Why Mathematics Is So Crucial for Philosophical Training according to Plato - Essay Example

Written above the door of Plato’s academy was the inscription: “Let no one unversed in geometry enter here”. Why is mathematics so crucial for philosophical training according to Plato? Plato – one of the most important educational theorists and curriculum developers in Western history, whose contribution to the theory and practice of mathematics education has had a profound impact over the ages, possessed educational interests and accomplishments founded on the grounds of mathematical education based upon proofs and facts. The entrance to the Academy he established in Athens, famously announced: “Let no one ignorant of geometry enters here.” - inscription above Plato’s Academy (Q1) Some of the admiring features of Plato’s Academy were: Tuition Free: The students of the Academy paid no set fee. It was expected that wealthier students would give gifts. It is believed that Dionysius II gave as his gift a sum equivalent to about a half-million dollars in American currency. Requirements: The Academy accepted only advanced students who possessed knowledge of geometry. Teaching Method: Plato lectured, utilizing his vast knowledge to present an organized body of information to his students. (Plato) According to Plato there are absolute truths, mathematics is the key. While statements about the physical world will be relative to the individual and culture, mathematics is independent of those influences. Plato had Four Basic Points regarding mathematical applications: 1. There is certainty. 2. Mathematics gives us the power of perception. 3. Though the physical applications of mathematics may change, the thoughts themselves are eternal and are in another realm of existence. 4. Mathematics is thought and, therefore, it is eternal and can be known by anyone. (Today we view mathematical ideas as free creations of the human mind. They are the tools we use to map the world. Experience is the key. Although absolute certainty is not possible, we can still attain accurate knowledge and reasonable beliefs about the world.) (Nick Strobel) In his classic volume, Republic, the mathematical sciences (arithmetic, geometry, astronomy, and harmonics) formed the foundation of Plato’s curriculum, which included Elementary, military training and higher education. When we elucidate Plato’s model of mathematical cognition and learning, we usually consider his mathematical curriculum, Plato may readily be seen as having put theory into practice on a scale unprecedented, in the history of mathematical education. (Stephen R. Campbell) Plato’s ideas of Mathematics in life and in education seem far less extreme than those touted by Pythagoras, as can be seen by reading Plato’s Laws. Mathematics was then considered the basis from which to move into philosophical thought and as such Plato proposed that studying mathematics should occupy the student for the first ten years of his education. This, he believed provided the finest training for the mind since they were then able to understand relations that cannot be demonstrated physically. Since clear logical thinking was prized not only in philosophical discussions but also in the political arena, Plato encouraged his students to train in mathematics because he thought that it encouraged the most precise and definite kind of thinking of which humans are capable. Plato’s Republic gives a different level of mathematical learning to the one just described. A learning reduced to the elementary, which was possibly inspired by the public pressure from the Romans who had a very different opinion of the worth of Mathematics in Education at that time. (J O’Connor and E F Robertson) Plato’s Mathematical Curriculum For Plato, the purpose of mathematics education was to feel the force of reason, and to explain and grasp mathematical notions with a solid ground based on logical theories. Since we are not extraordinary mathematicians, therefore we cannot give a full formal account of reason to justify our predictions. Unless we secure absolute validity at the price of absolute vacuity, we shall reach an end of arguing, where no more can be said, and we can only hope that the desire to know, and the ability to reason, will enable the person we are arguing with, to abandon opinions held as they are in the favour of his judgments held because they are true. But though that is the ultimate stage, there are many partial prescriptions and formalizations, which may enable him to see the error of his previous ways, and enable us also to detect flaws in our presentations, and see the whole issue in a more rational light. “Numbers are the highest degree of knowledge. It is knowledge itself.”- Plato (Q2) Plato was convinced by one of his colleagues that the cosmos operated in according with “a law of number,” and that it was within human potential to discover that law. It is no coincidence that Plato called his esteemed colleagues in the Academy to task to account for the “apparently inconsistent motions of the planets… by compositions of invariantly constant circular motions” (Vlastos, 1988, p. 362). For Plato, however, unlike today, the study of the quadrivium of mathematical sciences, to the exclusion of anything else, was a ten-year process of higher education for citizens (i.e., the guardian class) from the age of twenty to the age of thirty. This exclusive focus on the topics of arithmetic, geometry, astronomy, and harmonics served as “protreptic and proleptic instruments, positioning the student and providing hints for the work of completing the direction of thought by attending to ‘the things themselves’” (Wood, 1991, p. 525). Implication of Mathematics Education “There still remain three studies suitable for free man. Arithmetic is one of them.” – Plato (Q3) Plato’s theories and practices in mathematical education have exerted an influence unprecedented and unparalleled in the history. One may take issue with his ontology of transcendent Forms. The issues involved in explicating these ideas, in Platonic terms or otherwise, remain at the forefront of research in the philosophy of mathematics. For instance, in considering an infinite line as opposed to a finite line segment as a mental object, at what point and in what ways does imagination fail? At what point must we resort to propositional definitions and proofs? What happens between the notion of a potential infinity defined by n+1 for any natural number n, and the Platonic notion of an actual infinity defined by the set of natural numbers ? Questions such as these take on more contemporary relevance and immediacy for research in the psychology of mathematics education when considered from a cognitive rather than an ontological perspective. (Stephen R. Campbell) The Platonic Relation of Philosophy with Mathematics Philosophy, according to Plato, makes use of a method peculiar to it, which he calls ‘dialectic’ and that’s what he believed throughout his life. The exact nature of the Platonic dialectic is obscure, but this much is clear: philosophy proceeds by criticizing received opinions. Even mathematics, which is, the most developed of the sciences, is subject to philosophical criticism. According to Plato, mathematics rests on inarticulate assumptions, which influences the mathematical training and education to philosophically evaluate the results, as it is the philosopher’s task to bring these into the open and examine them critically. Judgment of mathematical assumptions are one way or the other based upon philosophy as philosophy is the highest form of inquiry, just because it alone involves no presuppositions. (Philosophy) “Geometry existed before creation.” – Plato (Q4) Plato’s extraordinary emphasis and perfection in mathematics and geometry has made scientists of today to research upon as to why mathematics is without exception the most important equipment for the problem under investigation. And so far as Plato promoted mathematics, all will agree that he had an understanding of its peculiar endeavours. Very few individuals have advanced mathematics as much as did Plato. But whoever considers it will surely find the judgment of such experts as Usener and Cantor justified: “Plato contributed infinitely more to mathematics than any one individual with every effort could ever have achieved,” that his lectures in the Academy “cannot be too highly valued for the history of mathematics,” and that beginning with a certain period “the Academy takes the leading rank in the history of Greek mathematics.” How greatly Plato valued the study of mathematics we know from one of his work, Republic, where he considered it as the best and even indispensable preparation for profitable philosophical studies and devoted much time to the study of mathematics than any other subject in the Academy. (Adam Alles) Republic emphasizes on a good education for the elite group in charge of Plato’s ideal society. Plato considered education in mathematics and astronomy to be excellent ways of sharpening the mind. He believed that intense mental exercise of this kind had the same effect on the mind that a rigorous physical regimen did on the body. Students at the Academy covered a vast range of subjects, but there was a sign over the door stating that some knowledge of mathematics was needed to enter -- nothing else was mentioned! Plato in particular loved geometry, and felt that the beauty of the five regular solids, he was the first to categorize, meant they must be fundamental to nature; they must somehow be the shapes of the atoms. (Michael Fowler) “He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side” – Plato (Q5) The Republic contains a passage, which presents this as a link to mathematical equations to which we are told never to strictly follow the perceptible presentations of mathematical relations. As we can clearly see from geometry, such a presentation can only give hints; can only be used as a symbol. Not even the greatest artist, a second Daedalus, could give an accurate, concrete picture of the mathematical concept by making a model of it or by making drawings of it with a pencil. And if we think of him as having executed his representations with the most painstaking care, we shall find that “a person, who is at all acquainted with geometry, would upon seeing them consider them most exquisitely executed, but he would find it ridiculous to take them seriously with the hope that in them he could grasp the truth of the relation of things equally large or twice as large, or of any other definite proportion.” Let us attempt to present even more clearly this imaginary example conceived by Plato himself. The triangle constructed of wood or metal bars, or drawn on the blackboard, is to realize the concept of the triangle, and is itself, because of its material form, not merely different from the triangle of the mathematician, but even the figure which is marked out by the immaterial lines of these forms is different (i.e., it is something no longer perceptibly concrete, but something which has come into being through the combining activity of the mathematician, and lies between sensibility and pure form)--at least such is Plato’s conviction. He is of the opinion that it is impossible to embody a thought accurately in a material form; Example is of three bars, which meet and form angles, extend in perfectly straight lines and meet in such a way that the sum of the angles is exactly equal to two right angles. (Adam Alles) When it comes to Plato’s philosophical views, he places the objects of mathematics, as mental elaborations remotely based on real quantity but proximately on the mind’s constructive activity. Like abstraction in general, it is a way of knowing in which the mind considers one aspect of a thing, leaving out of consideration other aspects of the same thing. Specifically, it considers the quantity of bodies, abstracting from their sensible qualities and motions. But the abstraction is constructive or completive as well as selective. The mind must add to the real foundation of the mathematical notion and complete its formal character, as it does with the notions of species, universal, time, and truth. The same would seem to be true of mathematical notions. In this new notion of mathematical objects real substance does not play an intrinsic role as the intelligible matter of quantity. Intelligible matter is within the order of quantity itself. Their proper subject of existence is the mind itself. They are not signs of anything in the external world. Hence mathematical terms cannot properly be predicated of anything real: there is no referent in the external world for a mathematical line, circle or number. Finally, mathematical notions are not false; but neither are they said to be true, in that they conform to anything outside the mind. (Armand Maurer) The most certain knowledge we have, the knowledge of mathematics, could not have come from sense perception: 1. In geometry we have access to perfect squares and circles, but no such objects exist in the material world. 2. We can know truths such as 2+2=4 without having to check our experience of the material world. The objects that we think about in mathematics must be real, since they are most certain. Since they could not exist in the material world, there must be another realm in which they exist that is even more real, the realm of forms. (Form) The requirements of object rationality afford adequate justification for adopting the Principle of Recursive Reasoning, Argument by Iterated Choice, and for there being infinite possibilities, and hence for various axioms of infinity; but there is an alternative view of the axioms, connected with meaning rather than rationality, which is also relevant to the completeness of theorems. The axioms of different geometries are often seen as implicitly defining different concepts of point and line. The axiomatic extension of the natural numbers to include the negative integers, the rational numbers, the real and complex numbers, can be viewed as articulating further our concept of number, as can the Axiom of Choice and other axioms of set theory, our concept of a set. Plato discusses the mystical nature of mathematics in Meno, The Republic, and The Laws. Plato gives us a division of the concept of human knowledge into a technical half and a theoretical half. He tells us that it is only by counting, measuring, and weighing that the individual branches of the first half can attain certainty, and overcome the vagueness of sense impressions and the uncertainty of the crude handicrafts which rely solely on sense-impressions. Number, quantity, and weight can be mathematically dealt with. But philosophical mathematics differs from the mathematics applied to these things; it (theoretical mathematics) deals with pure numbers, which consist of equal units, with surfaces and bodies not given in experience and whose motion is given only in thought. Only this pure mathematical theory possesses complete accuracy, and involves no contradiction. Plato’s philosophy of mathematics came from the Pythagoreans; so mathematical ‘Platonism’ as an ideal independent of human consciousness violates the empiricism of modern science. For Plato the ideals, including numbers, are visible or tangible. According to Plato, and many other mathematicians common knowledge and scientific knowledge are a posteriori. They came from observation of the material world. Mathematical knowledge is a priori – independent of contingent facts about the material world. Daily experience and physical experiment could conceivably be other than what they are, but mathematical truths will hold in every possible world, or so thought Plato. (Reuben Hersh) References Adam Alles, The Essence of Plato’s Philosophy: Lincoln MacVeagh, The Dial Press. Place of Publication: New York. Publication Year: 1933. Armand Maurer, Thomists and Thomas Aquinas on the Foundation of Mathematics: The Review of Metaphysics. Volume: 47. Issue: 1. Publication Year: 1993. pp 43 Forms, < http://www.anselm.edu/homepage/dbanach/platform.htm> J O’Connor and E F Robertson Michael Fowler, < http://galileoandeinstein.physics.virginia.edu/lectures/aristot2.html > Nick Strobel, Philosophy, The Encyclopedia of Philosophy, vol. 6 (Collier Macmillan Publishers, 1967), pp. 217-218 Plato, Reuben Hersh, What Is Mathematics, Really? : Oxford University Press. Place of Publication: New York. Publication Year: 1999. Routledge, The Conceptual Roots of Mathematics: An Essay on the Philosophy of Mathematics. Place of Publication: London. Publication Year: 2000 Q1, Q2, Q3, Q4, Q5 < http://www.pen.k12.va.us/Div/Winchester/jhhs/math/quotes.html > Stephen R. Campbell Read More