Gunky time and indeterminate existence

The paper criticizes an argument recently presented by Ross Cameron. The argument purports to show that, if time is gunky (that is, if there are no time atoms), and if changes in existence are underwritten by events of coming to be, then there are cases of indeterminate existence. The putative reason is that, if time is gunky, then events of coming to be cannot be instantaneous, and hence, changes in existence must be gradual, non-clear-cut. The paper argues that this argument conflates two different readings of \u201cevent of coming to be\u201d. Under one reading, the argument is unsound. Under the other, the argument is valid only if a further, nontrivial premise is added, which concerns the relation between time atoms, instants, and instantaneous events

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

The logic and topology of Kant's temporal continuum

In this article we provide a mathematical model of Kant?s temporal continuum that satisfies the (not obviously consistent) synthetic a priori principles for time that Kant lists in the Critique of pure Reason (CPR), the Metaphysical Foundations of Natural Science (MFNS), the Opus Postumum and the notes and frag- ments published after his death. The continuum so obtained has some affinities with the Brouwerian continuum, but it also has ‘infinitesimal intervals’ consisting of nilpotent infinitesimals, which capture Kant’s theory of rest and motion in MFNS. While constructing the model, we establish a concordance between the informal notions of Kant?s theory of the temporal continuum, and formal correlates to these notions in the mathematical theory. Our mathematical reconstruction of Kant?s theory of time allows us to understand what ?faculties and functions? must be in place for time to satisfy all the synthetic a priori principles for time mentioned. We have presented here a mathematically precise account of Kant?s transcendental argument for time in the CPR and of the rela- tion between the categories, the synthetic a priori principles for time, and the unity of apperception; the most precise account of this relation to date. We focus our exposition on a mathematical analysis of Kant’s informal terminology, but for reasons of space, most theorems are explained but not formally proven; formal proofs are available in (Pinosio, 2017). The analysis presented in this paper is related to the more general project of developing a formalization of Kant’s critical philosophy (Achourioti & van Lambalgen, 2011). A formal approach can shed light on the most controversial concepts of Kant’s theoretical philosophy, and is a valuable exegetical tool in its own right. However, we wish to make clear that mathematical formalization cannot displace traditional exegetical methods, but that it is rather an exegetical tool in its own right, which works best when it is coupled with a keen awareness of the subtleties involved in understanding the philosophical issues at hand. In this case, a virtuous ?hermeneutic circle? between mathematical formalization and philosophical discourse arises

Regions-based two dimensional continua: The Euclidean case

We extend the work presented in [7, 8] to a regions-based, two-dimensional, Euclidean theory. The goal is to recover the classical continuum on a point-free basis. We first derive the Archimedean property for a class of readily postulated orientations of certain special regions, “generalized quadrilaterals” (intended as parallelograms), by which we cover the entire space. Then we generalize this to arbitrary orientations, and then establishing an isomorphism between the space and the usual point-based R × R. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause” (to the effect that “these are the only ways of generating regions”), and we have no axiom of induction other than ordinary numerical (mathematical) induction. Finally, having explicitly defined ‘point’ and ‘line’, we will derive the characteristic Parallel’s Postulate (Playfair axiom) from regions-based axioms, and point the way toward deriving key Euclidean metrical properties

Some paradoxes of infinity revisited

In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirah ̃a, working with only three numerals (one, two, many) can help us to change our perception of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive ligh

A Meaning Explanation for HoTT

The Univalent Foundations (UF) offer a new picture of the foundations of mathematics largely independent from set theory. In this paper I will focus on the question of whether Homotopy Type Theory (HoTT) (as a formalization of UF) can be justified intuitively as a theory of shapes in the same way that ZFC (as a formalization of set-theoretic foundations) can be justified intuitively as a theory of collections. I first clarify what I mean by an “intuitive justification” by distinguishing between formal and pre- formal “meaning explanations” in the vein of Martin-Löf. I then explain why Martin-Löf’s original meaning explanation for type theory no longer applies to HoTT. Finally, I outline a pre-formal meaning explanation for HoTT based on spatial notions like “shape”, “path”, “point” etc. which in particular provides an intuitive justification of the axiom of univalence. I conclude by discussing the limitations and prospects of such a project

Brentanian continua and their boundaries

This dissertation focuses on how a specific conceptual thread of the history of mathematics unfolded throughout the centuries from its original account in Ancient Greece to its demise in the Modern era due to new mathematical developments and, finally, to its revival in the work of Brentano. In particular, we shall discuss how the notion of continuity and the connected notion of continua and boundaries developed through the ages until Brentano's revival of the original Aristotelian account against the by then established mathematical orthodoxy. Thus, this monograph hopes to fill in a gap in the present state of Brentanian scholarship as well as to present a thorough account of this specific historical thread