GÖDEL, RAMSEY, CARNAP AND WITTGENSTEIN

Ramsey, in the papers 1925 and 1926, printed in his posthumous 1931, gives a very brief but easily readable account of the three ‘schools’ of mathematical foundations. At the Königsberg conference of 1930, papers were read on the logicist, intuitionist and formalist views of foundations by, respectively, Carnap, Heyting and von Neumann. These papers had been advertised as taking place on the first day (the fifth of September), and they were printed in the second number of Erkenntnis (1931); and in the first Gödel volume Gödel’s reviews of them are reprinted (1932e, 1932f and 1932g). However, a fourth (and unadvertised) paper was read that day by Waismann, with the title “Das Wesen der Mathematik: der Standpunkt Wittgensteins”. It was, according to McGuinness (see page 21, Wittgenstein und der Wiener Kreis), received with respect, and regarded as representing a fourth ‘foundations school’. Unfortunately, Waismann did not send his text to the Erkenntnis editors, and as a result Gödel did not write a review of it. Yet he did, in fact, apparently, refer back to Waismann’s paper and its consequences in the first of two papers begun in 1953 and printed in Volume III as criticisms of Carnap. They were, indeed, two drafts (selected by the editors out of six) of an essay intended for the Carnap volume in The Library of Living Philosophers, but never included in it as a result of Gödel’s perfectionism.

The first draft printed is longer than the second, and opens with the words “Around 1930, R Carnap, H Hahn and M Schlick, largely under the influence of L Wittgenstein, developed a concept of the nature of mathematics which can be characterised as being a combination of nominalism and conventionalism”. Now whereas the 1932e review treated Carnap as an advocate of Russell’s logicism, this 1953 does seem to align him with Wittgenstein’s views, in so far as we can reconstruct them. The possibility that Carnap moved towards Wittgenstein’s views after the Königsberg conference (his own contribution listed in the Gödel references as 1931) may have something to do with an intemperate letter Wittgenstein wrote to him accusing him of failing to admit that he had borrowed his ideas (see the Nedo and Ranchetti volume and other sources; in particular, Malcolm quotes another letter that is also in Nedo and Ranchetti in his Wittgenstein, a religious point of view, written to Schlick about Carnap’s failings).

Wittgenstein’s views as recorded in the Wiener Kreis volume (pages 102 to 107, and Waismann’s ‘Anhang A’) do not lend themselves to being put in a nutshell. Nevertheless, one point stands out. In ‘Was in Königsberg zu sagen wäre’ Wittgenstein declares that in logic (in which he clearly includes mathematics) there are no concepts. Apparent concepts are no more than the chapter-headings of grammar. There is no super-concept of number which divides into various sub-concepts --- what we have is just a set of chapter-headings, as we might find in a grammar-book, and the syntax of what is described in these chapters has resemblances which lead us to use the one word “number” for all. Now traditionally (the distinction can be found in Ramsey’s 1925, and Gödel in his 1932e cites Carnap as drawing it in his Königsberg paper) logicism in general made two separate claims: that on the one hand the concepts and on the other hand the propositions of mathematics derive from logic, and Ramsey accuses the formalists (following Hilbert) of concentrating on the second and Russell of concentrating on the first. (The intuitionists he simply dismisses as prejudiced, and at the end of his Chapter IV calls their restrictions as to what is valid a Bolshevik menace. His account of intuitionism in his 1926 is much more interesting, and I discuss it ahead.) According to Gödel’s 1932e, Carnap had defended both claims at Königsberg, but in the 1953 draft which mentions Hahn, Schlick and Wittgenstein as well as Carnap, only the second claim, dealing with mathematical propositions, comes into question as a foundation for Carnap’s syntactical analysis. So I am tempted to guess that one very simple difference between Carnap’s views expressed at Königsberg and his later ones (in his 1934a, the Logische Syntax der Sprache, translated into English as his 1937) may be that he dropped any attempt to use his syntactical analysis to explain mathematical concepts.

I hope that is a clue that will be sufficient for researchers to find further differences between Carnap’s 1931 and his 1937, and if these shed light on Wittgenstein’s intemperate letter I shall be grateful for information about that, too. As I say in the third of these Gödel articles, I am far from clear as to whether Wittgenstein was objecting to a borrowing of his pre-1929 or of his post-1929 ideas.There was also a small early Carnap book on logic which might be relevant, and which I once owned but cannot trace, which I remember had Elizabeth Anscombe’s approval, and so possibly Wittgenstein’s as well (perhaps preferring Carnap’s original efforts to his borrowings).

As to a summary (see ahead) of the ‘syntax’ school offered by Gödel at the beginning of his *1953/9-III and at the end of his *1953/9-V, what intrigues me is its relationship to intuitionism. In his 1926, Ramsey, taking intuitionism a little more seriously, says that Brouwer would refuse to grant that either it was raining or it was not raining unless he had looked to see. In his 1932f, Gödel gives what appears to be a profounder basis for intuitionism, namely that Heyting, following Brouwer and Weyl, regarded mathematical objects as only existing in so far as they can be comprehended by human thought, which cannot verify the ‘nothing but truth or falsity’ of any proposition which it cannot ascertain to be actually true or actually false (a limitation which disposes of non-constructive existence proofs as well). What Ramsey says is simpler and clearer, and it accords better with Gödel’s 1953 summaries of the syntactical view: namely, that while the meaning of ordinary sentences is defined by semantical rules which determine under which circumstances they are true or false, there are also sentences which (like “it will rain or it will not rain”) can be known on syntactical grounds to be true in all circumstances, and are thus void of content. But true mathematical propositions are also true in all exterior circumstances, and therefore they too must be void of content and be purely syntactical in their justification. Gödel, of course, does not commit this non sequitur on his own account.

Ramsey simplifies the matter by pointing out that it is impossible to know a priori that the law of excluded middle is true and equally impossible to know this by experience, since experience is insufficient to definitively verify or falsify every possible potentially true or false proposition. He does not introduce the term “syntactical” to typify this point (for while it fits with the general tenor of his views, he does go out of his way, with his story about the Gogs, to defend mathematical meaning).

In my own view, these ‘boundary’ cases of ordinary language, sentences that are supposedly true in all circumstances, or false in all, like “it is both raining and not raining”, have nothing to do with the law of excluded middle as a principle of logic. One can easily look out of the window and find oneself in a Welsh mist (neither raining nor not raining) or in a Scotch one (both raining and not raining), or even in an Irish one (unable to make its mind up). In logic such geographical or meteorological possibilities can have no impact. Logic isolates itself from them. It deals only with clear-cut truth values. If the world were so very different from how it is that intermediate situations abounded, logic could change its rules --- but I believe there would still be a strong psychological bias for logicians to find a system of clear-cut truth values a desirable retreat from such real life hurly burly.

In his 1932f review of Heyting’s 1931, Gödel introduces an intuitionist distinction which, again, becomes clearer if one simplifies in the spirit of Ramsey. The distinction is between ‘propositions’ (Aussagen) and assertions (Behauptungen, symbolised by Frege’s assertion sign ├). Gödel says that Heyting concluded his lecture by distinguishing between these as follows: a ‘proposition’ expresses only an expectation or intention --- an aim, one might say --- for example, the mere ‘proposition’ that a number c is rational expresses the expectation that whole numbers a and b can be found such that c = a/b; while the assertion says that they have been found. The law of the excluded middle can therefore be classified as a mere ‘proposition’, expressing the expectation that any given mathematical statement [[“Satz” --- presumably a portmanteau combining “Aussage” and “Behauptung”]] can either be proved or reduced to a contradiction [[in effect, be disproved]]. Thus, the law itself could only be asserted if a general method could be specified whereby a decision could always be made in the case of any given individual Satz, but it can still be, let us say in English, propounded (which is all it actually needs to be). Gödel then quotes a rather subtle argument of Heyting’s which he interprets as grist to his own mill --- “and from this the author concludes that a proof that a [[given]] mathematical question is undecidable in principle lies entirely within the realm of the possible”.

My own view follows from my remark in parenthesis above: it is that this distinction between ‘proposition’ and assertion yields the intuitionist pass. For “assertion”, though expressed by the Frege sign, no longer means what Frege (let alone common sense) would have it mean. Frege’s concept of a sentence’s unasserted content (see the first section of the Begriffschrift) could certainly not be summed up by the word “aim”, while his concept of assertion expresses a speaker’s or writer’s acknowledgement of truth but does not specify that a content must be actually established as true before its assertion can be meaningful. Naturally, in his calculus Frege expects a configuration to be established formally before an assertion sign is put in front of it, but this does not detract from the meaningfulness of putting it where one asserts something ‘on one’s own responsibility’. A test case is Gödel’s XI of 1931: it was not, at that point, a theorem because it had not been proved. My reading of Begriffschrift is that Frege would have upheld the meaningfulness of Gödel’s setting an assertion sign in front of it. In Heyting’s system, on the contrary, a ‘proposition’ can only be elevated to counting as an assertion when it has been empirically ascertained, or formally proved, or when a method of constructing a proof for it has been specified. Very well --- admit that --- it still leaves room for the naive hope that, one fine day, a given ‘proposition’ will turn out to be a genuinely true assertion, or alternatively a false one. An opponent of intuitionism will believe this naive faith meaningful, an intuitionist will declare it meaningless, but a working mathematician can afford to make use of it in accepting the law of excluded middle as a mere ‘proposition’ rather than an assertion in the Heyting sense, and on these terms he has no need to reject what Ramsey calls “many of the most fruitful methods of modern analysis”. In other words, there is a perfectly proper sense in which the foundations of analysis are acceptably provisional, and so are other mathematical outcomes, of which queries raised by Gödel’s work on Cantor’s continuum hypothesis are examples.