Note 114 — Chapter I · THE MILESIAN SCHOOL

I have assumed that the word ἄπειρον means spatially infinite (though not in any precise mathematical sense), not qualitatively indeterminate, as maintained by Teichmüller and Tannery. The decisive reasons for holding that the sense of the word is “boundless in extent” are as follows: (1) Theophrastos said that the primary substance of Anaximander was ἄπειρον and contained all the worlds, and the word περιέχειν everywhere means “to encompass,” not, as has been suggested, “to contain potentially.” (2) Aristotle says (Phys. Γ, 4. 203 b 23) διὰ γὰρ τὸ ἐν τῇ νοήσει μὴ ὑπολείπειν καὶ ὁ ἀριθμὸς δοκεῖ ἄπειρος εἶναι καὶ τὰ μαθηματικὰ μεγέθη καὶ τὰ ἔξω τοῦ οὐρανοῦ· ἀπείρου δ’ ὄντος τοῦ ἔξω, καὶ σῶμα ἄπειρον εἶναι δοκεῖ καὶ κόσμοι. (3) Anaximander’s theory of the ἄπειρον was adopted by Anaximenes, and he identified it with Air, which is not qualitatively indeterminate.