Plato's Confusions , Platonic Solids

Plato's Confusions, Platonic Solids

In this report, I will try to provide an argument for Plato's approach to philosophy, where my main assertion will be that all Platonistic philosophies on the reality hidden behind the appearence are ultimately begging the question. I will try to support this assertion by several evidence from his book Timaeus, and I will mainly focus on his approach to mathematics.

Timaeus, as all other texts of Plato, is hard to read and almost impossible to understand at the first time. This is usually the case about Plato's texts, and Plato's philosophy in general, that they are difficult to follow by a critical mind of today. What he does is to presume weird assumptions, and then by use of valid argumentation and beautiful explanations, draw even weirder conclusions.

An explanation, by its very definition, must be convincing; but in Plato's case, to be able to follow the explanation, the reader has to be convinced already. So, if we read the text without asking any 'why?' questions and accept them true, then and only then are we able to acknowledge the plausibility of his logic. Plato himself once admits it: “I must endeavour to explain to you in an exposition of an unusual type; yet, inasmuch as you have some acquaintance with the technical method which I must necessarily employ in my exposition, you will follow me.”(Tim.53c) Here, of course, he means that the listeners are already studying in his Academy and are accustomed to his methods of thinking.

This kind of an argumentation is essential for idealism; because idealism in itself cannot be comprehended by human beings. It is the clearest expression of the egocentric ideology, and this part I will try to demonstrate during this report.

To begin with, I would like to give a brief summary of Timaeus. First of all, Plato's God is not a Creator in the sense that he creates everything from nothing. Instead, he “only imposes order and system on pre-existing Chaos.”¹ Hence, “he brought order from disorder, deeming that the former state is in all ways better than the latter.”(Tim.30a) After that, Plato's argument goes as the following: Everything that exists has a bodily form, i.e. it is visible and tangible. To be visible, it must contain fire, and to be tangible, it must be solid, implying the existence of earth. For both fire and earth to exist, we need a third element to conjoin them.(Tim.31bc) But three objects define a surface and the body of All is solid. Therefore, we need a fourth object and we have the equations air:water = fire:air and water:earth = air:water (Tim.32b). This is indeed a mathematical explanation of the four elements theory of Empedocle, but has no mathematical meaning though. It seems totally impossible to understand what these equations really mean. Then, Plato explains why the universe has the shape of a sphere (Tim.33b). But suddenly, he begins to explain why the universe has no need for eyes, ears, mouth, hands, legs and feets (Tim.33cd). It is in fact typical for a person to give more and more explanations when he/she himself/herself sees that his/her words are not convincing. Indeed, Plato becomes less and less convincing with his exuberancy. Next, having these four elements, “God began by first marking them out into shapes by means of forms and numbers”(Tim53d). Here, Plato introduces the five regular polyhedra, today often referred as Platonic solids. Afterwards, he assigns a solid to each element. To this point, that Plato's God obeys to a certain reasoning, I shall return later. Tim.57c onwards, Plato shortly explains why there are infinitely many kinds of physical objects and discusses sense perception, human (actually, man) physiology and how other animals and women are created from men; demonstrating why Aristotle, being his student, studied Physics and tried to really understand natural phenomena, as Plato's descriptions hardly make any sense.

However, Plato is historically the first philosopher to “attempt to describe natural phenomena by mathematical means and not by mythological means.”² Mathematical objects, being the objects of Reason and not of physical world, are suitable to his Ideas or Forms. Plato, following Parmenides' distinction between the Way of Truth and the Way of Opinion, distinguishes between the Being and the Becoming (Tim.28a). “Being is changeless, eternal, self-existent, apprehensible by thought only; Becoming is the opposite – ever-changing, never truly existent, and the object of irrational sensation.”³ By these properties, mathematics fits to Being; since a triangle itself cannot essentially be drawn on a piece of paper, but is more an object of abstraction of all the triangles drawn. Hence, Plato elegantly gives a mathematical description to natural phenomena. Moreover, in his “Academy, with the motto “Let no man ignorant of geometry enter” inscribed over its gate”⁴,geometry was a major subject in higher education.

Before entering a discussion on why Plato's idealism fails to explain natural phenomena, I would like to provide a few evidence on Plato's egocentricism and the reflection of it in his philosophy. First, I would like to recall the Allegory of the Cave in Plato's The Republic. In the allegory, Plato obviously refers to himself as the one who is able to see the reality, and not to someone else; and his justification is valid only when other people believe in the story beforehand. The Republic goes on defining philosopher kings, where we identify that Plato again means himself. He possesses the truth, he doesn't have to justify the truth of his assertions; and he invites us -though sadly, knowing that not every one is able to see the reality behind appearence- to his philosophy without offering any evidence of its truth, neither physical nor metaphysical.

Secondly, Plato is never interested in things which everyone can understand. Therefore, things that can be understood by everyone are, according to Plato, not worth studying; they are not true, not real. And hence his conclusion: “The physical world is an imperfect realisation of the ideal world and is subject also to decay. Hence the ideal world alone is worth of study.”⁵ Also, as we anticipated: “Only through strenuous, properly trained reasoning can the human mind discover Forms behind their ephemeral and shadowy images presented by the senses.”⁶ That's one of the reasons why he “disdained applied mathematics and protested the use of moving mechanical instruments in geometry.”⁷ Contrary to his teacher Socrates -who was known as teaching mathematics to a slave-, Plato's philosophy is not for the average person; because Plato feels a superiority in himself, to which point I will give more evidence later. But before that, I must, in this context, return to the story of the creation of the universe. Recalling the remark that even God follows a certain reasoning while creating the universe, we must definitely notice that God follows Plato's reasoning. So, Plato is quite sure that he has the true knowledge.

Having recalled the description of the creation, I would also like to point out the dodecahedron problem, which a careful reader might have noticed. I already mentioned about God's assigning a solid to each element; now, I would like to give a more extended summary of what Plato describes. He takes two triangles, and after certain procedures that were already known in his time, he constructs four of the regular polyhedra; namely, the pyramid (tetrahedron), the octahedron, the icosahedron, and the cube. The first defect of Plato's reasoning is that the dodecahedron cannot be constructed with his triangles. Furthermore, he knows that there are five regular polyhedra, but he also assumed that there are four elements but not five. Here, Plato's ego shows himself and he finds the solution: “And seeing that there still remained one other compound figure, the fifth, God used it up for the Universe in his decoration thereof.” (Tim.55c) The searchings of the curios reader are in vain, there are no other explanations for that. Plato does not even mention dodecahedron through the text again. This is, in my opinion, the clearest case that, according to Plato, if something does not fit into his story, then his assumption is that it was already a special case for the God in the Creation.

Lastly, Plato directly humiliates people who do not approach to philosophy in his way. After the creation of men, Plato depicts how other animals are created from men: “And the tribe of birds are derived by transformation, growing feathers in place of hair, from men who are harmless but light-minded”(Tim.91d), which is followed by “And the wild species of animal that goes on foot is derived from those men who have paid no attention at all to philosophy nor studied at all the nature of the heavens, because they ceased to make use of the revolutions within the head and followed the lead of those parts of the soul which are in the breast.”(Tim.91e) Here, Plato clearly identifies the people who are not interested in things he does, as inferior to him. He continues: “On this account also their race was made four-footed and many-footed, since God set more supports under the more foolish ones, so that they might be dragged down still more to the earth.”(Tim.92a) It is not incorrect, I suppose, to interpret this statement as an insult to the people who don't philosophise in Plato's way.

Taking the above discussion into consideration, I would like to point out the similarity between Plato's explanations about the creation of the universe, and the typical “God exists because the Bible says so.” arguments. In both of them, the argument begs the question, thus convincing only those who were already convinced. Platonic Idealism, and idealism in general, is based on strange assumptions which are declared as unquestionable. This is, in fact, the main problem of idealism. Interestingly, however, “Plato had criticized the earlier philosophers for their failure to indicate the Cause of the physical processes by which they explained the World.”⁸

On the other hand, as Plutarch report Plato's famous “God eternally geometrizes”⁹, Plato was the first philosopher to see the essential connection between mathematical knowledge and natural phenomena. His emphasis on mathematics made it possible for his followers to make effort on finding the right relations between them; whereof one of the longest lasting higher education institutions of Europe, the Academy (B.C.E. 387 – C.E. 529)¹⁰, was for centuries the leading institution of mathematics in Europe.

All in all, today's natural sciences owe a lot to Plato's writings, but only in the historical sense as his works encouraged debates and studies on the subjects he discussed, and not in the philosophical sense as his texts offer no help to today's scientists and thinkers.

1Bury, R.G. ; Introduction to the Timaeus in PlatoVII ; Page T.E. , Capps E. ; Rouse W.H.D. et al ; The Loeb Classical Library ; 1952 ; page 7

2Artmann, Benno ; Euclid - The Creation of Mathematics ; Springer ; 1999 ; page 306

3Bury, R.G. ; Introduction to the Timaeus in PlatoVII ; Page T.E. , Capps E. ; Rouse W.H.D. et al ; The Loeb Classical Library ; 1952 ; page 6

4Calinger, Robert ; A Contextual History of Mathematics ; Prentice-Hall ; 1999 ; page 103

5Kline, Morris ; Mathematical Thought from Ancient to Modern Times ; Oxford University Press ; 1972 ; page 44

6Calinger, Robert ; A Contextual History of Mathematics ; Prentice-Hall ; 1999 ; page 103

7Calinger, Robert ; A Contextual History of Mathematics ; Prentice-Hall ; 1999 ; page 103

8Bury, R.G. ; Introduction to the Timaeus in PlatoVII ; Page T.E. , Capps E. ; Rouse W.H.D. et al ; The Loeb Classical Library ; 1952 ; page 5

9Kline, Morris ; Mathematics: The Loss of Certainty ; Oxford University Press ; 1980 ; page 6

10Kline, Morris ; Mathematical Thought from Ancient to Modern Times ; Oxford University Press ; 1972 ; page 42-43