Este ensayo se centra en un breve pero significativo pasaje de Física, IV 13, 222a28-b7, en el que Aristóteles provee dos argumentos a favor de la infinitud extensiva del tiempo. El primero, argumenta Vigo, presenta su infinitud extensiva como dependiente de la infinitud del movimiento. El segundo argumento, en cambio, procede inmanentemente a partir de la consideración de las propiedades que el ‘ahora’ posee como límite que da cuenta tanto de la posibilidad de la delimitación como de la continuidad del (...) tiempo. El artículo también examina algunas consecuencias sistemáticas de este último argumento e intenta aclarar algunos acertijos que involucra desde el punto de vista metodológico. Tales dificultades subrayan algunos límites estructurales del intento de Aristóteles por llevar a cabo un tratamiento reductivo de las propiedades del tiempo consideradas como un modo del continuum dependiente de otros dos dominios más básicos: el movimiento y la magnitud espacialmente extensa. (shrink)

RESUMEN Este artículo apunta a someter a consideración si existe o no un fundamento histórico-filosófico para la hipótesis de una homonimia no azarosa entre "escala de tiempo" y "escala musical". Un examen minucioso de ciertos pasajes del diálogo Timeo muestra que Platón fue el primero en sentar las bases para el concepto de tiempo, definido como número, al incorporar el concepto de 'analogía' o proporción, el cual fue desarrollado por los pitagóricos en el marco místico y teórico de los principios (...) emergentes de la armonía tonal. La teoría pitagórico-platónica de proporciones subyace -como intentaré probar- en el concepto aristotélico de tiempo y, en cuanto tal, en la representación tradicional del tiempo como parámetro y escala del movimiento. ABSTRACT This article aims to unpack the query of whether or not there is a historical- philosophical basis for the hypothesis of a non-random homonymy between 'time scale' and 'musical scale.' A deeper look into certain passages of Plato's dialog Timaios, shows that he first laid the foundation for the concept of time, defined as number, by incorporating the concept of 'proportion' or 'analogy', which was developed by the Pythagoreans in the mystical and theoretical frame of the rising principles of tonal harmony. The pythagorean-platonic theory of proportion underlies -as I will try to prove- the aristotelian concept of time, and thus the traditional representation of time as a parameter and scale of movements. (shrink)

This paper reconstructs the relationship between the now, motion, and number in Aristotle to clarify the nature of the now, and, thereby, the relationship between motion and time. Although it is clear that for Aristotle motion, and, more generally, change, are prior to time, the nature of this priority is not clear. But if time is the number of motion, then the priority of motion can be grasped by examining his theory of number. This paper aims to show that, just (...) as numbers are generated by the soul, time is not presupposed by motion, but emerges through the soul’s articulation of motion. Time is co-constituted by the soul and motion. The now is the key to understanding both the contribution of motion and soul to the being of time. The now is part of the soul’s articulation of motion, and sets the stage for an act that distinguishes a unit from its underlying motion. The now, then, sets up an abstraction by which the soul generates the temporal number from motion. Reconstructing this account of abstraction allows us to formulate more strongly Aristotle’s claim to the ontological dependence of time on motion. The paper then gives a systematic overview of the relationships between the now and number in order to address the question of whether the now might be extended. It closes with an examination of the possibility that motion depends on time, and how universal time is possible. (shrink)

Perhaps the real paradox of Zeno's Arrow is that, although entirely stationary, it has, against all odds, successfully traversed over two millennia of human thought to trouble successive generations of philosophers. The prospects were not good: few original Zenonian fragments survive, and our access to the paradoxes has been for the most part through unsympathetic commentaries. Moreover, like its sister paradoxes of motion, the Arrow has repeatedly been dismissed as specious and easily dissolved. Even those commentators who have taken it (...) seriously have propounded solutions with which they profess themselves to be perfectly satisfied. So my question is: will Zeno's Arrow survive into the millennium just begun? (shrink)

Sydney Shoemaker argues that time without change is possible, but begs the question by assuming an, in effect, Newtonian absolute time, that 'flows equably' in a region in which there is no change and in one in which there is. An equally possible, relativist, assumption, consistent, it seems, with relativity theory, is that where nothing changes there is no time flow, though there may be elsewhere, where there is change. Such an assumption would require some revision of uncritical common thought (...) about time. Aristotle argues that there is no time without change but that time is not change. His arguments for the latter can be faulted both internally and again in terms of the same relativist assumption. From the Physics we can derive, though Aristotle himself did not, an argument that time is to change as geometrical space is to body: the thing itself in abstraction. (shrink)

I offer a novel solution to Zeno’s paradox of The Arrow by introducing nilpotent infinitesimal lengths of time. Nilpotents are nonzero numbers that yield zero when multiplied by themselves a certain number of times. Zeno’s Arrow goes like this: during the present, a flying arrow is moving in virtue of its being in flight. However, if the present is a single point in time, then the arrow is frozen in place during that time. Therefore, the arrow is both moving and (...) at rest. In “Zeno’s Arrow, Divisible Infinitesimals, and Chrysippus,” White suggests using an infinitesimal value as the length of the present. Contra Zeno, this allows the arrow to be moving in the present, rather than frozen in place. In this paper, I follow the basic outline of White’s solution but argue that his solution suffers from arbitrariness and a related theoretical artificiality in relation to the system of infinitesimals he invokes, viz. in relation to the hyperreal infinitesimals of nonstandard analysis. After arguing that any solution to the paradox must satisfy certain theoretical requirements, I examine White’s solution alongside two nilpotent solutions. One of these solutions is inspired by F.W. Lawvere’s Smooth Infinitesimal Analysis and the other is inspired by Paolo Giordano’s ring of Fermat Reals. I argue that Giordano’s nilpotents supply the best answer to Zeno’s paradox. (shrink)