Ancient Philosophy. 15. Plato, Science & Nature The Republic & the Timaeus. Some Questions from the Republic

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1 2 Some Questions from the Republic Ancient Philosophy What is "the vulgar utilitarian commendation of astronomy" (528e)? In what way does astronomy "compel the soul to look upward and lead it away from things here to higher things" (529a)? In what way could it be said to deal with "being & the invisible" rather than "things of sense"? 15. Plato, Science & Nature The Republic & the Timaeus 3 4 Idealization in Science 1 Falling Bodies: Predictions & Observations Question: At what velocity do bodies fall? Aristotle (in the real world): That depends on the weight of the bodies and the medium through which they fall The heavier the body (& the less dense the medium) the faster they fall Galileo (in a vacuum): That depends on how long they are falling They accelerate: The longer they fall, the faster they fall, independent of their weight Bodies accelerate (independent of their weight) until they reach terminal velocity Graphs from: Robert March, Physics for Poets 5 6 Idealization in Science 1 Idealization in Science 2 What is the figure of planetary orbits? Which answer is better? Aristotle s? There are no vacua & even if there were, the question is about ordinary objects falling through the air Galileo s? The medium is not part of the problem of falling itself It s just an impediment which complicates & obscures our object of study. Johannes Kepler: The orbit of each planet is an ellipse and the Sun is at one focus. Isaac Newton: Third Law of Motion: For every action there is an equal and opposite reaction. So, the sun & the planet act on each other and all the planets perturb one another s orbits. The orbits are not exactly elliptical

2 7 8 Plato s Mathematical Physics Two key features of Plato s views on nature lay the conceptual foundations for contemporary mathematical physics The theory of the elements The program of planetary orbits in astronomy Elements & Atoms Atoms & Elements Atomic theories claim that there are smallest (indivisible, ἄτομος) pieces of matter from Democritcus to Dalton (the origin of chemical atomism) & Einstein (whose explained Brownian motion as the result of molecular motion) Theories of elements, by contrast, identify the fundamental kinds of things that there are. Theories of Elements Democritus claimed that, fundamentally, there is only one kind of substance. All atoms are qualitatively the same. Anaxagoras, thought that ordinary objects were made up of parts of all kinds of things. Empedocles doctrine that there are four kinds of things was the view generally accepted by later thinkers. Empedocles Identification of Four Basic Elements 9 The Final Theorems of Euclid s Elements 10 Fire Air Earth Five existence proofs at the end of Bk. XIII (though stated as construction problems, what is constructible must exist) Prop. 13. To construct a pyramid [four-faced solid] Prop. 14. To construct an octahedron [eight-faced solid] Prop. 15. To construct a cube [six-faced solid] Prop. 16. To construct a icosahedron [twenty-faced solid] Prop. 17. To construct a dodecahedron [twelve-faced solid] And a proof that no other such solid exists Prop.18: No other figure, besides the said five figures, can be constructed by equilateral and equiangular figures equal to one another: Water Plato s Identification of the Elements with Regular Geometric ( Platonic ) Solids 11 Plato s Mathematical Analysis of the Elements 12 Fire Earth Air Water

3 13 14 Plato & His Ancient Rivals Analysis of Elementary Particles in Modern Physics: Murray Gell-Mann s Eightfold Way Plato & Empedocles Share the reduction of material things to four elements. Differ in this: For Empedocles, these are ultimate elements. For Plato, these material elements are reducible into mathematical components. Plato & Democritus Share in the reduction of the familiar (& sensible) to the unfamiliar (but comprehensible). Differ in the extent of the reduction: Compare [Plato s theory] with the best of its rivals, the Democritean. There atoms come in infinitely many sizes and in every conceivable shape, the vast majority of them being irregular, a motley multitude, totally destitute of periodicity in their design, incapable of fitting any simple combinatorial formula. If we were satisfied that the choice between the unordered polymorphic infinity of Democritean atoms and the elegantly patterned order of Plato s polyhedra was incapable of empirical adjudication and could only be settled by asking how a divine, geometrically minded artificer would have made the choice, would we have hesitated about the answer? Gregory Vlastos, Plato s Universe By the early 1960 s, physicists had discovered nearly a hundred sub-atomic particles. Murray Gell-Mann was able to construct a theory (the Eightfold Way) organizing baryons & some mesons into octets and other baryons into a decuplet. Only nine of the baryons had been observed when the theory was proposed in Gell-Mann s predicted tenth particle (the Ω- particle) was discovered in Group Theory (a branch of mathematics) is crucial to an understanding of the beauty & coherence of the theory. Gell-Mann won a Nobel Prize for this work in This, I claim, is in the spirit of Plato s analysis of the elements in the Timaeus The Octets & Decuplet of the Eightfold Way Background to Greek Astronomy The Babylonians had a long history of earlier observation. The meson octet. They had some ability to predict eclipses. But this was done by an algebraic method, not by geometric or mechanical modeling. Greek interest in astronomy dates back to the earliest Milesian physikoi. Anaximander already offered a mechanical model. The spin-1/2 baryon octet. But no one before Plato thought to attempt a precise account of the movements of the heavenly bodies. For a description of those motions, see the next two slides. In what follows, the word stars will be used as the Greeks used it, to name all heavenly bodies. The spin-3/2 baryon decuplet. Most look like points of light. Two (the Sun & the Moon) do not. In all these diagrams, the horizontal lines show strangeness; the diagonals show charge. The Motion of the Stars: 1. The Fixed Stars 17 The Motion of the Stars: 2. The Wandering Stars ( Planets ) 18 Most stars remain in the same place relative to one another (in constellations ) but move over the course of the evening. Counterclockwise to those facing North; Clockwise to those facing South, Rising in the East & Setting in the West. Because of the first-mentioned fact, they are called fixed stars. Most stars move as described above. Seven do not. These seven move generally with the stars, but not at the same speed. Equivalently, they drift generally eastwards through the constellations on their path (the Zodiac), making a complete circuit every month (for the Moon); every year (for the Sun, Mercury & Venus); or every several years (for Mars, Jupiter & Saturn). Three of them (Mars, Jupiter & Saturn) sometimes stop & even reverse direction for a short time (show stations & retrogradations).

4 Plato s Geometric Challenge to the Astronomers By the assumption of what uniform and orderly motions can the apparent motions of the planets be accounted for? (according to a tradition running back to Sosigenes (C2 AD) via Simplicius (C6)) In this, Plato proposes a new approach to astronomy ("saving the phenomena"), which would become an exact mathematical science. Two different research programs arose to meet this challenge The Eudoxean Program (from C4 BC) This program is adopted also by Aristotle. The Hipparchan-Ptolemaic Program (from C2 BC) The motions of the fixed stars are clearly circular. There is no reason to think that they do not move exactly as they appear to do by circling around the Earth. Each tradition posited circular motion as the appropriate underlying motion for the planets as well. This is not, however, part of the project as reported by Simplicius. 19 The Eudoxean Program Eudoxus (Εὔδοξος) of Cnidus ( ) was, for a time a student of Plato. He explained the motions of the planets by positing nested concentric spheres. The Sun & Moon had three spheres each: one rotated daily, explaining rising & setting, one monthly (or yearly), explaining the monthly (or annual) motion against the background of the fixed stars, another to explain miscellaneous motions (the deviation from the ecliptic). The other planets had four spheres each: one to explain rising & setting, one to explain the motion against the background of the fixed stars, two to explain retrograde motion. Later contributors to this tradition added more spheres; Aristotle proposed around fifty The Hipparchan-Ptolemaic Program The Hipparchan-Ptolemaic Program was an alternative to that proposed by Eudoxus. It was a response to the Eudoxean inability to explain planets variation in brightness. It offered an essentially mathematical model of the universe ideas which would make predictions about where the planets would be at any given time. There was no attempt to elaborate an account of the physics of the model. This made it different from the system of Eudoxus, for which Aristotle had offered a mechanical model of nested spheres. The main contributors to this program were: Hipparchus (Ἵππαρχος) of Rhodes (c BC) Ptolemy (Κλαύδιος Πτολεμαῖος) (c ) His work, The Mathematical Treatise came to be called The Great Treatise (Ἥ Μεγάλη Σύνταξις in Greek, al-magisti in Arabic, & Almagest in English). Hipparchus Foundational Ideas 1. Epicycles & Deferents (below left) The deferent was a circle which carried the center of another circle (the epicycle) which in turn was the path of the planet. This explained retrogradation as well as some minor variations in planetary motion. 2. Eccentric Location of the Earth (below right) Since planetary motion was not uniform in speed relative to the earth, Hipparchus suggested that maybe the Earth was not at the center of the planet s orbit. Planetary motion would be uniform in speed relative to the center of the orbit, but not relative to the Earth Ptolemy s Contribution to the Program Summary The Equant Point Placing the Earth away from the center of the orbit was not sufficient to explain the apparent variation in the speed of the planet. Hence, Ptolemy proposed the third (& most controversial) of the three devices for explaining planetary motion the equant point. The equant point was a point different from the center of the circle and from the location of the Earth, relative to which the motion would be of uniform speed. Deferents & Epicycles Eccentric Location of the Earth The Equant Point

5 The Medieval Continuation of the Program Medieval astronomy continued the Hipparchan-Ptolemaic System, but the demand for accuracy led to increasing complexity. Alfonso X, the Learned, King of Galicia, Castille & Leon, commissioned new astronomic tables based on Ptolemaic principles in C13. These Alfonsine Tables, completed in 1252, remained in use until 1551, when Erasmus Reinhold calculated his Prutenic Tables based on the principles of Copernicus. According to legend, when the principles of Ptolemy were explained to him, King Alfonso replied that, if he had been God, he would have made the world-system simpler. 25 Copernican Astronomy as a Continuation of the Platonic Program Nicolaus Copernicus (Mikołaj Kopernik, ) proposed a double motion of the Earth revolution around the Sun, making the Sun, not the Earth, the center of motion for all planets except the Moon. rotation on its axis This allowed him to get rid of the equant point major epicycles (those used to explain retrograde motion) This system was simpler than its geocentric predecessors. This system still meets the terms of the Platonic challenge It reducеs the observed motions to uniform motion. It even retains circular motion. Its novelty is to make the Sun the center of the uniform motions. Jan Matejko, The Astronomer Copernicus: Conversation with God 26 Two Theoretical Analyses of Retrograde Motion 27 In Ptolemy s model, retrograde motion occurs when the planet s epicycle moves it backwards faster than the deferent moves it forwards. Retrograde Motion In Copernicus model, apparent retrograde motion occurs as Earth overtakes an outer planet. 28 Hipparchan-Ptolemaic Deferent & Epicycle Analysis for Geocentric Astronomy Copernican Epicycle-free Analysis for Heliocentric Astronomy Kepler s Astronomy as a Continuation of the Platonic Program 29 Ellipses & Epicycles 30 Johannes Kepler ( ) continued the heliocentric approach proposed by Copernicus, but not yet generally accepted in the early C17; continued to try to meet the Platonic Challenge to reduce the motions of the planets to uniform motions; introduced a further innovation motion would be elliptical, not circular. An elliptical orbit can be produced by setting the periods of rotation of epicycle & deferent equal to one another.

6 31 32 Elliptical Orbits Kepler & the Uniformity of Motion His abandonment of the traditional preference for circles was forced by observational data. He was working with the data of Tycho Brahe, the best data then available. Kepler maintains a kind of uniformity of motion (though not of velocity) in his Second Law of Planetary Motion: A line joining a planet and the sun sweeps out equal areas during equal intervals of time. He replaced the circle with the next available conic section, the ellipse (with the Sun at one focus). An ellipse is a figure whose points lie equidistant from two fixed points (the foci). When the foci are close to one another, the figure is nearly a circle. Kepler s Platonism from the Mysterium Cosmographicum (1596) 33 Newton s Rejection of the Platonic Program 34 In this book I intended to demonstrate, that the all-good and almighty God at the creation of our moving world and at the arrangement of the celestial orbits used the five regular polyhedra, which from Pythagoras and Plato s times and up to now got so loud glory, and selected a number and proportions of celestial orbits, and also the relations between the planet motions pursuant to the nature of the regular polyhedra. The Earth orbit is the measure of all orbits. Around of it we circumscribe the dodecahedron. The orbit circumscribed around of the dodecahedron is the Mars orbit. Around of the Mars orbit we circumscribe the tetrahedron. The orbit circumscribed around of the tetrahedron is the Jupiter orbit. Around of the Jupiter orbit we circumscribe the cube. The sphere circumscribed around of the cube is the Saturn orbit. In the Earth orbit the regular icosahedron is inserted. The orbit entered in it is the Venus orbit. In the Venus orbit the octahedron is inserted. The orbit entered in it is the Mercury orbit. Newton was the first to abandon the Platonic Challenge, in his Principia Mathematica of He does not attempt to reduce the motions of the planets to a composite of uniform motions. Rather, he explains them as the result of a combination of uniform straight line (inertial) motion disrupted by outside forces primarily the gravitational attraction of the Sun but also that of other planets If the whole orbit of a planet were caused by the attraction of the Sun, its orbit would be an ellipse. But the planet & the Sun exert a mutual attraction, each rotating around the other, & the planets exert an attraction on each other. So, the orbits are only approximately elliptical. The very stability of the solar system remains controversial until the publication of Laplace s Méchanique Céleste ( ) Newton s Physics as a Continuation of the Platonic Program Quantum Mechanics Quantum mechanics offers mathematical models of physical systems that have great predictive accuracy, but lacks the kind of physical models familiar from Democritean philosophies of nature. Newton did not abandon Plato s mathematical approach to the philosophy of nature. He called his great book on physics Philosophiae Naturalis Principia Mathematica The Mathematical Principles of Natural Science. In this, he followed in the tradition of Galileo, who had said, The book of nature is written in the language of mathematics. The state of a physical system is expressible as a vector moving (as the system changes its state over time) determinately through a multidimensional phase space. Any observation of the system results in the immediate & discontinuous motion of the vector to one of several other positions in the phase-space, each of which is mathematically related to the position immediately before the observation How is this in the spirit of Plato? No one can imagine (or draw) this, though 3-dimensional approximations might be drawn. (Think of the Cave & the reflections in the water outside the Cave.) But those who prepare themselves (by the study of mathematics) can acquire some understanding of the quantum mechanical reality that underlies the world of experience.

7 37 38 The Timaeus & the Doctrine of Creation The Timaeus does not offer a creationist account of the world According to Timaeus (e.g., 28a), the present state of the world is a result of an intelligent being (the Demiurge, lit. "craftsman") imposing order on pre-existent matter The concept of creation has two semantic notes divine action exnihilation (coming into being out of nothing) This is probably best taken as the essential note» since only divine action can bring something into being out of nothing (cf. Aquinas, S.T. 1a, Qq ) It is not merely an historical, but an ontological claim It is explicitly rejected by many ancient philosophers» ex nihilo nihil fit (Lucretius, De Rerum natura, 1.149&205; 2.287) The Timaeus does offer an account that many Christians find congenial as it is one in which The world is designed by the Demiurge (a craftsman, cf. Gen 2:7) even if it was not exhnihilated by God & dependent on him for its continued existence Dualist Anthropology Plato s Legacy Man is composed of two independent substances body & soul. Platonist Mathematics The objects of mathematical knowledge are independently existing mathematica. Mathematical Physics Mathematics is the key to physics Semi-creationist Cosmology