Two Ways Of Looking At A Newtonian Supertask

A supertask is a process in which an infinite number of individuated actions are performed in a finite time. A Newtonian supertask is one that obeys Newton's laws of motion. Such supertasks can violate energy and momentum conservation and can exhibit indeterministic behavior. Perez Laraudogoitia, who proposed several Newtonian supertasks, uses a local, i.e., particle-by-particle, analysis to obtain these and other paradoxical properties of Newtonian supertasks. Alper and Bridger use a global analysis, embedding the system of particles in a Banach space, to determine the origin of the strange behavior. This paper provides a common framework for the discussion of both the local and global methods of analysis. Using this single framework, the areas of disagreement and agreement are made explicit. Further examples of supertasks are proposed to illuminate various aspects of the discussion.     

What Is A Newtonian System? The Failure Of Energy Conservation And Determinism In Supertasks

Supertasks recently discussed in the literature purport to display a failure ofenergy conservation and determinism in Newtonian mechanics. We debatewhether these supertasks are admissible as Newtonian systems, with Earmanand Norton defending the affirmative and Alper and Bridger the negative.     

Avoiding Infinite Masses

The examples of dynamic supertasks analyzed to date in the philosophical literature, in which both determinism and the classical laws of conservation of energy and momentum are violated, all share the important limitation of requiring material systems of infinite mass. This paper demonstrates that this limitation is not necessary. This has important consequences for the scope and meaning of such violations.     

On The Dynamics Of Alper And Bridger

Bridger and Alper (1999) maintain that the nonphysical featuresof the supertasks described by Pérez Laraudogoitia (1996) involving a system containing an infinite number of particles may be avoided by introducing, in a specific way, Hilbert space in classical dynamics. I argue that it is possible to interpret their proposal in two ways, neither of which is acceptable for the purpose for which it was introduced.     

A PROOF OF THE IMPOSSIBILITY OF COMPLETING INFINITELY MANY TASKS

In this article, I argue that it is impossible to complete infinitely many tasks in a finite time. A key premise in my argument is that the only way to get to 0 tasks remaining is from 1 task remaining, when tasks are done 1‐by‐1. I suggest that the only way to deny this premise is by begging the question, that is, by assuming that supertasks are possible. I go on to present one reason why this conclusion (that supertasks are impossible) is important, namely that it implies a new verdict on a decision puzzle propounded by Jeffrey Barrett and Frank Arntzenius.     

Indeterminism,classical gravitation and non-collision singularities

Until the present, the Newtonian theory of gravitation has only been studied in any detail through the usual, presupposed ontology of point particles. This paper shows that changing our ontology into one which makes use of continuous bodies (non-point particles) allows us to obtain in a simple way two important results relevant to the theory: (a) The Newtonian theory of gravitation is indeterministic in a way apparently unparalleled when non-point particle models of it are used. (b) In the Newtonian theory of gravitation it is possible to find non-collision singularities qualitatively different from the ones presented in point particle models.     

The staccato roller coaster: a simple physical model of the staccato run

I present a simple model of Grünbaum’s staccato run in classical mechanics, the staccato roller coaster. It consists of a bead sliding on a frictionless wire shaped like a roller coaster track with infinitely many hills of diminishing size, each of which is a one-dimensional variant of the so-called Norton dome. The staccato roller coaster proves beyond doubt the dynamical (and hence logical) possibility of supertasks in classical mechanics if the Norton dome is a proper system of classical mechanics with metaphysical import. If not, challenges raised against the metaphysical significance of the Norton dome are shown to be challenges against various arguments for the dynamical possibility of supertasks, and the staccato roller coaster clearly shows the importance of meeting these challenges. And the staccato roller coaster can provide, as well as interesting lessons, illuminating analyses of Burke’s (Mod Schoolman 78:1–8, 2000) attempt to refute the dynamical possibility of the staccato run and Pérez Laraudogoitia’s (Synthese 148:433–441, 2006) rebuttal of it.     

A Look at the Staccato Run

This paper considers a recent criticism of the physical possibility of supertasks which involves Achilles’s staccato run. It is held that the criticism fails and that the underlying fallacy can be linked with interesting developments in the modern literature on physical supertasks.     

On fair countable lotteries

Two reverse supertasks—one new and one invented by Pérez Laraudogoitia (Philos Stud 168:619–628, 2014)—are discussed. Contra Kerkvliet (Log Anal, 2016) and Pérez Laraudogoitia, it is argued that these supertasks cannot be used to conduct fair infinite lotteries, i.e., lotteries on the set of natural numbers with a uniform probability distribution. The new supertask involves an infinity of gods who collectively select a natural number by each removing one ball from a collection of initially infinitely many balls in a reverse omega-sequence of actions.     

Unlearning Aristotelian physics: a study of knowledge-based learning

A study of a group of elementary school students learning to control a computer-implemented Newtonian object reveals a surprisingly uniform and detailed collection of strategies, at the core of which is a robust “Aristotelian” expectation that things should move in the direction they are last pushed. A protocol of an undergraduate dealing with the same situation shows a large overlap with the set of strategies used by the elementary school children and thus a marked lack of influence of classroom physics training on this student's naive physics. The data from these two studies are pooled and elaborated into a “genetic task analysis” of how one might come to understand Newtonian dynamics as a more or less natural evolution from the naive state.     

Quantum measurements and supertasks

This article addresses the question whether supertasks are possible within the context of non-relativistic quantum mechanics. The supertask under consideration consists of performing an infinite number of quantum mechanical measurements in a finite amount of time. Recent arguments in the physics literature claim to show that continuous measurements, understood as N discrete measurements in the limit where N goes to infinity, are impossible. I show that there are certain kinds of measurements in quantum mechanics for which these arguments break down. This suggests that there is a new context in which quantum mechanics, in principle, permits the performance of a supertask.     

IS QUANTUM INDETERMINISM REAL? THEOLOGICAL IMPLICATIONS

Quantum mechanics (QM) studies physical phenomena on a microscopic scale. These phenomena are far beyond the reach of our observation, and the connection between QM's mathematical formalism and the experimental results is very indirect. Furthermore, quantum indeterminism defies common sense. Microphysical experiments have shown that, according to the empirical context, electrons and quanta of light behave as waves and other times as particles, even though it is impossible to design an experiment that manifests both behaviors at the same time. Unlike Newtonian physics, the properties of quantum systems (position, velocity, energy, time, etc.) are not all well‐defined simultaneously. Moreover, quantum systems are not characterized by their properties, but by a wave function. Although one of the principles of the theory is the uncertainty principle, the trajectory of the wave function is controlled by the deterministic Schrödinger equations. But what is the wave function? Like other theories of the physical sciences, quantum theory assigns states to systems. The wave function is a particular mathematical representation of the quantum state of a physical system, which contains information about the possible states of the system and the respective probabilities of each state.     

Zeno and flow of information

Although the current literature on supertasks concentrates largely on their supposed physical implications (extending the tradition of Zeno’s classical paradoxes of movement), in this study I propose a new model of supertask that explores for the first time some of their information-related consequences and I defend these consequences from a possible criticism.     

Dissociable neural systems support retrieval of how and why action knowledge

In everyday discourse, people typically represent actions in one of two ways: how they are performed or why they are performed. In the present study, we determined the neural systems that support these natural modes of representing actions. Participants underwent functional magnetic resonance imaging while identifying how and why people perform various familiar actions. Identifying how actions are performed produced activity in premotor areas that support the execution of actions and in higher-order visual areas that support the perception of action-related objects; this finding supports an embodied view of action knowledge. However, identifying why actions are performed preferentially engaged areas of the brain associated with representing and reasoning about mental states; these areas were right temporoparietal junction, precuneus, dorsomedial prefrontal cortex, and posterior superior temporal sulcus. Our results suggest that why action knowledge is not sufficiently constituted by information in motor and visual systems, but requires a system for representing states of mind, which do not have reliable motor correlates or visual appearance.     

NEWTON AND SPINOZA: ON MOTION AND MATTER (AND GOD,OF COURSE)

This study explores several arguments against Spinoza's philosophy that were developed by Henry More, Samuel Clarke, and Colin Maclaurin. In the arguments on which I focus, More, Clarke, and Maclaurin aim to establish the existence of an immaterial and intelligent God precisely by showing that Spinoza does not have the resources to adequately explain the origin of motion. Attending to these criticisms grants us a deeper appreciation for how the authority derived from the empirical success of Newton's enterprise was used to settle debates within philosophy. What I emphasize is that in the progression from More to Clarke to Maclaurin, key Newtonian concepts from the Principia (1687), such as motion, atomism, and the vacuum, are introduced and exploited in order to challenge the account of matter and motion that is presented in Spinoza's Ethics (1677). Building on this treatment, I use the arguments from More and Clarke especially to help discern the anti‐Spinozism that can be detected in Newton's General Scholium (1713). Ultimately, the Newtonian criticisms that I detail offer us a more nuanced view of the problems that plague Spinoza's philosophy, and they also challenge the idea that Spinoza seamlessly fits into a progressive narrative about the scientific revolution.     

A formal system for classical particle mechanics,its model-theoretic applications and space-time structure

In the history of Newtonian Mechanics physicists and astronomers did not rely on so-called inertial frames, indeed they were not able to identify such frames. So the usual neo-Newtonian formalism of Newtonian Mechanics contains some superfluous components. In the present paper I will formulate a formal system for classical particle mechanics in Leibnizian space-time, where a relation, a counterpart of the second law of motion, between force on bodies and derivative of their momentum will be defined relative to every, inertial or not, reference frame. And I will present a view that in the research process under Newtonian mechanics the accumulation of those models of the relation that satisfied some realistic conditions determined an additional structure of non-rotating frames to Leibnizian space-time.     

Emergence,Naturally!

The prevailing common assumptions about how nature behaves have their origins in the early Enlightenment. The notion of emergence does not sit comfortably within this framework. Emergence appears virtually impossible within a world determined by ineluctable and unwavering natural laws. But the variety and combinations inherent in living systems render physical laws indeterminate. The study of ecological dynamics suggests that processes rather than laws are what accounts for most order seen in the living realm. As a consequence, there are aspects of ecological dynamics that violate each of the Newtonian postulates. The dynamics of ecosystems suggest a smaller set of rational assumptions through which to view nature—an “ecological metaphysic.” Emergence appears as a rare but wholly natural phenomenon within the new rational platform. In addition, several apparent conflicts between science and theism that arose under the Newtonian framework simply vanish under the new perspective.     

Making sense of recommendations

Computer algorithms are increasingly being used to predict people's preferences and make recommendations. Although people frequently encounter these algorithms because they are cheap to scale, we do not know how they compare to human judgment. Here, we compare computer recommender systems to human recommenders in a domain that affords humans many advantages: predicting which jokes people will find funny. We find that recommender systems outperform humans, whether strangers, friends, or family. Yet people are averse to relying on these recommender systems. This aversion partly stems from the fact that people believe the human recommendation process is easier to understand. It is not enough for recommender systems to be accurate, they must also be understood.     

On the Algebraizability of Annotated Logics

Annotated logics were introduced by V.S. Subrahmanian as logical foundations for computer programming. One of the difficulties of these systems from the logical point of view is that they are not structural, i.e., their consequence relations are not closed under substitutions. In this paper we give systems of annotated logics that are equivalent to those of Subrahmanian in the sense that everything provable in one type of system has a translation that is provable in the other. Moreover these new systems are structural. We prove that these systems are weakly congruential, namely, they have an infinite system of congruence 1-formulas. Moreover, we prove that an annotated logic is algebraizable (i.e., it has a finite system of congruence formulas,) if and only if the lattice of annotation constants is finite.