On Ancient Greek Mathematics

Now, where did I leave my car keys......

As moderns, all we can do is gape in wonder at the intellectual achievements of the ancient Greeks. With the eclipse of Greece following their conquest by Rome, Greek genius descended unto stasis. In the West, their achievements would not be matched for over a Millenium.

Mathematics is the epitome of clarity of thought. The scientific method, although powerful, can only give us mere mortals a side shifted glance at true knowledge. To view knowledge full-on there is only one vision, and that vision is mathematics. Only mathematics can lead fragile humans to true knowledge. One plus one is always two, at least in base ten, regardless of context and intellectual stance. To think otherwise is to contemplate the mind of a madman, or perhaps a genius.   
   
The ancient Greeks were the first civilisation to undertake mathematics for its own sake regardless of any practical application. The ancient Egyptians had developed their own mathematics but it was subject to matters of state and engineering. It was a practical discipline and there is no evidence that the Egyptians had any abstract concept of mathematics that was not allied to the practical and mundane.

Although the Egyptians initially influenced Greek mathematics of the 7^th and 6^th centuries BC, the Greeks expanded and developed mathematical principles far beyond anything envisioned by the Egyptians. The Greeks, as far as we are aware, were the first ancient culture to undertake rigorous mathematical proofs to underpin their geometrical conjectures.     
   
In the realm of abstract mathematics, the Greeks were supreme for their time and managed to squeeze everything that could be possibly imagined from a straight edge and a compass. They achieved much even though their mathematics was limited by their crude number notation and the absence of the concept of zero. Like the Romans, the Greeks substituted letters for numbers. The system was additive and unlike the system we use today, the position of the numbers was not important. Thus, the development of algebra was denied to them. It can be argued that the development of algebra and its advanced offshoot/offspring, calculus, is the most majestic and exquisite branches of mathematics; they are certainly the most practical. And even though Greek mathematics, of the time, could not work with abstract numbers and notational substitutions, they did manage to develop a form of geometric proto-calculus under the auspices of the last great mathematician of Western antiquity, Archimedes.  The death of Archimedes in 235 BC, at the hands of a Roman soldier, signalled the end of Greek mathematics in any innovative sense. Archimedes’ calculation of the area of two-dimensional figures by the product of the infinitesimal anticipates the great insights of Newton and Leibnitz in the 17^th century.

Up until a hundred years ago, Euclid’s 14 books on geometry, ‘The Elements’, were an essential study for Victorian grammar school pupils. All of geometry is laid bare in these worthy tomes. These books are crammed with ancient mathematical wisdom and wonderful expositions of mathematical proofs. The basic geometrical axioms established by ancient Greek thinkers is probably one of their greatest legacies.

Most folk have heard of Pythagoras and the theorem that bears his name, although it is unlikely that Pythagoras ‘invented’ this theorem himself. Why he has become associated with this cardinal rule has been lost in the vast (nay extensive) mists of time. Pythagoras and his acolytes flourished in the 6^th century BC and were a rather a strange bunch. If they were existent today we would describe them as a cult. They espoused some rather odd ideas and they were obsessed with numbers. To the Pythagoreans, everything was related to number and they extended their mathematical insight into the realm of musical harmony. When one of their acolytes discovered irrational numbers (a number that cannot be expressed as a fraction) he was taken out to sea and drowned. In this way, they hoped to suppress this seditious and dangerous notion. With the discovery of irrational numbers, mathematics lost its impeccable perfection and symmetry at least in the eyes of the Pythagoreans. The discovery that ‘rogue numbers’ had an abstract existence was a serious blow to this rather weird aesthetic sect- they would never be the same again. I’ve discussed Pythagoras, and his followers, previously in this blog: here is the link.

Alexander the Great’s conquest of Persia and parts of India introduced the Greeks to Babylonian astronomy. During his excursions, Alexander founded a number of cities throughout the former Persian empire, the greatest, of course, was Alexandria in northern Egypt. Egyptian Alexandria became a seat of great learning and knowledge and attracted eminent mathematicians and philosophers throughout the Hellenistic world.

With the subjection of the Greeks by Rome and their integration within the Roman Empire, innovative mathematics ceased. The pragmatic/phlegmatic Romans had little time for abstract mathematical concepts. If maths could help with the building of straight roads or the manufacture of engines of war, all well and good, however, the Romans had no interest or aptitude for mathematics in general, especially in the abstract. And it is probably true that all Roman mathematics was wholly dependent and derivative from Greek ideas and principles.                            

Western mathematics would languish until invigorated by Indian mathematical concepts, transmitted through contact with the Muslims, during the early middle ages. The power of ‘Arabic numerals’, our current counting system, was quickly recognised by the scholars of the day, although the Catholic church was against it (O, what a surprise!). It allowed Western mathematics to develop beyond geometry and expand into new and exciting mathematical territory. From the Hindus, the West came to understand the concept of zero. The importance of this simple (?is it) concept cannot be underestimated and historically, together with the development of the positional decimal number system, represents perhaps the most important insight in the development of mathematical theory.

Perhaps, I will be so moved by zero’s noble and lofty countenance, that I will pen a post dedicated to its magnificence: I’ll put it on the list for future contemplation.