Abstract
In this paper I examine the fundamental views on the nature of logical and mathematical truth of both Frege and Carnap. I argue that their positions are much closer than is standardly assumed. I attempt to establish this point on two fronts. First, I argue that Frege is not attempting to defend metaphysical theses. Second, I argue that Carnap, where he does differ from Frege, can be seen to do so because of mathematical results proven in the early twentieth century. The differences in their views are, then, not primarily philosophical differences. Also, it might be thought that Frege was interested in analyzing our ordinary mathematical notions, while Carnap was interested in the construction of arbitrary systems. I argue that this is not the case: our ordinary notions play, in a sense, an even more important role in Carnap’s philosophy of mathematics than they do in Frege’s. Finally, I address Tyler Burge’s interpretation of Frege which is in opposition to any Carnapian reading of Frege.
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Notes
- 1.
This is the paper I presented at The Bucharest Colloquium for Analytic Philosophy: Frege’s Philosophy of Mathematics, Bucharest, Romania in May 2011. As a result of discussions at the colloquium, I significantly reorganized my thoughts on these subjects. This reorganization involved primarily my focus changing more or less exclusively to defending my interpretation of Frege (this was published as Lavers 2013). While there is some overlap between the present paper and the one just mentioned, the heart of the present paper concerns the forces that led to Carnap’s position differing from Frege (Sects. 3 and 4) are not at all present in Lavers (2013).
- 2.
- 3.
Erich Reck, in his (Reck 1997), argues that Frege is not a metaphysical realist in an argument that depends heavily on the use of the context principle. While I don't disagree that the context principle plays an important role, I wish to show that by looking at his criteria for a successful account of number, we can see that Frege's goal is not to establish realist theses.
- 4.
Both Carnap and Frege maintain that uses of terms by communities of specialists, prior to a systematization, stand in need of clarification. When I speak of ‘our ordinary understanding’ I meant this to apply equally to such groups of specialists.
- 5.
Frege’s views on fruitfulness are discussed in detail in Tappenden (1995).
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- 9.
Again see Beaney (1996).
- 10.
Nelson (2008) gives an interpretation of Frege, and specifically he focuses on Frege's views on analysis, according to which this task of identifying what is true of our ordinary notion is given primary importance. However, Nelson's position is based on a single quotation from Frege (1914/1979) in which Frege discusses the task of examining our ordinary notions. In this work Frege divides the task of the construction of the system into two subtasks. There is the merely preparatory subtask that involves examining our ordinary notions to see what is true of them. After this there is the step of providing a constructive definition, in which all ties to the pre-existing sense are severed. The emphasis that Nelson places on the first stage is out of keeping with the importance that Frege himself places on it. In fact, immediately after the quote that Nelson uses—where Frege talks about examining our ordinary notions—Frege says the following: “Work of this kind is very useful; it does not, however, form part of the construction of the system, but must take place before hand. Before the work of the construction is begun, the building stones have to be usable; i.e. the words signs, expressions, which are to be used must have a clear sense, so far as a sense is not to be conferred on them in the system itself by means of a constructive definition.” (Frege 1914/1979, p. 211) Frege reserves the term analysis for the first task, but it is providing the constructive definition that is the more important task.
- 11.
The difference in their respective attitudes to geometry will not be discussed in the present paper.
- 12.
In his 1931 address on logicism, published as (Carnap 1983), Carnap has a much more optimistic view as far as logicism is concerned. Here he defines the natural numbers as the numerically definite quantifiers and embraces Russell’s if-thenism with respect to he axiom of infinity (and Choice). Carnap, here, sees the major obstacle for logicism being a justification of Ramsey's simple type theory that does not involve Ramsey’s platonism. Also of interest concerning Carnap’s views on the nature of logical and mathematical truth are Carnap (1942, 1939/1955, 1950a, b).
Also, I do not want to make claims about Carnap’s conscious motivations. I do not want to say that Carnap held the position he did in order to remain maximally Fregean (as I am using the term). Nor, of course, do I want to imply that other thinkers not mentioned (such as Hilbert) had no significant impact on shaping Carnap’s philosophical views at this time. I merely wish to argue that Carnap’s position is maximally Fregean in the defined sense.
- 13.
There were of course other bifurcating pressures on logic at the time. Ought one accept type theory at all? If so, ought it be simple or ramified?
- 14.
I mean here an axiom asserting that there are infinitely many individuals, not a set with infinitely many members.
- 15.
Weiner (1990) argues that this is the principal motivation of Frege's logicism.
- 16.
See Carnap (1963b).
- 17.
Of course, Carnap calls the definition syntactic. However, Carnap uses the term ‘syntax’ at this time to include much of what we now call semantics (see Creath 1990).
- 18.
This phrase is taken from the forward to Syntax (pp. xv).
- 19.
Since I have discussed Carnap on ordinary notions in detail elsewhere, this discussion will be kept relatively brief.
- 20.
- 21.
Interestingly, Carnap agrees with Frege in holding that our ordinary notions may not be completely precise. In his response to Beth, Carnap claims that our ordinary notions may not be ‘univocal’.
- 22.
I am not aware of any evidence that Burge puts forward with respect to the claim that Frege is trying to ward off deflationary positions.
- 23.
Burge avoids going into a detailed comparison with Carnap, but takes it to be clear that there are obvious differences between their positions.
- 24.
Matters of ontology are treated in a quite confused manner in Syntax. For an argument to this effect see Lavers (2004). It is for this reason that I focus in this section on Carnap's post semantic views on matters of ontology.
- 25.
Austin translates ‘Satzes’ as ‘propositions’ rather than ‘sentences’. However, ‘sentences’ is a better choice here. Number words do not appear in propositions.
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Lavers, G. (2016). Frege the Carnapian and Carnap the Fregean. In: Costreie, S. (eds) Early Analytic Philosophy - New Perspectives on the Tradition. The Western Ontario Series in Philosophy of Science, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-24214-9_15
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