Abstract
In “Action without interaction” (2005) I showed that one might act on a physical system (there, a particle), without interacting with it, by the procedure of making it disappear. This paper presents further extensions and a critique of that result. These extensions show why physical actions without interaction are possible, while underscoring the philosophical fertility of a characteristic approach to the actual infinite inaugurated by Benardete.
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Notes
The gods must have the capacity to intercept particles in motion (for instance by detaining them or deviating them from their previous paths), of setting particles in motion at a certain velocity (e.g. by percussion) and of transferring particles from one place to another (although at uniformly bounded velocities. Let us say that, for every n, the particle q n that God n controls and which is not in the instant t = 0 on the X axis, which is where we are considering the movement, is in t = 0 at a distance (1/2n) from this axis). All this is guaranteed by the laws of particle dynamics (whether classical or relativistic) and is scrupulously respectful of the requirement (essential to my argument) that the particle world lines be continuous. For a particle world line W(t) (parametrized by the time t, measured in relativistic mechanics by an inertial observer) to be continuous means that it is continuous at every single point of that world line, i.e. if W(t*) is a point of W(t) then the particle will occupy points of the space belonging to an arbitrarily small neighbourhood of W(t*) for instants of time belonging to a sufficiently small neighbourhood of t*. Clearly, if the particles of a certain set move at uniformly bounded velocities (as occurs in my creation argument) their world lines will without doubt be continuous. In particular the world line of particle Q created by the gods and which only exists for t > 0 is continuous.
Someone might be tempted to criticize my models of action without interaction in this paper (not just the one in Sect. 1 but the ones in Sect. 2 as well) on the basis that they rest essentially on non-physical idealizations, that they have at best tenuous links to the real world and that consequences follow from them that run counter to the way things behave in the real world. Leaving the point about non-physical idealizations aside for the moment, the rest of the criticism would be valid if directed against (BL) (as I have already discussed in the main text of the paper) but beyond (BL) any such criticism would be unsound. As in much of the literature on the philosophy of science and the philosophy of physics, rather than saying something relevant in relation to the real world, in this paper I am more interested in saying something relevant with regard to two major theories proposed to explain it (classical or relativistic particle dynamics). Both dynamical theories have been refuted experimentally (not least by the mere fact of our world being a quantum world) but none of this has dissipated the sustained, ongoing interest shown by physicists and philosophers of physics in them. And my models’ relevance lies in that (as shown in Sect. 2) they manifest surprising forms of action without interaction in possible worlds governed by those dynamical theories (this is what I refer to when I use the word “possible” in the abstract of this paper) and that (here I’m thinking now specifically of the model here in Sect. 1), they might well be plausible arguments against (BF), as we saw above. The “ontological furniture” I suppose consists simply in particles and agents with the capacity to manipulate them (always in a dynamically admissible way, as we saw in the previous note) so that the accusation of non-physical idealization could only come from the fact that I also suppose a numerable infinity of particles and agents. The reason for this latter assumption lies in the fact that the other central interest of this paper is, as I said from the outset, to explore the consequences that follow from the original treatment of the actual infinite inaugurated by Benardete. If one deplores the idea of the actual infinite, then my paper is certainly not going to have a great deal to say (however, the conclusion needs to be read) but in my defence I would argue that there is nothing in classical (or relativistic) physics of particles that excludes on principle this kind of infinity. This last point has in fact given rise to a growing literature on dynamical supertasks in infinite systems, with interesting results (for a general exposition that links up with classical problems of the philosophy, see, for instance, Peijnenburg and Atkinson (2008)). The present paper may generally be included in this tradition.
If there were a sufficient number of people with similar psychokinetic powers the serious problem of nuclear waste would be solved. In such circumstances, nobody would deny the causal effectiveness of their power.
This is a case in which the counterfactual account of causality in Lewis’s version (1986) fails: it is not true that if the gods had not acted the system of particles would not have self-excited, nor is it true (and this is the point against Lewis) that if the gods had not acted the system of particles would have self-excited. Further, as the possible self-excitations of the base configuration are indeterministic processes whose probability is not defined, no probabilistic theory of causality will do it better. Indeed, I do not know of any other major theory of physical causation (with the possible exception of the Regularity account, a theory with its own problems, and lots of them) that functions in this case: neither transference theories, nor process theories or even conserved quantity theory seem to do so, although this is not something I shall be discussing here.
Beware! If, for example, equal and opposing external forces act on a particle, there is no change in its dynamic state, despite interaction taking place with the exterior. However, nothing like this happens in our case, unlike other examples that one might easily imagine and which, therefore, would only apparently be similar. The most famous of such examples is Benardete’s (1964) paradox of the gods, in which all we have is a man (a “particle”) at the point x = 0 who decides in t = 0 to walk to the right towards x = 1. In this case, for each n, God n will impede the man from passing the point x n = 1/n. He will be unable to abandon the point x = 0 despite the action of the force destined in principle to push him to the right (exercised, for instance, by the friction with the ground on which he stands) because the gods act collectively on him (see my 2003) by means of an equal and opposed force that keeps him at x = 0, preventing a change in his dynamic state. Here obviously the gods (and the ground) act with interaction. But this is not so in the first example I have just discussed in the Sect. 2 of this paper and which may be considered as a more subtle version of Benardete’s paradox in which, together with set of gods, not one, but a likewise infinite set of particles intervenes: this reveals new, contra-intuitive aspects of infinity.
To say that the gods may be substituted by machines programmed to act like them is not justified on the grounds that the gods don’t interact with the particles in my models. It is justified quite simply on the basis that, as I pointed out in a previous note, all the gods have to be capable of doing is essentially to prevent a certain particle moving from the place it is in or to provoke another specific particle to move in a particular way, and these are elementary actions that a machine could perfectly well be programmed to perform! In many circumstances, it is not even necessary for each machine to be programmed to be capable of acting individually exactly as the god it has replaced in order to ensure that, finally, the set of machines will effectively end up acting like the set of gods. By way of example, let us consider once again the base configuration in t = 0, where God n shall prevent, at any t ≥ 0, p n from abandoning point x n = 1/n (n = 1, 2, 3, …). The gods prevent the spontaneous self-excitation of this configuration for t ≥ 0 and in doing so there is no interaction, as the particles don’t move. However, this is not a universe in which the gods have to have psychokinetic powers (nor is it a universe in which one has to admit some kind of mind-matter dualism or some influence of the mind over matter) because the gods may be replaced by (very simple) machines that in effect end up acting like them. Machine M n that here replaces God n is simply a sufficiently rigid, heavy wall, placed for instance at the mid point between p n and p n+1, and joined sufficiently rigidly by its base to the other walls (this makes each wall permanently immobile). Utterly lacking any psychic power, the walls M n prevent the spontaneous self-excitation of the base configuration (p n can not be set in motion by p n+1 because they are separated by a permanently immobile wall) and in doing so there is no interaction as the particles don’t move. It may be useful to remember a propos of all this that the “gods” of Benardete are essentially a rhetorical (but highly suggestive, in my view) variant of the idea of the “infinity machine”, a term introduced by Black (1951) in the philosophical literature (although the idea itself is anterior). Replacing the gods with machines entails returning to the original “terminology”. After Black many examples of “infinity machines” programmed to a wide variety of things have been discussed in the literature (the best-known examples being probably the π-Machine and the Peano Machine, see Grünbaum (1967) for the standard exposition). In all cases, the infinity machines (like the infinite gods of Benardete) correspond exactly to numerable infinities of machines programmed to perform some very elementary actions. Nobody has ever suggested that in dealing with problems involving infinity machines one is assuming any type of mind matter dualism.
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Acknowledgements
I would like to thank two anonymous referees of Erkenntnis for some very helpful suggestions in the writing of this paper.
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Pérez Laraudogoitia, J. Physical Action Without Interaction. Erkenn 70, 365–377 (2009). https://doi.org/10.1007/s10670-009-9155-0
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DOI: https://doi.org/10.1007/s10670-009-9155-0