Sunday, April 22, 2012

How was Aristotle using mathematics to contribute to ancient and modern society?

Many consider Aristotle the first physicist (or at least the first we know of), because he was the first person in recorded history to make significant use of quantitative mathematics in understanding physical phenomena.

His formalization of logic has largely been preserved in modern classical logic, though we now have some expansions such as bivalent logic and Bayesian fuzzy logic. His quantifiers "there exists" and "for every" are the foundation of modern predicate logic, though his other quantifiers involving "essence" and "intrinsic" properties have not fared as well.

His account of empiricism is basically correct as a sort of first stab at the scientific method; he didn't understand statistics or experimental rigor, but his basic idea that we formulate theories and test them with observation was exactly right. This was a radical approach for his time; many other philosophers thought that all truth could be understood simply by thinking about it, without any input from the real world. Where Plato would try to answer a question by thinking about how it should be, Aristotle would actually go out and look. This is probably Aristotle's greatest contribution, though sadly it went underappreciated and underutilized for centuries. (Ironically, much of what held back Medieval science was a perverse admiration for Aristotle's results instead of Aristotle's methods.)

He also had a theory of mathematical ontology that greatly contrasted with Plato's and more closely resembled modern concepts of mathematical ontology. Where Plato imagined some strange concept of "Forms" in some "other world", Aristotle correctly appreciated that mathematical concepts are logical abstractions and generalizations idealized from real phenomena. 

Aristotle also had a theory of physics, which was of course wrong, but not as wrong as a lot of people seem to think. His equations for motion and gravitation, for example, are reasonably accurate approximations under normal conditions of friction and air resistance. Heavy things do fall faster in the real world, as long as you're not in outer space; and most things do stop on their own when you stop pushing them, because they are subject to friction. Aristotle noticed that there were exceptions (such as something launched from a catapult), and tried to come up with ad hoc theories to explain that---and these ad hoc explanations actually worked fairly well.

Similarly, his theory of four (plus one) elements is often mocked today because we have over a hundred elements; but I don't think Aristotle's concept of an "element" was what we call "elements" today. (If he'd consulted even a single metallurgist, they would have told him that gold and bronze aren't made of the same stuff.) I think a better word for it would be phase of matter. With that in mind, "earth, water, air, fire" actually maps quite nicely onto "solid, liquid, gas, plasma"---and recognizing that plasma is a distinct state of matter is actually an achievement modern scientists wouldn't grasp until the 18th century. Aristotle also theorized a fifth celestial element called quintessence, which he thought made up the Sun, Moon, and stars; on that he was just flat wrong, as celestial objects are made of the same stuff as everything else. (Indeed, "celestial" is a relative term; for someone standing on Mars, we are the ones in the sky.)

But overall, Aristotle was one of the most important thinkers who ever lived. Had he not existed, it's hard to say how long it would have been before someone else came along to think of empiricism, logic, and physics. Had it taken long enough, the course of human history could have been radically different---science and technology might have been held back hundreds of years.

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