Abstract
In this chapter we visit the subtle question of whether the pluralist is pluralist towards himself. We answer this question in two ways. The first is technical, and we develop this answer in Sects. 11.2 and 11.3. The second answer is general, and we develop this answer in Sect. 11.4. A reader could skip Sects. 11.2 and 11.3 without loss of coherence, especially if said reader is not wedded to a particular formal paraconsistent logic. The final two sections are for those who feel queasy from following the conceptual gymnastics of pluralism.
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Notes
- 1.
One complaint I have heard against this sort of work is that the ‘interesting models’ are not so interesting since they have not been used to tell us anything about other parts of mathematics. To answer this criticism, first: when Priest et al. claim that the models are interesting, they mean this in a specific sense. The uninteresting models are those that are trivial everywhere except for a small consistent part. The interesting models are ones that recover consistency after a certain inconsistent limit. The inconsistent limit might re-surface again later. The inconsistency is a fixed point (Priest 2002, 173). Therefore it is correct that this is not enough to interest other mathematicians. It is a completely legitimate demand, since acceptance and inclusion of new mathematical areas is only achieved through crosschecking. As things stand at present, paraconsistent logics and mathematical theories are generally treated as a mathematical curiosity. Regardless, it is philosophically important for distinguishing trivial from inconsistent mathematics. But there is a second answer: working out links with other areas of mathematics takes time and resources. There happen to be very few people working in this area. I am certain that as soon as one of them finds something interesting for others in the mathematical community, he, or she, will let us know. The time it takes is a feature of communication in the mathematical community, and how that works.
- 2.
On explaining something about paraconsistent logic to a Romanian ecological economist, he assured me that all Rumanians think this way. So the group might well include much more than only the paraconsistent logicians. In fact, this is one of the motivations for studying paraconsistent logic: paraconsistent reasoning is empirically observed, especially in philosophy classrooms, but also on the street in Romania.
- 3.
LP is Post non-trivial, and this is enough for coherence.
- 4.
I was torn about what to call it myself. On the one hand, the ‘collapsing lemma’ is more current, so it would be less confusing to use that name. On the other hand, the ‘bubbling’ lemma is more fun.
- 5.
The finite number of predicates is important for ordering minimally inconsistent LPs (Priest 2006b, 227). The lack of functions is needed for the proof of the collapsing lemma to go through. Of course, we can re-express functions as relations (Priest 2002, 173). ‘Predicates’, here, are predicates or relations. That is, a relation is a two-or-more place relation.
- 6.
There are more careful ways of saying this that are acceptable to a constructivist, but they are elaborate, and therefore, using them (since there are different versions of constructivism) would risk increasing confusion.
- 7.
It is plausible to ignore these if we realise a few facts. First, there are several interpretations of LP which we can make using the collapsing lemma. Second, philosophical considerations might not decide between all pairs of interpretations to favour one over the other. Moreover, third, there might not (yet) be any, non-ad hoc, philosophical or otherwise, means of making a determinate choice.
- 8.
In know of no discussion of trivialism which has degenerated into trivialism, except in moments of jest.
- 9.
One might think that I am being somewhat unfair, and ignoring a lot of philosophical activity. For example one might point out that Russell was much aggrieved by the paradoxes, and theorised a lot about them; and I should not ignore this since Russell’s investigation into the paradoxes shaped his philosophy and formal system. Moreover, some very important philosophical work has been done in looking very closely at Frege’s trivial theory – such as the work of Dummett, Wright, Boolos and Heck. I appropriate such activity, and call it pluralist! What is anti-pluralist is any accompanying revisionism. So, we should be careful about our interpretation of the intention behind the excellent work cited above, we might say that these philosophers engage in pluralist work despite themselves.
- 10.
We might come to this position by supposing, say, that ZF contains a contradiction. More precisely, we need a theory which is considered to be foundational to mathematics, we need for it to be a classical theory: allowing ex contradictione quodlibet inferences, and we need to be able to derive a contradiction from the axioms using the rules of inference.
- 11.
For a good discussion of trivialism see (Priest 2006a, 56–71).
- 12.
In (Priest 2006a), Priest writes that he is not completely satisfied with the argument from meaning, and thinks that his argument from physical survival is stronger. Note that in (Priest 2006a), Priest is arguing about trivialism in general, not about trivialism as a philosophy of mathematics. For the purposes here, the argument from meaning is both satisfactory, and the stronger argument.
- 13.
The trivialist will ‘hold’, in the sense of ‘assert’, any position. This is not the point. Trivialism in mathematics arises from the idea that mathematics is classical and there is a contradiction in mathematics, and therefore (under our old classical reasoning) all of mathematics is true, we then get to the meaninglessness of any particular mathematical statement, and wallow in our degenerate theory. There is a sequence to the reasoning, which gets us to the degenerate position. Once there, reasoning, as such, is impossible.
- 14.
The trivialist will, of course, agree that ‘\( \vdash \) PA 2 + 9 = 34 is false’, since the trivialist will agree to everything. The maximal pluralist will disagree that ‘\( \vdash \) PA 2 + 9 = 34’ is true. The quotation marks are important. The trivialist has to agree, and cannot disagree, except in quotation marks. This is all we need to distinguish the positions.
- 15.
If (what we suppose to be) two trivial theories have different languages, then they can be distinguished from each other, not otherwise. Some sentences will be true in one, but not recognizable in the other. I thank Priest for pressing me on this point at the Logica conference 2005.
- 16.
This is a paraphrase. What Mortensen (2010, 4) actually writes is: “The importance of inconsistent images is enormous, I think. Even sceptics who disbelieve in paraconsistency have difficulty in insisting that the inconsistent has no structure, when confronted with these examples [of images of inconsistent objects].” I do not think I have misrepresented him in my paraphrase.
- 17.
For example, we did not stop doing arithmetic when Russell discovered paradox in Frege’s reduction of arithmetic to logic. This is also evidence against trivialism.
- 18.
Azzouni (2007, 599) says something similar about the triviality of natural language. Accepting that the semantic paradoxes make natural language trivial, it is then clear that “no one actually makes any inferences on their basis [the basis of inconsistencies arising from the papradoxes], and so the body of purported knowledge that speakers (collectively) are building up, is not… tainted by such.”
- 19.
Note that Byers makes no mention of paraconsistent or relevant logics. I therefore assume that he is not advocating a paraconsistent point of view or anything of the sort. Nevertheless, in the quotations I cite here, and in many other places in the book, I found support for the position advocated in this book. I do not know what Byers reaction would be to the mention of paraconsistent logics.
References
Azzouni, J. (2007). The inconsistency of natural languages: How we live with it. Inquiry, 50(6), 590–605.
Byers, W. (2007). How mathematicians think; using ambiguity, contradiction, and paradox to create mathematics. Princeton/Oxford: Princeton University Press.
Mortensen, C. (1995). Inconsistent mathematics (Mathematics and its applications 312). Dordrecht: Kluwer Academic Publishers.
Mortensen, C. (2010). Inconsistent geometry (Studies in logic 27, Series Ed. D. Gabbay). London: Individual author and College Publications.
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Friend, M. (2014). Pluralism Towards Pluralism. In: Pluralism in Mathematics: A New Position in Philosophy of Mathematics. Logic, Epistemology, and the Unity of Science, vol 32. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7058-4_11
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