  When one turns from the Phaedo to the Republic, the notion of hypothesis appears in the two ‘upper’ segments of the line. They are distinguished in part by the fact that those considering their subject matter dianoetically still make use of hypotheses, whereas those using nous have arrived at an unhypothetical first principle and descending the line have destroyed the hypotheses. It seems that the mathematicians, for example, use discursive thought because they assume the starting points, i.e., the axioms or definitions, of their sciences. In this respect, they are like Socrates of the Phaedo positing the Forms to explain generation and corruption. In Plato's hands, scientific inquiry seems to emerge from reflection on the everyday material world. The step from conviction to dianoia, however, is taken by far fewer individuals than those who step from eikasia to conviction.