Saturday, February 26, 2022

Answering Aristotle - Index

I am reading Aristotle's Physics, and as I do with non-technical books, I try to write a topic sentence (here, "summary") for each section. The "chapters" in Aristotle are approximately the size of a good section. This is a list of those sentences. The chapter links, however, have me doing something different: when Aristotle makes assertions or predictions, and where I think that contemporary physics has something to say about them, I try to make some notes about that.

Book I
Chapter Summary
Although understanding something means we can reason from first principles, discovering these principles requires us to sort them out from the aggregate observations we are built to apprehend.
There must be more than one thing because the ways in which all reality can be made of one thing each require there to be multiple things.
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.
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.
All principles must be opposites that admit admixtures of opposites, and the properties of an object may consist of combinations of these principles.

Book II

Book III

Book IV

Book V

Book VI

Book VII


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).

[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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Tuesday, February 22, 2022

The Two Ockham's

In reading the first chapter of Bostrom's Anthropic Bias: Observation Selection Effects in Science and Philosophy,[1] his overview dealing with multiple worlds makes me feel as if there are two ways in which Ockham's Razor[2] are being used in science. That is, the general maxim of reducing the number of "entities" to a minimum is applied in two opposing ways, one of which is evident in anthropic reasoning. One of these ways is to reduce the number of actual things that you suppose to exist and the other is to reduce the number of postulates required to make predictions with a theory.

How does that work?

Fine tuning is a modern sin in theoretical physics. A theory that has a large number of free parameters, but only a few of which could lead to observed consequences, needs to have additional assumptions about those parameters. If these are unexplained, then the theory is fine tuned. This is generally felt to be a flaw because it is surprising that an otherwise successful theory should require a large number of ancillary assumptions -- 31 in the case of particle astrophysics[3] -- to predict the existence of the universe. Although not strictly a error, taking an elegant and insightful theory and clothing it in hand-me-down experimental parameters is a bit gauche.

This is where the anthropic principle comes in. This principle, in this case, states that the universe has to be the kind of universe where you and me can exist. We're pretty sure that we do. This constrains overly loose theories, theories that require fine tuning, so that they can make predictions. This method was famously used by Steven Weinberg to predict the approximate magnitude of the cosmological constant. This has also recently been used to try to shore up string theory as it has become looser, parameter-wise, than once thought. Anthropic reasoning is an end around fine tuning.

The expansion of possible string theories from five to an infinite number has made anthropic arguments possible in that each particular universe that would be associated with a string theory[4] (with different parameters) exists, and the reason why we are in this universe with these fundamental constants is not because of any finely tuned assumptions that we have to make, but rather it is because of the existing universes, we have to be in a universe that supports our existence. This can be true with parallel universes, sequential universes, and so on, just as long as there is an infinite reservoir and the proportion of those universes is a subset of the same transfinite cardinality of the reservoir.

This is what brings me to Ockham's razor. This is usually stated as "entities must not be multiplied beyond necessity" or "plurality should not be posited without necessity." If there is no necessity to postulate a soul in order to understand consciousness, then don't postulate a soul. Normally, you would think that this would exclude a string theory landscape, that in order to explain the values of the universal constants that we measure, we need to postulate not just a soul, but an infinite number of souls.

But that's just one way to think about it. Another one is Aquinas' maxim, "It is superfluous to suppose that which can be accounted for by a few principles has been produced by many." In this case, the anthropic principle is supported by Ockham's razor. We have literally reduced the number of assumptions we need to make from 31 ad hoc interpretations of experiments to a single proven principle.

Both cases have a claim to be following the spirit of Ockham, and to me neither is obviously right. At least, neither seems to be the better argument in all cases. So, we have a situation where the same, admittedly somewhat subjective[5], principle would require us to take opposite approaches to the same problem. How do we decide which to follow?

[1] Anthropic Bias: Observation Selection Effects in Science and Philosophy, Nick Bostrom. [Amazon]

[2] I prefer the ckh over the cc. How can Occam's Razor be named for Isaac of Ockham?

[3] See for example Tegmark, Aguirre, Rees, and Wilczek's "Dimensionless Constants, Cosmology and Other Dark Matters." [arXiv] See also Physics Frontiers 55: Multiversality.

[4] Anthropic Landscape of String Theory, Leonard Susskind. Extrad Dimensions in Space and Time [Amazon], Bars and Terning, Multiversal Journeys Series. See also: Physics Frontiers 35: The String Theory Landscape.

[5] But Ockham's razor is no less subjective than the beauty of a physical theory, and a lot of people give that a lot of weight.