Aristotle, the Being, the Infinite and Physics

Aristotle, the Being, the Infinite and Physics by Martha de Aratanha Thesis Advisor Prof. Doctor Tito Marques Palmeiro  Introduction This monograph compares Aristotle's thinking with 21st-century Physics. This comparison is generally considered derogatory for Aristotle, but we will try to show that the notions he developed are still meaningful, and show their potential importance to contemporary physics thinking. In summary, we shall make an effort to recover Aristotle for the twenty-first century. Greek thought "... operated the decisive passage from utilitarian technique and myth to pure and disinterested science; it first stated systematically the logical demands and the speculative needs of reason: it is the true creator of science as a logical system and of philosophy as rational consciousness and solution for the problems of universal reality and life." E. Zeller, Die Philosophie der Griechen in ihrer geschichtlichen Entwicklung, 1919. Italian translation by R. Mondolfo, La Filosofia dei greci nel suo sviluppo storico, 1943. Zeller-Mondolfo, I, 1, p. 99. [Greek Philosophy in its historical development] By virtue of the spirit which created Philosophy, that is, of its speculative nature, the Greeks qualitatively transformed the knowledge they received from the surrounding cultures such as the Egyptians, Babylonians, and Chaldeans. Each of these peoples had already developed mathematics, numbers, and astronomy but it was the Greeks who brought the speculative method to the field of logical, methodical and rational argumentation. The Greeks shrewdly postulated, 2500 years ago, that there was a reality behind that which we perceive through our senses. The Pre-Socratics, called physiologoi, investigated the physis to understand "how are the things from the indeterminate (απειρον, apeiron) born, through which process and why?" G. Reali, História da filosofia antiga, vol. I, p. 55. [History of Ancient Philosophy] Thus arose the Philosophy of Physis, which seeks to explain the totality of things, that is, all of reality, using the logical argument, the logos. It is important to note that the physiologoi affirm that element is "... that of which all things consist, from which they first come and into which on their destruction they are ultimately resolved, of which the essence persists although modified by its affections." Aristotle, Metaphysics, 983b 7-11. Therefore, "there is some one entity (or more than one) which always persists and from which all other things are generated." Aristotle, Metaphysics, 983b 17-19. This is the line that Aristotle will seek in his investigation of the Being. It is worth noting that this principle was called physis, meaning, as we shall see, principle, and only later did it acquire the modern sense of Nature. The Sophists relegated the investigation of physis to the background because they found impossible to solve the aporias resulting from the speculations of the Pre-Socratics, such as Zeno's paradoxes on motion. They inaugurate a period where philosophical inquiry shifts its focus to Man, which was deepened by Socrates and Plato, questioning ethical relations, societies, and laws. Aristotle takes an interesting position in this regard, for he not only continues research on Man, but also resumes Pre-Socratic research on physis. The concept of becoming is fundamental in his conception of physis, for it is the way he finds to get out of the aporia of the Eleatic school. In order to circumvent the impasse between the immobile one and the mobile plurality that we experience, discussed by Plato in his analysis of Parmenides, Aristotle develops the concepts of hylé (υλη, unformed matter), eidos (ειδός, form), dynamis (δύηαμις, potency), entelecheía (εντελεκεία, actuality) and energeía (ενεργεία, act). During the Renaissance, man goes back to researching Nature, and philosophical argumentation distances itself from the dogmas of faith and their revealed truths, but this time with a singular modification: it does so aided by instruments. With the invention of the telescope in 1610, man begins to extend one of his senses and “see” far beyond his normal vision. He was able to see another reality behind appearances. The question of aletheia, Aletheia, from the Greek αληθεια, means Truth (Bailly, p. 76). as a search for the truth behind appearances, will now be realized not only with pure thought, but also with the aid of instruments. Aristotle's thinking is neither superfluous nor trivial. His treatise on Physics is currently ignored or underestimated, but the concepts developed there were and are fundamental to today's physics. In particular, as Martin Heidegger said, Aristotle’s Physics is the hidden, and therefore never adequately studied, foundational book of Western philosophy. On the Being and Conception of Physis in Aristotle’s Physis B, 1, in Man and World, 9, nº3, 1976, p.224. His work comes to us through manuscripts that seem to be lecture notes, thus written from the memory and understanding of others. Moreover, the Greek spoken in Athens was especially brought to us through Latin translations and, from these translated into other languages, a process prone to introduce deviations and create misconceptions. One shall add to this the innumerable “amendments” and “corrections” made by commentators over the centuries. The problems of translating the Physics are many. The precise philological knowledge of that period’s Greek is not enough, a knowledge of Mathematics is equally essential. And many of the concepts, intuited and discussed by Aristotle, were only formally demonstrated by science centuries later. The purpose of this monograph is to show that Aristotle's revolutionary ideas are pertinent and current even in the twenty-first century and have not yet been fully understood. We shall start by explaining the concept of phenomenon, as understood in his time, but in the light of today's technologies. Next, we will show how Aristotle presents and develops the discussion about the Being, and how things arise from non-being, a discussion then ongoing for 400 years already. And we shall finally see, how he, with a genius’ insight, engendered a totally original conception of the Cosmos. Analyzing the theories proposed by the philosophers who preceded him, Aristotle constructs a new relation between hylé and eidos, and modifies the idea of telos, introducing the revolutionary concepts of entelecheia (actuality) and dynamis (potential). To this end, using mathematical concepts, he analyzes the differences between real numbers and natural numbers, and develops, from this discussion, the application of apeiron to the physical world of actual bodies. Moreover, he rejects the concept of place and void as being a rigid space, a “little box” that remains after a body leaves it. Finally, he analyzes what is time and asks in what way it is infinite. We shall see that all these concepts modified by Aristotle, which have been so misinterpreted, still resonate with the latest advances in present-day Physics. 2. Development 2.1 What is the meaning of the world of appearances (phenomena) What is the appearance of the world, the phenomena, this World that we see through our senses? The scientific answer will depend on our “visual apparatus”. Let’s try to understand what we mean by that. Electromagnetic radiation presents a wide range of frequencies. Today’s instruments can detect only a delimited range, between gamma rays, waves of ultra-high frequency (indicated on the left side of Figure 1) and ultra-low frequency long radio waves (indicated on the right side of Figure 1). The only difference between these several kinds of light is their wave frequency. Human beings only “see” in a range of frequencies called visible light; other live beings “see” the world in other frequencies. fig.1: Electromagnetic radiation and visible light. Visible light covers just a very small portion of the electromagnetic radiation spectrum. Human beings see their world using only frequencies between ultraviolet and infrared, as shown on Figure 1. With this we perceive colors, textures and volumes (Figure 2). Bees, flies, fishes and birds see the world using also the ultraviolet frequency. Their world looks like Figure 3. fig. 2: Light in visible light frequency fig.3: Light in ultraviolet frequency If we could imagine an animal looking around using X-ray frequencies, it would see the world under a different light, as in Figure 4. What if your “visual apparatus” perceived your surroundings using infrared, like snakes do? You would see the world like Figure 5. fig. 4: Light in X-ray frequency fig.5: Light in infrared frequency Therefore, depending on your “visual apparatus”, the world presents itself differently, and a different reality will be perceived. With all this, we do not reach the final answer about reality, for we can still descend to the level of atoms and electrons, pure energy or dark energy, so called by being “invisible” to us and our current measuring instruments.. The point to be emphasized here is that the Being, which is described in various ways, does not present itself equally to all living beings. The kind of access to “reality” that a living being has depends on the frequency of the electromagnetic radiation it sees. In short, it will only “see” what is “perceivable” by it, leaving another “reality” behind appearances. It may seem surprising to convene notions of science in a work of Philosophy, but Aristotle in his Physics is doing Science, and investigating how, with it, one arrives at the knowledge of the Being. The discussion of the Being is conditioned by what one can perceive about the world around us. Aristotle perfects, modifying, the works of other Greek thinkers that preceded him. He starts from the ideas synthesized and described by Plato in The Republic, when he discusses how to attain the knowledge of the Being, using three narratives: “the Analogy of the Sun”, “the Analogy of the Divided Line “ and “Allegory of the Cave”. The understanding of the Aristotelian thought construction is based, therefore, in the understanding of the precedent Greek thinking, that he analyzed and refuted. The Being: What does it mean The fundamental problem of the Philosophy of Nature is to explain how the bodies that we perceive come into being, that is, the things that surround us, our planet and the Cosmos or, as summarized by Auguste Mansion: “Voilá donc le problème fondamental de la philosophie de la nature: expliquer le devenir cosmique”. Mansion, Auguste. Introduction à la Physique aristotélicienne, p. 24. Using the book História da filosofia antiga, Reale, Giovanni. História da filosofia antiga, vol. I. as a source, we will briefly show how the question of the Being and the knowledge of it, was treated along the centuries. The first text to reach us with an attempt to explain the creation of the world, but still relying on the gods, was that of Hesiod, Theogony, first known text about mythical Greek Cosmogony (Hesiod, 750 - 650aC). with the creation of the Cosmos from Chaos. Next, the physiologoi, headed by Thales, so-called for attributing one or more elements to the prime and ultimate reality of things, affirmed that the principle of nature is matter. There is also Anaximander, who attributed the principle of all things to the apeiron. In sequence, Parmenides and Zeno, both Eleatics; and soon after, the atomists Leucippus and Democritus, among successive others. All of them, each in his own way, discussed how to arrive at the definition and explanation of the Being. Protagoras is considered to be the first sophist and, by his time, the discussion of this theme was paralyzed by the insurmountable paradoxes of Zeno of Elea. With the success of Sophists, the philosophy of Nature was set back, and “...contribuiu de modo essencial para pôr definitivamente em crise ... as pretensões da especulação de viés naturalista” […essentially contributed to jeopardize … the pretentions of naturalistic speculations]. Reale, Giovanni. História da filosofia antiga, vol. I, p.171. By the time of Socrates and Plato we read this age long debate in Plato’s dialogue Theaetetus or about knowledge that begins by analyzing Protagoras to understand if knowledge is sensation, to later on analyze two contradictory arguments: those of Heraclitus and Parmenides. On one hand, Heraclitus affirms that one cannot know something if that thing is always changing; in other words, if all is flux Theaetetus, 181a. “I think, then, we had better examine first the one party, those whom we originally set out to join, the flowing ones (τους ρέοντας), and if we find their arguments sound, ...” Bailly, p.1715, ρέος, péos, river, stream. Metaphysics, XIII, iv, 1078b 13-15. how can we know anything? On the other hand, in that same dialog, Socrates quotes Parmenides, to whom change is impossible. ...others teach the opposite of this, for example, ‘So that it is motionless, the name of which is the All [Didactic Poem, fragment 8],’ ... that everything is one and is stationary within itself, having no place in which to move. Theaetetus, 180e. Available at http://www.perseus.tufts.edu. Accessed on June 14, 2018. But then, how the One becomes the Multiple? Again, since “one” is itself quite as ambiguous a term as “existent”, we must ask in what sense the “All” is said to be one. (a) Continuity <συνεχες, synekés, continuous> establishes one kind of unity; (b) indivisibility <αδιαίρετον, adiaíreton, indivisible> another; …So then, (a) if a single continuum is what is meant by “one”, it follows that “the One” is many, for every continuum is divisible without limit. …Whereas (b) if “one” signifies an “indivisible”, then it excludes quantity; and therefore, the One Being, not being a quantum, cannot be “unlimited” as Melissus declares it to be; nor indeed “limited” as Parmenides has it, for it is the limit only, not the continuum which it limits, that complies with the condition of indivisibility. Physics, I, ii, 185b 7-19. Aristotle uses these words, continuous and indivisible, in a strict mathematical sense. In this passage, while arguing how the One becomes the Multiple, Aristotle discusses language and its ambiguities, and how Mathematics, which also articulates itself as a language, can be applied to the investigation of the Being in physis. Mathematics, for Aristotle, is not transcendent as in Plato, it is immanent to physis, and becomes an essential instrument to understand it. In this way, he extends the process of the divided line, including the discussion of the passage into infinity as a fundamental step for the understanding of the Being. Aristotle will return to a discussion that was already 400 years old, and divide it into two works: The Being susceptible to generation, corruption, and locomotion is investigated in the Physics, and the Being as immobile and eternal is investigated in the Metaphysics. In this monograph we shall analyze the first four books of the Physics, a work composed of eight books. In the first two, Aristotle discusses the general principles of the Being and establishes the definitions that will be used. He retakes Plato's Analogy of the Divided Line, Plato. The Republic, Divided Line Analogy, [509c a 511e], analyzed below. as we see in the opening lines of the Physics quoted below, stating that we can know the world from the things that are closest to us, and then go toward its principles. We must go from the phenomena accessible to our senses to the concepts that lead to principles, therefore, going from that which can be sensed to the intelligible. In all sciences that are concerned with principles or causes or elements, it is acquaintance with these that constitutes knowledge or understanding. For we conceive ourselves to know about a thing when we are acquainted with its ultimate causes and first principles, and have got down to its elements. Obviously, then, in the study of Nature too, our first object must be to establish principles. Now the path of investigation must lie from what is more immediately cognizable and clear to us, to what is clearer and more intimately cognizable in its own nature; for it is not the same thing to be directly accessible to our cognition and to be intrinsically intelligible. Aristotle. Physics, I, i, 184a 10-19. The introduction of Physics returns to Plato’s Divided Line Analogy. In the passage above it is very important to note that the word physis was translated as “Nature”; but for Aristotle it meant “in itself”, “the internal activity that makes anything what it is.” Sachs, Joe. Aristotle’s Physics: a guided study, p. 250. Only later, with Modernity, did it come to mean Nature, or the natural world. The search for the principle that generates all things is the search for the Being. Plato, in the Republic dialogue, uses three analogies to explain how to arrive at the knowledge of the Being. These three analogies describe the same thing: how to pass from the world of appearances to reality. These are the operations of the mind (ψυχη, psyché) Plato, Republic, VI, 511a. to arrive at the truth behind appearances. 2.2.1 The Divided Line Analogy Socrates, in the Analogy of the Divided Line, Plato, Republic, VI, 509d-5011e. exemplifies different degrees of clarity and reality, of how to go from the world of appearances to the Being. Let us imagine a line divided into four segments, where the first two consist of visible bodies accessible by the senses, of which, therefore, the mind can form an image (εἰκόνων, icon), and the latter two are of intelligible bodies, therefore accessible only by the intellect. Without entering into the merits of Plato's theory of Ideas and its developments, let us focus here only on the description of this analogy, which Aristotle will take as the starting point of his Physics. Visible World όρατόν: visible Invisible World νοητόν: intelligible supposition beliefs understanding intuitive science reflexions from bodies bodies themselves mathematical hypotheses based on bodies mathematical hypotheses based on mathematical hypotheses Allowing access to the non-hypothetical, according to Plato. In the first segment, Socrates poses that knowledge is analogous to the reflex of bodies in the water or in a mirror. In this segment knowledge is confused and obscure, just as the reflection of a body is unclear in relation to the body itself. Plato calls this knowledge supposition, conjecture (εἰκασία, eikasía, representation). In the second segment, Socrates makes an analogy between knowledge and the very bodies we see in the world around us. Seeing the bodies proper is sharper than seeing their shadows, as in the previous segment. In this segment, knowledge is less confusing, much sharper in contours. Plato calls this knowledge belief (πίστις, pístis, conviction), Republic, 511e. for there is still no questioning of what we perceive. In the third segment we make use of reasoning, and the mind "starting from the hypothesis and, discarding the images from the previous segment, goes on only with the aid of ideas." Republic, 510b. This is knowledge though the causes, where the mind seeks to know the causes of what is visible through reason, but still starting from things we sense. This is how geometers proceed when formulating hypotheses based on the design of a triangle but, abandoning the drawing, begin to formulate hypotheses only with ideas. We can call this knowledge comprehension, understanding (διάνοια, diánoia). Republic, VI, 511e. In the fourth segment, Socrates continues the path to the knowledge of the Being using the power of dialectics (διαλέγεσθαι δυνάμει, dialégestai dynámei); Republic, VI, 511b: “…making from the hypotheses steps to go to that which does not admit hypotheses, which is the principle of everything and, attaining that, descend, fixing itself in all the consequences that follow, until arriving at the conclusion, without using any data from our senses, but moving from ideas to ideas and ending with ideas.” this is the understanding achieved by generating hypotheses from other hypotheses, without needing the support of actual bodies. We can call this knowledge intuitive science, reason, intellection (νοῦς, nous). Republic, VI, 511d. In this last segment the Being ceases to be accessible by the senses, and becomes accessible only by the mind, using the reason and power of dialectics. Aristotle in the first lines of his Physics is just stating this, as we have seen above. 2.2.2 How Mathematics helps us to reach the Being Later on, in the Republic, Socrates affirms that it is necessary to teach Calculus and Arithmetic to the guardians, ... until they attain to the contemplation of the nature of number, by pure thought... for facilitating the conversion of the psyche itself from the world of generation to essence and truth. Plato, Republic, VII, 525c. This requirement is imposed because "Geometry ... is the knowledge of the eternally existent, not of what is generated or destroyed", Plato, Republic, VII, 527b. meaning that Mathematics deals with eternal truths, and abstracts itself from physical matter. Therefore, calculus, arithmetic, and geometry are recommended by Plato to develop the capacity for abstraction necessary to intuit the Being. The decisive point is that the Being is only attained by the mind, not being material or perceivable by our senses; and for this task Mathematics provides the skills of abstraction. In his Physics, Aristotle makes a distinction between what is physical and what is purely mathematical. Also criticizing Plato, he states: Now the exponents of the philosophy of “Ideas” also make abstractions, but in doing so they fall unawares into error; for they abstract physical entities, which are not really susceptible to the process as mathematical entities are. Physics, II, ii, 193b 37 – 194a 2. Aristotle criticizes Plato, distancing himself from the concept of Ideas and Forms. ... Nature...can neither be isolated from the material subject in which it exists, nor is it constituted by it. Physics, II, ii, 194a 13-15. Nature (φύσις) is composed of hylé (formless matter) and eidos (form), and physicists must always consider them together. To separate them is to neglect the movement and the change that takes place in things, like birth, growth and qualitative changes. Aristotle considers nature as causal, and so subject to change; therefore, not immutable like Mathematics, but is eternal in another way. Physics is the domain of what moves, but as Nature can be expressed by Mathematics, there will come a certain point from which Physics will become immaterial, and not accessible to the senses, only to the mind. In parallel, Mathematics studies bodies that, although having no existence separate from the natural bodies, can be studied abstracting this movement that we call Nature. We quote Aristotle: Physicists, astronomers and mathematicians, all have to deal with lines, figures and the rest. But the mathematician is not concerned with these concepts qua boundaries of natural bodies, nor with their properties as manifested in such bodies. Therefore he abstracts them from physical conditions; for they are capable of being considered in the mind in separation from the motion of the bodies to which they pertain, and such abstraction does not affect the validity of the reasoning or lead to any false conclusions. Physics, II, ii, 193b 32-36. Mathematics abstracts itself from physical bodies but its development interacts with that of Physics. Many mathematical discoveries were caused by problems from Physics. For example, the development of Fourier Series Theory and Theory of Partial Differential Equations due, in chronological order, to the problem of heat propagation in a wire, problems of Electromagnetism (Maxwell), problems of elasticity in structures, etc. And vice-versa, new discoveries in Mathematics facilitated the advancement of solutions in Physics, for example, the application of Riemannian geometry to Physics in the twentieth century. Or as Paul Dirac states, Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematicians find interesting are the same as those which Nature has chosen. Paul Dirac. The Relation between Mathematics and Physics, 1939. Proceedings of the Royal Society (Edinburgh) Vol. 59, 1938-39, Part II pp. 122-129. Still related to Mathematics, it is necessary here to explain a very important specific point in the development of Aristotle’s thought in the physis: it is the distinction between mégethos, magnitude belonging to an ordered continuous, non-enumerable set, as for example, the line of real numbers; and arithmós, a concrete unit belonging to a discrete, ordered, enumerable set, such as the set of natural numbers. The difference between a continuous set (μέγεθος, mégethos, magnitude) and a discrete set (αριθμός, arithmós, number) is crucial. The real numbers (1; ; 1,67; 2,3; 2,345; 2,35678; Ω; ...) form a continuous set (mégethos), for wherever the line of real numbers is cut off, there one finds a real number, in other words, the set of real numbers is infinitely divisible. The same does not happen with natural numbers. The set of natural numbers (1, 2, 3, 4, ...) is formed by discrete, isolated units, so one goes from one to the other in discrete steps. And you can always add another number, it can grow indefinitely. So the set of natural numbers is infinitely increasing. In short, they are different kinds of infinities: the set of real numbers is infinitely divisible, and the set of natural numbers is infinitely increasing. This distinction will be important to understand how Aristotle will apply the different infinities to the Physics of natural bodies. Another difference between mégethos and arithmós is the counting unit. To count discrete bodies, for example, horses or men, we use one horse or one man as the unit of counting. That means we automatically have a counting unit, one man or one horse. On the other hand, to measure a continuous quantity, we need to arbitrate a standard with which to measure this quantity. For example, we can measure an extension in meters, feet, or miles, but for that we need to establish a standard to measure that length, and that choice is always arbitrary. This distinction between mégethos and arithmós will be important to understand the discussion about movement, time and the relation between them, as we will see below. These two different infinities are going to be formally worked out by Cantor in the nineteenth century: Cantor’s work between 1874 and 1884 is the origin of the set theory. Prior to his work, the concept of a set was a rather elementary one that has been used implicitly since the beginning of mathematics, dating back to the ideas of Aristotle. Available at https://en.wikipedia.org/wiki/Georg_Cantor. Accessed on June 14, 2018. The Set Theory proposed by Cantor in 1874 was criticized by important mathematicians of the time but became a fundamental theory in modern mathematics as it unifies propositions of several areas of Mathematics in a single theory (likewise, around the same time, Physics unified several areas). Cantor has proved that there are infinities of different “sizes”: natural and rational numbers are infinite and countable, and real numbers, though they are also infinite, are not countable. By definition, a set A is countable or enumerable if and only if there is a bijection between A and a subset of the natural numbers. In Mathematics, it has been noted that there is a great qualitative difference between a potential infinite sequence of discrete elements and the succession of points of a line segment, what is called a continuous line. In the first case we can always add one more element, taking another step to the next element. A sequence is infinitely extensible. In the case of the continuum it does not make sense to speak of the following element: between a certain point and a posterior one, as close as you wish, it is always possible to find an intermediate point, and so on, to infinity. A continuous segment is infinitely divisible. This second type of infinity raises great questions about actual infinities, since it starts from a given whole (the straight line) which can contain an infinite number of elements. Infinity in act seems to be a necessary property of the continuum. Available at https://pt.m.wikipedia.org/wiki/Infinito. Accessed on June 14, 2018. Cantor's work opened a schism among mathematicians, for many did not admit the concept of hierarchies between infinities since it generates several paradoxes. These issues were only framed by Gödel, 60 years later. It is important to note that in Mathematics this discussion is known as the Continuum Hypothesis, and that it is the subject of numerous controversies even today. The Continuum Hypothesis, which says that the set of real numbers has no voids, depends on whether an extremely powerful axiom is true: that is the Axiom of Choice or its equivalent known as the Zorn’s Lemma. Behind it is the ability to perform operations on infinite sets without recourse to the intuitive recursive method. Gödel's Theorem is one of the most profound results in Mathematics. He demonstrated that if Mathematics is inconsistent, that is, if there is a proposition that is both false and true when the Axiom of Choice is taken as true, then it will also be inconsistent if we assume it to be false. The Continuum, which Aristotle calls mégethos, allows him to determine the place occupied by a body, and is the focal point of his physis. The Continuum Hypothesis depends directly on the Axiom of Choice. This difficulty, the difference of the infinite natural numbers to the infinite real numbers, is perceived and discussed by him. In an absolutely brilliant way, using none the modern mathematical tools, he was able to intuit the implications and complications, and apply them to the Being. Surprisingly, Aristotle will apply both infinities to Physics in an original way, and will influence the development of Mathematics for the next centuries. How Language helps us to reach the Being Aristotle understands that our knowledge passes through language, involving the problem of how to express the world around us. He often reminds us of the importance of precision in the use of concepts in philosophical discussions. First it is necessary to define the word that is being discussed to make sure that we are talking about the same thing, otherwise we will be discussing parallel concepts. He always investigates in what different ways a term is said, to arrive at an unequivocal language accuracy. For example, when he investigates what apeiron means (unlimited or infinite), he analyzes minutely in how many different ways we use this term, Physics, III, iv to viii. and then analyze its meaning in Physics, because the inadequate or imprecise use of language leads to errors that we do not even detect. In this context, we will analyze some concepts redefined by Aristotle in his Physics. Hylé: the hylé (υλη) is a substrate (υποκειμένη υλη, hypokeiméne hylé: substrate) devoid of form (eidos). Only after acquiring eidos, hylé turns into something material, an ousia (substance). Ousia: is the subject that acts and is acted upon, the body that exists as actual matter, signifying the hylé that acquired form. Energeia: the word energeia Dictionnnaire Grec-Français, Bailly, p. 676. is the process of the movement of all things in the Cosmos; it is the Being-in-act, force in action that may not come to term; as opposed to dynamis, the force in potency. Dynamis: potency (δυναμις), innate tendency of anything to be at work. “A potency in its proper sense will always emerge into activity when the proper conditions are present and nothing prevents it”. Sachs, Joe. Aristotle’s Physics: a guided study, p.252. Metaphysics, 1047b 35 – 1048a 16. Entelecheia: ousia and energeia merge into one single idea of entelecheia, being-in-act-striving-to-persist-towards-an-objective. The word entelecheia, invented by Aristotle, is the fusion of the idea of persistence (endelecheia), purpose (telos) and effort (echein), is the active and effective state of the Cosmos. In the Metaphysics, the meaning of energeia and entelecheia converge [1047a 30-31; 1050a 21-23]. It is the existence of an ousia, and it is also the existence of the movement of the Cosmos, it is the effort to persist in existence. These concepts are important in understanding the development of Aristotle's thinking, and for this reason we will discuss them in more depth in the sequence. Aristotle’s revolutionary concepts Aristotle’s approach is revolutionary for introducing a Pure Mathematics reasoning, devoid of mysticism, to analyze problems that were secular already. These new interpretations will change the way of knowing and apprehending reality, in other words, the Being. This is how it associates hylé and dynamis, movement and apeiron, hylé and apeiron, dynamis and entelecheia, time and movement, as we will see below. Hylé, Eidos and Telos (formless matter, form and purpose) Aristotle, in his Physics, retakes from Plato’s Timaeus Plato. Timaeus. 50c. “The matter to be molded, he adds, must be entirely formless before the forms are impressed on it”. the concept of formless matter, the formless matter from which everything comes, and to which everything returns, as we see in the passage ...although they go so far with us as to recognize the necessity of some underlying subject, yet in truth the “great” or “small” Great: infinitely increasing; and small: infinitely divisible. They are two different types of infinities. of which it consists can only be equated with our “matter” ... Physics, I, ix, 192a 9-11. and will consolidate this in the hylé (υλη) concept, a “matter-substract” before acquiring “form” (ειδός, eidós). First, Aristotle analyzes whether nature is matter, and whether it is formed by the elements water, air, fire, or earth, as postulated by the early philosophers. He then argues that nature is more form (eidos) than matter (hylé). Finally, he postulates that nature is form (eidos). Let's see how he develops this argument. Aristotle begins, as always, analyzing what other philosophers have said before him: “Now some hold that the nature and substantive existence of natural products reside in their material on the analogy of the wood of a bedstead ...” Physics, II, i, 193a 10-13. The physiologoi argued that if we buried a bedstead and the sap was still present in the wood, a tree shoot would emerge from it, not a bedstead. With this analogy, they affirmed the elements as the principle of physis. But Aristotle countered: “This then is one way of regarding nature (physis)”, Physics, II, i, 193a 28-30. that is, as a substrate formed by one or more eternal elements, while everything else entered and exited from existence. But on the other hand, Aristotle argues that we can think of Nature as form (ειδος, eidos), that is, “the kind of thing it is by definition”. Physics, II, i, 193a 30-31. Aristotle here introduces the concept of dynamis (potency), presenting an analogy between art (τέχνη, techne) and nature (physis) For as we give the name of “art” to a thing which is the product of art and is itself artistic, so we give the name of “nature” to the products of nature which themselves are natural. And as in the case of art, we should not allow that what was only potentially a bedstead and had not yet received the form of a bed had in it as yet any art-formed element, or could be called “art”, so in the case of natural products; what is potentially flesh or bone has not yet the “nature” of flesh or bone until it actually assumes the form indicated by the definition that constitutes it the thing in question, nor is this potential flesh or bone as yet a product of nature. Physics, II, i, 193a 32 - 193b 4. So the wood, which is potentially a bedstead, only becomes a bedstead after a acquiring form through the techné. So it also happens in physis, Aristotle argues, for only after the hylé acquires form is that the products of nature become. Before that, hylé exists only in potential (dynamis) and is not accessible to our senses. With this, then, he will postulate another way of defining Physis in relation to his contemporaries, considering it immanent: These considerations would lead us to revise our definition of nature as follows: “Physis is the distinctive form or quality of such things as have within themselves a principle of motion, such form or characteristic property not being separable from the things themselves, save conceptually”. Physics, II, i, 193b 3-6. This definition emphasizes form, at the same time including matter and movement. Hylé acquires concreteness and purpose only after finding a form, before it exists only in potential (dynamis). In other words, something immaterial and unlimited (hylé) becomes concrete, countable, leaves the potential existence for concrete existence, with form. It leaves the apeiron, indiscriminate and unaccountable, for the countable, leaves the infinite not enumerable to the enumerable infinite, leaves the real numbers and appears as a natural, countable number. Eidos and hylé are characteristics of the Cosmos, independent from the mind. Available at https://plato.stanford.edu/entries/aristotle. Accessed on April 25, 2018. By connecting the hylé to something in potential and further on to the apeiron, Aristotle is modeling a dynamic and constantly moving Cosmos that follows a cosmic time, independent of the duration experienced by us humans, as Spinoza analyzes in his Letter XII to Louis Meyer, also known as the Letter on the Infinite. Letter no. 12, to Lodewijik Meijer, 1663. Aristotle goes on to inform us that physis is etymologically equivalent to genesis, and is even used as a synonym: See comment in Physics, p.114. Also in Latin, na-tura derives from na in nascor and na-tivitas. ... nature, then, qua genesis proclaims itself as the path to nature qua goal. …for that which is born starts as something and advances or grows towards something else. Towards what, then, does it grow? Not towards its original state at birth, but towards its final state or goal. Physics, II, i, 193b 17-19. Therefore, nature is form, stating that physis itself has in form its principle of motion. We see that Aristotle completely transforms the definition of previous thinkers and introduces his own, strongly based on the concepts of Mathematics. Causes Ενα μεν ουν προτον αιτιον…, “These are the prime causes…” The question about nature, which takes up and transforms the understanding of the physiologoi, will be accompanied by an important discussion about our knowledge of it. Aristotle will analyze how we claim to know things. To understand one thing, we must seek to know the “how and why” of this thing. Physics, II, iii, 194b 19-21. In Physics, therefore, one must investigate “the how and why” of things that enter and exit existence, and investigate the essential constituents of motion, so as to trace back to the principles of bodies. Physics, II, iii, 194b 21-23. Aristotle seeks the causes of things, and their principle of motion. We have next to consider in how many senses “because” may answer the question “why”. For we aim at understanding, and since we never reckon that we understand a thing till we can give an account of its “how and why”, it is clear that we must look into the “how and why” of things coming into existence and passing out of it, or more generally into the essential constituents of physical change, in order to trace back any object of our study to the principles so ascertained. Physics, II, iii, 194b 16-24. Aristotle then establishes and names the four causal determinants Physics, II, iii, 194b 23-36. for physical bodies to exist. He argues that "... the existence of material <ενυπάρχοντος, enypárchontos, substrate> for the generating process to start from is one of the essential factors we are looking for." Physics, II, iii, 194b 24-27. Where matter is an immanent cause to the body, in contrast to a movable cause external to it. Then, he presents the formal cause, for "naturally, the thing in question cannot be unless the material has actually received the form <ειδος, eidos> or characteristics of the type, conformity to which brings it within the definition of the thing we say it is ..." Physics, II, iii, 194b 27-30. Aristotle considers matter as “something deprived of” (στέρησιν, stéresin, privation) and looking for a form (ειδος, eidos). Physics, I, ix, 192a 4-7. It is only after receiving a form that something becomes countable and therefore, it can be named. In the sequence, he presents the efficient cause, Then again, there must be something to initiate the process of the change or its cessation when the process is completed, such as the act of a voluntary agent (of the smith, for instance), or the father who begets a child; or more generally the prime, conscious or unconscious, agent that produces the effect and starts the material on its way to the product, changing it from what it was to what it is to be. Physics, II, iii, 194b 29-33. The efficient cause, in physis, are the local and causal conditions that allow the emergence of that phenomenon. And lastly, it presents the final or end cause, And lastly, there is the end or purpose, for the sake of which the process is initiated. Then there are all the intermediary agents, which are set in motion by the prime agent and make for the goal, as means to an end. Physics, II, iii, 194b 33-36. With these four causes, the existence and behavior of any body is totally determined. Of all of them, two stand out: the substrate (hylé) and the form (eidos), where form synthesizes the formal cause and the final cause. In other words, for Aristotle, the form of a thing is what brings it into existence, it is what makes it, that is what he calls the formal cause; but form is also the purpose, in which a thing is realized, he calls this the final cause. The form both causes a body to arise in existence, and impresses upon it its purpose, the “what is it for” of that body. Aristotle concludes that form (eidos) is nature itself (physis), because it is the internal origin of energeia (entity in act), in other words, "form is the activity that makes each thing what it is." Aristotles’s Physics: A Guided Study. Joe Sachs, p.58. And this activity is causal and relational. Everything in nature is at the end, middle and beginning of a chain of events, where all bodies affect and are affected indefinitely. As Sachs states, "Aristotle's teleology does not impose the human idea of purpose on non-human nature, but recognizes that all natural beings are whole and act only to preserve that wholeness and fulfill their potencies." Aristotles’s Physics: A Guided Study. Joe Sachs, p.58. Or yet, as the German researcher Zeller rightly states that "The most important feature of the Aristotelian teleology is the fact that it is neither anthropocentric nor is it due to the actions of a creator existing outside the world or even of the mere arranger of the world, but is always thought of as immanent in nature." Zeller, Eduard, 1883/1955, Outlines of the History of Greek Philosophy, rev. by W. Nestle, trans. L. Palmer, London: Routledge, §48. Aristotle states that things are drawn to their final cause. And everything has its own particular attraction. En sus Eticas objeta Aristóteles a Platón el que nadie se rige en sus acciones por la Idea de Bien en sí, sino por el bien humano los hombres, por el bien animal los animales, por el bien vegetal los vegetales. Cada cosa tiene su bien, al que realmente tiende y por el que tiene apetito real, que se sacia con la real posesión del bien, de su bien. La auténtica causa final tiene que ser término de una apetencia real, y su posesión ha de ser real, para causar una real satisfacción, para que el ser se note ya en el fin, en el término de sus aspiraciones. Por este motivo, Aristóteles adoptará como definición de causa final; “aquello hacia lo que todas las cosas salen disparadas”; y el bien moral, el bien humano, objeto de sus Eticas se alcanzará, y es comparación suya, como alcanza el arquero con sus flechas el blanco, la presa. Siete Modelos de Filosofar, J. D. García Bacca, p.20.   [In his Ethics Aristotle objects to Plato that no one is governed in their actions by the Idea of Good in itself, but for the human good men, for the animal good the animals, for the vegetable good the vegetables. Everything has its own good, to which it really tends and for which it has a real appetite, which is satisfied with the real possession of the good, of his good. The authentic final cause must be the end of a real appetite, and its possession must be real, to cause a real satisfaction, so that the being fulfills its purpose in the end, in the term of his aspirations. For this reason, Aristotle will adopt as definition of final cause; "That towards which all things are drawn"; and the moral good, the human good, object of his Ethics will be reached, and it is his comparison, as the archer reaches with his arrows the target, the prey.] Therefore, Aristotle's innovation is to merge the idea of desire, or attraction, which moves things to their end (τελος, telos) with the concepts of dynamis, entelecheía and energeia, which we will see below. For him, in Nature everything happens in a relational way and according to its own needs. But, if that is so, what about “luck” or chance that we encounter in our daily lives? Aristotle, analyzing Chance, comes to the conclusion that it happens when two causal lines intersect. For example, a person goes out to buy bread, following a natural human desire, and is struck by a tile falling from the roof, which also follows the natural falling motion, which would happen even if the man did not pass by at that moment. Both movements are natural and inherent to the bodies involved, independent of each other. Motion, Hylé and Apeiron The scope of Physics, the science of Nature (epistéme tes physis), covers all bodies that have the tendency to move and change. Everything that exhibits natural movement exists by its own principle (physis). The study of Nature revolves around properly understanding what is motion. Motion is something absolutely comprehensive, including both living beings, animals and plants, and the Cosmos. For Aristotle, motion means change or locomotion; such change can be generation or corruption (birth and death); change of qualitative status (eg. white to black) or quantitative (eg. small to large); and locomotion, displacement from one place to another. In Book III of his Physics, chapters i to iii, Aristotle leads us through an investigation about Motion. On one hand, we have a potential mover, and on the other hand, we have a potential mobile; one is the maker (ποίησις, poíesis) of the order of action, and the other is the sufferer (πάθησις, páthesis) Physics, III, iii, 202a 25. of the order of passion, and what is common to both is the movement (κίνησις, kynesis). The mover’s realization of the potentiality of moving takes place in the mobile; and the mobile’s potentiality of being moved, is realized in its motion. We are not going to enter into the discussion of the 'first mover', for it belongs to the scope of eternal things, and is dealt with in Metaphysics. Aristotle concludes that motion is the realization (entelecheia) of the potentially active and of the potentially passive, and this realization happens in the mobile. Physics, III, iii, 202b 27-28. Continuing, Aristotle addresses the question of apeiron. Why does he deem it important to discuss this notion in a book on Nature? What would be implicit in Nature that requires a study on the unlimited? Everything in the Cosmos is in motion, and Aristotle begins Book IV of his Physics, stating that The study of nature is concerned with extension, motion and time; and since each one of these must be either limited or unlimited..., it follows that the student of Nature must consider the question of the unlimited, with a view to determining whether it exists at all, and, if so, what is its nature. Physics, III, iv, 202b 30-35. Aristotle will analyze the preceding arguments, given that “all philosophers of repute who have dealt with Physics have discussed the “unlimited”, and all have regarded it as in some sense a “principle” of actually existing things.” Physics, III, iv, 203a 1-4. The Pythagoreans and Plato viewed the unlimited as ousia, not as an attribute. Physics, III, iv, 203a 4-6. (Οι μεν, ωσπερ οι Πυθαγόρειοι και Πλατων, καθ αυτο, ουχ ως συμβεβηκός τινι ετερω αλλ ουσίαν αυτο ον το απειρον: Others, like the Pythagoreans and Plato viewed the unlimited or undetermined as existing as substance ‘ουσίαν’) The physiologoi, who postulated the elements as principle, considered the unlimited an attribute of those. Anaxagoras and Democritus considered that the elements themselves were unlimited, each giving the apeiron quite different definitions. So, asks Aristotle, in what way do the unlimited exists? And here we are talking about Physics as an investigation of actual bodies. We see that he begins by analyzing what other philosophers have said before him about apeiron, however when in analyzing language, that is, in what ways they spoke about the apeiron, he will overturn the arguments of his predecessors. First he distinguishes between the Whole and the Complete, and then presents his definition. He begins by refuting Parmenides’s self-contained whole, for he argues that the Whole “is that outside which there is nothing”, therefore it does not admit recursion, which we find, for example, in counting natural numbers in an infinite sequence. He then rebuts Melissus’ all-encompassing whole by arguing that "we can say that nothing is complete if it does not come to an end, but an end is a limit, so everything that is complete comes up against a limit." Therefore, the apeiron is neither Melissus’ all-encompassing whole nor Parmenides’s self-contained whole; apeiron is not that which is All, nor that which is Complete. Aristotle proposes a new definition of infinity as “that which is beyond any definable limit”; its essence is incompleteness. The unlimited, then, is the open possibility of taking more, however much you have already taken; that of which there is nothing more to take is not unlimited, but whole or completed. Physics, III, vi, 207a 7-9. Aristotle comes to a startling conclusion when he associates hylé with apeiron, for the “unlimited” is really the “material” from which a magnitude is completed, and is the potential (entelecheía), though not the realized, whole. It is “divisibility without limit” in the direction of reduction or converse expansion, and is not any determined whole in itself, but only as the unlimited and “material” factor of the whole which is constituted as such by the limiting and “formal” factor. As “unlimited”, then, it is embraced and not embracing. Therefore qua unlimited it is unknowable, since “material” as such, is formless. Physics, III, vi, 207a 21-25. The apeiron is the hylé “divisible without limits” in both the reduction and the expansion process. It is not in itself a determinate whole, but only the unlimited hylé factor, which, when limited by the eidos factor, constitutes the whole. While apeiron, it does not encompass, but is encompassed by eidos. The apeiron is really the hylé with which extensive things are made, in other words, it is the totality in potency (dynamei), but not in actuality (entelecheía). By this, Aristotle solves the problem of the two infinities in physis: the infinite of the continuum is the infinitely divisible and immaterial hylé, is the infinite in act; and the infinity of recursion is the constant becoming of the Being, it is the infinite in potency. Dynamis, energeía and entelecheía Dynamis is translated as potency, or potentiality. This concept is intertwined with that of apeiron. For Aristotle, physis is motion, and motion is always relational, that is, something moves something else, ad infinitum. In Book III, he defines motion (or change) as the passage from potential existence (δύηαμις, dynamis) to actual existence (εντελεκεία, entelecheía): since something is currently a and potentially b, change is the process (ενεργεία, energeía ) which ends by realization of the capacity to be b. Therefore, change (or motion) is essentially incomplete while it lasts; the potential, in the process, progressively loses its characteristic; but when it is transformed into complete actuality (entelecheía), the change was completed. The entelecheia (being-in-act being itself) is the natural state of the Cosmos whose manifestations are the ousia, the actual body (the entelecheia of a material potential) and the motion (present throughout the Cosmos). In other words, the entelecheia (being-in-act being itself) of the potential of something material is the ousia (actual body). An actual body is the fusion of hylé, which is immaterial substrate in potency, with the Being-in-act, which is eidos, form. And the entelecheia (being-in-act being itself) of a potential while potential is motion. Aristotle reaches the most surprising conclusion of his Physics when he defines what is the apeiron. It is never current, except in the sense that we say that “the day” or “the Olympics” are current, are happening; because as potentiality, it is analogous to hylé (“matter” without form), which never exists as a body, as a determinate unit. Physics, III, vi, 206b 12-15. The Cosmos is always in the unlimited process of actualising itself. The Being is not static, it is being (present participle of the verb to be); the Being becomes unlimitedly, infinitely, eternally. Aristotle solves the arithmós infinity problem using the concept of potentiality, that of how things arise and disappear from existence, with the hylé following the movement of physis, being limited by eidos, acquiring form and telos, becoming the Universe infinitely, being. And he solves the mégethos infinitude problem with the infinitely divisible hylé, existing in act. Place (topos) and Void (kenon) The Greek word topos can mean “place” (in Latin, locus), or “space” (in Latin, spatium). This generates several developments: the word locus implies motion, whereas the word spatium implies geometric space, delimited area. These different concepts imply different spatial representations. Why does Aristotle think it necessary to discuss the place of a body? All bodies in nature are able to move from one place to another, and this local mobility is a universal phenomenon that must be investigated. He realizes that some things go up, are drawn up, while others come down, are drawn down. The cosmic “locality” seems to be a reality distinct from its content, and Aristotle begins to sketch the idea of attraction: an attraction to the outer limits of the Cosmos, and another attraction to the center of the Cosmos. Now these terms – such as up and down and right and left – when thus applied to the trends of the elements are not merely relative to ourselves. ...in Nature each of these directions is distinct and stable independently of us. “Up” or “above” Always indicate the “whither” to which things buoyant tend; and so too “down” or “below” always indicate the “whither” to which weighty and earthly matters tend, and does not change with circumstance ... Physics, IV, i, 208b 18-23. Both directions are active and positive. The bodies seem to be attracted to their natural places, where they can remain at rest, “and that all the elemental substances have a natural tendency to move toward their own special places, or to rest in them when there”, Physics, IV, iv, 211a 4-6. and “we must recognize that no speculations as to place would ever have arisen had there been no such thing as motion or change of place”. Physics, IV, iv, 211a 15. Aristotle complains that his forerunners did not leave any argumentation, nor even formulated any inquiry on this subject. Therefore he begins the investigation of “place” starting from the common notion we have of it: To begin with, then, the phenomenon of “replacement” seems at once to prove the independent existence of the “place” from which – as if from a vessel – water, for instance, has gone out, and into which air has come, and which some other body yet may occupy in its turn; for the place itself is thus revealed as something different from each and all its changing contents. Physics, IV, i, 208b 2. And he even mocks Hesiod, One might well conclude from all this that there must be such a thing as “place” independent of all bodies, and that all bodies cognizable by the senses occupy their several distinct places. And this would justify Hesiod in giving primacy to Chaos where he says: “First of all things was Chaos, and next broad-bosomed Earth”; since before there could be anything else “room” must be provided for it to occupy. For he accepted the general opinion that everything must be somewhere and must have a place. Physics, IV, i, 208b 28-34. Hesiod found necessary to create a place, Chaos, to place Gaia. Aristotle then enumerates the characteristics of what is a place: Physics, IV, iv, 211a 1. the place of a body is not part of the body itself, but it is what surrounds it; the proper (immediate) place of a body is neither greater nor smaller than it; the body can leave the place where it is, therefore it is separable from it; and finally, every place is subject to be attracted up or down. Place, therefore, is not undifferentiated. But to get out of our current conception, and to respond to what is “place”, Aristotle examines four possibilities. He begins by investigating whether place should be form (morphé), for surely that which embraces seems to suggest “form”, for the surface-boundaries of the enclosed body and the surrounding vessel coincide. Certainly “place” and “form” are limits, but not of the same body, for the form determines the body itself, but the place determines the surrounding vessel. Physics, IV, iv, 211b 10. Place, then, is not form. This investigation continues by asking whether place should be matter (hylé), for if we consider “place” as undisturbed by the changes happening in it, and also as undifferentiated in itself, and as a continuum, we can compare it to matter, hylé. This “place” may seem like permanent “matter”. In the case of matter, we argue so because where there was air, we now have water; but in the case of place we say that it exists because in the same place that we had air now we have water. However, matter is neither separable from the body nor involved by it; but place is separable from the body and surrounds it. Physics, IV, iv, 211b 30. Place, then, is not matter. Should place be some kind of dimensional extension between the points of the container? Since we see that the content may change several times, while the container remains unchanged, “the imagination pictures a kind of dimensional entity left there, distinct from the body that has shifted away.” Physics, IV, iv, 211b 15-18. Place, therefore, is not a fixed dimensional entity. Aristotle concludes that place must be the very inner surface of the container containing the body, for things in motion are actualizing their potentialities, or being moved by other bodies, Physics, IV, iv, 211a 18. and things are surrounded by a certain surface that fits around them - Such then – the inner surface of the envelope, namely – is the immediate place of a thing. Physics, IV, iv, 211a 29. ...then it is “in” its inner surface of this continent as its immediate and proper “place”, - which inner surface is neither a part of the content nor dimensionally greater than it, but equal to it; for when things touch each other their surfaces coincide. Physics, IV, iv, 211a 32-34. Aristotle concludes that place do not precede bodies, but depends on them. Here then is the immediate place of a body - the inner surface of the envelope, Physics, IV, iv, 211a 28. which moves with it, and adjusts dynamically to it. For example, when a stone falls into a lake, passing through a body of water, the place of the stone is the portion of water that surrounds it dynamically as it falls. Aristotle therefore examines and rejects the notion of rigid geometric space, where a body comes out, and leaves an empty fixed dimensional entity, and states that “for what lies between point and point of the continent is the body, as existing independently of it.” Physics, IV, v, 212b 23. This extraordinary notion was only resumed in the 20th century, as stated in La Physique nouvelle et les quanta, by Louis de Broglie: Une de ces hypothèses (de la physique classique) était que le cadre de l’espace e du temps dans lequel nous cherchons presque instinctivement à localiser toutes nos sensations, est um cadre parfaitement rigide et déterminé où chaque événement physique peut être rigoureusement localisé indépendamment de tous les processus dynamiques qui s’y déroulent. ...La localisation exacte dans l’espace et dans le temps est une sorte d’idéalisation statique qui exclut toute evolution et tout dynamisme. ...Tout autre est le point de vue de la physique quantique... La Physique nouvelle et les quanta, Louis de Broglie, Flammarion, 1937, p.6-7. [One of these hypotheses (of classical physics) was that the space-time framework in which we almost instinctively seek to locate all our sensations, is a perfectly rigid and determined framework where each physical event can be rigorously located independently of all the dynamic processes that take place there. ... The exact location in space and time is a kind of static idealization that excludes all evolution and dynamism. …The point of view of quantum physics is different. ...] As we saw above, Aristotle had already reached this conclusion, in his Physics, 2500 years ago. Following the study of motion, he addresses the issue of vacuity, or void. The central questions will be: Does the void (κενον, kenon) exist? How does it exist? Is void a “place with nothing inside”? Aristotle refutes the existence of void as something self-existing and self-constituted. The arguments for and against its existence are analogous to those about “place” ...inasmuch as the believers in its reality present it to us as if it were some kind of receptive place or vessel, which may be regarded as full when it contains the bulk of which it is capable, and empty when it does not. Physics, IV, vi, 213a 15-17. According to Aristotle, those who denied the existence of void, like Anaxagoras, have only demonstrated that air is a physical substance, inflating balloons and showing their resistance to compression. Physics, IV, vi, 213a 24. So, those who affirmed the existence of emptiness only said that things full of air are empty, because they did not perceive the presence of air, "and, because they regard perceivable substances (σωμα αισθητόν, soma aisthetón) as the only things that exist at all." Physics, IV, vi, 213a 29-31. Furthermore, they affirmed that the void was necessary for motion, claiming that motion would not exist without it, because it would be impossible for something full to receive a body. Neither side really addressed the issue. Aristotle claims that those who deny the existence of the void need in fact to demonstrate that: there is no such thing as a dimensional entity, other than that of material substances, existing in such detachment and actuality (ενεργεία, energeía) as to intervene and break the continuity of a body. Physics, IV, vi, 213a 31-35. Abridging the discussion presented in his Physics, Book IV, chapters vi to ix we summarize that for Aristotle there is no discontinuity in the Cosmos. Aristotle replies that one body can only move into what is full if at the same time another body is moved from there. He claims that a homogeneous and undifferentiated void could not cause motion, claiming that a body in the void would not seek its natural place. But in nature all bodies seek (or are attracted) to their natural place, so the existence of the void is not physically logical. Another argument used by Aristotle to refute the existence of the void is based on the speed of bodies in a medium. The denser the medium, the longer a body will take to cross it, so a body goes slower in a dense environment. Likewise, the lower the density of a medium, the faster a body will cross it. If the vacuum density is zero, how fast would this body go? Aristotle argues that since the density of the void is zero, the velocity of that body would be infinite, and that would be impossible for actual bodies, so it is not mathematically logical for a vacuum to exist. Aristotle and Time (χρόνος) The concept of time presents many difficulties. Aristotle analyzes Time in his Physics, Book IV, chapters x to xiv, and asks himself what is its nature, putting forward some initial considerations. First, he notes that time is connected to motion, and motion occurs in extension (μέγεθος, mégethos). Since extension is a continuous quantity, time and motion are also continuous quantities, and it is only through motion that we perceive the passing of time. Therefore, this perception needs a human-perceivable body to be real. In other words, we need to notice an ousia going from one place to another, to perceive the passing of time. But how can time exist, if no part of it exists? As follows, some of it is past and no longer exists, and the rest is future and does not exist yet... how can we conceive of that which is composed of non-existents sharing in existence in any way? Physics, IV, x, 217b 36 - 218a 4. Moreover, …if anything divisible exists, ...either all its parts or some of them must exist. The present now is not part of time at all, for a part measures the whole, and the whole must be made up of the parts, but we cannot say that time is made up of “nows”. Physics, IV, x, 218a 5-7. And carries on, arguing that time cannot be a succession of “nows”, as a line A line does not exist concretely, it exists only in the mind that imagines it. is not a succession of points, Points have neither shape nor dimension, are dimensionless objects, are immaterial. stating For we must lay it down as an axiom that there can be no next “now” to a given “now”, any more than the next point to a given point. This is demonstrated in Book VI, i. Aristotle continues in his reasoning, stating that time is something that passes, but motion or change occurs in what moves, whereas time is everywhere and is in relation to all things, Now the most obvious thing about time is that it strikes us as some kind of “passing along” and changing; but if we follow this clue, we find that, when any particular thing changes or moves, the movement or change is in the moving or changing thing itself, or occurs only where that thing is; whereas the “passage of time” is current everywhere alike and is in relation with everything. Physics, IV, x, 218b 9-15. He will also analyze the differences between time and motion, noting that: movement occurs in things, and locally where they are, while time is everywhere; the movement may be fast or slow, but not time, since fast and slow are defined by time; time cannot measure time, as if it were a distance or a qualitative change. “Plainly, then, time is neither identical with movement nor capable of being separated from it”. Physics, IV, xi, 219a 2-3. Having said that time is not a succession of “nows”, nor identical to motion, Aristotle begins to outline his theory. As we have seen, time is associated with motion (kynesis) and change (metabolé). But in what way time and motion are connected? For us, humans, it is only through the perception of motion that we can say that time passes, that is, "... when we are aware of motion we are aware of time, ..." Physics, IV, xi, 219a 4-5.. Aristotle reasons that ...since anything that moves, moves from a “here” to a “there”, and magnitude as such is continuous, movement is dependent on magnitude; for it is because magnitude is continuous that movement is so also, and because movement is continuous so is time; … Physics, IV, xi, 219a 11-15. As we saw above in 2.2.2, to measure any continuous magnitude, we need to establish a cut in this continuum in order to measure it. Motion is a continuous magnitude, so to measure it we need to establish an anterior and a posterior, and an arbitrary standard for the length traveled. This is also the case with time, as …when we distinguish between the extremes and what is between them, and the mind pronounces the “nows” to be two – an initial and a final one – it is then that we say that a certain time has passed; … Physics, IV, xi, 219a 25-29. Or, as well summarized by Fernando Rey Puente, ...para Aristóteles, o tempo em sua infinitude constitutiva não pode ser pensado diretamente, mas apenas indiretamente enquanto um intervalo de tempo determinado a partir da nossa percepção do anterior-posterior no movimento. Rey Puente, Fernando. Os Sentidos do Tempo em Aristóteles, p.189. [... for Aristotle, time in its constitutive infinity cannot be thought directly, but only indirectly as a time interval determined from our perception of the anterior-posterior in motion.] Aristotle concludes that in the same way that we measure a continuous quantity defining the anterior and the posterior, to then measure it using an arbitrary standard, we do the same with time. Time is also continuous as it follows magnitude, and it is only due to motion that we can establish a before and after, to then measure it; therefore, time is the measure of motion between two “nows”. In other words, time is something that measures this motion, that is: time is the measure of motion. Physics, IV, xii, 221b 9. "Time, then, is not motion, but that by which motion can be numerically estimated." Physics, IV, xi, 219b 4-5. This has implications in today's physics. Aristotle chooses as a measure of time the uniform circular motion, that is, "the uniform rotation will be the best standard, since it is easiest to count." Physics, IV, xiv, 223b 19-20. For to us humans, the counting of the days through the rotation of the Earth is the most natural; and by extension, the counting of years through the translation of the Earth around the Sun, by being cyclic, provides us with a standard for time measurement. Moreover, he argues that we need to be aware, because when we talk about time we are referring to the measure of time rather than time itself. Physics, IV, xii, 220b 1-6. Aristotle goes on comparing time to place, for time embraces existing things in the same way as place. And since what exists in time, exists in it as number (that is to say, as countable), you can take a time longer than anything that exists in time. So we must add that for things to exist in time they must be embraced by time, just as with other cases of being “in” something; for instance, things that are in places are embraced by place. Physics, IV, xii, 221a 28-30. Concluding that things affected by time always deteriorate, Aristotle intuits the principle of entropy, arguing that nothing improves or becomes more perfect over time, it is always the opposite that happens with things “in time”. Physics, IV, xii, 221a 30-33 e 221b 1-4. But there are things that are not in time. "From all this, it is clear that things exist eternally, as such, are not in time; for they are not embraced by time, nor is their duration measured by time." Physics, IV, xii, 221b 4-6. He gives as an example of eternal things mathematical objects, such as the incommensurability of a square’s diagonal, which exists eternally, and therefore is not in time. But is time itself limited? Certainly not, because if motion is eternal, time is also eternal. Physics, IV, xiii, 222a 32-33. For that which enters and leaves existence must be embraced by time, “…for there must be some time great enough to exceed the time of their duration and therefore the time which measures their being”. Physics, IV, xiii, 221b 29-32. Therefore, we have a finite time, perceptible when cut from the continuum by our perception, and an infinite time that embraces all things countable (and uncountable) and not perceptible by our senses because it is infinite in potential.  Aristotle and current Physics 3.1 Aristotle... In what way do Aristotelian concepts fit current Physics? Research on the Universe is just beginning, but we continue to “see” a spherical Universe in all directions. Today our modern telescopes reach 13 billion light-years from Earth, that is a sphere much larger than Aristotle could see. He did not know what existed outside Earth, but he described a spherical Cosmos with fixed stars, for that was what he observed, and that is what we observe today. We have seen above that Aristotle describes two forces of attraction: one for the center and one for the outer limits of the Cosmos, for place is not undifferentiated. The introduction Physics, I, i, 184a 10-22. of Aristotle’s Physics begins by reaffirming the method of Divided Line Analogy to arrive at the understanding of the Being. Aristotle's physis is constructed from the assertion that the knowledge of Being, and by analogy all the laws of Physics, must be deduced from actual observations. The Cosmos, whose properties Physics proposes to study, is essentially immanent motion, there is no room for transcendence. Aristotle, with his study of physis, precedes by centuries the answer of Laplace to Napoleon when asked where the considerations about God came in his construction: "Sire, such a hypothesis was not necessary." Available at https://pt.wikipedia.org/wiki/Pierre-Simon_Laplace. Accessed on June 17, 2018. In this line, the first mishap Aristotle wants to avoid is that of defining something by the simple negation of something else. It is thus, for example, with two fundamental discussions that are basic in the construction of Modern Mathematics: the discussion about void and the discussion about infinite. These two discussions are highly pertinent and relevant, and Aristotle took on the task of looking into them, which he did in part with tools created by earlier thinkers (Parmenides, Anaxagoras, Tales) and above all with what he himself developed. For Aristotle, the Cosmos is that which IS, in the broadest sense of the verb to be. This implies that there is not something outside it, because that something would have to BE, and therefore, would be in it. Aristotle is faced with a dilemma: should he accept that the notion of Being is purely intuitive, or is there any way to go deeper, based on physical experience, with the idea of what Being means? The discussion about void and infinity is interwoven with the discussion of what the Being is in physis. He will seek to redefine the intuitive idea of Void, and in order to do so, he will give a definition of body, and of a place occupied by the body. The latter being defined almost by mathematical reasoning. In its definition of place, it will be fundamental the definition of envelope, of border, of limit, which will depend on the idea of mathematical continuum. Let's see how he does this, step by step, starting with the definition of Continuum. The idea of infinity may be the first sophisticated mathematical idea that has shown itself to mankind. In fact, the idea of infinity extends the idea of the Being, as an important step beyond what we might consider intuitive. Suppose now a fable, primitive Man when he began to count, probably followed these steps: first he associated a unit of something with a unit of something else, he attached a gesture to a concept. A finger is, it exists, just as a horse exists, so a finger could indicate a horse. Two fingers would indicate two horses. Man began to count. The need to count larger quantities naturally leads to a recursive process, which leads to the idea of infinity. If my recursive process does not stop, I'm building ever-larger amounts. A thousand horses is a very large number of horses. A million horses is an even larger number, and I can always keep on counting. Aristotle will discuss two types of infinity: the infinity of recursion (arithmós) and the infinity of magnitude (mégethos). After discussing the first notion of the mathematical infinite, as a quantity that cannot be limited, which we have seen, naturally follows from the idea of recursion (1, 2, 3, 4, 5 ...), Aristotle discusses the idea of continuum, the fact that wherever one makes a cut in the line of the real numbers one will find a real number. In fact, the process of dividing a segment into as many whole and equal parts as we want, proposed by Thales of Miletus, indicates that the cardinality of rational numbers is the same as the natural numbers. Figure 1 below gives an indication of this, while figure 2 gives the modern proof of the fact. Fig.1: Thales' theorem shows that a line can be divided indefinitely into as many smaller segments as one wants. fig.2: By associating pairs of integers with rational numbers in their reduced form, Cantor has proved that both have the same cardinality and both are countable. By simply following the arrows one can understand this. Therefore, unlike real numbers, rational numbers are countable. Clearly Aristotle is one step ahead. For him, the definition of the continuum of real numbers is beyond that allowed by the mere use of recursion. However, the correct formulation of the continuum idea, as already mentioned, would only be achieved at the end of the nineteenth century, with Dedekind’s cuts, “Every location on the number line continuum contains either a rational or an irrational number. Thus there are no empty locations, gaps, or discontinuities.” https://en.wikipedia.org/wiki/Richard_Dedekind Cantor, “…Cantor proved that the set of real numbers is ‘more numerous’ than the set of natural numbers; this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciate the importance of one-to-one correspondences in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countable infinite) sets and non-denumerable sets (uncountable infinite sets).” https://en.wikipedia.org/wiki/Georg_Cantor and the use of the Axiom of Choice. “Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection; this makes it possible to directly define a choice function. For an infinite collection of pairs of socks (assumed to have no distinguishing features), there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice.” https://en.wikipedia.org/wiki/Axiom_of_choice Aristotle does not have these tools in arguing about topos, but his thinking is absolutely clear when he discusses the existence of the limit of a bounded increasing sequence A sequence, by definition, has an infinite number of terms. For example, real numbers are an unlimited sequence. of magnitudes and that amongst all the possible superior bounds to this sequence there must be a lower bound. Physics, III, vi, 206b 3-14. Using current and non-formal language, he closely approximates the modern definition of the boundary of a growing and limited sequence. This idea of continuum will suggest the idea of a body’s place not as a position given by dimensions in space, but as the smallest region containing that body, which coincides with the largest region occupied by that very same body. There is no room in Aristotle for this “inner place” and this “outer place” to be different. There are no empty spaces in nature, just as there are no empty spaces between the real numbers. Another surprising result that emerges from the Physics is how Aristotle relates the notions of the two infinities to the concrete physical world. From the discussion of the two types of infinity, he concludes that the apeiron, the unlimited is the hylé (immaterial substrate), limited by eidos. When considering the continuous infinite of magnitude, mégethos, where all points of the line of real numbers are present in act, Aristotle associates it with the Being-existing-in-act, that is, the hylé permeating the entire Cosmos, immaterial, formless and in search of an eidos. When considering the increasing infinity, that which has no limit because it is continuously becoming, Aristotle associates it to the hylé encompassed by eidos, entering and leaving existence in a constant becoming, in an infinite becoming. To summarize: the infinite of the continuum (mégethos) exists in act, while the increasing infinity (arithmós) exists in potential. Based on the inference that the Being-in-act, in Aristotle, is the continuum, behaving like the set of real numbers, we will analyze in what way Physics reasoning can influence Mathematics. Still in this line of thought, it should be noted that we still use false or true binary logic, 0 or 1; but it would be possible to imagine a mathematical logic more compatible with the physical world. A logic based on the continuum of mégethos instead of the discrete one of arythmos, where innumerable possibilities are likely to happen. Taking Aristotle as an inspiration, we affirm that "when going from the false to the true, one assumes a discreet logic, but when going from white to black, which is the change we see in physis, infinite shades of ash-gray rest between the two colors ". Therefore, we could go beyond false/true logic, and it might be possible to open a new field in Mathematics. ...and current Physics Studying this Cosmos, beyond this light visible by humans, resulted in the construction of telescopes for all electromagnetic frequencies. Today, telescopes operate everywhere in the invisible part of the electromagnetic spectrum. We can observe phenomena ranging from high-frequency gamma rays (photons operating at the highest observable frequency), with a wavelength of 10-11m, to low-frequency radio waves with a wavelength of twelve meters in length, crest to crest. Astrophysics for People in a Hurry. Neil de Grasse Tyson, p.163. If we did not have telescopes operating today at this wide frequency range, we would say that the space between the galaxies of the Universe was empty! The Cosmos observable through telescopes became much more interesting: gamma-ray telescopes see the explosion of giant stars, breeding new particles; ultraviolet and X-ray telescopes see emissions in the vicinity of black holes, and super black holes in the center of galaxies; Infrared telescopes see the birth of stars in the depths of galaxies’ gas clouds; Microwave telescopes see the cosmic background radiation; Radio telescopes see the gas between galaxies. We get this vision of the Cosmos only after “seeing the invisible”, for none of this would be visible to us without the aid of instruments. We found that the Cosmos conceived by Aristotle as being composed of concentric spheres only got bigger, because we can now see 13 billion light-years in all directions, and with each technological advance, we see a little further. If we took into account only the mass of the Universe visible by our current instruments, the universe would be contracting due to the powerful action of gravitation. That is not what is observed. The galaxies are moving apart so fast that in a few billion years we will no longer observe those closest to us. The amount of visible matter does not explain the expansion found in the Universe. The fact that the galaxies are drifting away faster than initially thought was the first direct evidence that a repulsive force permeates the universe, opposing gravity. Therefore, speculation has been made about the existence of dark matter and dark energy to try a new modeling, with recent calculations revealing that dark energy would account for 68% of all mass-energy in the universe, dark matter comprises 27% , and ordinary matter a mere 5%. Astrophysics for People in a Hurry. Neil deGrasse Tyson, p.107. It is presumed that dark matter is not even matter, for it only interacts with matter through gravity. It is assumed that dark energy is a quantum effect, dubbed “vacuum” energy. It is an ocean of virtual particles, a prediction of quantum physics that manifests as a push toward the outer limits of the universe, acting against gravity. Astrophysics for People in a Hurry. Neil deGrasse Tyson, p.74. A notion proposed by Aristotle, as we saw in 2.3.5, when the notion of Topos was discussed. Recently, gravitational waves, suggested by Poincaré and predicted by Einstein, were detected. First observed on September 14, 2015, originated by the collision between two black-holes, 1.3 billion de light-years away. Available at https://en.wikipedia.org/wiki/Gravitational_wave. Accessed on July 5, 2018. Those waves are a subtle perturbation in the curvature of the space-time observed on Earth, due to super explosions of great energy. Gravity for Newton is an “action-at-distance” due to the attraction between masses, or coefficients of inertia; whereas Aristotle states that the Cosmos is not homogeneous, and the masses are drawn to their natural place. This is just another way of trying to explain the gravitational attraction, which, according to Aristotle, in addition to attracting to the center also attracts to the outer limits of the Cosmos. In addition, we could say that hylé are free particles, which under great pressure first become light elements such as hydrogen and helium within younger suns, and progressively heavier elements are formed inside older stars under larger pressures. In the same way, we could say that eidos are the internal laws of physis, since the physical bodies become after receiving their eidos, but since everything is causal, and the place is not undifferentiated, favorable conditions are necessary for that to happen: free electrons subjected to pressures inside a star or a black hole, for example. The observable universe presents us with many mysterious events. Vacuum energy, dark matter, dark energy, black holes, massless particles that undergo the action of gravitation ... All these events have not yet been explained. Today the evidence about the universe points to something permeated by energy. In the twenty-first century, we cease to discuss what the Being is, and we begin to discuss what the Universe is, but with this change of focus we lose an important philosophical dimension: that of investigating Nature through concepts, as opposed to the current tendency to work with operational notions, with temporary validity. Physics today still comes up against the imponderable. What is beyond the observable Universe is an unknown and a mystery, today and in the time of Aristotle. Therefore, it is necessary to rethink physics in a conceptual way, as Aristotle had showed us. References ANGIONI, Lucas. Aristóteles, Física I – II. Campinas: Editora Unicamp, 2009. ARISTOTLE. Physics. Londres: Harvard University Press, Loeb Classical Library, 1957. ARISTOTLE. Metaphysics. http://www.perseus.tufts.edu/hopper/ ARISTÓTELES. Metafísica, Ensaio introdutório, texto grego com tradução e comentário de Giovanni Reale. Vol. 2. São Paulo: Loyola, 2009. BAILLY. 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