How to learn mathematics by reading Plato

Plato's writings are famous for their logically rigorous and conversational tone. For the mathematically-inclined out there, the dialogues resemble in many ways usual mathematical proof-writing. To be more specific, Plato begins with axioms (assumptions), and develops a purely logical argument¹ which he concludes with or without a contradiction.

For example, in the Republic (Book I, 331e) the idea of being just is elaborated upon. Beginning with an axiomatic constructions following Simonides:

Polemarchus: "That it is just to give to each what is owed."

Then Plato (incarnated in Socrates' figure) demands clarification, like a good mathematician would do in sight of shaky grounds:

Socrates: "Well, it certainly isn't easy to disbelieve a Simonides [...] However, you, Polemarchus, perhaps know what on earth he means, but I don't understand."

Previously, Socrates had raised an issue with the current definition that Polemarchus defends:

Socrates: "Everyone would surely say that if a man takes weapons from a friend when the latter is of sound mind, and the friend demands them back when he is mad, one shouldn't give back such things, and the man who gave them back would not be just [...]."

In other words, the current definition gives an undesirable corollary. This hints at the fact that our definition of what is just is necessarily wrong! Socrates continues:

Socrates: "Then Simonides, it seems, means something different from this sort of thing when he says that it is just to give back what is owed."

And Polemarchus reveals something crucial:

Polemarchus: "Of course it's different, by Zeus, [...] For he supposed that friends owe it to friends to do some good and nothing bad."

Polemarchus has secretly instrumentalized the notion of what is just by setting a logical condition (not an obvious one) on his argument; that is, that one owes to his friend if one has something that belongs to him and that something shall be used to do some good. But we never agreed on that, and we haven't even defined what it is for some action to be good or bad! Moreso, Socrates poses an interesting question:

Socrates: "Now, what about this? Must we give back to enemies whatever is owed to them?"

Hence Polemarchus faces the need to twist his definition around, forcing it to adjust to our desirable conclusions: we shall not give back to enemies what is owed to them, save for "some harm", obviously! Hence Polemarchus adds another condition to being just, to giving back: that it must be "fitting". To summarize:

Definition: To be just is to give back what one owes, assuming the following conditions hold:
   (1) One has something that belongs to someone else,
   (2) What is given back shall be used to do some good, and
   (3) It must be fitting for that specific "someone else".

Whew! In any case, Socrates eventually causes the definition to be "depurated":

Polemarchus: "[...] I no longer know what I did mean. However, it is still my opinion that justice is helping friends and harming enemies."

This is easier to handle, in terms of deriving logical conclusions from it. Socrates asks for clarification yet again:

Socrates: "Do you mean by friends those who seem to be good to an individual, or those who are, even if they don't seem to be, and similarly with enemies?" [...] "But don't human beings make mistakes about this, so that many seem to them to be good although they are not, and vice versa?"

Polemarchus: "They do make mistakes." 

Socrates: "So for them the good are enemies and the bad are friends?"

Polemarchus: "It looks like it."

Socrates: "Yet the good are just and such as not to do injustice?"

Polemarchus: "True."

Socrates: "Then, according to your argument, it's just to treat badly men who have done nothing unjust?" 

A Q.E.D. or ∎ would finish off this section well enough, for all I know! Polemarchus struggles with accepting that his definition is essentially wrong:

Polemarchus: "Not at all, Socrates, [...] For the argument seems to be bad."

Socrates finally suggests, again like a good mathematician, to fix our definitions:

Socrates: "[...] Let's change what we set down at the beginning. For I'm afraid we didn't set down the definition of friend and enemy correctly." 

Let us recap. It all began with an apparently harmless assumption, which in fact required further clarification and detail. In mathematical terms, this is equivalent to stating an axiom with "undefined" vocabulary, like postulating the existence of the real numbers without defining number first. After adequate adjustments were made (sometimes, as is the case, a clear sign of bad foundations), the axiom was restated in a much more succinct form. This reduced form finally allows us to transparently derive an "undesirable conclusion" from it, a logical fallacy in technical terms.

This is in fact a dialectical version of "proving the inconsistency of a theory". As far-removed as this might seem, we encounter such fallacious arguments every single day on TV, newspapers, discussions, and a long etcetera. Reading Plato can help mathematicians the same way it can help regular folk in their daily lives: freeing us all from the tyranny of words.



Footnotes—

• [1] One might argue that Plato is not always "logical", and I would in fact agree that his philosophy can contain deep non-explicit assumptions. Still, he writes mostly logically coherent arguments, as is the case here.

Bibliography—

• Plato. The Republic of Plato. Translated by Allan Bloom. Introduction by Adam Kirsch. New York: Basic Books, 2016.