## CD choice #6

Time goes so very fast: it is Martha Argerich’s 75th birthday today. Here is perhaps still my favourite recording of hers, from 1980. Completely stunning Bach playing. I can’t put it better than Jessica Duchen has:  “Martha Argerich presents intense, minor-key Bach: fierce but pure, finely argued and rhetorical yet never losing the rhythm of dance; expressive but contained and articulated with beautiful clarity. Her intensity is completely compelling from the very first note of the C minor Toccata onwards: she demands, seizes and holds attention at every moment. Her personal vision displays the emotional darkness at the heart of these pieces without any suggestion of mannerism or gimmick.”   If you don’t know this disc, you have missed what is surely one of the greatest ever recordings (and it’s now immediately available for streaming if you have an Apple Music, or similar, subscription.)

But oh how we wish that Argerich had recorded more Bach over the years …

## The English art

We drive to and from Cornwall slowly each year, staying over for a night somewhere, visiting National Trust gardens en route — this year Coughton Court and Killerton as we drove down, Knightshayes and Hidcote on the way back to Cambridge. If the weather is bad we’ll take the tour of a historic house — but they are only shells of the homes they once were, while their grounds are bursting with life and colour in May.

The making of gardens is surely one of the art forms the English do best, and care about the most.

## Postcard from Cornwall

In Cornwall again, back in St Mawes. As lovely as before. The photo was taken walking along another little bit of the South West Coastal Path this morning, this time along the cliff-tops between Portloe and Pendower. On the whole, kind weather. Too kind, at any rate, to be sitting indoors, reading much serious stuff: that, and logical blogposts, can wait until I return to Cambridge, far too soon.

Posted in This and that | 1 Comment

## TYL 2017?

The Teach Yourself Logic 2016 Study Guide is linked here at Logic Matters but also at my (decidedly sparse) academia.edu page. Rather startlingly, the latter link has now been followed up over 50K times. Who knows how much impact the Guide really has. Still, I occasionally get appreciative emails (and equally cheeringly, I don’t get protests from colleagues complaining bitterly that I am leading the youth astray). So, hopefully, TYL 2016 is doing some good in spreading the logical word.

That’s the plus side. The downside is that, given it indeed seems to be used quite a bit, I suppose I should keep on updating the Guide. The year is already rattling by, and I guess I should soon start turning my mind to the time-consuming business of (re)reading around logic books old and new, familiar and less familiar, as background homework for producing TYL 2017. So if you do have suggestions for improvement, and in particular suggestions of recent books I should really take a look at over the next few months, do let me know sooner rather than later.

So many logic books, so little time …

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## Kurt Gödel, Philosopher-Scientist #5

The next paper in Kurt Gödel, Philosopher-Scientist is by Paola Cantù, on “Peano and Gödel”. The headline claim is that Gödel’s philosophical notebooks indicate that he had read Peano (and in particular Peano’s contributions to the Formulaire des mathématiques /Formulario Mathematico) rather carefully — and that Gödel was alert to the important differences too between Peano and Russell.

Unfortunately, when it comes to trying to spell out carefully what Peano’s views were (e.g. about the nature of functions), how Russell’s differed, and exactly what Gödel’s response comes to, I found Cantù less than ideally clear.

Here’s just one example where Cantù seems muddled. Peano defines the “classe nullo” $\Lambda$ to be the class of objects which are common to every class (“classe de objecto commune ad omne classe” Formulario, §6). Peano also defines the iota function (Formulario, §7) such that $\iota x$ is the singleton class of $x$ (the class of $y$ such $y = x$). And then he adds a [partially defined] inverse iota function which undoes the effect of the iota function. I can’t typographically invert an iota here, so I’ll follow the later Peano and write $\overline{\iota}$ instead: and then the idea is that, if $a$ is a class other than $\Lambda$ such that any two members of it are equal (i.e. if it is a singleton class), then ${\overline{\iota}a}$ = ${x}$ iff ${a = \iota x}$. Note, by the way, Peano’s ${\overline{\iota}}$, i.e. his inverted iota, is thus significantly different from Russell’s inverted iota!

Peano then shows, inter alia, something of this shape: $\Lambda = \overline{\iota}K$, where ‘$K$’ is in fact just one obvious way of characterizing the singleton of $\Lambda$. The finer details of ‘$K$‘  don’t matter.

So far so good. But Cantù comments on the latter rather trivial result as follows:

The definition of  $\Lambda$ by means of the inverse iota operator allows us to define it as an individual object rather than as a set: nullo is the element associated to a set that contains all the $x$ such that $(a) x = a \land \neg a$, i.e. the elements that, for any property, satisfy that property and its contradictory.

This is surely muddled (even forgetting about Cantù’s symbolic foul-up, though to be honest that doesn’t inspire confidence). For ${{\Lambda} = {\overline{\iota}K}}$ no more defines $\Lambda$ as an individual object in any sense that contrasts with its being a class than would do the close equivalent  ${\Lambda}$ = ${\overline{\iota}}{\iota}{\Lambda}$$\Lambda$ is still a class (though of course, it also an element — an element of its own singleton). Peano’s result which Cantù quotes doesn’t make nullo an element associated to a set containing just those things satisfying some contradictory condition (associated how?); rather it still is, as originally defined, a class

To move on, there is a quite separate issue for Peano when he later makes use of the inverted iota operation applied to classes defined by conditions not guaranteed to determine singletons. How are we to understand ‘${\overline{\iota}a}$’ if ${a}$ is not a singleton? Good question. Cantù gives an entry from Max Phil where Gödel (a) holds that Peano commits himself to the idea that there is a contradictory null object (Unding), and “$\overline{\iota}a$ when ${a}$ has no elements [or] several elements, is this null object”, but (b) Peano can avoid this obscure doctrine. It is not clear, however, that Peano is countenancing a null object in this sense  [which is not to be confused with the perfectly good class $\Lambda$!]. Gödel’s alternative isn’t clear either, and Cantù’s discussion of it here is not easy to follow. But I am not minded right now to sit down with more of the Formulario to try to work out what’s going on — fun though it is to decode Peano’s simplified Latin!  So for the moment, I’ll have to leave things in this unsatisfactory state.

## “Multiversism and Concepts of Set”

An interesting paper here by Neil Barton.

Posted in Phil. of maths | 2 Comments

## Kurt Gödel, Philosopher-Scientist #4

After the two introductory papers, the volume we’re taking about divides into two parts: “Close Readings of some of Gödel’s Philosophical Remarks (Max Phil)” and “New Readings in Gödel’s Philosophy”, and in each part the papers are ordered alphabetically by author rather than thematically or in some sensible reading order. We’ll just have to dive in.

So the next paper is by Éric Auderau, “Gödel: From the Pure Theory of Gravitation not Newton’s Absolute”. I don’t propose to comment on this paper. It claims, inter alia, that remarks in Max Phil show that Gödel was thinking about the his cosmological model with a rotating universe three years before he was invited to contribute to the Schilpp volume on Einstein, and that more light is thrown on how Gödel regarded his models by these remarks. Annoyingly, however, Auderau doesn’t bother to translate the German of the various remarks he quotes from  Max Philso I can’t assess their significance.

Julien Bernard then contributes a paper “From the Physical Existence of Tuples to Quantum material prima: Gödel revives some Leibnizian ideas on Physics Within the Frame of Contemporary Physics of Matter”. This concludes, helpfully but disappointingly, with nearly six pages of remarks from Max Phil. Helpfully, because the remarks appear in a parallel text with translations (albeit rather shaky ones). Disappointingly, because the remarks are — shall we say? — not very impressive. There’s rather little sign on this evidence that we are going to learn much from Gödel the philosopher once he strays from logic and mathematics.

Here’s the second remark Bernard quotes (his translation):

Analogy:

Force of Gravity : Inertia = God : Devil

The force of gravity governs the sky [‘Himmel’, surely better rendered ‘the heavens’] and tries to destroy every multiplicity. It would cease only if matter was concentrated in a single point. Profession is only for men. There are very fewer [sic!] different types of men than of women. Adherence, cohesiveness is also part of chemical forces. What is heat? A “frenzy-force” that consists of an idleness? Light = an effect of love and profession? Light is actually nothing physical, because the identity of colour is no more a physical property.

At first blush, the remarks about men and women seem mad! But that’s unfair — for Gödel’s preceding remark actually gives another analogy(!) according to which electrons = men, nuclei = women, magnetic force = professional life, inertia = idleness. But decoded like that, what are we left with but some banalities (there are more kinds of nuclei than electron) and some bad philosophy (colour is not physical so light is not physical — contestable premiss and non-sequitur)? Oh dear.

Ok, some of the other remarks recorded here are in rather better shape. In particular, Gödel raises the question of when a pair of things form some physical compound entity that is more than (as we would say) a mereological fusion — I think Gödel would regard mere fusions as logical fictions. Bernard discusses this “problem of the compound” a little, looking over his shoulder towards Leibniz. But we don’t get a sharp answer from Gödel. And you’d never guess from Bernard that this sort of question has been done to death in contemporary analytic metaphysics — so no effort is made to place Gödel in relation to positions marked out in recent discussions.

## Kurt Gödel, Philosopher-Scientist #3

We are talking about the second paper of this volume, namely “Kurt Gödel’s Philosophical Remarks (Max Phil)”  by Gabriella Crocco and Eva-Maria Engelen, and are turning to its third and final section titled  “The content of the Philosophical Remarks (=Max Phil). First insights”.

(3) Crocco and Engelen say that the philosophical notebooks were started “as a realization of an ethical approach. Self-perfection and self-admonition are part of this approach. They seem to be very much in the tradition of Meditations by Marcus Aurelius or Goethe’s Maxims and Reflections. At the same time Gödel discusses philosophical problems from the view point of different academic disciplines and has a kind of 17th century metaphysics as an outline in mind.” But our authors don’t actually give us any quotations illustrating the first, ethical, aspect of the (early?) notebooks. They turn instead to a remark from the end of notebook IX which, they say, “gives us some precious insights into the nature of his philosophical concerns and tasks”. Here’s their translation:

Remark Philosophy: My work with regard to philosophy shall consist in an analysis of the uppermost concepts (the logical and psychological ones); in other words what has to be done after all, is to write down a list of these concepts and to think of their possible axioms, theorems and definitions (including the application on the empirically given reality of course). But in order to be able to do so, one has first to have acquired a sense of what one can assume through (halfway understood) philosophical reading and the writing down of philosophical notes. On the other hand, the understanding of an axiomatics will in turn increase the understanding of philosophical authors [thus an interplay from above and below at which the right proportion is important]. A replacement for the reading of works by philosophers is the reading of some other good books accompanied by a precise analysis, the learning of languages [Hebrew, Chinese, Greek?] and the precise definition of the occurring words and concepts.

This surely must be a rather ropey translation: ‘application on’? ‘halfway understood’?? ‘at which the right proportion’??? And “good books accompanied by a precise analysis” [“accompanied” sounds as though we have to add the analysis as readers] actually renders “guter Bücher mit genauer Analyse” which surely means good books which have, i.e. already include, accurate/precise analysis.

The general thought, however, seems to be this: Engaging charitably with the philosophers we read — those, at any rate, who aim to offer a systematic metaphysics with an account of the “uppermost concepts” and basic principles governing them — must also mean thinking things through for ourselves, trying to reconstruct a coherent story of those concepts and principles: and deepening our understanding of those concepts and principles on the one hand and of the philosophical texts on the other hand involves a two-way process. But that is, surely, a very familiar reflection. And equally for the thought that there isn’t a sharp distinction to be drawn between engaging with canonical ‘philosophical’ texts and with other works by analytically reflective mathematicians, logicians, physicists, biologists, psychologists, etc.: that is surely entirely familiar too. Which isn’t at all a complaint about Gödel’s own reflections here (after all, these are familiar truths, not fallacies!). But it is perhaps odd for Crocco and Engelen to pick out this remark — the only one from Max Phil which they actually quote here — as offering especial insight into Gödel’s thinking: on the contrary, we’d surely be rather surprised to find any half-competent twentieth century thinker explicitly denying these thoughts.

At the end of their essay, Crocco and Engelen do add a page or so mentioning four subjects which they say feature importantly in the later notebooks which they have worked on. (1) A theory of concepts where concepts can apply to themselves (so not a stratified type theory). (2) Self-referentiality. (3) Time. (4) Issues about definitions of mathematical notions. However, for the moment, they only hint here how the discussions might go:  hopefully we’ll find out a lot more from later papers in this book.

## Kurt Gödel, Philosopher-Scientist #2

After John Dawson’s very introductory piece there follows another introductory paper, “Kurt Gödel’s Philosophical Remarks (Max Phil)” by the editors of the volume, Gabriella Crocco and Eva-Maria Engelen. All the essays in the book, by the way, are in English — though the prose has very obviously not always been well checked by a native speaker for naturalness.

This paper starts with a preamble (1) on ‘Gödel’s philosophy and his scientific oeuvre’, followed by a central section (2) giving what the authors call a technical description of the Max Phil. Their concluding section (3) is on ‘The content of the Philosophical Remarks (=Max Phil). First insights’.

(1) We already know, e.g. from Hao Wang and from Gödel’s unsent response to the Grandjean questionnaire, something about Gödel’s negative philosophical views — we know what he was against. In particular, Gödel found himself fundamentally out of sympathy with the prevailing philosophical ideas he encountered at the Vienna Circle. What is a lot less clear is Gödel’s positive philosophy (to use our authors’ phrase). “I was a conceptual and mathematical realist”, he wrote in his draft reply to Grandjean: but what exactly does that mean?

We find other thoughts that appealed to Gödel in that page from the Nachlass, where he epitomizes  “My philosophical viewpoint” in fourteen points, given in translation by Wang (A Logical Journey, p. 316) and repeated here in an improved transcription and revised translation. But this is, let’s be honest, pretty strange. For example, we read (according to our authors):

1. The world is rational.
2. There are other worlds and rational beings, who are of the other and higher kind.
3. The world in which we now live is not the only one in which we live or have lived.
4. The higher beings are connected to the other beings by analogy, not by composition.

Taken cold, without further elaboration, it is very difficult to know what to make of such remarks — and what else is currently published of Gödel’s work doesn’t help a great deal, as  Crocco and Engelen themselves say. So the great interest of working on the Max Phil notebooks, they maintain, is in seeing if we can indeed reconstruct a systematic positive philosophical viewpoint from them (even though the remarks are often brief, allusive, and written only for Gödel’s himself). We shall see.

One quick remark about this section. Our authors mention as one of Gödel’s technical achievements that are, in Wang’s words, “disproofs of philosophical hypotheses of the age”,  the “proof of the independence of the axiom of choice from the axioms of set theory”(!) which “shows that strong realism in set theory, represented by the axiom of choice, is no less coherent than constructivism.” Oh dear. Gödel proved a consistency result not the independence result. But taking that as an unfortunate slip, the more serious point remains that a constructivist who is unhappy about the coherence of impredicative ideas, and hence about the coherence of classical impredicative ZF is hardly going to be mollified by a classical proof of the relative consistency of ZFC, is she?

(2) At the end of the Collected Works there is a catalogue of the Nachlass, which dates but otherwise says very little about the Max Phil notebooks. The second section of the present paper tells us a little more,

Fifteen out of sixteen notebooks survive, numbered 0, I-XII, XIV and XV (notebook XIII dating from 1945-1946 having been lost by Gödel himself in 1946). The character of the notebooks changes somewhat as time goes on, and it is suggested that we can consider them as falling into four groups.

Group A contains the first notebook of 80 pages, dating from 1934 to 1941,  together with notebooks I and II, 157 pages dating from 1937 to 1942 (plus another 40 pages added to notebook II devoted to ‘time management’). The first notebook contains notes on lectures Gödel has attended, and the first three three contain work plans, thoughts about working methods, “maxims for the conduct of life as a successful thinker and researcher”, as well philosophical remarks scattered across a range of topics.

Group B contains notebooks III to VIII, whose pages are consecutively numbered from p.1 to p. 680. These date from the beginning of 1941 to late 1942 and are more purely philosophical. Group C contains notebooks IX to XII, numbered separately but amounting to another 463 pages, dating from late 1942 to mid 1945 and continue in much the same pattern though there are now remarks too on relativity and quantum mechanics, and on the idea of force in physics and biology.

Group D contains the two late notebooks, XIV dating from July 1946 to May 1955 (128 pages), and  a short XV from May 1955 onwards (just 33 pages).

So all in all, that’s a significant number of pages: though I got no sense at all of how much material (when transcribed from the shorthand) is on a  typical page. Our authors add some bar charts giving the relative occurrence of headings like “Bemerkung”, “Bemerkung Philosophie”, “Bemerkung Grundlagen” and so on — but not terribly helpfully as the categories Gödel uses are pretty sweeping, and it seems that in many of the notebooks two or three headings cover the large majority of entries. The “technical description” of the Max Phil therefore doesn’t give us much of a steer about content. So what do our authors go on to say about this in their third section?

To be continued

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## “The 8000th Busy Beaver number eludes ZF set theory”

In case you haven’t seen it: this blog post by Scott Aaronson (on a new paper co-authored with Adam Yedidia), written with his usual clarity and zest, is fascinating!