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‘The demonstration with the slave in the Meno proves nothing. The discussion is rigged. Socrates has the answers from the outset, so he can lead the slave into giving the answers.’ Discuss.

Posted by Philosophy Foundation on December 5, 2010

Meno asks Socrates to “somehow show that things are as he says”; to show that “…we do not learn but that which we call learning is recollection.” (81e) In response Socrates asks a slave boy to come over to them and he proceeds to question the boy about geometry in order to demonstrate to Meno that he is not teaching him but that the boy is “recollecting things in order” (82e). But is the discussion rigged such that Socrates is really giving him the answers by knowing them himself and then feeding the boy the answers by a series of carefully constructed leading questions? The discussion is indeed littered with leading questions and in some cases Socrates does tell the boy answers in some barely disguised questions, and on a first reading one is certainly left with a feeling that maybe he is just telling him the answers. All readers of Plato are familiar with the Socratic method known as the elenchus and the process of dialectic that Socrates employs with his interlocutors, but readers of Plato are also familiar with the, sometimes, disingenuous use of the dialogue form. Many of Socrates’ opponents or collaborators in the dialogues are made to agree with Socrates for the purposes of the discussion when we – the readers – often feel that objections need to be made; the answers Socrates’ interlocutors give often seem rigged by Plato to go in the direction that he wants. This can seem acceptable in some cases because Plato is simply using the dialogue to expound his ideas and the artificiality of the responses is not relevant to the philosophical point he is making. However, in the case of the demonstration with the slave boy in the Meno, it seems that it matters very much because, as Socrates repeatedly says to Meno, he is at pains to make sure that the boy answers with opinions that are his own. This is important because he must be able to show, for the success of his demonstration, that the boy can reach a conclusion at the end of the discussion that he was unable to reach at the beginning. If Socrates simply tells him this conclusion in the interim then his demonstration would have failed. How can we respond to this concern?

 

To say that Socrates ‘leads the boy to give the answers’ is vague and unclear so I would like to draw a distinction between two relevant, but competing, ways that can be meant by this phrase. Firstly, we can mean that Socrates ‘tells the boy the answers’, as in “What is p? Is it not q?”[1] Secondly, there is a more complex sense I call the poria-sense[2] that to feed the boy the answers is to ‘provide a path’ from ignorance, through aporia, to eureka. I will argue that it is this second sense that Socrates is using when he feeds the answers to the boy at the crucial stages of the discussion, and that, not only does this sense not lead the boy to the answers in a way that is detrimental to his demonstration, but that the poria-sense of ‘leading’ is at the heart of the method that he wants to demonstrate – what I will call the ‘poria-method’. With the demonstration he will attempt to show that, via the poria-method, the boy can reach true belief about something he has no knowledge of; but the method has a still greater role in the demonstration: Socrates suggests that it will form part of how the boy will then go on to reach knowledge. I will outline what the poria-method is as this is central to Socrates’ method and therefore central to the argument that the demonstration proves something. 

 

The demonstration as a whole falls into three parts: the first part ending where the boy reaches the false conclusion that a double size figure would follow from a double length side (82e); the second part ending with the boy’s aporetic moment where he declares that he does not know on what line the 8-foot figure would be based (84a), and the third part ending with the boy correctly identifying the diagonal line on which the 8-foot figure is based, line EH (85b). Socrates uses these different stages of the inquiry to identify significant stages of the boy’s progress through the problem to which he deliberately draws Meno’s attention: the first part is where the boy thinks he knows the answer but is ignorant; the second part is where the boy realises that he is ignorant, and the third part is where the boy finally reaches insight and true-belief about the correct answer. The important move that we shall be interested in – and to which I believe Socrates must not tell the boy the answer – is the conclusion: the line on which the eight foot figure is based, indicated ostensively by the boy when he answers Socrates’ question: “Based on what line? – This one.”[3]

 

So, does Socrates tell the boy the answer at 85b? We can eliminate the kind of disguised-answer-leading-question that is found at 83b and 83c[4] as nowhere in the discussion does Socrates covertly reveal the answer (the diagonal of the square with the side of 2) in this way with his questioning. But to show exactly in what way he illuminates the way to the answer, and to show that this is not counter to his aim, I will now have to say something about the poria-method Socrates uses that I identified earlier: how he provides a path without telling the boy the answer.

 

Showing the path: the ‘poria-method’

 

The example chosen by Socrates for discussion with the slave boy is a geometrical one and the conclusion he leads the boy to is a valid conclusion and it is reached by a series of deductive steps. The boy is therefore able to reason logically in the demonstration, but there are a variety of ways that this could be taught or facilitated, some of which involve telling the student the answer. Socrates’ questions facilitate the boy to apply his understanding of concepts to move him autonomously (searching and selecting for himself) by inference towards eureka (insight) and through aporia (perplexity). But how does Socrates facilitate this? He does it by the introduction of salient concepts and then his questions allow the boy to ‘join the dots’ using the concepts he has learned to bring him to the answer. By the expression ‘join the dots’ I mean that the boy has to make the necessary connections between the various concepts that he is considering and therefore he has to make judgements about the selection of the concepts he is searching through and the connections he subsequently has to make. In a word, we might call this process ‘inference’. It is this crucial aspect of the method that preserves the boy’s autonomy for Socrates to justifiably be able to claim that he has not been given the answer and that the boy has answered with his own opinions. Socrates’ questioning is such that the answers to previous questions provide the foundations for the answering of later ones. Socrates’ job, through the use of questioning, is to activate the boy to perform the operations I have just described, and Socrates does this by setting problems to which the relevant (and acquired) concepts can then be applied.

 

I will now take a closer look at how concepts are introduced; problems set for the application of the concepts; and then the candidate-answers tested in the light of these new, relevant concepts. In the first part the concept of ‘square’ is introduced (82b), followed by a discussion of the concept of ‘size’ (82c) and then the concept of ‘double’ (82d); a problem is then set (82d-e): “Socrates: Come now, try to tell me how long each side of this will be. The side of this is two feet. What about each side of the one which is its double?” The boy is, however, missing an essential concept in order to give the right answer at this stage: he is lacking the concept of ‘area’[5]. Socrates therefore sees it as his role to provide the boy with the relevant concept (83b) and then to enable the boy to test the answer that has been given in light of the new concept. When the boy does this he discovers that the original answer is contradicted by the new – area-informed – answer (83c). A process of logical elimination (83c-d) then enables the boy to see, rightly, that the answer must lie between a 2-foot side and a 4-foot side and he concludes, reasonably but incorrectly, that it must be ‘three’ (83e). I say ‘reasonably’ because 3 must be the only known number to the boy between 2 and 4. This is the boy’s second attempt to answer the question. But when this yields a 9-foot figure (83e) the second answer is also falsified as a result of Socrates’ testing of this second answer. The boy is consequently ‘numbed’ into aporia: “Socrates: But on how long a line [is the 8-foot figure based]? Try to tell us exactly, and if you do not want to work it out, show me from what line. – By Zeus, Socrates, I do not know.” Now, at this point, he cannot see how to proceed: he is without a path in the usual manor of a Socratic dialogue. He knows the line must be between 2 and 4 but he also knows that it cannot be three so no other options appear open to him. What does Socrates do to ‘provide a path’ in, what is an unusual extra step in his elenchus method, where he proceeds beyond aporia? Socrates’ own word for what follows is that the boy will come out of perplexity by “searching along with me.”[6] (84c) He then says, “I shall do nothing more than ask questions and not teach him.” (84c-d) The word ‘searching’ here implies the character of the last stage: that of ‘discovery’. Socrates needs to preserve the element of discovery for his demonstration to succeed because of my earlier argument for the necessity of the boy’s autonomy in the process. He begins the third part by recapping that the combination of the four 2-foot squares gives a square four times the size and not twice the size (84d-e); he then introduces the final relevant concept: that of ‘the diagonal’. Following the introduction of this final, crucial concept Socrates asks a series of questions (85a), which do not contain any disguised answers. The boy is only able to answer these questions because he has understood the conceptual content thus far, and he has understood the conceptual content because he has been applying the concepts as he has proceeded through the inquiry. The boy therefore has to search his own understanding and select his answers appropriately enabling him to say, in the last stretch, not “yes”, “no” or “certainly”, but substantive answers that appear nowhere in Socrates’ questions: “four”, “two”, “double”, “eight” and finally, very much in the spirit of inferential discovery, “this one” (85a-b).

 

The extraordinary culmination of this inquiry is that the answer is reached at 85b when Socrates asks the boy, “[the 8-foot figure is] Based on what line?” and he answers with, “This one.”[7] But the answer is not explicitly stated[8]. There are two reasons for this: one reason is the parallel process the reader is undergoing whilst reading the dialogue and the other is the fact that Socrates tells Meno that the boy has only true-beliefs and not knowledge at the end of the inquiry. It will later emerge that Socrates thinks knowledge is ‘true-belief plus an account’ (98a). The slave boy demonstrates that he understands the concepts and the reasoning employed in the inquiry but he does not provide accounts – that is: he does not explain why he gives the answers he does. If any accounts are given then they are given by Socrates[9]. So, the demonstration is a ‘demonstration’ in two senses: 1) it demonstrates to Meno that the boy can indeed acquire true-beliefs about something he was ignorant of without being told the answer; in other words, it demonstrates at least part of Socrates’ point (it has not yet proved that this is recollection) and 2) it only demonstrates – as in ‘shows’ – the boy’s understanding in the context of the discussion with Socrates. Meno only witnesses the boy’s true-beliefs and understanding, the boy himself cannot provide an account of his true-beliefs with definitions and explanations etc. One would not know he had true-beliefs without having also witnessed the demonstration. But this is sufficient to meet the problem of the paradox of inquiry, because if the boy can reach true-beliefs without knowledge then there is a sense in which an inquiry can yield something significant (i.e. true-beliefs) without having knowledge of the object of inquiry (Socrates: “So the man who does not know has within himself true-opinions about the things that he does not know?” 85c). This is intelligible in the light of a distinction between knowing (exhaustive, propositional definitions with justificatory and explanatory accounts) and showing (witnessed demonstration of true-beliefs in an applied context). As far as I can tell this is not made explicit in the dialogue but is what we are left to infer from the demonstration itself (the showing) and his later definition of knowledge at 98a: true-belief plus an account (the knowing).

 

Teaching without teaching

 

Several times in the dialogue Socrates says that he is not teaching the boy. There is a sense in which he is, and a sense in which he is not. Consider the following senses of ‘teaching’: 1) teaching as the transmission of a body of information and 2) teaching as the activation of the application of salient concepts with a given problem (the poria-method). From the student’s point of view the first of these is passive and it is possible for a student to demonstrate the successful transmission of a body of information without the correct understanding of the concepts and principles involved (for example with a recited-answer or memorised essay). The second of these styles of teaching is active and therefore must include understanding in order to move from one stage to another. It is a collaborative and interactive process where the student and teacher make different kinds of contributions but where they both must make contributions for the inquiry to progress. And, crucially, 1) also involves the telling of answers and then the testing of them with questions, whereas 2) includes only the method of ‘providing a path’. The analysis I have provided above I think shows that Socrates is teaching but only in the second sense of teaching: he is activating the boy to search for the answer.

 

Hypothesis: entertaining without accepting

 

In his Metaphysics Aristotle said, “It is the mark of an educated man to entertain a thought without accepting it.” This describes the method of hypothesis, which Socrates introduces as an explicit method in the Meno (86e) and although it is introduced after the demonstration with the slave boy I think it is used in the discussion, though in a special way. At two points in the discussion Socrates elicits an answer from the boy, which they then go on to test. First they test the conclusion reached at 82e (that a double-sized figure must be based on a double-sized line) in the light of the new idea of ‘area’ (83a-c) with which they reach a different conclusion (that a double sized line would give a figure four times the size), and this shows that the first conclusion must be false. If we call the conclusion at 82e the hypothesis then at 83a-c they test and falsify the hypothesis. A second answer (hypothesis-2) is reached at 83e (“Three”) which is also subjected to a test which again falsifies hypothesis-2 (“So the 8-foot figure cannot be based on the 3-foot line” [as this gives a 9-foot figure]) But it is important to point out that Socrates and the boy have different notions of what the hypothesis is – and this is the ‘special way’ I outlined earlier in which Socrates uses the hypothesis-method. The boy believes it to be ‘the answer’ – in Aristotle’s terms, he has ‘accepted it’ – but Socrates treats it only as a ‘candidate-answer’ to which he will invite the boy to apply a test. When the test is then applied the boy comes to realise that ‘the answer’ was wrong.  In so doing Socrates has brought the boy to a new attitude towards answers, that they can be entertained without yet being accepted: he is learning the method of hypothesising, not by being told about it but by actively engaging with it. To return to the distinction between ‘knowing’ and ‘showing’, he cannot explain this method, or even identify it, but he can do it and can, presumably, apply it when it is necessary to do so on future occasions.

 

Does the teacher need to know the answer?

 

I have been addressing the common concern raised in regard to the Meno that Socrates has the answers at the outset and is then only able to lead the boy to give the answers because of this and that this shows that the demonstration is fixed. We have already considered the extent to which – and sense in which – Socrates leads the boy to give the answers, but now I would like to look at the relevance of Socrates’ having the answers. Although it is probably the case that Socrates does know the answers before the discussion begins in this particular instance, the poria-method employed does not require that he have the answers; he could have done this just as well without the answers at the start, arriving at the answer himself only at, or, as he approaches the conclusion. All that is necessary for Socrates to enter into the teacher/facilitator role is that he understands how to approach the problem or how to see the way forward; it is not necessary for him to have the answers to be able to do this. As I have shown, understanding how to approach the problem involves, among other things: identifying the salient concepts for introduction to the student; understanding how to test candidate-answers; being able to identify the next step along the path, or, to see what follows from what has been so far established. One objection to this may be that, in order to do all of this, it will be necessary to have the answers so that one is able to identify the salient concepts, and so that one is able to understand how to test candidate-answers etc. But this would mean that no problems could be solved without prior knowledge of the answers and therefore there would be no problems as any answers to them would be known. It is absurd to suggest that there are no problems[10] and equally absurd to suggest that there are no problems that have been solved[11], so it must be possible to identify a path to the solving of a problem without having to have the answer to the problem. In order to test this idea in the context of the Meno one could try the following thought-experiment: re-read the demonstration with the slave boy and assume that Socrates does not know the answer to the problem. Could the discussion unfold as it does with this assumption in mind? Are there any moments where you think he would need to know the answer in order to ask the questions that he does? I would like to propose that the demonstration is able to be read with a ‘supposed naïve Socrates’ and that this shows that the method throws a veil over the status of the teacher in regard to the searched-for answer. This makes it a method independent of the teacher’s prior possession of the answers. All that is necessary for the teacher to know is the concepts that are needed to be introduced to progress through the problem, but this knowledge is distinct from knowledge of the answer per se; this is knowledge of the context of the problem not knowledge of the answer to the problem.

 

Evidence from the reader’s progress

 

In order to establish that the demonstration with the slave boy is not artificial and therefore rigged to support Plato’s conclusion, there is the evidence of a parallel with the reader of the dialogue. If they too experience all the stages of cognitive progress that the slave boy experiences then is that not sufficient to show that the demonstration does what it is supposed to do and that it is drawn from a real process rather than simply a rhetorical literary device? This, of course, says nothing of how it is done or what the process is, but just that it is not artificial. A working through of the dialogue by a reader clearly shows the reader going through all the same stages, especially if the reader is not familiar with geometry. What is of note in the parallel-of-the-reader is that the answer at 85b is left without an explicit account and I believe that this is for a reason: that Plato is inviting the reader to progress further than the demonstration goes and to venture forward towards knowledge. The problem that Plato sets for us, the readers – just as Socrates set problems for the boy – is to discover the ‘silent account’[12]for ourselves. This problem is only set by implication, so, we have the further task of identifying that the problem has been set.

 

From true-belief to knowledge

 

For the true belief to become knowledge the slave boy would need to be able to provide an account such that he would be able to defend what he believed from attacks with reasons[13]. If he had knowledge he would continue to believe what he believed over time and would be able to explain and justify what he believed to others. This is what Socrates means when he says that knowledge is true belief ‘tethered by an account’ (98a). But, at the close of the demonstration, the slave boy would find doing all the above difficult even though he has reached a true-belief. So how does he move on? How does he up-grade his cognitive state to the status of knowledge? Socrates suggests that repeating the process in various ways would achieve this: “If he were repeatedly asked these same questions in various ways… in the end his knowledge about these things would be as accurate as anyone’s… and he will know it without having been taught but only questioned… [Will he not] find the knowledge within himself?” (85d) Presumably, this will mean something like the following: as the boy approaches the problem from different and various perspectives the conceptual ‘tethering’ becomes stronger and firmer and therefore knowledge is eventually reached. Although, at this stage, it should be pointed out that Plato only says, “his knowledge about these things would be as accurate as anyone’s.” He does not say that his knowledge would be complete and exhaustive. One thing that can be said about the poria-method is that it is a repeatable method that does not simply repeat the procedural steps algorithmically. As Plato suggests, it is progressive, and repeated uses of the method improve one’s understanding of the concepts and problems one is considering, whether or not this can ever be called knowledge and whatever the extent of the knowledge is. On repeated uses of the method one may also find multiple paths to the solution and thereby enrich one’s understanding.

 

I have said a great deal about what Socrates is doing during the demonstration but what is it, exactly, that the student is learning in being subjected to the poria-method, apart from how to solve the particular problem he is faced with? What also emerges from the demonstration is that the boy is not learning a body of knowledge he is learning a way to approach learning itself. What is significant about Plato’s inchoate idea that the method of the demonstration is repeatable and able to bring the student to knowledge is that the process is a meta-attitude to learning. That is to say, the boy is learning to learn by observing the poria-method as it is modelled for him by Socrates. He is not taught the method propositionally but assimilates it by imitation, or at least, he would over time. The method is the tool that the boy can then use to approach learning any subject, and, Plato hopes that this would be as true of virtue as much as of geometry, though much more would need to be said about this conceit. However, presumably – and less controversially – the poria-method can be used with any subject, problem or subject-area that would include clear, logical, inferential thinking. And whatever the empirical content of a subject, most subjects of learning require good, clear thinking in the handling of the subject matter. So, I think, therefore, that the poria-method is a universal method of learning not limited only to the a priori sciences (maths and geometry) but applicable to any subject that requires clear, logical, inferential thinking. This is, I think, the gift Plato confers on us with his poria-method. So, far from proving nothing, the demonstration gives us a universal method of learning to learn.

 

Conclusion

 

It is essential that Socrates be shown not to have told the slave boy the answer – the conclusion at 85b – and I have shown this by a detailed analysis of the text revealing that there are no disguised-answers at the crucial moments of the discussion. Next, I have shown that Socrates shows a path towards the answer using a method I have called the ‘poria-method’. This is an active, collaborative process that tests candidate-answers with set problems, and then moves, stepwise, from what has been established to what has yet to be established by the introduction of salient concepts; leaving the student with an autonomous role of selection, connection and judgement (inference) so that he engages his understanding to search for the answer which he seeks to discover for himself. Socrates is a teacher only in as far as he provides the conditions for the process of the poria-method to take place. Socrates also employs a method of hypothesis with the boy to test candidate-answers where the boy learns to ‘entertain an idea without accepting it’ in order to move the inquiry forward. All this goes to show that Socrates does not give the answer and, given that we need to show that Socrates does not give the answer to prove the success of the demonstration, we can conclude that the demonstration proves that the slave boy (and the reader) can go from ignorance about the side of the 8-foot figure to, at least, true-belief about the side of the 8-foot figure. Through the demonstration the slave boy is able to show that he has true-belief though he cannot account for it. But, finally, and perhaps most importantly, the demonstration has also furnished us with a critical method of learning that can be applied in many educational contexts and could play a crucial role in any attempt to gain knowledge.


[1] Examples of this in the text are at 83b: “How big is it then? Is it not four times as big? – Of course.” And 83c: “Then, my boy, the figure based on the line twice the length is not double but four times as big? – You are right.”

[2] Poria being the ancient Greek for ‘path’ and a-poria therefore meaning ‘without a path’ – the latter more commonly translated as ‘perplexity’ or ‘confusion’.

[3] See footnote 9 for a full, explicit account of the answer.

[4] See footnote 2

[5] There is a sense in which this answer seems reasonable when one is without the concept of area: if you want to double the figure then it might seem reasonable to double the length of the side, as that is what you are using to derive the size of the figure.

[6] My italics.

[7] Socrates then informs the reader, with his next question, which line it is that the boy has pointed to: “That is, on the line that stretches from corner to corner of the 4-foot figure? – Yes.”

[8] The silent account: The diagonal slices the area of the 2-foot side square in half; if the area of the 2-foot side square is 4 feet then the triangle within it is 2 feet, given that it is half; if there are 4 x 2-foot triangles in the enclosed larger square EHFG then the enclosed square must be 4 x 2 feet, which is 8 feet; So, the area of the enclosed square EHFG is 8 feet, based on the diagonal of the 2-foot line figure.

[9] Not explicitly, only implied justifications through the questions asked.

[10] Goldbach’s conjecture, for example.

[11] Euclid’s proof for the infinity of prime numbers, for example.

[12] See footnote 9

[13] Such as the account at footnote 9.

 

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A Socratic Method for Problem Solving

Posted by Philosophy Foundation on December 1, 2010

This is a method for problem solving that I have developed over some years inspired by Socrates’ demonstration with a slave boy in Plato’s Meno. It is a method I have found to be highly effective with a 100% success rate so far. It provides strong motivation in uncertainty and encourages collective intelligence and collaborative effort.

Have the children sit around a board in a horseshoe shape. Set a problem with an answer such as a logical problem but follow this model.

  • Tell the children that the problem may or may not be solvable. It is for them to decide.
  • Tell them that they can ‘give up’ if they want to but that you will only consider the class to have given up when there is nobody in the classroom who wants to ‘have a go’.
  • Emphasise that if they give up then no answers will be given by you: they will have to live with the inconclusiveness of having given up.
  • Make sure you that each attempt is witnessed by everybody in the room (by drawing on the board for example).
  • Strongly encourage them to get up and have a go even if they don’t solve it and keep reminding them that each go will in some way help the others by providing clues.
  • DO NOT INTERFERE OR TRY TO SHOW THEM THE ANSWER. Offer no advice but only tell them where they have broken the rules or stipulations. Say nothing more.
  • Answer no questions. If they ask questions then direct them to the board to try it out.

Provide a clue after a set number of goes but always try to find the clue in what has already been done by the others. Point out that the clue is not proof that the problem is solvable. It may be a red herring.

  • Make the clue as minimal as possible with as little explanation as possible (for example simply point to an attempt that includes an important clue).
  • Do not put anyone on the spot but try to choose people who have not yet had a go.
  • If they solve the problem then congratulate everybody who had a go and ask the person who solved it where they found their clues (very probably from other attempts).
  • Ask those who are familiar with the problem to remain silent for the task.

Variations depending on the problem:

  • You may decide to provide them with paper.
  • You may decide to let them talk it through with each other.
  • You may decide to have them solve it in silence simply by watching each other.
  • If the problem involves something on the board then all attempts should be recorded for all to see throughout the process.

(Originally published in Teach Primary magazine)

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Tell them the answer!

Posted by Philosophy Foundation on December 1, 2010

Reasoning is like joining the dots. It is all about making connections, step by step, between different points in a chain to reach a conclusion. Sometimes, in order to establish a route, you need to know the starting point and the end point. So, it can sometimes be logically fruitful to tell your class the answer but to withhold the reasons why it is the answer. Explaining the answer then becomes the task for the class and they have a clear end-point to navigate towards. To make things a little more challenging you could give them a choice of answers with the stipulation that they must justify the answer they choose. This is a process-orientated style of teaching rather than goal-orientated. Try the following puzzle in this way and see how the children deal with it much better than if you simply give them the problem, which will often leave them very confused. Use this puzzle with the problem-solving procedure I described in an earlier issue of Teach Primary (Vol. 4.6) ‘Any answers?’ (included on this site as ‘A Socratic Method for Problem Solving’).

The Way to Larisa

You are standing at a fork in the road and you are trying to get to the city of Larisa but you don’t know whether to take the left fork or the right fork. There are two brothers at the fork and you know that one of them always lies and the other always tells the truth. What single question could you ask them that will reveal the correct path to take?

The answer is: ‘Which way would your brother say to take?’ I shall leave it to you and your class to say why this is the correct question to ask!

(Originally published in Teach Primary magazine)

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