Konstancja Duff playing Schubert

Another post today, again spreading the word — this time not about a maths result I chanced to stumble across, but about a young pianist (who happens to be a recent Cambridge philosophy student, and who is now studying for a Masters in Performance at the Royal College of Music). Again, I just chanced to come across Konstancja Duff’s SoundCloud page, and recognising her name I started listening in particular to her performance there of the Schubert G Major sonata. And then I continued listening, and listened again. It is a very good, serious and reflective (philosophical!) performance of one of Schubert’s masterpieces. I thought it rather remarkable, enjoyed it a great deal, and hope you will too.

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Colouring natural numbers, colouring real numbers

If you have lots of objects lined up in a row, and only a relatively small palette of colours to paint them with, then you’ll expect to be able to find some patterns lurking in any colouring of the objects.

Here’s a famous and lovely combinatorial theorem to that effect due to Van der Waerden.

For any r and k, there is an N big enough so that, however the numbers 1, 2, 3, … N are coloured with r colours, there will be an monochromatic arithmetic progression of numbers which is k long.

For example, suppose you have r = 7 colours, and put k = 4, then there is a number NW(7, 4) such that, it doesn’t matter how you colour the first N or more positive integers with 7 colours, you’ll find an arithmetical sequence of numbers a, a + e, a + 2e, a + 3 which are all the same colour. As is so often the way with numbers that crop up in this sort of combinatorics, no one knows how big W(7, 4) is: the best published upper limit for such numbers is huge.

Here’s a simple corollary of  Van der Waerden’s theorem (take the case where k = 4, and remark that  a + a + 3e  = a + e + a + 2e)

For any finite number of colours, however the positive integers are coloured with those colours, there will be distinct numbers a, b, c, d the same colour such that a + d = b + c.

So far so good. But now let’s ask: does this still hold if instead of considering a finite colouring of the countably many positive integers we consider a countable colouring of  the uncountably many reals? In other words, does the following claim hold:

(E) For any \aleph_0-colouring of the real numbers, there exist distinct numbers abcd the same colour such that a + d = b + c.

Or since a colouring is a function from objects to colours (or numbers labelling colours) we can drop the metaphor and rephrase (E) like this.

(E*) For any function f : \mathbb{R} \to \mathbb{N} there are four distinct reals, abcd such that f(a)  = f(b) = f(c) = f(d), and a + d = b + c.

So is (E*) true? Which, you might think,  seems a natural enough question to ask if you like combinatorial results and like thinking about what results carry over from finite/countable cases to non-countable cases. And the question looks humdrum enough to have a determinate answer.   No?

Yet Erdős showed that (E*) can’t be proved or disproved by ZFC. Why so? Because (E*) turns out to be equivalent to the negation of the Continuum Hypothesis. Which is surely a surprise. At any rate,  (E*) is the most seemingly humdrum proposition I’ve come across, a proposition not-obviously-about-the-size-of-sets, that is independent of our favourite foundational theory.

Make of that what you will! — but I just thought it was fun to spread the word. You’ll learn more, and be able to follow up references, in an arXiv paper by Stephen Fenner  and William Gasarch here.

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Gödel book related …

  • Do please check that your library by now has a copy of the second edition of An Introduction of Gödel’s Theorems (it really is significantly better than the first edition, with a lot of changes throughout). Having put in the work to improve the book, I’d like you/your students to have the best version available!
  • I’ve just updated the corrections page for IGT2 (thanks to Richard Baron for spotting a few more, fortunately minor, errors). 
  • I have also updated the page on what to read before/after/instead of IGT2.
  • I have further updated the notes Gödel Without Tears (the first three episodes  have now been revised).
  • In updating those notes I’ve removed a few unnecessarily fancy asides — the material appears in a different guise in the newest set of exercises for the book (which will you will find here). Indeed my plan as I work through updating the GWT notes is, at the same time, to add  exercises on the topics of revised episodes as I go along.
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The Higgins

The Early Ploughman, Samuel Palmer

We hadn’t been to The Higgins since the now united galleries and museum in Bedford re-opened last year after a really major refurbishment. The building is beautifully renovated    (or mostly so — the café is not particularly attractively laid out), and the museum displays look terrific. In particular, there is currently a very enjoyable small exhibition A National Art: Watercolour & the British Landscape Tradition. This is drawn entirely from the gallery’s own collection, and includes works by Samuel Palmer (including the etching above), Turner, and Cotman, alongside twentieth-century watercolours. Some very fine pieces. All very well worth a visit, and indeed a re-visit.

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Introducing Homotopy Type Theory

Yes, Homotopy Type Theory is the latest, greatest, thing (we are told). Yes, a free book is available, following on from a major year-long program at the Institute for Advanced Study at Princeton in 2013-13, and this will tell you lots about the current state of play. And yes, you too started the book and found it pretty impenetrable. What on earth is going on?

Help is at hand.

Robert Harper at CMU ran a grad course last semester, ‘Introducing Homotopy Type Theory’. Notes written up by his students are online. So too are videos of his lectures (use the “stay on the web” option if visiting the site on an iPad). This all looks a pretty good way in if you are still curious about the HoTT phenomenon.

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Gödel Without (Too Many) Tears, 2014 version

I’ve made a start updating the notes Gödel Without Tears.

The previous version of the notes was downloaded nearly three thousand times in the last twelve months: so it certainly seems worth putting in the effort to produce a better version, and to get the notes to integrate better with the new edition of my Gödel book.

Some months ago, I rashly said I might try to run some kind of informal online course based on GWT. But unexpected pressures on my time have made that impossible. I’ll only be able to continue updating the notes at irregular intervals. However, you can leave comments on the GWT page to report typos or unclarities. And if you (or your students) post more substantive queries in a sensible form on math.stackexchange then almost certainly someone (quite possibly me!) will answer them.

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Edmund de Waal at the Fitz

Image linked from Apollo Magazine’s article ‘Fragile Histories’ by Jon Sanders

There’s a wonderful small exhibition ‘On White: Porcelain Stories from the Fitzwilliam Museum’ by Edmund de Waal at the Fitz until Sunday 23 February 2014. There are three works by de Waal himself, and a series of cases in which he chooses some favourite pieces of porcelain from the museum’s collection, and comments illuminatingly on their significance and strangeness. A delight if you are in Cambridge and want to escape the madness of the town centre during the sales.

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A Christmas card

Gentile Da Fabriano (c. 1370 – 1427), Navity, Uffizi

All good wishes for a happy and peaceful Christmas

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Intro to Formal Logic: seventh time lucky?

My Introduction to Formal Logic was published in 2003, and CUP’s initial print run was rather large, so I didn’t get the chance to correct the inevitable typos and thinkos until a reprint in 2009. By that time, needless to say, there were quite a few little presentational things I wanted to change, so I slipped in a load of minor rewritings too. This revised version has been reprinted a number of times. (Oh yes, but of course, I’m making an absolute fortune …)

Along the way, Joseph Jedwab kindly sent me an embarrassingly long list of further errors in the revised printings (I have to put my hand up to having introduced quite a number of these in making the “improvements” in the second printing: thankfully, they were nearly all minor typos). Eventually I had the opportunity to make the needed further corrections, and I’ve just picked up the seventh printing from CUP bookshop. I hope this latest version is a heck of a lot cleaner than the previous ones. Fingers crossed.

A revised printing is not a new edition with a new ISBN, so I’m afraid you can’t put in a bookshop request for the seventh printing and be guaranteed to get one. But eventually the new version will propagate through the distribution system, and jolly good it is too. Or at any rate, reading through while making corrections and looking for any that Joseph Jedwab had missed (none, as far as I could find), I found I didn’t actually hate the book. Distance lends enchantment, eh?

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TYL, #19: the Teach Yourself Guide reorganized and updated

Sooner than I was planning, there’s now yet another update for the Teach Yourself Logic Guide. So here is Version 9.4 of the Guide (pp. iii +  72).  

The main change — though it is a significant one, which is why it is worth propagating this new version ahead of schedule — is that the Guide has been reorganized to make it easier to navigate, and hopefully less daunting. Topics on the standard “mathematical logic” curriculum (of interest of mathematicians and philosophers alike) are now separated more sharply from topics likely to be more specialized interest to some philosophers. I’ve also added comments on books by Devlin, Hodel, Johnstone, and Sider.

As I’ve said before, do spread the word to anyone you think might have use for the Guide. As always, there’s a stable URL for the page which links to the latest version, http://logicmatters.net/students/tyl/. You can reliably use that link in reading lists, or on your website’s resources page  for graduate students, etc.

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