Aristotle flourished before the great flowering of Greek mathematics that gave us Euclid, Ptolemy, Apollonius, and Aristarchus. In his day, mathematics amounted to just arithmetic and simple geometry. In spite of the famous Pythagorean theorem that the square constructed from the hypotenuse of a right triangle is equal in area to the sum of the squares constructed from the other two sides, the historic reality of the Pythagorean movement had more to do with number mysticism, other superstitions, and curious injunctions like “don’t eat beans” than it did with real mathematics.
I think Aristotle was entirely right to conclude that arithmetic and simple geometry were of little use for explaining change in the natural world. I’ve characterized his physics as grounded in a kind of semantic inquiry that Aristotle pioneered. We are not used to thinking about science this way, as fundamentally involved with a very human inquiry about the meaning of experience in life, rather than predictive calculation. For Aristotle, the gap between natural science and thoughtful reflection about ordinary experience was much smaller than it is for us.
Aristotle invented the notion of cause as a semantic tool for expressing the reasons why changes occur. Aristotle’s notion is far more abstract than the metaphor of impulse or something pushing on something else that guided early modern mechanism. Even though the notion of cause was originally developed in a text included in Aristotle’s Physics, the “semantic” grounding of Aristotelian physics places it closer to logic than to modern physical inquiries.
I think the discussion of the kinds of causes could equally well have been grouped among his “logical” works. In fact, the form in which we have Aristotle’s works today is the result of the efforts of multiple ancient editors, who sometimes stitched together separate manuscripts, so there is room for a legitimate question whether the discussion of causes was originally a separate treatise. We tend to assume that there must be something inherently “physical” about the discussion of causes, but this is ultimately due to a circular argument from the fact that the more detailed version of it came down to us as part of the Physics (there is another, briefer one that came down to us as part of the Metaphysics).
Since Hume and especially since the later 19th century, many authors have debated about the role of causes in science. Bertrand Russell argued in the early 20th century that modern science does not in fact depend on what I have called the modern notion of cause.
More recently, Robert Brandom has argued that the purpose of logic is “to make explicit the inferential relations that articulate the semantic contents of the concepts expressed by the use of ordinary, nonlogical vocabulary”. I see Aristotelian causes in this light.
I want to recommend a return to a notion of causes in general as explanatory reasons rather than things that exert force. This can include all the mathematics used in modern science, as well as a broader range of reasons relevant to life. (See also Aristotelian Causes; Mechanical Metaphors; Causes: Real, Heuristic?; Effective vs “Driving”; Secondary Causes.)