Why we use mathematics in economics

Economics Nobel winner Jean Tirole put it succinctly as follows:

In a mostly auto-generated translation via Twitter’s web interface, this says “[we] use mathematics not because we’re smart but because we’re not smart”.

I agree wholeheartedly. Using mathematics in our work in economics (and in so many other areas of research) allows us to stand on the shoulders of giants and use their smarts. It’s on us to make good use of this powerful tool, honed over the centuries by so many brilliant people. Criticisms of using mathematics in economics are pointless; criticisms of using mathematics badly in economics are valuable.

Big news on the Wolfram language

Wolfram Alpha has been freely accessible on the Web for some time now. It allows anyone to do some of the work that Mathematica can do, in a browser. Now, it looks like the new Wolfram language, a very ambitious project to open up programming to many more people, is also going to be freely available on the Wolfram “cloud”. Details in this post by Wolfram himself. When I get some time freed up, I want to play with this!

Mathematicians’ good thought habits

Once in a while, I come across a gem of a post that I can file away to use in my teaching and advising, not to mention my constant struggle to think better and more productively as an economist (actually, in any area where I happen to be trying to think well). Here is one such gem, by Jeremy Kun. Enjoy! I am pretty sure you will find it enlightening even if (perhaps especially if) you are not thinking mathematically and consider those who do crushing bores. You’ve been warned!

Three links (OK, four) on teaching math

I came across this NPR post today via a post on Facebook. This led me to remember this one and that one, from a recent post and an older post I made myself on FB. The last one of these three links leads to a great discussion of how becoming good at math is a path open to everyone, but it takes hard work. Older people in a position to advise / teach younger ones should be keenly aware of the last one, and help their students/children/tutees adopt a growth mindset.

Art and science, and some economics, too

The pursuit of beauty and economic theory: friends or enemies?

I read this post on Aeon by Frank Wilczek this morning. I posted about it on Facebook but doing so did not make me stop thinking about it. I want to put some of these thoughts down here, as they relate to economic theory, what draws me to it, and whether the draw of beauty in the mathematics used in economic theory detracts from the theory’s value to society. First of all, I invite you to follow the link above and come back to read the rest of my thoughts.

Wilczek has published a book recently, A Beautiful Question: Finding Nature’s Deep Design (Google Play link, Amazon link). I have not read it to the end yet, but I hope I will over the winter break. His discussion in the Aeon post immediately made me think about the art of mathematics and how it motivates me (and many others) to do economic theory and guides our attempts to do it.

Empires built on math

In my early years studying for my doctorate, I thought of math as a collection of empire-building tools for empires of the mind (castles in the sky might be another apt name). Economic theory, heavily math-laden, becomes in this view a galaxy far, far away where each theorist builds an empire (or at least a few starships) to impose order on the universe. Once you have laid down your assumptions, you then have a solid foundation for building and you are the emperor of your theoretical creations, as long as you can write down your mathematical arguments correctly.

This view is not far from Asimov’s galactic empire fiction. Indeed, Asimov himself thought that some sort of social physics exists, and if we can discover its laws, we could predict the unfolding of societies over vast, even galactic, scales. For economics, this perspective explains at least some of the psychological attraction theorists feel towards their castles in the sky. This attraction is particularly prominent for me when I am working on general equilibrium theory, which most of the profession has left behind.

Ethereal but irrelevant?

Good question! Let me first tell a story about logic and the demise of the Hilbert program in mathematics. Then I will talk about the fragmentation of economic theory, this one brought about by the confrontation of the theory with empirical tests, such as they are (famously much harder in the social sciences than in physics).

Hilbert was a prominent German mathematician. His famous and eponymous program was an attempt to make all mathematics utterly rigorous by putting it on a formal basis and to prove within that framework that mathematics is consistent. (Perhaps the best way to understand “formalization” in this sense, besides the obvious which is that it should be totally understandable to a computer, is by thinking about something von Neumann said: “Young man, in mathematics you don’t understand things. You just get used to them.” —source)

Hilbert’s program came undone when Kurt Gödel came up with his stunning, deep, and worldview shattering (for mathematicians) Incompleteness Theorems. The gist of these theorems is that if you build all of mathematics on a finite list of axioms, then you cannot prove by using this list of axioms all true theorems of mathematics and you cannot prove that the mathematics you have founded on these axioms is consistent (contains no contradictions).

Coming as it did in 1930, this devastating development seems to be a symptom of the fragmentation of European culture, which was soon to produce a horrible war. But what does it have to do with economic theory? Well, it tells us that the theoretical castles are indeed based on air, not stone foundations.

What’s worse, that’s not all that ails theoretical empire-building in economics. While there are plenty of valid criticisms to aim at empirical research in economics, its basic import is that grand, unified economic theories perform badly. This has led to the growth of “behavioral economics”. (Once again, let me point out how silly this name is: isn’t all economics behavioral? — I prefer psychological economics). However mainstream this field has become, it does have a shattering impact on beautiful theory-building that aims at wide applicability.

We contain multitudes and that can be beautiful too!

Well, it was good enough for Walt Whitman. How about we apply it to the entire society, whose functioning is the domain of study of economic theory, broadly conceived? Does the fragmentation I just talked about make further theory development ugly?

Many social scientists have written on this issue. To me, it seems that a fragmented field of theories can still be an object of beauty in the mind of the theorist. Think of it as a kaleidoscope (a word that literally means “a device for showing beauty”). This is how I attempt to keep my motivation up when I am struggling with long, messy arguments trying to prove something in complicated mathematical models of society that invariably leave my head spinning after only a couple of hours of effort.

Another good thing to say about the fragmented collection of models that constitutes modern economic theory is that it has produced some quite literally life-saving innovations (kidney exchanges, which came from a deeply mathematical field of economic theory called mechanism design theory). A good and easily readable account is Al Roth’s recent book Who Gets What — and Why.

Some others to read on related matters

In conclusion, here are some thoughtful works to read carefully that I constantly feel guilty for not having time to read carefully enough.

Daniel Little has a website portal and a blog about what it means to do social theory, even more widely construed than the widest view of economics.

Dani Rodrik has a recently published book in which he “argues that economics can be a powerful tool that improves the world―but only when economists abandon universal theories and focus on getting the context right”, as the book description on Amazon states. Another book I have started — so I really should wrap this post up now and get back to reading, even without having convinced myself that this gigantic post is as well written as possible! Of course it’s not. But I move on anyway. Ars Longa, Vita Brevis. Or, in the original language of Hippocrates, taken from the last link:

Ὁ βίος βραχύς, ἡ δὲ τέχνη μακρή, ὁ δὲ καιρὸς ὀξύς, ἡ δὲ πεῖρα σφαλερή, ἡ δὲ κρίσις χαλεπή. In English: Life is short, and art long, opportunity fleeting, experience perilous, and decision difficult.

More on “mathiness” in economics

Paul Romer has a new post about “mathiness”, this time in financial economics. Right at the top of the post, he includes two links to blog posts by Tim Johnson about how mathiness was used to obfuscate what the math says in finance, so that official investigations into the latest financial crisis, the one that started in earnest in 2007, would miss something. In Romer’s words, “People in finance used math to hide what they were doing.” The Johnson posts are rich in material from the beginnings of probability theory, incidentally, and surprised me with a connection to Aristotle and how thinking about money as a universal measure showed people the way to apply math to physics; Johnson also connects finance to a notion of justice. Fascinating stuff! I hope to carve some time out to delve in this more deeply.

H/T for the Paul Romer link: Mark Thoma, here.

Topology counterexamples online

I just stumbled upon this page that presents a website that has explanations and visualizations of topological spaces that have certain combinations of properties. I found this via a post by Jennifer Ouellette on Google Plus and I thought it definitely merits inclusion here. Fascinating work, and if you are an expert, you can help improve it.

Improving scientific software

For open-source software used heavily in the sciences, there is a problem with giving the proper incentives and recognition to developers. A recent article in SIAM News called Quo Vadis, Scientific Software correctly points out the main problem with current practice.

Current practice involves researchers who use scientific software reinventing the wheel very frequently. As the authors of the article point out, if you try to publish a paper in a journal of mathematics and claim some theorems in your paper, you will not be allowed to omit the proofs. Yet, we often see such things as a paper that uses some mathematical software but does not include the code written for the calculations included in the paper.

I have always insisted that the graduate students whom I advise include any simulation code they wrote and used in the appendix of their dissertations. More people need to do this.

From the problem of not sharing code readily in scientific work there also follows the problem of researchers reinventing the wheel. You may know of a paper that did a simulation you would like to do but don’t know the code for the simulation, so you have to write your own to do something very similar. The article points out some efforts to build open-source libraries of code already written by practicing scientists who use computation in their work but also mentions a related problem: there is no established system for contributors to such libraries to receive professional credit. This problem reduces the incentive for a young scholar to contribute good code to a library that could benefit the entire scientific community.

It seems to me there is a strong argument in favor of solving the credit-giving problem and stopping the current inefficient practice surrounding scientific software. The linked article concludes with some eminently sensible proposed solutions. I highly recommend it.

On Karush of the Karush-Kuhn-Tucker theorem

People like Paul Krugman would call this type of post “wonky”. You have been warned.

In preparation for teaching Mathematics for Economists I for graduate students tomorrow, I decided to see what the Web has on Karush, of the Karush-Kuhn-Tucker theorem. I should have done it earlier! Even though I had been writing Karush-Kuhn-Tucker in my lecture notes almost from the first time I taught mathematics for economists, I did not have much to offer my students on the mystery of why Karush’s work was never published. Why I never searched online about it before I do not understand. Nevertheless, I hit a goldmine in my search today: a paper by R.W. Cottle that offers a lot of fascinating details on Karush and Kunh-Tucker, along with photos of the protagonists and excerpts of letters between Kuhn and Karush, when Kuhn belatedly discovered Karush’s master’s thesis, which pretty much anticipated the Kuhn-Tucker theorem, and hastened to give credit to Karush to fix the historical record. And how did Kuhn discover that Karush had done this? By reading Akira Takayama’s massive 1974 book on Mathematical Economics.

Call me a hopeless nerd, but I find this fascinating. A bit of human interest to liven up all the constrained maximization we do for our work, in one way or another, every day.

The URL where I discovered the paper is
http://www.math.uiuc.edu/documenta/vol-ismp/41_cottle-richard.pdf