada Lovelace, Charles Babbage, and the Difference Engine. It’s a bit hard to navigate, to get to all three installments of “Lovelace and Babbage vs. The Economy” you have to click on the “Economic Model” link in the sidebar. The upside is, though, that you’ll get to browse through Sydney’s many intriguing links and finds that follow the strips. If there wasn’t work to be done, I’d probably trace her steps and learn all kinds of fun and interesting things about these two pioneers of computing!

The BBC Techlab has a 6-panel strip (colored) by Sydney up here. A pity that Ada is only “guest-starring” there.

Several of the commenters on my previous post on motivating the study of logic in my intro class have suggested that one important aspect of logic is the precision it affords, and hence the usefulness of logic in avoiding ambiguities. So I tried to find some nice examples of where ambiguity in natural language—and the resulting different interpretations—can have important consequences. (I’m still looking for examples, especially form philosophy!) I happened upon a paper entitled “Syntactic Ambiguity” by Paul Conway, which gives some very nice actual examples from law. I picked one of the examples that can be dealt with in propositional logic (no quantifiers used yet).

a is a cube in front of b, or a tetrahedron in front of b, or to the left of b.

That’s ambiguous between^*

(Cube(a) ∧ FrontOf(a, b)) ∨
(Tet(a) ∧ (FrontOf(a, b) ∨ LeftOf(a, b))

and

(Cube(a) ∧ (FrontOf(a, b) ∨ LeftOf(a, b))) ∨
(Tet(a) ∧ (FrontOf(a, b) ∨ LeftOf(a, b))

Here’s the real-life example from the above paper:

In R v. Casement, Sir Roger Casement was charged with high treason contrary to Treason Act, 1351 (Eng.). It was alleged that during World War I he incited British subjects who were prisoners of war in Germany to renounce their allegiance to the King. The statute declared that treason was committed ‘… if a man do levy war against our Lord the King in his realm, or be adherent to the King’s enemies in his realm, giving to them aid and comfort in the realm, or elsewhere, and thereof be properly attainted of open deed by the people of their condition: …’. The charge alleged adhering to the King’s enemies elsewhere than in the King’s realm, namely in the empire of Germany. The defence unsuccessfully submitted that the Crown had failed to prove an offence in law. ‘The contention is that those words “or elsewhere” govern only the words “aid and comfort in the realm” and have no application to the words “be adherent to the King’s enemies in his realm.’

I believe that part of the reason that the trial and conviction caused such an outcry, aside from the fact that Casement was famous as a humanitarian exposing human rights abuses in the Congo and Peru, was that it wasn’t clear if the original document of the Treason Act contained the last comma or not.

^* A third reading would be

(Cube(a) ∧ FrontOf(a, b)) ∨ (Tet(a) ∧ FrontOf(a, b)) ∨ LeftOf(a, b)

but that isn’t a possible reading of the clause in the Treason Act.

Yesterday’s mail contained my copy of Logicomix: An Epic Search For Truth, a graphic novel by Apostolos Doxiadis and Christos Papadimitriou with art by Alecos Papadatos and Annie Di Donna. It is scheduled to be released in the US on September 29, but amazon.ca apparently already had it. The UK edition is now sold out (a second printing is scheduled to be in stores October 2). It’s a compelling read for everyone interested in logic and its history, or in Bertrand Russell, or in intelligent graphic novels.

The main story arc consists in Russell giving a lecture on “The Role of Logic in Human Affairs” at “an American university” (looks like Berkeley) just after the start of WWII (September 1939). In the lecture, he tells the story of his own life, how his quest for finding certain truth led him to a study of the foundations of mathematics, discovering logic, writing Principia Mathematica with Whitehead, meeting Wittgenstein, the inter-war years with the Tractatus, the Vienna circle, and Gödel—and his personal life. There’s a lot about madness and logic, the conflicts within logicians between their work and their passions, about struggle and failure. All this is interleaved with a frame story in which the authors of the book discuss what they’re trying to do in the book, explain some mathematical details, and reflect on the story that Russell’s telling his audience, ending with a … well, I don’t want to give it away.

The book is very well done overall: it’s an engaging read, the art is great, the logic and philosophy are accurate for the most part. There’s a lot of license taken with historical details, but that usually makes for a better story. My favorite is the barfight between adherents of Poincaré and of Hilbert at the 1900 International Congress of Mathematicians. And, truth be told, you don’t have to take much license with many of the characters in this story to make them compelling—think Wittgenstein:

There was only one thing that really bothered me: they claim—not just in the story but also in the otherwise informative background section at the end of the book—that Hilbert sent his son Franz off to an asylum when Franz was 15, that Franz spent the rest of his life there, and that Hilbert never visited him. But at least according to Constance Reid’s biography of Hilbert, a) that happened when Franz was 21, b) he was in treatment only until 1917, and c) thereafter lived with the Hilberts again. I’m also no great fan of the title (it’s about as unimaginative as “LogBlog” is for a logic blog). But: I am a fan of the book. Finally a logic book for the coffee table! I might even assign it for a history of logic class. Get your copy, it’s even pretty reasonably priced at about $15 on amazon.com.

The website has a preview and some additional information, including nice pictures of the original locations. Feel free to reply with your own opinions, speculations, and queries about the historical details and I’ll see if I can fact-check them…

UPDATE: There’s a book tour which includes a stop at the LA public library, with Zlatan Damnjanovic (USC Philosophy) on Oct 7 and one at MSRI (pronounced “misery”, the Mathematical Sciences Research Institute in the Berkeley hills) with Paolo Mancosu (Berkeley Philosophy) on Oct 19.

UPDATE: More on the logic and madness theme here.

Here’s an interesting new-ish paper on the issue of gender differences in mathematics aptitude:

Janet S. Hyde, and Janet E. Mertz. Gender, culture, and mathematics performance. PNAS vol. 106, no. 22, (June 2, 2009).

Abstract: Using contemporary data from the U.S. and other nations, we address 3 questions: Do gender differences in mathematics performance exist in the general population? Do gender differences exist among the mathematically talented? Do females exist who possess profound mathematical talent? In regard to the first question, contemporary data indicate that girls in the U.S. have reached parity with boys in mathematics performance, a pattern that is found in some other nations as well. Focusing on the second question, studies find more males than females scoring above the 95th or 99th percentile, but this gender gap has significantly narrowed over time in the U.S. and is not found among some ethnic groups and in some nations. Furthermore, data from several studies indicate that greater male variability with respect to mathematics is not ubiquitous. Rather, its presence correlates with several measures of gender inequality. Thus, it is largely an artifact of changeable sociocultural factors, not immutable, innate biological differences between the sexes. Responding to the third question, we document the existence of females who possess profound mathematical talent. Finally, we review mounting evidence that both the magnitude of mean math gender differences and the frequency of identification of gifted and profoundly gifted females significantly correlate with sociocultural factors, including measures of gender equality across nations.

H/T: Justin Snider

Volker Peckhaus has a new entry in the Stanford Encyclopedia on the influence of Leibniz on the development of logic in the 19th century.

I got Herbert Feigl’s Theorie und Erfahrung in der Physik from the library, and on the front flyleaf there’s a handwritten dedication to Karl Menger that reads “Herrn Professor Menger ergebenst überreicht vom Verf., 13. VI. 1929.”

Wow. Four students (Sean Geggie, Martin Have, Anders Nissen, Mikkel Vester) at the University of Aarhus, Denmark, constructed a Turing Machine tape read/write assembly in LEGO. This was a final project for the course Embedded Systems – Embodied Agents, taught by Ole Caprani of the LEGO Lab at Aarhus. On their blog Lego of Doom, they describe the initial idea as follows:

The “Turing Machine” will need to traverse a track of some kind, reading marks on the track and altering them. Reading and altering bits on a track entails three things: Detecting that the machine is over a cell, detecting the state of the cell and altering the state of the cell. These three problems will each be solved by careful application of various sensors. Many solutions exist for each problem and much experimentation will be needed to find out which yields the most stable results.

Architecturally speaking, the Turing Machine robot could be in contact with a PC via bluetooth connection. Via this connection, the robot could leave some calculations to the PC which would send back instructions. One example of this could be that the PC handles all the “boring” Turing Machine calculations while the robot itself could be in charge only of cell and bit state detection as well as motor control.

The main part of this project would be getting the robot to accurately read and set the bits on the track. The implementation of the turing machine itself is trivially accomplished. The “meat” of the project is embedding the program in a machine that actually performs computations in a physical environment.

They did all that and then they made an awesome movie about it:

[youtube=http://www.youtube.com/watch?v=cYw2ewoO6c4&hl=en&fs=1&]

HT: Francesco Berto/Andrea Sereni

The new journal of the Association for Symbolic Logic, the Review of Symbolic Logic, started up in 2008. Two of the papers in that first volume were selected for the Philosopher’s Annual, vol 28, which each year “attempts to select the ten best papers in philosophy published in each year”. They are:

• Thomas Forster, The Iterative Conception of Set, Review of Symbolic Logic 1:1 (2008), 97-110
• Penelope Maddy, How Applied Mathematics Became Pure, Review of Symbolic Logic 1:1 (2008), 16-41

Only the Phil Review also had more than one (viz., three) papers selected. The selected papers also include another logic paper:

• Fabrizio Cariani, Marc Pauly & Josh Snyder, Decision Framing in Judgment Aggregation. Synthese 163 (2008), 1-24

You can read the papers online (and free) at the Philosopher’s Annual website. Congratulations to all the authors!

In response to the petitions mentioned recently, the UK government has issued an apology. The statement in full, as published on the 10 Downing St website:

2009 has been a year of deep reflection – a chance for Britain, as a nation, to commemorate the profound debts we owe to those who came before. A unique combination of anniversaries and events have stirred in us that sense of pride and gratitude which characterise the British experience. Earlier this year I stood with Presidents Sarkozy and Obama to honour the service and the sacrifice of the heroes who stormed the beaches of Normandy 65 years ago. And just last week, we marked the 70 years which have passed since the British government declared its willingness to take up arms against Fascism and declared the outbreak of World War Two. So I am both pleased and proud that, thanks to a coalition of computer scientists, historians and LGBT activists, we have this year a chance to mark and celebrate another contribution to Britain’s fight against the darkness of dictatorship; that of code-breaker Alan Turing.

Turing was a quite brilliant mathematician, most famous for his work on breaking the German Enigma codes. It is no exaggeration to say that, without his outstanding contribution, the history of World War Two could well have been very different. He truly was one of those individuals we can point to whose unique contribution helped to turn the tide of war. The debt of gratitude he is owed makes it all the more horrifying, therefore, that he was treated so inhumanely. In 1952, he was convicted of ‘gross indecency’ – in effect, tried for being gay. His sentence – and he was faced with the miserable choice of this or prison – was chemical castration by a series of injections of female hormones. He took his own life just two years later.

Thousands of people have come together to demand justice for Alan Turing and recognition of the appalling way he was treated. While Turing was dealt with under the law of the time and we can’t put the clock back, his treatment was of course utterly unfair and I am pleased to have the chance to say how deeply sorry I and we all are for what happened to him. Alan and the many thousands of other gay men who were convicted as he was convicted under homophobic laws were treated terribly. Over the years millions more lived in fear of conviction.

I am proud that those days are gone and that in the last 12 years this government has done so much to make life fairer and more equal for our LGBT community. This recognition of Alan’s status as one of Britain’s most famous victims of homophobia is another step towards equality and long overdue.

But even more than that, Alan deserves recognition for his contribution to humankind. For those of us born after 1945, into a Europe which is united, democratic and at peace, it is hard to imagine that our continent was once the theatre of mankind’s darkest hour. It is difficult to believe that in living memory, people could become so consumed by hate – by anti-Semitism, by homophobia, by xenophobia and other murderous prejudices – that the gas chambers and crematoria became a piece of the European landscape as surely as the galleries and universities and concert halls which had marked out the European civilisation for hundreds of years. It is thanks to men and women who were totally committed to fighting fascism, people like Alan Turing, that the horrors of the Holocaust and of total war are part of Europe’s history and not Europe’s present.

So on behalf of the British government, and all those who live freely thanks to Alan’s work I am very proud to say: we’re sorry, you deserved so much better.

Gordon Brown

Next week it’s back to the classroom for me, and I’m teaching intro logic again. I’ve been thinking a bit about what to do on the first day, especially in the “why you should take this course” department. There’s the obvious reason: it’s required (at least for philosophy and CS majors). So I’m really talking about “why you should want to take this course”. And here, the textbooks usually don’t do such a good job. First there’s the “you’ll learn how to think correctly and identify logical errors” line. The examples there are usually a valid and an invalid syllogism, examples that I suspect anyone with any chance getting a decent grade in the class can already identify as good and bad instances of reasoning. Second, there’s the “important applications in logical circuit design” story. But, honestly, any logic design course can cover the logic they need for combinational circuits in a week. Third, there’s the “taking this course will train your analytic and abstract thinking skills”. Ok, maybe, but that’s not really a good selling point.

So I’m looking for concrete, real-life examples where some of the things that you learn in a formal logic class are useful: examples that are relatively easy to describe, where it’s obvious that these are “really relevant” to whatever discipline they’re taken from, and where you can reasonably claim that you need to be able to deal with a formal language, understand relations and multiple quantification, or use logical methods like formal proofs or model-building techniques to avoid errors or solve a problem.

One of the examples I think I’ll use is SNOMED CT. That’s a health-care terminology database (aka an “ontology”) with over 300,000 concepts organized by over 1,000,000 rules. These rules could be formulated in a fragment of first-order logic (some description logic suffices, I’m not sure which). One example I’ve seen mentioned here is this: In SNOMED CT, an leg amputation is defined as a procedure with method amputation and procedure-site-direct lower limb structure; and a toe amputation as a procedure with method amputation and procedure- site-direct toe structure. Now SNOMED CT also knows that the toe is a part of the lower limb, so that if a procedure happens in the toe, it eo ipso happens in the lower limb. Therefore, a toe amputation is also a leg amputation. But of course you wouldn’t want a surgeon to take off your entire leg if you have a gangrened toe! On the other hand, if you have a pain in your temple, then since the temple is part of the head, you have a headache, and you do want SNOMED to know that. So here you need all kinds of logic: you need a formal language in which to express these concepts and relations, it needs to be expressive enough so that you can express everything you want to express, you need logical methods to tell you a) what follows from SNOMED (queries), b) wether SNOMED is consistent, c) where the errors are and how to remove them. (I learned about SNOMED CT from Frank Wolter’s talk at the Logic Colloquium, “Mathematical logic for life science technologies“.)

Of course, all of this is just a particular case of the various important applications of logic in AI and databases, but I thought it was a nice example that wasn’t just a toy database. Also, I like the “mistakes that logic helps avoid or correct” flavor.

I’d also like examples like that from philosophy and mathematics. For mathematics I was thinking of talking about Cauchy’s “erroneous” proof of the uniform convergence theorem, and pointing out the importance of the order of quantifiers. That has the problem that (as we know from Lakatos) Cauchy didn’t really overlook the necessary requirement of uniform convergence, and also it might be a bit too difficult (to explain in a short amount of time). For philosophy, I thought of maybe using Skorupski’s argument for the principle of moral categoricity from Ethical Explorations, which I found in a post by Doug Portmore on PEA Soup. I like it because it’s simple, and recent, and from ethics, which is often considered by students to be next to the opposite of logic( as far as courses are concerned).

Do you have other ideas? Better ideas? Ideas applying in other disciplines?

I think it would be nice to have an example where a famous mathematician or philosopher committed a more-or-less elementary logical error that can be diagnosed or avoided by formalization.

Exciting new entry in the SEP on Hermann Weyl, by John Bell.

David Johnston, of the University of Victoria Philosophy Department, has just released three apps for the iPhone (and iPod Touch), which will be of interest to students (and teachers) of introductory logic courses:

Logic 100 These utilities for truth-functional logic allow you to check syntax, construct truth tables, and test for consistency and validity. Notation can be set to match any logic textbook.

Syllogism These utilities for categorical logic allow you to construct syllogisms, test them for validity, and display their Venn diagrams.

Logic 101

This app helps you construct derivations based on the system SD from The Logic Book. It checks the syntax of each line and automatically applies derivation rules. Completed derivations, including line justifications, can be emailed directly from the app.

I guess we’ll have to be more vigilant about students having cellphones on them when they take a logic exam! But, in the words of Hans von Ditmarsch, “anyone who gets people to do logic while waiting for their bus, wasting time otherwise, …, deserves praise!” Read more about these apps on hatzicware.com, try them out, and let us (and him) know what you think!

Incidentally, these apps are versions of David’s Logician’s Toolkit, which lets you do all these things inside a Java applet on his website. Useful especially if you use the Logic Book.

As you probably know, logic pioneer Alan Turing invented the Turing machine model of computation, proved the undecidability of the halting problem and (independently of Church) the undecidability of the decision problem, and played an important role in the work at Blechley Park that broke various German ciphers during World War II. He was also gay, and committed suicide following his criminal conviction for “gross indecency” and the chemical castration he was forced to undergo. There are now two petitions circulating, calling for a formal apology from the British Government for Turing’s treatment: one for British citizens and an international petition.

I had to look up a Russell quote the other day, and that’s when I noticed that many of his books — including the Foundations of Geometry, Our Knowledge of the External World, Introduction to Mathematical Philosophy, Analysis of Mind, Principles of Mathematics, Mysticism and Logic, and Principia Mathematica (annoyingly, only vol. II) — are available in their full glory through Google Books. There are lots of other gems, including Hilbert’s Grundlagen der Geometrie, the Tractatus, etc. But beware: the Google metadata are unreliable, to say the least (see Geoff Nunberg on Google Books: A Metadata Trainwreck).