Wilczek on Golden Age of Physics

Standard Model

Seven.

I made major advances in my conceptual understanding of the W(y) model yesterday, to the point that I used it as a point of joy to keep me happy while watching the Eagles game. Anyway, now that the conceptual barrier that yesterday’s work knocked down, I am now understanding math at a new level of abstractness. Once a process becomes an object. That’s the sign of having a concept completely clarified. Being able to USE the mathematical structure. Its a great feeling, although somewhat rare.

I am turning back to those writings on Dimensional Analysis (Bridgman, 1921, Barenblatt, 2003, Hornung, 2006). This is the “conceptual scheme” that I have to clarify for everyone.

Purpose: Why am I writing about Dimensional Analysis?

Dimensional analysis underpins Harrison White’s W(y) Model, but this style of reasoning is very rarely taught in the sociology. Dimensional analysis has high academic credentials– it is the theoretical underpinning of much of the studies of turbulence– aeronautics, and other such applied engineering physics. This is not a trendy, faddish subject–it has been a central component of the physics “mentality” for almost a century if not longer. (Bridgman, 1921 is still considered a central work) A more realistic version of strategic action, based on “condensed-matter” or “few-body” principles could use the skills developed in these fields to develop better models of of how social action occurs within groups extending beyond the dyad, but smaller than the amorphous crowd. Patterns of interaction amongst social actors/players in the range of 5-10 members.

We will be Wittgensteinians here– rather than defining what “dimensional analysis” is, I would rather give examples of the types of problems that dimensional analysis deals with. My sources are Hornung and Barenblatt, who both work in the field of fluid dynamics. (at Cal-Tech and Cambridge, respectively)

This is Barenblatt’s work:

http://www.amazon.com/gp/reader/0521533945/ref=sib_dp_ptu#reader-link

“Scaling” is the title of the book, and it is the style of “qualitative” mathematical analysis that I have always enjoyed immensely. (Polya, 1954, Martin Gardner, Hofstadter, 1979, Davis & Hersh, 1983, Feynman, 1983, Borges, Nabakov, and large numbers of others) This is the style of mathematical reasoning that will be used more in the future, since there have been recent drastic advances in the interactivity between the person and the computer that allow us to exploit our spatial intuiton in more dimensions more readily.

I am typing out the table of contents for references:

Introduction

Chapter 1. Dimensional Analysis and Physical Similarity
1.1 Dimensions
1.2 Dimensional Analysis
1.3 Physical Similarity

Chapter 2. Self-similarity and Intermediate Asymptotics

Chapter 3. Scaling laws and self-similar solutions that cannot be Obtained by Dimensional Analysis

Chapter 4. Complete and incomplete similarity. Self-similar solutions of the first and second kind.

Chapter 5. Scaling and transformation groups. Renormalization group.

Chapter 6. Self-Similar Phenomena and Traveling Waves

Chapter 7. Scaling Laws and Fractals

Chapter 8. Scaling Laws for Turbulent well-bounded shear flows at very large Reynolds numbers

If you are trying to understand phenomena that operate at multiple scales, then it seems that this line of research is the way to go. This is basically my rationale for wanting to learn this field. Is this justifcation enough? So far so good. Also, I could seriously see myself working for a good long time working on this stuff. I could even see myself teaching this stuff on the side. Its close to the scientific aesthetic I am trying to capture. It is more of an MIT style than what we usually see in Sociology. But I have to be around those types of ideas in whatever I end up doing. (A boat I missed at Haverford — the non-linear physics stuff.)

So a glance at the Table of Contents shows us that some of the “key words” associated with Dimensional Analysis are:

DIMENSION
PHYSICAL SIMILARITY
SELF-SIMILARITY
INTERMEDIATE ASYMPTOTICS
COMPLETE AND INCOMPLETE SIMILARITY
TRANSFORMATION GROUPS
RENORMALIZATION GROUP
SELF-SIMILAR PHENOMENA AND TRAVELING WAVES
SCALING LAWS AND FRACTALS
TURBULENCE
SHEAR FLOWS
REYNOLDS NUMBER

Many of these themes are directly related to our work on turbulence, extreme events, non-equilibrium dynamics, non-linear systems, etc. So why not use the tools made by the best scientists working in these fields, since they directly involve the types of processes that we are interested in?

Barenblatt begins his work:

“Applied mathematics is the ART of constructing mathematical models of phenomena in nature, engineering and society.” Every model is based on a certain “IDEALIZATION” of the “PHENOMENON”. The model is not the phenomenon itself, but an approximation of that phenomenon. This is one of the useful pieces of information to know when thinking about modeling in the sciences. Whenever you make a mathematical representation of a situation in the real world, you are by necessity providing a description that has less information than what is being copied. For instance, a truly “accurate” map of a region would be as large as the map itself. So a situation can be understood at multiple levels of “focus” that each are somehow all true, and that cohere together. There is of course no “God’s eye” view, and that is the challenge that physics has been dealing with for the past 100 years (usually attributed to the rise of General Relativity and Quantum Mechanics at the turn of the 20th century).

The social sciences have dealt in many of these issues of observer-dependence, interaction effects, and so on, in both the “qualitative” and “quantitative” branches of sociology in various guises. (It is staggering the amount of literature there is on interaction effects, across the political spectrum, etc. and across the social sciences.) The approach of Barenblatt’s “applied mathematician” is half-way between the two versions of researchers. On the one hand, the applied mathematician uses numbers and counts things, and is thus “quantiative” in this sense. On the other hand, the applied mathematician draws upon his or her store of mathematical techniques and knowledge (we could call it a cultural “toolkit”) to create mathematical representations of particular phenomena, real or ideal. This is the ART that Barenblatt refers to above, and there is no one single method of calculating this aspect of mathematical practice.

For J

Jay,

Since you asked earlier today, I will be giving you an explanation of my thesis topic as clearly as I can. It is overly long, but I wanted to be sure that I am clear about why I am doing this, and why it interests me. Remember. I am getting a Phd in Sociology, but I am studying a lot of mathematics, computer science and physics because many of the same models that are used in these fields are also used in the social sciences. The goal of writing a dissertation is to demonstrate that you can perform a piece of research, and that you can prepare it a professional manner and in accordance with all the political correctness that this requires, yahdi, yahdi. Certain skills (especially mathematical skills) are more valuable for a PhD candidate to have than others, and so in order to work on these skills (which I will describe below)  I decided to focus on the mathematical aspects of a certain branch of sociology called “social network theory”.

Why social network theory you ask?

Practical Reason: Basically because it is the most fun thing that I could make a pretty “good” living in the sense of relatively high enjoyment/fulfillment and without sacrificing my scientific principles.  Also, I like the “aesthetics” of networks and diagrams, and doing the little visual maneuvers is kind of fun. (sort of like drawing football plays)  Also learning social network theory “forces” me (in a good sense) to learn all this awesome computer programming, and mathematical visualization stuff (Processing, Java, Wolfram Mathematica). I’m playing a bit of catch up since I missed out on the sciences as an undergraduate, so I worked extra hard to learn at least conceptually some important parts of theoretical physics, and I am now up to Calculus III, which is something of a game-changer– i.e. if you can make it here, the coolness starts to take off. In other words, Calculus I & II are like black-and-white TV, while Calc III is like color TV. A whole new dimension of opportunity opens up. There is little downside, only up. The hardest part is getting to that level III. That’s why I am so big on these mathematical visualization tools, and part of my dissertation will demonstrate how this works, since I believe that other people may find my particular method useful.

In a sense it is the opposite of mountain climbing a mountain. Let’s say I want to climb Mount Everest, and  so I train for 20 years and I make it. I am at the top of the mountain. Now, although I’ve never been to the top of Mount Everest myself, my sources tell me that it is a really unpleasant place to be a while. Its not Hawaii. So no one goes to Mount Everest for the stay, but rather because they can say they scaled the biggest baddest mountain on the planet. That in and of itself is a major achievement, but you can’t use that actual experience to scale an even bigger mountain. Once you’ve done Everest, you’re finished and that’s all. Math is more like climbing a mountain, and then realizing that the mountain you just scaled is even nicer than the realm you just came from. At least I found it is this way for me. Also, unlike Everest, you don’t really leave the place you were (i.e. you are still at the base of the mountain with your family and all your loved ones), but you just see more than you did before, and then you have a new base from which to understand old ideas in new ways. It is really nice, and once you pick up a new math skill, you don’t lose it. So in that way it is more like Yoga than mountain climbing. You never “master” Yoga in its entirety, but you can “master” the practice of Yoga, in the sense that it becomes something you do, rather than forcing yourself to do. Well I am at the level of “mastering” mathematics in the former sense, of having good control over certain problems than I hadn’t before. It is a nice place to be.

So, because I now have a variety of programming, mathematical and statistical skills, learned over the past 5-7 years or so (both in and out of class).  I’ve had something of a mathematical renaissance in my life of late. Remember in college and high school when I didn’t like math and science, well I’m like that, only the exact opposite. So right now, even though I am a sociologist, I am more interested in understanding the nuances and subtleties of mathematical reasoning right now than anything else, and I am doing this through the mathematical modeling of social networks.

Since I want to work in this field, I wanted to find the best thinker (i.e. the Yoda) in the field. In social network analysis we have Harrison White, who was Kontopoulos’ (another Yoda) teacher back at Harvard in the 1970′s, and is still teaching up at Columbia. His work is similar in some respects to that of Alex Pentland (who Jim Warren likes) on the topic of “signaling.” See the wiki:

http://en.wikipedia.org/wiki/Harrison_White

Notice the badass looking diagram on the old-school projector. That clearly identifies him as a master in the ancient arts. The “style” is called the “parameterizing” style, and it just makes you a much smarter problem -solver.  So I am doing the sorts of models you see on the board, but I have a couple of new twists on White’s models, mainly because of the way I am exploiting today’s advances in graphics. One of these models, which is called the W(y) model, is the focus of my dissertation, which I will explain below.

There is another great scientist who influenced me – an LA guy of all things!- Cal Tech’s Richard Feynman. He’s one of my new heroes even though he died 20 years ago. The only way I can explain him is that, for me, anyway, he is a cross between Doc Hamer  and Hugh Hefner. He kind of looks like Hugh Hefner he lived a relatively West Coast lifestyle, and he was known for being the coolest and funniest physicist, but still he was a physicist. He was really high profile, and probably the second most famous physicist of the 20th century behind Einstein, at least in the United States.  Sort of like being really cool and wicked uncool at the same time. But like Doc Hamer, he was a really good teacher with the classic “physics” sense of humor, and there is a lot of good Youtube footage of him and stuff. For instance:

Anyway, he invented these things called Feynman diagrams, and these were part of a major scientific project that won him a Nobel Prize and helped give a more intuitive short-hand for extremely complicated mathematical calculations. They look cool, and they are very minimalist/Zen looking…Mathematical Elegance:

http://en.wikipedia.org/wiki/Feynman_diagram

Well, I basically made something like a “Feynman Diagram” for social network analysis, even if not as great. This is a major achievement for me personally. Imagine writing a screenplay that you had been working on for years. The outlines are there, some of the connections are made, the plot is two thirds completed, and so forth. Then one day it is completed, even if every single word is not yet in place, you know that it is a legitimate piece of work. It may not be “Gone by the Wind” or “Dirty Work”, but it also isn’t “Dracula Dead and Loving It.” Well, I feel this way about the visual representation of a market that I just developed. I don’t know if anyone else will find this interesting, but I can assure you that I find this way of making calculations useful, and even more importantly, this will greatly help me to solve other problems later on. This may not sound particularly exciting to you, but believe me, when you have been working a conceptual problem for 3+ years, this is a great feeling.

The real test is that they should help me explain to my family the basic outlines of my project. I think I am at the point when I can explain it to my family and friends willing to make a small effort to understand it.

“White Flags”: A Visual Toolkit For Understanding Complex Social Processes (working title)

Translating from jargonese to English: “White Flag” is a little title I made up as a play on words- White for Harrison White (who developed the original model), and “Flag” because the shapes I use to explain the model look like little flags. A lot of times a great scientist (White) comes out with an interesting theory that very few people can understand. Often a graduate student will take it on him or herself to clarify aspects of the work for a larger audience, or fill in some of the missing details. That’s basically what I’m trying to do. In 2002, White wrote an amazingly challenging book called Markets from Networks, which provided a model for understanding production and financial markets, but it is very hard to get through, unless you are fully committed to understanding it. This was the book that I was constantly reading while at home with Mom & Dad, and scribbling out formulas and training my programming skills with.  Well it took me about 3 years of background reading, and 6 months of dedicated reading and model building on my computer,  I am now at the point where I think I can use the model in an effective way. Even more importantly, I have developed a visual toolkit (java applets) that allow people to develop an intuition about the mathematics taking place.

Now what’s great but also somewhat overwhelming about studying White is that by studying his work, you have to learn all of this other “more fundamental” science by necessity. (otherwise you won’t fully appreciate technical depth of his work) So even if some of insights are incorrect, he is reasoning at such a high level that figuring out the logic gives you such a good workout that makes you a better thinker in general. Since I’ve spent so much time figuring out the logic of the model and organizing many of the formulas into a more intuitive framework, I figured I would make this visual toolkit that I used available to the general public.

How did I get into this stuff?

Let’s go back to when we were kids. What did I do when I was bored– I did all that memorizing Presidents dates, football statistics, lotta math, looking at maps, right?  The math that I do when I was bored. Like calculating things and playing with numbers and calculators.  Well that’s sort of what I am doing, except that I am looking at the automobile industry as a case study.

I am focusing on the automobile industry for a number of reasons (roughly in order of importance):

1- Outstanding set of data- This is really important in graduate school, because sometimes you want to study something obscure, but if there is weak information available it can take too much time to make it worth your effort. So for instance, while I was thinking about studying the robotics industry, the field is too new and too small for there to be consistent long term data set that allowed me to perform the types of mathematical tests that I am interested in performing.