Abstract
Wittgenstein gave a clearly erroneous refutation of Russell’s logicist project. The errors were ably pointed out by Mark Steiner. Nevertheless, I was motivated by Wittgenstein and Steiner to consider various ideas about the natural numbers. I ask which notations for natural numbers are ‘buck-stoppers’. For us it is the decimal notation and the corresponding verbal system. Based on the idea that a proper notation should be ‘structurally revelatory’, I draw various conclusions about our own concept of the natural numbers.
Saul Kripke passed away on September 15, 2022. He had made his final revisions on this paper and submitted it on May 5, 2022. It is among the very last things on which he worked. Kripke’s tribute to Mark Steiner in the following note is all the more moving because now the philosophical world mourns his own loss. (The Editors)
Kripke’s footnote:
This paper is dedicated to Mark Steiner and his memory. Those of us who knew Mark are aware of the personal, as well as the intellectual, loss we felt at his passing. It was his work in Mathematical Knowledge and later discussions with him on these issues that gave rise to this paper and related material.
The present paper is based on a lecture delivered at the Jowett Society, Oxford University, on January 28th, 2016. Previous (and longer) versions were given at Harvard University (as the Whitehead Lectures), UCLA, UNAM, the Hebrew University (on the occasion of a conference in honor of Mark Steiner), at seminars at Princeton University and the Graduate Center, CUNY, and elsewhere.
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28 June 2023
This book was inadvertently published with the addition of the editor’s name, C. J. Posy, as co-author of the chapter. His name has been removed now and the author’s name Saul A. Kripke has been updated in the chapter.
Notes
- 1.
This definition is somewhat contrary to ordinary usage. It would make, say, north-south a single direction, the same for east-west, etc. (here regarding each direction as projected onto the plane).
- 2.
A definition of finiteness will be given below. In the early days of set theory, finding definitions of finiteness (without presupposing the natural numbers) was considered an important topic.
- 3.
In contemporary ZF set theory with the axiom of regularity, a surrogate for the Frege-Russell definition of cardinality was proposed by Dana Scott (1955). The cardinality of a set is the set of all sets of least rank in 1-1 correspondence with it.
- 4.
Wittgenstein says that this defines the ‘cardinal numbers’ in Principia. Aside from other criticisms of his views, it is hard to see why he thinks Principia does not allow a ‘cardinal number’ to be 0. In III-13 Wittgenstein seems to say that 0 can be defined as one of 1, 1+1, etc. Steiner simply replaces Wittgenstein’s formulation with S…S0, where S is successor and presumably the sequence of S’s can be empty.
- 5.
Actually, formulae of first-order logic with identity. Steiner rightly criticizes Wittgenstein for his neglect of the need for identity (p. 50). It is possible that Wittgenstein’s neglect of identity is a carry-over from his view in the Tractatus that identity can be eliminated in a proper logical symbolism. See Tractatus (Wittgenstein 1922) 5.53-5.534. Steiner also notes that Wittgenstein’s conditionals need an antecedent of the form (x)~(Fx∧ Gx). It is not so easy to explain this omission.
- 6.
However, it takes more work to extend this sort of first-order analysis to multiplication (and it is harder to argue that the Principia definition reduces to the first-order definition). For exponentiation it is quite impossible, as is explained in the text.
- 7.
In stating everything in terms of primitive recursion, I have somewhat altered Steiner’s presentation. See Steiner (1975) pp. 47-48. His presentation (following Quine 1963) is set theoretic and is based on a set theoretic definition of iterates of a function. But Gödel (1931) showed that primitive recursion can be carried out in first-order arithmetic with only + and ∙ as primitive, and the usual logical symbols.
If one wishes to include the definition of multiplication as repeated addition, the set theoretic definition should be followed. (It can also be argued that it is simpler and more direct than the formulation used in Gödel’s result.)
- 8.
However, if x≠0, it may be simpler to define xy directly: if s1 is a set of cardinality x and s2 is a set of cardinality y, xy is the cardinality of the set of all functions from s1 to s2. This definition is independent of the choice of s1 and s2 and easily applies when x1 and x2 are not finite.
It remains the case that exponentiation can be defined in terms of iterated multiplication as stated above, which would also allow us to define super-exponentiation in terms of iterated exponentiation in the same way, etc.
- 9.
In Benacerraf’s terminology ‘intransitive counting’ is simply the recitation of the designations for the numbers in the appropriate order. ‘Transitive counting’ is the use of these designations to count objects. In fact, given a definition of intransitive counting, one uniformly gets a definition once one has decided whether the first element of the progression denotes zero or one.
- 10.
Actually, Russell thinks that taking natural numbers as primitive might be sufficient for pure mathematics, but not for ‘daily life’, where we want to have ten fingers, two hands, etc. (1919, p. 9). But of course this is not really accurate. See e.g., the discussion above of the numbers of roots of an equation. If the objection is valid, it applies also to pure mathematics.
- 11.
I don’t think that this formulation is too sophisticated. Ordinary uses of decimal notation would take such a condition for granted.
It would be simpler, of course, to define the finite sequence as ranges of a function indexed on sequences 0, …, n (where n means, S…S(0)), and 0 and successor are already known, say, as in Principia. But I wish to emphasize that the notions used do not depend on any prior concept of natural number.
- 12.
Here, strictly speaking, if one puts things this way, one would have to introduce expressions as objects of the language (say, by Gödel numbering, or perhaps taking them as primitive).
Alternatively, and probably more simply, having defined +, ∙, and exponent for numbers, one could introduce polynomials in ten by definition. Also, one could introduce finite sequence numerals by definition –including the definitions of the ordering, successor, addition and multiplication (with the school definitions) – and prove that they are equivalent to the polynomials. That is, that every sequence version is equal to the corresponding polynomial version and that the definitions of addition, multiplication, successor, and the ordering are the same in both versions (i.e., yield identities). This appears to be what Steiner has in mind. See also Steiner, pp. 62ff on definitions (as actually introduced notation, not merely abbreviative definitions, together presumably with the result that the extension is conservative).
- 13.
I am assuming that her notion of when one has a de re belief should coincide with my idea of a ‘buck-stopping’ designation of a number. In fact, I myself originally formulated the problem in terms of de re beliefs about natural numbers. Mark Steiner persuaded me to reformulate it in terms of ‘buck-stoppings’.
- 14.
Clearly, he doubts or even denies the existence of a set of all sets in 1-1 correspondence to a given set, exactly as in standard set theory. The expression ‘the idea’ may express an intensional notion which suggests the proposed response to the Hambourger objection.
- 15.
Actually, properties are ‘propositional functions’ (in one variable). It is somewhat obscure to me what these were supposed to be, and Russell himself seems to have changed his mind about ‘propositions’ and ‘propositional functions’ over the years.
- 16.
Bernays gives further axioms for the system in Bernays (1941). One need not accept his proposal to adopt ‘global choice’ in Part 2. ‘Global choice’ as opposed to ‘local choice’, is very implausible, unless one believes something stronger, such as V = L.
- 17.
All these ideas have occurred to me only rather lately. They were not included in the original talk, nor in any other version. No doubt they would have to be investigated in detail to see if they work. Independently of Oberschelp and not in the present connection, I myself had the idea of taking proper classes to be urelements. But later I saw that Oberschelp had already published a paper with this proposal.
- 18.
That is, the last three digits all represent numbers less than one thousand, the preceding three digits (if any) represent the number of thousands less than one million, etc.
- 19.
Actually, a fancier rhetorical form of English somewhat imitates the French, as in the Gettysburg address, ‘Four score and seven years ago…”. But the usual English would be ‘eighty-seven’, whereas the French expression in terms of twenties is usual, not rhetorical.
- 20.
- 21.
I would like to thank Yale Weiss and Oliver Marshall for editorial assistance and helpful suggestions. Special thanks go to Romina Padró for her comments and for working with me on the lecture delivered at the Jowett Society and on the final version. This paper has been completed with support from the Saul Kripke Center at The City University of New York, Graduate Center.
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Kripke, S.A. (2023). Wittgenstein, Russell, and Our Concept of the Natural Numbers. In: Posy, C., Ben-Menahem, Y. (eds) Mathematical Knowledge, Objects and Applications. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-031-21655-8_8
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