Philosophers before Socrates (470 BC–399 BC), the first philosophers in the early antiquity, are customarily mentioned as pre-Socratic. Socrates was an Athenian philosopher credited as one of the founders of Western philosophy, and as being the first moral philosopher of the Western ethical tradition of thought. An enigmatic figure, he had no writings, and is known chiefly through the accounts of classical writers writing after his lifetime, particularly his students Plato and Xenophon. Plato’s dialogues are among the most comprehensive accounts of Socrates to survive from antiquity, from which Socrates has become renowned for his contributions to the fields of ethics and epistemology. It is this Platonic Socrates who lends his name to the concepts of Socratic irony and the Socratic method, or elenchus. However, questions remain regarding the distinction between the real-life Socrates and Plato’s portrayal of Socrates in his dialogues. Socrates exerted a strong influence on philosophers in later antiquity and in the modern era. Depictions of Socrates in art, literature, and popular culture have made him one of the most widely known figures in the Western philosophical tradition.
The statement “I know that I know nothing” is often attributed to Socrates, based on a statement in Plato’s Apology. The conventional interpretation of this is that Socrates’s wisdom was limited to an awareness of his own ignorance. Socrates believed the best way for people to live was to focus on the pursuit of virtue rather than the pursuit, for instance, of material wealth and happiness. He always invited others to try to concentrate more on friendships and a sense of true community, for Socrates felt this was the best way for people to grow together as a populace. His actions lived up to this standard: in the end, Socrates accepted his death sentence when his accusers thought he would simply leave Athens.
It is worthwhile to mention a few mathematicians after Plato who himself can be considered as one. In the Greek-speaking Alexandria, the capital of Ptolemy Kingdom,7 Euclid (325 BC–270 BC) wrote an encyclopedic treatise on all fields of mathematics, having written the integral knowledge of accumulated wisdom in Greece before. Being the best known among many, he may be regarded as the Father of Geometry. His remarkable book Elements provided an axiomatic logically coherent framework to discuss geometry on a two-dimensional flat plane involving parallel lines, similar and isosceles triangles. Euclid had the longest lasting impact of any ancient Greek mathematician as his textbook remained in use until the 20th century over 2,200 years later. Little is known about his life or how much of Elements was based on his own versus others’ discoveries.
It is remarkable that it was not until the 19th century that somebody discussed geometry on a curved surface such as on a sphere where parallel lines meet and that the angles of a triangle do not add to 180°, and so the Euclidean axioms fundamentally changed. This underlines just how simple and persuasive Euclid’s presentations were.
There was a mathematician physicist Archimedes (287 BC–212 BC) in Syracuse, Sicily. He was one of the leading scientists of classical antiquity and generally regarded as the greatest mathematician among the ancient Greeks. He partially anticipated modern calculus by the concept of sum of infinitesimals. He made an accurate estimate of π to two decimal places as π ≃ 3.14 and discovered that the area of a circle is given by πR2, the volume of a sphere by 4πR3/3.
In order to determine whether the King’s crown had been made of pure gold, he enunciated the principle that weight is reduced under water by the weight of water displaced. He is alleged to have run naked into the street shouting “Eureka” after making this discovery while taking a bath, and went on to find that the crown-makers had indeed cheated the King.
Archimedes developed the idea of exponentiation to write large numbers. In mechanics, he contributed substantially to hydrostatics and to understanding the principles of levers. He invented a screw pump that is still used today. He designed a huge ship, Syracusia, the largest ship ever built at the time in which an Archimedean screw was used as the bilge pump.
Much more recently, Galileo described Archimedes’ accomplishments as ‘superhuman’ and the Fields Medal, the most prestigious prize in mathematics, is emblazoned with his image.
Despite protective orders by superiors, Archimedes was killed erroneously by a Roman soldier at the age of 75.
In Alexandria, Egypt, there were Hero (10 AD–70 AD) and Ptolemy (100 AD–170 AD). Hero was an engineer and mathematician. He invented an aelipile (or Hero engine), which is the precursor of the steam engine which prompted the field of thermodynamics in the 19th century AD. Hero is responsible for Hero’s formula in mathematics, which provides the area of a triangle from its side lengths. He also discovered methods for computing square roots and cube roots. Ptolemy was a mathematician, astronomer, and geographer. He wrote Almagest, the only surviving ancient treatise on astronomy. In it, he famously suggested a geocentric Ptolemaic Solar System, an idea which was generally accepted for 1,200 years until the work of Copernicus. He was certain that the Earth does not move, an idea firmly adopted by the Catholic church as Galileo found out to his peril. Ptolemy also contributed significantly to cartography, producing maps of Eurasia and Africa, which can be admired at the Royal Geographical Society in London. His maps are surprisingly good given their age, although not surprisingly they do contain entertaining mistakes when compared to any modern map.
To summarise Chapter 1, we started with the atomist Democritus and have described the accomplishments of over a dozen ancient Greeks, all with extraordinary intellects. The main achievements of antiquity were in philosophy including ethics, morals, and politics. These ideas remain a major influence in modern philosophy.
The overlap of science and modern ideas is necessarily less because the technology necessary to do useful experiments had not been developed since there were less data available. This is probably why the ancient Greeks depended more on philosophical speculations.
Nevertheless, modern physics has evolved in a Darwinian style from the atomism of Democritus and his school, who showed such remarkable prescience. Greeks admired Democritus, who was depicted on the 10-drachma coin before the switch to Euros. Of the ancient Greek mathematicans, Euclid and Archimedes founded the subject and set example for all later discoveries in mathematics.
Given all the intellectual progress between 600 BC and 100 BC, it can seem disappointing that only relatively small advances were made in the next 1,500 years.
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1T. L. Carus, On The Nature of Things (in English) (Enhanced EBooks Publishing, USA, 2015), Book I.
2Aristotle did not define the prime mover exactly and considered it as a general concept for the force that set the planets and universe in motion.
3W. Heisenberg, Physics and Philosophy: The Revolution in Modern Science (Penguin Books, New York, 1962), p. 71.
4C-h. Park and Y-k. Kim, Plato’s Timaeus in Korean (Seo Gwang Publishing Company, Seoul, Korea, 2000).
5Twelve is considered to be the number of heaven.
6S. Greenblatt, The Swerve: How the Renaissance Began (W.W. Norton & Company, New York, 2011).
7Even now there are more Greek speaking people in Alexandria than those in Greece.
Chapter 2
God’s Plan
Immediately following the ancient Greeks, we should discuss Titus Lucretius Carus (99 BC–55 BC) who is considered to be the intellectual heir to Epicurus. He was a Greek Epicurean at the time of Julius Caesar (100 BC–44 BC) and the author of the remarkable, long poem De Rerum Natura (On the Nature of Things) written in Latin. Lucretius must have admired