Logicism, Neo Logicism, Caesar

[Oliver Marshall]

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>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Subject: Re: Logicism
Message-ID: <B4CBD46C-7BCE-4CB0-848A-0274C7B3314B at uconn.edu>
Content-Type: text/plain; charset="utf-8"

> On Aug 12, 2020, at 10:59 PM, Deutsch, Harry <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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