Nicomachus seems to divide his exposition on Arithmetic into three parts: viz. its natural superiority, ontological priority, and the posteriority of foregoing Mathematical sciences that are subject to it.
Firstly, he describes Arithmetic as being the origin, root, and mother science of all and any mathematical conception that has or could ever exist. Indeed, he says: “… it existed before all the others in the mind of the creating God like some universal and exemplary plan”. He goes on to assert that – because of this – it served as the archetypal design for the ordering of material creations to attaining their proper ends.
Next, he transitions into examining the ontological priority of Arithmetic. We learn that it abolishes other sciences but is itself not abolished by them. He demonstrates, with appeals to natural philosophy: “For example, ‘animal’ is naturally antecedent to ‘man’, for abolish ‘animal’ and ‘man is abolished; but if ‘man’ be abolished, it no longer follows that ‘animal’ is abolished at the same time…. Hence arithmetic abolishes geometry along with itself, but is not abolished by it”.
Finally, he concludes that the ontological posteriority of the rest of the Quadrivium implies the older with themselves but are not implied by the older. Nicomachus affirms this by saying: “Conversely, that is called younger and posterior which implies the other thing with itself, but is not implied by it”. He goes on to give different examples: how musician implies man, but man is not implied by it, how horse implies animal, but animal is not implied by it, etc. He ties all this together conclusively by teaching: “For how can ‘triple’ exist, or be spoken of, unless the number 3 exists beforehand, or ‘eightfold’ without 8? But on the contrary 3, 4, and the rest might be without the figures existing to which they give names”.
EAR