Plato's mathematics

Plato's Mathematics (Platonism) from Wikipedia

Platonism reports as realism suggests that Greek mathematical entities were abstract with no causal properties. The entities may have been eternal and unchanging. This view of numbers was claimed by mathematicians in the ancient Classical Greek era as well as medieval and modern mathematicians. The term Platonism offers e parallel to Plato's belief in a World of Ideas typified by Allegory of the cave: the everyday world was imperfectly approximate to an unchanging, ultimate reality. The modern topic requests modern scholars read the ancient math texts as the texts were written in historical time periods. Plato's cave and Platonism had meaningful implications, not just superficial connections, because Plato's ideas were preceded and influenced by Pythagoreans of ancient Greece, and Egyptians that used the same form of exact arithmetic, the scaling of rational numbers into exact unit fraction series. Ancient Near East may have believed that the world was literally generated by numbers, but only the numeration systems defined by exact rational numbers provided the desired clarity.

A modern problem of mathematical platonism suggests where and how do the mathematical entities exist and how do we know about them. There a world completely separate from our physical one which is occupied by the mathematical entities. This is the theoretical realm. How can any one gain access to two abstract worlds, one theoretical and one practical, and discover truths about math entities? One answer might be ultimate ensemble a theory that postulates structures that exist mathematically also exist physically. To see aspects of the world of numbers through ancient eyes read Plato's Republic:

Plato spoke of the ancient mathematical world in his life-time by asking in the "Republic", 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 Plato's works at various times rebelled against practical versions of classical practical mathematics. Plato himself and Greeks copied Egyptian fraction abstract unities several ideas were 1,500 years older. For example, a hekat unity (64/64) recorded in 1950 BCE was divided by 3, 7, 10, 11, 13 (and n as needed) in the Akhmim Wooden Tablet. Ahmes 300 years later divided 100 hekat written as (6400/64) by 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100 in RMP 47. Ahmes replaced one hekat by 320/320, 320 ro, in RMP 35-38. In RMP 38 320 ro was multiplied by 7/22, with the answer returned to 320 ro by multiplying by 22/7.

Thirdly Ahmes used an unity form

53/53 = 2/53 + 3/53 + 5/53 + 15/53 + 28/53

in RMP 36. Taking all three method, each allowed Egyptian, Greek, Arab and medieval scribes to not get lost in unit fraction calculations. A fourth meta method developed proofs.

Egyptian, Greek, Arab and medieval weights and measures defined region-wide economic units within localized economic systems that spanned across 3,700 years of the Ancient East East. Middle Kingdom Egyptians used scaling methods, such as:

4/13 by LCM 4 to 16/52 = (13 + 2 + 1)/52 = 1/4 + 1/26 + 1/52.

with 13 + 2 + 1 recorded in red.

Arabs, and Fibonacci scaled difficult rational numbers, like 4/13 by two LCMs, considering

4/13 to 1/4, with remainder 3/52 scaled by 1/18 to obtain a final

1/4 + 1/18 + 1/468

series, thereby maintaining the unit fraction system a few more years.

The oldest unit fraction system formally began in 2050 BCE, and ended in Europe in 1454 AD when the Liber Abaci fell out of favor. The unit fraction system completely ended in 1637 AD when the Arab world introduced modern Arabic script, destroying linguistic connections to the very old economic system.

Godel's platonism postulated a mathematical intuition that allowed perceptions of mathematical objects, but not the precise mathematical language that describes the object. This view resemblances things Husserl said about mathematics, and supports Kant's proposed idea that mathematics can be 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 may infer opinions that amount to nuanced versions of Platonism. These ideas are best described as neo-platonism.

Modern neo-platonic points of view provide unclear templates to decode ancient Greek and Egyptian finite arithmetic texts. To explicitly decode Greek arithmetic, algebra, and geometry the Greek word multitude is understood as ancient scribes reported a finite least common multiple scaling idea. To Plato and mentor Egyptians multitude m scaled rational numbers n/p by LCM m to (n/p - 1/m) = (mn - p)/mp, setting (mn-p) = 1 whenever possible. The Greek, Arab and medieval method connects to the oldest Egyptian scaling method that scaled n/p by LCM m to mn/mp before the best divisors of mp were recorded in red summed to numerator mn in the RMP, Kahun Papyrus 2/n tables and all hieratic uses.