Abstract
At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.
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Notes
The notion of a model of arithmetic is that of a completed infinite totality. This, however, does not undermine the finitary character of Parsons’ proposed route to arithmetical knowledge, since we do not need to be able to know that the structure formed by a sequence of strings of strokes is a model of arithmetic; we only need to be able to know through intuition that the axioms of arithmetic are true when interpreted over a domain of strings. The knowledge acquired is mathematical because the sequence of strings is isomorphic to the sequence of natural numbers, but we do not need to know that this is the case.
It will not do to say that being black is not an arithmetical property, since this presupposes knowledge of arithmetic.
Although for a criticism of this contention see Resnik 2000, pp. 222–226.
This is not completely implausible. After all, one might think that the types of tokens which are subject to vagueness phenomena can themselves form a sorites series.
Parsons formulates (5) in a slightly different way, giving central relevance to our knowledge of the recursion equations for \(f\). But since \(f\) is being defined, what seems to be relevant is not whether we can know the recursion equations for \(f\) but whether we can see that the recursion equations for \(f\) capture a well-defined function.
Although he does not really distinguish between the two readings, Gentzen expounds both of them in his 1935.
A referee suggested the following thought experiment to strengthen the analogy. A subject is shown on a screen an immense number of stroke signs, asked to look away from the screen, and informed that perhaps a stroke has been added but perhaps not. She is then asked to look again at the screen and judge whether a stroke has been added. If the subject is unable to reach a decision, one might be tempted to conclude that, because of vagueness, the subject is unable to rule out the presence of rogue elements verifying \(Sx = x\).There is an important difference between the case described by Parsons and the case described in the thought experiment: in the former, the string to which we are adding a stroke is not, ex hypothesi, a string of strokes for some particular \(n\), whereas in the latter it is. This means that the string featuring in the thought experiment is not vague in the sense that Parsons has in mind when he talks about imagining a stroke string vaguely. Whether there is nonetheless a sense in which the thought experiment presents us with a case in which, due to vagueness, we are prepared to admit rogue elements is an issue which we can set aside for current purposes. For what matters is that there is a distinction between the situation in which one is unable to rule out that \(Sx = x\) because of the possible presence of rogue elements and the case in which one is unable to do so because the imagined string is vague in the sense Parsons has in mind.
This option is reminiscent of the Kantian line of thought of pure intuition as exhibition of a concept: we exhibit the concept of string of strokes in intuition, and see that it is extendible. But surely this notion of intuition is not analogous to perception in the sense Parsons wants and needs.
It has to be stressed that the claim, endorsed by Parsons and Tait, that the notion of iteration cannot be given in intuition is plausible only as long as we conceive of intuition in the way Parsons does. On the Kantian view, the notion of iteration is given in intuition through the a priori intuition of time. This also undermines Galloway’s (1999, p. 105) claim that intuition of iteration is ‘an idea that has its home [...] in conceptions of intuition that stress intuition de dicto, and make no room for the idea of intuition de re’ since Kantian intuition is mostly and typically intuition de re.
As Hale and Wright (2002, p. 107) also notice.
Tait (1981, p. 526) considers not only numbers but also \(k\)-tuples of numbers for fixed values of \(k\). For simplicity, we restrict attention to the situation where we are dealing with numbers, but our discussion easily generalizes to the extended case.
For if \(\mathsf {ZF}\) is consistent, for every \(n\) there is a proof of \(\lnot \mathrm{Prov}_{\mathsf {ZF}} (n, 0=1)\) in \(\mathsf {PRA}\). By taking the proof which associates with each \(n\) the proof of \(\lnot \mathrm{Prov}_{\mathsf {ZF}} (n, 0=1)\) with the least Gödel number, we have obtained a construction which would count as a proof of the consistency of \(\mathsf {ZF}\). The (not uncontentious) assumption here is that such a proof would clearly be non-finitistic.
Thus, Tait writes: ‘We have already explained that \(f :\forall x F(x)\) should mean \(fa :F(a)\) for arbitrary \(a :A\)’ (1981, p. 536). Niebergall and Schirn (2003, pp. 59–63) suggest another possible reading of Tait’s proposal concerning what it is to have a finitary proof of \(\forall x F(x)\). According to this reading, his proposal would amount to endorsing some restricted version of the \(\omega \)-rule. As a referee pointed out, however, Tait is quite explicit in considering such a rule not finitistically acceptable. For a different view on the finitistic acceptability of the \(\omega \)-rule, see Ignatović 1994.
As usual, a sentence can be considered as a special case of a theory.
The nature of the equivalence turns out to be a delicate matter. See Kaye and Wong 2007 for details.
As opposed to the standard version expounded by Hilbert and Bernays in the Grundlagen der Mathematik (1934), which makes use of propositional logic.
Once we do this, we also no longer need the recursion equations for addition and multiplication.
Of course, Gödel’s second incompleteness theorem might be taken as showing that the second part of the Cartesian project cannot be carried out.
In particular, then, the idea is that any form of strict finitism (also known as ultra-finitism or ultra-intuitionism)—roughly, any position according to which the natural numbers are closed under addition and multiplication but not under exponentiation—would be an instance of such a kind of radical sceptical doubt and should thereby be excluded. In what follows, we will focus on positions that, although taking the natural numbers to be closed under exponentiation as well as addition and multiplication, would still be considered instances of radical sceptical doubt by Tait. For some remarks suggesting that strict finitism itself might not be that radical, see Gaifman 2012, Sect. 2.1.
This is the class of functions obtained starting with 0, addition, multiplication and exponentiation, and closing under composition and bounded primitive recursion, i.e. primitive recursion with the added stipulation that the function under definition cannot grow faster than any other function previously defined.
The conjecture appears in a message posted on the Foundations of Mathematics mailing list on 16 April 1999. See http://www.cs.nyu.edu/mailman/listinfo/fom for the archives of the list.
The point that Tait’s claim of minimality for \(\mathsf {PRA}\) is threatened by the existence of weaker non-trivial systems is also made by Richard Zach (2001) in his PhD thesis (Section 4.6), which also contains extensive discussions of Parsons and Tait. The finitistic credentials of \(\mathsf {EA}\) are investigated in Ganea 2010.
E.g. in the case of \(\mathsf {EA}\), the Kalmar functions arbitrarily restrict primitive recursion to bounded primitive recursion.
This is a controversial issue. For example, Parikh (1971) argues that if one takes feasibility seriously, one has to recognize as effective a class of functions which is known to coincide with the class of Kalmar elementary functions. Feasibility issues might therefore provide grounds for restricting iteration to bounded primitive recursion.
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Acknowledgments
Many thanks to Alex Oliver, Charles Parsons, Erich Reck, William Tait and two anonymous referees. An earlier version of this material was presented at the History and Philosophy of Infinity Conference at the University of Cambridge. I wish to thank Benedikt Löwe for inviting me to the conference and the members of the audience for their valuable feedback.
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Incurvati, L. On the concept of finitism. Synthese 192, 2413–2436 (2015). https://doi.org/10.1007/s11229-014-0639-3
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DOI: https://doi.org/10.1007/s11229-014-0639-3