Our Mathematical Universe

It was recently brought to my attention that Russell Howell, Professor of Mathematics at the Christian academy of Westmont College in Santa Barbara (California) has taken notice of my article "Fundamental Flaws in Mark Steiner's Challenge to Naturalism in The Applicability of Mathematics as a Philosophical Problem" (2003). He only quotes (in fact somewhat misquotes) a single isolated comment from my article and ignores almost everything else it says, even though the ignored material contradicts his argument in ways he curiously neglects to mention. His paper, "Does Mathematical Beauty Pose Problems for Naturalism?" was published in the 2006 issue of the online Journal of the Association of Christians in the Mathematical Sciences (yes, Christians have an association for everything).

I won't bother discussing the rest of his essay, since my article against Steiner already rebuts the same thesis Howell defends, and more than adequately in my opinion. I am only interested here in Howell's lame mischaracterization of my arguments, insofar as he quotes me at all. Since I'm not the actual target of Howell's article, I'm only mentioned on page 9. He brings me up only when discussing Maxwell's use of a particular heuristic to discover electromagnetic radiation: by combining the assumption of a "conservation of charge" with mathematical descriptions of electrical systems that were already empirically established, he calculated (in effect) that energy should be leaking away from electrical systems (he was right: it was being converted into radio waves).

Howell is impressed that Maxwell did this by tinkering with mathematical descriptions, though as I argue in my article on Steiner, mathematics is nothing more than a precise language. It is thus no more amazing that tinkering with mathematical descriptions can discover the truth than tinkering with ordinary English descriptions can do so. Maxwell combined descriptions of conservation with descriptions of current and developed a new hypothesis worth testing. This is no different than combining "my wallet just went missing" with "pickpockets are often about" and developing therefrom a new hypothesis about what happened to my wallet. No one would conclude from this that the universe was anthropocentrically designed so I could discover pickpockets.

I suppose Howell misses this point because he thinks the word "mathematical formalism" conveys something special. But a "mathematical formalism" is still nothing more than a sentence in a language that describes an observed or hypothesized fact. It is only "formal" in the sense that it is descriptively more precise than, say, ordinary English. But it is otherwise no different. And insofar as anything that behaves consistently can be described using language, the fact that the universe can be described using the language of mathematics is simply a trivial consequence of the obvious, not a sign of anything deeply anthropocentric about the universe (as Howell might notice if he ever tried explaining how a universe would be incapable of being described mathematically). I say a lot more about these sorts of facts in my Critical Review of Victor Reppert's Defense of the Argument from Reason (2004) and in my book Sense and Goodness without God (chapters II.2 and III.5.4).

But now back to the point...
 

Getting It Right

Here is how Howell (mis-)quotes me:

Maxwell rightly picked the simplest imaginable solution first, which due to human limitations is always the best place to start an investigation, and which statistically is the most likely [as] simple patterns and behaviors happen far more often then complex ones. [Thus] Maxwell's moves [that] anticipated EM radiation [were] therefore a natural conclusion from entirely Naturalist assumptions.

Howell immediately responds:

But with such language Carrier plays into Steiner's hands. Picking a simple solution in accordance with human limitations is precisely analogous to using the number ten as a means of unlocking secrets to the universe. It is quintessential anthropocentrism. Because of Carrier's background in history, one wonders if it is difficult for people who were not trained in science to appreciate how absolutely uncanny is the continued use of mathematical formalisms by physicists.

That's the sum of his argument against me. Let's begin with the fact that Howell altered his quote of me in ways he did not indicate. What I actually said was (putting the material he omitted now in bold):

Maxwell rightly picked the simplest imaginable solution first (e.g. that it all went one place, rather than several), which due to human limitations is always the best place to start an investigation, and which statistically is the most likely (simple patterns and behaviors happen far more often than complex ones—since Maxwell's day, again, the discoveries of Chaos Theory have changed that assumption, but again only after vast amounts of empirical evidence confirmed and thus justified the change in our assumptions).

That Maxwell's moves anticipated EM radiation was therefore a natural conclusion from entirely Naturalist assumptions. Charge was going somewhere, which we knew because the descriptions of charge behavior that we had, which were empirically well-grounded, left out and thus entailed the disappearance (or spontaneous appearance) of charge, which begged for an explanation. Maxwell hypothesized such an explanation by making some simple and obvious changes to the descriptions that accounted for this discrepancy--changes to the way the pattern of behavior was described that allowed inclusion of another element to that pattern. The changes he made were the simplest ones he could make that didn't invalidate but instead preserved the predictive success of the existing descriptions, while also bringing them into line with conservation laws. And the changes he made were still, in fact, hypothetical. They could have turned out wrong, and many tinkerings with these equations, by him and others, no doubt preceded this success and failed. But on Naturalism, his final guess was a smart one, and one likely to succeed. So we should not be amazed that it did.

With a closer look you can see how Howell's omissions (which he fails to indicate with ellipses or even address in his response) reveal how off-base his remarks are. To begin with, of course, we do not have to assume the universe is anthropocentric to believe in conservation of charge, or to suspect that the charge disappearing from our equations can most likely be recovered in our descriptions if one thing rather than several is responsible for the discrepancy. To start with the hypothesis that charge is disappearing from our equations due to a single phenomenon (like Maxwell's hypothesized "displacement current") is reasonable in any universe, whether anthropocentric or not, because a single explanation will always be more likely than several explanations just "happening" to cause the same effect at the same time. Such an assumption is also the easiest place for a human to start--hence that Maxwell would start there is fully understandable.

Howell conflates these two explanations into one, ignoring my statistical argument and pretending I only offered an argument from human limitations. He also gets wrong the import of both explanations, by ignoring what else I say, here and in the rest of my article. As one can see from the full and correct quote, I did not argue that Maxwell succeeded because he chose a "simple solution in accordance with human limitations" but that this human limitation is what caused him to try a simple explanation first. As I allude already in the quote above (and give specific examples of elsewhere in the same paper, which again Howell ignores), this tactic does not always work, a fact that actually refutes Howell's thesis. For instance, we "tried" a simple chemistry of four basic elements first. But it turns out there are over ninety.

The reason we tried four elements first is the same reason Maxwell tried to find only one cause first. The cause he was looking for was of the spontaneous appearance and disappearance of electrical charge in electrical systems as then described. Maxwell hypothesized that this 'disappearing' charge was never actually leaving the system, which entailed that, instead, energy had to be leaking from the system, in one way or another. And he hypothesized only one leak: which he called "displacement current," and which we now recognize as "electromagnetic radiation" (i.e. light and radio waves). But in chemistry this same tactic (of trying the simple solution first) failed to align with reality, as it often does not.

Hence it was entirely possible that charge was not conserved (after all, we now know matter is not), just as it was also possible that charge was being conserved but that energy was leaking from electrical systems in two completely different ways at the same time (or three or ten or twenty). Maxwell guessed it was one, and got lucky. But his luck is not surprising, since statistics favor the simple answer even in a blindly operating, undesigned cosmos--for obvious reasons: absent deliberate design, the more complex a system, the more improbable it is (as advocates of Intelligent Design are always reminding us). The improbable is not impossible, just less frequent, but that still means we will luck out more often if we start with the simpler hypothesis and work our way up from there. And though the causes of individual events are always incredibly complex, constantly repeating events are generally the result of the predominance of a few simple causes. Only a cosmic puppeteer could make it otherwise.
 

Reality Isn't Pretty

Thus Howell's contention that simple systems imply anthropocentrism is baseless--and in fact a little bizarre. Since his thesis entails we should expect an abundance of simple systems only in an anthropocentric universe, Howell apparently thinks if we found a completely unanthropocentric, undesigned universe, it would be fundamentally more complex than the one we are in. That certainly sounds perverse to me. And rather indefensible.

Nevertheless, there will always be complex systems, as simple systems will randomly and catalytically combine and interact even in an unplanned universe. In fact, most of reality is an immensely complex fabric of interacting systems, which individually are simple but in aggregate are not. However, since humans are really only good at solving the relatively simple problems, the reason we have discovered so many "simple" laws is that these are the kinds of laws we have most often been looking for, and are most able to find. Meanwhile, most of the universe is actually governed by "laws" so complex we have made little progress in predicting even commonplace phenomena, like earthquakes, or the weather, or even, in most cases, human behavior (a note to the nit freaks: I only use "law" here, and throughout, to mean a consistently repeating behavior of matter and energy, in keeping with modern metaphor).

Consider another passage from my article that Howell entirely ignored:

But isn't it at least the case that scientists have found a successful scientific method in focusing on 'beautiful' and 'convenient' mathematical theories? Not really. Though that has been an effective heuristic for getting at simple and focused problems in comprehensible ways, this is simply the result of human limitations: we have to start small, and solve simple problems first, in the few ways we know how and are best at. But if we were to rely solely on this heuristic, most of the greatest scientific discoveries would never have been made. Far from a "beautiful and convenient" chemistry of four elements, we discovered in the end an incredibly ugly, messy, and inconvenient Periodic Table of over ninety elements and counting (never mind the mind-boggling complexity of the Standard Model of particle physics); far from the "beautiful and convenient" planetary theory of Copernicus, the paths and velocities of the planets are so ugly and inconvenient that we need supercomputers to handle the messy intersection of Newtonian, Keplerian, Einsteinian, Thermodynamic, and Chaotic effects, and even then they are not always entirely accurate in their predictions on astronomical scales of time (like thousands and millions of years).

I think this argument should be extended beyond particular examples to the whole of science, and that is what I will do here, to make even clearer the argument I make against Steiner, and am making now against Howell.

Take Newton's formulas for motion and gravity (which some people inaccurately call "Newton's laws"). Many have thought these are beautifully simple (though in practice they typically require the application of calculus, a method of mathematical analysis so complicated many humans can't even learn it), but we should not let their "beauty" distract us from the fact that nothing in the real world obeys them. Even apart from the fact that Einstein found Newton's formulas needed to be much more elaborate and complex, and even apart from the fact that the laws of thermodynamics and quantum mechanics complicate the application of simple equations like Newton's to real-world cases, even setting all that aside, any competent scientist will tell you that if you run the same experiment several times, e.g. dropping an apple from a fixed height, you will get different results every single time. We only find Newton's laws of falling bodies in this discordant data by averaging experimental results out and rounding them off. Yet in reality, a falling apple will sometimes fall faster, sometimes slower, and this will be noticed more the more precisely you measure its fall.

Why? Because the world is an extremely messy, complex place. The moon's gravitational effect on a falling apple, for example, is constantly changing, as is the sun's gravitational effect, and Jupiter's, and so on, and even the earth's, as magma and continents and oceans and masses of air are always on the move, and even the rotation of the earth is always changing, while friction against the apple in the air will constantly change in response to variations in temperature and pressure, and even the apple's shape and mass will constantly change (as it gets dented from repeated dropping or squeezing, and emits olfactory molecules, and collects or sheds dust, and absorbs or evaporates moisture, and even as light bounces off of it, and cosmic rays pass through it, and now radio waves, and on and on), and so on (a complete list of variables would be immense).

Consequently, Newton's equations for motion and gravity only apply to ideal situations, which never in fact exist. That humans choose to focus on the ideal as a means to get a handle on the complexities of the real world is a product of human limitations. But this means Newton's laws are essentially human fabrications. We made them simple on purpose. Because we needed them simple to be useful. The universe, however, is never that simple. It never anthropocentrically conforms to our ideals. This does not mean there is no objective truth to Newton's laws. Rather, it means their truth is similar to that of Euclid's geometry. As we now know, there are non-Euclidean geometries, and in fact the real world obeys them far more frequently (another example of things turning out way more complicated than humans first thought). But Euclid's geometry often works well enough.

Why? Because Euclidean geometry is a description of what necessarily follows for any system that conforms to its axioms, as in fact Euclid logically demonstrated. So the more closely a real system fits those axioms, the more closely Euclid's "laws" will describe that system. His geometry thus becomes a useful tool, provided we are willing to overlook all the little ways it never quite works. For example, no circle we draw is ever exactly perfect, so in the real world, the Euclidean law of circumference (2[pi][r]) will always be wrong, by some tiny amount. The choice to overlook this law's failure is a human choice, not one the universe makes. The universe is quite content with wonky circles.

That a system conforming to Euclid's axioms will also conform to Euclidean conclusions is a product of the fact that the conclusions are already inherent in the axioms. That humans have to engage tremendous labor to discover these consequences of those axioms is another example of human limitations, but since these consequences follow from those axioms in all possible universes, even universes that have nothing anthropocentric about them, the success of Euclidean geometry has nothing to do with the universe being anthropocentric. Instead, it has everything to do with our willingness to use such an imperfect tool to describe and predict a messy world, and even then this tool only works well enough when some part of the world just happens to almost conform to Euclidean axioms. When it doesn't, we try something else, whatever we find that happens to work. Hence if nothing ever conformed to Euclid's axioms, we would instead be talking about a geometry based on some other set of axioms, whichever set did occasionally conform to the world, at least near enough to be useful. Since every possible universe will have some geometry that describes it, it's just silly to act surprised when one does.

Newton's laws operate the same way. Like Euclid, Newton began with axioms. The most fundamental of these are more correctly called Newton's Laws of Motion, which were not mathematical formulas, but hypotheses stated in plain English (or Latin, as the case may be: for how they are stated in English see Newton's Axioms of Motion). Newton then argued that if these three axioms held (in conjunction with certain other conditions on a case-by-case basis), then certain consequences followed regarding the motion of objects in the universe, and this is where all his mathematical formulas come from. What is generally overlooked is that, unlike the conclusions of Euclid's geometry, Newton's three axioms don't suffice to generate any of the mathematical equations that are sometimes referred to as Newton's laws of motion and gravity. Those equations only follow when a huge number of additional assumptions are introduced, which have the deliberate effect of keeping the math simple. Those additional assumptions amount to hidden axioms, and these, like Euclid's axioms, never perfectly describe anything in the real world, and frequently don't even come close. Thus, reality is not making Newton's formulas "beautifully simple." We are.

If we chose to, we could build immensely complex (and thus hideously ugly) formulas describing the motion of objects, using the same three fundamental axioms, by incorporating all the incidental factors that change from moment to moment. And yet those ugly laws would be more accurate than all the familiar Newtonian formulas everyone finds so pretty. For example, we could add air pressure to the equations. We could add elements pertaining to the position and velocity of the moon and sun. We could add to Newton's laws variables pertaining to magma displacement in the earth's core, the absorption and evaporation properties of falling bodies, and whether KCET is broadcasting today and how far we are from its transmitters. But we choose not to.

Why? Because the simplest equations are good enough for most human needs. But the universe didn't choose that. It clearly prefers the reverse. Contrary to Steiner and Howell, the universe did not anthropocentrically choose the simple and "beautiful" Newtonian equations of motion. Rather than choosing to obey the simple equations, the universe chose to have bodies always falling according to the most complicated and ugly Newtonian equations imaginable. In fact, apples fall according to mathematical formulas fully beyond any human ability to discover much less work out and employ. But by sticking with the simplest equation, we get results "good enough" for us. And still, only in some cases. Sometimes we need messier equations, but even then we never end up with an equation that exactly describes what will happen. We always choose the simplest equation we can get away with. That has nothing to do with the universe being anthropocentric. It has only to do with humans being practical. Humans thus chose to break down the complex world into simple component behaviors, to make it easier on us. But the universe couldn't care less.

Thus it seems to me that Howell ignores the real world. When he asks whether my "background in history" makes it "difficult for people [like me] who were not trained in science to appreciate how absolutely uncanny is the continued use of mathematical formalisms by physicists," I see things quite the reverse: Howell's armchair obsession with numbers and formulas seems to have taken him so far from reality that he can no longer see how messy the real world actually is. He fails to see how we are the ones who broke everything down into its simplest components, so we could try to get a handle on complex phenomena by finding their simplest roots. Maxwell was likewise looking for the simplest elements underlying the behavior of electrical systems, and luckily stumbled on some. Not surprisingly, either, as it is inherently probable (especially in a mindless, unplanned universe) that any complex event will be composed of a large array of simpler ones, and that a repeating phenomenon will more often be the result of the repeating of fewer causes than the repeating of several.Hence there is nothing about Maxwell's discovery of electromagnetic radiation that "is precisely analogous to using the number ten as a means of unlocking secrets to the universe" (Howell does not argue that there is anything special about the number ten, he means only that such a thing would be anthropocentric if that were true). Simplicity of component parts just isn't the sort of thing that is unexpected on naturalism, nor is the human tendency to test simpler theories first, or to look for the simplest repeating causes.
 

In Sum

What the likes of Steiner and Howell fail to recognize is that there is nothing uncanny about the success of languages. We invented them to be successful. Mathematics is simply more precise than (for example) English, and thus scientists prefer mathematics as a language because they prefer being precise. That is a human choice. The only way the universe could thwart us in that choice is if it stopped behaving consistently, so that precision of description would no longer produce any benefit. Yet if there were any consistency at all, then more precise descriptions would still be more successful than less precise descriptions, and thus even in such a bizarro universe mathematics would work better than ordinary languages, and thus scientists would still prefer the math. Howell can only get us to a universe that couldn't be more usefully described mathematically by making that universe so fundamentally devoid of consistent behavior as to make life impossible there.

Sure, we can then ask why our universe behaves so consistently, but once we do that we are no longer talking about mathematics or beauty or even anthropocentrism. For merely being consistent is neither uniquely mathematical nor always beautiful, and such a thing does not entail any concern for man. Proposing a rule like "consistency only results from someone's desire" would simply beg the question against any evidence to the contrary. The consistency of the universe can logically have any number of explanations, and naturalists have many plausible hypotheses on the table, none of which are any more bizarre than "God did it" (a God, we must suppose, who is himself inexplicably consistent). And true, we can also ask why humans get so emotional about simple or economic mathematical equations, but I already have enough to say about that in my article on Steiner (though for more on why humans find simplicity beautiful, and why this is useful, check the index in my book Sense and Goodness without God for "beauty" and "simplicity").

I've said enough on Howell's abuse of my article. I suppose I should also chastise Howell for his arrogant assumption that I have no training in science. I happen to be a historian of science (evidently something he had not imagined). I'm a long-standing member of the History of Science Society and am completing my dissertation in ancient science. I have also formally studied mathematics all the way to the level of calculus. I have a semester of college credits in electronics engineering from the Navy, and worked in the field of electronics and sonar for the Coast Guard, studying for the latter the physics of sound. I also took a seminar in experimental laser physics, and majored in the sciences in high school (all my electives were science courses, receiving perfect grades in physiology, ecology, physics, biology, and other subjects), and in college I took (and received top grades in) several university-level courses in the sciences (including cultural and physical anthropology, theoretical and laboratory geology, and statistical mathematics). I have even published on a scientific subject in a peer reviewed journal (involving, incidentally, mathematics).

But such idle presumption doesn't bother me. Perhaps he would be happy to qualify his psychological hypothesis in just such a way as would exclude me from those who have what he wants to call "training in science," and then still claim that I fail to appreciate how successful mathematical descriptions of the universe have been. Such a conveniently qualified hypothesis would still be falsified by the fact that I clearly do appreciate that. I just understand it better than Howell seems to. My real-world experience may have something to do with that, though I don't presume to know. All I can say is that Howell sure sounds like someone whose ass is firmly in the armchair. Real scientists, for example, work out the kinds of observations that would falsify their hypothesis (in Howell's case, what the universe would actually look like if he was wrong), and Howell does not seem to have done this.Be that as it may, Howell's paper looks like the work of a hack to me. Deliberately ignoring all arguments and evidence contrary to your view in the very article you claim to be addressing, misrepresenting your opponents (especially by altering your quotations of them), conflating their arguments into a straw man, attacking irrelevancies while avoiding the most substantial points, and then pretending you've attacked something pertinent to your thesis, are all the marks of hack philosophy. But let anyone compare my article with his and judge for themselves.