Abstract
Michael Dummett presents a modus tollens argument against a Wittgensteinian conception of meaning. In a series of papers, Dummett claims that Wittgensteinian considerations entail strict finitism. However, by a “sorites argument”, Dummett argues that strict finitism is incoherent and therefore questions these Wittgensteinian considerations.
In this paper, I will argue that Dummett’s sorites argument fails to undermine strict finitism. I will claim that the argument is based on two questionable assumptions regarding some strict finitist sets of natural numbers. It will be shown that strict finitism entails none of these assumptions. Hence, the argument does not prove that the view is internally incoherent, and consequently, Dummett fails to undermine the Wittgensteinian conception of meaning.
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Notes
Intuitionists, on the other hand, hold that surveying a proof is within our capacities if we can in principle do so. There are some difficulties concerning the “in practice/in principle” distinction between SF and intuitionism. Thus, other attempts to draw the line between SF and intuitionism were made (Wright 1982). Nonetheless, as Dummett’s argument is at stake, his own traditional definition of SF will be the one discussed in this paper.
For further discussion on this point see Craig (1982).
The quantifiers range over natural numbers. A similar formulation is suggested by Magidor (Magidor 2007). Mawby formulates the first condition somewhat differently (Mawby 2005):
$$ a^{\prime }.\sim \exists n\ \left[n\in S\&\left(n+1\right)\notin S\right] $$Conditions a and a′ are equivalent in classical but not in intuitionistic logic (Heyting 1966, chapter 9). Magidor’s formalisation captures Dummett’s definition more accurately and hence will be used here,
There are other versions which are more plausible and enable strict finitists to construct prominent numbers such as (243112609-1)243112608. Nonetheless, in order to discuss Dummett’s claim, I will consider only his limited version of SF. For other conceptions of SF, see Mawby (Mawby 2005).
See for example (Wittgenstein 1964, pt. III 3–12, 45–47, 49, 51, 54, 57)
A similar explanation is suggested by Mawby (Mawby 2005, chapter 3).
The construction of real numbers is more complex. Owing to reasons of simplicity, I will confine the discussion to natural numbers.
Magidor also argues that in principle, we cannot know of such a number k which is apodictic while 2k is not. However, the justification of her claim seems suspicious. Magidor justifies the claim that we cannot know of such a number k because if ‘apodictic’ is a vague predicate, “then the predicate ‘apodictic’ is undecidable in the sense that there are some numbers n for which we do not know and are not in a position to know either that they are apodictic or that they are not apodictic” (Magidor 2007, p. 409). While this claim is correct, it does not justify the claim that we cannot know of an apodictic number k such that 2k is not apodictic. Compare the predicate ‘apodictic’ with the vague predicate ‘short’. We know that a person whose height is 100 cm is short and that adding 1 cm would not make the short person not-short. Still, we know that a person who is 200 cm tall is definitely not short. The fact that the borders of a vague predicate F are undecidable does not entail that we do not know of a value x such that Fx is true and that F(2x) is false. A detailed criticism of Magidor’s objection appears in Saad (Saad 2014, Appendix)
This claim can be proved by induction. 1 is an apodictic number. Thus, for every set S which is apodictically-closed-under-addition, if n is a member of S, then (n + 1) is a member of S. It can therefore be proved by induction that if n is a member of S, then every natural number which is bigger than n is a member of S. Hence, S cannot be weakly finite.
References
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Acknowledgments
I am grateful to Peter Hacker, Jonathan Berg, Ofra Magidor, Jonathan Erhardt, and an anonymous referee for their helpful comments.
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Saad, A. On the Coherence of Wittgensteinian Constructivism. Acta Anal 31, 455–462 (2016). https://doi.org/10.1007/s12136-016-0284-1
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DOI: https://doi.org/10.1007/s12136-016-0284-1