Monday, December 20, 2021

Geometric Irony

This morning on my drive to work, I was listening to an old (May 2018) episode of Conversations with Tyler, the second half of which was a discussion between Nassim Nicholas Taleb and Bryan Caplan on the problems with education. The recurrent examples of these problems are poetry and geometry, at least one of which probably scares you. Their issue wasn't that there's anything inherently wrong with poetry and geometry, in fact they think that both are perfectly good hobbies that everyone should have (Taleb, I should say, was a little more tolerant of your poetry classes, but not that much). But they worry about the practical problem that very few students will pursue a career in them, and students graduate from school at eighteen or twenty-two having forgotten approximately 100% of the poetry and geometry that they "learned"1 in class and having no idea what kind of career will suit them. There aren't many poets in the world, and there are even fewer that have learned to convert verse into cash with ab efficacy sufficient to fill a refrigerator. And what fifteen year olds need to do is to sample the possibilities of how they can fit into the world, like plumbing and customer service.2

The liberal arts are best left to Sunday afternoons on the porch and autumn walks in the park.

I had finished the first half (where Tyler Cowen interviewed Taleb) and started on the part where Taleb talked with Caplan about his book The Case Against Education (both segments included a lot of talk about Taleb's books, especially Antifragile),4 when it was time to stop and get my cup of coffee.2 I pulled out a paper on the history of the renormalization group and a pad of Bristol board, and started the day off by working on a cartoon for class and a bikini girl for Instagram while waiting for the caffine to get me attentive enough to read an academic paper. While I was drawing the cartoon, which is about integrating to find a volume, a local antique dealer, J., came by and we talked about finding volumes a little bit.



The cartoon, I hope you can see,5 shows the Riemannian process behind the integral for finding the volume of the pyramid. J. saw the point right away ("what about the steps?"), and pulled out a problem for his store. The way he'd have to solve the problem was, he said, to weigh a stick of butter, then to carve a scale model out of it, weigh again, and then do some ratios. This is a very good method.6

When he was in Versailles and saw a structure whose somewhat triangular shape he wanted to reproduce for his store, where he would put featured paintings at one point, antiquities on another, and books on the third. The shape was an equilateral triangle with the tips cut off (so, a hexagon), where the long edges had a slight inward curve. He'd add some walls and french doors as well, but what he wanted to know is that if the distance between next nearest points was 16' and the size of the cut was 4', what is the volume of concrete required to build the 1' thick base of the structure? I calculated it out with a little geometry, the quadratic formula, and a rather annoying bit of vector calculus (which was overkill).7 Then C., the old Airforce master sergeant, came by and we talked about history books and historical novels until the larger group started coming in. J. took the sketch and calculation, I went back to drawing, and some girls sat behind me watching me draw the leopard print on the pinup's bikini.

Then, when I got back into my car to get to work, I listened to the next fifteen minutes of the discussion, where Bryan Caplan and Nassim Nicholas Taleb continued their discussion on how unlikely it was that you'd end up using things like geometry or art in your day to day life, and thought about the wisdom of their words.

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1 In fact, students lose the factoids they memorize for high school and undergraduate texts with a half life of about two weeks. So they'll always "remember" something from your class.

However, it might be the wrong thing. I remember talking to a high school friend just a year or two after graduation who was sure that Lamrkian evolution was correct, because he'd read it in the HS biology textbook. He even remembered the specific example: proto-giraffe mommies stretched their necks to reach higher leaves, and so their babies had longer necks. And then this repeated over generations until giraffes were the long necked freaks of nature we see in zoos today. This was in fact in the book (I remembered it, although I never studied in high school, so I don't know why), but it was there as a historical contrast to Darwinian evolution.

2 And if you know me, especially if you knew me twenty years ago, you might remember my old rants against mindless education. And I still feel that way. I think, on the whole, Caplan and Taleb are correct. I'm not a fan of the pyramid schemes of psychology, but education kids love them, especially Maslow's heirarchy of needs. Poetry, art, and music are offerings to the sacraficial altar of self-actualization at the top of the pyramid. Caplan, and to a lesser extent Taleb, want the schools to focus more on the bottom steps of the pyramid, helping kids build the skills to keep themselves fed, housed, and safe. Those are the fertile soil for spiritual growth.

On the other hand, whenever I hear someone say "it would be better to teach kids EXCEL instead of calculus," I think, "You have to do the calculus before you use EXCEL."

If you're cool, then you know Iggy Pop's feelings are also in tune with Caplan and Taleb from reading his liner notes on the reissue of Raw Power (which I did, of course).

...if you both like Iggy Pop and are literate (which I do and am, of course), which is not guaranteed (and that's the way he likes it; check out his liner notes).

3 I own a copy of Caplan's book and now I want a copy of Taleb's (I've read two others), but I'm about 40 books behind over the course of the pandemic. I usually read at least 50 books a year, but have gone down to a little over 30 each in 2020 and 2021.

4 I have a long commute which interacts with a complex morning routine, so this is a very simplified version. I've discussed the jalapeno boudin kolaches elsewhere.

5 Hopefully I'll get a way to put images up here.

I did! Ha! Although the screw ups on the cartoon make me nervous. I've spared you the pinup, which is beautiful, but you're not authorized for that kind of titillation. Bring a note to me during office hours from your psychiatrist saying that it's mentally safe for you to view such things, and I'll give you a link.

6 Before there was a lot of computer time available, and even when computers were reasonably slow, this was how experimentalists would do numerical integrations on their data. They'd plot out the spectrum on their plotter, very carefully cut out the shape of the specturm, and then weigh it. That weight would be compared with the weight of the paper and the untis on the axes to find the integral.

This is exactly J.'s butter technique.

7 If you are Nassim Nicholas Taleb and you really do enjoy doing geometry on your porch on Sunday afternoons, then this is a good little problem for you.

Monday, December 13, 2021

Answering Aristotle I.2 - There Is More Than One Thing I

This one was very confusing, especially in trying to figure out what Aristotle means by a "principle" -- or what the translator translates as "principle." Part of this section's difficultly (and the next) is that in arguing for a particular position, Aristotle isn't laying his argument down precisely, but is responding to other philosophers of the day. This makes it hard to pick out what is essential in his argument that ther is more than one thing and what is an accidental argument because it is meant as a refutation. My discussion will talk a lot about my understanding of qunatum field theory, which is certainly wrong, and propose that string theory is a counterexample to Aristotle's main assertion, even if I can't be sure if all of the arguments Aristotle puts forward here even apply.

The following is the best one sentence summary I could come up with for Physics I.2:

Physics I.2 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.


Aristotle is agnostic about the kind of thing that is the one thing. Is it a substance, a property, an element?* It doesn't matter. If there is only one of them, then he asserts there will be a contradiction. Although most of his arguments are directed against Parmenides and Melissus, they are of a kind: find what seems to be a logical contradiction, e.g. the skinny man is fat, that comes from some assumptions. One that he asserts are that something cannot be finite and infinite at the same time.

These do not feel very compelling. For example,"...so there will be a substance as well as a quality, in which case it is twofold..." feels, at least in the translation, as if there is some confusion here. The two things are of such different kinds that I don't know how you can call them "two things," really.

What this brings to mind, though, is string theory. In the standard model of particle physics, you have a number of fields corresponding to two kinds of particle: bosons and fermions. The fermions are leptons (electrons) and quarks, which constitute matter. The bosons, photons, gluons, and W & Z particles, constitute the fundamental forces of nature (sans gravity), the connections between matter particles, in a way.

String theory makes all of these particles one kind of thing.

And because we can envision these particles as excitations in their corresponding fields. That is, whenever the electron field gains energy, a new electron is born. This is a little bit weird to think about in fundamental physics, because we don't have an independent concept of an electron field. But in condensed matter physics, we do have strong classical ideas about the meaning of some of the fields that appear in matter. We know about sound waves, we know about spin waves (magnetization waves). These waves are continuous and extend through the body. They have standing states, just like the standing waves on a string or a membrane that you might be able to envision.

And they're quantum mechanical.

The physics of these waves are describes as excitations in their corresponding fields. A spin wave is an excitation in the local magnetic polarization (magnetization) of a ferromagnet, and these excitations can only happen for certain multiples of a fundamental oscillation mode, just like the vibrating string. However, unlike the vibrating string,** there is a minimum energy necessary to excite a single vibration, and increasing the amplitude of the vibration requires additional quanta of that vibration mode. The amplitude of the spin wave is an integer number of of these quanta.

How much energy is this minimum energy? A ferromagnet (like your refrigerator magnets) is a material whose atoms' magnetic moments tend to align so that there is a net magnetic moment of the material. The minimum energy of a spin wave is exactly the amount of energy required to take one of these aligned moments and flip it 180 degrees. If you flip a spin like this, it can propagate through the material by successive mutual flips between neighbors. And we can examine the behavior of these spin flips, and their interactions with defects and oscillations

The spin wave and spin flip are two aspects of the same thing: a magnon.

This is the kind of quantum mechanical dualism you're used to, but it also shows the dualism between particle and field. The spin wave is an excitation over the entire field of atomic spins and the particle is an excitation at a single point, and depending on what we're investigating at the time.

This is how I still envision fields.

The difference between this and fundamental fields is that there's no substrate for the quantum field. There is no aether serving the role of the electromagnetic field that has some property that we excite electromagnetic waves in, and whose interactions with electrons are particulate photons. There is just the electromagnetic field. The same is true of the gluon field and weak fields. However just as water waves give you an analogy for water waves, the atoms in a material give you an idea about how a field works.

And it's more than a simple analogy. Many of the big "verifications" of high energy theorists that we've seen in recent years, Dirac and Majorana fermions, for example, are coming out of materials and metamaterials research in condensed matter. Experimentalists can construct systems with the correct symmetries to realize the particles. It's like creating universes on demand, universes that contain the thins you want to find.

In material, it's even easy to see something that becomes difficult to envision: how do these (at least) 25 fundamental fields of the standard model superpose over the entire universe? In our crystal lattice, quasiparticle fields correspond to different properties of the atomic and material structure. Lattice vibrations become phonons. Magnetic exicitations are magnons. And there are many others, but they all have this character of being related to properties of the collective properties of the material.

You can view string theory in a similar way: there is this fundamental structure of the string, and the 25 fields are all different manifestations of the properties of the string. Is it open or closed? How does it vibrate? In string theory there is just one kind of thing, the string, and since the different manifestations of this kind of string are all conceptualized as excitations in universal fields. So, there may be only a single, universal field.

Both of these interpretations, every particle is a string or every particle is an excitation in the stringy field, would count as Aristotle's "one principle."* String theory is a monist theory, the kind that Aristotle tries to disprove here. I don't think that the arguments he propounded in Physics I.2 really refute string theory, partly because many of them are arguments against specific philosophers and partly because many of them have mistaken logic of Greek mathematics.

For example, if I change a line in Physics I.2 to read "if there is a continuous fundamental field, then immediately it must be many fields because anything continuous must be divisible," then we have string theory exactly as I described it. But, string theory is logically consistent. It is also logically coherent. There is no obvious logical problem with string theory as mathematics. It's only possible problem is correspondence: even though it's currently the best guess at a unified theory of the world, it may never be shown to actually predict anything. But although string theory might not be correct, and there may even be no theory of everything,(*3) the field theoretic structure has both the continuity (say, the electromagnetic field) and the divisibility (say, the photon) built in. In some way. And in some way, it is a counterexample to Aristotle's assertions against monism in Physics I.2.

So, it seems to me, string theory refutes these assertions by Aristotle.

Monism is at least possible.

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* In the translation the, Aristotle is arguing against the idea that there is only one "principle," whether that principle is "a substance, a quantity, or a quality." Democritus has an "infinite number of principles," because his atomic theory had atoms of "all shapes" -- and there are no limits on the number of shapes there could be (If I recall, Epicurus would have a limited number in the form of regular polygons, when he finally got around to being born, if I recall). Heraclitus, apparently, had zero principles. So the wording here made life difficult for me: I wanted a principle to be something like Newton's Law, rather than a substance like water or air.

** I think. There probably should be a quantum mechanical description of transverse mechanical waves on strings, although I don't know what its use would be, or how you'd do an experiment to detect it.

This would be like building up the vibration of a guitar string by adding transverse vibrations to individual atoms, one at a time. That would probably make them phonons.

(*3) Although I should go through these arguments on the TOE page for Wikipedia before I say that.

Friday, December 10, 2021

Answering Aristotle I.1 - The Basic Process

This may be too fine grained an approach, but I thought I'd go through Aristotle's Physics, section-by-section, to see what was in it. This is in part prompted by my earlier post on doing an infinite number of things in a finite amount of time. I used to have a copy in a portable paperback edition mixed up with other things, but I've lost it. So, I decided to pick up a copy that I like (Oxford World's Classics) instead of one that I could afford in school (50 cents, used). I will make an attempt to work through it here.

Since the sections are so short, I was worried about whether there would be enough in them to discuss. I generally like short-sectioned books, but I like them around 3-5 pages, but these are about 1-2 pages each. I'm having a bit of trouble with another book I'm reading (Lectures on Phase Transformations and the Renormalization Group) for the same reason: the sections are so short that often they don't have an independent point, which makes my note taking difficult. I like to reflect over each section and write a topic sentence for it before moving on. For example, section 1 of book I of the Physics would be

I.1 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.


And do that over and over a hundred times per book. I only rarely go back to them. I thought what I'd do here is to extend this a little by discussing how Aristotle's insights hold up, how they compare to what I've been told they are, and so on.

There are two things that Physics I.1 brings to mind. The first is the actual evolution of physics, which in some places follow Aristotle's insight and in some placed doesn't. The other is that this discussion reflects the advice of Arnold Arons on the teaching of physics.

* * *

When I wrote about operational definitions (or will write about in the past), I used an example from Arnold Arons, most likely, that describes how the concept of temperature arose in physics. This is a relatively new idea, and we know exactly how it developed. And it definitely follows Aristotle's process. "The natural way" to proceed is to "start with that which is intelligible to us and then to move toward what is intelligible to the thing in itself." That is, we start with what we perceive about the world, and then we try to use that to determine the way the world actually works.

The first instance is temperature. This is a concept we all have a fairly intuitive understanding of, right? Well, not really. We have an understanding of "hot" and "cold," which was always a fairly ill-defined idea until Galileo. In order to construct a notion of temperature, we need to define a reliable way to compare "hot" and "cold," which is quite difficult. If you hold a book that has been sitting in a room for a long time, it feels neither hot nor cold, but if you touch marble, it feels cool. Finding a common understanding under such conditions is difficult. At the turn of the 17th century, Galileo invented his thermoscope, an instrument that held a glass bulb containing air and suspended in water that would rise and fall with changes in the state of the air (both the temperature and pressure state variables would cause these changes). It was only qualitative, but it was the first way in which our subjective idea of hot and cold could be related to the internal state of the things we called "hot" and "cold."

It would be another hundred years before Fahrenheit constructed reliable thermometers based on the relative thermal expansion of air to that of mercury or alcohol. This allowed a science of thermodynamics and a theory of engines to rise, but it did not tell us what temperature is. What was needed for that was the kinetic theory of gases, a statistical examination of the motion of air molecules. This would wait for another 200 years, after the atomic theory of matter was accepted and probability theory was on a sound footing. The temperature of the air became the average kinetic energy in the translational motion of its molecules. Which is not what your feeling, your apprehension, of hot and cold is about.

"Hot" and "cold" is about the rate of energy transfer from a material to you, which is why your book feels neutral and the marble feels hot. But, this too is explained by statistical mechanics. So, our basic ideas, the categories of our experience, led us to discover the idea of a measurable temperature, which in turn allowed us to discover what this meant to the air, and finally to even explain what our experience is really measuring.

This mirrors the point of Aristotle's Physics I.1 exactly.

I was going to offer a second example of the kind, the nineteenth's century's development of the idea of energy, which displaced the "imponderable fluids" of the 18th century (caloric, etc.). I think the story would further support Aristotle.

A counterexample, however, might be the late 20th century's search for fundamental particles. Here, the big minds theorized the existence of fundamental particles, but rather vaguely based on precise theories, and provided the material experimentalists, who then searched for them with amazingly powerful and expensive machines. At meetings, you would see maps of the parameter space, regions blocked off from where different experiments could measure. Experiments verified, experiments falsified, but experiments didn't drive the science. And neither did our perceptions. I cannot see how this follows Aristotle's program, although perhaps a longer view could make a good story of it.

It seems though, for most of its existence, physics followed something close enough to that program.

* * *

The other comparison that this brought to mind was an educational one. Maybe two, in fact. The first from Arnold Arons and the other from Edward Redish, although many of these insights I've seen elsewhere.

One of the more interesting admonishments of Arons' Teaching Introductory Physics is his insistence that concepts come before names [2]. This is part of his Socratic attempt to build students' physical intuition. The idea is to use identify the need for a concept, to start using the concept, before naming it. Even going to the point of admonishing students who use the term (e.g., "energy") before it is fully defined. Naming things gives people a feeling of understanding when they do not, and it relieves them enough that they ignore the rest of what's being said ("oh, that's energy -- let's get back to the important things, like "Hearthstone"). But you'll notice, this teaching style mirrors Aristotle's Physics I.1.

* * *

So it looks like Aristotle's approach to physics looks like the same approach physicists usually use both to investigate phenomena and to teach physics. This, at least, is a good sign for the rest of the book, despite its reputation.

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[1] Most of this comes from Arnold Arons' Teaching Introductory Physics Part III: Introduction to the Classical Conservation Laws.

[2] This comes mostly from part I. I just skimmed Arons and couldn't find what I remembered. Is it Knight's Five Easy Lessons? Can't find it there, either. Probably Arons.

[3] He may also have said many of the same things in his Teaching Physics with the Physics Suite, which is also good (despite much very particular advice relating to the Physics Suite). The references for the articles are:

0. Redish, E.F., "Using Math in Physics: Overview." [arXiv]
1. Redish, E.F., "Using Math in Physics: 1. Dimensional Analysis." [arXiv]
2. Redish, E.F., "Using Math in Physics: 2. Estimation." [arXiv]
3. Redish, E.F., "Using Math in Physics: 3. Anchor Equations." [arXiv]
4. Redish, E.F., "Using Math in Physics: 4. Toy Models." [arXiv]
5. Redish, E.F., "Using Math in Physics: 5. Functional Dependence." [arXiv]
6. Redish, E.F., "Using Math in Physics: 6. Reading the Physics in a Graph." [Not Yet Published]
7. Redish, E.F., "Using Math in Physics: 7. Telling the Story." [Not Yet Published]

I deleted this section for now.