I am writing a post a day. I want to do a series on classical liberalism. The corner stone of classical liberalism is Aristotelian logic which appeared in Ancient Greece. it is beneficial to understand the culture that spawn formal logic. The picture from wikicommons shows a bust of Pythagoras in the Vatican.
Ancient Greece was itself an ancient culture founded on the remains of even older cultures.
The Greek world was not a single nation. It was a collection of tribes wrapped around the Aegean Sea. The culture would have included the Western shores of modern Turkey.
Greek mythology included epic tales of travelers and heroes and long with a few colorful fables. The Greek Religion presented a Pantheon of Gods locked in balanced conflicts.
Many of the tribes were quite fearsome. Notably the Spartans was a tribe of warriors. It is said that the soldiers would kill peasants for practice. Greek historians claim that in Battle of Thermopylae 300 Spartans held off an attack of a hundred thousand Persians.
The Ancient Greeks had a penchant for fanciful thought. For example, Epimenides of Crete (circa 700 BC) is said to have said that all Cretans are liars.
But, if the statement is true, than it must be false because Epimenides was Cretan.
He must be a liar!
Isn't it clever!?
The liar's paradox is an example of the reflexive paradox. The reflexive paradox occurs whenever a system includes a self-reference and an negation.
The most common example of the reflexive paradox is: "This sentence is false."
The sentence negates itself.
Isn't that clever!?
I believe that the statement "This sentence is true" is as paradoxical as the negative version and it is dependent on a self-reference for verification.
There are thousands of versions of the reflexive paradox.
Pythagoras Confronts the Hypotenuse
Many of the early Greek Philosophers including Thales, Anaximander, Anaximenes and Pythagoras lived in Ionia which is in modern day Turkey. It is likely that the philosophers had connections with Egypt as the cultures had a common interest in triangles.
Pythagoras was a mystic who loved numbers. He is most famous for his work with musical theory. Pythagoras is said to have noted that strings which are of certain proportions tend to make harmonious sounds. His work is the basis of musical scales.
Legend tells that the Pythagorean school held mythical notions about numbers and much of their philosophy was closer to numerology than mathematics.
Pythagoras attempted to express phenomena as ratios of two whole numbers and that he was driven to distraction by proofs that showed that the hypotenuse of the unit square could not be expressed as a simple ratio.
In mathematics, one uses the term "rational" for numbers that can be expressed as two whole numbers such as 1/2 and 2/3. One uses the term "irrational" for numbers that cannot be expressed as a ratio such as the square root of two.
I don't know how to write mathematical formulas in markdown. I think I will end with a picture from The Creative Commons
x is the hypoteneuse (aka diagonal line) crossing a rectangle. The picture shows how to find it.
The pythagorean theorem was named for Pythagoras as he is known to have struggled with the observation that many numbers cannot be expressed with rational numbers.
Philosophers throughout the ancient world made great strides in mathematics and science; however, their attempts to describe the natural world kept running against paradoxes and contradictions.