Abstract
In this paper I argue, contra Mortensen, that there is a case, namely that of a degenerate topos, an extremely simple mathematical universe in which everything is true, in which no mathematical “catastrophe” is implied by mathematical triviality. I will show that either one of the premises of Dunn’s trivialization result for real number theory –on which Mortensen mounts his case– cannot obtain (from a point of view “external” to the universe) and thus the argument is unsound, or that it obtains in calculations “internal” to such trivial universe and the theory associated, yet the calculations are possible and meaningful albeit extremely simple.
For Christian Edward Mortensen, long-distance mentor, in his 70th birthday.
This paper has been written under the support from the PAPIIT project IA401015 “Tras las consecuencias. Una visión universalista de la lógica (I)”, as well as from the CONACyT project CCB 2011 166502 “Aspectos filosóficos de la modalidad”. I thank Charlie Donahue and Chris Mortensen for useful comments on previous versions of this paper, as well as to the referees for saving me from at least a couple of embarrassing mistakes. Diagrams were drawn using Paul Taylor’s diagrams package v. 3.94.
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Notes
- 1.
And sometimes an incredulous stare means a lot of debate, because according to some, trivialism deserves no stare at all; cf. [5, p. 252].
- 2.
I am not alone on thinking that mathematics might have place for triviality. Priest in [15] considers models of arithmetic with (atomically) trivial objects in which, among other principles, neither the transitivity of ‘\(=\)’ nor the substitutivity of identicals hold. That work was developed independently of Dunn’s result, but the ideas serve to block it. Priest models are examples that no mathematical catastrophe needs to follow from a trivial arithmetical object. Nonetheless, I do not aim to compile here all the ways to block Dunn’s result, so acknowledging Priest’s work is enough for my purposes.
- 3.
At the eleventh hour prior to publication I was referred to two notions of relative triviality close to mine: In [18], negation-triviality (all the negations of a theory hold) is defined and called quasi-triviality, and in [7] ‘quasi-triviality with respect to i and C’ means that a contradiction with degree of complexity i implies all the formulas of a class C. Thanks to María del Rosario Martínez-Ordaz for the pointers.
- 4.
- 5.
Actually, all what is needed is the “functional” version of this principle, that is, when \(P(\ldots \tau \ldots )\) is a sentence free of logical connectives. The “transparent” version is when \(P(\ldots \tau \ldots )\) is any sentence. Classically, the functional and the transparent versions of the principle are equivalent, but in general they are not in inconsistent mathematics. See [10, Chaps. 1 and 2] and [11].
- 6.
The extra resources to get full triviality are described in [11, p. 205].
- 7.
Clear introductions to category theory in general, and topos theory in particular, can be found in [8].
- 8.
For those who might wonder of a definition in terms of objects and morphisms: a terminal object in a category C, denoted ‘\(\mathbf 1 _\mathbf C \)’, is an object such that for any object X there is exactly one morphism from X to \(\mathbf 1 _\mathbf C \). The dual notion, initial object, denoted ‘\(\mathbf 0 _\mathbf C \)’, the categorial version of an empty set, is an object such that for any object X there is exactly one morphism from \(\mathbf 0 _\mathbf C \) to X.
- 9.
But there is an alternative, dual reading of \(\nu \) as false, so M would be rather the anti-extension of the predicate f, etc. See [10, Chap. 11].
- 10.
In fact, the usual non-degeneracy axiom states that terminal and initial objects are not isomorphic.
- 11.
The contrast between internal and external perspectives of a mathematical universe are already familiar in standard set theory. For example, up to some ordinal in the cumulative hierarchy one can see from the outside that there are infinite elements in the universe, but within the universe (up to that rank) those elements do not form yet a set, so within the universe there is no infinite set yet.
- 12.
Although I think it is, and this paper would be an argument for that, but I will not press this point further. See [3] for an example of how interesting things become for Platonists when triviality is taken a bit more seriously.
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Estrada-González, L. (2016). Prospects for Triviality. In: Andreas, H., Verdée, P. (eds) Logical Studies of Paraconsistent Reasoning in Science and Mathematics. Trends in Logic, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-319-40220-8_5
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