Entropy and uncertainty

In research for this blog post, I came across an old favorite by Cosma Shalizi that links back to this other post. Shalizi says (in the former):

I doubt it helps matters that many statistical physicists are in the grip of sub-Bayesian ideas about maximum entropy

In the latter:

One of the ideas in physics which makes no sense to me ... is that statistical mechanics is basically an application of Bayesian ideas about statistical inference. On this view, the probabilities I calculate when I solve a stat. mech. problem --- say, the probability that all the molecules of air in this room will, in the next minute, be found at least three feet above ground level --- are not statements about how often such events occur. Rather, they are statements about the strength of my belief that such things will occur. Thermodynamic entropy, in particular, is supposed to be just the information-theoretic, Shannon entropy in my distribution over molecular states; how much uncertainty I have about the molecular state of whatever it is I'm dealing with. 

Here's an (unfair) way of putting it: water boils because I become sufficiently ignorant of its molecular state.

He also has a paper.

I've never been one to get into the Bayesian-frequentist argument, which much like the interpretations of quantum mechanics, seem to be a waste of my time (and are basically the same underlying issue). That it's a waste of my time does not imply it is a waste of your time if you happen to be a masochist who's into philosophical arguments that never go anywhere.

I would completely concur with the paper's result. Additional measurements should never cause you to become more uncertain on average. But that means entropy should decrease if you equate it with an ideal observer's uncertainty about the system.

Shalizi's solution is to abandon the idea that entropy represents an ideal observer's uncertainty about the state of the system.

And I'd agree. This doesn't really affect anything I've said on this blog since the basic mathematics is all still the same, and I've never implied Bayesian updating except when talking about belief in a theory.

Is information equilibrium disruptive?

Matthew Yglesias has a good article at Vox on how Tesla is not really disruptive in the true sense of the term. In reading the very good definition of disruptive, I thought -- hey isn't information equilibrium disruptive to economic theory? Here's Yglesias:

The core idea of disruptive innovation is that successful companies tend to become obsessed with getting better and better at serving their existing high-margin customers. 

Those customers provide the profits, so they get studied closely and provided with more and more services and features to ensure their continued loyalty. Eventually a new competing product comes into the marketplace that is generally cheaper, simpler, and, in an abstract sense, inferior. This is the disruptor. What makes it disruptive is that even though the incumbent company could, technically, copy the disruptive new product, it can't bring itself to actually do so, because that would involve undercutting its existing profit margins. 

Instead, the tendency is for the incumbent to reassure itself, accurately, that the new product simply does not meet the needs of the most valuable customers and to continue focusing on them. 

The problem for incumbents is that over time the new disruptive product tends to get better and better, eating away at a bigger and bigger share of the incumbent's market share. At the same time, because the disruptive new product was designed from the beginning to be cheaper, simpler, and lower-margin it manages to enlarge the market far beyond its previous size.

Let's tell a story of disruption in the free market of ideas.

Economic theory is obsessed with better serving their high status customers: the political elite. To that end, they've gotten very good at producing models that seem very rigorous and serve their interests from banks (controlling the economy via monetary policy) to politicians (fiscal policy that involves giveaways to corporations or people depending on the political persuasion).

A new "competing" theory (information equilibrium) enters the marketplace of ideas that is simpler and in an abstract sense inferior to existing economic models. In the information transfer framework there literally is just one equation, it considers most macro aggregate movements to be noise, and it gives up on understanding economic agents.

Of course economics Phd's could take easily take it up, but that could undercut their existing status relationships and institutions.

Since the information equilibrium model is so easy to use (again, literally one equation), it enlarges the market of people capable of performing economic analyses. Where previously you needed a Phd in economics to set up a DSGE model (unless you are John Handley), now any high school graduate with a bit of calculus can join in.

So is it disruptive? Well, information equilibrium hasn't disrupted anything yet. Will it be disruptive? Maybe!

Different types of emergence

I think Daniel Little misunderstands phase transitions here, but it gives me an excellent excuse to talk about the meaning of "emergence". Little says:

What seems to be involved here is a conclusion that is a little bit different from standard ideas about emergent phenomena. The point seems to be that for a certain class of systems, these systems have dynamic characteristics that are formal and abstract and do not require that we understand the micro mechanisms upon they rest at all. It is enough to know that system S is formally similar to a two-dimensional array of magnetized atoms (the "Ising model"); then we can infer that phase-transition behavior of the system will have specific mathematical properties. This might be summarized with the slogan, "system properties do not require derivation from micro dynamics." Or in other words: systems have properties that don't depend upon the specifics of the individual components -- a statement that is strongly parallel to but distinct from the definition of emergence mentioned above. It is distinct, because the approach leaves it entirely open that the system properties are generated by the dynamics of the components.

Well, yes and no. What is being discussed here by Solé are universality classes of phase transitions. Near a phase transition, the correlation length of the order parameter can go to infinity meaning the system becomes "scale free" and depend on some pure numbers (critical exponents). Here's an example with magnetism. Two of the best studied universality classes are the ones containing the Ising model and percolation theory. In plain English what this means is that wildly different systems (with wildly different "micro theories") can have identical behavior (i.e. don't depend on micro theory scales) near a phase transition if they have the same critical exponent.

I should hand the mic to Cosma Shalizi [1]:

... over the last half century or so, the statistical mechanics community has devoted much of its research energy, and its pedagogy, to the theory of phase transitions, such as those between solids and liquids, or liquids and liquid crystals, or between magnetic and non-magnetic materials. ... I don't mean that this is bad for statistical mechanics, because phase transitions are important and the theory developed around them is one of the jewels of the field. But phase transitions have the weird property of "universality". In the vicinity of the critical point, the behavior of the system comes to depend only on a few key parameters, so that any two systems in the same "universality class" are quantitatively similar near the transition, even if they are otherwise as different as chalk and cheese. If what you are interested in is this behavior near the critical point, then, you can get away with analyzing or simulating ridiculously over-simplified models, if only you get their universality class right. The implicit lesson is that details don't matter, and results on toy models should generalize directly to real systems. (Of course, details can matter a lot, even with toy models.) I can't make myself believe it's coincidence that so many of the people active in econophysics come from a background in the theory of critical phenomena.

This represents one way of interpreting "emergence": elements of a macro theory that don't depend on the micro theory are emergent because you are near a phase transition.

Now universality classes aren't just for phase transitions; probability distribution also have universality classes. There's one that many more people are familiar with as well: the Gaussian (normal) distribution universality class. The central limit theorem is directly related to the universality of the normal distribution. There are other distribution universality classes as well [pdf] (related to something I did here).

Instead of universality appearing as you approach a phase transition, the normal distribution appears as the number of samples becomes large regardless of the underlying distribution (well, regardless as long as it has a well-defined mean and variance).

We can say that the normal distribution is "emergent" and the details of the micro system (underlying distribution) don't matter. Coupled with the law of large numbers (that the average of a large number of trials approaches the expected value), this is one of the things I mean when I say something is emergent.

Central limit theorems and the law of large numbers is also one of the ways dimensional reduction can be achieved. Instead of depending on the complex dynamics of the micro theory (at least N-dimensional with N >> 1), the macro theory depends on a smaller number (dimension) of state variables d << N. An example of this is the ideal gas law. The complex properties of the underlying molecules (micro theory) are relevant only as bulk constants to be measured (macro theory).

Let's shift gears to a couple of different types of "emergence" where the details of the micro theory don't matter as much. Near an equilibrium, many complex theories simplify. In physics, near an energy equilibrium, many theories look like a harmonic oscillator (e.g. a pendulum). This applies from plasmons to pogo sticks. I talked about this more here, but the basic idea is a combination of effective field theory and "Taylor expansions" (actually, effective operator expansions). If your system obeys a symmetry, then it will look the same as any system that obeys that symmetry for small enough perturbations around the equilibrium.

There's one more type of emergence that I talk about: macro degrees of freedom or forces that don't exist unless you have more than one of the micro theory degrees of freedom. Examples include nearly all of solid state physics ("electron holes" don't exist without a filled Fermi sphere and phonons don't exist without a lattice), and all of thermodynamics. Entropy does not exist without a large number of degrees of freedom (enough so that the fluctuation theorem isn't significant impact). Therefore entropic (pseudo)forces -- such as the pseudoforce that smooths out a distribution of atoms through diffusion -- do not exist for individual degrees of freedom. I've talked about how sticky prices (nominal rigidity) may be such an entropic force -- it looks like individual prices aren't really sticky at all, but in aggregate they might be.

...

So here is an incomplete list of emergence mechanisms:

• Unversality classes of phase transitions. Near points of drastic changes from one equilibrium to another, very different complex models can look the same.
• Unversality classes of probability distributions. Complex agents can obey simple equations that are largely independent of the agents when the system is "statistically large".
• Small peturbations around an equilibrium (effective field theory). For example, rational agents are the first order effective theory near an economic equilibrium.
• Entropic forces. Emergent dynamics that derive from a large number of degrees of freedom and entropy maximization that could include things like nominal rigidity.

...

Footnotes:

[1] One complaint Shalizi has about "econophysics" is that this phase transition stuff is just a small part of physics:

Let me also complain that there isn't enough physics: the repertoire of ideas taken from physics is very impoverished. Basically, we see random walks, power laws, and spin systems over and over again.

It's complicated: alternative approaches to economics

Here's an article at Medium on "complexity economics". I am generally sympathetic to alternate approached to economics -- it would be hypocritical not to be. And of those alternate approaches, the "complexity economics" coming from the Santa Fe institute has a lot of interesting ideas. However, the approach seems to suffer from some of the same issues as a lot of alternate approaches. I've taken four quotes and offer a response. Some of these are my own alternate views (derived from the information transfer approach), and some are more mainstream criticisms.

I.

To see how complexity economics works, think of the agents in the economy – consumers, firms, banks, investors – as buying and selling, producing, strategizing, and forecasting. From all this behavior markets form, prices form, trading patterns form: aggregate patterns form.

Duncan Black came up with a great phrase for people who pass off moralizing, ideology, conventional wisdom, and facile analysis as insight: very serious people. I think this term applies to some of those who see economics as a complex system. Sure, it is a complex system. We all see that. But there are three things "complexity" is sometimes used to excuse:

1. That economics is too complex to try to model with math (see here or here)
2. That models should be more complex than the limited (macro) data can support (see here)
3. That models should add in effects regardless of the evidence that such an effect improves the model

Complexity economics seems to be guilty of the latter two (#2 and #3). For example, here's an agent algorithm from one agent based model:

There isn't enough macroeconomic data out there to fit all those parameters and exercise every pathway to justify the conclusion:

In other words, the Central Bank must navigate in a narrow window: too little is not enough, too much leads to instabilities and wildly oscillating economies. This conclusion strongly contrasts with the prediction of DSGE models.

Something closely related to this is the desire to add things to agents regardless of whether they are empirically justified. For example, take the statement that we should think of the "agents in the economy ... as buying and selling, producing, strategizing, and forecasting". However, it seems that a simple backward-looking martingale gets inflation expectations (but oddly not inflation itself) about right. So is forecasting really important?

John List [pdf] asserted that his field experiments demonstrated "a tendency for exchange prices to approach the neoclassical competitive model prediction after a few market periods". However, a random agent model does just as well at describing the data. Sure, we observe that people think about stuff. We experience it ourselves. I am supposedly thinking [or have supposedly thought] about these words I've selected. But is this important empirically? And if a rational agent model is indistinguishable from a random agent model -- and both describe the data -- how is a boundedly rational agent going to fare?

Yes, we know some things about human behavior. But we don't really know if these things we know have an impact on aggregated economic data. Richard Thaler says that behavioral is only recently catching on -- after more than thirty years. Maybe that's because behavior doesn't have a large effect. And maybe some of the behavioral "cognitive biases" and deviations from "rationality" we see are just due to the wrong model. Maybe rational agents are emergent from irrational agents.

In short, until you figure out what works, you don't really know what works. Therefore you shouldn't assume that just because you observe it, it is important. A simple example: just because you observe that a pair of dice are purple with gold dots does not mean that the colors purple or gold are important to the distribution of dice rolls.

II.

Conventional economics asks how agents’ behaviors (actions, strategies, forecasts) would be upheld by – would be consistent with – the aggregate patterns these cause. It asks, in other words, what patterns would call for no changes in micro-behavior, and would therefore be in stasis or equilibrium.

This seems to be a Nash equilibrium, but as Noah Smith says the idea of equilibrium seems to have lost  its meaning entirely:

So how about “equilibrium”? The word used to refer to a situation where prices adjust in order to clear markets, so that supply matches demand. Later, game theorists came up with “Nash equilibrium,” named after mathematician John Nash, which refers to a situation where everyone is responding optimally to everyone else in a strategic situation. Other concepts proliferated, and so by now the word has lost all meaning entirely. When economists say “equilibrium,” what they really mean is “any solution to any equations I decide to write down.”

Therefore contrasting the complexity economics approach with the conventional economics approach via "equilibrium" (e.g. "We knew we wanted to create an economics ... where the economy was always forming and evolving and not necessarily in equilibrium.") really doesn't pin it down at this point.

The information transfer framework defines a new kind of equilibrium (information equilibrium) that closely matches the original definition where supply matches demand (in this case, the information entropy of the supply distribution matches the information entropy of the demand distribution). However, the framework allows for "(information) non-equilibrium", i.e. non-ideal information transfer where information is lost in the market.

III.

The standard, equilibrium approach has been highly successful. It sees the economy as perfect, rational, and machine-like, and many economists – I’m certainly one – admire its power and elegance. But these qualities come at a price. By its very definition, equilibrium filters out exploration, creation, transitory phenomena: anything in the economy that takes adjustment – adaptation, innovation, structural change, history itself. These must be bypassed or dropped from the theory.

One of the problems I have with alternate approaches to economics is that they seem to assume a bunch of specific effects have been left out but don't provide any evidence that those specific effects are important. The difference is between saying "this model of the Hydrogen atom leaves out the fact that the nucleus can move" and saying "this model of the Hydrogen atom leaves out effects on the order of ~ m/M".

Why is this important? Here's an example. The naive neoclassical model of employment predicts an employment rate of 100%. The labor market clears. That isn't true exactly, but it's a really good approximation!

The matching model or natural rate hypothesis represent tiny corrections: instead of 100%, you have 95%. And the fluctuations are on the order of a few percent, so all of these transitory and adjustment phenomena are really just perturbations to the neoclassical model. 

While the deviations from rational agents and perfectly clearing markets are "small" from a theory standpoint, they tend to get a lot of attention. News doesn't tell stories about how supply met demand again via the price mechanism, and real people are hurt by sustained unemployment. But whatever your alternate theory is, to first order it'll give neoclassical results. How often do you go to the grocery store and find they are out of most of the things on your list? (If your answer is frequently, you are probably going to Trader Joe's, which doesn't really count as part of the free market.)

In the information transfer framework, I've started with leaving out nearly every aspect of human behavior -- that humans are effectively unconscious atoms wandering an economic state space. That this does a good job at describing economic field experiments tells you something about how important specific behavioral, transitory and/or adjustment phenomena are.

PS Here is Noah again on rationality:

Then there’s “rationality.” When some economists say “rational,” they mean that people simply pursue their desires. Others use the word to refer to a modeling technique called rational expectations, which says that people’s beliefs about the world match up with the economic model itself. Still others use it to mean Bayesian rationality, which is a way of revising one’s beliefs after considering the evidence.

IV.

Instead of assuming agents were perfectly rational, we allowed there were limits to how smart they were.

Again, assuming they are not smart at all leads to a similar result to assuming they are perfectly smart, both of which compare favorably to date. So is bounded rationality important? 

...

Maximum entropy distributions (reference)

I've mentioned maximum entropy distributions and constraints in a couple of places before, but I think it's been mostly in comments. There's a handy table on Wikipedia that gives the maximum entropy ("least informative") distributions given various constraints:

It's a bit long, so that isn't the whole table. The basic idea is that if you know something about your system, such as its mean and variance, you can look up the maximum entropy distribution that corresponds to that knowledge (a normal distribution). If you know the expected lifetime of something, the maximum entropy distribution is an exponential distribution with a parameter that is (one over) that expected lifetime. The Pareto distribution is the maximum entropy distribution where you can measure an average value of log x. This is useful for systems that cover several orders of magnitude, like incomes. That incomes seem to be drawn from Pareto distributions (at least out in the tails) should immediately make you think that the process that produces them doesn't depend too critically on the microfoundations. If someone from the top 1% of income or wealth says their income or wealth depends on their hard work, you can respond: Then why is the distribution of incomes in the top 1% seem like it is Pareto distributed?

One quibble with this chart is that the uniform distribution is the maximum entropy distribution given a constraint that the random variable has a maximum and minimum value (continuous) or covers a finite number of states (discrete). The former is useful for e.g. budget constraints in economics -- you can spend up to X dollars means a probability density of 1/X -- a minimum of 0 and a maximum of X.

List (2004) field experiments with random agents

As I mentioned in this post, I am attempting to run some simulations using random agents and comparing with the results of List (2004) [pdf]. I've attached my Mathematica code at the end of this post.

In the "symmetric" case (called S12 by List), he gets average prices under 14 plus or minus 2 dollars, averages about 7 sales, with the total buyers' profit of between 14 and 20 dollars and total sellers' profit in the same range:

Here are my results for random agents averaged over all five "market periods":

My random agent model is largely in line with List's field experiment. One advantage the simulation has over the field experiment is that I can run it 1000 times. If you do that, you can see the biases between the results of List (2004) and the random agents. The random agents appear to get one to two additional sales (the higher profits and variance seem to derive entirely from this fact):

There are a couple of possible ways to explain this. My first reaction was that maybe I was seeing the endowment effect in action: sellers would value their good a bit more than their "seller's card" told them to because they had the good in their possession. The effect would be fewer sales than the utility (reservation price) on the cards actually showed. However, the better answer is probably that the original experiment was timed.  My random agent model proceeds until no more sales can be made (the highest reservation price of the remaining buyers is below the lowest reservation price of the remaining sellers). In the field experiment (and the random agent model), it takes longer for the last possible buyer and seller to find each other and negotiate (randomly choose) a price. This is the basis of matching models. As you can see in the graphic above, removing one sale and the average profit from one sale from the results brings the List (2004) and the random agents closer together.

I think this is pretty powerful evidence that experiments like List (2004) don't really demonstrate the claim that there is "a tendency for exchange prices to approach the neoclassical competitive model prediction after a few market periods" (per List's abstract). The fact that agents programmed with reservation prices making random offers after randomly encountering each other produces an almost identical result means that something like Gary Becker's irrational agent model could explain the data just as well -- if not better -- than the neoclassical model.

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Appendix

One thing to note is that List's Table 2 actually just assigns a random permutation of the same set of reservation prices to the buyers and sellers in the experiment (i.e. good experimental procedure). Instead of doing this, I just randomized the first round buyer reservation price data from List (2004) each time a round was executed.

The Mathematica code:

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Update 1 September 2016

Causal entropic forces are also relevant -- it explains the tendency to move towards the equilibrium, not away without resorting to it being random chance as above.

Simulations with supply, demand and prices

One thing I wanted to clarify is that when I said in this post that increasing supply increases prices, it doesn't always increase prices. A great example (or natural experiment) is the case of Magic cards:

The first thing the company had to do was to bring the price of the cards back down, so the average person could buy them again. They did this by dramatically increasing the supply.

The key factor is the speed of the increase. If the increase  is such that demand can catch up (e.g. a slow increase, as for housing), then you get a steadily rising price. Here's an example (here the IT index k = 1.3, so demand D ~ S^1.3 and P ~ S^0.3) of the "general equilibrium" solution:

And now here's the same thing, but adding a faster increase in the supply for a short period:

The brief period of rapid increase in supply causes a dip in the price ("partial equilibrium"), which then returns to the trend increase.

For more details on general and partial equilibrium, you can see the paper.

People skills

As Noah has shifted from criticism of economics to defense of the status quo (becoming part of it as an Assistant Professor at Stony Brook [nice school; did a talk there on my postdoc search]), his defenses have gotten a bit more strained (e.g. here). Take this for example:

More tech companies are hiring economists: https://t.co/2cTDUVeool Because there's lots of econ that actually, you know, *works*.

— Noah Smith (@Noahpinion) April 8, 2016

But according to the article:

Much of [economists'] value to tech firms is in helping to connect the engineering side with the business side. 

“Economists are trained in the intuition and goals of business, but they are also comfortable with data, statistics and the technical aspects of running a business,” Athey said. “A successful economist is multilingual.”

Which makes me think of this:

Supply and demand experiments

There is a new working paper on field experiments in economics from Omar Al-Ubaydli and John List. I am in the process of looking into this literature review, but after 1) figuring out the List (2004) that the only figure comes from is neither of the List (2004) papers cited in the references and 2) pulling out the reservation prices from the correct List (2004) and coming up with this histogram [1]:

... I decided to write my first Tweetstorm.

I pulled some quotes from the working paper as well. These two are consistent with the information transfer framework:

"Chamberlin found that prices were volatile, and that they converged to a below-equilibrium price, implying less-than-fully-efficient exchange." 

"In particular, many behavioral anomalies that were regularly detected in laboratory experiments, such as the endowment effect (Knetsch, 1989)"

And this is what I was commenting on in this post:

"the key advantage offered by the laboratory experimental methods pioneered by Vernon Smith was the ability to artificially control how much traders valued the commodities being traded (known as inducing values), as this allowed researchers to accurately estimate demand and supply schedules ..."

Anyway, there will be more to come.

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Footnotes

[1] I swear this is just John List is giving us the finger.

The mathematics is not the issue here, Dude

Wishy thinking.

I had a conversation on Twitter with Mike Sankowski, who thought I might have misread an essay at Aeon about how economics was a pseudoscience obscured by math, and perhaps I did.

I had linked to an earlier post I had written about using math. My point was that pseudoscience can happen with or without math. Therefore it's not the math that's the problem -- it's the wishy thinking, ideology, or unquestioned assumptions.

But after my conversation with Mike and a re-read of the article I realize there was a thread that I missed -- because I don't see math that way. Here are some quotes from the author as well as quotes from economists selected by the author:

But the ubiquity of mathematical theory in economics also has serious downsides: it creates a high barrier to entry for those who want to participate in the professional dialogue, and makes checking someone’s work excessively laborious. Worst of all, it imbues economic theory with unearned empirical authority. 

... mathematics in economic theory serves, in McCloskey’s words, primarily to deliver the message ‘Look at how very scientific I am.’ 

Krugman named economists’ ‘desire ... to show off their mathematical prowess’ 

When mathematical theory is the ultimate arbiter of truth, it becomes difficult to see the difference between science and pseudoscience

...

I take it you've read through those quotes. Now change math to something challenging to learn but that you've conquered in your own field and re-read them. Try putting French, oil painting, or drafting in the place of math. 'Look how very artistic I am.' 'desire ... to show off their French prowess'. English probably creates a higher barrier to entry for much of the world to participate in the economic dialog than math.

The thing is that if you know math, none of these things are true. If you know math, it doesn't imbue things with empirical authority. If you know math, checking work isn't excessively laborious. If you know math, it isn't difficult to tell the difference between science and pseudoscience.

If you know math, the people you might want to impress with your mathematical knowledge probably also have that mathematical knowledge. I can assure you that impressing a high school student with my math skills doesn't give me a sense of pride. Teaching one how to use math does. No one who knows economics-level math is going to be impressed with your economics-level math. I have never been impressed by math [1], but I have been impressed the insight math communicates. If you have an insight into human nature, I am impressed with the insight, not the vocabulary you express it with. Anyway, any time hear people say economists try to impress people with their math skills it makes me chuckle. Those skills could only be impressive to people who don't have them.

The insight here is that math is seen by the mathless as 1) a barrier to entry, 2) a pure signalling strategy, 3) difficult, and 4) a veneer of respectability, empirical accuracy, etc. It's not seen as legitimate or necessary. This is belied by this quote:

Fortunately, non-experts also participate in the market for economic theory.

Imagine the variants:

Fortunately, non-experts also participate in nuclear reactor design. 

Fortunately, non-experts also participate in commercial aircraft design.

The difference is that the educational and experience barriers to entry in the latter two are seen as legitimate. Why the difference?

I came up with an a good analogy ...

When technology is applied to cell phones, it's seen as a gee whiz factor (even if they monitor your GPS location), but when technology is applied to voting (e.g. the Diebold voting machine controversy in the US) it's seen as a barrier to transparency.

There are two factors here. First, we intuitively understand how voting works (or at least think we do). Second, we see the inner workings of voting as more important than the inner workings of a cell phone.

Because we feel we should understand how voting works [2], and it's important to us, technology is a obfuscating barrier. Because no one cares how a cell phone works [3], technology is seen as a wonder.

I think this is what is happening with math. Everyone thinks they should intuitively understand economics, and money is important to them. Therefore the math is an obfuscating barrier. No one thinks they should understand quantum field theory, and it's results don't impact our day to day lives much, so math is just seen as part of the wonder. I think that's why physics blogs are different from economics blogs.

The technology and the math are not the issue here. It's the gulf between the desire to understand and the capacity to understand something important.

The thing is that mathematics is behind some of the greatest advances in understanding in physics. And sometimes the math came before the intuition. Newton left some of the calculus out of his book because he felt more people would understand what he was talking about if he just used trigonometry. But Newton understood it in terms of calculus. Heisenberg confused everyone with his matrix mechanics, but it got the answers right. Quantum mechanics became more broadly accepted when Schrodinger showed how it works with differential equations that were more commonly used in physics at the time. However, most modern physicists understand quantum mechanics in terms of matrix elements (Dirac showed how the two fit together). Einstein's work led to tensor fields and differential geometry becoming a bigger part of physics. [Update: see comments below. Einstein's insight into general relativity came from Minkowski's mathematical representation of special relativity as a 4D space-time.]

In these cases, it was the lack of understanding of math that was the barrier to initial understanding. Later on, when things became well understood, quantum physics became the subject of popular books. I imagine that if some quantum device was to be used to encrypt your bank account information in the 1930s, people would have been up in arms about physics being just a veneer of respectability over some kind of Ponzi scheme. And that's the crux: economics isn't well understood, so it's not yet amenable to transparent talk and clear diagrams. But it deals with employment and money, so it's important to people.

It's completely understandable that people are angry and want to forego the math to see what's really going on.

...

PS I have a solution: nihilism. Macroeconomic policy doesn't seem to be that important to actual outcomes according to the information equilibrium framework, and most of the macroeconomics coming from the pros seems to be wrong. If you find the math to be obfuscating, just realize that if you were to get through it, there's not much you're missing out on in terms of policy-relevant knowledge. If you find the math to be obfuscating, just realize you can ignore macroeconomics and expect zero impact on your life.

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Update

A response that gets into Wittgenstein and Plato from Tom Hickey.

Also, I think I should have kept to solely The Big Lebowski references instead of combining them with ones from The IT Crowd.

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Update 12 April 2016

Noah Smith has weighed in favorably on the Aeon article, so you might be interested in an different take. While I agree with the points made by Pfleiderer and about Lucas, neither have anything to do with math (Pfleiderer's point about chameleon models is Holbo's two step of terrific triviality and Lucas tried to evade empirical discipline, respectively). And my opinion of Paul Romer's mathiness claim has been made known here. Put simply economists do not understand limits in the context of extant reality.

The subtitle of the article reads:

By fetishising mathematical models, economists turned economics into a highly paid pseudoscience

However it has nothing to do with math, but rather politics and uninformative data.

...

Footnotes

[1] At least in the service of real world applications. People who are good at pure math still amaze me. Take Terry Tao for instance. My math skills, meager as they are compared to the likes of most theoretical physicists (part of the reason I didn't go the postdoc route), nearly entirely derive from my intuition about the physical systems the math represents. If I understand the system, the math follows. If I don't, the math is hard. It's really like any other language. If I know what I'm talking about, the words flow easily. If I don't, then they don't.

[2] I say feel because many of us don't actually know how it works. In presidential elections there's the whole elector business. But even in Washington state, there were people who were going to mail in their ballot for the primary for the Democratic nominee and not attend the caucus. The primary doesn't count for Democratic delegates in Washington.

[3] One of my favorite facts is that the GPS in your cell phone depends on Einstein's theory of general relativity to work, which is behind the accurate predictions of the big bang theory. Couple that with the knowledge that I'm sure there are young Earth creationists who use the GPS on their cell phone.