  What about the pupils of Plato and Aristotle? As discussed in the second paragraph of section 5 above, Eudemus, who wrote a series of histories of mathematics never mentions Pythagoras by name. Arguments from silence are perilous but, when the most well-informed source of the fourth-century fails to mention Pythagoras in works explicitly directed towards the history of mathematics, the silence means something. There are only two passages in which Pythagoras is explicitly associated with anything mathematical or scientific by pupils of Plato and Aristotle. First, Aristotle’s pupil Aristoxenus reports that Pythagoras “most of all valued the pursuit (pragmateia) of number and brought it forward, taking it away from the use of traders, by likening all things to numbers” (Fr. 23). Zhmud translates  pragmateia as “science” (2012a, 216) so that he has Aristoxenus attributing the invention of the science of number to Pythagoras but, while Aristoxenus does use pragmateia to mean science in some contexts, it more commonly simply means “pursuit” (Huffman 2014b, 292). Here surely it must mean “pursuit,” because Pythagoras is presented as taking it away from the traders and we can hardly suppose that the traders were engaged in the theoretical science of arithmetic. Moreover, Aristoxenus explains what he means in the final participial phrase. He is not ascribing rigorous mathematics with proofs to Pythagoras but rather says that Pythagoras was “likening all things to numbers”. This is consistent with the moralized cosmos of Pythagoras sketched above in which numbers have symbolic significance. The second important passage is Plato’s pupil Xenocrates’ assertion that Pythagoras “discovered that the intervals in music, too, do not arise in separation from number” (Fr. 9). Xenocrates is being quoted here in a fragment of a work by a Heraclides (Barker 1989, 235–236), perhaps Heraclides of Pontus. There is controversy whether the quotation of Xenocrates is limited just to what has been quoted in the previous sentence or whether the whole fragment of Heraclides is a quotation of Xenocrates. Burkert (1972a, 381) and Barker (1989, 235) argue that it is probably just the first sentence that Heraclides ascribes to Xenocrates, while Zhmud would include at least a second sentence in which Heraclides presents Pythagoras as pursuing a program of research into “the conditions under which concordant and discordant intervals arise” (Zhmud 2012a, 258). If the second sentence is accepted then Xenocrates clearly presents Pythagoras as an acoustic scientist. It seems most reasonable, however, to accept only the first sentence as belonging to Xenocrates. If the quotation from Xenocrates does not break off at that point, there is no other obvious breaking point in the fragment and the whole two pages of text must be ascribed to Xenocrates. The problem with ascribing it all to Xenocrates is that Porphyry introduces the passage as a quotation from Heraclides, which would be strange if everything quoted, in fact, belongs to Xenocrates. If just the first sentence comes from Xenocrates, then all he is ascribing to Pythagoras is the recognition that the concordant intervals are connected to numbers. It is easy to assume, as Zhmud does, that Xenocrates is saying that Pythagoras was the first to discover that the concordant intervals are governed by whole number ratios but Xenocrates’ remarks need not mean this. Xenocrates’ comments might well come from a context like that in the fragment of Aristoxenus, above, i.e., a context in which Pythagoras is presented as likening all things to numbers and arguing that numbers in some sense explain or control things. In such a context Xenocrates would not be making the point that Pythagoras discovered the whole number ratios but rather that he found out that concords arose in accordance with whole number ratios, perhaps from musicians (who discovered them first not being the issue), and used this fact as another illustration of how things are like numbers. Thus, the fragments of Aristoxenus and Xeoncrates show that Pythagoras likened things to numbers and took the concordant musical intervals as a central example, but do not suggest that he founded arithmetic as a rigorous mathematical discipline or carried out a program of scientific research in harmonics.