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.

### And Yet…

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.

Foghorn – Priceless!

Even more than the fellow on Smothers Brothers back in the sixties, the one who said, “Velly Interesting,” I might make a number of comments on this matter before David cancels the blog out of pure frustration at such esoteric intellectualism. Like everything else Mathematics has its orderliness and its disorderliness, the latter being the inability (temporary, I think), to provide the algorithms which explains everything, the theory of everything. However, there is a math for chaos, developed from the study of fractals. It is from the math for complexity that we come to understand how dynamic systems can be examined mathematically. The butterfly effect, wherein a butterfly flaps its wings in Brazil and a few weeks later we have a tornado in the USA. I think it was Galileo Galilei who said that Mathematics was the way to go to understand the universe or something to that effect. He was not thinking as the gentleman that you mentioned Bart. He was seeing the orderly precision as the reflection of Divine Perfections. The scientific method as Alfred North Whitehead pointed out really took its rise from the orderliness of the universe that God had created as something that could examined and explained. Thus, many of the early scientists, of both the Catholics and the Reformed, were not only believers; they were committed people of faith. After all, faith is really trust based on evidence, and, where the evidence is not available, it is the wherewithal to extrapolate from the known to the unknown with the expectation that it, too, will have some kind of orderly, mathematical precision to it. The problem is that our present scientific method suffers from the paralysis of analysis; it lacks the wherewithal to handle two or more apparently, antithetically, apt realities like the essence of our digitally real world of 01, 0110, et. al. To put it another way, the particle and wave realities of nature need a mathematical formula and theory that can work synthetically, experimentally, and rationally with a both/and like a plus and a minus, a positive and a negative, a high and a low, not to mention the jagged edges of seeming disorder (to which there is a mathematical orderliness just like everything else, with the proviso that it might be an order that virtually exceeds our grasp). I open a book, Synchronicity. written by Allan Combs and Mark Holland, and… Read more »

Pythagoras is certainly a fountainhead for this discussion. With you, I agree that math springs forth from the mind of God. God created a universe in which math plays a significant part.

The platonic ideals were perfect versions of flawed reality. Aristotle’s view that what is “is” and that there is no ideal that informs existence is more faithful to actual reality while perhaps not adequately considering the invisible, spiritual realm. Math as model is effective. Math as perfect description is very difficult to accomplish. But reality as the testbed of math and math as the underlying language of science generally serves human epistemelogical, deductive discovery awfully well while dependence on theological thought often strains or warps our ability to actually observe and reason about reality. That revelation and miracle perhaps have no corollary or place in scientific endeavors shouldn’t either surprise or offend any more than the “Preacher’s” turning of the phrase “under the sun” in Ecclesiastes in my opinion. Similarly, through semantic precision we can distinguish between commenting on the shared genetic structure that includes DNA and RNA (and therefore leading to a discussion of observed mutation and natural selection) and the inductive leap of faith to the assumption that God isn’t necessary if we can imagine a purely materialistic process for emergence of species. The latter might even be possible without ruling out the possibility of a God who creates everything. Which implied he participates spiritually in a material world he claims to have created. I mean, since God according to Jesus himself IS spirit and they that worship God must worship in spirit and truth, it kind of is a “duh” moment that there is some manner or conduit for the spiritual being that creates a physical, material world to affect it while having no requirement for a physical nature. But I digress and I would note this (in the spirit of the Scottish Commonsense Realists): what is, is. Math can be viewed as either being present in nature or providing a very rich language for describing nature. It doesn’t require a physical manifestation to be encountered and understood (though usually it is more easily comprehended via analogy to physical reality.) Therefore math in some rather interesting ways is an analog of how an invisible God not only invests in a physical reality, but also hints at how God’s nature sits just beyond the edge of direct human discovery in a land where duality disguises the nature of such things as waves and particles. Math perhaps is one of the most direct instruments for discovering awe in nature.… Read more »

That’s the problem with some of these fields–they try to boil down the sum total of universal explanations and human existence into their fields as if a single field holds the answer.

There’s a reason why we have mathematics, science, sociology, philosophy, theology, etc. as different fields–they all speak to different aspects of the Truth; but when singled out as the definitive and only field, then it misses a lot.

Side note: I’m really interested to know what their weather example was. From what you said here, it seems it has to do with forecasting…

The biggest problem is actually the complexity of the equations for three dimensional meteorological motion. We dumb down the formulas, essentially, through various assumptions that make them easier to compute. In other words, our biggest issue is that computer processing power has not caught up with the needs of the equations.

(I own a BS in Meteorology from the U. of Oklahoma–with a minor in math, btw 🙂 )

Hey, Mike. I remember that you had that meteorology degree. They said that math doesn’t work well to predict the weather. They said that flaws in the math prevent us from being able to make forecasts that go out further than a few days.

I’m simply making the point that there’s a difference between human behavior and the weather. The weather, I think, behaves mathematically. We can say that the math is complex. We can say that we have insufficient computing power to do all of the math in just the right way. We can observe that we don’t have a complete set of observations as initial data. I know that meteorologists like it when we pilots call in PIREPs, and I think that some commercial airplanes are set up to auto-report the weather through which they pass, because the data gathering for the weather is only at a relatively few (compared to the surface area of the earth) locations.

I know that we have a large number of surface observations here in the US, but I think, if my information is correct, that the closest actual skew-T chart (taken from a radiosonde, not computer-generated by interpolation) to my location here in Farmersville is sent up from the FWD office of the NWS. So, am I correct in saying that, as far as deep analysis of the atmosphere goes, we only get soundings at a few locations?

Also, meteorologists in this area have told me that one of the problems that makes DFW forecasting more difficult than, say, Chicago forecasting is that sometimes our weather comes out of the Chihuahuan Desert in Mexico, where the observations are not nearly so well distributed as they are in the USA.

In any event, I think both you and I would say that human behavior defies mathematical analysis because it is not, in its nature, a mathematical phenomenon. Weather trips up those trying to forecast it long-term not because the unfolding of the weather is not mathematical but because the math is very difficult and demands things (measurements, computation power) that we do not have available to give to the effort.

Many of my physics professors would wonder aloud in class regarding the impossibility of physical existence without the support of something beyond physics. The one atheist physics professor I had (the others were Catholic, Protestant, and Muslim) regularly expressed his lack of ability to understand why things were the way they were. Science goes so far and then hits a proverbial brick wall. It’s like looking into a black hole and expecting to see something. Math (and by extension logic) is merely a means for quantifying, qualifying, and categorizing what we know (or think we know) and generating meaningful conclusions from it. It is analogous to the nature of God over and against his creation. First, there is God. Second, there is that which is not God. And so we have bivalence in our ability to reason. Morality is descriptive of God’s nature. Inasmuch as we have a sense that some things are right or wrong, we can see that we have been created with a sense of who God is outside of the existential world. It is “personal” as you say precisely because God is personal and has made his image bearers morally personal as well. Being born disconnected from a personal relationship with God means that we have been born with a frustrated sense of morality. The noetic effects of sin distorts our ability to behave with a moral intent that resonates God’s morality as well as distorts our ability to reason well. It distorts our perceptions and desires by frustrating us with apparently unsolvable moral dilemmas. If there were no sin, the ethical calculus would work well. The equation would always balance and we would have no doubts regarding our motives and course of action. Since there is sin, the math never works out well. We can usually come up with something that makes the most amount of sense, but even if we do the right thing, we must contend with the fact that our hearts weren’t completely pure in the matter. If we understand at least this, then we understand our desperate need for God. Our confidence is not in our ability to understand or do the right thing. Our confidence is in God to do the right thing in and through us as we live in a state of confession to him that we cannot do the right things on our own accord. So the… Read more »

“The mystery of math is ultimately ethical.”

I like that!

C.S. Lewis addressed the notions of Natural law (mathematical) and Human law (ethical) from a philosophical perspective.

He argued that Natural law is nothing more than a description of how things are. Laws of gravitation simply say things like “More massive items have greater ability to pull on other bodies than less massive items.” This does not dictate their behavior but instead only describes it. I would guess from this perspective math reigns as the most objective descriptive language we have.

Human law, on the other hand, describes how are are supposed to behave. It attempts to require but does not; neither does it really describe. Human law primarily establishes an ideal and nothing more. Unsurprisingly, math can do nothing more than give best guess and statistical analyses of behavior that exists independent of any Human law.

It must be able to do a wee bit more. After all, Einstein sets forth e=mc2, and the USA invests big bucks in the working out of that equation and, presto, we have one of the more powerful weapons in history. I would say that is a bit more than being merely descriptive. It must in some sense be a reflection of reality, especially since we can build on such axioms and algorithms to make advances in science and technology. Philosophically, the idea that it is merely descriptive comes up wanting in the face of the advances that have been made. On the other hand, it still leaves us with more to explain than we can understand. Just consider this: How does one account for atomic explosions from the more maximum side of the universe as well as those from the zero point?