We’ve discussed a few modern areas of research in mathematics: the search for a combinatorial proof of the density Hales-Jewett theorem lead by Timothy Gowers and some of the open questions about the number . A fair question is why should people spend their time (and the money of the university/government/companies/etc.) on solving such impractical problems? Shouldn’t they be working on a cure for a disease? Even if you enjoy doing math for math’s sake, you probably have relatives who wonder why anyone would do basic research which isn’t directly related to solving a real world problem!

A few days ago President Obama gave a speech at the National Academy of Science in which he gave a good answer to this question:

As Vannevar Bush, who served as scientific advisor to President Franklin Roosevelt, famously said: “Basic scientific research is scientific capital.”

The fact is, an investigation into a particular physical, chemical, or biological process might not pay off for a year, or a decade, or at all. And when it does, the rewards are often broadly shared, enjoyed by those who bore its costs but also by those who did not.

That’s why the private sector under-invests in basic science – and why the public sector must invest in this kind of research. Because while the risks may be large, so are the rewards for our economy and our society.

No one can predict what new applications will be born of basic research: new treatments in our hospitals; new sources of efficient energy; new building materials; new kinds of crops more resistant to heat and drought.

It was basic research in the photoelectric effect that would one day lead to solar panels. It was basic research in physics that would eventually produce the CAT scan. The calculations of today’s GPS satellites are based on the equations that Einstein put to paper more than a century ago.

— President Obama

We should point out that President Obama did make one mistake. He forgot to mention that Einstein’s theory of relativity (on which GPS technology depends) itself depends on the work of mathematicians which came even earlier. In fact, in a recent article by Alicia Dickenstein in the Bulletin of the American Mathematical Society talks about how in the first page of Einstein’s handwritten notes for his paper on general relativity he writes about how importantly his work depends on the mathematics done by pure mathematicians (who had no idea it was useful in the “real world”!). Dr. Dickenstein tells the story like this:

These are Albert Einstein’s words on the ﬁrst page of his most important paper on the theory of relativity:

“The theory which is presented in the following pages conceivably constitutes the farthest-reaching generalization of a theory which, today, is generally called the “theory of relativity”; I will call the latter one—in order to distinguish it from the ﬁrst named—the “special theory of relativity,” which I assume to be known. The generalization of the theory of relativity has been facilitated considerably by Minkowski, a mathematician who was the ﬁrst one to recognize the formal equivalence of space coordinates and the time coordinate, and utililzed this in the construction of the theory. The mathematical tools that are necessary for general relativity were readily available in the “absolute differential calculus,” which is based upon the research on non-Euclidean manifolds by Gauss, Riemann, and Christoffel, and which has been systematized by Ricci and Levi-Civita and has already been applied to problems of theoretical physics. In section B of the present paper I developed all the necessary mathematical tools—which cannot be assumed to be known to every physicist—and I tried to do it in as simple and transparent a manner as possible, so that a special study of the mathematical literature is not required for the understanding of the present paper. Finally, I want to acknowledge gratefully my friend, the mathematician Grossmann, whose help not only saved me the effort of studying the pertinent mathematical literature, but who also helped me in my search for the ﬁeld equations of gravitation.”

So, indeed, he was not only paying homage to the work of the differential geometers who had built the geometry theories he used as the basic material for his general physical theory, but he also acknowledged H. Minkowski’s idea of a four dimensional “world”, with space and time coordinates. In fact, Einstein is even more clear in his recognition of the work of Gauss, Riemann, Levi-Civita and Christoffel in [7], where one could, for instance, read, “Thus it is that mathematicians long ago solved the formal problems to which we are led by the general postulate of relativity.”

Einstein’s work in 1917 also laid the foundations for lasers. If you want a good read, chack out the article at the following link:

http://www.phytem.ens-cachan.fr/telechargement/Optique_Quantique/Kleppner_Coef_Einstein.pdf

I think that research for the sake of knowledge is a noble pursuit, and I hope that we continue to allow the great minds of our world to just work on their ideas to enrich our understanding. We never know if it may be useful in the future some way other than the insight that we gain at the present.

Amanda,

Thanks for the link!

This is a good article. I’ve always wondered about this. I have heard a lot of people myself criticize basic research.

I found a discussion page that has some interesting comments on this topic and ethics in mathematical research. Just some things to consider for the future mathematical researchers out there.

If you want to see the discussion, check ou this link.

http://www.physicsforums.com/showthread.php?t=188319

I believe that each researcher in their field of study has to make up their own mind as to what and why they are doing their work. Others may or may not find it useful, and someone’s work can be used in ways that they probably did not foresee. I think that if you are passionate and gifted in a subject you should find a way to pursue that passion. That’s what makes a person feel that their life has purpose.

I really enjoyed reading through the discussion Amanda posted. There really are a lot of arguments people can make for either side of this argument. Ethics in mathematics is a much more extensive debate than I was previously aware of. In S.E. Bromberg’s book, Evolutionary Ethics, he said, “to posit that mathematical principles lay at the foundations of evolving ethical systems would land even a good theory on already contentious grounds.” I agree that while relationships between the two do exist, it is the individual, not the math, in which the ethical debate should be made.

I think the main purpose of this article is very interesting and holds valid arguments about the necessity of research in all aspects of science. However, one of the things about the post that stands out most to me is something that I have always loved about math: that mathematicians are needed everywhere. I always think about something that Dr. Cross said to me in advising one day when I asked what some of the main things are that math majors end up doing for careers. He said, “whenever students come in and ask me what they can do with a math degree, I respond, what do you want to do?” His point was obviously that you can do almost anything you want with math, which is what I think is so great. And, I think this post just reemphasizes this point. There are almost undoubtedly mathematicians behind every bit of the research Obama is referring to, even though the research involves varying realms of ideas and science.

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