Surreal Time and Ultratasks

This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible

Leibniz's Laws of Continuity and Homogeneity

We explore Leibniz's understanding of the differential calculus, and argue that his methods were more coherent than is generally recognized. The foundations of the historical infinitesimal calculus of Newton and Leibniz have been a target of numerous criticisms. Some of the critics believed to have found logical fallacies in its foundations. We present a detailed textual analysis of Leibniz's seminal text Cum Prodiisset, and argue that Leibniz's system for differential calculus was free of contradictions.Comment: 19 pages, 1 figure. See http://www.ams.org/notices/201211. arXiv admin note: text overlap with arXiv:1205.017

Against Pointillisme about Geometry

This paper forms part of a wider campaign: to deny pointillisme. That is the doctrine that a physical theory's fundamental quantities are defined at points of space or of spacetime, and represent intrinsic properties of such points or point-sized objects located there; so that properties of spatial or spatiotemporal regions and their material contents are determined by the point-by-point facts. More specifically, this paper argues against pointillisme about the structure of space and-or spacetime itself, especially a paper by Bricker (1993). A companion paper argues against pointillisme in mechanics, especially about velocity; it focusses on Tooley, Robinson and Lewis. To avoid technicalities, I conduct the argument almost entirely in the context of ``Newtonian'' ideas about space and time. But both the debate and my arguments carry over to relativistic, and even quantum, physics.Comment: 37 pages Late

A puzzle about rates of change

Most of our best scientific descriptions of the world employ rates of change of some continuous quantity with respect to some other continuous quantity. For instance, in classical physics we arrive at a particle’s velocity by taking the time-derivative of its position, and we arrive at a particle’s acceleration by taking the time-derivative of its velocity. Because rates of change are defined in terms of other continuous quantities, most think that facts about some rate of change obtain in virtue of facts about those other continuous quantities. For example, on this view facts about a particle’s velocity at a time obtain in virtue of facts about how that particle’s position is changing at that time. In this paper we raise a puzzle for this orthodox reductionist account of rate of change quantities and evaluate some possible replies. We don’t decisively come down in favour of one reply over the others, though we say some things to support taking our puzzle to cast doubt on the standard view that spacetime is continuous

A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint

The Legitimate Route to the Scientific Truth - The Gondor Principle

We leave in a beautiful and uniform world, a world where everything probable is possible. Since the epic theory of relativity many scientists have embarked in a pursuit of astonishing theoretical fantasies, abandoning the prudent and logical path to scientific inquiry. The theory is a complex theoretical framework that facilitates the understanding of the universal laws of physics. It is based on the space-time continuum fabric abstract concept, and it is well suited for interpreting cosmic events. However, it is not well suited for handling of small, local topics as global warming, local energy issues, and overall common humanity matters. We now forward may fancy theories and spend unimaginable effort to validate them, even when we are perhaps headed in a wrong direction. For example, in our times matters of climate changes are debated by politicians based on economical considerations that are as illogical as they come. The venerable paths of scientific method developed during centuries by prominent scientists and philosophers has been willingly ignored and abandoned for various and prejudiced purpose. Contact email: [email protected]

Indispensability Without Platonism

According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of metaphysics, 'Aristotelian (...) realism', in order to identify the truthmakers of mathematics. We consider in particular as a case study the indispensability to physics of real analysis (the theory of the real numbers). We conclude that it is possible to run an indispensability argument without Quinean baggage

Body, Habit, Custom and Labour

Theories in the modern age in philosophy, as well as in the discourse of the social sciences, are pervaded with the presuppositions of the dualisms of mind and world, theory and practice, private and public. These theoretical dualisms make it impossible to have an account of the interconnected nature of the experience of individuals and societies. The philosophical theoretical vocabulary to take account of the relations between these dualisms has been effaced with the legacy of Cartesian dualism. I argue that through a conceptual analysis of the body, as has been posited by Maurice Merleau-Ponty, and the related concepts of habit, custom and labour, we can reclaim some concepts that allow a mediation of these dualisms. In this article, I make a conceptual analysis of the epistemic, metaphysical and social–political interrelations between these concepts and argue for the relational role they play in our philosophical theoretical discourse

Leibniz, the Young Kant, and Boscovich on the Relationality of Space

Leibniz’s main thesis regarding the nature of space is that space is relational. This means that space is not an independent object or existent in itself, but rather a set of relations between objects existing at the same time. The reality of space, therefore, is derived from objects and their relations. For Leibniz and his successors, this view of space was intimately connected with the understanding of the composite nature of material objects. The nature of the relation between space and matter was crucial to the conceptualization of both space and matter. In this paper, I discuss Leibniz’s account of relational space and examine its novel elaborations by two of his successors, namely, the young Immanuel Kant and the Croat natural philosopher Roger Boscovich. Kant’s and Boscovich’s studies of Leibniz’s account lead them to original versions of the relational view of space. Thus, Leibniz’s relational space proved to be a philosophically fruitful notion, as it yielded bold and intriguing attempts to decipher the nature of space and was a key part in innovative scientific ideas