Aristotle, the Pythagoreans, and Structural Realism

I Two basic QUESTIONS OF PHYSICS ARE: what is the world made of, and why do these constituents do the things they do? These two questions are closely related. If certain kinds of things are the ultimate constituents of the world, it can only be because their characteristics explain what we observe to be the case. Further, if it were possible to grasp laws or necessary truths that explain absolutely everything, there would be no need to appeal to any kind that underlies those truths or laws; such a move would do no theoretical work. Nonetheless, in the West most scientists have offered explanations that appeal to both what the basic things or stuff are, and the fundamental features they possess or laws they obey. This approach goes back to Aristotle, who insisted that explanation in physics requires identifying both a material substrate and a formal basis for what is to be explained. He argued that this holds in regard to explaining both the existence of substances and the fact that they bear certain attributes. In order to explain both why a substance exists and why a substance does what it does, one must appeal to its essence, as expressed in a definition that includes both matter and form. (1) An event or attribute (such as anger) is to be accounted for both by pointing to the persisting substrate (the blood around the heart) and the form it takes on (a kind of boiling that is the result of perceived anger). (2) Metaphysics 1 tells the story of how, in fits and starts and to varying degrees, earlier thinkers came to realize that explanation demands identifying all four causes, including both the material and formal causes. Aristotle's earliest philosophical predecessors, the Milesians, are credited with offering explanations on the basis of matter. (3) Aristotle criticizes their accounts as radically incomplete on the grounds that matter alone cannot account for all things and their characteristics. For example, it cannot account for goodness or beauty. (4) While the Milesians are said to have identified certain kinds of stuff as basic, Aristotle takes other predecessors, the so-called Pythagoreans, to have done the same for number. Aristotle associates them with the Milesians, insofar as he understands them to give at least some numbers the status of matter. The explanatory strategy of explaining derivative kinds on the basis of the characteristics of the basic kinds is the same; their dispute with the Milesians concerns what basic characteristics are most explanatory. (5) Aristotle is dismissive of the Pythagorean notion of numbers that are not quantities of substances: "[A] number, whatever it is, is always a number of certain things, either of fire or earth or of units." (6) Aristotle dismisses the ontology of the Pythagoreans by indicating the confusion of positing as basic what must inhere in a substrate. Perhaps it is a similar reaction to the notion of a number ontology that has led some contemporary scholars to deny that Philolaus (Aristotle's likely source for Pythagorean number ontology) could have possibly thought that all things are made of number. (7) Aristotle takes the Milesian explanatory strategy to be one of accounting for things on the basis of their being made up of certain stuffs, which, if not identical with the perceived constituents of familiar objects, are at least conceivable along the same lines. Today there are few, if any, neo-Milesians. As Planck wrote: "the physical world has become progressively more and more abstract; purely formal mathematical operations play a growing part, while qualitative differences tend to be explained more and more by means of quantitative differences." (8) Mathematical features are formal. Nonetheless, Planck himself did not follow the path of Aristotle's Pythagoreans. He felt compelled to posit a kind of substrate to physical reality, even though he cautioned that imagining that substrate as like the stuffs or particles familiar from sensation can be highly misleading. …