Preserving Plato

Plato’s Meno contains several instances in which Socrates imparts knowledge to his interlocutor’s slave. These instances belie Socrates’ claims that he is simply eliciting a recollection of mathematical knowledge. In imparting knowledge, however, Socrates demonstrates that mathematical inquiry requires the presence of a teacher. By defining more precisely the role that the teacher has to play in Socratic learning, we can reformulate the dialogue in such a way as to preserve Plato’s mathematical epistemology. This reformulation requires that we recognize that the teacher has an auxiliary role to play in Socratic inquiry and that Socrates curtails the amount of geometrical knowledge he makes explicit in the dialogue.

Some of the mathematical knowledge that Socrates imparts is trivial. In describing the area of a rectangular figure he says, “If it were two feet this way, and only one foot that way, the figure would be once two feet?” After the slave acknowledges this claim Socrates continues, “But if it is two feet also that way, it would surely be twice two feet” (82d). In so many words, Socrates has told the slave that the proper way to calculate the area of a rectangle is to multiply its length by its width. Socrates had previously stipulated that the ability to speak Greek was the only prerequisite for the slave’s participation in geometrical inquiry (82b). Meno later affirms that no one has taught the slave geometry (85e). The slave’s rapid acknowledgement of an item of geometrical knowledge (i.e. the area of a rectangle is its length times its width) therefore bears two interpretations: (1) Socrates has imparted that item of knowledge to the slave or (2) in depicting such a figure and describing it Socrates recommends that item of knowledge so strongly to the slave’s recollection that he has no means to resist admitting it as a truth.

These two interpretations are not mutually exclusive. Part of the difficulty in adjudicating the role that Socrates plays in the Meno may derive from a lack of logical precision. Option (2) might be construed as an analysis of the term “teaching.” If such an analysis is correct, then we must take the “teacher” to be an auxiliary albeit necessary role in Socratic inquiry. He or she is to commend items of knowledge to the student’s consideration in such a way that the student’s intellect is overwhelmed by the truth of the proposition he or she is considering. Such a definition of “teaching” does not preclude the possibility that “knowledge” amounts to “recollection.” In this case, the slave admits the proposition that the area of a rectangle is its length times it width not on the authority of Socrates per se, but rather because he is confronted with an arresting mental depiction of a rectangle. The slave would not have acknowledged that item of knowledge without the mediating role of Socrates. Yet once that role has been fulfilled, the slave’s intellect is subdued by logical necessity and not by didactic fiat.

In addition to commending items of knowledge to their students’ recollection, the Socratic teacher also has a role to play in exposing ignorance. When he is first asked to identify the length of the line that would double a square’s area, the slave confidently asserts that such a line would be double its original. By illustrating just such a square, Socrates proves that the slave has proposed a method of quadrupling rather than doubling the original square’s area. After he has been brought to acknowledge this, the slave is asked again how long the required line must be, to which he responds, “By Zeus, Socrates, I do not know” (84a).

At this point in the dialogue Socrates suggests that he and his interlocutors have reached an epistemological tipping point. “You realize, Meno, what point he has reached in his recollection,” says Socrates. He continues, “even now he does not yet know, but then he thought he knew, and answered confidently as if he did know, and he did not think himself at a loss, but now he does think himself at a loss, and as he does not know, neither does he think he knows.” Just as the Socratic teacher is to commend mathematical truths to the unfettered consideration of their students, so too are they to expose assumptions that conflict with those truths. Here again, false assumptions are undermined not because of Socrates’ authority, but because Socrates has directed the student’s attention to a higher authority—that of mathematical proof. By forcing the student into a state of aporia, or perplexity, Socrates has accomplished two things. First, he has diminished the student’s propensity to speak falsehoods confidently. Secondly, he has imbued the student with a desire to rectify his ignorance. In short, Socrates has “taught.”

Once the slave’s ignorance has been exposed, Socrates stipulates the constraints under which their inquiry will proceed. “I shall do nothing more than ask questions and not teach him,” he tells Meno (84d). He instructs his interlocutor to “Watch whether you find me teaching and explaining things to him instead of asking for his opinion.” In a literal sense, he never transgresses these constraints. The conclusion toward which the dialogue’s slave interlude moves—the length of the side of a square that doubles that square’s area is its diagonal—proceeds through a series of diagrams, rhetorical questions, and succinct assents. What troubles the reader interested in maintaining Plato’s contention that knowledge is recollection is how rhetorical the questions and how succinct the answers seem to be. Socrates constructs a diagram of a square of area 16 units, bisects each constituent square 4 feet in area by its diagonal, and commends the resulting square to the slave as one of area 8 units, or double the area of the original 4-foot square (84d-85d). We can be forgiven for thinking that Socrates is doing all the work here.

Consider, for instance, the point in the dialogue at which Socrates and the slave are on the verge of reaching their conclusion regarding the doubling of a square’s area. Socrates has bisected the constituent squares of his 16-foot square. He then asks the slave how large the figure enclosed by the 4 bisections is, to which the slave responds, “I do not understand” (85a). To this interjection Socrates replies, “Within these four figures, each line cuts off half of each [constituent square of area 4 feet], does it not?” This seems to us a rhetorical bridge too far. The diagram has been composed. Lengths have been allocated to each constituent part of the figure. Trivially, it seems that a thorough examination of the figure would lead the slave to the conclusion that the bisections of the constituent squares yield a 16-foot square whose area has been bisected, i.e. a square of area 8 feet. Once we have admitted that the teacher has an auxiliary role to play in mathematical inquiry, we would be more sympathetic to the manner in which Socrates proves his claim that knowledge is recollection if he were to commend the completed diagram to the slave’s unaided reason. Stated simply, Socrates makes too much explicit.

It is conceivable that this objection stems from the literary and stylistic constraints under which Plato was working. Perhaps the logical necessity of the final diagram is made explicit more for Plato’s audience than Socrates’ interlocutor. Still, we would feel more comfortable if the character of Socrates confined himself to the composition of diagrams and the allocation of numerical values. These would be the premises from which the slave’s unaided recollection would proceed to geometrical conclusions. Once we have stipulated that it is the Socratic teacher’s role to provide such premises, the slave’s arrival at the proper geometrical conclusion would provide more convincing proof of Plato’s epistemological claims. Such a view casts the “teacher” in the same role as the projectionist at a movie theater. His role is one of explanation rather than explication, which is provided by the audience’s unaided reason.