4. Greek Philosophy and Mathematics

I find the Greeks surprising: they developed a system of thought that was completely different from polytheism while they were still a polytheistic society.

In faith-based societies, life is governed by fate and magic. Nothing needs to be explained—things happen because of fate, or the will of the Gods, or some other magical explanation [more on this here].

Therefore, when Socrates says that wisdom is to know that you know nothing, it is a paradigm shift. He is not saying—I don’t know the will of the Gods, or I misinterpreted the oracle, or I am not sure where my fate will lead me; he just says I do not know. In fact, he was tried and executed for corrupting the youth of his day because he stopped people and asked them how they knew what they knew.

This is best explained by Plato’s allegory of a cave in ‘The Republic’. He says that if you were to imagine that prisoners were tied in a cave such that all they could see was the wall opposite them, and they could not see anything else—all they would know of the world through observation would be the wall opposite them.

Now, if someone put puppets in front of a fire behind them and made dancing forms on the wall opposite, they would think that those large, dancing forms were what being were like. But this is vastly different from what beings are really like. Therefore, our knowledge of the world is limited by the limits of our observation and perception.

Imagine what this implies:

  • We are limited by our senses.
  • We are limited by our ability to perceive.
  • We are limited by our experience (age, sex, geography, class, economic position, ethnicity, etc.)
  • We are limited by what society at the time can perceive, in terms of openness to ideas, technologies for observation, contact and communication with other cultures, etc.

Here, it seems that Plato is only talking about the limits of observation, and not the veil of biases that surround us that do not allow us to observe something without preconceived notions. He says that if the men in the cave were taken outside and allowed to see real humans, they would not be able to value the knowledge produced inside the cave again. They may even be considered mad once they returned to the cave and reported what they saw.

But I wonder, when the prisoners went outside the cave and saw humans in their true form—would they recognize that this was the true form of the human, or would they think that these were mutant versions of humans because they did not resemble the shadow forms dancing in the light? [more on this later]

The bias towards observation can also be seen in Greek mathematics. Now, we all know that Pythagoras gave us equations such as:

a2 + b2 = c2            

for hypotenuse triangles, and the Greeks made major advancements in geometry and mathematics.

But did you know that the equations were not written in this form or in any form we consider mathematics today? The equations were written discursively and proved by actually filling two squares, one with side a and one with side b, into a square of side c (a square of side x is x2). The a square and the b square were actually cut into smaller and smaller squares and fit into the c square. All mathematical proofs by the Greeks were in this form [Read more about this here].


Fig. This is the Greeks calculating the value of pi by fitting smaller and smaller squares in a circle (Source: Heaton, 2015).


  1. Heaton, Luke 2015. A brief history of mathematical thought: Key concepts and where they come from. Robinson, Croydon, UK.
  2. edX classes ‘Western Civilization: Ancient and Medieval Europe’ and ‘Question Reality! Science, philosophy, and the search for meaning’.