Is Plato contradicting himself in the Republic?

Question

In Book VII of the Republic, Plato claims that geometry and arithmetic will lead to eternal/pure truth. However, as we know the eternal/pure truth are Forms. Knowing the forms is knowing the pure truth. However, what we engage in, in mathematics, is not the forms. He actually says that mathematics are based on unproved assumptions that that is why it is still nothing like the Forms. So what is going on here? Am I misunderstanding him?

Answer 1

The objects of mathematical knowledge are, as you indicate, not Forms. And the hypothetical nature of mathematical knowledge also removes mathematics from the unhypothetical first principle, the Form of the Good, acquaintance with which yields absolute, incorrigible knowledge.

But mathematics is an inescapable rung on the ladder to such knowledge. Plato distrusts and despises the senses; and what mathematics does, indispensably, is through its intense abstractness to deflect the mind from the shifting, changing, unstable sense-based world, of which there can be no reliable or accurate knowledge, to a world of stable, unchanging objects. This is how Plato sees the objects of mathematical thought.

It is in this pure state of mind, cleansed of thought about the sense-based world, that the mind is able eventually to pass beyond mathematics to grasp the unhypothetical first principle. Plato offers no plausible account of how this is to be done. We are asked to trust to a flash of insight, of immediate apprehension, of final intuitive insight.

In fairness Plato cannot describe the exact process non-metaphorically because he has not himself experienced it. He can only conjecture what it will be like, and so resorts to metaphor.

We can rightly question whether there is any unhypothetical principle to grasp by intuitive insight or in any other way. But given his view of the epistemological vacuity of sense-based knowledge, Plato finds (I think) a perfectly coherent and indispensable role for mathematics in the ladder to unhypothetical, absolute and incorrigible knowledge. Abstractness is the key.

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Reading

R.C. Cross & A.D. Woozley, Plato's Republic : A Philosophical Commentary, Macmillan, 1994, ch. 10 'Mathematics and Philosophy;.