Here is another bit of Carnap scholarship from Reck:
From Frege and Russell to Carnap: Logic and Logicism in the 1920s
There is a gap in Carnap's autobiography in relation to a major transition which he made in his thinking about his core programme of applying logic to science,
In the early twenties Carnap has taken a lead from Russell in terms of the orientation of his philosophical program, and is studying Principia Mathematica.
His "Abriss der Logistick" is a logic text based on Russell's Theory of Types (with the ramifications and the axiom of reduction dropped). This was not published until 1929,
But when Carnap comes to Logical Syntax, in the early 30s, he has adopted a position which is much more like that of Hilbert, which is quite a substantial movement. The Frege/Russell approach is sometimes called universalism because it is not pluralistic with respect to the logical systems, they think in terms of one logical system. More importantly, after the logical system is set up you are supposed to define concepts using explicit definitions.
Hilbert's was more pluralistic, and for Hilbert an axiom system was used to give an "implicit" definition of mathematical concepts.
In his autobiography Carnap gives us little information about how this transition in his thinking occurred.
Reck provides an interesting story on what was going on.
Another source of detail about what was happening to Carnap's ideas on logic may be found in a paper by Goldfarb:
On Godel's Way In: The Influence of Rudolf Carnap
The puzzle for me about how Hilbert's influence came to bear on Carnap is resolved by the mention of two men who Carnap does not speak of in his autobiography. The first is Heinrich Behmann, who was an associate of Hilbert's and with whom Carnap "closely collaborated". The other is Fraenkel, whom we have just come across in the Fraenkel-Carnap problem.
It sounds as though at this time Carnap was heading in the direction of adopting axiomatic "implicit" definitions, and was undertaking theoretical investigations to underpin the legitimacy of such methods. The question of categoricity is of course relevant in this connection, one might take the view that axioms do provide a good (if implicit) definition of mathematical concepts only if they are categorical.
In this connection it appears that Carnap was working on a book to be called "Allgemeinen Axiomatik", which never appeared. Goldfarb's paper suggests that this may be because when Godel entered the field with his PhD proving the completeness of first order logic, this exposed the weakness of the methods by which Carnap had been approaching similar problems. Hence Godel's book was never published.
RBJ
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Good. Fascinating. You are making such good work in tracing the continental history of logic! (I have been recently sort of revising some of my notes in what I call the "Oxford" history of logic -- I note that there are such good Wykeham professors that one may not need travel! -- just kidding). But a point about the 'axiom'. Indeed, as Jones has noted elsewhere, this will be, if I read Jones alright, a bit of the bait that Carnap set for Quine, unintentionally. For part of the early criticism by Quine against Carnap deals with the 'implicit' definition thing and the 'axiom' thing -- so here is where it may connect back to Oxford logic -- (At Oxford they never cared for Principia Mathematica until Strawson got seriously about Quine's points in "Methods of Logic" -- I am speaking, roughly, about Grice's generation and circle). This is Grice about 'axiom' in WoW:22-23. The two pages that I call "My Hilbertian Grice", so I hope _I_ can add Behmann to the picture (Grice deals with Principia Mathematica explicitly in Strand 6 of WoW). But in 1967, Grice is 'caricaturing' the formalist -- and there is a lot of talk of 'axiom':
ReplyDeleteHe is speaking of 'formal devices' by which he means the vocabulary of -- in this order:
FD ==== ~ /\ \/ ) (Ax) (Ex) (ix)
and proposes the Formalist as wanting to construct "a system". This 'system' is constructed 'out of' or "in terms of" as Grice prefers, the FDs (formal devices). This is a system. But a system of what. "A system of formulas" -- 'very general formulas' -- "Such a system may consist of a certain set of SIMPLE formulas" -- the axioms, but Grice does not care to call them thus --. The point is that these axioms "MUST be acceptable IF the devices" -- the FDs -- "have the MEANING" (rather vague use of 'meaning' here seeing that this may just involve the syntactics) "that has been assigned to them". Less "simple" formulas "can", it is hoped, "be SHOWN [demonstrated] to be acceptable IF the members of the original set [of 'simple formulas' or axioms] are acceptable." Taking into consideration that algorithms are not to come by with respect to first-order predicate calculus with identity he adds, as far as the propostional-calculus segment is concerned, "if ... we can apply a decision procedure, we have an even better way".
When Strawson caricaturised the formalist in his "Introduction to Logical Theory" he would typically assume a more irreverent language, or an irreverent language simpliciter. He would say that talk of 'meaning' here is totally unwelcomed. To compare the formal logician with a painter will do -- only insofar as we realise that the ABSTRACT (indeed formalistic painter) is NEVER wanting to 'imitate nature'! I love to think of Hilbert as being the butt of Strawson's joke, if he only would have cared to give more precise references to his diatribe in favour of "Traditionalism"!