Peano Systems and Internal Relations

(Logic, )

ecent and less recent metaphysics has contained a good deal of discussion that revolves around internal relations, where a relation R is internal if and only if the essential properties of its relata entail it. One might maintain, given the numbers 2 and 3, their essential properties (e.g., their numerosity) entail the internal relation, ``being greater than'', which holds between them. However, in the last decade, the idea that we can dispense with internal relations has caught on. Being R an internal relation, any comparative claim of the form aRb is made true by the essential properties of the individuals they relate, Fa and Gb. The previous thesis belongs to the philosophers of reductionist fashion, those who, for nominalistic reasons, wish to reduce the internal relations in aRb's claims to the essential properties of a and b (Simons, 1994) (Lewis, 1986). Throughout this essay, we will argue the opposite: the essential properties F and G of a and b are not always enough to make true comparative claims of the form aRb, where R is a strong internal relation. According to Johansson (2004), strong internal relations are defined in terms of mutual existential dependence. a and b are mutually dependent if and only if a is dependent upon b and b dependent upon a. Hence, a and b are strongly internally related if and only if a cannot exist if b does not exist, and vice versa. Using the latter notion, we offer a counter-example to the reductionist thesis that comes from Peano Systems, where the strong internal relation being greater than and the essential properties of natural numbers are both necessary as truth-makers for comparative claims of the form aRb.

Chair: Larissa Bolte

Time: 15:20-15:50, 11 September 2021 (Saturday)

Location: SR 1.006

Remark: (Online Talk)

Antonio Freiles

(University of Italian Switzerland, Switzerland)

My name is Antonio Freiles and I am MA student at the University of Italian Switzerland (USI). My interests lie in the intersection between Metaphysics, Logic and Philosophy of Mathematics.

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