This and that
Oh dear. That’s very embarrassing. I spotted a horrid thinko at the top of p. 77 of printed version of Beginning Mathematical Logic. I can hardly believe that I wrote, concerning infinite binary strings and real numbers between 0 and 1, that “different strings represent different reals”. Ouch. So replace the para numbered “2.” by
Note too that a real number between 0 and 1 can be represented in binary by an infinite string. And, by the same argument as before, for any countable list of reals-in-binary between 0 and 1, there will be another such real not on the list. Hence the set of real numbers between 0 and 1 is again not countably infinite. Hence neither is the set of all the reals.
I’ve updated the online PDF, and uploaded a corrected file to Amazon which will take a couple of days to work through the system). And I’ll start a corrections page for those who have the first printings of the book.
I’m not going to fret about every minor typo. But I will correct major mistakes that could mislead the reader (and then, when I do, I’ll take the opportunity to correct any smaller errors I know about). I won’t add new content though: that can wait until a second edition …!
Update Amazon reports that changes are now live: so a book ordered from now on should be the very-slightly-revised version (dated 18 Feb on the verso of the title page).
Among the newly published books in the CUP bookshop today, here’s one that could well be of interest to some readers of this blog, namely Sharon Berry’s A Logical Foundation for Potentialist Set Theory.
It is, as is now the default for new CUP monographs, published at a ludicrous price. However, it is good to report that there is a late draft downloadable from Berry’s website here (though a Word document, sad to relate, so some of the symbolism is a bit gruesome).
The headline news is that Berry advocates a version of potentialist set theory — as she nicely puts it, “the key idea … is that, rather than taking set theory to be the study of a single hierarchy of sets which stops at some particular point …, we should instead interpret set theorists as making modal claims about what hierarchy-of-sets-like structures are possible and how such structures could (in some sense) be extended.” The virtue of this idea is supposed to be that we can avoid problems that arise from assuming that the height of the universe is fixed (giving us baffling questions along the lines of “fixed how?”, “why stop there?”). There are already potentialist set theories out there in the literature; but Berry gives a new version which depends on deploying a certain natural generalization of the logical possibility operator. Then,
I show that, working in this framework, we can justify mathematicians’ use of the ZFC axioms from general modal principles which (unlike those used in prior potentialist justifications for use of the ZFC axioms) all seem clearly true. This provides an appealing answer to classic questions about how anyone (realist or potentialist) can satisfyingly justify use of the axiom of replacement.
The main work is done in the first two Parts of the book, a bit over a hundred pages. This is indeed interesting stuff. And there is a shorter but attractive exposition of some of the key ideas available here, in a paper jointly by Sharon Berry and Peter Gerdes. (See also and compare Tim Button on potentialism in the second paper of his trilogy linked here.)
The third Part of Berry’s book, another hundred pages, looks beyond set theory, and “turn[s] to larger philosophical questions”. That’s perhaps a mistake, though; from what I have read it might have been better to take some of the Part II topics in a slightly more relaxed and expansive way, increasingly accessibility, and then kept only the most immediately set-theory relevant sections on Part III.
My eye was caught by this new paper ‘Against Fregean Quantification’ by Bryan Pickel and Brian Rabern. From the Abstract:
There are two dominant approaches to quantification: the Fregean and the Tarskian. While the Tarskian approach is standard and familiar, deep conceptual objections have been pressed against its employment of variables as genuine syntactic and semantic units. Because they do not explicitly rely on variables, Fregean approaches are held to avoid these worries. The apparent result is that the Fregean can deliver something that the Tarskian is unable to, namely a compositional semantic treatment of quantification centered on truth and reference. We argue that the Fregean approach faces the same choice: abandon compositionality or abandon the centrality of truth and reference to semantic theory. …
Now, the treatment of quantification in my IFL is Frege-flavoured, in the way that Benson Mates’s classic treatment was. So I need to work out whether I should feel challenged by the arguments here. At a first pass, my hunch is not. But I’ll certainly put this on my list of things to worry about when I get back to reworking IFL for a third-and-last edition! And if inspiration strikes, I may return to it here rather sooner.
The paper’s headline news is also available in a somewhat lugubrious 27 minute YouTube video.
Fifteen years ago, as I mentioned here, I chanced on this photo of Monica Vitti on the web. A sudden jolt from the past, as I had a framed copy of that very shot on my wall as a student for five or six terms.
About the same time, I saw L’Avventura again after a gap of many, many years. I’m not sure why, but I wasn’t really expecting the film to stand up after four decades, and had expected that it would seem too mannered and pretentious. But I was completely bowled over anew: bleak but stunning still.
The image of the young woman who played Claudia in L’Avventura and Vittoria in L’Ecclise, so beautiful and so touching, haunted and entranced many of us who watched her back then. And yes, it was an image. But still, a moment of real sadness to hear today of Monica Vitti’s death.
Beginning Mathematical Logic is now available as a paperback, ISBN 1916906338. I’d hoped to get it finished well before the end of 2021; but these are strange times, when spirits can droop and concentration levels plunge. Still, I’ve made it to the finishing line.
I can quietly update the PDF and the text of the printed paperback if and when I hear about typos or minor thinkos. So do let me know about anything you notice. And it’s the very nature of the enterprise that the Guide will no doubt someday need revising, and I’ll no doubt start an addendum webpage in the interim: so suggestions for more substantial future improvements will still be very welcome.
The paperback is only available from Amazon. But as I’ve explained before, using their basic-level print-on-demand service (without the potential for expanded distribution to bookshops) keeps the price to a rock-bottom minimum, £4.99/$5.99. Amazon refuseniks still can download the PDF!
I might be raising a glass this evening …
She was asked in an interview about her most memorable concert experience. And Elisabeth Brauß responded “It’s always a few notes, a phrase, a shared moment on stage, during which you know why you’re doing it all, at least as long as the music sounds … That can happen in every concert.”
But it must, I think, have been more than a few notes that were memorable for her last Monday at Wigmore Hall, more than a phrase. The opening pieces alone were a wonder and a delight. Elisabeth started with a sequence of Scarlatti sonatas, beautifully chosen to make (as it were) an emotionally satisfying meta-sonata. And she played with such sensitively controlled phrasing, bringing out the depths particularly of the second B minor sonata, true to the music, so individual yet without a moment’s self-indulgence. The audience knew that they were listening to something very special. My very hard-to-please piano-teacher sister was “blown away”. And the way Elisabeth herself responded at the end of sequence and then smilingly took the applause before she settled to compose herself before launching into the Mozart, rather suggested that the shared Scarlatti which she been evidently been so wrapped up in would be as memorable for her as for her audience. Just outstanding, as long as the music sounded.
But see and hear for yourself, on YouTube for some weeks.
When we had our loft extension built a couple of years ago, we had to have the hallways (re)decorated from top to bottom; and so the shabby old William Morris wallpaper needed to be replaced. And because some of the spaces are narrow and potentially dark, we went for a very pale version of Morris’s Poppy. Perhaps we played over-safe; but it is still a delight.
I have always been interested in and hugely admired William Morris. And my Christmas present to myself was this quite beautifully produced book, edited by Anna Mason, and published by the V&A last year to mark the 125th anniversary of his death. There are consistently interesting essays on the man (as a designer, writer, political activist, conservationist), and on the art (from the early painting to the Kelmscott Press, via the furniture, tapestry, wallpaper and more). The many, many illustrations are just terrific. So this is a joy, and very warmly recommended.
I was trying to write a piece about why Elisabeth Brauß’s playing is so extraordinary, but rather failing. So I must return instead to a more familiar theme here (which regular readers will be as tired of as I am!). But with many thanks to those who e-mailed in last-minute comments and corrections, I’ve one last time gone through a complete version of Beginning Mathematical Logic: A Study Guide, making many corrections. You can download a final, final, draft version here (all x + 184 pages of it). I hope you like the wise addition to the very last page …
There are no major changes since the previous online version, but there’s been a fair amount of minor tinkering at the level of changing punctuation, adding a very few more footnotes, etc. It made a remarkable difference having a printed proof copy in book form to work from: all kinds of minor glitches hit the eye in a way they don’t onscreen or even in a stack of print-out.
So the plan is that I send off to get another proof copy for a final typographical check. And then, all being well, we are good to go, and there should be another Big Red Logic Book to buy by the end of next week. Start saving your pennies.
And if you’ve been meaning to let me know about some error you spotted in a previous version, do let me know if it is still there in this current version. But don’t delay!
Added Jan 28 Refresh your browser to download a version which has got the index linking corrected.
Added Jan 29 Half a dozen misleading internal links removed (see comments), and one typo corrected. Keep ’em coming …
Added Jan 30 Today’s version has improved handling of internal links; and silly thinko about cumulative hierarchy corrected.
Added Jan 31 Some tiny changes to make a few pagebreaks in Chapter 12 fall more nicely. One additional reading added to end of Chapter 9. And at this point, unless some egregious error is pointed out, I’m calling a halt to further tinkering. Famous last words.