St. John's

The Irrational Line in the Meno

01 July 2025

Class: Pedagogy Preceptorial — Summer 2025

In the Meno Socrates performs a geometry demonstration with one of Meno's slaves. This slave has had no education and is presumed to be totally ignorant of geometry before this demonstration. In this demonstration Socrates asks the boy to tell him how long a line is needed to produce a square with an area of eight feet.¹ This line would have a length of √8 or 2 × √2 or ≅ 2.828427. An exact decimal value of the length of the line cannot be provided because the quantity is irrational. It cannot be expressed as the ratio of two integers. The Greek word for such a number would have been alogos which we might translate as inexpressible or unspeakable. In this essay, I want to explore why Socrates asks the slave the length of a line which is irrational. Why does Socrates produce the line without giving its length? Why does Socrates not acknowledge that he produced the line without stating its length? And how do the answers to the questions help us understand the broader topic of the dialogue, virtue, and if it can be taught/learned or otherwise acquired?

I will begin with a short summary of the Meno up until the demonstration. The Meno opens with Meno asking Socrates if virtue can be taught or acquired in some other way. Socrates says that he cannot answer this question because he does not know what virtue is. At Socrates prompting Meno attempts to tell Socrates what virtue is. Meno manages to describe some individual virtue(s) but not virtue itself. Socrates explains this to Meno by comparing virtue to color or shape. One might describe various shapes or colors without describing the essence of them that makes it appropriate to group them together in a category. Meno in response makes the debater's argument "How will you look for it Socrates, when you do not know at all what it is? How will you aim to search for something you do not know at all? If you should meet with it, how will you know that this is the thing that you did not know? Socrates rejects this argument by relaying a story told to him by some priests. He says that the soul is immortal and that this means that the soul has seen all things here or in the underworld, thus that what men call learning is just recollection. Meno asks Socrates to demonstrate to him that all learning is just recollection. Socrates then begins the geometry demonstration. Note: Most translations include diagrams that may not have been present in the original text. I will proceed as if these additions were not present.

82[b] …

Socrates: Tell me, boy, do you know that a square figure is like this? [1]
Boy: I do.

82[c]

Socrates: Now, a square figure has these lines, four in number, all equal? [2]
Boy: Certainly.
Socrates: And these, drawn through the middle, are equal too, are they not?[3]
Boy: Yes.
Socrates: And a figure of this sort may be larger or smaller?[4]
Boy: To be sure.
Socrates: Now if this side were two feet and that also two, how many feet would the whole be? Or let me put it thus: if one way it were two feet, and only one foot the other, of course the space would be two feet taken once ?[5]
Boy: Yes.

This opening portion opens several questions, first, to what scale did Socrates draw the figure? This matters because if the lines are not to scale, then the lines become symbols or images of a sort, representing a line which may represent a quantity. The quantity may be alogos unspeakable, but the line is still a way of communicating it. The second question is: why include units in the description of the figure? This matters because the trick of Socrates demonstration relies on the quantity associated with the unit being a number for which the square root is irrational. If the unit can be changed and the quantity changed accordingly then there is the possibility of the demonstration being defeated. And lastly why does Socrates ask the boy/ the reader to imagine the figure which is two feet by one foot, a square with a length of 2 feet on one side and 1 foot on the other is an impossibility. Why would Socrates ask about a figure that cannot be?

I imagine that this third question has two possible answers. The first is rather simple, that Socrates has not said square and he has not asked the boy to imagine an impossible figure. I think this unlikely. Socrates may not have said "square," but he has implied it. Given that Socrates, in his discussion with Meno, moved him to a state of perplexity in which he was forced to admit that he knew nothing of what virtue itself is. Socrates here, by asking an impossible thing may be priming the boy and even the reader to think that producing the line which produces the square of area eight is also impossible to produce in the reader a stupor similar the one produced in Meno. This stupor produced first in Meno and later in the boy seems to be an essential step in Socrates methods of pedagogy and because of this I believe that Socrates is asking for an impossible figure to be imagined as sort of trick to make the boy or the reader believe that the line which produces the square of double area is also impossible.

The second question, why specify a unit of length for the lines of the figure is more confusing. I see that for the sake of precision that one might want to specify a generic unit. I.e., "Now if this side were two units of length and that also two, how many units would the whole be?" I don't know if the Greek language would have supported such an expression and lacking the capability I could see Socrates settling for using a specific unit. This settling though introduces a way to ignore the trouble of the irrational number, a single Greek pous (foot) is made up of 16 daktyloi (fingers) If Socrates had instead asked Now if this side were 32 fingers and that also 32 fingers, how many fingers would the whole be?" the answer would be 1024 fingers. And the square of double area would be a square of 2048 fingers which also has an irrational square root.²

The relationship between the length of the lines that Socrates produces and the quantity that those lines/lengths represent is at the core of my questions on how to communicate an unspeakable or irrational number.

The dialogue continues referencing the figure that Socrates has drawn,

82[d]

Socrates: But as it is two feet also on that side, it must be twice two feet?
Boy: It is.
Socrates: Then the space is twice two feet?
Boy: Yes.
Socrates: Well, how many are twice two feet? Count and tell me.
Boy: Four, Socrates."
…

Socrates here seems to imply that the figure he has drawn is actually 2 feet long. In the previous passage Socrates had asked "if this side were two feet and that also two …" implying that the figure had lines of some length other than 2 feet. I am left wondering what the actual length of the lines is. I believe this is important because if the lines are of some length other than 2 feet than the lines are signs of some sort. They don't refer to themselves but to something else perhaps a line of the specified length or perhaps to the quantity corresponding to the specified length. The word "two" and the symbol "2" might both refer to each other but they also seem to refer to some quantity of more than 1 but less than 3 but yet also whole.

82[d] ctd

Socrates: And might there not be another figure twice the size of this, but of the same sort, with all its sides equal like this one?
Boy: Yes.
Socrates: Then how many feet will it be?
Boy: Eight.
Socrates: Come now, try and tell me how long will each side of that figure be. This one is two feet long: what will be the side of the other, which is double in size?"

Here we begin the portion of the dialogue that likely asks the boy to do the impossible, Socrates asks the boy how long each side of a square will be if that square has an area of 8 feet. Would Socrates accept an answer of "approximately 2.828"?Would Socrates accept an answer of "Two times the square root of two" or even "the square root of 8". We may never know because the boy never offers any of these answers. The boy first says that a line with a length of 4 feet will produce a square with an area of 8 feet. His reasoning being that a line of double length will produce a square of double area. Socrates proceeds to show him that the line of length 4 will in fact produce a square with an area of 16 feet.

The boy then suggests that a line with a length of 3 feet will produce a square with an area of 8 feet, Socrates proceeds to show him that the line of length 3 will in fact produce a square with an area of 9 feet.

The boy now has his torpedo fish moment. The boy has offered two guesses and been wrong both times. He knows that both 2 and 3 are incorrect answers, 2 being too small an answer and 3 being too large an answer. The boy likely understands that items like bread can be broken into portions but if he has no formal education he might not know about decimal numbers or fractions. That given, I believe that Socrates was expecting the boy to continue. At 86b Socrates after confirming with Meno that the boy has had no training in geometry tells Meno,

86b "…
Socrates: I think so too Meno. I do not insist that my argument is right in all other respects, but I would contend at all costs in both word and in deed as far as I could that we will be better men, braver and less idle, if we believe that one must search for the things one does now know, rather than if we believe that it is not possible to find out what we do not know and that we must not look for it."

If the boy had guessed 2.5, he would have again been wrong but this time he would have been wrong in a new way! He would have guessed a value that is smaller than the true value. This sets a range in which the length of the line must exist. The length must be more than 2.5 feet and less than 3 feet. Continued guessing in this range would narrow it further and would eventually if somewhat arduously lead to an approximate answer.

At the beginning of the geometry demonstration Menos attempts to tell Socrates what virtue is have failed and they seem to fail in a way similar to the way in which the geometry demonstration has concluded. While Meno can describe various virtues he can not describe virtue itself. While Socrates cannot state the length of this line, he has put forth a method for reliably producing a hypotenuse that will produce a square that is double the area of a previously given square. For virtue it was the common part, the general case, that could not be put into words. For geometry it is the length of this particular line that is alogos, that cannot be expressed.