Year: 2012

Amethe von Zeppelin

I’ve had Carnap’s The Logical Syntax of Language on my shelves for over forty years. I can’t say it was ever much consulted; but I’ve been reading large chunks of it today, in connection with Gödel. Carnap’s book is often credited with the first statement of the general Diagonalization Lemma, and I wanted to double check this. (The credit seems to be somewhat misplaced in fact, but that’s for my next post!)

Reading the early pages of the book, I’ve been struck by how good the 1937 translation seems. Well, I can’t vouch for its accuracy — for I don’t have the German to check — but it’s very readable, and seems entirely sure-footed with the logical ideas (though that of course could be because of careful oversight by Carnap). The translation is by Amethe Smeaton, who is warmly thanked in the Preface to the English Edition. But who was she?

A Countess von Zeppelin, no less. (But no, not the Countess who later threatened to sue Led Zeppelin for illegal use of her family name.) Which makes me wonder: what led the Countess to becoming a translator, and why did she become involved in this project. What was her background that made her an apt choice for translating this book of all books?

She’d translated before, a history book by Paul Frischauer, Prince Eugene: A Man and a Hundred Years of History, first published in German in 1933, translated in 1934, and still in print. Later she translated Schlick’s Philosophy of Nature (1949), Walter Schubart’s Russia and Western Man (1950), Bruno Freytag’s Philosophical Problems of Mathematics (1951), and a book by Karl Kobald Springs of Immortal Sound (1950), on places in Austria associated with composers. She also co-translated Werner Heisenberg’s Nuclear Physics (1953). That’s a rather remarkable catalogue! And she wasn’t “just” a translator: she was competent enough to be asked to review a group of logic and philosophy of science books for Nature in 1938 (she writes the composite review in a way that indicates she was very much up with developments).

But that is all I’ve been able to discover. Amethe von Zeppelin (that seems the name she most used) was, at least for a period, seemingly very knowledgeable about the then contemporary developments in logic and the philosophy of science, and it seems she had very wide intellectual interests (if we can assume a Countess von Zeppelin could at least choose which translations she wanted to take on). I’d like to know more. But, as Mrs Logic Matters opined, it seems she is another woman whose history has been written in water. Or is there someone out there who can help fill in her picture a little?

Somewhat gappy Gödel

When planning and actually writing my Introduction to Gödel’s Theorems, I intentionally consulted other books as little as possible, trying to reconstruct strategies and proofs from memory as far as I could. I thought that would be a good discipline, and rethinking things through for myself would help me to explain things as clearly as possible. Well, people have indeed said nice things about the clarity of the resulting book, though my writing policy did mean I made a few nasty mistakes, some caught by pre-publication readers, and others corrected in later printings. I blush to recall ...

Anyway, now that I am working on a second edition, I want to spend the next month pausing to have a look at how others have handled the First Incompleteness Theorem. The basic shape of my book is fixed (after all, I’m doing another edition of IGT, not writing another different book, fun though that would be to do): but I might well get inspiration for how to make local improvements. So how do others state the Theorem? And how do they prove it? I’m planning to write up notes on expository themes and variations as I go along, and will post them here in due course.

The natural place to start is with Hilbert and Bernays. Slight problem: I don’t have German (yep, schoolboy Greek was all well and good, but hasn’t exactly been useful). Does anyone out there have detailed notes of how they prove the First Theorem in §5.1 of their second volume? I’d be really delighted to hear if so! Otherwise, there is a French translation, so I suppose I ought to do battle with that. Not that my French is any good these days either …

Until I can get hold of a translation of Hilbert and Bernays, then, the expository tradition for me will really have to start with Kleene’s wonderful 1952 Introduction to Metamathematics. I’m looking forward a lot to dipping into that again. Then I counted over forty other books on my shelves which give more or less detailed proofs of the First Theorem: hours of fun ahead, obviously. But I’ve started, yesterday and today, by rereading Gödel’s 1931 and his 1934 Princeton lectures. I have to blush again. I’d forgotten, for example, just what Gödel didn’t prove in 1931. ‘We shall only give an outline of the proof’ that every recursive relation is (as we would now say) representable in P;  ‘the entire proof’ [that for any predicate expressing a recursive property/relation there is an equivalent arithmetical predicate] ‘can be formalized in the system P‘ [it can be, no doubt, but this isn’t proved].

In fact, the situation is this. There are in fact two significantly different results stated in the 1931 paper, the ‘semantic’ and the ‘syntactic’ incompleteness theorems. The first requires that we are dealing with a theory which can express primitive recursive relations and is sound; the second version beefs up the first assumption and requires a theory which can represent p.r. relations while weakening the second assumption to require only that the theory satisfies the syntactic condition of omega-consistency. The first, ‘semantic’ theorem is sketched in Gödel’s §1, the official ‘syntactic’ version is the topic of the central §2 and §3. Look carefully, however, and while there is (enough material for) a full proof of the underplayed semantic version, the proof of the ‘syntactic’ result (the result that is normally meant when people talk now about the First Theorem) is in fact gappy, gappier than I’d really remembered.

Meanwhile, I see over 200 people have downloaded the draft of the first eight chapters of the second edition. Thanks to David Auerbach and Seamus Bradley for some amazingly speedy comments. Keep ’em coming folks!

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