There’s a popular conception that mathematicians are unworldly, and that mathematics is, at its heart, walled off from the real world, its pursuit a form of escapism that takes the pursuer into a realm of pure, abstract thoughts.
Certainly, that’s a general sense of mathematics that I held for many years. Yes, like all my fellow mathematicians, I always knew that mathematics – all of it – arose, directly or indirectly, from real world problems, and that any branch of mathematics having any discipline-internal significance almost always turns out to have real-world applications. But neither of those was why I did mathematics. For most of my life as a mathematician, I simply did not care about the history or application of what I was doing. It was all about the chase – the search for new knowledge in a beautiful domain.
Early on in my career, when more politically active colleagues urged me to boycott conferences and workshops funded by NATO (a big issue back in the 1970s), or to avoid applying for research funds from commercial or military sources, I essentially turned a deaf ear to what they were saying, and got on with the work that interested me.
As a mathematician working in axiomatic set theory, with particular foci on the properties of sets of large infinite cardinality and on undecidability proofs, I felt fairly confident that nothing I did would ever find practical application, so for me the issue was purely one of where the money came from to support my research. I felt “clean,” and not under any moral pressure regarding potential unethical uses being made of my work.
True, I was aware that the famous early twentieth century mathematician G. H. Hardy had made the same claim about his work in number theory, yet in the mid-1970s his work found highly significant application in the design of secure cryptographic systems. But I felt that a similar outcome was unlikely in the case of infinitary set theory. (I am no longer quite as sure about that; I speculated about possible applications of Cantor’s set theory in my June column.)
I think we all have to address the morality-of-possible-applications question about our work as mathematicians at one time or another. Some, from Archimedes to Alan Turing, have actively engaged in military research; others try to avoid any direct contact with commercial or warfare-related activities.
The rise of math-based corporations such as Google that form a large and influential part of today’s global world, and the closely related growth of the modern, math-driven security state, as iconicized by the NSA, make it impossible to maintain any longer the fiction (for such it always was) that we can pursue mathematics as a pure activity, separate from applications, be they good or ill.
The uncomfortable fact is, we are in no different a situation than manufacturers of sporting guns who deny any agency when their product is used to kill people. (Yes, people pull the trigger, but as comedian Eddie Izzard pointed out, “the gun helps.”)
If we want to be able to maintain that our work will not play a role in someone’s death, torture, or incarceration – or in someone else achieving enormous wealth and power – our only option is to not go into mathematics in the first place. The subject is simply way too powerful as a force – for good or for evil.
Shortly after September 11, 2001, I was asked to join a research project funded by the U.S. intelligence service. For me, that was my crunch time. The work that led to that invitation was an outgrowth (described in my 1995 book Logic and Information) of my earlier research in mathematical logic and set theory. Like it or not, I was already in deep. To say no to that invitation would have been every bit a positive action as to say yes. Sitting on the fence was not a possibility. I was a mathematician. I’d already made the gun.
As the Google founders Larry Page and Sergei Brin eventually discovered, “Do no evil” is a wonderful grounding principle, but the power of mathematics renders it an impossible goal to achieve. The best we can do is try to make our voice heard, as many mathematicians and nuclear physicists did during the Cold War, who spoke publicly about the massive scale of the danger raised by nuclear weapons.
Finding out (as I have over the past few weeks) that the work I’d done over the past twelve years – for various branches of the U.S. government (intelligence and military) and for commercial enterprises (in my case, the video game industry) – was part of a body of research that had been subverted (as I see it) to create a massive global surveillance framework, I felt I could not remain silent.
Not because I felt that I, as an individual, did anything of significance. I worked on non-classified projects, and made no major breakthroughs. I was a very tiny cog in a very big machine. (If “they” are keeping an eye on me, they are definitely wasting our tax dollars!)
But I did take the money and I did do the work. I don’t regret doing so. The fact is, I’d made the crucial choice long before 2001; back in my youth when I decided to become a mathematician.
Those of us in mathematics education have always told our students that math is useful. In today’s world more than ever, we cannot at the same time pretend it is free of moral issues. Agnosticism is not an option (if it ever really was). To say or do nothing is inescapably a positive act, just as significant as saying or doing something.
We humans have created our mathematics, and used it to help shape our world. Now we have to live in it. Not only are we the ones who bear a large responsibility for that world, we are also, by our very expertise, the ones who (in many fundamental ways) understand it best. (It often seems that only the mathematically sophisticated really appreciate that an American is more likely to die in his or her bathtub than from a terrorist attack, and that more people died on the roads due to increased traffic during the time after 9/11 when all flights were grounded than did in the Twin Towers attack.)
So, to return to the question implicit in my title, “What is mathematics used for?” Douglas Adams provided the answer: “Life, the universe, everything.” With such reach and power comes responsibility.
FOOTNOTE: For a more personal take on the above issues, see the interview I did on June 21 on Shecky Riemann’s Math Tango blog.