In my last post, I explained how I think the modern evolutionary picture of the universe cries out, at least to me, the existence of a mind behind it all. In this post, I want to suggest that the intelligibility of the universe is improbable enough that a reasonable response is to think that it was intended to be understood.
Suppose I turn around one day in my kitchen and I see a mouse at the far end. I might like to get the mouse to cross my kitchen, follow me through the hall, where I will open the front door and have the mouse go past me, outside, at which point I will close the door again. That's a rather unlikely journey for a mouse to take. It's not going to do it on its own. And I don't speak mouse, so I can't negotiate. And if I come anywhere close to it, it's going to run and hide.
While I can't get the mouse to do the whole thing, I can get the mouse to move a couple of feet. If I keep my distance but toss some cheese towards it, it will come my way a foot or two to get the cheese. Step 1. Now I move into the hallway, laying a path of cheese bits at regular intervals. I open the front door and put a nice big cheese reward on the front porch. Then I get out of the way, and watch as the mouse follows the trail. Done!
This is how the history of math and science seems to me. How did we little mice make this journey from caveman ignorance to discovering the Higgs boson, or proving Fermat's Last Theorem?
Plimpton 322 is a 3700-year-old Bablylonian cuneiform tablet that contains a table of Pythagorean triples, numbers a, b, c that satisfy a2 + b2 = c2. Finding the length of the hypotenuse of a right triangle comes up in all sorts of practical applications, like carpentry and surveying, so it's no surprise the ancients were interested in the problem. The solution is certainly not obvious. I would assume that the path to it began with tables of measurements, and from there someone noticed the pattern and came up with the famous equation.
But imagine if the formula were 15 pages long, containing logarithms, trigonometric functions, sums of infinite series, integrals, etc. There's no way the ancients would have discovered it. But why isn't the world all that way? Why isn't every problem so complicated we couldn't possibly make any headway on it? The mouse will never move if every piece of cheese is miles away. But this formula is remarkably simple. And important.
G. H. Hardy, in his book, A Mathematician's Apology, calls it the most important theorem in the history of mathematics, and I would agree. It's not just that's it's both ancient and useful: it has also been a seed for new ideas far removed from its origins. One can look at it as a number theory problem: find all solutions where a, b, c are all integers. Generalizing this to higher exponents was the content of "Fermat's Last Theorem," and the pursuit of a proof of this has led to numerous new and useful ideas. In analytic geometry, it is used to define the Euclidean distance from the origin to a point. It generalizes to higher dimensions. Tweaking the formula opens up the door to other geometries, in particular the space-time geometry of relativity theory in physics.
This is so often the case. A discovery in one area leads to developments in others. The development of non-Euclidean geometries and Gauss's work on the curvature of surfaces in the 1820's led, in the 1850's, to Riemann laying the foundations for what would become known as Differential Geometry. This was a work of pure mathematics, but it turned out to be the natural language for Einstein's general theory of relativity: gravity can be understood geometrically as the curvature of space-time.
That has been the history of mathematics and other physical sciences. Once we find a piece of cheese, it turns out that if we look around, there's always another piece of cheese not too far away. I think of the work of Joseph Fourier, who struggled to create a mathematical theory of heat. Once he had his equation, the "heat equation," which is a second-order linear partial differential equation (PDE), he found that he could come up with individual trigonometric solutions, but that to have a useful theory he would need to expand a general function as an infinite series of such solutions. This was a ground-breaking idea, and now there are large books written about Fourier analysis, containing Fourier series, Fourier transforms, etc. The idea, which originated in physics, has turned out to be enormously important in many fields of pure mathematics.
In 1960, Eugene Wigner, a Nobel prize winner in physics, published an article entitled, The Unreasonable Effectiveness of Mathematics. Why should so much of the universe be ruled by second-order PDE's? As Wigner points out, "the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and [...] there is no rational explanation for it." (Link to article.)
I recall being struck by this mysteriousness even before I became a Christian. Newton's law of gravitation, an enormously successful theory, by which we still send satellites to distant planets, is summed up easily: the force of gravity between 2 bodies is proportional to the product of the masses, and inversely proportional to the square of the distance between them. Really? The mechanism of gravity is so simple? And it turns out that electrical attraction and repulsion follows the same kind of law. Because these were so simple, many problems in physics were reduced to calculus problems that could be solved with a bit of clever calculation. And so the Scientific Revolution took off.
One could play this game all day, making James-Burke connections between discoveries. My point here is that the very fact that we can do this shows how at every point in history there have been new discoveries to be made. People who have learned what others have done before them are poised to find out new things. At no point have we run aground, where every problem we look at is so intractably difficult we must surrender. There is always another piece of cheese in sight.
I loved the Jacob Bronowski series, The Ascent of Man, when I was a child. When we look at the history of science, as he did, from its humble beginnings to where we are now, it is hard not to see it as something like a ladder we have been ascending. He liked to focus on us human beings ascending, hoping that the values of scientific enquiry---openness, honesty, humility---would lead mankind to a morally as well as technologically improved future. Here, I want us to consider and wonder that there is a ladder at all.
Again, this is no proof of anything. But doesn't the physical universe seem like it's a giant puzzle to be understood? And generation after generation have been playing at it, making wonderful progress, but never coming to the end of the game. Some people will simply shrug their shoulders and say, "OK, that's the way it is." But, to me, the fit between the universe and our ability to reason, make analogies, experiment, and calculate mathematically, seems like looking at the coasts of western Africa and eastern South America. That they look like they fit together like jigsaw puzzle pieces could just be "the way it is." Or there could be an underlying deeper reason for it to be that way.