Answering Aristotle I.4 - There Cannot Be an Infinite Number of Things

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Physics I.4 The number of kinds of things must be finite since the infinite is unknowable and a finite body cannot be composed of an infinite number of finite bodies.

Again Aristotle is arguing against someone to prove his point. In this case, Anaxagoras. Aristotle presents this as since "it is impossible for something to come into being from non-being," so everything that exists is made up of smaller constituents. And if everything comes from smaller constituents, then there cannot be a smallest object -- it, itself, would need to be composed of even smaller things. Aristotle, disagrees. Although there must be more than one thing, or even one kind of thing, there cannot be an infinite number of things, let alone an infinite number of kinds of things that the finite things we see are composed of.

It is a little difficult to keep track of things here, because sometimes Aristotle seems to be talking about the number of things that exist; at other times, he seems to being talking about the the number of kinds of things that exist; and at still other times, he seems to be arguing about the number of different properties that a thing can hold.

Aristotle's argument that there cannot be an infinite number of things has five parts, of different quality:

(1) The infinite is unknowable.

This is rather technical in the sense that it makes a strong point about the limitations of what we can perceive in thought. However, it is not really true that "if an object consists of an infinite number of things and forms, its nature is unknowable." This is a little strong, since something that consists of an infinite regularity could be understood in principle without apprehending its nature as a whole. That is in fact what we do in science. We don't perceive the entire array of atoms in a crystal lattice. For the most common measurements of the lattice, from X-ray diffraction, we don't even directly look at the regular array. Instead, we look at the regularities in the array of atoms making up a sample, and learn about its constituents that way.

(2) If an object has a finite size, then its parts must also be finite.

Zeno would be proud. This is a classic piece of the Dichotomy. However, Aristotle didn't buy Zeno's paradoxes any more than we do. In fact, many of his reasons are good enough to refute Zeno in any but a pedantic way (today, even the pedants should understand these paradoxes are refuted even at the more fundamental levels*). The parts of a homoeomerous substances could be made of ever smaller parts if you take a continuum model or even an infinitesimal model of matter.

(3) That some of every constituent is present in every object is inconsistent.

The best way I can present this argument is that if an object has a finite number of (kinds of) constituents, and if those constituents are differentiated and their relative preponderance determines the object's nature, then at some point when you reduce the size of a sample below a certain volume, you will have a sample of the object with an insufficient amount of one of its constituents to be the same kind of thing as it reduces in size. The divided objects cease to be of the kind of the original object. This would be equally true with random fluctuations in small samples.

(4) Nothing material can be extracted from the smallest instance of a substance.

Once you reach the limit above, you cannot make a smaller copy of the thing you're dividing. This is a direct response to Anaxagoras.

(5) An infinite collection must be both divided and connected simultaneously.

The infinite number of things that Aristotle is talking about are the constituents of other things.

Now that I try to explain them, I don't think any of them are very good. I think many of the reasons he discussed earlier are much better than his discussion here, but the conculsions he came to were wrong. And ironically, on this one Aristotle is right.

Nothing is composed of an infinite number of things, to our knowledge.[2] It seems like there are a finite number of kinds of things that something can be composed of at the smallest level (standard model) and even a finite number of possible elements something can be made of (periodic table). So although a copper ball has an impossible number of atoms in it (maybe 10²⁵ or so), it is still a finite number, and although the atoms that make up the ball have constituents, they have a small number of constituents (say, around 250).

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[1] Although, as philosophers, they do continue to find value in them. See Salmon, Zeno's Paradoxes and Sainsbury, Pardoxes. [Amazon]

[2] But there could be preons inside of quarks (but probably not). And if there are preons inside of quarks, then what's inside the preon?

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