# Greek Math, A 'Strange Detour'?

1. Feb 2, 2014

### zoobyshoe

What do you think of this take on Greek math? I have never heard such a sentiment before.

"Recorded mathematics begins in the Orient, where, about 2000 B.C., the Babylonians collected a great wealth of material that we would classify today under elementary algebra...

…It may be that the early discovery of the difficulties connected with 'incommensurable' quantities deterred the Greeks from developing the art of numerical reckoning achieved before in the orient. Instead they forced their way through the thicket of pure axiomatic geometry. Thus one of the strange detours of the history of science began, and perhaps a great opportunity was missed. For almost two thousands years the weight of Greek geometrical tradition retarded the inevitable evolution of the number concept and of algebraic manipulation, which later formed the basis of modern science."

What Is Mathematics?
Richard Courant and Herbert Robbins, 1941

2. Feb 3, 2014

### SteamKing

Staff Emeritus
Although the Greeks could express numbers which had colossal magnitudes, their notation did not easily lend itself to numerical calculation. Regular Greek numeration crapped out when magnitudes reached 10000. Archimedes had to invent a special notation which would allow him to discuss numbers many orders of magnitude beyond 10000.

http://en.wikipedia.org/wiki/Greek_numerals

http://en.wikipedia.org/wiki/The_Sand_Reckoner#Naming_large_numbers

For all of Archimedes' cleverness, his notation still hobbled one intent on doing numerical calculations. The Greeks' difficulty was one shared by many ancient civilizations, and the development of Arabic numerals and the means (algorithms) to calculate with them hardly occurred overnight. Still, the Greeks were able to lay out very complex, but subtle, shapes when they constructed their temples, like the Parthenon. A number of years ago, there was an article in Scientific American about the discovery of one set of building plans for a Greek temple:

L Haselberger, "The Construction Plans for the Temple of Apollo at Didyma," Scientific American 253.6 (1985) 126-32

The average temple contains a number of visual illusions which make the structure appear to be more pleasing to the eye, and it is a great testament to the craft of the temple builders of how painstaking their methods were.

3. Feb 3, 2014

### AlephZero

Early number systems also had the limitation that they did not recognize zero, or negative quantities, as "numbers".

This crippled the development of algebra or any form of symbolic computation, because almost every general algorithm for solving a problem resulted in a collection of special cases, depending on whether intermediate results were positive or negative (using modern terminology).

For example Euclid's elements has geometrical proofs of several results in elementary algebra, like $(a+b)^2 = a^2 + 2ab + b^2$ and $(a-b)^2 = a^2 - 2ab + b^2$, But with no concept of negative numbers those results were treated as independent of each other, and both proved separately from first principles.

But the other hand, we now know the Greeks knew a lot more geometry, and a lot more ways of doing what we would now call technical drawing, than what was in Euclid. Unfortunately much of the original literature is lost. The Arabs collected a lot of it, and translated it, but the Western Crusaders weren't very interested in mathematics. We have a good idea how much was lost, from some of the Arabic library catalogs that have survived, but a list of book titles doesn't tell you much about the contents.

4. Feb 3, 2014

### The Dog Star

The Greeks were the first civilisation as far as we know to use infinities in any context. They had uncountable infinities and countable infinities although as noted before, they did not really appreciate their use. Later the concept was further explored by Arabian scholars who had read material from Greek archives such as in Alexandria, and later still the Indian mathematicians introduced a concept called all or atma with 3 infinities:
Enumerable: lowest, intermediate, and highest
Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
Infinite: nearly infinite, truly infinite, infinitely infinite

That which was infinite based on some trade of ideas I have no doubt between the Persians the Greeks and the Arabians as they were then. Ultimately this lead to an infinity being something to do with all there is, but it was a long old process drawn out between something we know and something we can not because it is somewhat all and divine. Of course most know about Cantor and his infinity and the later scholars but they were as always standing on the shoulders of giants, although they were of course finitely tall.

I think 0 was introduced first by the Persians or might be Babylonians but I may be wrong on that they are responsible for that damnable base 12 time system though 24 hours in a day etc. :)

In reference to the above he's very much right Algebra for example is an Islamic/Arabian word although the Babylonians were the first to use it as far as is known.

It's perhaps also interesting to note that the ancient Egyptians were the first to use 3 dimensional "vectors" to build things, the first noted occurrence of a 3d vector with a weight and hence gravitational consideration is recorded on one of the pyramids. It shows how the distance and amount of force will let you work out how to pull a resultant block up an incline with a resultant force equal or greater than it and on average how many men that would take given the usual strength of those pulling. Which you have to admit for a pre mathematically sophisticated civilisation 5000 or whatever years ago or so is quite impressive.

Last edited: Feb 3, 2014
5. Feb 4, 2014

### zoobyshoe

Thanks for the responses, everyone!

I googled to find the geometric representations of AlephZero's squared binomials and found this site:

http://pespmc1.vub.ac.be/POS/Turchap11.html

Where the cause of the limits of Greek math is laid on the doorstep of Platonic thinking:

"THE GREEKS' failure to create algebra is profoundly rooted in their philosophy. They did not even have arithmetic algebra. Arithmetic equations held little interest for them: after all, even quadratic equations do not, generally speaking, have exact numerical solutions. And approximate calculations and everything bound up with practical problems were uninteresting to them. On the other hand, the solution could have been found by geometric construction! But even if we assume that the Greek mathematicians of the Platonic school were familiar with arithmetic letter symbols it is difficult to imagine that they would have performed Descartes' scientific feat. To the Greeks relations were not ideas and therefore did not have real existence. Who would ever think of using a letter to designate something that does not exist? The Platonic idea is a generalized image. a form a characteristic: it can be pictured in the imagination as a more or less generalized object. All this is primary and has independent existence an existence even more real than that of things perceived by the senses. But what is a relation of line segments? Try to picture it and you will immediately see that what you are picturing is precisely two line segments, not any kind of relation. The concept of the relation of quantities reflects the process of measuring one by means of the other. But the process is not an idea in the Platonic sense: it is something secondary that does not really exist. Ideas are eternal and invariable, and by this alone have nothing in common with processes.

Interestingly. the concept of the relation of quantities, which reflects characteristics of the measurement process, was introduced in strict mathematical form as early as Eudoxus and was included in the fifth book of Euclid's Elements. This was exactly the concept Descartes used. But the relation as an object is not found in Eudoxus or in later Greek mathematicians: after being introduced it slowly gave way to the proportion which it is easy to picture as a characteristic of four line segments formed by two parallel lines intersecting the sides of an angle.

The concept of the relation of quantities is a linguistic construct, and quite a complex one. But Platonism did not permit the introduction of constructs in mathematics: it limited the basic concepts of mathematics to precisely representable static spatial images. Even fractions were considered somehow irregular by the Platonic school from the point of view of real mathematics. In The Republic we read: "If you want to divide a unit. learned mathematicians will laugh at you and will not permit you to do it: if you change a unit for small pieces of money they believe it has been turned into a set and are careful to avoid viewing the unit as consisting of parts rather than as a whole.'' With such an attitude toward rational numbers, why even talk about irrational ones!

We can briefly summarize the influence of Platonic idealism on Greek mathematics as follows. By recognizing mathematical statements as objects to work with. the Greeks made a metasystem transition of enormous importance but then they immediately objectivized the basic elements of mathematical statements and began to view them as part of a nonlinguistic reality, "the world of ideas." In this way they closed off the path to further escalation of critical thinking to becoming aware of the basic elements (concepts) of mathematics as phenomema of language and to creating increasingly more complex mathematical constructs. The development of mathematics in Europe was a continuous liberation from the fetters of Platonism."

6. Feb 5, 2014

### Hornbein

Greek mathematics was geometric. Algebra was invented by the Islamic nations (Al Gebra) and didn't catch on in Europe until after Isaac Newton. There was a time when the Islamic countries were the most modern in the West.

But the axiomatic method has its advantages. And how were the Greeks supposed to know that algebra would win in the end? So I don't find it at all strange.

7. Feb 5, 2014

### Tobias Funke

I think this is a pretty common belief. From Mathematics in Western Culture