Monday, January 3, 2022

Answering Aristotle I.3 - There Is More than One Thing II

Physics I.3 The definition of a whole cannot be found in the definition of its parts, so that things exist does not mean that there is an existence that they are a part of.

This reminds me of my first encounter with Aristotle, which happened to be when I was taking a graph theory course as an undergraduate. In the Posterior Analytics, Aristotle propounded that an argument that had to be made case by case wasn't a true derivation, or "...we often fall into error because our conclusion is not in fact...universal in the sense we thing we prove it so." When we prove a proposition case-by-case, instance-by-instance, "then the the demonstrations will be true of the individual instances...and will hold in every instance of it, yet the demonstration will not be true of this subject commensurately and universally." Aristotle's example, as so many it the text, is geometric: if the proof is shown for isosceles, scalene, and equilateral triangles, or if it is true of acute, right, and obtuse angles, then even though it is true for every triangle or for every angle, it is not true of triangles themselves because the proof does not follow from the nature of the triangle.

Aristotle always seems to say that kind of thing.

There is something reasonable about this. If there is a property of an object or a situation, then it should be derivable from the qualities of the object. If you need to bring in other conditions, then maybe it's the combination that has the property. Humans don't look like bushes, but humans wearing camouflage clothes and makeup can. The property of bush-likeness is a property of the clothing, not the man. Furthermore, I could create a situation where I found tofu dishes everyone likes: some people like tofu hot dogs, some people like tofutti, and even I like gelatinous tofu with beef in a szechwan sauce. However, people don't like the tofu, they like the flavorings around the tofu. Very few people eat plain dehydrated tofu bars.

But remember, as I was reading this, I was taking a graph theory course. It was taught by the math department, but the course was a required course computer science majors, but it did use proofs.* Proof courses often had themes that were not explicit in the content, techniques for proofs that are useful in the field. For probability theory, it was condition and uncondition. For graph theory, proof by cases. For example, if you can prove that a process works for all even number cases, then you can do the same for all odd number cases, then you can say the process works for all cases. This was even true of the four color theorem, which I think we will all say is a property of maps -- it is not a different property for different kinds of maps.

So I think that Aristotle has a point, but he's taken it a little too far: proof through cases, if you can demonstrate that you have exhausted all the cases, can still show that a derived property is a property of the system.

Some of what Aristotle is saying in I.3 has a similar feel.**

Unfortunately, that feel is very hard to follow. Aristotle talks about qualities like "paleness" as if paleness is to humans as quarks are to protons, and he does so with the object to show that there must be more than one thing. He does so mostly while arguing against positions that aren't clearly defined, so it feels like he's not arguing directly -- but since he doesn't restate the positions he is attacking.

Here Aristotle feels he has successfully defended against two propositions:

(1) Non-being has being.
(2) There exist indivisible magnitudes.

both of which have some merit, today.(*3) That non-being might have being means something like that the vacuum has some properties, but today we think that gravity is the theory of space-time, that mass can bend its fabric. Furthermore, quantum gravity is an attempt to find the quantum mechanical properties of this fabric. We think that this is true. The second is the atomic hypothesis, which wouldn't be accepted until the late 19th century, and today the standard model is full of indivisible, but not immutable, magnitudes.

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* "Pure math" courses are proof-based somewhat like high school geometry, they aren't the endless calculations of algebra and calculus classes. You are expected to demonstrate that something works, or is a kind of mathematical tool, using logic. When the courses were required for other majors, computer science and math ed., they ended up being very, very simple.

** I thought I would give an example here of Aristotle saying something similar in I.3 as I quoted in the Posterior Analytics, but the prose is quite convoluted, and the long, ellipsis-laden quotes I can find in the Physics say almost the same thing.

(*3) Giving examples like this and those I did in I.2 are not what I wanted to be doing here.

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