Ontology and metaphysics

We are discussing infinity, and the views of Frege, Russell, Hilbert, Carnap and of course Grice.
And even Friedman.

I was thoroughly shocked by what JL culled from the Wikipedia entry on "Finitism" and to this I will shortly respond, but first I must correct an aspect of my previous posting, in which I described Carnap's move from Universalism to Pluralism as a movement from abstract to concrete.
Since the universalisms of Frege and Russell were both ontologically not entirely abstract this was not a good move (on my part).

The terminology differs from one philosopher to another, but in Frege a relevant distinction is between concept and object, and Frege's logicism involved taking numbers (and generally extensions of concepts) as objects, which he later seems to have thought a mistake.
In Russell, we have "individuals" rather than objects, and these do not include numbers (they are more concrete than that perhaps), but arithmetic does depend on there being infinitely many of them, which Russell, despite being a logicist regarded as contingent.

A better way of describing Carnap's move, which does fit into the way which he talks of it, is from the analytic to the synthetic.  For Carnap (though both Frege and Russell had their doubts), the Universalistic systems of Frege and Russell were both entirely analytic (which entails, for Carnap, consisting entirely of logical truths, in his "broad sense") and so therefore must be any claims obtained in those systems by the approved method of adding definitions and working within the same deductive system.

So to formalise empirical sciences (involving synthetic claims) Carnap adopts Hilbert's idea of deductive systems (rules and axioms) as providing implicit definitions of the ontologies they describe, aided and abetted perhaps by Hilbert's view that consistency suffices to establish existence (though I suppose this is irrelvant for Carnap who would. at least by the time of "Semantics and Empiricism", regard this doctrine as puporting to settle meaningless "external " questions).
The distinction between analytic and synthetic then derives from classifying rules and axioms into two groups, one of which determines meanings (and on the basis of which analyticity is determined), the other group expressing physical laws.  A synthetic truth is then one which is determined by the rules and axioms altogether but not by the logical rules and axioms alone.

With this amendment let me now move to the apparent relevance of the Wikipedia article on "Finitism" which JL has brought to our attention.
The two startling revelations which JL pulls out of this are that both Hilbert and Friedman were finitists.

With both of these I must take issue.

Let us take Hilbert first.
The purpose of Hilbert's "programme" was to save mathematics in the broadest sense, including "Cantor's paradise", from the intuitionists.
His idea was to defend mathematics involving so called ideal elements by obtaining a "finitary" proof of its consistency.
Hilbert was not himself a finitist, but sought a consistency proof for non-finitary mathematics which would be acceptable to finitists (though presumably a dyed-in-the-wool finitist would, after accepting the proof of consistency, stiill reject the formal system as "not mathematics").
What the Wikipedia article says about Hilbert seems to be unobjectionable but JL takes matters a step too far when he infers from it that Hilbert was a finitist.

When it comes to Friedman, my gripe is with Wikipedia.
Friedman has devoted an extraordinary amount of talent and energy to showing that there are interesting parts of mathematics (inventing a whole new branch of mathematics in the process) which demand large cardinal axioms, hence going well beyond the ontological extravagances known to Cantor and Hilbert.
Friedman might therefore be said to be an uber-infinitist.
In taking Friedman to be a finitist JL can claim to have been mislead by the Wikipedia article, which does give an inaccurate description of one of Harvey Friedman's claims.

Wikipedia states "However, by Harvey Friedman's grand conjecture most mathematical results should be provable using finitistic means.", which certainly makes him sound close to a finitist.
But when we follow the link to the "grand conjecture" we find it quoted as:

"Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. EFA is the weak fragment of Peano Arithmetic based on the usual quantifier-free axioms for 0, 1, +, ×, exp, together with the scheme of induction for all formulas in the language all of whose quantifiers are bounded."

The qualification "whose statement involves only finitary mathematical objects" is crucial.
Friedman is not talking about most of mathematics, he is just talking about most of (published) arithmetic.

None of this is terribly important to the matters under discussion, but my suggestion that Carnap's pluralism was partly inspired by Hilbert makes no sense unless we understand Hilbert as ontologically liberal.

Carnap's ontological position is, so far as I know, original and a substantial departure from precedent, even in positivist circles.
Kolakowski identifies four key features of positivist philosophy one of which is "nominalism".
In Carnap we find an abhorrence of "metaphysics" (though this may not be in Carnap what you might have thought), and though one might have expected nominalism to flow from this, Carnap's conception of metaphysics rejects nominalism no less than platonism.
Carnap tells us that the negation of a metaphysical claim is also metaphysical, and that we can distinguish ontological claims according to whether they are made in the context of some conventional ontology, and hence may be established or refuted using those conventions, or are made in vacuo and are then meaningless and no more refutable than confirmable.

The resulting ontological liberalism is not dissimilar to that which we seem to find in Grice, though in Grice I am not aware of a comparably incisive philosophical underpinning.

It seems to me also that it is more in touch with the workings of ordinary language than philosophy generally has been, for it surely is the case that most ordinary ontological discourse belongs to some context which more or less determines the ontological criteria involved.

Going back to the stars and their infinity, it seems to me that in ordinary language to talk of the infinity of stars may have a meaning quite different to that which either a mathematician or a physicist might assign to it, and be compatible with presently received opinion among cosmologists.
It is arguable that the claim asserts no more than that the stars are so numerous that there is no practical possibility of counting their number.
There may be even in the pedestrian assertion something like the poetic licence which allows us to talk of infinite sorrow, joy, beauty.

Roger Jones