Logicism, Neo Logicism, Caesar

[Kapantais Doukas]

Dear participants to this tread,

I think that there is sense in which the answer to Joe Shipman's question is a trivial one. I.e. the amount of PA truths that are truths of logic is maximal. All PA truths are truths of logic. And not only these, but all mathematical truths are truths of logic as well. This is what logicism would have defended back then and at times nowadays too. Of course, there is a catch in this kind of attitude. In order to make "logic" yield PA, or anything extending first order predicate calculus, one needs to strengthen it considerably with respect to what minimalists like, e.g., Quine, have the inclination to narrow it down to. Be it Law V of the Grundgesetze, be it HP, be it Frege's proposal in the Appendix of the Grundgesetze, or/and an axiom positing infinite totalities, reducibility, mathematical induction, etc., one needs to introduce means that are-one way or the other-controversial with respect to their "logical" nature or even controversial per se. This parameter put aside, there is also a historical dimension to the question. The fact the that many of these proposals have been overruled because they lead on to contradictory or problematic theories. So, and as I have suggested in a previous post, the answer to Shipman's question is conditional and cannot ignore these philosophical and historical parameters. Primarily the philosophical ones. To put it in other words, and from the other end of the spectrum, if one is not willing to allow into "logic" anything that exceeds first order predicate calculus, the answer to Shipman's question is: none! No mathematical truth (and PA truth) can be made into a truth of logic. These latter people would have denied that any PA truth is a truth of logic, even in the case that, e.g., the Grundgesetze were not contradictory.

Best,
Doukas



From: FOM <fom-bounces at cs.nyu.edu> On Behalf Of Oliver Marshall
Sent: Monday, August 17, 2020 3:10 AM
To: fom at cs.nyu.edu
Subject: Re: Logicism, Neo Logicism, Caesar

Dear Marcus and other interested parties,

Have you seen this paper by Nathan Salmon, who argues that Frege should not and does not entertain HP as a foundation, because of the Julius Caesar problem properly understood?

Best, Oliver

https://link.springer.com/article/10.1007/s11098-017-0927-0

This article offers an interpretation of a controversial aspect of Frege's The Foundations of Arithmetic, the so-called Julius Caesar problem. Frege raises the Caesar problem against proposed purely logical definitions for '0', 'successor', and 'number', and also against a proposed definition for 'direction' as applied to lines in geometry. Dummett and other interpreters have seen in Frege's criticism a demanding requirement on such definitions, often put by saying that such definitions must provide a criterion of identity of a certain kind (for numbers or for linear directions). These interpretations are criticized and an alternative interpretation is defended. The Caesar problem is that the proposed definitions fail to well-define 'number' and 'direction'. That is, the proposed definitions, even when taken together with the extra-definitional facts (such as that Caesar is not a number and that England is not a direction), fail to fix unique semantic extensions for 'number' and 'direction', and thereby fail to fix unique truth-values for sentences like 'Caesar is a number' and 'England is a direction'. A minor modification of the criticized definitions well-defines '0', 'successor' and 'number', thereby avoiding the Caesar problem as Frege understands it, but without providing any criterion of number identity in the usual sense.




Today's Topics:

   1. Re: Logicism (Rossberg, Marcus)


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Message: 1
Date: Sat, 15 Aug 2020 16:10:50 +0000
From: "Rossberg, Marcus" <marcus.rossberg at uconn.edu<mailto:marcus.rossberg at uconn.edu>>
To: Foundations of Mathematics <fom at cs.nyu.edu<mailto:fom at cs.nyu.edu>>
Subject: Re: Logicism
Message-ID: <B4CBD46C-7BCE-4CB0-848A-0274C7B3314B at uconn.edu<mailto:B4CBD46C-7BCE-4CB0-848A-0274C7B3314B at uconn.edu>>
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> On Aug 12, 2020, at 10:59 PM, Deutsch, Harry <hdeutsch at ilstu.edu<mailto:hdeutsch at ilstu.edu>> wrote:
>
> I wonder what Frege would have thought of the derivation of the Peano postulates in FA (without the help of Basic Law V). Would he think we only have a ?free standing? epistemic right to HP, or would he see this idea as a sign of psychologism obtruding in matters of logic?  Harry

In The Foundations of Arithmetic, Frege entertains HP as a foundation (and sketches a derivation of PA from it), but rejects it because of what we now call the Julius Caesar problem.  That?s where we?re all getting it from.

In a letter to Russell, after the discovery of the antinomy, Frege says that the problem with HP is the same as with Basic Law V, so founding arithmetic on HP isn?t an option. It?s a tantalizing remark, since he doesn?t spell out what he takes the problem to be.  Did he think HP was inconsistent or what?

Marcus

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