Sunday, August 31, 2014

Search Theory And Idle Capacity

Through Mainly Macro, a blog about macroeconomics from a New Keynesian perspective, I discovered this paper by Pascal Michaillat and Emmanuel Saez. The associated slides are a good summary. The paper is very hardcore economics theory paper, aimed a a very hardcore economics question - why does unemployment lurch around the way it does? It's a difficult question. Why did people gainfully employed in 1928 suddenly find themselves without jobs for 11 years? No matter what your opinions  are on this matter, this paper will be of a benefit to you for clarifying how that cause got into the economy as a whole.


This paper takes a search theory perspective on the subject. This improves on the repeatedly cited Barro model because it allows Saez and Michaillat to use supply and demand to analyze the situation. In addition, they are able to use this model in a compartmentalized way. For instance, in this paper, they just use aggregate demand, but the model can be relaxed with different models of demand. Different ways of finding prices are examined in this paper, from crude price fixing to Nash bargaining. As far as I can tell, one could take any pricing mechanism and get a new version of this model as a result. Risk and uncertainty are abstracted out, meaning this is not a model of - say - the 2007-8 financial crisis (which in this model is just "some demand shock"), but rather of how the crisis got into the economy. Since prices are affected by risk and the pricing mechanism in this model is arbitrary, this would be a good place to put that. For instance, there is a great deal of difference (in productivity) in hiring different people for a given position, even a "low skill" position. How is the model affected by increasing uncertainty of hiring?


I don't know much about the relevant precursor model. I will say that as a mathematician, I think that the name "disequilibrium" is a bad choice of words. The word "equilibrium" means that there is no change over time, the use comes from mechanics and has been extended by physicists and mathematicians (and economists!) in appropriate ways. When Barro, Fisher, etc. call a model "disequilibrium", they mean that the economy is not assumed to rapidly respond to changes in the way that the old fashioned Alfred Marshall model. That means that this paper is equilibrium analyses of a disequilibrium model, which is comprehensible only if you already know what this is all about. I am against this word, antidisequilibrium. But unfortunately antidisequilibriumism has little chance against established use.


Also, economists draw charts sideways. The variable x (market tightness, illustrated on slide 9 and 10) is the independent variable and quantity of a good is the dependent variable (with a maximum capacity k). They can't help it, Alfred Marshall did it and people had to do it to match him, then their students had to do it to match them, etc. But surely even economists can see that this is worse than Hitler and that the only use for such charts is discovery of landmines? Can't economists and publishers band together to rid of us this evil?

Why are they doing that? I've seen this show and I have no idea...

Because there is already a good explanation of the bones of the model (from a New Keynesian perspective) on the blog I found this paper on, I'll end by talking about myself. Though you can't tell from this post (in which I talked about famous crises), I have a really hard time thinking about economic models in relationship with the world I am in directly. When I read the excellent textbook The Spatial Economy I found myself thinking a lot more about Sun Tzu than the highway system. When I read this paper, up until section 5 (the empirical part) I found myself thinking a lot more about feudal societies (such as the Tokugawa Shogunate) than modern fluctuations. It's strange, because the modern ones have actual time series and other empirical data - which is what I do! Perhaps it is just more fun.

Thursday, August 28, 2014

Feynman On What "Science" Is


In celebration of the decision of CalTech to put The Feynman Lectures On Physics online, I thought I'd reproduce something he wrote in his wonderful little lecture QED: The Strange Theory of Light and Matter. The full lecture can be found here. As far as I know, this is the only lecture on quantum field theory aimed at a popular audience. I first read this book years ago, and I often find myself using Feynman's visualization tricks often. For instance, Feynman's visualization of complex numbers as being little clocks on points helped me make sense of complex analysis. I've found that even the obvious fact that \( e^z \) can pass from positive to negative without hitting zero can be a stumbling block for people raised on real numbers

Naturally, this lecture series starts out with a sort of mission statement and declaration of purpose. Reviewers prefer to skip these, instead reviewing the book/lecture/etc that they want instead of what they are really getting. And I'm sure that some will be disappointed that this book doesn't go into detail about, for instance, the mathematics of path integrals. But this throat clearing is an important part of engaging with an audience, even an audience that wants to see you. In so doing, Feynman actually produces an interesting thesis on the philosophy of science:

"The Maya Indians were interested in the rising and setting of Venus as a morning 'star' and as an evening 'star' - they were very interested in when it would appear. After some years of observation, they noted that five cycles of Venus were very nearly equal to eight of their 'nominal years' of 365 days (they were aware that the true year of seasons was different and they made calculations of that also). To make calculations, the Maya invented a system of bars and dots to represent numbers (including zero), and had rules by which to calculate and predict not only the risings and settings of Venus, but other celestial phenomena, such as lunar eclipses.

In those days, only a few Maya priests could do such elaborate calculations. Now, suppose we were to ask one of them how to do just one step in the process of predicting when Venus will rise as a morning star - subtracting two numbers. And let's assume that, unlike today, we had not gone to school and did not know how to subtract. How would the priest explain to us what subtraction is?

He could either teach us the numbers represented by the bars and dots and the rules for 'subtracting' them, or he could tell us what he is really doing: 'Suppose we want to subtract 236 from 584. First, count out 584 beans and put them in a pot. Then take out 236 beans and put them on one side. Finally, count the beans left in the pot. That number is the result of subtracting 236 from 584.'

You might say, 'My Quetzalcoatl! What tedium - counting beans, putting them in, taking them out - what a job!'

To which the priest would reply, 'That's why we have the rules for the bars and dots. The rules are tricky, but a much more efficient way of getting the answer than by counting beans. The important thing is, it makes no difference as far as the answer is concerned: we can predict the appearance of Venus by counting beans (which is slow, but easy to understand) or by using tricky rules (which is much faster, but you must spend years in school to learn them).'"

I think this is a wonderful metaphor for mathematical modeling, and it is a very good approach to teaching students. In my teaching, I very frequently encounter students completely uninterested in my subject - it would be more correct to say that I occasionally find students interested in math. What I do to engage the students is to encourage them to think that what I am teaching is not abstracta to be vomited onto a test, but tools they can use as scientists and engineers. I notice that I get much more engagement when I do this (it seems to also get higher grades, but I haven't done a regression or anything...).

Incidentally, Feynman earlier admits that his history of QED is a Whig history, and mentions a couple of other issues in then contemporary philosophy of science. Feyerabend's claim that he was philosophically ignorant was always complete horseshit, based on Feyerabend's unwillingness to confront new scientific and philosophical difficulties. For instance, Feynman's anti-foundationalism shows up in a later part of this section - to him, it's modeling all the way down. I don't know whether this is trivially true or exaggerated. Just wanted to beat that dead horse a little more.

Tuesday, August 26, 2014

Foundations and Other Unneccessary Things


The economist John Maynard Keynes once said "Madmen in authority, who hear voices in the air, are distilling their frenzy from some academic scribbler of a few years back.". Practical men, to paraphrase, are usually under the spell of popular philosophy. A few weeks back I did a post on Wittgenstein's criticism of the logicist program. I concentrated on a technical aspect, he pointed out that the interpretation of quantification over infinite sets is left open (that is, there are multiple models for given sets of axioms), therefore the alleged foundations of mathematics don't specify a specific mathematical language. Modern mathematicians admit this, but don't care. I didn't go into as much detail about a stronger, but more philosophical, criticism. Principa Mathematica , Die Grundlagen der Arithmetik, etc claimed to be the foundations of mathematics, but if we found an error in them (and an error was found in Grundlagen), then we would dispose of the book and not mathematics. In other words, in practice, there is nothing special about axioms that make them "below" theorems. Mathematics, and Wittgenstein argues science and even more life in general, is more like a hyperbolic tower where everything leans on everything else than an inverted pyramid where everything leans on the bottom stone. I bring up Keynes because I realize now that there is no way to read this and not be affected. I may or may not be a Wittgensteinian, but he has affected how I see things in a fundamental way. I must keep this in mind when I enter into "foundational" controversies.


After my Jaynes post, I did a bit of re-reading of his big book. What is the value of Cox's Theorem? What makes it superior to the usual Kolmogorov Axioms? To the extent that Cox and Kolmogorov disagree, so much the worse for Cox (as far as I can tell). Kolmogorov's axioms are deliberately vague as to interpretation. They are models for statements about normalized mass or subjective valuations of probability. Cox's theorem is no shorter or more intuitive. I don't think that the interpretation that the functional equations are about subjective degrees of belief is any more suggested than in the Kolmogorov axioms (that is, it isn't at all). Why? We can interpret f to be "the sand in this unit bucket outside this set", then recognize that being outside the outside is being inside, etc. Therefore, Cox isn't any better a foundations for subjective probability than Kolmogorov.

Azazoth

Cox's theorem isn't strong enough to constrain countable unions, which means that if it was The Real foundation of probability then it would run into strange problems. As I said in the Wittgenstein post, mathematicians like to deal with the infinite by making it as much like the finite as we can without risking contradiction. Countable additivity is a way of doing this. If you have half a bucket of sand and half a bucket of sand, then you have (half plus half equals) one bucket of sand. That's additivity in a nutshell. But if you add up infinitesimal (that is, limits of smaller and smaller scoops of) grains of sand (in a limiting procedure), what happens? In countable additivity, you get a bucket of sand - lucky you. In finite additivity, the answer isn't defined. There's no reason to think that you wont add up bits of sand and get Azazoth. In other words, you give up the ability to compute probabilities.


The problems are even worse for the Bayesian, because finite additivity isn't consistent with conditionalization (hat tip: A Fine Theorem). Since finite additivity is all Cox's Theorem gives you, clearly it needs to be made more robust! (Unlike, say Kolmogorov's Axioms) Obviously, I strongly disagree with that paper's thesis that de Finetti gave "compelling reasons" to abandon countable additivity, and regard de Finetti's examples of "intuitive priors" as bizarre. (Also, I find A Fine Theorem's Kevin Bryan's arguments even weaker. It isn't obvious to me that his hostile description of frequentist consistency is induction in any sense, much less a bad one...). The famous Bayesian Jaynes must have at least sensed this, because he was always combatitively pro-countable unions. But is his foundation for himself a castle built on sand? The answer is obvious to me, Jaynes just never cared about such things, thought it was a merely technical problem without deep import to general theory (he says in the appendix that the only difference between his approach and Kolmogorov's was that Kolmogorov took an infinity first method and him an infinity last).

Dr Fine, Dr Howard and Dr Howard in deep philosophical debate

This issue might be worth maintaining low level controversy about it, and Kolmogorov put it in the right place - as a questionable but reasonable assumption. An "axiom" as we mathematicians say. Sure, countable additivity is so useful and clearly correct in so many contexts that giving it up seems like giving up your legs. But science is multithreaded being, and intellectual controversy often ends in clarification. But in the Cox framework, finite additivity isn't a theorem, it's just a quirk of not constraining our function enough. It just doesn't feel like enough to me, it seems to me that if Kolmogorov, Doob et al were wrong they must be wrong in a much deeper way. Anyway, that's enough about countable probability.

As I said from the outset, it seems obvious to me that axioms are philosophical matters and arguing about them gets you into nothing but a Wittgensteinian language game. But there are differences between Kolmogorov and Cox about finite additivity (and whether functional equations are more intuitive than measure theory). So maybe there is some, small content there. Therefore, I will now e-beg for answers. Tell me about the wonders of Cox's Theorem, internet! I'm all ears!

Saturday, August 23, 2014

Quick Review: Phoenix Wright - Ace Attorney



Well, you can't spend all your time working hard. Even the sick, dying and people in dire poverty have to entertain themselves. Even though I have a plethora of projects, a few ideas that need exploration to become projects and of course there's all the time I ought to be improving myself in some way. Let me share with you some of what I spend that time doing:


The Ace Attorney series is a set of video games, members of a genre called visual novels. A visual novel is sort of like the old choose-your-own-adventure books or, more directly, adventure games where the main form of puzzle is choosing dialog. I don't have much experience with this genre of games, in fact this series is basically the only ones I've played. Even then, I've only played the core part of the series the Phoenix Wright trilogy. From the little bit of research I did, it seems that Ace Attorney is a bit more like an adventure game in that there are many puzzles other than choosing dialog. The idea is that a visual novel will make up for its reduced emphasis on traditional video game design to tell a more compelling story. Again, I haven't played much, but I expect that the quality of the stories varies wildly.

P Mason

In essence, these are murder mystery stories where you are cast in the role of the great detective yourself. If you've seen or read Perry Mason, you'll find the basic goals familiar. You have the clues, you have to solve the case and you have to force the killer into a dramatic courtroom confession. This participation, a fact unique to its video game format, is what sets the series apart. Well, that and its off the wall humor.


The legal system in the game is vaguely based on the Japanese legal system (very judge oriented, confessions by the defendant aren't compelling, prosecutors take even the concept of losing a case to be a personal insult, etc.) and beyond that fairly absurd. I'd like to read a lawyer's comments, but what I'd really like to read is a philosopher of law talk about the series. The reason is that a good lawyer has a lot of knowledge of one kind of law, just like a good surgeon has a knowledge of one species, but a philosopher of law would be better to comment on an arbitrary legal system - and the legal system of the Ace Attorney series is often very arbitrary!


Being a series of stories about murder, betrayal, decade long schemes, endless lies, corruption at high levels of government and innumerable small ills that accompany the big ones, the series is mostly a comedy. The characters are as over-the-top and broadly drawn as anything Dickens wrote. David Simon once said "Murderers lie because they have to; witnesses and other participants lie because they think they have to; everyone else lies for the sheer joy of it...". And so it is, everybody is filled with secrets, sometimes things embarrassing only to them and sometimes issues that are more comprehensible. One murder involved a low rent tokusatsu studio. One young fan refused to tell what he saw because he was traumatized. Not by seeing a murder, but because he thinks he saw his favorite hero be defeated! Truly we live in trying times.


Another recurring element in the series is mysticism. Not dark, esoteric lore like Eric Voeglin or more intellectual fare like the Shin Megami Tensei series. Possession and spirit channeling are major elements of the series. This seems to be a trademark of producer and man in charge Shu Takumi, since he also made the excellent puzzle game Ghost Trick which is about ... well, ghosts. If this bugs you, then so it goes.


Complaints, I have a few. The series is sometimes too linear. Many of the times I've failed to present evidence it was because there are two problems with testimony and I was going to present them in the wrong order. The series doesn't have a lot of replay value, especially since you'll spend time asking everybody everything the first run through. Well, anyway, back to work - this time for real!

Personal Note: Posts full of dull pictures of scientists streak broken! Now I will be less self-conscious.

Wednesday, August 20, 2014

Pseudo-Review: Physical Cosmology And Philosophy

The first place I ever read about Physical Cosmology was Jim Peebles's Principals of Physical Cosmology, a good case study in the genre "textbook". If I were to publish any real philosophy of science, I would certainly place the chapter gently, but firmly explaining what is wrong with fractal cosmology ideas high on the list of things to analyze.

Imagine my surprise when a philosophy section in a local used book store, a section usually barren of anything interesting, contained a collection essays Physical Cosmology and Philosophy. I haven't read every essay yet, so this is a tentative pseudo-review. Most of the writers are pursuing their own hobby horses, and the editor can't be accused of homogenizing them. I don't know how he made decisions with respect to ordering. Many writers talk about deep issues such as the interpretation of quantum mechanics and singularities in general relativity, but there's no essay introducing those concepts for non-experts.

G Gamow

I skipped the first essay, so that the book starts for me on page 51, with the second essay - Modern Cosmology, written by George Gamow. Gamow is a very good writer, and the essay is sufficiently hedged and popular that it hasn't aged terribly bad. Good enough for philosophers, but Cosmology today is a very lively and data-driven field, so it might not be best for people who want hard equations. This is continued by an essay by W B Bonner on the cosmological principle. Big Crunch and steady state theories were once popular, and one shows up in this and several other essays. These are no longer of great interest, and won't be without a significant theoretical and empirical revolution.

 Trurl

There was one essay that annoyed me. Richard Swinburne makes an extremely crude argument from design for the existence of God, quite inferior to what I expect are his abilities. I think he is out of his element. His argument is, of course, subject to all the objections of Hume and his successors. He certainly fails to give any reason to suspect that The Ancient of Days rather than Trurl is behind things.

There are many essays by people with non-standard views on cosmology, frequently airing their methodological views on why the evidence isn't as strongly pro-standard view as it might seem. This is to be expected, but I think that the standard view could be expounded and defended. As it is, nobody is given the chance to answer these arguments.

In summary, this is a highly variable book that shouldn't be read by those who are looking for a quick introduction to a field and it's possible philosophical implications, but to those who are learned enough not to fall for everything they read and willing to put up with some old fashioned views and ... well, I guess that's not too many people anyway.

Tuesday, August 19, 2014

Bayesian Theory and the Supernatural: Edwin Jaynes's Perspective

E Jaynes (top left)

Edwin Jaynes was a physicist, statistician and probability theorist. He believed these fields to be inexorably linked, that they were all extended forms of logic applied to experimental experience. He was, in essence, a hardcore logical positivist who fully embraced Positivism's subjective implications. Obviously, Jaynes was a Bayesian. For him probability was just the formal thinking through of beliefs. He was inspired by Gibbs's approach to statistical mechanics, which involves (most importantly) an ensemble defined by what experimental parameters you can measure or control - that is, subjectively. What you can't control is considered to be maximally random given what you can, that is, the entropy is maximized. This is all done before the data was looked at, which is to say a priori. Therefore, the distribution over the ensemble is called the "prior" distribution. Gibbs could be confident in this approach, he had physical and mathematical arguments that gave him seemingly reasonable priors. Beyond these priors, we can use Bayes rule to change our beliefs when we get data. Therefore, this method is called "Bayesianism".

T Bayes

Jaynes felt that all of science could be done with just those mathematical arguments to do all of science. He was inspired by Claude Shannon's information theory, which made Jaynes realize that there was a certain universality about Gibbs's arguments beyond their physical basis. There have been objections. I want to emphasize that these are objections to the basics of the theory. Obviously, with enough unmotivated tricks and lowering of standards, any philosophy can be saved (it isn't as clear to me that any theory can be, but philosophers of science seem to think so). Anti-Bayesian Cosma Shalizi notes that the core method always gives exponential distributions, which is very worrying. One can do different things to get heavy tails, but it's not obvious to me that these are well motivated in Jaynesian theory. We were promised that the Gibbs Approach To Statistical Mechanics was the path to science because of the universality of Shannon Entropy theorems. How much of that can we get rid of before we are sold a bill of goods?

 
Psychic Phenomena

I would like to open a line of inquisition that I think goes deeper than technical matters. I will do so in a specific context: parapsychology. In many situations, Bayesian solutions do not appear to give us what we want, evaluations of whether a phenomenon is real or not. The reason is not just subjectivism, but personal subjectivism (if that makes sense).

A young person who reads history may be surprised how many Victorian thinkers were deeply interested in the supernatural, and not just the usual crowd. Henry Sidgwick, an economist, philosopher and moralist, was a founder of the Society for Psychical Research and his successors included the psychologist and philosopher William James and the physicist William Crookes. Ian Hacking has shown that the philosopher, mathematician, statistician, physicist and co-founder of experimental psychology Charles Sanders Peirce invented the modern concept of randomization partly to deal with practical theories in psychology (namely, "Is there a threshold below which weights cannot be distinguished?" is answered "No."), and it was applied widely to para-psychological research, where it has been a powerful force for skepticism. The classic randomization set up is described by R A Fisher to test whether a lady could tell whether tea or milk was added to a cup first (the answer, incidentally, was "Yes."). This simple test was perceived by Fisher to have deep lessons for experiments in general. Randomization is a part of (note: not the foundation, as Hacking above makes clear) Deborah Mayo's error statistics approach to frequentism, because randomization is a kind of error probe. And in this case, it gives us what we say we want, an answer to the question" Are the alleged phenomena distinguishable from blind chance?".


My objection to Jaynes concerns Chapter 5 of Probability Theory: The Logic Of Science, "Queer Uses For Probability Theory". In this chapter, Jaynes takes up the subject of psychic phenomena, as an example of how probability theory can be used to evaluate subjective degrees of belief. This is an important subject. In truth, Professor X cannot tell whether tea or or milk was put first in the cup, but Dr Bristol can. Of course, Professor X claims otherwise. His claim might be honest, as many magicians fool themselves eventually. RA Fisher would say, let's see how they do in the same test, and he'd compare the likelihood ratio, etc. Jaynes's examination is different. Jaynes does and must say that we should start by evaluating our internal degree of belief in the manner of IJ Good:

"Our brains work pretty much the way this [Bayesian] robot works, but we have an intuitive feeling for plausibility only when it's not too far from 0 db. We get fairly definite feelings that something is more than likely to be so or less than likely to be so. So the trick is to imagine an experiment. How much evidence would it take to bring your state of belief up to the place where you felt very perplexed and unsure about it? Not to the place where you believed it - that would overshoot the mark, and again we'd lose our resolving power. How much evidence would it take to bring you just up to the point where you were beginning to consider the possibility seriously?

We take this man who says he has extrasensory perception, and we will write down some numbers from 1 to 10 on a piece of paper and ask him to guess which numbers we've written down. We'll take the usual precautions to make sure against other ways of finding out. If he guesses the first number correctly, of course we will all say 'You're a very lucky person, but I don't believe it.' And if he guesses two numbers correctly, we'll still say 'You're a very lucky person, but I don't believe it.' By the time he's guessed four numbers correctly - well, I still wouldn't believe it. So my state of belief is certainly lower than 40 db."

Jaynes estimates that he has a belief degree of -100 db in psychic phenomena. These so-called psychics often ask us to keep our minds open, to not do what Jaynes is doing. Why should we not oblige them, at least in a formal manner? It seems that it would be better dialectically if they still couldn't convince us even if we gave them the benefit of the doubt. In the long run, experiments will eventually overcome any initial degree of belief, but strong disbelief can make it take a very long time. Jaynes goes on to give himself another out: "In fact, if he guessed 1000 numbers correctly, I still would not believe that he has ESP...". He describes an experiment that supposedly gives very strong evidence of psychic phenomena. But it would not convince Jaynes. "[The psychic] will then react with anger and dismay when, in spite of what he considers this overwhelming evidence, we persist in not believing in ESP. Why are we [Jaynesians], as [the psychic] sees it, so perversely illogical and unscientific?". Jaynes argues that the data can never prove psychic phenomena to Jaynes, because all a high likelihood (that is, data improbable to be produced by chance) does is increase the probability of error and deception.

But this is not what we say we want. How do we know that if this idea were not pursued that it wouldn't end up giving you error statistics, perhaps error statistics constrained by Bayesian constraints? I think that it would, though I don't have much argument for it. Further, how does this rule in cases of mistaken disbelief. If I had a mathematically intimidating (but physically naive) argument from statistical mechanics that it shouldn't matter whether tea or milk is put first, I could and should use this out to disbelieve Dr Bristol as much as Professor X. The fact of her complete success is meaningless, since I can simply say that she managed to deceive me. Write mathematically intimidating but ill founded models, and bend the evidence to their will? Perhaps this is what economists do all day (joke)!

What is "wrong" here is that the Bayesian solution is for yourself. If you are a firm that is evaluating its goods and you want to convince yourself they are good, then Bayesianism works well. Good (and Savage and Friedman, etc.) broke their teeth on the statistical problems of WWII, many of which were of this sort. But one advantage of the error statistics approach is dialectical - it is for convincing others. At the end of the experiment, Dr Bristol and Sir Fisher agree. When particle physicists discovered the Higgs Boson, they used (among other things) p-value analysis and other frequentist techniques. This is because they are in dialog with themselves about what the ultimate laws of physics are, not loners trying to make their priors a little sharper. Minimax estimators are the ultimate expression of the Hegelian philosophy (even worse joke)!


Don't make the mistake that dogmatic adherence to only the strictest of frequentist ideas (for instance, banishing stopping times) is the only path forward! In fact, the man who designed the lady drinking tea experiment described had some very strange ideas about probability. Problems formally equivalent to interior monologues happen all the time. Bayesian theorists have made many contributions to science. I mean to say this as a corrective, not as a declaration of purpose. It's an easy mistake to make that just because one side seems wrong, the other must be right. Instead, what I think is right is to let one hundred flowers bloom, but only enjoy them at the appropriate time. For instance, if optional stopping is desirable, there is no reason to report statistics unstable in their presence. If broad agreement is necessary then there is no reason to report "results" that depend on unmotivated priors. This is obviously correct, but sadly totally informal. What is to be done?

Monday, August 18, 2014

The Mandelbrot Approach To Statistical Mechanics

B Mandelbrot (gasp, color!)
 
The mathematician Benoit Mandelbrot is most famous for coining the term "fractal" and being one of the first to treat them as anything more than theoretical curiosities. He was widely known to be in the first rank from a young age, working some of the best scientific institutions in France and the United States. Fans of mathematical genealogy will be pleased to learn that he was sponsored at the Institute of Advanced Study by John von Neumann himself! He spent most of his mathematical life working at IBM, where he used and developed both the tools (computers, visualization, statistics) he would use for the rest of his life and found the problems that interested him the most. Mandelbrot was a "visual thinker", more so than most, and his most famous book The Fractal Geometry of Nature was a best seller at least partly because he was able to expound his mathematical view of the world without dense equations. I'd recommend it, but it has been quite some time since I've even so much as leafed through it.

Strangely enough, this post isn't a place for going into Mandelbrot's most famous views and discoveries, the things that will make him immortal. Instead, I want to highlight some really interesting applied mathematics work he did at IBM. Last post, I sort of mentioned a few approaches to statistical mechanics, two of which got big time names behind them the Boltzmann (single set-up, large number of particles) and Gibbs (ensemble, ergodic system) approaches. I mentioned that there are two other approaches, both avoiding interpretation as much as possible in different ways. The last approach mentioned was to do straight up dynamical systems. This approach is very rigorous, very slow, and open to complaints that we no longer believe in dynamics smooth in the very small (in other words, it's still classical but nature is quantum). Good work continues to be done in this field, such as this description of a deterministic random number generator by Fields Medalist Artur Avila. The other method was to avoid dynamics as much as possible, postulate the existence of free energy functions and derive what you need from there. This method is highly visual and very fruitful, but has certain scientific limitations because, as mentioned before, free energy functions are not always analytic. What I completely failed to mention is that Mandelbrot made some huge contributions to this field

Wrong kind of field

The first contribution is this paper, which is very excellent. It concerns a statistical interpretation of the free energy function approach, making it more clear that this really is a kind of statistical mechanics. This is in many ways a precursor to Jaynes's method, but doesn't have the same subjective implications because the probabilities aren't conceived as coming from a subjectively defined ensemble. In fact, it is very important to Mandelbrot that the estimators will converge on the truth, which he calls "self-consistency". The paper shows that many thermodynamic variables of interest (such as temperature) are sufficient statistics of whatever it is that underlies them. That is, if you know the volume, temperature, mass, et cetera of a system, you can't learn more by tracking the particles or otherwise zooming in. One remarkable about this paper is its inversion of our usual values. We usually think of the randomness and chaos of statistical mechanics as being the interesting part, the entropy, the second law. In this approach, it is the measurability (the zeroth and first laws) of these systems that has the deep implications and powerful theorems!

This work is extended in this paper and its sequels, in which several more classical concepts are introduced. This paper is, in many ways, better than the first, being more clearly written, but the first is more focused and impressive. It contains an interesting "uncertainty relationship": the better energy is estimated, the worst (inverse) temperature is estimated. This is a classical uncertainty relationship, a consequence of the linearity of Laplace transforms. This paper also pushes a more philosophical line, which is interesting in itself. Mandelbrot was obviously impressed by the positivists, and isn't afraid to let a little subjectivity get into his arguments as a result. I will suspend judgment about this.

I think this part of Mandelbrot's work is of the very highest quality, but I will remain agnostic about judging it against the other methods, especially due to problems about phase transitions, etc. I think his contributions as being good enough that I will call this approach The Mandelbrot Approach, placing him along side Gibbs and Boltzmann.

Saturday, August 16, 2014

A Few Approches To Statistical Mechanics

J C Maxwell (this is gonna be another of those black and white men posts...)

I've been collecting some thoughts recently about statistical mechanics, after repurchasing some old books. There are several approaches to statistical mechanics, which can have different seeming implications. I am not going to attempt to carefully delineate them, theorems from one may be used in another. I am going to just list a few thoughts, no real substance yet, but maybe in the future there will be more. There's tons to think about.

R Clausius

The first and oldest is no interpretation at all. More to the point, in the original purpose of statistical mechanics was to get classical thermodynamics out of Newton's Laws. One could avoid this, try to derive thermodynamic relations by presuming that they can be derived by working with analytic free energy functions. These functions are tractable and have beautiful and suggestive visualizations. This technique was used by James Clerk Maxwell and the earlier Gibbs - Gibbs's massive monograph "On The Equilibrium Of Heterogeneous Substances" is basically a formalization of this method. This method is also covered in many an engineering textbook. One could try to stick to these guns, but there are severe limitations on this method. As outlined here, this theory gives poor result when it comes to phase transitions and other physically important phenomena. In addition to it's limitations, it gave a somewhat false view of it's theorems. For instance, Maxwell thought that the early, static approach to the second law of thermodynamics was necessary to avoid perpetual motion. But I've discussed this before.

Inspired by his work on Saturn's rings and with an eye on extending preliminary results due to people like Bernoulli and especially Clausius pictured above. Instead of trying to use global results (more on this later), Maxwell concentrated on the whole distribution. Conceptualizing a gas as being made of hard spheres that impart velocity on each other. Maxwell showed that there was only one stable distribution with certain symmetry properties (there have been other derivations since). By considering the average case, Maxwell was able to develop statistical laws. Thus was the birth of statistical mechanics.

L Boltzmann

The first of the three big approaches I'll call the Boltzmann approach. Boltzmann was a physicist who worked in what he felt to be the tradition of Maxwell (in several of his polemical papers, he uses the Maxwell name as a weapon). I do think his thinking was along these lines, but I wouldn't mind being informed of differences. Boltzmann's approach was based on very large assemblages. This forces the dimension of the subject very high, allowing one to use concentration of measure results to argue that the system will spend almost all of its time in a specific state. Concentration of measure results are direct rather than asymptotic, making them useful for machine learning (hint hint!). A quick introduction to concentration of measure can be found here. The dynamics are important (frequently, Markov Models are used or found), but not as central as the concentration of measure results. One can use these methods to find concentration of measure results, which makes this approach really interesting to me.

Because the measure of the actual system really is concentrated and only the actual system enters into the result, this approach is very objective. For instance, entropy is a semi-objective property. Entropy is the count of microstates compatible with the observable macrostate (some subjectivity comes in how we define microstates). There's no obvious connection to information theory in this approach, and being objective has nothing to do with Bayesianism. In this approach, ensembles are a calculation device justified by ergodic theorems and equilibrium arguments

J W Gibbs

The second of the approaches I will call the Gibbs approach. This approach is ensemble based. Instead of just considering the actual system, every possible system - subject to certain constraints. Entropy is the spread of the ensembles. These constraints are often more about our measuring systems than the system itself, which leads to subjectivity. That entropy is subjective makes this sit well with Bayesian approaches. Bootstrapping and other resampling methods have their origin in this kind of thinking.

Again, though dynamics can be important, they are secondary to information theoretic arguments. Arch-Bayesian Edwin Jaynes is famous for his dogmatic, chapter and verse defense of this approach. How does the use of ensembles sit with the supposedly Bayesian likliehood principle? Aren't we supposed to only consider the actual results? I don't know.

I don't know how the Gibbs approach relates to the Boltzmann approach, especially out of equilibrium. Sorry.

H Poincare

The third approach is the dynamical systems approach. This approach is very, very powerful and underlies the others. However, it basically never has answers. As an example, it was 103 years before even one dynamical system was directly found that obeys the above mentioned Maxwell distribution. At the time of Maxwell and Boltzmann this was the uncontroversial foundation of all things, but these days quantum mechanics and quantum field theory are known to be more basic. The first big result in this field is Liouville's Theorem. Other important results are proofs of chaotic behavior, ergodic theorems, etc. I've talked about my favorite learning device for learning this theory in this post. This is the best approach if you like your science extremely rigorous and moving at the speed of pitch.

Personal Note: had a conversation about application of Kelvin Inversion to economics and statistics (!!!) can't remember a single thing. Look for this later.

Wednesday, August 13, 2014

Contra Feyerabend

P Feyerabend

I've mentioned before that I don't like the work of Paul Feyerabend. Some groundwork for what is going to be done here. First of all, there is no complete Feyreabend philosophy. He didn't want a system, and didn't build one. This is not a criticism. If you want to critique him you have to critique individual papers and critiques. I doubt that Feyerabend could be so dense that everything they wrote was completely wrong, so I'm just going to stick to a few broad notions that he is famous for. I'm not going to, for instance, attack his historical work or explore his opinions on philosophy of mind. Instead, I'm going to go after the notion of epistemological anarchism and some of the arguments for it.

First of all, there is the notion that every observation is theory laden. This is either false or trivial, depending on how widely you let the notion of theory stretch. If you let the light tickling your retina be a theory, go ahead. But there is nothing special then about being "theory laden". No reason to suspect there is anything even subjective about them. Modern data-driven, nonparametric learning theory allows one to do explicit, quantitative mathematics without being "theory laden" in the way important to philosophers. I recently wrote a classifier to try to count the number of little buggers on some seagrass leaves. The data was collected totally independent of any input from me, it was already here by the time I joined the project. I don't even know what the schmutz is, biologically speaking. But despite having no theory of what they should look like, I managed to successfully classify them and other kinds of objects on the leaves using tools that didn't require me to have that knowledge. Theory-ladenness can matter, but it it isn't a necessary truth or even central to science.

L Valiant (nothing but pictures of black and white men in this post)

A simpler criticism can be made. Theory-ladenness is based on a false, "Good Old Fashioned AI" view of knowledge. When I walk through a room with a table, I don't and don't have to represent the room in my mind and trace my whole path through it before I walk. What I really do is much simpler to process and more fallible. You can see Leslie Valiant's excellent book Probably Approximately Correct for more on this subject.

S Wright

There is a deeper critique of epistemological anarchism, that it is doesn't give you what you say you want. There is nothing wrong with taking the 'let ten thousand flowers bloom' approach. But Darwin informed us that not all flowers are created equal. Seawall Wright introduced the very useful idea of a fitness landscape, evolution is an optimizing procedure on this landscape. Having many peaks and noise is not a flaw of this theory, just parts of it. The mass of beasts and scientists wander around this landscape, compelled to optimize. The difficulty with epistemological anarchism is that it uses the vague, equivocal notion of "incommensurability" as a hammer to flatten out the landscape. A totally flat landscape is what Feyerabend needs, a highly peaked landscape has far different properties (Stuart Kauffmann has done fascinating work in this area).

What we actually deal with is much harder work, people doing a lot of work actively trying to make better epistemologies. Science is a dirty, complicated, thing with complex dependencies, not a flat "everything is as good as everything else" field. One could say that the different notions of better ("unbiased" vs "efficient", for instance) make the choice of what one wants a bit subjective. But this is not anarchism, not a simple democracy. It's a hierarchical bureaucracy with republican elements at best. It is even possible that the dynamics is objective, as seen in statistical mechanics, but I won't defend that notion since.

A real example will hopefully help clarify. I was talking recently with a friend who is a bit of a biblical scholar. He mentioned that the discovery of some old papyrus falsified the work of many old time German historical critics that claimed it wasn't written for centuries after we now have documented evidence of their existence. According to Feyerabend, the historical critics should have said "We aren't interested in carbon dating. In this department, we use philology.". This isn't what they do. It's the epistemological anarchist thing to do, declare freedom from physics. It isn't what people do, not in the real world. So epistemological anarchism goes.

I've already mentioned the severe criticisms of the notions of democracy that were supposed to be what epistemological anarchism were defending. The problem is that real democracy, like real science, is much more complex, more peaked, more dynamic than the flat philosophical politics that Feyerabend defended.

So, what is left? Feyerabend left us a lot of criticism of over ambitious philosophy of science arguments. I've read philosophers call the discovery of the Higgs Boson in high energy physics "bad science" because it didn't exactly match their pre-conceived notions of what a good scientist should do (they used p-values at some point). Perhaps Feyerabend will help philosophers avoid hubris. This modest achievement doesn't leave the working scientist with much. That's only because there isn't much to it.

Tuesday, August 12, 2014

Wendy's Old Fashioned Scientific Terms

I was reading an old book, and I don't mean to name names but it's definitely an instruction in why we need mathematical economics! Bertrand Russell once said "A good notation has a subtlety and suggestiveness which at times make it almost seem like a live teacher.". It doesn't follow that a bad notation is like a quack spreading confusion, but it is true and one can find it in old books like diseased mosquitoes preserved in unholy amber.

A Marshall

The confusion is around the economic notion of the "short run" and "long run", terms that I think were introduced by Alfred Marshall. Marshall, the old empiricist, noticed that in the beginning of the industrial revolution people tended to be immiserated by machines, then later they (not necessarily the same people, but "the people" all the same...) became exponentially wealthier because of the fruit of the capital. This was probably already known by other economists before him, but I plead ignorance about their. This was one of many things that were noticed by Marshall. Marshall was a talented monetary theorist in addition to everything else, developing the cash-balance (that is, demand oriented) version of the quantity theory of money. In so doing, he must have noticed that a bit of inflation now can give a temporary boost, but in the long run inflation tends to hurt. The modern version of this observation - which is now called "dynamic inconsistency" - got Finn Kydland and Edward Prescott a Nobel Prize.

Marshall put these two empirical eggs in one basket, perhaps wrongly. He divided the economy into two different "runs", the "short run" where inflation boosted things and new machines immiserate workers and the "long run" where better capital is the source of most wealth and low inflation is good. This is very misleading! What Marshall was after was the fact that the economy can be modeled by - in a way, "is" - different dynamical systems, some where some things are held still and others where others are held still. I think this is well known in economics, but don't feel like looking it up. There will never come a day in which we are suddenly out of the short run and into the long, or vice-versa. It is always 40 years from 40 years ago, the long run of our fathers is already here. It will always be this time tomorrow in 24 hours. The right way to think about these issues, perhaps the unique one, is the perspective of dynamical systems.

Marshall chose the term "short run" and "long run" because of the empirical facts, he was observing the industrial revolution turn from a force that throws workers on the streets to one that feeds them all healthily. You argue that he was after and dimly perceived an idea like a Kuznets Curve, but not well or forcefully. Kuznets showed that there exists some economic dynamics that can produce Marshallian runs. It doesn't take much thought to realize that there are far too many dynamic paths for just one, Malthus believed that one would get an exponentially curve during a growth phase, followed by exponential decay in the famine phase- like the following:


Since this is periodic, it completely lacks anything like Marshall's simplistic idea of runs. I'm not saying this particular dynamics is the true dynamics of the wealth of nations - in fact there is absolutely no empirical evidence that it is. What I mean is that the dynamics that Marshall talked about are not fundamental, but the terms make it seem that they are. In fact, the dynamics questions in economics are right now at the forefront of the field, both in theory and in empirical matters. Thomas Piketty's best selling book is, at root, about the empirical underpinnings of dynamic models of the economy - what features must a model have and which must it lack to resemble reality.

Mislead by Marshall's terms, several economists and writers on economics have made daffy mistakes. As I already noted, in the above book, Henry Hazlitt notes without irony that it took more than 40 years for Marshall's long run to get to workers in the garment industry. Hazlitt strangely takes this as vindication. Perhaps it is for one notion, but the theory he puts around it is filled with confusion. The notion was fundamental to the worldview of economist Joan Robinson. She thought that the greatest task facing economist was to turn idealized periods into real time, in other words to do dynamics. Sadly, she contributed nothing to this task

There has been much written about dynamics and equilibrium. A modern has no excuse for making the excuses of the esteemed ancestors written about in the previous paragraph. The book reviewed below is a good primer for the theory of dynamics in general.

Monday, August 11, 2014

Abduction



I've been meaning to watch this for some time. Abduction is the process of going from effects to causes. The name is due to the American philosopher CS Peirce. I wonder if it really is different than deduction and induction, since there are special cases where it clearly isn't, such as learning circuits.

Thursday, August 7, 2014

Quick Review: Dynamics - The Geometry of Behavior

Given that something is in some state, how will that thing change? This is the problem of dynamics, and if it seems vague part of it is that dynamics is really that general. Born for applications to physics, this theory is now used everywhere, including economics. We start by making assumptions about continuity and such in order to use the powerful tools of calculus. These assumptions are the same assumptions that give us a nice geometric picture of an evolving system. Each state of the system is a point in space. We try to figure out the geometry of how the system moves around, and what doesn't change about that geometry as things move. This is the methods of topology and catastrophe theory. We can go around looking for "equilibria". An equilibrium is a state where if you start there, then you stay there. Close to an equilibrium, the system will be approximately linear. There are only a few possible general shapes a linear system can give you. You can use these to figure out where the system will be in the long term. This becomes the method of equilibrium analysis and chaos theory.

G Galileo

The study of dynamics was begun by Galileo, all previous thinkers had been forced to remain in the field of statics. The field remained informal and many of the theorems implicit until the work of the French physicist and mathematician LaGrange formalized the calculus of variations. Later, the English physicist and mathematician Hamilton redid the same thing from a different angle (interesting to note what Hamilton was doing, he was convinced there would be a deep connection between theoretical optics and basic physics ... and he found it!). Unfortunately, these analytical methods rarely result in explicit global solutions

H Poincare

 There was a revolution in these techniques brought out by the great French mathematician Poincare, who developed methods of analyzing classes of solutions. These methods use a new science of topology (he called it "analysis situs"), looking for things left unchanged by the kind of evolution we often see in systems. At the same time, Russian mathematician Lypunov was discovering the methods of linearization and equilibrium analysis. These methods dovetail nicely with each other.

A Lorenz Attractor

In the usual presentation, all of these techniques are heavily analytical, involving tremendous and difficult mathematics. This is perhaps a little strange, after all the whole theory works because there are a) only a few configurations one can have around an equilibrium (for equilibrium analysis/chaos theory) or b) only a few possible globally interesting shapes you can get (in topology/catastrophe theory). Christopher Shaw and Ralph Abraham's book Dynamics: The Geometry of Behavior is the book that uses this possibility. This book is very good for learning the heart of the theory, avoiding merely technical distractions. It is entirely geometric, taught without equations. I feel that from this book the very important subject could be learned by an ambitious high school student. Those technicalities must be dealt with in the long term if you want to become an expert, but this is a great first book.

R Abraham

The book is divided into four parts - it was originally multiple books. The first part is an excellent introduction to dynamical systems and a treatment of periodic systems. This part is interesting, though an economist might wish they spent more time on stable equilibria. They seem to regard them as fairly uninteresting, but as the above linked discussion makes clear, considering basins of attraction of a stable equilibrium can be good science. This is discussed in the third part, but I think it is more fundamental. The second part treats chaotic systems - which are extremely interesting to me personally. Most discussions of chaos focus on the consequences treated in the last part of this section, such as unpredictability, fractal structure and noisy spectra. These results are very clearly presented in an easily understood and demystifying way. Multiple equilibria are treated in the next section, which is good, but I again feel it should have been treated earlier. Finally, catastrophe theory is treated. Since much of my understanding of this theory comes from this book and Vladimir Arnold, I feel unable to comment on what I don't have a mastery of. I certainly feel as though I learned a lot from this section.

Very highly recommended, especially for self-teaching and especially to visual learners. However, be warned that Ralph Abraham's later books do not reach this standard.

Wednesday, August 6, 2014

A Warning



Don't fear those who abuse the legal system! Sometimes it works out okay! This is an old story at this point, but a good one.

Tuesday, August 5, 2014

Hempel's Paradox

Hempel's Paradox is yet another problem with induction. Hume taught us that much of what we consider to be induction is fallible and probabilistic, rather than certain. But this raises a whole host of difficulties, difficulties that Hume - who lacked training in mathematics and statistics - dimly perceived. Statisticians from CS Peirce to Bruno de Finetti have tried to firmly grasp what Hume offered, and found much to disagree with each other even within this. Bruno de Finetti was a Bayesian - perhaps the Bayesian. In the frequentist/error statistics tradition from CS Peirce to Deborah Mayo, have attempted to use probability in a way that has been called ampliative, they amplify our knowledge. Deborah Mayo is explicit on this point, a severe test teaches us something new. There is another tradition running from Frank Ramsey to de Finetti and through to modern Bayesian theory, that probability is a form of logic - and therefore not ampliative. I don't mean that probability is founded on logic - nobody doubts that probability is a branch of mathematics and statistics is applied mathematics. What is at stake is what statistics could teach us even in theory. The question "Is probability a form of logic?".
C Hempel

Carl Hempel added a new wrinkle to this debate with his raven paradox. How would you go about testing the proposition "All ravens are black."? Look for an albino raven? Save your time, there's a much easier way. Obviously, this is equivalent to testing the proposition "If something is a raven, then it is black.". This is equivalent to saying "If something is not black then it isn't a raven.". So, look at your shoes. They're not black and not ravens. So that's some evidence. And hey, your fingernails aren't black either, are they? Thinking about it, how many van Gogh paintings aren't black? How many dots in a Seurat painting? The world is mostly evidence that there are no white ravens!

There are, of course, solutions for those who hold, with de Finetti, that probability is a form of logic. The technique is to use the size of the sets involved. There are so many non-black non-raven things in this world that the weight of observing one is low. That is, we do learn that there are no black ravens by drinking white milk, but it is weak evidential milk. This is a bit counter-intuitive, since the eye color of Chuck Berry's eyes seems to be completely unrelated to ornithology. In addition, this response gives rise to odd ducks (perhaps odd ravens...), like Laplace's argument about the probability of the sun rise.

This argument is popular with those who accept a Bayesian view of probability as something more than a sometimes useful tool. What is the frequentist/error statistic point of view? It might seem extreme, but it actually makes good sense. Observations of ravens gives us an estimate of the ratio of black ravens to the count of ravens. Obviously, this ratio will be near one. But what the frequentist test gives is error bars, confidence intervals. These confidence intervals can overlap, be useless or shrink around one. In the last case, the frequentist/error statistician will converge on popular opinion. But what of the non-white non-raven? Since a frequentist does not accept that tests are closed under logical operations, they don't accept them as tests. They are only interested in tests of the object in question.

In my view, this makes error statistics less expressive, but less paradoxical than Bayesian statistics. Which you need depends on the practical situation, and so I say let ten thousand flowers bloom. Incidentally, the philsopher Willard Quine also wrote on this subject. Quine's solution was to claim that non-black things didn't form a natural kind. This is worse than accepting the paradox, since it will soon mean natural kinds aren't closed under any logical operation, yet we're supposed to use logical analysis and probability theory to learn... Good luck writing a classifier with that in mind!

Sunday, August 3, 2014

Quick Review: Elementary Particles and the Laws of Physics

Or, The First Annual Paul Dirac Fan Club!
P Dirac

This is actually a pair of lectures from physicists Richard Feynman and Steven Weinberg entitled "The Reason for Antiparticles" and "Towards the Final Laws of Physics". They were on topics inspired by the great English physicist Paul Dirac, one of the people who codified Quantum Mechanics and one of the first to deal with quantum mechanics and relativity. The excellent biography The Strangest Man is about his life and work, so ... go read that too. Dirac is widely believed to have suffered from Asperger's Syndrome, so people with an interest in that topic might find it interesting to read about one person's life.

R Feynman


Feynman's work is pedagogical, it explains the modern view on antiparticle that evolved out of Feynman's elucidation of Dirac's work on relativistic quantum mechanics. This is a really good lecture, very well presented. Many of the most important laws of physics, such as the Pauli Exclusion theorem and the  spin-statistics theorem (and therefore, virtually all of chemistry...), are direct consequences of relativistic quantum mechanics. Feynman talks about Dirac's style, which he describes as trusting his equations. There's been some interesting history of science work on Dirac's attachment to projective geometry, which was essential to his understanding of relativity. H S M Coxeter wrote several articles on this idea for classical (that is, non-Quantum) relativity, such as "A Geometrical Background for de Sitter's World" and elsewhere. Feynman aims his lecture at a fairly high level, knowledge of basic quantum mechanics and relativity, but this is fairly well presented. Familiarity with Bra-Ket notation and space-time diagrams should be enough when it comes to physical theory. More important is the level of mathematical imagination it requires. Feynman puts the interpretation into easy to understand levels, a process he compares to Maxwell's mechanical models of electrodynamics, but it will really help if you're already used to this kind of discussion. This is too bad, since this is a very important topic, especially for chemists, and could use better popularization. Of course, it would take at least three times as much time and space... If you're capable of doing math at an advanced undergraduate level, I highly recommend this talk for a deep understanding of the parts of this theory most relevant to non-specialists. I'd particularly recommend it to chemists for the proofs of the Pauli Exclusion Principle and the Spin-Statistics Theorem. Particularly excellent is his discussion of the proof (first arrived at by Dirac) that if there is one magnetic monopole then magnetism is quantized everywhere, essentially as a consequences of path independence of a particular integral that comes up naturally in the theory.

S Weinberg

Steven Weinberg's lecture is more philosophical. He describes the symmetries and physical intuitions that lead to QED and then later theories, and takes care to be critical of insecure notions. Weinberg is careful to note the actual consequences of knowing the ultimate laws of physics and disclaims overblown Laplacian conclusions, distancing himself from what Daniel Dennett would call "greedy reductionists". This part has less diagrams, less physical intuition (it is after all, being compared to Feynman) and more speculation, but Weinberg also covers a much more ambitious topic. Feynman doesn't discuss, for instance, Dirac's model of anti-particles, the Dirac Sea, just constructs the modern version. This is a good discussion, but still a textbook topic. Weinberg explains why the Dirac Sea was abandoned for the modern theory in a footnote (it only works for spin-1/2 particles), and moves on to more speculative theories. Weinberg doesn't cover any topics not of interest to specialists, but it is good for young people wondering if they want to become specialists. This area, the idea of final laws of physics, is an area of interest to Weinberg and is better covered in Weinberg's book Dreams of a Final Theory. Weinberg's connection to Dirac is mostly from Dirac's Platonism, Dirac's suggestion that the final laws of physics will be mathematically beautiful. Weinberg points out that much of our skepticism of proposed roads to final theories - such as String Theory - comes from their mathematical inelegance, such as renormalization difficulties or lacking non-perturbative forms. This is a nice observation and obviously true, but I don't know what to make of it philosophically.