Plato's Meno

The selection in chapter one of Plato's ``Meno'' Socrates claims to prove that an uneducated slave-boy actually knows geometry. However, it is quite apparent that this slave-boy is actually being fed leading questions. Socrates does nothing more than ask the correct next question. This type of leading questioning is not unlike the questions asked by professors and instructors when lecturing a new topic. The student had no conscious knowledge of the subject matter at hand before the questioning; the questions were asked and, more importantly, linked in correct succession to arrive at a conclusion. Hence, one actually has no knowledge of x before the exposure to x. The act of asking questions does nothing to prove the knowledge of x.
 

The Divided Line Theory of Knowledge

In this case, we have, at the lowest level, Images. In this grouping, there are such things as shadows, reflections, memories and dreams. These are things that are not real at all. They are really bad copies of the original object. What is this original object? It's an entity existing in the second level, or the physical phenomena level. Here we find entities that exist in real life-the world of things. In this level, we might find the automobile, a lake, mountain, a watch, or just a cylindrical object. These things make possible the entities in level A. For example, a mountain by the lake, might cast a reflection or shadow of itself on the water. All of these things account for the existence of the visible (B, A) level.

The intelligible section is made of the last two levels. These are things that we can know; they are the precursors (or concepts) of their shadows cast on the lower levels. In the level of mathematics, it consists of those people which take certain hypothesis to be true. For example, consider the person who knows that a circle is made of 360°, or that a square is made of lines which are of equal lengths, and that a line, in turn, is made of an infinite number of points. At this level, one understands the existence of the rules and their relationships, but has yet to truly understand and conceive of the reasons for these rules. And finally we are left with the zenith of this understanding, which is true knowledge. In this case, we understand and know the relationships' reason for existence. We can fully conceptualise the point.

Putting it all together, let us take a modern day example to clarify the above. Consider the computer: it's in handy use everywhere, hence a prime example for drawing an analogy. At the lowest level we have the documentation for using a computer. It's a copy of what the regular computer user should do. It's not always accurate, and it's very cheap (cheaper than a computer.)

At the level of the physical phenomena, we have the average computer user. This person is not a rocket scientist, or brain surgeon, he just logs on, run's a few programs, types his papers, browses the web etc. He doesn't understand how the computer works, but doesn't really need to, in order to get his way around in the computer.

In the level of maths, we have someone who works at a computer as a programmer. This is his job; he doesn't understand many of the behind the scenes reasons why he must programme something in a particular manner. He doesn't question why something might be done this way. He just sees that it's been done in a particular manner before, and it worked, so he's just following the rules.

At the highest level, that of reason, you have the zen computer programmer. It might be his day job or not. But either way, he embellishes in code, even when he comes home at night from work. He knows why something was done the way it was, and attempts to find simpler more efficient ways of doing it. He attempts to change the rules.

Innate Knowledge

Very much like the average computer user described in the previous section, true knowledge doesn't exist until the person knows what the rules are behind the conclusion. And even so, one is only in the mathematical level of knowledge. To the true knower, one must know (as Plato himself noted in the Divided Line Theory) not only that a relationship exists between two bodies, but why the relationship exists. Although the slave-boy comes to know that there is a relationship, he doesn't understand how this relationship works; he does not understand how the relationship comes to be, and conversely, its significance in other areas of the maths.

Let's analyse the first line of questioning. Plato draws for the slave-boy a square subdivided into four smaller squares. He then asks the boy about the square: are the lines equal? Can the square be of any size? He then asks that if the square's sides are two feet each, then what is the area of the whole. Before allowing the boy to continue, he gives an example: if the square were one foot, by two feet then the whole would be two feet ``taken at once.''

In the example, he is setting the rules of the game. Very much like the manual on the computer software saying `press this button to turn the computer on.' Then the user presses the button to turn the computer on, he doesn't immediately know computers. It just means that he's following the rules. The boy, through the line of questioning, is being made aware of the rules - the relationship. In answering the question of the area of the square, he is using what he has just learned (the relationship between one and two when taken as a whole, which equals two) to apply it to the original question: twice two when taken as a whole to equal four.

When Plato extends the example to apply Pythagorean's Theorem in determining the length necessary to fill up the area of double the original square (eight square feet) the boy answers incorrectly. He says that the boy claimed he did not know anything about the original questions before, and now he does. So once again, the boy knows nothing regarding the pythagorean theorem and Plato claims that he will remember.

Plato resumes his line of (leading) questions to show to the boy that twice the area of a square doesn't imply twice the lengths of the sides. Let's make a quick note to return to later that Plato believes that the boy is better off for knowing that he was wrong; lest he go off telling the world (incorrectly) that twice the area of the square denotes twice the lengths of the lines. So Plato continues to draw four squares of the same size of the original. And places them such that they make up a larger square. Then draws four lines (the diagonals of the smaller squares) to make up the answer. He divides the smaller squares in half, since the large square is four times the size, and it should only be twice the size. Now you have a square in the middle made of of the diagonals which is twice the area of the original square. Interesting, the boy notices that the square is the centre is the square that we were trying to figure out in the first place... and it's lengths are the diagonals of the smaller square.

When Plato continues with his questioning, it seems like the boy's figured out this relationship between the diagonal of the smaller squares, and the area of the larger square. However, let's pause a moment and consider what has actually occurred. Plato has been conducting the questioning, drawing the lines in correct order, so that the evidence may move towards ``as any damn fool can see.''

Conclusion

In this manner, the child was nothing more than the inexperienced user of this analogous computer. He didn't know anything before, and he still knows nothing now. All he has done is found out the rules of the playing field. He will now obey those rules and apply them to other such examples. Just as if the user would go to his neighbour's computer... he would recognise the power button and push it accordingly. This selection is, in all actuality, a decent demonstration of how one isn't born with innate knowledge, and that one acquires it through experimentation, and exploration. In fact this is the most effective method of teaching something to someone. Asking leading questions, allowing them to have gestalts, so that the subject matter remains in perminantly.

Plato did, however, leave us with a bit of important information, Socrates says:

"We have certainly, as would seem, assisted him in some degree to the discovery of the truth; and now he will wish to remedy his ignorance, but then he would have been ready to tell all the world again and again that double space should have double side... but do you suppose that he would ever have enquired into or learned what he fancied that he know, though he was really ignorant of it, until he had fallen into perplexity under the idea that he did not know, and had desired to know?"