Zeno of Elea

This article attempts to elucidate the phenomenon of time and its relationship to consciousness. It defends the idea that time exists both as a psychological or illusory experience, and as an ontological property of spacetime that actually exists independently of human experience.

I examine the reasons Aristotle presents in Physics VIII 8 for denying a crucial assumption of Zeno’s dichotomy paradox: that every motion is composed of sub-motions. Aristotle claims that a unified motion is divisible into motions only in potentiality (δυνάμει). If it were actually divided at some point, the mobile would need to have arrived at and then have departed from this point, and that would require some interval of rest. Commentators have generally found Aristotle’s reasoning unconvincing. Against David Bostock (...) and Richard Sorabji, inter alia, I argue that Aristotle offers a plausible and internally consistent response to Zeno. I defend Aristotle’s reasoning by using his discussion of what to say about the mobile at boundary instants, transitions between change and rest. There Aristotle articulates what I call the Changes are Open, Rests are Closed Rule: what is true of something at a boundary instant is what is true of it over the time of its rest. By contrast, predications true of something over its period of change are not true of the thing at either of the boundary instants of that change. I argue that this rule issues from Aristotle’s general understanding of change, as laid out in Phys. III. It also fits well with Phys. VI, where Aristotle maintains that there is a first boundary instant included in the time of rest, but not a “first in which the mobile began to change.” I then show how this rule underlies Aristotle’s argument that a continuous motion cannot be composed of actual sub-motions. Aristotle distinguishes potential middles, points passed through en route to a terminus, from actual middles. The Changes are Open, Rests are Closed Rule only applies to actual middles, because only they are boundaries of change that the mobile must arrive at and then depart from. On my reading, Aristotle argues that the instant of arrival, the first instant at which the mobile has come to be at the actual middle, cannot belong to the time of the subsequent motion. If it did, the mobile would already be moving towards the next terminus and thus, per Phys. VI 6, would have already left. But it cannot have moved away from the midpoint at the very same moment it has arrived there. This means that the instant of arrival must be separated from the time of departure by an interval of rest. I show how Aristotle’s reasoning applies generally to rule out any continuous reflexive motion or continuous complex rectilinear motion. On my interpretation, however, the argument does not apply to every change of direction. When, as in the case of projectile motion, multiple movers and their relative powers explain why the mobile changes directions, distinct sub-motions are not involved. Aristotle holds that such motions cannot be continuous, not because they involve intervals of rest, but because they involve multiple causes of motion. My interpretation of the Changes are Open, Rests are Closed Rule allows us to make better sense of Aristotle’s argument than any previous interpretation. (shrink)

NOTA PRELIMINAR: o texto a seguir representa a versão ampliada (e corrigida conforme as indicações dos pareceristas) do artigo homônimo, publicado na revista Archai em 2014. Por algum problema técnico, acabou sendo publicada, na época, a primeira versão, sem as melhorias sugeridas pelos avaliadores. Eis, então, a versão ‘definitiva’ do artigo “Zenão e a impossibilidade da analogia”: -/- A reductio ad absurdum foi elevada por Zenão de Eléia a único método que permitiria vislumbrar a verdadeira realidade, invisível tanto aos sentidos (...) quanto à maneira de pensar comum. Mostrando uma certa continuidade com os filósofos anteriores, não só na busca de um procedimento para que a especulação pudesse avançar, como também na mesma rota de afastamento daquilo que é mais próximo, conhecido e particular (visível) em direção àquilo que é menos conhecido, distante e universal (invisível), para dizê-lo em termos aristotélicos, Zenão utilizou-se de argumentos aporéticos como único caminho possível para poder entrever o ‘reino do Ser’. Esse, com efeito, é invisível não somente aos nossos sentidos, como também ao nosso raciocínio ordinário. Eis que somente a ‘via do não-ser’, a única que poderia ser trilhada após Parménides, como diz Wolff, nos permite ter uma ideia, por quanto vaga, daquilo que é ‘verdadeiramente invisível’. Tão invisível ao ponto de ser inalcançável até mesmo pelo pensamento. (shrink)

The article uses Zeno’s metrical paradox of extension, or Zeno’s fundamental paradox, as a thought-model for the mind-body problem. With the help of this model, the distinction contained between mental and physical phenomena can be formulated as sharply as possible. I formulate Zeno’s fundamental paradox and give a sketch of four different solutions to it. Then I construct a mind-body paradox corresponding to the fundamental paradox. Through that, it becomes possible to copy the solutions to the fundamental paradox on the (...) mind-body paradox. Three of them fail. But one of them – the Aristotelian one – gives us an interesting hint. Finally, this hint is pursued somewhat further and through comparison with Zeno’s fundamental paradox, the impossibility of a solution to the mind-body problem is shown again. The main new point of this article is the comparison of the mind-body problem with Zeno’s fundamental paradox. The article is a revised english version of an article published in: Allgemeine Zeitschrift für Philosophie, 23, 1998, p. 61-75. (shrink)

لا شك أن مفارقات زينون في الحركة قد تم تناولها – تحليلاً ونقدًا – في كثيرٍ من أدبيات العلم والفلسفة قديمًا وحديثًا، حتى لقد ساد الظن بأن ملف المفارقات قد أغُلق تمامًا، لاسيما بعد أن نجح الحساب التحليلي في التعامل منطقيًا مع صعوبات الأعداد اللامتناهية، لكن الفرض الأساسي لهذا البحث يزعم عكس ذلك؛ أعني أن الملف مازال مفتوحًا وبقوة – خصوصًا على المستوى الرياضي الفيزيائي – وأن إغلاقه النهائي قد لا يتم في المستقبل القريب. من جهة أخرى، إذا كانت فكرة (...) وجود العالم ذاته – مصدر مشكلاتنا الفلسفية – لا معنى لها إلا في إطار نسقٍ من التصورات التي تُصاغ بالضرورة في لغة، ومن ثم يُصبح العالم كيانًا كوّنته أساليبنا اللغوية في وصفه، فإن ما يواجهنا دومًا هو السؤال الكانطي المُعضل: إذا كانت اللغة ليست مرآة عاكسة للواقع الفعلي – أو للشيء في ذاته – وإنما للواقع كما تدركه الذات وتؤسلبه، ألا يعني ذلك أن وجودنا ذاته ووجود العالم مجرد فكرة تقطع الصلة بين العوالم الممكنة والعالم الفعلي؟. (shrink)

We outline Brentano’s theory of boundaries, for instance between two neighboring subregions within a larger region of space. Does every such pair of regions contain points in common where they meet? Or is the boundary at which they meet somehow pointless? On Brentano’s view, two such subregions do not overlap; rather, along the line where they meet there are two sets of points which are not identical but rather spatially coincident. We outline Brentano’s theory of coincidence, and show how he (...) uses it to resolve a number of Zeno-like paradoxes. (shrink)

Aspetti filosofico-matematici dei celebri paradossi di Zenone di Elea.

The perception of spatial bodies is at least in part a perception of bodily boundaries or surfaces. The usual mathematical conception of boundaries as abstract constructions is, however, of little use for cognitive science purposes. The essay therefore seeks a more adequate conception of the ontology of boundaries building on ideas in Aristotle and Brentano on what we may call the coincidence of boundaries. It presents a formal theory of boundaries and of the continua to which they belong, of a (...) sort which allows a resolution of certain Zeno-style paradoxes. The theory proves to be applicable not only in the cognitive science field but also in regard to problems relating to the ontology of geographical and geopolitical boundaries. (shrink)

Putting all of his faith in the power of abstract reason, Parmenides argues in his poem that genuine knowledge can only involve being, and that non-being is literally unspeakable and unthinkable. Using only the premise that "what is" is and what "is not" is not, he proceeds to deduce the nature of reality. The reality he arrives at bears no resemblance at all to the world we experience around us through our senses. -/- When starting out on a rational inquiry, (...) according to Parmenides, there are only two logically coherent possibilities: either you begin your inquiry with the premise that the subject of your inquiry exists or you begin with the premise that it does not exist. (shrink)