## What is the mission statement of the mathematician?

In the past five years, I’ve been learning a lot about how mathematics is used in the “real world”. It’s fascinating, thought provoking, exciting, and truly scary. Moreover, it’s something I rarely thought about when I was in academics, and, I’d venture to say, something that most mathematicians don’t think about enough.

It’s weird to say that, because I don’t want to paint academic mathematicians as cold, uncaring or stupid. Indeed the average mathematician is quite nice, wants to make the world a better place (at least abstractly), and is quite educated and knowledgeable compared to the average person.

But there are some underlying assumptions that mathematicians make, without even noticing, that are pretty much wrong. Here’s one: mathematicians assume that people in general understand the assumptions that go into an argument (and in particular understand that there always *are* assumptions). Indeed many people go into math because of the very satisfying way in which mathematical statements are either true or false- this is one of the beautiful things about mathematical argument, and its consistency can give rise to great things: hopefulness about the possibility of people being able to sort out their differences if they would only engage in rational debate.

For a mathematician, nothing is more elevating and beautiful than the idea of a colleague laying out a palette of well-defined assumptions, and building a careful theory on top of that foundation, leading to some new-found clarity. It’s not too crazy, and it’s utterly attractive, to imagine that we could apply this kind of logical process to situations that are not completely axiomatic, that are real-world, and that, as long as people understand the simplifying assumptions that are made, and as long as they understand the estimation error, we could really improve understanding or even prediction of things like the stock market, the education of our children, global warming, or the jobless rate.

Unfortunately, the way mathematical models actually function in the real world is almost the opposite of this. Models are really thought of as nearly magical boxes that are so complicated as to render the results inarguable and incorruptible. Average people are completely intimidated by models, and don’t go anywhere near the assumptions nor do they question the inner workings of the model, the question of robustness, or the question of how many other models could have been made with similar assumptions but vastly different results. Typically people don’t even really understand the idea of errors.

Why? Why are people so trusting of these things that can be responsible for so many important (and sometimes even critical) issues in our lives? I think there are (at least) two major reasons. One touches on things brought up in this article, when it talks about information replacing thought and ideas. People don’t know about how the mortgage models work. So what? They also don’t know how cell phones work or how airplanes really stay up in the air. In some way we are all living in a huge network of trust, where we leave technical issues up to the experts, because after all we can’t be experts in everything.

But there’s another issue altogether, which is why I’m writing this post to mathematicians. Namely, there is a kind of scam going on in the name of mathematics, and I think it’s the *responsibility of mathematicians* to call it out and refuse to let it continue. Namely, people use the trust that people have of mathematics to endow their models with trust in an artificial and unworthy way. Much in the way that cops flashing their badges can abuse their authority, people flash the mathematics badge to synthesize mathematical virtue.

I think it’s time for mathematicians to start calling on people to stop abusing people’s trust in this way. One goal of this blog is to educate mathematicians about how modeling is used, so they can have a halfway decent understanding of how models are created and used in the name of mathematics, and so mathematicians can start talking about where mathematics actually plays a part and where politics, or greed, or just plain ignorance sometimes takes over.

By the way, I think mathematicians also have another responsibility which they are shirking, or said another way they should be taking on another project, which is to educate people about how mathematics is used. This is very close to the concept of “quantitative literacy” which is explained in this recent article by Sol Garfunkel and David Mumford. I will talk in another post about what mathematicians should be doing to promote quantitative literacy.

Amen! Great post! Any chance you want to run for a spot in the AMS Council?

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The mathematical community and maybe specifically the AMS needs to take on more responsibility for the people we send out into the real world labeled as “mathematicians” and their actions. But more than that we need to send these people out better armed. If the AMS were to define more clearly what a mathematician is and what a mathematician’s ethical obligations are (analogous to the Hippocratic oath for doctors), then it would make it much easier for individual mathematicians to speak out when they need to because they will know that they have their professional colleagues behind them.

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I agree with much of what you say here, although I think calling out misuses of mathematics is, perhaps, only half the equation. I think another part of the equation is to work on increasing understanding of both models and mathematics. I do often wish that mathematicians played a larger role in math education. I think that there is quite a large disconnect today, in the US, between the math educational world and the mathematical world. This disconnect leads to all kinds of crazy things being taught in math classes that vary from “missing the point” to “downright wrong.”

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Regarding the article linked, I would have to strongly disagree. That same argument can be made for just about every class/subject before college. Since when did “real life” become life at a finance job? People struggle with math earlier than other subjects because it’s one of the only subjects where young kids have to problem solve instead of memorize. If there are early troubles, then the environment contributing to mental development needs to change, not the content–though teaching of the content is probably poor too. Plus, I think it’s easier to go from “abstract” to “real life” than the reverse. Why do people feel the need to answer the common student question, “When are we ever going to use this?”?

I sense a difficult-to-describe general cultural problem with this whole mindset, where everything is geared toward furthering college apps/resumes/etc. God forbid someone learned about redox reactions or cyclic groups as an investment banker. I do think you make some good points though, and there is enormous divide between math and non-math types.

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I totally disagree with the Mumford & Garfunkel article as well. If the problem is that we teach mathematics badly, the solution should not be to teach something else and call it mathematics. (Is there any reason at all to believe we would teach that better? Will we really be serving these kids better by teaching data & finance & engineering badly but calling it mathematics?)

In an earlier post about jobs, Cathy mentioned that the hedge funds hire mathematicians and don’t really care if they know the financial stuff. Learning that is easy (ish) if you’re a mathematician, but not vice-versa. Learning *lots* of stuff is easier once you know the mathematics well, but not vice-versa. So “How to fix our mathematics education” is not to change *what* we teach, but to change how we teach it.

I have a long-ish response to the article written by a former colleague (and a mathematician who has spent more than 25 years in high school classrooms, so actually has some insight into things like what motivates kids. Hint: it’s not data and budgets). But I don’t want to post something that long in the comments, so I’ll leave it at this.

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I would agree with both of you that it doesn’t make sense to *replace* our mathematics education with applied math. There’s definitely something to be gained to learn purely abstract logical reasoning. However, I do think we are not preparing children (or grownups!) for our current culture by restricting mathematics to something that is almost by definition abstract. We really should just teach both.

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The long-ish response is now posted as a blog post, so I’ll link it here.

http://patternsinpractice.wordpress.com/2011/08/31/response-to-a-nyt-editorial/

One of the problems with the NYT article is that it sets up this straw man as “math with no meaning” as what is advocated by the current Common Core Standards. But that’s not the case at all, and it’s *never* been the case that we tried to teach mathematics that way. Every good mathematics teacher (and curriculum) throughout history has known that the right way to teach math is to use a mixture of pure abstraction and concrete motivating problems. But the motivating problems are not the mathematics.

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I was wondering, Cathy, if you were interested in calling out any favorite practitioners of the aforementioned scam, i.e. who abuse the badge of mathematics to wield undue trust?

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I love this post, and I wrote one for mathematicians from the other side on my blog, which I called “Equations don’t govern reality”, because I have met a lot of mathematicians that tend to place a lot of confidence in models derived from physical theories. In fact, they tend to trust physicists’ results as much as they trust mathematical ones, which is a baaaaaaaaaad idea.

But back to your point, I run into this in meteorology all the time. Folks use a massively complex physical model, and treat the results as if they were the real atmosphere, despite the fact that they “know” all of the errors present in the model and its assumption. Somehow, though, the field moves forward.

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Hi,

I think you raise an important point. I believe that part of the problem is related to the use of the term “model.” About a century ago, because of great advances made in physics, the term “theory” was much more in vogue. I think this was better, because the idea of theory

emphasized the underlying assumption. A theory was better, the less assumptions it had

and the more elegant these assumptions were. There was a difference between hard sciences and soft ones.

In order to raise the status of softer sciences, the more egalitarian word model has come into use. An undergraduate student in differential equations might not see any difference between, say, Newton’s laws of motion and the logistic model for population growth. One

popular undergraduate text in differential equations (Edwards and Penney) offers unironically a model explaining how infinite populations of alligators arise in finite time.

If mathematicians want to make a difference, they should resist the language of the modelers who are always trying to coopt us. But it is just too hard politically. My university has just introduced a mathematics requirement for all students. It’s name is “the math modeling

requirement.” (This is usually fulfilled by taking Finite Math.)

Nets

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FWIW, the Big Idea article generated some interesting comments here: http://slashdot.org/story/11/08/16/0142226/The-Post-Idea-World, and I think a lot of them have a good point. The feeling that everything significant has already been invented is not new to this era, and progress keeps going on. News and feedback are certainly faster than ever though.

I had seen the Garfunkel and Mumford article and had quite mixed feelings. I agree with improving quantitative literacy and relating it to life skills people need but are often forced to learn the hard way, especially personal finance and some data/stats and computing. It’s very important and schools generally do nothing now. But I would not want to take away from of the traditional high school math curriculum for this — just a bit of reorganization and modernization perhaps. It’s one thing to ask “how often do most adults encounter a situation in which they need to solve a quadratic equation?” but in fact circles and ellipses are everywhere, and the idealized trajectory of a projectile is a perfectly natural thing to study in school. I just don’t see it as at all a loss if you spend a bit of time studying that as a kid and go on to do something where you never need that any more.

Definitely agree models need to be used with their assumptions and limitations in mind.

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I have reposted this on my Google+ page and am going to tweet it. I would argue that economists need a similar mission statement…intellectual honesty is really important.

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Here is, from the May Notices of the AMS, an example of a mathematician (John Ewing) confronting “mathematical intimidation.”

http://www.ams.org/notices/201105/rtx110500667p.pdf

May his paper and your blog post be an inspiration to the mathematical community.

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