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.

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.

Sunday, November 28, 2021

Critical Thinking v Problem Solving II - Problem Solving from Start to Finish

In order to explore problem solving and critical thinking a little more, I would like to talk about problem solving. Mainly, I'll talk about a 1963 paper by Slagle, who programmed the first symbolic integrator using punch cards. And a very clever algorithm. It was able to solve 52 of 54 integrals on what sounds like a Calculus II final at MIT* using a ladder of proximate goals reaching up to an ultimate goal, the solution of the integral. It needed, to mimic some aspects of human problem solving by implementing two sets of rules: those that are always good and those that are heuristics and sometimes break down. But it does get to the answer. This isn't so surprising in itself, since commercial products, e.g., Mathematica, have been doing so for decades, but the fact that it could be done with a 1959 mainframe computer and with so few rules is an astounding fact. I will use this algorithm as the backbone for how to think about problem solving.

The general way that people solve math problems is by a searching method.** You start with what you're given and then you search the space with known rules and inspiration and try to move forward, one step at a time. From time to time you get stuck, and then you take a few steps back and try another likely path. One of the differences between an expert and a novice is how they react to a setback: a novice usually chooses one way of doing a problem, then never steps back.*** An expert tries several tactics, until they complete the task. Matthew Shoenfeld's work quantified this behavior, and private discussions make it clear that many people consider this searching, a la Polya,(*4) almost a moral imperative of reasoning.

Slagle's system worked on a method of proximate goals. That is, you start with a main goal, where you'd like to get to, and then as you work towards it, you sometimes identify a subgoal. When you do, you keep track of that subgoal. Sometimes as you work, you can reach a point where several directions are possible. In these cases you keep all of the goals, and place them in your tree. You don't try to work all of the goals together. Instead, you use various methods to assign the goals priority. Some goals can be reached through automatic processes, processes that are guaranteed to move you toward the main goal. You do these first. Others are heuristics, and you look to those only when there is no automatic option. Furthermore, for heuristics, you need to judge how costly each one is in the given situation: how hard it is compared to the goal. And you try the least costly first, as long as its line remains the least costly. When you find a way to reach the main goal, you have solved the problem.

This should remind you of the method that people use to solve problems in the real world, as observed on Betamax by Shoenfeld: you search the space with known rules, trying to move forward, and when you get stuck, you take a few steps back and try a likelier path.

There were three types of methods used by the program to solve the integrations problem. The first was a short integration table, a list of "Standard Forms." If you came to a standard form, then you had effectively solved the problem. In effect, the rest of the edifice is built to put a non-standard form in the image of a standard form. The second set of methods were those that always improved the situation. Whenever you came to one of these you tried it, and then checked it against the standard forms. These had no deviations and didn't really require any interesting tracking. You tried one after another until there were no obvious steps to try. Finally, there were the heuristics. These are rules that sometimes improve the situation, but sometimes do not (sometimes, trying them makes matters worse). When you came to this point, the program would try all of the applicable variants, and then assess the character of each try to judge which is the best way to proceed. And as I said, it worked in 52 of the 54 cases, and the other two could not be solved because the IBM 7090 didn't have enough memory for more entries in its integral table.

This goal-directed reasoning is what I'll use as a paradigm for problem-solving. I often characterize it as a tangram in my classes: physics gives you a set of tiles with which to form the shape of the solution. The tiles are limited, but the forms are infinite.(*5) Arranging the tiles into the correct form is called thinking.

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* At MIT, it's part of 18.01 Single Variable Calculus. 'most everywhere else, it's Calculus II - Methods of Integration -- with some idiosyncratic identifier cataloged by the inane, parochial system used by universities (a college catalog is a definitive, smack-down, irrefutable argument against expert judgement, although listening to the faculty senate is even better). I first learned of this algorithm from a very good lecture by Patrick Winston from 6.034, Reasoning: Goal Trees and Problem Solving. A short description of the algorithm can be found in Winston's Artificial Intelligence textbook.

** See Mathematical Problem Solving, Matthew Shoenfeld. I would like to show some diagrams from his work, but at this time I don't have a way to use images.

*** A college freshman thinks that problems should take 2-3 minutes to solve, and literally thinks that a problem that takes more than 10 minutes is impossible.

(*4) See, e.g., How to Solve It, George Polya.

(*5) Well, not really. Firstly, of course, there is room for disagreement about how many tiles there are, and what constitutes a different tile. I am sure there are many fewer than the 60-odd list of things the engineering college says it wants the students to learn (about one per 25 minutes of instruction), since "solve a quadratic equation," "Newton's Third Law," and "Kepler's Laws" are all very different types of things. Also, I think that there aren't technically an infinite number of solutions. I am 95% sure there are only seven one-dimensional kinematics problems for one process on one object (and 100% certain for uniformly accelerated motion). Well, I guess, then, if I allow an infinite number of processes and an infinite number of objects, I could end up with an infinite number of kinematics problems. Probably. And these can be sutured onto more complicated dynamics problems, which unlike kinematics, are really physics.

Wednesday, November 17, 2021

How Long Does It Take to Do an Infinite Number of Things?

Infinite processes shred our intuition to Hell like nothing other than probabilities and relativity. This was the infernal currency of Zeno of Elea, who proposed devilish paradoxes in order to prove the conjecture that movement is impossible. Zeno would construct a simple scenario that would show that everyday aspects of life, like chasing a tortoise or shooting an arrow. We really only have accounts from other authors, basically Aristotle, who disagree with him. Some of these arguments intertwine ideas about space and time, the arrow argues that an object is motionless at any instant, and both the dichotomy and Achilles and the Tortoise make arguments about the impossibility of doing an infinite number of things. I will eventually move on to an example to calculate the total time that it takes for a ball to bounce an infinite amount of time, but I will first describe the dichotomy using Philocetes' Arrow as a story (rather than Aristotle's bare-boned description from the Physics [1]).

Philocetes looses an arrow from the Bow of Heracles at the Trojan prince, Paris. In order for the arrow to strike the Son of Priam, first, it must fly half the distance between the heroes. But, it is clear that in order for an arrow to travel half way to the midpoint before it can get to the midpoint. And it has to travel half way to the point before that, ad infinitum. Therefore, in order to move from one spot to another, no matter how close, you have to move an infinite number of times in a finite amount of time to get anywhere, so the arrow never flies and Trojan War never ends.

This is paradox because we do move, but at least the first time your hear the argument, you don't have a good reason why it is wrong. The arrow must move, but logically it cannot. And this is applicable to every kind of motion. Before you can eat your Wheaties in the morning, you have to get out of bed and get into the kitchen. But, you have to move through an infinite number of small separations to get there.

And you can't do an infinite number of things before breakfast.

Or can you?

Most people think that the invention of Calculus resolved Zeno's paradoxes. This is certainly true in the case of the dichotomy: Leibniz used and even dirtier trick with infinity than Zeno did. Leibniz built calculus out of the idea of an infinitesimal to align with his cosmological ideas. An infinitesimal is a chunk of the universe that is smaller than the smallest division, basically the reciprocal of infinity. The integral calculus would define the distance that the arrow must travel as a sum of all of the infinitesimal chunks of space between Philocretes and Paris. The nature of infinitesimals is that they are smaller than the smallest fraction, there are an infinite number of them between each rational number. The infinity of the infinitesimals is that of the real numbers (the continuum) and the infinity of the dichotomy is that of the rational numbers (countable). So, if you can construct a theory of motion that adds up all the infinitesimal points, it will automatically encompass the infinity of halves used by Zeno.

This solve the dichotomy, but it does so indirectly with an end around.

By subsuming the motion of the dichotomy into a single, continuous process that can be analyzed separately, we show that motion can exist. We solve the riddle by changing the problem, though. However, this leaves Zeno's premise unchallenged: an infinite number of processes take an infinite amount of time. What I'd like to do here is take on the premise that an infinite number of discrete, sequential processes needs to take an infinite amount of time. You could do the same thing with the dichotomy, as well, but since that is an arbitrary partition of a single, continuous process, which I feel is a little different.

The specific question I ask is: how long does it take a rubber ball to stop bouncing? The physics here is quite simple. It can be done with kinematics using the simplest of deflection theories: the coefficient of restitution. The model uses the simple rule that the velocity of the rebound of an object is proportional to its original speed, and that proportionality (the coefficient of restitution) remains the same after each bounce. The duration of the air time of the ball is given by uniformly accelerated motion. The sum of a sequence of such bounces will lead to an infinite series with a known sum (thanks again, calculus), and this sum will be finite.

Where can this go wrong? Well, it's not in the assumption of uniform acceleration. Yes, it's not quite true, but it's pretty accurate at low velocities for short times, which a rubber ball acts in. If we complicate the problem by adding in air resistance, that will give us a slightly more accurate estimate at the cost of an annoying integral (no thanks, calculus). This accuracy will give us a time that is strictly smaller than the uniform acceleration version by giving us a factor similar to the coefficient of restitution itself. The significant assumption that would break this analysis, if it were relevant, would be that the time of the bounce itself will be the same each time the ball hits the ground if the bounce is modeled on an elastic restoring force, which is probably the best model available. Even though this will be small, at some point it will be larger than the air time per bounce, and since it remains the same, adding an infinite number of them would create an infinite time for the bounce.

But, to answer the basic question, can an infinite number of processes be completed in a finite amount of time, eliminating the time of the bounce is justifiable.

So what happens in this case? Well, from basic kinematics, we find that the time of an individual flight is proportional to the initial speed of that bounce.* Since the initial speed of each process is the coefficient of restitution is just the initial speed of the previous process, the duration of the subsequent process is scaled down by the sane proportionality,

tn = r tn-1 = rn t0

which means that flight is scaled down by a power of the coefficient of restitution.
When these are summed, we find an infinite series in powers of the coefficient of restitution that has a known sum: the inverse of one less the coefficient [ 1/(1-r) ]. So, the total time the ball bounces is finite if r < 1 (which is must be unless it is gaining energy from the environment somehow).

So, an infinite number of bounces takes a finite amount of time

t = 1/(1-r) t0.

This is a reasonable answer because if r = 1 the bouncing goes on forever and if r = 0 it stops after the first flight. This should be the same result you'd find if you were to sum the time to travel each segment of the arrow's path, but here we have distinctive processes represented by the flights between bounces. Our hero Leibniz has defeated Zeno of Elea's Satanic dichotomy.

So the Trojan War terminates, and you can do an infinite number of things before breakfast.



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* The proportionality constant is 2/g.

[1] All of my Aristotle is missing. Most of what I know about this comes from Sainsbury's Paradoxes, >although I was using the Stanford Encyclopedia of Philosophy.

Wednesday, November 10, 2021

Critical Thinking vs. Problem Solving I - How Are They Different?

I've spent a lot of time thinking about critical thinking and problem solving. Nominally, my course is required to include a critical thinking component by the university so that it order to satisfies a distribution requirement. Over the past five years, I have never really been able to discern the difference between critical thinking and problem solving, a more common term for what we "teach" in physics. Some papers explicitly say that problem solving is a kind of critical thinking, [1] others say that it's a separate skill. [2] I don't really know the proportion. I originally just went for a rather strict problem solving format from the University of Minnesota* that I feel shows me how a student is thinking about problems,and based on Force Concept Inventory scores, I implemented it well (gain around 35-45%).

The students hated it, but my job isn't to be adored.

However, one spring day in 2020, I was informed that my course was to be reviewed for its critical thinking component.** When reviewing a course, all I have to do is to submit a product for each student that shows their critical thinking skills. This product can be a test question, a homework problem, a paper. I have a section on the tests where students individually provide an explanation of how to solve a problem, and I felt that this would do. It is a little stilted. What do you want to do? "Find the velocity." How are you going to do it? "Use conservation of energy." How do you represent that? "1/2 m v^2 = 1/2 m u^2 + mgh." It shows exactly what the student is thinking and how they utilize the data. I felt it would be a good way to show the development of the students' problem solving capabilities over the semester.

But I was given a rubric, and the rubric told me that I was wrong.

These are some highlights from the rubric:

Explanation of Issues. Problem is stated clearly and described comprehensively, including all relevant information.
Evidence. Viewpoints of experts are questioned thoroughly.
Context. Thoroughly analyzes assumptions and carefully evaluates the relevance of contexts when presenting a position.
Student's Position. Position is imaginative and other's views are synthesized within it.
Conclusions. Conclusions are logical and reflect the evidence and perspectives in priority order.

Try that with conservation of energy.

For some of this, expanding the selection from just he planning phase of the problem solving process would probably do. For others, it seems irrelevant. In fact, some of the categories seem to be completely irrelevant to the course ("Influence of Context and Assumptions" is the full title). But, looking at the rubric for the curriculum component, I feel at minimum it requires a term paper, and probably a thesis. The school implicitly takes the side that problem solving is not a part of critical thinking.

Critical thinking, as described by the rubric, is really separate from problem solving.

However, I still think that there should be some overlap. I think in come coming posts, I'll talk about what I think problem solving and critical thinking are, possibly in several posts each, and then I'll talk about some specific problem-solving tools for first year physics students.

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* That, I think, they don't use it any more.

** And just after I wrote this, they told me that this was the evaluation of the engineering students' "written communication."

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[1] Willingham, D., "Critical Thinking: Why Is It So Hard to Teach?" American Educator (2007).

[2] Pasquinelli, E., M. Farina, A. bedel, and R. Casati, "Naturalizing Critical THinking: Consequences for Education, Blueprint for Future Research in Cognitive Science." Mind, brain, and Education 15, 168 (2021).