The Convergence Approach to Benardete’s Paradox

Philosophia https://doi.org/10.1007/s11406-023-00612-5 The Convergence Approach to Benardete’s Paradox Jon Pérez Laraudogoitia1 Received: 28 April 2022 / Revised: 24 November 2022 / Accepted: 8 January 2023 © The Author(s) 2023 Abstract The paper analyses Benardete’s paradox of the gods from a more general perspective (the convergence approach) than several of the most important proposals made to date, but in close relation (and sharp contrast) with them. The new theory, based on the notion of limit, is systematically applicable in different possible scenarios involving a denumerable infinity of objects. In particular, it reveals in what way ω-consistency can be compromised in an otherwise consistent description of such "infinitary" situations. Keywords Benardete’s Paradox · Infinity · Limit · Convergence · ω-Consistency Benardete’s (1964) paradox of the gods has been the source of growing discussion in the literature. In its original formulation it reads as follows: “A man decides to walk one mile from A to B. A god waits in readiness to throw up a wall blocking the man’s further progress when the man has travelled ½ mile. A second god (unknown to the first) waits in readiness to throw up a wall of his own blocking the man’s further progress when the man has travelled ¼ mile. A third god … &c. ad infinitum. It is clear that this infinite sequence of mere intentions (assuming the contrary-to-fact conditional that each god would succeed in executing his intention if given the opportunity) logically entails the consequence that the man will be arrested at point A; he will not be able to pass beyond it, even though not a single wall will in fact be thrown down in his path. The before-effect here will be described by the man as a strange field of force blocking his passage forward.” (pp. 259–260). The situation involves an infinity of objects (walls) that are capable of interacting with one another (the man). The paradox is that this interaction (which has nothing to do with gravitation) is not by contact. And one of the ways of dealing with the * Jon Pérez Laraudogoitia 1 Departamento de Lógica Y Filosofía de La Ciencia. Facultad de Letras, Universidad del País Vasco, C/ Paseo de La Universidad 5, 01006 Vitoria‑Gasteiz, Spain 13 Vol.:(0123456789) Philosophia problem consists of directly analysing this actual infinity, extracting consequences from it. This is what Hawthorne (2000) does, for example, thereby attempting to explain what the solution is. He argues that the paradox shows the falsity of what he calls the "Change Principle": "If x is the fusion of y’s and y’s are individually capable only of producing effect e by undergoing change, then x cannot, (without the addition of some non-supervening causal power), produce effect e without undergoing change." (p. 630). Although he gives no explanation of what "change" refers to, it is clear from the context that he means change in the physical state. The fusion of the gods is thus supposed to be able to prevent the man’s movement (indeed, the man cannot reach any of points x = 1/2, x = 1/4, …, x = 1/2n, …) without change, which no god alone can do. This solution to the problem is essentially the same as the one Benardete seems to advocate: the fusion of the gods’ causal power creates the strange force field to which the latter refers. I shall call it the mysterious fusion solution (MFS). An alternative approach is to begin from experience we have of finite sets of interacting objects and generalise some feature of the former deemed essential to the infinite case. There are two significant versions of this approach in the literature, both of which are widely supported: the causal finitism and the unsatisfiable pair diagnosis. It is not my intention here to expound these two positions (the reader’s knowledge of which is presupposed), but simply to mention the parts of them that are relevant to the problem at hand (Benardete’s paradox). According to causal finitism, nothing can be affected by infinitely many causes and, in addition (according to one of its refinements), excludes "cases where an infinite number of things each has the power to contribute to some effect,[ …] even if they do not actually do so" (Pruss, 2018, p. 144). This directly blocks the infinite system in which the paradox of the gods is formulated. The unsatisfiable pair diagnosis (Shackel, 2005) blocks in a different way. It considers that the fusion of the gods cannot have causal effects (nor the fusion of the walls, contrary to Laraudogoitia, 2003) because the man can only interact (collide) with specific and well-determined walls, as is necessarily the case in systems with a finite number of objects. Consequently, if no god-m (m > n) stops the man at respective point x m = 1/2m controlled by the former, it does not follow that the latter has not passed point x = 0 rather that, on the contrary, he has done so (since only a wall could stop him) and is now approaching point xn = 1/2n (where god-n stops him).1 Symbolically: ∀n(¬∃m(m > n)the man is stopped at xm → the man is stopped at xn ) (1) Yet we knew from the very outset, given the conditions of the situation, that: ∀n(the man is stopped at xn → ¬∃m(m > n)the man is stopped at xm ) (2) 1 This is basically the condition that Caie (2018) calls "Only Walls", and is also taken for granted in his analysis of Benardete’s paradox. 13 Philosophia Conjunction of the two gives: ∀n(the man is stopped at xn ↔ ¬∃m(m > n)the man is stopped at xm ) (3) Finally, as has been stipulated: the set of points on line {x1 , x2 , ..., xn , ...} have no f irst member (4) (3) and (4) form an unsatisfiable pair of conditions. The conclusion follows that it is impossible for the gods to fulfil their plan, and the paradox disappears. 1 The Convergence Approach Against the background of the scheme of options discussed above, the proposal herein to analyse the paradox of the gods can be better understood. It shares with causal finitism (CF) and the unsatisfiable pair diagnosis (UPD) the intuition that it is the experience we have of finite sets of interacting objects which should provide us with the key to generalising to the infinite case. Nonetheless, it differs from them in what this key is. The key for CF and UPD is ontological; it is linked to a certain conception of causal action that is considered generalisable. The key for the convergence approach (CA) is purely methodological (or, if preferred, purely formal): the infinite system’s behaviour (time evolution) must be "as similar as possible" (in a formal sense to be specified below) to that of a finite system. In more suggestive language, an infinite system’s time evolution should be understood as the limit of a finite system’s time evolution when the latter’s number of components tends towards infinity. The most obvious understanding of a system’s time evolution is the idea of its spatial configuration’s evolution: the history of each of its pa (...truncated)