  Zhmud supports Pythagoras’ position as genuine mathematician rather than someone interested only in number symbolism by pointing to gaps in the development of early Greek mathematics. Although there is no explicit evidence, Pythagoras is the most likely candidate to fill these gaps. Thus between Thales, whom Eudemus identifies as the first geometer, and Hippocrates of Chios, who produced the first Elements, someone turned geometry into a deductive science (Zhmud 2012a, 256). Similarly, Hippasus’ experiment with bronze disks to show that the concordant intervals of the octave, fifth and fourth were governed by whole number ratios is too complex to be a first attempt so that once again someone must have discovered the ratios in a simpler way earlier (Zhmud 2012a, 291). In each case Zhmud suggests that Pythagoras is that someone. Finally, the study of proportion ties together airthmetic, geometry and harmonics and Zhmud argues that, although there is no explicit fourth-century evidence, later reports which assign Pythagoras the discovery of the first three proportions (Iamblichus, Commentary on Nicomachus’ Introduction to Arithmetic 100.19–101.11) are likely to go back to Eudemus (2012a, 265–266). Such speculations have some plausibility but they highlight even more the puzzle as to why, if Pythagoras played this central role in early Greek mathematics, no early source explicitly ascribes it to him. Of course, some scholars argue that the majority have overlooked key passages that do assign mathematical achievements to Pythagoras. In order to gain a rounded view of the Pythagorean question it is thus appropriate to look at the most controversial of these passages.