Jon Perez Laraudogoitia

Linnebo recently presented a puzzle about infinite sums according to which the net weight marked by a spring balance depends on the order the weights are placed on it when these weights form a conditionally convergent series. The aim here is to show that Linnebo’s analysis has an interesting limitation that has gone unnoticed: its conclusions only follow when conditional convergence is considered on the basis of a well-ordered sequence of operations. The consequences otherwise are even more puzzling. There is however a way to avoid them, but only for magnitudes (which certainly do not include weight nor the like), for which a continuity principle is available (such as the principle of continuity of motion, associated with the position of a material object). This also has interesting implications regarding the metaphysical adequacy of the standard mathematical concept of convergence for a whole class of infinite series, so-called doubly infinite series. Cases are shown here in which this concept of convergence is metaphysically adequate (useful), and cases where it is not.

With the classical distinction between context of discovery and context of justification considered by many to have been overcome, heuristics (understood in a broad sense) has increasingly rekindled the interest of philosophers of science. Building on this trend, a heuristic approach to the Voigt transformation (based on Rescher’s Aporetics) is first presented – a topic on which there seem to be no precedents in the literature. Second, the value of this approach is defended from a philosophical (and, indirectly, pedagogical) viewpoint. By using this approach, several conceptual links in the theory of space-time can be highlighted (links which go unnoticed in classical hypothetical-deductive methods leading to the Voigt transformation). Third, Rescher’s aporetics also suggests a heuristic path from the previously obtained Voigt transformation to the Lorentz transformation. Taken together, the proposed heuristic route enables the structure of the conceptual process underpinning the following chain of links (obviously not logical, but heuristic) Galileo ⇒ Voigt ⇒ Lorentz to be presented in an original and unified form. According to this route, an interesting sense is thus established in that the Voigt transformation occupies a natural place between the Galilean and Lorentz transformations.

Shackel recently defended a revised formulation of what he calls ‘the nothing from infinity paradox’. In contrast to other well-known and unchallenged examples of ‘nothing from infinity’ in the literature, it will be argued here that the model he proposes (which would be far more surprising than the others if correct) fails to work as such. Quite the contrary, it is argued here that the form of dynamic evolution underlying it is actually rather trivial. The basis of this criticism contains an elementary foundation that is not contingent on conflicting theoretical assumptions and/or their debatable applications. It is built on the following commonplace: if we wish to determine how a system of particles evolves, the particles must be tracked. As trivial as this may seem, Shackel fails to do so, and in not doing so, is mistaken.

There is broad consensus (both scientific and philosophical) as to what a rigid body is in classical mechanics. The idea is that a rigid body is an undeformable body (in such a way that all undeformable bodies are rigid bodies). In this paper I show that, if this identification is accepted, there are therefore rigid bodies which are unstable. Instability here means that the evolution of certain rigid bodies, even when isolated from all external influence, may be such that their identity is not preserved over time. The result is followed by analyzing supertasks that are possible in infinite systems of rigid bodies. I propose that, if we wish to preserve our original intuitions regarding the necessary stability of rigid bodies, then the concept of rigid body must be clearly distinguished from that of undeformable body. I therefore put forward a new definition of rigid body. Only the concept of undeformable body is holistic (every connected part of an undeformable body is not always an undeformable body) and every connected part of a rigid body in this new sense is always a rigid body in this new sense. Finally, I briefly discuss the connection between this conceptual distinction and the dimensionality of space, thereby enabling it to be supported from a new and interesting perspective.

With the classical distinction between context of discovery and context of justification considered by many to have been overcome, heuristics (understood in a broad sense) has increasingly rekindled the interest of philosophers of science. Building on this trend, a heuristic approach to the Voigt transformation (based on Rescher's Aporetics) is first presented - an issue on which there seem to be no precedents in the literature. Second, the value of this approach is defended from a philosophical (and, indirectly, pedagogical) viewpoint. By using this approach, several conceptual links in the theory of space-time can be highlighted (links which go unnoticed in classical hypothetical-deductive methods leading to the Voigt transformation). In particular an interesting connection with the Lorentz transformation becomes apparent.

Achilles and the tortoise compete in a race where the beginning (the start) is at point O and end (the finish) is at point P. At all times the tortoise can run at a speed that is a fraction F of Achilles' speed at most (with F being a positive real number lower than 1, 0 < F < 1), and both start the race at t = 0 at O. If the trajectory joining O with P is a straight line, Achilles will obviously win every time. It is easy to prove that there is a trajectory joining O and P along which the tortoise has a strategy to win every time, reaching the finish before Achilles.

This paper presents three interesting consequences that follow from admitting an ontology of rigid bodies in classical mechanics. First, it shows (in Sects. 4 and 5) that some of the most characteristic properties of supertasks based on binary collisions between particles, such as the possibility of indeterminism or the non-conservation of energy, persist in the presence of gravitational interaction. This makes them gravitational supertasks radically different from those that have appeared in the literature to date. Second, Sect. 6 proves that the role of gravitation in supertasks of this kind may be highly non-trivial. Third, (in Sects. 7 and 8), the gravitational supertasks found in the first part enable us to show that indeterminism of classical mechanics with gravitation is much more general than has been supposed until now. This result is especially interesting from the philosophical viewpoint, as it links the scope of indeterminism in a theory like classical mechanics directly with the nature of its ontological hypotheses.

While the problem of the philosophical significance of Riemann’s theorem on conditionally convergent series has been discussed in detail for some time, specific versions of it have appeared in the literature very recently, over which there have been widespread disagreements. I argue that such discrepancies can be clarified by introducing a rather conventional type of composition rule for the treatment of some infinite systems (as well as supertasks) while analysing and clarifying the role of the concept of continuity by stripping it of the excesses to which its application by the Leibnizian tradition has led. The conclusion reached is that the indeterminacy associated with conditional convergence has a clear philosophical significance, but no fundamental ontological significance.

This paper shows that, in Newtonian mechanics, unstable three-dimensional rigid bodies must exist. Laraudogoitia recently provided examples of one- and two-dimensional homogeneous unstable rigid bodies, conjecturing the instability would persist for three-dimensional bodies in four-dimensional space. My result proves that, if one admits non homogeneous balls or hollow spheres, then the conjecture is true without having to resort to tetra-dimensionality. Furthermore, I show that instability also holds for at least certain simple classes of elastic bodies. Altogether, the laws of classical dynamics actually lead to the existence of unstable material bodies belonging to the three types of entities accepted therein: point particles, rigid bodies and continuous deformable bodies. A whole range of forms of indeterminism which, until now, has not been considered in the literature. I end with a new conjecture on the connection existing between all these forms of instability and the dimensionality of space.

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.

It is common knowledge that the Aristotelian idea of an unmoved mover was abandoned definitively with the advent of modern science and, in particular, Newton’s precise formulation of mechanics. Here I show that the essential attribute of an unmoved mover is not incompatible with such mechanics; quite the contrary, it makes this possible. The unmoved mover model proposed does not involve supertasks, and leads both to an outrageous form of indeterminism and a new, accountable form of interaction. The process presents a more precise characterization of the crucial going-to-the-limit operation. It has long been acknowledged in the existing literature that, theoretically, in infinite Newtonian systems, masses can move from rest to motion through supertasks. Numerous minor variations on the original schemes have already been published. Against this backdrop, this paper introduces three significant additions: 1) It formulates for the first time a limit postulate for systematically addressing infinite systems; 2) It shows that an Aristotelian unmoved mover is possible in systems of infinitely many particles that interact with each other solely by contact collision; 3) It shows how interaction at a distance can emerge in systems of infinitely many particles that interact with each other solely by contact.

Several paradoxes of infinity have recently featured in this journal involving gases distributed in a denumerable infinite series of compartments. I shall demonstrate in this paper that:a) None of these new paradoxes applies where the gases comply with both Boyle’s law and Avogadro’s law. As several of these new paradoxes expressly require compliance with Boyle’s law, it is unclear, in principle, as to whether there is a plausible model of gas that is able to uphold them all.b) Notwithstanding a), any of the above paradoxes can be reinstated by acknowledging that there are two distinct, non-equivalent concepts of ideal gas. Indeed, the various infinity puzzles actually enable a distinction to be made between the two concepts.

There is broad consensus (both scientific and philosophical) as to what a rigid body is in classical mechanics. The idea is that a rigid body is an undeformable body (in such a way that all undeformable bodies are rigid bodies). In this paper I show that, if this identification is accepted, there are therefore rigid bodies which are unstable. Instability here means that the evolution of certain rigid bodies, even when isolated from all external influence, may be such that their identity is not preserved over time. The result is followed by analyzing supertasks that are possible in infinite systems of rigid bodies. I propose that, if we wish to preserve our original intuitions regarding the necessary stability of rigid bodies, then the concept of rigid body must be clearly distinguished from that of undeformable body. I therefore put forward a new definition of rigid body. Only the concept of undeformable body is holistic (every connected part of an undeformable body is not always an undeformable body) and every connected part of a rigid body in this new sense is always a rigid body in this new sense. Finally, I briefly discuss the connection between this conceptual distinction and the dimensionality of space, thereby enabling it to be supported from a new and interesting perspective.

This paper introduces a new infinite paradox. The main novelty is that it poses problems of causality in a very different form from to the one in use until now. By means of a probabilistic generalization, the paradox shows that the disposition to act according to a specific plan is not always necessary to derive causal effects in Benardete-type contexts involving infinity. It also suggests that, in such cases, the explanation for those causal effects requires a propensity interpretation of probability.

In a recent article, L. Angel ([2001]) argues that if we do not implement Newtonian physics adding to it a certain usual type of boundary condition, then this leads to the rejection of what he calls the P principle: ‘the composition of contact interactions does not create a noncontact interaction.’ Here I shall demonstrate that this conclusion does not follow. However, as will be made clear, this in no way diminishes the interest or importance of the model introduced by Angel in his paper. 1 Introduction 2 The ‘impact without contact’ argument 3 Taking self-excitations seriously 4 Some interesting implications.