The mathematical sciences studied by the apprentice philosopher in The Republic are only a prelude to the crowning stage of the philosopher’s education, namely, the study of dialectic, leading to the intellectual apprehension of the Form of the Good. Mathematics is, indeed, the indispensable tool if students are to rise above a transient physical world and the partial information they can derive from delusive senses. Through the study of geometry, the mind reaches a conceptual understanding of plane and solid figures before theorising about objects in space and eventually mastering the hidden harmony of sound and time itself. After ten years of training in mathematics between the age of 20 and 30, the philosopher is, by then, ready for the vision of the Good, just like the prisoner could finally look straight at the sun, after his slow ascent from the darkness of the cave. He has reached ‘the summit of the intellectual realm as the man who looked at the sun was of the visual realm.’ [532b]
Earlier on in the dialogue, Socrates refused to give a definition of the supreme good, arguing that it should be thought of as a ‘revelation’. He feels, once again, uneasy about giving a straight answer to Glaucon when his friend requests a definition of dialectic. Socrates tells him evasively that ‘what you’d see would no longer be an image of what we are talking about but the truth itself.’ [533a] The word ‘dialectic’ is originally derived from the Greek word meaning ‘art of discussion’ but there is a notable difference between the art of persuasion practised by the Sophists, ‘dianoia’ which corresponds to the practical use of logical arguments in solving mathematical problems (knowing how?) and ‘noesis’ which is the exercise of pure intelligence (knowing why?) as illustrated in the highest section of the simile of the line. The reluctance of the master of ‘dialectic’ to define his own expertise is, in fact, a sign of humility as Socrates is prepared to admit that the mind can never claim absolute certainty ‘though one can claim that there is something of the kind to see.’ [533e] The Form of the Good does not totally reveal itself to the human mind but only the method of dialectic can give some access to its inner nature. Even the mathematical sciences still have ‘some hold on reality’ and ‘can never wake up and look at it’ [533c] with the clarity of pure intellectual vision.
Who, then, qualifies as a ‘dialectician’? Socrates ventures an explanation: he is a man ‘who can take account of the essential nature of each thing; and in saying that anyone who is unable to give such an account of things to himself [through logical reasoning] or to other people [through discursive reasoning] has to that extent failed to understand them.’ [534b] The last point is a most important proviso or condition as students who fail to master the dialectical method will not become Guardians and will have to be content with a secondary role as Auxiliaries. Dialectic is, therefore, ‘the coping-stone [or pinnacle] that tops our educational system; it completes the course of studies and there is no other study that can rightly be placed above it.’ [534e]