From Krugman’s blog today, via Deane Yang’s FB feed:

Math is a friend of mine. There have been a number of occasions in my life when doing the math on an economic model has led me to conclusions very different from my preconceptions.

But I have always been able, after the fact, to find a way to express in plain English what the math is telling me. If you resort to math to justify what looks like a very foolish claim, and you can’t find a plausible way to express that justification in plain English, something is wrong.

I disagree. The reason we use mathematical formalism is *exactly because* it expresses things that can’t be said precisely in English, or any other natural language.

We can, should, and do utter English sentences that *paraphrase* mathematical assertions; but that’s not the same thing.

Possibly useful analogy: “Music is a friend of mine. There have been a number of occasions in my life when a piece of music has conveyed to me a powerful emotion or sensation. But I have always been able, after the fact, to express in plain English the way the music sounded and the way it made me feel.”

If someone told you this, you would say “NUH UH,” and I think the same response is due Krugman here.

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I’m not sure I buy the analogy. Music doesn’t have the property that if one plays one note incorrectly, one ends up with a completely incorrect emotional response at the end of the piece. I think all that Krugman is arguing against is conclusions formed by following a mathematical computation blindly, without having the accompanying conceptual understanding to keep the argument robust against errors or breakdown of hidden assumptions.

I’m not convinced that’s what Krugman’s arguing, but if he is, then I agree with him.

What you’re saying is is more like “English sentences and mathematical formalism must both be present and must bolster each other, each using its particular virtues.” I read Krugman to be saying that the mathematical argument, once constructed, can be

replacedby something in “plain English.” Music and mathematics are different in many ways, including the one you mention, but are alike in their insubstitutability.Since I expressed support for Krugman’s post, I obviously disagree with you. Although I’m probably oversimplifying, I think a key difference is whether you are doing mathematics or you are using mathematics as a tool in another field such as economics or finance.

For us mathematicians the power of the language and notation is crucial to our work, and we certainly wouldn’t want to give any of it up. But even there all of us have encountered people who mistakenly believe they understand something only because they have memorized the language and notation. We often determine whether a student understands something or not by trying to get them to express their knowledge in “plain English” as well as mathematical notation.

But when mathematics is used as a tool, things get a little more complicated. There are the mathematically skilled who actually build the models, and there are the people who need to use the models. I believe that a model is useful and effective only if the mathematicians who built the model can explain the behavior of the model in terms of the basic principles of the application domain (finance, economics, physics,…). In practice, a model is implemented numerically and analyzed by feeding into it different possible inputs and studying the corresponding output. A model is not useful unless you understand what you observe, not just in terms of mathematical formulas but in terms of ultimate meaning of the input and output parameters. And this you should be able to explain in plain English to someone who understands the application if not necessarily all the mathematics.

Andy Kalotay and I developed a valuation model for mortgage-backed securities. Over the years, we have often been surprised by the behavior of the model, but always found clear financial explanations for the unexpected behavior. We now tell people that we know a model is good, if you learn new insights about the thing you’re modeling from the model itself. That’s of course the standard criterion for good models in physics but is rarely cited in finance.

Nobody in finance and economics should ever say, “trust the mathematics” or “trust the model”. As far as I’m concerned, that’s one of the reasons why Wall Street got itself into so much trouble.

From what little I know about traffic modeling, I understand that various mathematical models for what happens in traffic flow sound counterintuitive when translated into “plain English”, but accurately predict what real drivers do.

I like Krugman a lot, but he’s not perfect and I’m not so interested in determining what he might mean by his principle in general. On the other hand, he does address a valid concern in economics, the same concern expressed by Serge Lang in his attack on the political scientist Samuel Huntington. Namely, that it is easy for scientists to get caught up in formulas and lose track of what’s actually important in their field. Also, in the specific case at hand, I think that Krugman makes a fair point.

Krugman’s real beef is with fellow Nobel Laureate Robert Lucas. In a speech in 2009, Lucas said that stimulus spending should be divided into two parts, creation of new money and extra government activity. His advice to the government is to implement the first part, because the economy needs it right now; and skip the second part, which in his view doesn’t help the economy. In his critique of extra government activity, he asked whether it would be worthwhile without creation of new money. He doesn’t think so, and he made some cryptic comments that sound like Ricardian equivalence. Wikipedia tells me that it’s the principle that people won’t be moved to spend a tax cut, or a suppressed tax increase, if they think that taxes will be higher later to compensate.

Krugman, as is well known, wants much more stimulus spending. He wrote some vitriolic blog posts about Lucas. He said that Ricardian equivalence is grossly overplayed, and that Lucas’ reasoning doesn’t work with or without Ricardian equivalence. In response to that, David Andolfatto piped in not to say that Lucas is right — in a postscript he said that he doesn’t truly think that Lucas is right — but that Lucas is right in a certain kind of idealized model of the economy. Moreover, Andolfatto argues that in this thought experiment, Lucas is right without Ricardian equivalence.

As I said, I think that Krugman makes a fair point. It looks like Andolfatto is talking about something much less important, professional decorum and idealized mathematical models, than what Krugman and Lucas are talking about, which is what the federal government should actually do. What’s also strange about this contrast is that Krugman and Lucas are ivory tower faculty, while Andolfatto is a Vice President of the St. Louis Federal Reserve. (Although admittedly Andolfatto used to be university faculty as well.)

On the other hand, I am interested in the general issue of formulas vs “plain English”, even though I’m somewhat less interested in Krugman’s stand on that general issue. I have to say that it’s a big cliché in economics, and also in physics and probably every area of science, to accuse the other side of losing track of ideas in favor of formulas. Even though it’s also a valid concern. At some philosophical level I mostly agree with Jordan. We need formulas as an extension of English to properly express modern ideas.

I just have two reservations. One is that the best human thinking can’t only come from formulas; there is always also a good supporting explanation in prose. Likewise with music (or any form of art) I do not like the position that art just speaks for itself and that you should only listen to music with no supporting explanation.

My other reservation has to do with the way that the cliché accusation is used by scientists. Even though the accusation is not completely unfair to formulas and models, it is unfair to mathematics in the broad sense and to mathematicians. I wouldn’t want to play into the cliché by arguing that formulas are better than prose ideas after all. Instead I would point out that of course mathematicians also care about conceptual ideas, and that there are many important math papers that have a lot of prose and not all that many formulas. There are also math papers that are weak as mathematics precisely because they are too formal. Mathematicians are are also human and we face the same tension between human thought and formalism, just like other scientists.

I strongly disagree with Greg’s comment that what the federal government should actually do is more important than idealized mathematical models.

A particular action of the federal government is ephemeral. In a thousand year, no one will remember it. A theorem is forever.

I didn’t mean to compare federal policy to all mathematical models of all kinds. The issue here is whether economists should talk about the economy, or whether they should talk about some abstract models that aren’t really all that exciting as mathematics. Yes, I believe in the idea of doing mathematics for the ages — what other defense do I really have for my own research. But I also believe in specialization of labor. If an economist wants to do mathematics, then more power to him, but it should be important as mathematics. He also shouldn’t call it economics unless it’s also important as economics.

I’ve met more than a few economists who want to be doing mathematics.

It may not pass your standards of beauty, but some of it is nontrivial. Basically, the way they operate is that they prove theorems about or at least study numerically various simple models of the economy. About these models, they can say what is going on with absolute (or near absolute) certainty. It is also possible to run one’s mouth off about what the federal government should do. That is something which it is does not seem to be possible to say anything about with certainty. There are controversies and one can’t do both sides of the experiment. The hope of my friends who are economists is that gradually they will know more about all kinds of models and be able to reach conclusions about whatever kind of model the economy is. In mathematics, we view as more important the things we can actually prove over the things we only conjecture. It is a shame if division of labor makes this impossible for other disciplines. What we know for a fact about the role of Ricardian equivalence may be much more important than what two Nobel prize winning economists happen to think the federal government should do right now.

I think the whole question depends very much on how much you trust your models. If you are very confident of them, then you can let the mathematics take over, however surprising the results. Quantum mechanics is one of the best examples of this. But in economics you have to make so many simplifications that the can-it-be-explained-in-plain-English test is (I would as a non-expert imagine) a useful one. It’s basically a Bayesian idea: if you get a highly counterintuitive result from your model, then if the prior probability of the model’s being correct isn’t close to 1 the result is probably a sign that the model is not correct, whereas if the probability is close to 1 then the result is probably a sign that something counterintuitive is indeed going on.

I also don’t buy the music analogy because music isn’t about justifying statements.

I have nothing against an abstract economic models in and of themselves. I agree that they sometimes lead to interesting mathematics, and I agree that there is some intersection between significance in economics and significance in mathematics. It’s also not as simple as that a model is “right” or “wrong”. An abstract model is not just a topic in mathematics, but also a tool that a scientist can keep on the shelf for some future purpose. For instance, I’ve studied square ice. As a statistical mechanical model of real ice, the model is obviously “wrong”. It nonetheless deserves study not just in mathematics proper but also in mathematical physics.

That said, there is a bias in science in general in favor of interesting questions over important questions. Sometimes this bias is healthy. Arguably in pure mathematics it’s always healthy and it’s a reason that I like pure mathematics. But sometimes it leads to a disconnect between science and its potential utility.

An example that is personally important to me is pinky RSI. There has been a lot of study over the years of the ergonomics and performance properties of QWERTY vs Dvorak keyboards, or modern keyboards that are adapted to the shape of the human hand in more radical ways. With the result that the ergonomics field largely missed an enormously consequential, entrenched decision: The assignment of 16 computer keys, fully 1/4 of a standard keyboard, to the right pinky finger. There is too little incentive to write a research paper about this standard, because it’s a no-brainer that it’s a bad idea. Unlike the problem that Dvorak solved, it isn’t an interesting question in combinatorics.

I see that Krugman asserts

“Mathematical explanation” + “good explanation” => “English explanation”

and (as a corollary)

“Mathematical explanation” + “no English explanation” => “not a good explanation”

but I don’t think Krugman asserts

“English explanation” + “no mathematical explanation” => “still a good explanation”.

Of course, depending on how much we discount the future importance of the theorem, arguing about policy might still have a higher present utility. :D

A few more remarks:

1. I don’t know economics very well, but I agree with a lot of the discomfort expressed above about economists trying too hard to do mathematics and then claiming that what they’ve proved has something to do with the real world.

2. I find financial models (e.g., Black-Scholes) much more convincing, because they are much less ambitious, more easily tested. It is much to understand what the assumptions are and quantify how they fall short of reality.

3. It should be emphasized that good models cannot be built without a rather solid grasp of the underlying mathematics, including the language and logical rigor. I have never found anyone who knows only the “plain English” explanation of things and is able to develop a model or understand properly how a model works. I have, however, encountered people who don’t know the math but know the application well enough to validate or find flaws in a model by using it as a black box. It is in this sense that ‘plain English” is needed, because such people need to be taken quite seriously (they have real experience, say, trading bonds and know what happens in different circumstances). If you can’t convince such a person using “plain English” that your model is modeling honest real phenomena and no spurious phenomena, then that person is justifiably skeptical.

4. I have to say that my experience working with non-mathematicians on Wall Street has given me a very different perspective on a lot of things, and it has radicalized my views on many things related to math, especially how we teach it. I don’t think most math teachers understand how to teach math properly as a useful tool, as well as a body of rigorously derived knowledge, to students who do not have the same level of math skills as professional mathematicians.

5. I also think mathematics has been badly misused in Wall Street (for example, in a failed effort to quantify risk exposure), and this was a not insignificant factor in causing the bubble and the terrible aftermath. I believe that this was partly due to mathematicians focusing too much on the mathematics of their models and too little on the “plain English” interpretation of the models.

Nets, you were obviously being flippant here. I still like your comment, because it expresses exactly why I find myself unable to give up my career as a research mathematician for a more practical career.

I would agree with Gowers here. If there is any truth to what Krugman is saying, it is because economics is still a very soft discipline, in which mathematical arguments are often wielded at least in part for reasons of prestige. We are not in a realm where assumptions are actually being tested against data.

At the same time, what economics may need is checks against reality, rather than checks against verbal language and conventional wisdom.

Yet again – if there is one thing that bothers me the most about the way some economists (not Krugman) write for the general public is how they use unsupported paradox to create the impression of a brilliant argument: :”Naive idealists, unaware of the law of unintended consequences, may think that being infected with cholera and the pest will affect their health negatively. In fact, economists, looking at the figures, know that

the very opposite is true. [New paragraph, change in topic]“Is .95 plain english? Or is it 95%? Or ninety-five percent?

What about “one plus one equal two” or “…zero, modulo 2″? And “topological groups which are manifolds are Lie groups”?

More seriously, it may be useful to distinguish “plain english” from “simple”, though the two are probably highly correlated in practice. We may assume complex explanations are better expressed with efficient/mathematical notation/language.

The question of how complex our explanation should be has various aspects. First how short can a proof be made? This is proof complexity. And the added complications that even if proof-length and psychological complexity correlate, this may not be perfect, i.e. some short proofs may be hard to remember.

Also from complexity theory we have the concept of interactive proofs and PCP. Under some circumstances we have relatively short/simple proofs, if we are convinced by true proofs and by some false ones too (\eps-many ones).

Oracle Turing machines are also relevant to modeling belief, the verifier may consider certain facts coming from experts/oracles as true.

In particular perhaps the “plain english” proofs/explanations that Krugman has in mind are “by fiat”: an oracle just computes the solution and accepts/rejects accordingly, or we assume the prover’s statement is always true and the verifier always accepts.

A nice point that Krugman makes is that explaining should be easy “after the fact”. So an apparent paradox is resolved: researchers must still use mathematics, difficult reasonings, to communicate to one another, if only to be able to research by themselves. There is a tension between taking a single decision, i.e. trusting a proof, and developing the capability to find the proof, “understanding it”. We believe searching is harder (P!=NP), so requires more complication, therefore more mathematics.

The issue of understanding is not just for academic researchers, people often want to understand, and they develop trust in complicated manners, based on this and other factors.

Then game theory is basic to model how trust is developed, how much complexity we should give to our explanations, and eventually how to build our political systems, which are based on optimizing decisions with distributed knowledge and computational power, and individual interests themselves depending on individual understanding. (The currently hot economic theme of asymmetric information comes to mind.)

All this to say that simplifying to “mathematics versus plain english” is perhaps misleading, it may be better to think of a whole range of possibilities -with various dimensions/aspects.

PS: The complexity perspective also applies to music and arts, a picture, or a piece of music contains alot of information, processed by highly specialized brain structures, and it seems reasonable that translating to english may be infeasible. But this is certainly an interesting issue: c.f. Scott Aaronson’s student Andrew Drucker on mental arithmetic using visual pattern recognition, http://people.csail.mit.edu/andyd/home.html , and the whole theme of approximating continuous by discrete, and then compressing that information.

The preceding debate has made no reference at all to that fact that economics is an attempt to explain the aggregate behaviour of large numbers of human beings. Its ability to do this is clearly pretty limited, if the failure to even imagine the current economic crisis happening, never mind to develop policies to deal with it, is anything to go by.

The maths on its own is pointless – it’s only useful when it’s able to develop workable theories about how human beings respond to their environment. Economists’ success in doing this is so far patchy at best.

To quote John Adams: “I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce, and agriculture, in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain.”

Of course, it is not clear what the termination point of this iteration process is. From empirical data, Youtube videos of cats may be involved.

Okay here are two examples. The actions of Alexander the Great were ephemeral concerns of war and politics. But the general economic principles laid down by his contemporary Xenophon in his works such as Oeconomicus and Cyropaedia are immortal knowledge! :-)

Given the arms-length relation of contemporary punditry to reality, I think the successor to porcelain might be politics again.

but the union of the Greek and Persian world brought about by Alexander gave rise to Ellenism, which strongly influenced European society

I believe the confusion arises from his use of value-laden terminology. The phrases “very foolish claim” and “plausible way” are unclear: is it “foolish” because it is counterintuitive? Empirically unfalsifiable? Already known (for all intents and purposes) to be untrue? What does he mean by “plausible”? What Krugman believes to be “plausible” may not be what I believe to be plausible.

I apologize, but I must take issue with your assertion that “the issue here is whether economists should talk about the economy, or (…) talk about some abstract models that aren’t really all that exciting as mathematics.” If you are asserting that (academic) economists should only examine issues related to the macroeconomy (inflation, unemployment, government spending, etc.), then you are misinformed as to what economists do. Most economists would agree that economics could be defined as the science of human choice under scarcity. From this lens, economics can give insight as to why couples choose to divorce, why people choose to have children, and other topics that are only tangentially related to the economy.

Abstract modelling is what economic theorists spend their time doing. Almost all theorists never throw a single dataset at their models (in fact, many models are too complex to have numerical solutions at all), and are content with just generalizing them by relaxing assumptions. Whether or not they are exciting as mathematics to mathematicians is not really a concern of economic theorists. The economist creates an ice sculpture, the mathematician sees H2O.

There’s also the issue of theoretical econometrics, which many economists spend all their time doing. There isn’t anything to do with the “economy” there either, and I doubt the work there impresses mathematicians.

You have the definition backwards. Modern economics looks at individual choices; modern macroeconomics aggregates these individual choices, hence the term “microfoundations”. You are also grossly misinformed as to the field’s successes and failures. Moreover, governments typically do not listen to academic economists, since ideology trumps logic almost every time when it comes to politics.

If you want to inform yourself, see “How did Paul Krugman get it so Wrong?” by John Cochrane. Google it.

Yes, that might secretly have been my point.

[...] If you read the report, which I haven’t had time to really do yet, you will notice how few equations there are, and how many words. I’m not saying that you need equations to explain math, but it sure helps when your goal is to be precise. [...]

I’m not sure what you mean when you say “modern economics looks at individual choices”. Certainly, many leading papers in macro (especially in growth, but in other fields, too) start by assuming that each sector or even the aggregate productive sector is Cobb-Douglas. While many agree that microfoundations are important, the modeling approaches of various fields are pretty ecumenical and still involve plenty of “macro” assumptions, which is fine.

And, of course, the ultimate goal — at least of many economists — /is/ to gain a better understanding of aggregate behavior of economies and societies. Do you really take issue with that aspect of Jonathan’s claim, AE?

I read Cochrane’s tirade and I feel perplexed rather than informed. At many points, I’m not sure whether to feel that he’s wrong or just that I don’t understand him. I’ll be generous and call it the latter. He discusses economics in a clipped style that doesn’t work for me. To be fair, Cochrane understandably felt provoked by Krugman. Maybe some of his writings that aren’t angry rebuttals are easier to understand.

For what it may be worth, Krugman has another blog post that fills out some more of his views on the relationship between mathematical models, verbal reasoning, and economics, in particular saying some things which are complementary to the quote that started this discussion.

Great catch, Mark. This one makes it sound like Krugman’s view is not actually very different from my own. (Which is meant to be evidence for my view, not his!)

Agreed.

Unbelievably good discussion, it’s fantastic that we have Fields Medal winners in mathematics commenting on Krugman’s thoughts on explaining in English vs explaining in math.

I agree with plm above that “simplifying to “mathematics versus plain english” is perhaps misleading, it may be better to think of a whole range of possibilities -with various dimensions/aspects.”

In general popular highbrow popular journalism, perhaps most egregiously the New Yorker, seem terrified to imply that a simple model with a little algebra and a few graphs might be necessary to get an elementary understanding of some of these issues. A sad commentary on the current consensus of what it means to be educated. Calculus is almost 400 years old but we can’t depend on a current citizen to be comfortable with graphs and algebra.

In the preface of his textbook, “Macroeconomics” MIT economist and IMF chief economist, Olivier Blanchard, writes that he makes arguments in three ways: algebraically, graphically and in English, “stating the intuition behind the results.” I think this approach deepens understanding and prevents gross errors.

Blanchard also writes that “I .. never present a theoretical results without relating it to the real world” and “macroeconomics is not an exact science but an applied one where ideas, theories, and models are constantly evaluated against the facts and often modified or rejected. His actions and those of Krugman and Delong show that they agree that economics is an applied science, important because it helps us deal with pressing important problems. Those economists who differ and expect the public to fund “pure”, “unapplied” economics should not be surprised to find the public reluctant to do so.