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.