  It is striking that a very similar picture of Pythagoras emerges from the evidence for his cosmology. A famous discovery is attributed to Pythagoras in the later tradition, i.e., that the central musical concords (the octave, fifth and fourth) correspond to the whole number ratios 2 : 1, 3 : 2 and 4 : 3 respectively (e.g., Nicomachus, Handbook 6 = Iamblichus, On the Pythagorean Life 115). The only early source to associate Pythagoras with the whole number ratios that govern the concords is Xenocrates (Fr. 9) in the early Academy, but the early Academy is precisely one source of the later exaggerated tradition about Pythagoras (see above). One story has it that Pythagoras passed by a blacksmith’s shop and heard the concords in the sounds of the hammers striking the anvil and then discovered that the sounds made by hammers whose weights are in the ratio 2 : 1 will be an octave apart, etc. Unfortunately, the stories of Pythagoras’ discovery of these relationships are clearly false, since none of the techniques for the discovery ascribed to him would, in fact, work (e.g., the pitch of sounds produced by hammers is not directly proportional to their weight: see Burkert 1972a, 375). An experiment ascribed to Hippasus, who was active in the first half of the fifth century, after Pythagoras’ death, would have worked, and thus we can trace the scientific verification of the discovery at least to Hippasus; knowledge of the relation between whole number ratios and the concords is clearly found in the fragments of Philolaus (Fr. 6a, Huffman), in the second half of the fifth century. There is some evidence that the truth of the relationship was already known to Pythagoras’ contemporary, Lasus, who was not a Pythagorean (Burkert 1972a, 377). It may be once again that Pythagoras knew of the relationship without either having discovered it or having demonstrated it scientifically.  The relationship was probably first discovered by instrument makers, and specifically makers of wind instruments rather than stringed instruments (Barker 2014, 202). The acusmata reported by Aristotle, which may go back to Pythagoras, report the following question and answer “What is the oracle at Delphi? The tetraktys, which is the harmony in which the Sirens sing” (Iamblichus, On the Pythagorean Life, 82, probably derived from Aristotle). The tetraktys, literally “the four,” refers to the first four numbers, which when added together equal the number ten, which was regarded as the perfect number in fifth-century Pythagoreanism. Here in the acusmata, these four numbers are identified with one of the primary sources of wisdom in the Greek world, the Delphic oracle.  In the later tradition the tetraktys is treated as the summary of all Pythagorean wisdom, since the Pythagoreans swore oaths by Pythagoras as “the one who handed down the tetraktys to our generation.” The tetraktys can be connected to the music which the Sirens sing in that all of the ratios that correspond to the basic concords in music (octave, fifth and fourth) can be expressed as whole number ratios of the first four numbers. This acusma thus seems to be based on the knowledge of the relationship between the concords and the whole number ratios. The picture of Pythagoras that emerges from the evidence is thus not of a mathematician, who offered rigorous proofs, or of a scientist, who carried out experiments to discover the nature of the natural world, but rather of someone who sees special significance in and assigns special prominence to mathematical relationships that were in general circulation. This is the context in which to understand Aristoxenus’ remark that “Pythagoras most of all seems to have honored and advanced the study concerned with numbers, having taken it away from the use of merchants and likening all things to numbers” (Fr. 23, Wehrli). Some might suppose that this is a reference to a rigorous treatment of arithmetic, such as that hypothesized by Becker (1936), who argued that Euclid IX. 21–34 was a self-contained unit that represented a deductive theory of odd and even numbers developed by the Pythagoreans (see Mueller 1997, 296 ff. and Burkert 1972a, 434 ff.). It is crucial to recognize, however, that Becker’s reconstruction is rejected in some recent scholarship (e.g., Netz 2014, 179) and no ancient source assigns it even to the Pythagoreans, let alone to Pythagoras himself. There is, moreover, no talk of mathematical proof or a deductive system in the passage from Aristoxenus just quoted. Pythagoras is known for the honor he gives to number and for removing it from the practical realm of trade and instead pointing to correspondences between the behavior of number and the behavior of things. Such correspondences were highlighted in Aristotle’s book on the Pythagoreans, e.g., the female is likened to the number two and the male to the number three and their sum, five, is likened to marriage (Aristotle, Fr. 203).