Abstract
Mathematical pluralism notes that there are many different kinds of pure mathematical structures—notably those based on different logics—and that, qua pieces of pure mathematics, they are all equally good. Logical pluralism is the view that there are different logics (consequence relations), which are, in an appropriate sense, equally good. Some, such as Shapiro (Varieties of logic, Oxford University Press, Oxford, 2014), have argued that mathematical pluralism entails logical pluralism. In this brief note I argue that this does not follow. There is a crucial distinction to be drawn between the preservation of truth (simpliciter) and the preservation of truth-in-a-structure; and once this distinction is drawn, this suffices to block the argument. The paper starts by clarifying the relevant notions of mathematical and logical pluralism. It then explains why the argument from the first to the second does not follow. A final section considers a few objections.
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Notes
As does Shaprio (2014).
On intuitionist mathematics, see Dummett (2000).
On which, see Bell (2008).
Microcancellation follows. Take f(x) to be xa. Then, taking x to be 0, Microaffineness implies that there is a unique r such that, for all i, \(ai=ri\). So if \(ai=bi\) for all i, \(a=r=b\).
For an overview, see Mortensen (2017).
For more on the following, see Mortensen (2010).
And one can set things up in such a way that this does not imply that \(90=0\).
Which is not to say that all mathematics are interested in them. Most mathematicians are interested in only parts of mathematics. It is to say that these theories have a mathematical structure which warrants mathematical interest, and so interests some mathematicians.
A referee of a previous draft objected that this begs the question against a mathematical monist. These theories are not legitimate. The institution of mathematics is just making a mistake about what pure mathematics really is. This is actually beside the point here. I am not arguing for the truth of mathematical pluralism. I am arguing that logical pluralism does not follow from it. However, I do indeed think that mathematical pluralism is true. To suggest that mathematicians are mistaken as to what is legitimate mathematics strikes me as philosophical hubris.
For discussion, see Priest (2019), §8. One might also suggest that the matter was clear within set theory itself. For, given that, say, the Continuum Hypothesis (CH) is independent of ZFC, mathematicians investigate the theories ZFC+CH and ZFC+\(\lnot \)CH. However, in this case, one might suggest that this is not really pluralism, since both investigations can be subsumed under classical model theory. In a similar way, it is often pointed out that the internal logic of a topos is intuitionistic logic. However, this is established in standard category theory, using classical logic.
I use the word ‘structure’ here in the way that is standard in mathematics, namely a complex of objects and the relations and functions between them. This has absolutely nothing to do with mathematical structuralism as a philosophy of mathematics, as one referee wondered.
For more on this, see Priest (2006).
‘Euclid’s treatise contains a systematic exposition of the leading propositions of elementary metrical geometry.’ Rouse Ball (1960), p. 44.
Though if this were a response in a philosophical debate, it might be a prelude to a discussion of which of these logics is right; but that is another matter.
On the variety of logical pluralisms, see Priest (2006).
Indeed, Shapiro (2014) takes it to do so.
One thing it cannot mean is that if \(A\vdash B\) then the conditional \(T\left\langle A\right\rangle \rightarrow T\left\langle B\right\rangle \) holds, where T is a naive truth predicate. For then trivialism in the shape of the Curry Paradox follows. However, there is no bar to truth preservation in the following form: if \(A\vdash B\) then \(T\left\langle A\right\rangle \vdash T\left\langle B\right\rangle \)—or a number of others. For a general discussion of the matter, see Priest (2010), §13.
See Priest (2016), §7.4.
For a fiction in which the internal logic is a paraconsistent logic, see ‘Sylvan’s Box’, Priest (2016), §6.6.
I note that there are versions of fictionalism about mathematics which treat mathematical claims as tacitly prefixed by an operator such as ‘According to the fiction F...’ (See Balaguer (2011) and Leng (2018).). This kind of fictionalist might well think of the prefix ‘In structure \(\mathfrak {A}\)...’ as ‘In the fiction F...’. The account given here should, then, be very agreeable to this kind of mathematical fictionalist. However it is not committed to such fictionalism.
On applied mathematics, see Priest (2016), 7.8.
As a referee noted, in his (2018) Williamson argues that applied mathematics poses a challenge for non-classical logic. This is not the place to discuss the cogency of his argument; it is irrelevant to the position being argued here, which is quite compatible with there being one true logic, and that logic being classical.
If there were a pluralism in applied logic, this might arguably imply logical pluralism of the kind in question. However, at least so far, the parts of mathematics that get applied (number theory, analysis, various geometries, probability theory, etc.) can be seen, in the usual way, as fragments of classical ZFC. So there is as yet no case to be made for logical pluralism here.
It has always seemed to me that most of those who endorse such a view simply confuse what is true and what is held to be true.
On truth pluralism, see Pederson and Wright (2013).
See Priest (2006).
Many thanks go to Colin Caret, Hartry Field, Teresa Kissel, and Stewart Shapiro, for helpful comments on an earlier version of this essay. Thanks also go to three anonymous referees of the journal. A version of the paper was given at the conference Anti-Exceptionalism and Pluralisms: from Logic to Mathematics, IUSS, Pavia, March 2019. I am grateful to the audience there for a number of helpful comments and questions.
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Priest, G. A note on mathematical pluralism and logical pluralism. Synthese 198 (Suppl 20), 4937–4946 (2021). https://doi.org/10.1007/s11229-019-02292-9
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DOI: https://doi.org/10.1007/s11229-019-02292-9