Plato's mathematicsPlatosMathematics2008-09-25 08:38:522009-12-26 13:47:47Definition% this is the default PlanetMath preamble. as your knowledge
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Plato's Mathematics (\PMlinkexternal{Platonism}{http://en.wikipedia.org/wiki/Philosophy_of_mathematics}) from Wikipedia
Platonism is a form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers, in the modern time period. The term Platonism is used because such a view is seen to parallel Plato's belief in a World of Ideas (typified by Allegory of the cave): the everyday world can only imperfectly approximate of an unchanging, ultimate reality, a topic that requests modern scholars to read the ancient math texts. Both Plato's cave and Platonism have meaningful, not just a superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular Pythagoreans of ancient Greece, and Egyptians that used the same form of arithmetic. All ancient scribes in the Ancient Near East believed that the world was, quite literally, generated by numbers.
The modern problem of mathematical platonism is this: precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, which is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One answer might be ultimate ensemble, which is a theory that postulates all structures that exist mathematically also exist physically in their own universe. Yet, to see the world of numbers through ancient eyes, read Plato's Republic:
Plato spoke of mathematics by: "How do you mean?
"I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument. You know how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply, taking care that one shall continue one and not become lost in fractions.
That is very true.
Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible, --what would they answer? "
from Chapter 7. "The Republic" (Jowell translation).
In context, chapter 8, H.D.P. Lee translation, reports the education of a philosopher containing five mathematical disciplines:
1. arithmetic, written in unit fraction 'parts' using theoretical unities and abstract numbers;
2. plane geometry, and,
3. solid geometry consider the line to be segmented into rational and irrational unit 'parts';
4. astronomy;
5. harmonics, that include music.
Translators of the works of Plato rebelled against practical versions of his culture's practical mathematics. However, Plato himself, and Greeks, had generally copied 1,500 older Egyptian fraction abstract unities, one being a hekat unity scaled to (64/64) in the Akhmim Wooden Tablet, thereby not getting lost in fractions.
Godel's platonism postulates a special kind of mathematical intuition that may let us directly perceive mathematical objects. This view bears resemblances to things Husserl said about mathematics, and supports Kant's proposed idea that mathematics is analytic-synthetic distinction: conceptual containment (synthetic), A priori, and a posteriori (philosophy). Philip J. Davis and Reuben Hersh suggest in 'The Mathematical Experience' that most mathematicians act as Platonists, even though, if pressed to defend the position carefully, they may retreat from this formalism taken from the philosophy of mathematics.
Mathematicians sometimes hold opinions that amount to nuanced versions of Platonism. These ideas are sometimes described as neo-platonism.
Modern neo-platonic points of view have not provided clear templates to read Greek or Egyptian math. For example, the Greek arithmetic and geometry idea of multitude has not been described in terms of rational number conversion to unit fraction series methods. Hence, for the near future, Plato and Euclid's idea of multiple is of modern research interest only.