Will the Real Geometry of Nature Please Stand Up?

Is fractal geometry “the geometry of nature”? I was asked this question recently in an email from someone who had watched the PBS video Hunting the Hidden Dimension that I worked on, and appeared in, a few years ago.

It would have been easy to simply reply “Yes,” and for many audiences I would (and have) done just that—for this was by no means the first time I had been asked that question, or others very much like it. But the context in which this recent questioner raised the issue merited a less superficial response. So I wrote back to say that there is no such thing as the geometry of nature, or more generally, the mathematics of W, where W is some real world domain.

The strongest claim that can be made is something along the lines of “Mathematical theory T is the best mathematical description (or model) we currently have of the real world domain (or phenomenon) W.” But even then, this statement is less definitive than it might first appear: In particular, what do we mean by “best”?

Best in terms of understanding? (If so, then understanding by whom?)

Best in terms of building something in W? (If so, then building out of what, using what tools, and for what use?)

Best in terms of teaching someone about W? (If so, then teaching what kind of person in terms of age, background, education, motivation, etc.?)

Slightly edited and extended, the next few paragraphs are what I wrote back to my correspondent:

Nature is just what it is. Mathematics provides various ways to model our perception and experience of reality. Different parts of mathematics provide different models, some better than others. Fractal geometry provides one model that seems to accord with our observations, measurements, and experiences. But so too do the cellular automata models on which Steve Wolfram bases his “New Kind of Science.”

Many of us think fractal geometry does a better job than cellular automata in helping us understand the natural world by virtue of its nature, but that reflects an assumed patterns/relationship conception of what constitutes science.

I would prefer to call Wolfram’s framework a computational theory (of the world), rather than science. But the distinction is, I think, purely one of the meaning we attach to the relevant words (particularly “science”).

Both approaches can be said to begin by looking at how nature works, but the moment you start to create a model, you leave nature and are into the realm of human theorizing. From then on, the only available metrics are (1) degree of fit to observations and measurements, (2) degree of utility, and (3) degree to which we find the model’s assumptions reasonable.

There is lots of slack here.

In (1), what are we observing and measuring? (They are often entities created by those very mathematical theories, e.g. mass, length, volume, velocity, momentum, temperature, etc.)

In (2), how do we define utility? Doing stuff, building stuff, understanding stuff, teaching stuff, or something else? (Each with the various audience/use/purpose caveats I raised earlier.)

Then there is (3). Unless we make some initial assumptions, we cannot get a theory off the ground. And make no mistake about it, we do begin with assumptions. Not arbitrary ones, to be sure—not even close to being arbitrary. For the resulting theory to be fully accepted (as a plausible explanation or model), it has to accord to any and all the available facts, and it has to be falsifiable—it should make claims or imply conclusions that we can attempt to prove wrong.

For instance, a mathematical theory that implied 3 = 4 (as an identity of integers) would be immediately rejected.

What about a theory that implies 0.999… = 1.0, where those three dots indicate that the decimal series continues for ever? According to the widely accepted, standard definitions that mathematicians use to provide meaning to the concept of an infinite sequence of decimal digits, this identity is correct. Indeed, it can be proved to be correct, starting from the reasonable, plausible, and accepted basic principles (axioms) for the real number system.

Most university math students learn about the framework within which 0.999… is indeed equal to 1.0. (Though many of the popular “proofs” you come across are not rigorous.) As a result, many mathematically educated people will state, as if it were an absolute fact of the world, that 0.999 = 1.0. But that is not true. The identity holds because we have made some assumptions about how to handle infinity. It’s easy to overlook that fact. So let me provide a further example where it may be less easy to miss an underlying assumption.

Graduate students of mathematics are introduced to further assumptions (about handling the infinite, and various other issues), equally reasonable and useful, and in accord both with our everyday intuitions (insofar as they are relevant) and with the rest of mainstream mathematics. And on the basis of those assumptions, you can prove that

1 + 2 + 3 + … = –1/12.

That’s right, the sum of all the natural numbers equals –1/12.

This result is so much in-your-face, that people whose mathematics education stopped at the undergraduate level (if they got that far) typically say it is wrong. It’s not. Just as with the 0.999… example, where we had to construct a proper meaning for an infinite decimal expansion before we could determine what its value is, so to we have to define what that infinite sum means.

It turns out that there is an entire branch of mathematics, called analytic continuation theory, that provides us with a “natural” meaning for (in particular) that sum. And when we calculate the value using that meaning, we arrive at the answer -1/12. See this Wikipedia article for a brief account.

Incidentally, just as with the 0.999… example, you will find purported “intuitive proofs” floating around, among them this video that went viral earlier this year, but those arguments too are not rigorous.

Both frameworks, the one that yields a value for 0.999… and the one that produces a value for 1 + 2 + 3 + … , satisfy all the requirements of being reasonable, plausible, consistent with the rest of mainstream mathematics, and useful (in studies of real world phenomena, including physics). If you accept one, you really cannot reasonably deny the other. Rather, you have to accept the implications they yield, even if they at first seem counter to your expectations.

True, neither identity accords with our experiences in the physical world, since those experiences do not involve any infinite quantities or processes. (So there is nothing to accord with!)

One of the things surprising examples involving infinity remind us of is that mathematics is not “the true theory of the real world” (whatever that might mean). Rather, mathematical theories are mental frameworks we construct to help us make sense of the world. They survive or wither according to the degree to which they continue to accord with our real world experiences and to prove useful to us in conducting our individual and collective lives.

To return to geometry. For most people, throughout human history the geometry of the world experienced was planar Euclidean geometry, which accords extremely well with our everyday experiences.

But for the global air traveler (such as long distance airplane pilots), and for the astronauts in the International Space Station, spherical geometry is “the geometry.” In still other circumstances (for the most part, physics and cosmology), hyperbolic and elliptic geometries are the best frameworks.

For the artist trying to represent three dimensions on a two-dimensional canvas (or the movie or video-game animator trying to represent three dimensions on a screen), projective geometry is the best framework.

Picking up on my opening example, when you adopt a geometric perspective to try to understand growth in the natural world, you find that fractal geometry is the most appropriate one to hand.

And, finally, when you adopt a geometric perspective to try to make sense of social life in today’s multi-cultural societies, you may find that higher dimensional Euclidean geometries seem to work best, as I explain in this video (30 minutes) taken from a talk I gave at a conference in New Mexico earlier this year. (The relevant segment starts at 3:20 and ends at 11:00.)

The fact is, there is not just one geometry, and there is no such thing as “the geometry of W,” where W is a real world phenomenon or domain.

Likewise for other branches of mathematics we develop and use to understand our world and to do things in our world.

This means that, whereas, within mathematics there are “right answers,” when you apply mathematics to the world, that certainty and accuracy is only as good as the fit between the mathematics (as a conceptual framework) and the world.

And now we are back, more or less, at the topic of my previous Devlin’s Angle post. It merits a second look. Given the nature of the modern world, with mathematical models playing such a major role, with major consequences (in banking, information storage, communication, transportation, national security, etc.), we should not lose track of the fact that mathematics is not the truth.

Rather, it provides us with useful models of the world. As a result, it is a powerful and useful way of making sense of the world, and doing things in the world.

This distinction was not particularly significant for anyone growing up in the 20th century and earlier. Back then, there was usually no danger in viewing mathematics as if it were the truth. But it is an absolutely critical distinction to keep in mind for those coming of age today.

That New Mexico talk video I referred to a moment ago was in fact from a conference on middle school mathematics education, and was an attempt to raise awareness among middle school math teachers of the need to make their students aware of the way mathematics is used in the world they will live in and help shape, emphasizing not only mathematics’ strengths but also its limitations.

When you think about what is at stake here, much of the current debate (largely uninformed on the opposition side) about the Common Core State Standards resembles nothing more than two elderly bald men arguing over ownership of a comb.

In the case of the UK’s Falkland’s War of 1983, where this analogy originated, both sides appeared equally stupid. The sad aspect to the CCSS debate is that the level of ignorance (or malicious intent) on the “Stop” side forces many well-informed teachers and mathematics learning experts to devote time to the debate, lest ignorance prevail and our kids find themselves unable to survive in the world they inherit. (What the debate should focus on is how to properly implement the Standards. There be dragons, and someone needs to slay them.)

WORTH LISTENING TO: American RadioWorks has just aired an excellent radio documentary about the Common Core, in which we hear from real teachers who have been using it, both in states where it has been implemented according to plan and others where the implementation has been modified.

“Will this (mathematics) be of any use?”

Readers who follow me on Twitter will have noticed many tweets on the recent revelations about illegal NSA surveillance. Here is why I think that none of us in mathematics and mathematics education can ignore that debate.

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.