Plato's Meno: Recollection and Doubling the Square

Socrates draws a square with sides of 2 feet and asks one of Meno’s slaves, a boy born in the household who speaks Greek, how large it is. The boy counts 4 square feet^[1].

Socrates then asks for the side of a square twice as large, 8 square feet. The boy answers at once that the side must be doubled^[1].

A side of 4 feet gives 16 square feet, four times the original. The conversation that follows, in Plato’s Meno, is Socrates’ illustration of the claim that learning is recollection^[1,2].

Meno’s question

The dialogue begins with Meno asking whether virtue is acquired by teaching, by practice, or by nature^[1]. Socrates answers that he knows nothing at all about virtue, and that without knowing what something is he cannot know what it is like^[1].

Meno later compares Socrates to the torpedo fish, which numbs anyone who comes near it^[1].

The paradox of inquiry

Meno then asks how anyone can look for something they do not know, and how, on finding it, they could tell that it is the thing they did not know. Socrates restates the objection as a dilemma^[1].

Put this way, knowledge by inquiry seems impossible^[2].

Learning as recollection

Socrates answers with a teaching he attributes to priests and priestesses and to poets such as Pindar^[1].

This offers a way out of the dilemma. When we inquire, we both do and do not already know what we are looking for^[2]. To show this, Socrates questions the boy^[1,2].

Doubling the square

The boy’s answers, in order, give these squares^[1].

After the second guess the boy says he does not know. Socrates points out to Meno that the boy used to think he knew and now does not even think so, and asks whether giving him the torpedo’s shock has done him any harm^[1].

Socrates then adds three more squares of 2 feet to the first, making a square of 4 feet out of four small squares. In each small square he draws the line from corner to corner, and the four lines enclose a square in the middle^[1].

Each of these lines cuts its small square in half, and the middle square is made of four such halves. Four halves of 4 square feet make 8 square feet, twice the given square^[1].

Asked for the side of this square, the boy may point to the line instead of giving a number. It is the line the learned call the diagonal, and the boy agrees that the double square is the square on the diagonal^[1].

Its length is $2\sqrt{2}$ feet, about 2.83 feet, since $(2\sqrt{2}{)}^2=8$. No fraction of whole numbers gives it exactly.

What the boy had in him

Socrates insists that he taught the boy nothing and only asked questions^[1]. The true opinions were already in him and were stirred up as in a dream, and if he were asked the same questions often and in different forms, he would in the end know as well as anyone^[1].

The questions and figures are the occasion for the boy to recollect what he learned before. The doctrine is an early version of the thesis that some knowledge is innate^[2].

Socrates adds that he would not insist on every detail of the argument. He would defend one claim, that we are better, braver and less helpless if we think we ought to inquire^[1].

Knowledge and true opinion

Near the end of the dialogue Meno wonders why knowledge should be preferred to true opinion, or why the two should differ at all^[1]. A guide with a true opinion about the road to Larissa gets you there as well as one who knows the way^[1,3].

Socrates says that this fastening is recollection^[1]. Why knowledge is worth more than mere true belief is still discussed under the name of the Meno problem^[3].

How the dialogue ends

The dialogue never settles what virtue is. It ends with the suggestion that virtue comes to the virtuous by the gift of God, and with Socrates’ remark that the matter will not be known for certain until they first ask what virtue itself is^[1].

References