PBS’s NOVA has a fascinating recent episode entitled The Great Math Mystery. The “mystery” explored in the episode is the question of whether mathematics are ontologically real or are merely a way that we classify and think about the universe. In simpler words, where is math? Is it something in your head, or is it something out there?
Appearing in the episode is a colorful MIT physicist named Max Tegmark who claims not only that math is ontologically real (let’s call this “Mathematical Platonism”…uh…since that’s what it is), but that math is the only thing that is real (a step beyond Mathematical Platonism to “Mathematical Monism”). As Tegmark says it, “Our physical universe doesn’t just have some mathematical properties, but it has only mathematical properties.” According to Max Tegmark, Reading and wRiting can all be boiled down into a single ‘R’—it’s all aRithmetic.
Is this true?
Let me begin my saying that, although Tegmark is a militant anti-theist, and he sometimes infuses his writings with his grinding of this particular axe, there’s no logical necessity that an utterly mathematical universe could not be a universe created by God. Whether my readers believe this to be true or false, the fact that I believe it to be true demonstrates that I do not labor to overthrow Tegmark because I believe that my faith rises or falls based upon the outcome. Rather, apart from my theism (although perhaps not entirely uninformed by it), I find Tegmark’s description of reality to fall far short of what we know.
This Much Is True
Although the math mystery is certainly abstract—esoteric, even—it nonetheless represents a very interesting question. How is it that mathematics in its most rarified forms, although we can trace the creation/discovery of it to various individuals throughout history, has succeeded so well in describing the universe? Is this just luck?
I mean, it’s not hard to believe that something like ‘5’ is connected closely with reality. At least, we can conclude safely that “fiveness” is a property of many real things (like the sets of fingers on each hand typing this). But what about the imaginary numbers, for example? How is something like i (the square root of -1) to be considered just as “real” as the natural numbers? And indeed, by standard mathematical parlance, such numbers are called “imaginary” as opposed to “real” numbers.
But we have accomplished astounding things with math. Using math, we can roll back the precise positions of the heavenly bodies back to the candidate dates for the birth of Jesus and speculate concerning the star of the magi. Using math, we can analyze and duplicate the tones of a Stradivarius. Using math, we can create virtually unbreakable codes for our communications. Using math, we can speculate about the nature of the universe on the smallest and the largest scales. Using math, Paul DePodesta did something truly important: changed Major League Baseball forever.
Math does amazing things.
And it proves to be true and useful. Math predicts. Math explains. Math corresponds to reality. We know that this is true, we just don’t know why.
Tegmark’s hypothesis is intriguing, but it suffers from some problems. The NOVA broadcast sought to mark out a few. I found their weather example to be poorly chosen. Our analysis of the weather is actually, in my opinion, a pretty formidable mathematical accomplishment. Weather forecasts fail not so much because the mathematical principles involved are flawed or mysterious as because our measurement systems are not nearly ubiquitous or accurate enough to acquire the necessary set of initial data to get highly accurate long-term forecasts. Other aspects of the NOVA critique, however, ring more true. We can affirm some of these and add others to them
First, math actually has not succeeded in describing the universe around us at the smallest or largest scales. At least, it hasn’t done so without requiring some faith from its followers that it will yet fill in the gaps. There must be “dark matter” not because we have observed it, but because math must be wrong if it isn’t there, and we cannot conceive that math might be wrong (yes, your Algebra test in eleventh grade had some wrong math on it, but that was because you applied it wrongly, not because it was flawed in and of itself). Math has not only taken us to the imaginary numbers, but also to string theories and singularities and Gödel’s Incompleteness Theorems and so forth. Some of these things seem to range beyond what is testable in the universe and a few of them, quite frankly, are downright counterintuitive. Enough of the formerly counterintuitive things have been proven right to give some people an utter, unshakeable faith that ALL of the mathematical quandaries will, given enough time, resolve themselves in this way, but it is faith nonetheless.
Second, some of the aspects of reality that give every appearance of being real also seem to be utter non-mathematical. NOVA cited human behavior. I’d add in the behavior of most higher animals. If math could predict human behavior, Tegmark and his MIT colleagues would stop writing books and spend a few days on Wall Street instead, after which they’d never have to work again.
What’s more, in a question related to the problem of consciousness, although math can tell us how music works and can sometimes predict what combinations of notes people will find pleasing, it has proven woefully inadequate to explain what it is to find satisfaction in Dvorak’s New World Symphony. Math can describe refraction of light at various frequencies but cannot convincingly quantify the human reaction to Edvard Munch’s “The Scream.”
Are these things less real just because they are less mathematical?
I don’t see how something like morality can be utterly quantified. I believe in something not too terribly far from Mathematical Platonism, but I reject Mathematical Monism. I think that ethics represents at least one other monad. There is “right” and there is “wrong.” These concepts, like higher mathematics, are ubiquitous across the full scope of humanity. Even if human beings disagree as to which actions are right and which are wrong, they generally do not disagree that these categories exist and they generally draw the same conclusions about what it means for something to be right and what it means for something to be wrong. Furthermore, every human being affirms some standard of right and wrong, and every human being violates her or his own ethical standard. Ethics is a ubiquitous system that reflects something real about our experiences of the universe.
When it comes to human behavior, ethics actually performs better than math at predicting what people will do. Knowing something about someone’s character puts people into a position of being able to predict—sometimes very well—what people will do in various situations. Math is excellent at telling us after the fact what people have done (see “Exit polling”), but it often fails miserably at predicting what people will do (see “Campaign polling”).
Third, beyond the rudimentary understanding of math that the NOVA program demonstrates to be present even in fairly primitive forms of animal life, it is difficult to demonstrate a reproductive advantage that would dispose unassisted Darwinian evolution to produce in us a reliable ability to discern the higher mathematical nature of the universe. I think we can all agree, in fact, that we learned pretty convincingly in high school that a mastery of higher mathematics actually puts one at a reproductive disadvantage. 🙂 Of course, one could argue that the development of higher mathematical skills is merely incidental to other, more reproductively relevant skills, but then why would we have any reason to trust that those skills are actually reliable?
Fourth, the connection between the “real world” and “mathematics” sometimes seems to flow the other direction: The world—humanity, even—has given birth to math rather than math giving birth to the world. Consider, for example, decimal mathematics. Of course, one can perform math in any number of bases, but we generally perform decimal math, primarily because we normally have ten digits on which to count. Computers, on the other hand, perform binary math, not because base 2 math is purer or better, but simply because electronics know on-off states rather than sets of fingers on pairs of hands.
Often, in the way that we experience it, math is shaped to fit us, rather than our being shaped to fit math. If the interface through which we engage mathematics is so obviously an adaptation that we have created, how can we be so certain that the underlying concepts are not adapted to our preferences or aptitudes as well?
A Conclusion a Christian Would Reach
Of course, one explanation of the Great Math Mystery is that both the nature of the universe and the brains we use to try to understand it are the creations of a common Creator who chose to enable our minds to be able to perceive the ontological nature of the universe in which we live. Math is the product of a God who loves order. Ethics is the product of a God who loves holiness. Uncertainty and incompleteness are products of a fallen, sinful order. Mystery is a product of our finitude.
Math reveals the ordered side of the universe. Ethics reveals the personal side of it. Together they point to the divine nature of our universe. There is reading and there is writing, and in them we find a glorious array of human thought and experience, both mathematical and otherwise. Much of what happens in reading and writing is impervious to mathematics. Much of what happens in the fourth ‘R’—religion—is equally non-mathematical. Well-rounded understandings of the universe, in my opinion, would do well to follow Stephen Jay Gould in acknowledging these various separate magisteria (I will not assert that they do not at all overlap) and in resisting the urge to consolidate them all into some unified theory regardless of how one might have to truncate reality in order to do so.