Weinstein v. Krugman v. Orzel (Mathematical Elegance Death Match)

I strongly disagree (not even having read Krugman's essay!). It is absolutely not "anti-scientific" to say that we need to back away from looking for a clean mathematical theory of everything. Science isn't about mathematics -- consider biology or sociology. Economics doesn't have to be about pure mathematics. It needs to be connected to reality.

Freshwater economics especially is far off track. I don't know how to fix it. It is like string theory, if you believe Smolin's account. If you are a computer scientist, imagine if a group of theorists broke off and started considering some crazy computational model that has nothing to do with real computers but was just mathematically elegant. They cite each other, hire each other, market themselves well to politicians with buzzwords and phrases like "exponentially faster," "interdisciplinary" and "end of Moore's Law." How can you put a stop to this collective delusion? Whole departments have been subverted already.

If you want to be an academic imperialist, you need to know what you are talking about. Physics tools aren't going to resolve the problems of economics any more than they do biology. Searching for a clean theory of everything for economics is many, many times more difficult from what the physicists are trying to do. Physicists are free to look at smaller and smaller scales, whereas economists are stuck at the macro scale: big enough to be complicated, but too small for statistics to work perfectly.

The problem is closer to string theory not agreeing with experiment, and the string theorists arguing that experimentalists are not real physicists and should be ignored. It has been said that the Chicago School of economics does not consider econometrics to be real economics because it does not use optimization under constraints. Professor Krugman's point is that building elegant mathematical models on flawed assumptions gives you the expected result. Consider a spherical cow.

I do not think that he is arguing for a retreat from elegant mathematical models, as he creates more of them than humanly possible, but he is arguing that one should be aware of their limitations. Surely this is not a foreign idea to a physicist.

I think there's a lot to be said for the idea that economists weren't blinded by their models - they were blinded by the money. As in, their own income. The economists and bankers in the position to actually do something (and not just talk about doing something) chose to design their risk analysis modeling so as to maximize their own (and their clients, etc) short term profits. Ones that chose to maximize accuracy instead were probably quickly unpopular. It would be like being a member of Bush's intelligence community in 2002 and saying Saddam Hussein was unconnected to 9/11 while everyone else is ginning up a war: your opinion would not be wanted, and soon neither would your services.

There is a long series of comments on Krugman's article running on the Fortnow/GASARCH blog Computational Complexity.

E.g., we have Reinhard Selten's aphorism: Game theory is for proving theorems, not for playing games.

This I take to be the substance of Krugman's critique.

One consequence is that "real" economics, and "real" science too, are increasingly done outside of academia.

That is why (as I wrote on the Fortnow/GASARCH blog):

It is striking that (especially in biology where nowadays the highest-level maths are found) the most prosperous sectors of the academic ecosystem are evolving to resemble Wall Street trading groups ... rookies and veterans mixed together ... learning on-the-fly in a dynamically evolving and highly competitive research environment.

I'm not saying it should be this way ... nonetheless that is now the way that the dynamic sectors of academia are working.

S = Supply
D = Demand
E = Econoimics
GI = Government Interference
C = Chaotic

S + D = E

S + D(GI) = CE

Skullsike, have you have never been involved in any of the (numerous) human enterprises in which unregulated markets lead inexorably to social and ecological disaster?

Here I have in mind enterprises like salmon fishing, drug manufacturing, coal mining, and medicine.

No, I don't want to live in a world in which the seas are empty of fish, the skies are choked with heat-trapping CO2, Florida is underwater, meth addicts fill the streets, the Appalachians are a gravel parking lot, and quack MDs outshout the evidence-based MDs.

... and that's why I don't want to live in a world in which Chicago-school free-market economists have regulatory and political power.

Your assumption of disaster is just that an assumption(or more like rhetoric or unsupportable scare tactics),the system will correct itâs self when the undesirable outcomes start to effect the system itself (and they will), government intervention may seem justifiable but when you have intervention, you have what is called tampering, there will be a chaotic outcome and cannot balance the pure form - the use of a simple Deming red bead experiment will show you the result of tampering by increasing the variation!

Oh by the way I am in the evil Oil & Gas exploration and production!

Skullsike, we can all be grateful ... that ideologues of the free-market persuasion were not ascendant at the Constitutional Convention!

"Checks and balances? We don't need no stink'n checks and balances!" :)

Seriously, isn't it striking that the rationale for the American Constitution, as set forth in documents in the Federalist Papers, makes little or no reference to economic or religious dogma? ... these documents, and the political planning they reflect, are instead an extended meditation upon the practical realities of human nature ... and the prudent checks and balances upon that human nature that are necessary to secure (a reasonable approximation to) security, freedom, and prosperity.

If Milton Friedman---or any academic economist---had written the Federalist Papers, it is very doubtful (IMHO) that the Republic would have survived! :)

I am talking about economics and you are still in the government frame of mind!

So they do not reference economic or religious dogma, wouldn't that tell you something!!!! (check and balance the government fine - stay out of the market and the church)

I am just showing what happens when you tamper with a system good or bad, it changes the system and the pure model, I could care less of your ideology!

Skullside, may I respectfully recommend that you read Chapter XIV of Mark Twain's Roughing It ... specifically the passage that begins "Mr. Street was very busy with his telegraphic matters" ... for Twain's common-sense and to-the-point remarks on the intimate symbiotic relation between checks-and-balances Republican government and commercial enterprise. :)

Dave,

What if I interpret the Krugman paragraph that you quoted (beginning "It's much harder to say...") as equating economics with fluid dynamics?

Fluid dynamics guys definitely "live with messiness", Navier-Stokes equations notwithstanding. They "acknowledge the importance of [apparently] irrational and often unpredictable behavior" in their flows, they "face up to the often idiosyncratic imperfections" of real fluids. Whether a fluid "theory of everything" is a long way off depends on whether you're satisfied with a great microscopic description (got it) or would like an actual coherent theory of macroscopic flows (nope). I mean... we still use wind tunnels.

I'm suggesting that Krugman isn't telling us to get rid of math or rigor, but to abandon the expectation that the right theory will be beautiful and simple. Maybe it will be chaotic, complex, unstable, and [arguably] ugly... but nonetheless mathematical.

Is this a reasonable alternative interpretation of Krugman? Or do you think he clearly abjures math entirely, rather than just scorning the idea that the solution will be simple math?

Sam: the point is that your dogma that economics should not be amenable to elegant mathematical analysis is as wrong as the dogma that it isn't. Baby, bathwater, tub is the wrong way to go. And do note that this is not saying anything about Krugman's argument about the state of economics, part of which I agree with and part of which I think he failed to articulate in a coherent manner. The part I'm arguing with is what to do in the future. You argue that you know the answer (no math!) but this seems as wrong headed as the opposite.

I thought as much - Never Mind!

Tell Huck Finn hi!

Deinst: I do think that Krugman's main point is the attacking of the Chicago school. However, when it comes to how economics should move forward his words don't sound like they encourage a path which might involve (the horror) difficult mathematics. That's a slippery slope in my book. (Actually I'm just trying to channel Eric Weinstein :) )

Robin: My hope is that this is Krugman's view. But language is a slippery slope and if you read carefully looking for his suggestions, they really aren't anything more than "throw up your hands!"

Plus, no one has come back and said, look what Weinstein is saying, applying gauge there here is completely B.S. That would be the way to argue that what I'm saying is wrong. Instead, the line of attack seems to be "economics feels like it is messy" therefore "economics must be messy."

I would also point out that the figure accompanying the article has equations which, as far as I can tell, would never appear on a blackboard in the fashion given. This, in my opinion, is an even bigger sin. Fake equations, the gall!

Some points (from a former Wall Street rocket scientist who was trained as a systems engineer):

1) Virtually all human systems have inherent instability. The space shuttles' two catastrophic failures did not indict the theoretical (or experimental) physicists and mathetmaticians -- it indicted the engineers and managers who ran the project. See Feynman's comments on the Challenger disaster for an excellent discussion of this: http://www.ralentz.com/old/space/feynman-report.html

If one replaces "loss and vehicle and of human life" with "loss of financial system stability," it becomes an excellent read!

2) Because of this instability, there can never be a 100% certain fail safe shut down to an OPERATING failing system. This may be one reason why airplanes crash a lot more than trains. It's also why, despite RIMM's best efforts, we all find ourselves removing our blackberry batteries from time-to-time. Life is filled with examples of this.

3) It preposterous to blame theoretical academics for the mistakes of practictioners. One can find quite a few academics (John Taylor at Stanford; the Austrian School etc.) whose work would have predicted instability and problems. (Note that Roubini and Nassim are not on this list!)

4) At the end of the day, one word both explains and predicts these financial accidents. That word is LEVERAGE. Perhaps Krugman's best point might be that the theoretical models resulted in overconfidence on the part of practitioners -- which resulted in excessive leverage. (But I don't see him saying that explicitly?) Without excessive leverage, the mess would have never happened. Conversely with insufficient leverage, the ROE of these enterprises would have been a lot less during the good years. Bottom line: it's all about leverage....

This week Paul Willmott gave a cogent interview on CNBC ... this from a guy whose credentials in research, teaching, and practicing real-world economics are unassailable.

Willmott argues is that high-frequency trading has destroyed our individual and collective ability to assess risk and assign value ... obviously this wasn't supposed to happen ... but it did.

The technical discussions on the Willmott forum are enlightening too. These professional traders don't even try to defend the axioms of EMT. Why is that?

The Willmott Forum's article archives contain wonderfully interesting (and funny!) first-person accounts of the math/reality interface in economics ... Ed Thorp's articles are a good place to start.

I originally went to the Willmott Forum because it's a wonderful venue for discussing mathematical issues relating to the efficient simulation of stochastic processes on high-dimension state-space manifolds (a venue natural to both QIT and economics) ...

... but I stayed because it is strikingly apparent that (nowadays) successful academic research groups are evolving to resemble successful computer trading groups ... e.g. "simulate the system ten times; invest (in instruments or experiments) once" ... "mix senior experience with junior creativity" ... "rapidly training new members is half the battle".

Dave,

I think you have grossly misunderstood Krugman's point. His point is not that mathematical models are bad, or that they're not even useful. His point, as near as I can tell, is merely that we should not mistake our models for reality, and be taken in by the internal beauty of the model before comparing it against nature.

From that article, and from the extra notes he put on his blog here:
http://krugman.blogs.nytimes.com/2009/09/05/a-few-notes-on-my-magazine-…

...I gather that he's just saying that if/when we do achieve an full, accurate, and predictive description of economics, it's not going to be a terribly simple or terribly beautiful, in a mathematical sense. More likely it's going to be a hodgepodge of special cases.

My personal view is that this does, indeed, seem rather likely. We may have some mathematically-beautiful approximations to the true behavior, but the true behavior seems likely, to me, to always end up being very messy. My reasoning for this is simple: human minds are built together by the very patchwork process of evolution, with huge numbers of very specific mechanisms built-in, as well as significant variation in those mechanisms across different people. Since the fundamental units of economic behavior are so complex, then, economic behavior as a whole is liable to be complex. This doesn't mean it's indescribable, or that we shouldn't try, but it does mean that it's going to be hard.

"gather that he's just saying that if/when we do achieve an full, accurate, and predictive description of economics, it's not going to be a terribly simple or terribly beautiful, in a mathematical sense."

But this is where I have an issue. He is putting his conclusions about what a theory will look like before the conclusions which should arise from testing his theories (messy, elegant, whatever)! My point is not whether there is or is not an economic model which is elegant, but that the kind of reasoning which starts by assuming what the theory will be is wrongheaded and dangerous. Now my own personal opinion may be that indeed economics will be messy and full of special cases, but putting my opinion before the actual testing of the theories seems wrong.

Indeed, while I'm very sympathetic to the idea that there may be no way to construct a large or master theory of economics, I think the arguments put forth are weak. Complexity, by itself, for example, is not a reason that there cannot be elegant theories. Indeed I would argue that our current understanding of physics is something like this: we have no real idea what happens at very short length scales (high energies) but we have been able to develop effective field theories which work wonders. Indeed it seems that there is all sorts of hell that could be breaking out at small scales and still our physical predictions at a larger length scale are very precise. Renormalization is a wonderful way in which a very elegant theory emerges from what may not be so elegant at it's basic level. The fact that TQFTs describe the infinite limit of fractional quantum hall systems is elegant, but those systems are far from "simple" and are definitely dirty and complex (try writing down the wavefunction!)

Again, I think the problem I have is not with Krugman's harping on mathematics being used to gussy up bad reasoning, nor with some of his arguments about why the freshwater economists are wrong, but that he explicitly talks about what should be done and in doing so presages that any solution should not be elegant or mathematical.

Jason Dick: Since the fundamental units of economic behavior are so complex, then, economic behavior as a whole is liable to be complex. This doesn't mean it's indescribable, or that we shouldn't try, but it does mean that it's going to be hard.

Golly ... does this mean that predicting economic behavior is of a similar algorithmic complexity class to predicting protein structures (both living cells and modern economies being complex systems that dynamically evolve under strong selection pressures)?

If so, that's GREAT NEWS!

But it does raise a very important question ... will real-world science remain an academic discipline?

Because isn't it arguably the case that real-world economics used to be an academic discipline ... but isn't any longer ... the academic community having fallen too in-love with formal proofs based upon unrealistic axioms? :)

Krugman has responded to this on his blog:

Third, on an interesting point raised by Discover (via Mark Thoma): wonât we eventually have a true theory thatâs as beautiful as the full neoclassical version? Well, one thingâs for sure: we donât have that beautiful final theory now, so the current choice is between ideas that are beautiful but wrong and a much messier hodgepodge. But my guess is that even in the long run it wonât be all that neat. Discover suggests general relativity versus Newtonian physics; but a better model may be meteorology, which as I understand it starts from some simple basic principles but is fiendishly complex in practice.
Actually, let me put it this way: the economy is a complex system of interacting individuals â and these individuals themselves are complex systems. Neoclassical economics radically oversimplifies both the individuals and the system â and gets a lot of mileage by doing that; I, for one, am not going to banish maximization-and-equilibrium from my toolbox. But the temptation is always to keep on applying these extreme simplifications, even where the evidence clearly shows that theyâre wrong. What economists have to do is learn to resist that temptation. But doing so will, inevitably, lead to a much messier, less pretty view.
So be it.

I think you've misrepresented Krugman in your sylogism.

Krugman is not asserting (3):

"(1) there is a very beautiful and mathematically sophisticated theory of economics with the efficient market hypothesis as its cornerstone, (2) these theories appear to have not done well as a framework for understanding the current economic crisis, thus (3) we should abandon any search for elegant and mathematically sophisticated theories in economics."

Rather he is saying that the failure occurs at (2), in fact he simply states what around 99% of participants in the financial markets know:- the efficient markets hypothesis is wrong because securities prices don't conform to it.

Krugman's argument is really:

(1) there is this really neato theory built on a model that
(2) doesn't match the real world and we know it

Therefore the theory is wrong, time to look for another.

That is completely different from abandoning the search for a valid theory.

Read the last paragraph of Krugman's article, it puts forward a research program to do exactly that.

Hey JM, See Martijn's comment just before yours! It's hard to read that an not think that Krugman is _forcing_ a view that the answer must be messy. Where is his evidence for this beyond "humans are complex?" Why is he putting this conclusion before the actual theory is ever developed?

Put it this way: suppose there was an elegant theory for some chunk of economics that is not obvious. If everyone thinks like Krugman are we going to find this theory?

Dave says: Krugman is forcing a view that the answer must be messy. Where is his evidence for this?

Hmmmm ... in any discipline in which the questions are messy, isn't it a pretty solid heuristic that the answers are messy too?

From this point of view, the key mistake of academic economics is pretending that economic questions are simple, not that economic answers are simple.

I find selection supremely elegant but the phylogentic tree it generates bewilderingly complex and messy.

I don't know much about economics. But, it strikes me that this relates to the general question of decidable problems. Elegance aside, I have no reason to think that we can model a system where sentient individuals of the system are aware that they are being modeled and know what their model exactly is. The more "elegant" (what is that, anyway? Something with a short description?) a model is, the more accessible it will be.

Perhaps regular Joe's may be forever modeled, but the real giant economic players are free to reinvent themselves at any time.

Eric Weinstein: I find selection supremely elegant but the phylogenetic tree it generates bewilderingly complex and messy.

LOL! Eric, that is a wonderful observation!

Let's see ... if we really believe that "behind every messy answer there is a messy question" ... aha! ... howzabout this for the messy question (or is it just deep?) ... "What is evolutionary fitness?" :)

I am not an economist but from what I have seen economics is running into a similar issue that physics and biology have been running into increasingly.

Dealing with large complex systems dominated by nonlinear feedbacks between coupled parts is typically dealing with a nonequilibrium situation, regardless of your model.

Physics & chemistry, despite some early steps, have still not created a firm foundation for nonequilibrium thermodynamics with the usefulness and elegance of the classical theory. 30 years of bifurcation and chaos theory give us no unified theories on this issue. Too many papers are like "here we have an equation, we will look at the linearized, infintesimal perturbation of it...look a bifurcation!" etc. etc. People forget these are only valid for small perturbations and not far from equilibrium.

Biologists and ecologists are dealing with ecosystem dynamics which do not imply a steady state---I mean beyond the simple Lotka-Volterra models that predict cyclic predator-prey populations to phenomena like trophic cascades. We still can't even say if ecosystem diversity necessarily begets stability in systems where out-of-equilibrium is the norm not the exception. One of the best pieces of advice on mathematical modeling in the sciences was given to me by ecologist Jakob Weiner:

"I liken the emphasis on mathematical elegance and rigor among modelers to a football coach saying -'Wouldn't it beautiful if the players' movements on the playing field were synchronized?' Yes, it might look very nice, but the team would not win many games. Visually it might be great, but it would be terrible as football....remember, the model is not the object of study"

Economics I think will have to come to grips with out-of-equilibrium situations and a coherent theory of events like shocks that you see in modern fluid dynamics. I think financial contagion research since the East Asian Crisis is making good steps in this direction.

As an aside, I have always found the climatologist/economist debates interesting. In criticizing each other they seem to fail to realize that they are both stuck studying large-scale complex systems and that the faults they point out in each other could conveniently be reflected back on their own discipline.

@Dave
I think Krugman's argument is not anti-scientific. Rather it says that it does not make sense in proving the working of a human mind, mathematically. Unlike quantum physics, economics deals more with confidence and emotion. Irrational exuberance plays a major role in economics than in physics, primarily because people are irrational and they don't obey laws as bosons and fermions do.

Telling that Krugman writes just-so story with no concrete claim does not pass because you are in blogosphere and you have to adapt your writing in such a way that it is readable by everyone. For example, this article appeared in the science blog: http://scienceblogs.com/pontiff/2009/09/the_most_depressing_results_in…, which is again something that I can write. Nothing personal. The point is that you cannot expect a completely wonkish economic white paper to appear as a news paper article in the same way that seminal science and technology whitepapers do not appear in science and technology. If you want some serious Krugman paper, read this: http://ideas.repec.org/a/mcb/jmoncb/v11y1979i3p311-25.html

Lord Phat: 30 years of bifurcation and chaos theory give us no unified theories on this issue.

So, maybe we should reassess the value of the "Arnol'd"ian gifts that this line of mathematical research did give us ... for example, efficient symplectic integrators ... for sure, researchers like David Shaw are doing this (with outstanding success).

"mathematician-level computations"

Everyone knows mathematicians don't do computations. That would require arithmetic.

As for computer science (real CS), I'm not sure how predicate calculus applies to financial models.

Yeah, tongue in cheek... somewhat ;)

Got to agree with many others here... I think Dave's conclusion is incorrect.

Krugman is arguing not that we need to "throw up our hands". I'll go farther than Krugman: the premise that mathematically elegant 'physics like' models are a fruitful approach in economics is wrong (silly IMO). Economics is a complex adaptive system (like ecology or evolutionary biology) and trying to devise 'laws' for apparent system-wide properties is bound to lead to, at best, approximations which sometimes hold. It just doesn't really work that way. Fortunately we have other sorts of models (agent based for example), which are mathematically messy in the classic sense.

[Ever try to describe an algorithm in math? The translation is always possible (in theory), but even simple algorithms can end up being terribly convoluted math.]

As for some of the language you object to: "Uncertainty" and "irrationality". Krugman makes a minor writing mistake here, since these words are terms of art in economics. It is only a minor issue though, since the every-day meaning isn't far off. What he is saying is that economic theories/models need to include uncertainty and irrationality (bounded rationality) before they can hope to be correct. He is specifically advocating for progress in the field!

I find selection supremely elegant but the phylogentic tree it generates bewilderingly complex and messy.

"As for computer science (real CS), I'm not sure how predicate calculus applies to financial models."

Heh. Machine learning?

I know everyone thinks I'm wrong but I'll try to get my point across once more. I am attempting to argue that the problem is that PREJUDGING a solution because you "think" the system is messy, is a good way to not make progress. The whole goal of important research, I think, is to get your brain out of the mode of "hey this is what my intuition tells me" and into "how can I change the way we view and understand the problem." Krugman's piece satiates those who believe that there is no tractible path forward for complex adaptive systems (which I think does a bit of injustice to economics proper. Certainly the base agents are complex but this does not at all imply that the system is complex. A transaction, for example, where I buy or sell, is not, by itself, a complex adaptive system.)

Plus no one has answered Eric's query: define inflation without involving gauge theory please :)

Dave, you make some good points ... one way to boil down the issue is to ask "Is economics a theorem-proving mathematical discipline, a hypothesis-testing scientific discipline, or a design-and-operate engineering discipline?"

The answer of course is "all three" ... and the great trick is to keep all three communities talking to one another.

Then the point of Krugman's column is plain common sense -- "Running the economy is a job for engineers".

And this helps us understand why China's economy is showing sustained exponential growth: of the fourteen members of China's ruling politburo, all fourteen have engineering degrees.

I think Dave misunderstands Krugman because he's never studied economics. If you know the basic Arrow-Debreu model on which the vast majority of theoretic macroeconomics is built, you know that it's principal selling point is that it is a "beautiful", "perfect" model in which each and every element has a clear economic interpretation. Because A-D is such a well oiled machine it is easy to use.

What I think Krugman is really trying to say is that economists are lazy. They prefer to rely on an "out of the box" theoretic structure and adapt their arguments to fit this structure, rather than go through the excruciating process of developing new theory. (Among the less inquisitive members of the profession there is a large group that think that the simplified version of this model -- i.e. what's taught in intermediate micro -- is the "economic way of thinking".)

Of course this means that certain assumptions underlying the favored structure become so ubiquitous that they are invisible to members of the profession. For example, the Arrow-Debreu model assumes perfect liquidity of all assets -- and you will find that very few economists are capable of imagining a world where assets can be illiquid -- they have avoided building the theoretic structure that would allow them to think about this in a macroeconomic environment.

Keynes tried to reform the profession, but instead was coopted and his ideas were forced into the A-D framework, so post-Keynesians also have difficulty thinking seriously about liquidity -- except in the very narrow context of the zero lower bound.

While the final product of some new theory might end up being beautiful once its been worked over for several decades, you can be pretty sure that the first draft of any new theory is going to be pretty ugly. As long as economists insist on reacting in disgust to ugly theory, they will continue to prevent the development of new economic theories. As I understand Krugman this was his point and I think he's right.

Dave, when you state "Put it this way: suppose there was an elegant theory for some chunk of economics that is not obvious. If everyone thinks like Krugman are we going to find this theory?" I think you miss the point that people trained in a certain way of thought perceive "beauty" from that context. The "elegant theory" you are talking about will look ugly to people trained in A-D, even though three generations from now people may find it stunning. In short, the fact that the new theory represents a different way of thought will make it ugly to people who don't want to think differently -- or to admit that framework on which they've based their careers is fatally flawed.

So what you're saying econ grad, is that I'm missing the point because when Krugman says "beauty" and "mess" I've got exactly the wrong concepts in my head. I like it :)

I do maintain however, that the idea that one needs to go "messy" before emerging with "beauty" isn't, a priori, a good way to make progress in any scientific enterprise. If there are economists who really swear that anything outside of the Arrow-Debreu model (as I'm sure there are) is messy, then, well the field has bigger problems. An insistence on comparing theories to reality, admitting limitations, and not biasing your research based upon prior art beyond the limit of where it has shown its validity...well these are all things that one needs if one wants any form of economic science.

(And yes I've never studied economics as everyone can certainly tell! Have you looked at the video by Eric Weinstein I linked to in the article. Still waiting for someone to say which gauge theory is not needed to reason about price indices :) )

Heck, isn't "messiness" largely in the eye of the beholder?

At first, people thought that Euclidean geometry became messier when it was deformed onto Riemannian manifolds ... but after (roughly) one generation this feeling passed ... in accord with a famous passage in Numerical Recipes (that shows a keen understanding of human cognition):

"After awhile, your adversaries will give up trying to think of alternative hypotheses, or else they will grow old and die, and then your hypotheses will become accepted. Sounds crazy, we know, but that's how science works!" (emphasis as in the original).

Nowadays we think that Riemannian frameworks are actually simpler than Euclidean frameworks (and isn't a similar evolutionary process well underway with respect to Kählerian versus Hilbert QM frameworks?).

So I am going to split the middle -- I agree with Dave that subtler economic axioms may well be needed, and I agree with many posters that the resulting economic calculations will be difficult, and are likely to yield answers that upset long-cherished academic preconceptions.

A lot of these comments (and perhaps the article, I haven't read it) appear to imply that economics needs a huge infusion of fuzzy systems theory. Economics ultimately concerns how many different types of people behave. Which means many different reasoning processes at work. Sounds like informal multi-valued reasoning of a sort to me.