At futurism.com, there is a brief article explaining why the universe is mathematical, by saying, essentially just, that’s what we invented math for, to explain the universe. But this isn’t really an answer to the question. Theists have long used the lack of an answer to that question as an argument for God:
- Mathematics is inherently mental, therefore a mathematical universe can only come from a Mind;
- Our universe is mathematical;
- Therefore, our universe can only come from a Mind.
I’ve addressed this argument before, as presented by Russell Howell and Mark Steiner, for example. But I think those articles of mine are clunky and confusing. And discussions that ensued from them led to better ways to distill everything to clearer essentials. So here is an article to replace all of those.
The Steiner-Howell Silliness
Of course, right from the start, Steiner and Howell’s argument is formally invalid. Just because X can only be described with a certain kind of language, it does not follow that X cannot exist or be what it is without a language to describe it. In other words, they are confusing the tools of description with the thing described. Of course all languages in this sense are the invention of a mind. That’s why no descriptive languages existed before people did. Which is rather an argument against there being a god. After all, if there has always been a Mind, why aren’t we all speaking the same one language He invented? Why aren’t we just born knowing a language; why do we have to spend years learning one as a child? And why did we have to invent all the languages? And when we did, why did we come up with completely different languages? (The Tower of Babel tale is a silly myth. So if you are still wearing that tinfoil hat, you need to learn some science.)
We invented our languages specifically to accomplish the task of describing what exists. That a language succeeds very well at that task is indeed due to humans being very clever in perfecting language as a tool of description, but that’s as far as it goes. There is no evidence any language was invented by a Divinely Clever Mind. And that languages were intelligently designed by people to work well at describing what exists, is not the result of the universe being ‘suspiciously describable.’ It’s just the inevitable consequence of every godless universe that would ever contain intelligent life. Because all such universes are describable. So the probability of finding ourselves in one if a god doesn’t exist, is 100%.
To be taken in by this Steiner-Howell argument is like being amazed at what a miracle it is that Earth had oxygen, just what people needed to breathe—forgetting that we evolved to breathe what was here, not the other way around. Half the life on this planet evolved to breathe what was around before oxygen came along: carbon dioxide. And we would have evolved to breathe that, too, if that had remained the only breathable gas around. So there is nothing miraculous about the convergence of oxygen and oxygen-breathers. Just so, there is nothing miraculous about the convergence of a describable universe and a tool for describing it.
There is no reason to expect that any universe would be somehow “undescribable.” Any godless world capable of sustaining life will logically, necessarily contain quantities and ratios of quantities. There is no logically possible world that doesn’t contain those but does contain life without a god creating or maintaining it. And mathematics is just a language for describing and analyzing quantities and ratios. Therefore, all atheist universes will be mathematical. There is really nothing more to explain. But let’s unpack this further and see what it means.
Mathematics as a Language
Math is just a language, no different from any other (like English or German) except in two respects: component simplicity and lack of ambiguity. This is, in fact, all that distinguishes mathematics from any other language. And it differs in those two respects because we wanted it to: we designed mathematics as a language specifically so we would have a language with those two properties.
Component simplicity is a defining attribute because math is a language we created for describing the simplest elements of complex patterns: primarily, quantities and relations. Every “word” in mathematics refers to a very simple pattern (like a number or a function or a quantifiable relationship). Unlike English, where most words refer to very complex patterns (like “tree” or “theorem”). Of course, mathematical sentences and paragraphs can be immensely complex. But this does not change the fact that the words are all simple, literally as simple as any words can be. And that is what makes mathematics what it is, as opposed to English or German.
And the lack of ambiguity defines mathematics as a language because every mathematical word is deliberately defined by humans in an absolutely precise way, leaving no room for optional interpretations or ranges of vagueness. This is what math was invented for: to escape the perils of imprecision and vagueness, which are a feature of other languages. Thus, while in English “tree” can refer to a plant or a hierarchical list of computer files, and while applying the word “tree” might run into uncertainty when we are looking at something that straddles the properties of a tree or a bush, in mathematics no such problems arise—because we humans made sure of it. We wanted one, so we created one: a language without ambiguity. And we called it mathematics.
Some people might restrict the definition of mathematics further, thus distinguishing math from logic, by stating that mathematics is a language dealing solely with quantities, relations, and functions applied thereto, whereas everything that is simple and precise but not a quantity, relation, or function falls into the subject of logic instead. But many mathematicians regard logic as a special subset of mathematics, since anything said in a language of logic can also be described in the language of mathematics. However, in practice, logic is used as a bridge for extending the defining advantages of mathematics (simplicity and precision) to normal language and ordinary human thought. In that respect, logic straddles the spheres of math and ordinary language. Either way, though, mathematics is definable, either broadly as a language characterized by component simplicity and lack of ambiguity, or narrowed further to just the domain of describing quantities, ratios, shapes, and other physical patterns of arrangement.
And thus that we can invent such a language and find it useful, simply means we live in a universe where there are quantities, ratios, shapes, and patterns, that are composed of simple and precisely definable components. No God needed. That we would evolve the ability to invent languages to describe things is easily explained just on its rudimentary survival advantage. And it’s inevitable that any godless universe we would find ourselves in would be constructed of quantities and ratios of things.
How We Got There
Mathematics has to be learned. And before it could be learned by anyone, it had to be invented. And humans went hundreds of thousands of years without inventing it, and thus without learning it. But the more we tried to describe our world with precision, the more we discovered by trial and error that we needed a language that can describe quantities and ratios, and shapes and patterns, without ambiguity. Because that’s what every physical universe will consist of: quantities and ratios and shapes and patterns of things. In fact, all randomly generated universes without gods in them will consist of nothing but that (and all the things those then construct and manifest).
As I wrote in Loftus’s anthology The End of Christianity (pp. 301-02)
[H]umans evolved to understand the world they are in, not the other way around. Even then, the universe is so difficult to understand that hardly anyone actually understands it. Quantum mechanics and relativity theory alone try the abilities of someone of above average intellect, as do chemistry, particle physics, and cosmological science. Thus neither was the universe designed to be easily understood nor were we well designed to understand it. We must train ourselves for years, taxing our natural symbolic and problem-solving intelligence to its very limits, before we are able to understand it, and even then we still admit it’s pretty darned hard to understand.
If you have to rigorously train yourself with great difficulty to understand something, it cannot be said it was designed to be understandable. To the contrary, you are then making it understandable by searching for and teaching yourself whatever system of tricks and tools you need to understand it. Our ability to learn any system of tricks and tools necessary to do that is an inevitable and fully explicable product of natural selection; that ability derives from our evolved capacity to use symbolic language (which is of inestimable value to survival yet entails the ability to learn and use any language—including logic and mathematics, which are just languages, with words and rules like any other language) and from our evolved capacity to solve problems and predict behaviors (through hypothesis formation and testing, and the abilities of learning and improvisation, which are all of inestimable value to survival yet entail the ability to do the same things in any domain of knowledge, not just in the directly useful domains of resource acquisition, threat avoidance, and social system management).
Thus the actual intelligibility of the universe is not at all impressive, given its extreme difficulty and our need to train ourselves to get the skills to understand it—indeed, our need even to have discovered those skills in the first place: the universe only began to be “intelligible” in this sense barely two thousand years ago, and we didn’t get much good at reliably figuring it out until about four hundred years ago, yet we’ve been living in civilizations for over six thousand years, and had been trying to figure out the world before that for over forty thousand of years. Given these facts (our universe’s actual intelligibility), [the intelligent design of the universe] is actually improbable: the probability of the degree of intelligibility we actually observe is 100 percent if there is no [intelligent design], but substantially less [likely on the assumption it was designed to be understood].
It is almost certainly far less, since a God could easily have made the world far more intelligible by making the world itself simpler (as Aristotle once thought it was), or our abilities greater (we could be born with knowledge of the universe or of formal mathematics or scientific logic or with brains capable of far more rapid and complex learning and computation, etc.), or both, and it’s hard to imagine why he wouldn’t. God gave us instead exactly all the very same limitations and obstacles we would already expect if God didn’t exist in the first place.
In fact, we are actually pretty badly designed to correctly understand our universe, which is why it took us hundreds of thousands of years of failing at it before we ever got anywhere near good at the task. So rather than looking like we were intelligently designed to understand the universe, we look exactly like we weren’t.
Which looks exactly like there is no God.
Hmm. I wonder what the simplest explanation is for why the universe looks exactly like a universe with no god in it? Three guesses. (See my article on Bayesian Counter-Apologetics.)
What About Mathematical Beauty?
As I also wrote in that Loftus anthology, “Humans evolved to see beauty in certain properties of the universe (including the beauty of languages, efficiency, and puzzle solving, three skills of incalculable value to differential reproductive success), not the other way around.” That’s why we should conclude “the universe was not designed to be beautiful,” but rather “we were designed to see it as beautiful—by natural selection.” Steiner and Howell don’t get this. They think these are supernatural signs of God’s design. After all, they argue, why would the mathematical solutions humans find beautiful happen to align with the mathematics that’s correct? Surely only gods could arrange that!
This is of course all myth. They claim scientists have found a successful scientific method in focusing on ‘beautiful’ and ‘convenient’ mathematical theories. But that’s not true. 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. And we have the inclination toward simple and tidy solutions, solutions that don’t have unknown complexities or uncertainties, because our brains evolved a cognitive bias toward exactly those properties, which is actually not wise. Preferring simple solutions, preferring everything be certain and answered, makes our lives easier in a primitive context (and thus has some survival value), but isn’t actually a sound method for acquiring true beliefs about the world. If we were to rely solely on this heuristic, most of the greatest scientific discoveries would never have been made; and we’d be stuck believing numerous hopelessly false things—like Plato’s “beautiful” mathematical theory of the elements, or the many “beautiful” mathematical systems of astrology and numerology.
Far from a “beautiful and convenient” chemistry of four elements whose atoms could be described with a simple and perfect geometry, we discovered in the end an incredibly ugly, messy, and inconvenient Periodic Table of over ninety elements and counting (never mind the bizarre complexity of the Standard Model of particle physics that underlies the Periodic Table), which form in turn countless molecular elements and mixtures, whose geometry is mind-bogglingly complex and not even precisely definable in any way that would have pleased Plato’s senses. And far from the “beautiful and convenient” planetary theory of Copernicus (which was in fact such a bad and inaccurate model, it actually was substantially less successful in a predicting planetary positions than Ptolemy’s geocentric model; and geometrically was literally false), the paths and velocities of the planets are so ugly, complex, 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).
Apart from a mistake-prone preference for simple and tidy solutions that we acquired from our clunky hominid ancestry, any remaining notions of mathematical beauty are culturally invented and learned, not biologically innate. What works well, was decided should be well regarded; and consequently, we learned to react well, to what works well. That’s why equations that are nightmarishly ugly to most people, are regarded as “beautiful” and “elegant” only by mathematicians. That’s not innate. It was trained into them. And what is innate, is a reverence for simplicity and tidiness, and delight at unexpected solutions, all features we evolved for completely different purposes, and which in physics, as I’ve noted, can often lead us astray.
Christians Suck at Probability
In truth, you can use any method you want for discovering facts—including picking ideas out of a hat. As long as the result bears out in empirical tests, it is acceptable. I do not have to “assume” that there is anything mystically efficacious, anything metaphysically rational, about picking ideas out of a hat. I can still do it, simply because it is easy, and I know I don’t have to trust any results until they bear out in tests anyway. And the fact is, even such a totally random method will produce successes. And only the successes will get published and thus heard of.
A Steiner or a Howell would then come along, see that all the scientific discoveries in print came from picking ideas randomly from a hat, and foolishly conclude that there is some mystical power inherent in hats to produce knowledge of the universe. They would say the universe had to be Hatrocentric to explain this phenomenon. But they would be wrong. They would have forgotten to consider the hundreds of hatpicked ideas that fell by the wayside. Thus, even if it were the case that scientists have been using a heuristic that was contrary to the metaphysical assumptions of physicalism, it would neither follow that they were acting irrationally (since they need not assume their heuristic has a metaphysical basis—we do what is easy and engages us, what we know how to do well, because we’re human) nor that the universe was somehow metaphysically linked with that heuristic (since even a totally random method will score hits, and only successes then survive in the historical record).
But additionally, not only would their fallacious reasoning fool them into believing in the supernaturally god-designed powers even of hats, the actual heuristics that impress them are not random hatpicks, nor contrary to the metaphysical assumptions of physicalism.
Howell and Steiner are especially amazed and astonished that James Clerk Maxwell was able to discover electromagnetic radiation simply by analyzing some mathematical equations describing the behavior of electrical currents. He found that the equations when combined described a system that was losing electrical charge, but without describing where it was going. This could have been accomplished with plain English as well; there was nothing special about mathematics here, except in the one thing we designed mathematics for: to be precise with only simple components. Thus making things like disappearing quantities more obvious.
That’s why it is no more amazing that tinkering with mathematical descriptions can discover the truth about a system they describe, than that 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; much less deduce therefrom that the English language was intelligently engineered into the structure of the universe. The inference is silly.
To start with the hypothesis that charge was disappearing from any systems described by both those equations would have been wise in every possible universe; because that’s what they described. Maxwell simply noticed something about those systems as described that others missed; and all he used to notice that, was the description of those systems, based solely on human observations, using a language humans invented to precisely describe what they observed.
And then, to start with the hypothesis that charge was disappearing due to a single phenomenon (like Maxwell’s hypothesized “displacement current“) is also reasonable in every possible universe, whether godless 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. Because the latter requires more coincidences, and thus is necessarily less likely—certainly in every possible godless universe, where no intelligence can mess with random chance. Thus, such an assumption is not contrary to godless physicalism; it’s efficacy is entailed by godless physicalism.
Such an assumption is also the easiest place for a human to start—hence that Maxwell would start there is fully to be expected, and would be in every possible universe, godless or otherwise. And being the first place anyone would start, statistically most discoveries will occur upon that very assumption, simply because those are the explanations we are most commonly looking for. And therefore they will always be the ones we most commonly find. And the method will often still fail, as it did with Plato’s “first try” at a chemistry of four simple elements. But, again, we ignore and thus forget to count all the millions of “simple” explanations scientists tried that ended up being false, leaving us to be “amazed” at the successes.
The reason we tried four elements first is the same reason Maxwell tried to find only one cause of a missing current first. The cause he was looking for was of the spontaneous disappearance of electrical charge in contemporary descriptions of electrical systems. Maxwell hypothesized that this ‘disappearing’ charge was never just vanishing; which entailed that, instead, energy had to be leaking from the system, in one way or another. And he first hypothesized only one leak: which he called “displacement current,” and which we now recognize as “electromagnetic radiation” (i.e. light and radio waves). And still it wasn’t declared a confirmed science until his guess was verified in empirical tests. 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. This will be true in every godless universe imaginable. 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. In other words, only in a world made by a God could simplicity fail as a heuristic.
Thus Steiner and Howell’s contention that simple systems imply an anthropocentric universe is baseless—and in fact a little bizarre. Since their thesis entails we should expect most things to be reducible to an abundance of simple systems only in an anthropocentric universe, they apparently think if we found a completely unanthropocentric, undesigned universe, it would be fundamentally more complex than the one we are in. That hardly sounds logical to me.
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, precisely owing to our limitations. Meanwhile, most of the universe is actually governed by “laws” so complex we have made little progress in predicting even commonplace phenomena governed by them, like earthquakes, or the weather, or even, in most cases, human behavior.
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 give up even learning 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, for instance 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. It never really obeys Newton’s Laws. 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; and it does correctly describe a part of what is actually going on.
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, of (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. Because we are so limited, we couldn’t handle the real math.
If we chose to, we could build immensely complex (and thus hideously ugly) formulas describing the motion of objects, using the same three 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 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 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 limited.
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.
But Where Do All the Calculators Go?
I’ve written on the ontology of mathematics, and “numbers” specifically, already. Just look up “numbers, nature of” in the index of Sense and Goodness without God. As I already explained there, and just explained above, mathematics is simply a language for describing patterns (such as shapes and structures). See Resnik’s Mathematics as a Science of Patterns for a thorough peer reviewed defense of this point (which grounds my own mathematical realism, Aristotelian structuralism). But because people have a reification bias, they think what they imagine as disembodied, actually is or can be disembodied, when in fact it is always embodied: by their brain, the scaffolding that makes imagining anything possible, which they just can’t see (see The God Impossible).
This is what causes mathematical superstitions like the mistake of Platonizing numbers, imagining numbers are cosmic things “out there” somewhere, that they can’t possibly be human inventions, and must exist in some cosmic Mind Box that we mystically access with our psyches. People do that intuitively, often unaware of other theories that are far more scientifically and empirically plausible, such as Nominalism or Formalism or the Aristotelian metaphysics of mathematics that I defend in Sense and Goodness. Another example of a human attraction to simple explanations that misleads us.
There is no difficulty reducing numbers to nonmental things. There is no need of Plato’s silly metaphysics. Even so-called Platonic naturalists admit there is no way immaterial things (like Platonic numbers) can cause anything to happen (like our being aware of them), yet that renders Platonic naturalism incoherent. If immaterial abstract objects cannot cause us to know about them, how is it that we know about them? Answer: Because they aren’t immaterial. (See “abstraction and abstract objects” in the index of Sense and Goodness without God).
In Defining Naturalism, I addressed the question of how “numbers” can exist without a mind. Actually, numbers are words, and thus always the invention of a mind. But like all words, they can refer to things outside of minds. In the case of numbers, what they refer to is quantities and ratios. And quantities and ratios are real physical things. All universes unmade and uninhabited by any God or Mind will contain quantities and ratios. This is logically necessarily the case, unless you include as a “possible universe” the complete lack of any extension of space or time or substance whatever. But that describes a literal nothingness (even more nothing than empty space; because we are talking about the absence even of space), which should, honestly, be accounted the opposite of a universe. Nothing is the absence of a universe.
So when you decide thus to sensibly exclude the absence of a universe in the set of all possible universes, you cannot avoid the outcome: all universes contain something (space, or time, or material, or any combination thereof), and that something always exists in actual and potential quantities. If there is any amount of space, there is therefore a quantity of space; and that quantity of space can be divided, and it can be expanded, and thus quantities of it always exist in ratio to other quantities of it—if not actually, then always potentially.
And in Aristotelian physicalism, that’s all there is: actual and potential physical objects. And the potential for an object always exists necessarily in any actuality that can potentially be reshaped into it. If there is a two inch radius of space, that space inalienably contains the potential to become a four inch radius of space. Even if no physical means can be found to stretch it, even if it never is so stretched, the very existence of any quantity of space entails the potential for there to be more of it. At the very least, the logical potential exists. And as such, it can be referred to with words, which refer to that logical potential.
Thus, as soon as there is any quantity of godless space (or of anything whatever: any quantity of godless time; any quantity of godless bosons or leptons or fermions), all of mathematics is logically entailed. You can refer to the logical potential of that space to be reshaped, doubled, or split, of it to be stretched to infinity, or crushed to an infinitesimal, of it to be duplicated, or deleted. It would be logically impossible for you not to be able to deduce all of those potentials, and thus the entirety of any mathematics, from even the tiniest quantity of the most godless anything. You might be too dull or lazy to accomplish that deduction in practice. But if you were at all smart and industrious enough, there would be nothing to stop you figuring it out eventually. And at no point would any God or Great Mind have to exist for that to be the case.
What About Numbers I Just Made Up?
In Defining Atheism I pointed out that The Teapot Atheist couldn’t grasp this, and instead committed a common philosophical error, the error of not thinking things through, when he said “I have never encountered 326,519,438.004 objects.” Yet in a sense he just did: in his physical, reducibly nonmental brain. “Nor is there anything physical about it.” Yet there is. The word—that number he just contrived—refers to physical facts.
The bulk of it (the “326,519,438” part) is a count of distinguishable objects. Any godless objects can exist in that quantity. In this universe, they certainly do (this universe is well large enough for such a quantity to physically exist somewhere; indeed, practically everywhere). But even if he picked a quantity far greater than might actually exist in this universe, the potential still exists. This universe, by containing actual quantities, also contains all potential quantities: just by adding the quantities there are to each other until you get to the quantity you are looking for. Just by duplicating the contents of this universe enough, you get there. And once something actually exists, the logical potential of duplicating it exists—because there is no possible way to argue that duplicating it is logically impossible (and yes, even infinities can be duplicated; the results are just different than for duplicating finite quantities).
And the rest of that number The Teapot Atheist came up with (the “.004” part) also refers to a physical fact, actual or potential. Being a fraction, it refers to a ratio of two quantities, and quantity is, again, a fundamentally physical property: space, time, matter and energy, all necessarily entail that property. A fraction like he proposes can refer to some actual fact (odds are, such a ratio exists between some two objects in this universe—because there are so many objects in this universe to stand in ratio to each other) or a hypothetical fact (such a ratio can exist between two reducibly nonmental objects, without requiring anything irreducibly mental, e.g. I could cut two wires right now that have that ratio between them, even if, implausibly, no such ratio existed in this universe before). Thus, the potential ratio always exists. As soon as there is any actual quantity that exists. And no Mind is needed for quantities of things to exist.
By analogy, there are no unicorns, either, yet the word “unicorn” still refers to a (hypothetical) physical fact: unicorns exist if the physical (and thus reducibly nonmental) entity described by that word exists. And if they don’t exist, the thought of them exists in the brain that thinks it, and as long as the brain doing that is reducibly nonmental, then by the law of transitive property, so is the idea of a unicorn. QED. What about things no one has thought of yet, but that are logically possible? They exist potentially. What does it mean to potentially exist but not actually exist? It means a physical universe is capable of producing such things in the right conditions (as long as there are logically possible conditions that would have that effect), yet those conditions still do not require anything irreducibly mental.
If I have a gold ring, a gold cube potentially exists. Because I can mash it into a cube. But I require no supernatural power to do that. I don’t even require the physical ability to do it; the ability remains logically possible (because you cannot demonstrate it to be logically impossible). Nothing irreducibly mental need exist for a ring of gold to be potentially a cube of gold. Thus, potentially existing things are also not irreducibly mental. Hence the same follows for a quantity like 326,519,438.004: if not referring to an actual quantity (like the distance between two stars in ratio to some arbitrary unit of distance), then it refers to a potential quantity, and potential things are not irreducibly mental, nor require the irreducibly mental (Sense and Goodness without God, pp. 125-26).
The Teapot Atheist was still perplexed at how he can think of a number but not what it describes. But that should not be perplexing. Because a potential ratio can be a ratio of anything possessing the property of quantity. Hence, obviously, when we think solely of the ratio, we leave blank what it is a ratio of. We thus “abstract” the ratio from its particular instances, and the resulting impression is of a number divorced from any physical fact. But that’s an illusion, if we take it as anything other than a formalism of physical computation. For example, “x:y = 326,519,438.004:1” where x and y = “wires or sticks or pounds or persons or…[ad infinitum],” there being too many possibilities to state or imagine all at once, so we don’t. This was explicitly stated and explained by Aristotle over two thousand years ago. Someone really ought to get the memo.
Even when abstracted, a word like 326,519,438.004 is meaningless unless it describes some actual or hypothetical ratio between physical quantities. The blank must be potentially fillable. Otherwise numbers would be meaningless sounds. (I discuss this mistake more broadly in Sense and Goodness without God, pp. 31-32, essential to which is the whole discussion of pp. 29-35). If, on the other hand, 326,519,438.004 “exists” independently of our physical minds (or any other physical computer), and also independently of any actual or even potential physical quantities, then it would certainly be supernatural. Because there could then be no other explanation for how or why it existed at all. Failure to face the consequences of that fact can only make naturalists look ridiculous. But as it happens, numbers are just human words that refer to actual or hypothetical quantities, and all quantities potentially exist in any actual quantity of anything. Failure to face the consequences of that fact can only make supernaturalists look ridiculous.
Hence no mathematical number or concept has any meaning, if it cannot describe some logically possible physical structure.
Negative numbers for example, are subtraction operators (e.g. physical losses or debts) or relative position or direction indicators (e.g. negative temperatures describe the relation of one physical quantity to another: how much colder the air is than water at its freezing point, for example; negative altitude, how much farther below sea level an object is located; etc.). So negative numbers also describe real physical things, potentially or actually. Can you have less than nothing? Yes, when you owe money. Or travel not zero feet toward your destination but a negative ten feet toward it—which simply means ten feet away from it.
And yes, this means even imaginary numbers correspond to potential (even real) quantities or ratios of physical things, as does every cardinality of infinity or infinitesimal.
Imaginary numbers all reduce to multiples (quantities) of a singular quantity: the square root of negative one. And that refers to a rotation operator in physical space (actual or hypothetical, i.e. potential). So-called “imaginary” numbers are actually real numbers rotated out of a given number line. Read An Imaginary Tale for the whole story on this. Or read or listen to Kalid explaining it at Better Explained (he does an awesome job at it, IMO). And yes, that means imaginary numbers, aren’t really imaginary. They manifest in real physical systems, like the behavior of electromagnetic fields (corresponding to a geometric space-time shape).
Transfinites, similarly, are physically realizable. As a simple example, the only reason the axiom of infinity is accepted in mathematics is that an extension of a set by unending iteration is conceivable; and it is only conceivable because we can imagine a physical operation carrying it out, such that nothing would logically prevent it continuing (even if something might contingently do so), and we can sum that series over an infinite timeline. This is already entailed physically by Relativity Theory, which informs us that light can traverse an infinite distance in both space and time instantly—relative to the photon’s reference frame, and hence from the POV of the four dimensional structure that thus results. And sorry, William Lane Craig, this means actual infinities are not only metaphysical possible, they are actually realized in our universe.
Indeed, there is at least one actual physical fact that is infinite, and it’s a fact from which infinity is axiomatically constructed in set theory: the number of empty sets there are is always infinite in any definable region. For example, the number of empty spaces in the solar system we can potentially put a border around, or the number of dimensionless points we can potentially count along the edge of a common ruler. Though this derives from the potential fact (of our counting, for instance), the physical fact (that which we would be counting) is actually physically there (not potentially there, but in fact actually there). Hence though counting it up would be a process, it’s not as if our counting creates the things counted. The things to be counted are already there whether we count them or not. And those things are infinite in quantity. Hence actually infinite sets exist.
Thus, in any physical universe, anything mathematicians can intelligibly imagine will refer to some potential physical fact (like endlessly counting the same stone, or rotating outside of and back to a linear number line).
Why Physics Is Always Mathematics
The laws of physics are always mathematical, because what we mean by “physics” is a study of the quantitative relations and behaviors of things in nature. All one needs is physical objects, behaving in consistent ways. That will always be describable mathematically. In every possible universe. Including the godless ones.
Consider the Submersion Law of Hydrostatics. This was discovered and deductively proved by the Sicilian engineer Archimedes over two thousand years ago. And none of the premises from which he formally deduced it was “God exists.” Instead, the premises consisted solely of descriptions of physical objects and their consistent behaviors. And from that alone the mathematical laws of hydrostatics could be logically deduced. It only remained to test the predictions of those laws, to see if the premises (and thus the conclusion) held in nature, and not just in Archimedes’ imagination. Lo and behold, they did.
Archimedes’ law could be stated today as: the weight of an object submerged in water will equal its weight in air minus the weight of an equal volume of water. Mathematically: W[air] — (V[object] x D[water]) = W[underwater]. The Weight of an object in the open air, minus the product of the Volume of the object and the Density of water, equals the Weight of the same object underwater. Notably, when the water weighs more than the object, the product of this equation is a negative number, so the object actually has negative “weight” under water. Which means instead of falling down, it falls up. In fact, it keeps falling up until the weight of the object equals the weight of the volume of water it displaces. Thus explaining why objects float. And exactly predicting how high they will float—as everyone observes at the docks, as a boat gets lighter, it rises higher above the water.
Now, is it surprising and unexpected, is it a “mystery,” that we can describe how objects behave in water with mathematics? With the equivalent of a single equation, Archimedes was able to explain not only why objects float, but he was able to predict that even submerged objects would weigh less, and exactly how much less they would weigh. He was thus led to an amazing discovery about the physical universe, all just by manipulating a mathematical equation. Of course Archimedes wasn’t using equations in our sense, but he was using geometric equivalents. Modern equations are just a way of simplifying the notation, just like shorthand is a way of simplifying sentences in English. The equation is describing the same physical geometrical facts that Archimedes was. It’s just a different way of writing it down.
Archimedes formally deduced this Submersion Law of Hydrostatics from the axioms of geometry (established by Euclid) and a single physical assumption, called Archimedes’ First Postulate:
Let it be supposed that a fluid is of such a character that, its parts lying evenly and being continuous, that part which is thrust the less is driven along by that which is thrust the more; and that each of its parts is thrust by the fluid which is above it in a perpendicular direction if the fluid be sunk in anything and compressed by anything else.
In other words, all Archimedes assumed are three physical facts: that a fluid maintains it’s volume under pressure, but transmits all pressures placed upon it; that greater pressures overcome lesser pressures; and that all things are pulled downward at a constant pressure (in other words, that a constant gravity exists). In his treatise On Floating Bodies Archimedes formally proves all the basic mathematical laws of hydrostatics from the axioms of geometry and this single postulate. (He relies on only one other postulate regarding centers of gravity, but that second postulate also follows from the axioms of geometry, as he proves separately in another treatise called On the Equilibrium of Planes).
This means Archimedes proved that the laws of hydrostatics will be true in every possible universe in which the axioms of geometry and his first postulate are physically true. He thus proved that mathematical laws will exist in every physical universe that has those same physical properties: three spatial dimensions, a gravitational force, and physical fluids and solids. Wherever those three things exist, his laws of hydrostatics will exist, and will be described in the same mathematical terms. And yet nowhere in his proofs do we find anything about minds or gods or anything at all except physical facts: the physical facts of fluids and solids and gravity, and the physical facts of physical space.
In fact, Archimedes’ proof shows that even in a purely physical universe it would be logically impossible for fluids to behave in any other way than mathematically. And we can see why. As long as every liter of water pushes as hard as every other, and as long as a stone block immersed in that water pushes by any constant amount, then nothing will prevent this system from following the mathematical behavior Archimedes deduced. In other words, as long as the weights and volumes remain constant, the mathematical law results. Always. So all you need are the weights and volumes. But you don’t need a Great Mind to have those. You just need physical objects in physical space.
This refutes one of the two things that Steiner and Howell think are strange about the universe: they claim it’s strange that physical objects would somehow obey something so inherently “mental” like mathematical laws, but as Archimedes’ proved, that’s not strange at all. In fact, it’s exactly what we should expect. Because every physical universe has quantities of things that behave in quantizable ways, every physical universe will obey mathematical laws. Because that’s what mathematics is: a description of quantities and their ratios. Those laws will differ as the physical properties of the objects in a given universe differ, but no matter how you change things, you will always end up with mathematical behavior. Because one thing particular about mathematics as a language is that it deals in quantities, and physical things always exist in physical quantities.
This also refutes the second thing Steiner and Howell think is strange about the universe: that we can discover physical facts of the universe by simply manipulating mathematical symbols and equations. Archimedes discovered that any system with these physical properties, will behave in a particular way, as described by this equation. No one else really noticed that before then. But he could find it, because once he correctly described the system, that new conclusion follows necessarily from that description. Thus it is not surprising that by understanding the pattern of behavior this physical system exhibits, Archimedes was able to discover and predict something he didn’t know before about that system, such as the fact that submerged objects weigh less, and how much less. Because all he was doing was describing a pattern of behavior, and then deducing further the consequences of that pattern. Just like I did to discover a pickpocket stole my wallet.
Mathematics was just the language Archimedes used to describe that pattern. The pattern itself is entirely entailed by the physical facts, and thus only requires a physical universe. Since using descriptions of physical systems to discover new things about them is not surprising when we do it in English, it shouldn’t be surprising when we do it in mathematics. Mathematics is just a more precise language.
Back to That Artificial Simplicity
Now for the crucial plot twist…
Archimedes was wrong.
That is, he was only sort of right. Though he correctly deduced this mathematical law, and used it to correctly predict the behavior of solid objects in water, his deduction was based on premises that aren’t entirely true. They are only approximately true. Or only sometimes true. Like Euclidean geometry: Archimedes’ conclusions are true, for all universes where the premises entailing those conclusions are true; and for all parts of universes, if those premises are not true in every part.
Archimedes could prove those premises empirically (such as the behavior of fluids, the force of gravity, the geometry of three dimensional space, and so on). And they do hold in most cases. But in general they are actually false. Water actually can be compressed, just not by an amount that would have been visible to Archimedes; and water does change its density, in response to such factors as salinity and temperature; and not all water weighs the same, for example water whose molecules are excessively isotopic will weigh more than an average liter of water; likewise, Archimedes had the wrong geometry, as we now know from Einstein, space is actually curvy and not perfectly flat as his predecessor Euclid assumed, but this curvature was also too small for Archimedes to see (actually, he knew this: he was aware the earth was a sphere and therefore the surface of any container of water will be a curve and not a plane, but he noted that the curvature is so small it won’t affect the results at any scale that mattered to him, therefore he could pretend the surface of every body of water was planar); and so on.
We could list countless other factors. And when all these correct premises are brought in, Archimedes’ Law gets incredibly complicated and is no longer anywhere near as simple as that equation I just described. A “correct” equation would have to account for all these other things, from how the curvature of space-time affects the system, to how water can vary in density and compression all across the surface area of any immersed object—or even the properties of solids, since water can interpenetrate an object; some surfaces are hydrophobic or hydrophilic; an object’s density can change in reaction to water, or to changes in pressure or temperature; and so on.
The physical universe does not behave anywhere near as simply as our physical laws would have it. We just choose to ignore all the little things that don’t make enough of a difference for us to care about, and thus we reduce everything to a few simple premises like Archimedes did, and from this we get a simple mathematical law. But this is a human invention. It’s an idealization, a fiction of our own devising. It isn’t a complete fiction, as the law will correctly describe most of the physical systems we usually deal with, just not with complete precision, only with enough precision to suit our needs.
This is an important thing to remember, because people like Howell and Steiner constantly overlook it: though we prefer simple and beautiful mathematical theories, reality is always vastly complicated and ugly. Howell, for example, thinks it’s strange that our ideals of mathematical simplicity and beauty help us discover the truth about the world. But in fact, we are using those ideals to construct idealizations, not actual correct descriptions. We make our fictions simple and beautiful, like Archimedes’ Law, and are content with that because it works well enough. But reality doesn’t obey Archimedes’ Law. There is no real system anywhere in this universe that does. Instead, taking into account all those mitigating factors (only some of which I listed), the actual behavior of solid objects immersed in a fluid would only be correctly described by an equation so nightmarishly ugly and complicated, requiring hundreds of inputted variables that we will never actually know the truth of, that it is certainly beyond any human ability ever to construct, much less easily comprehend.
Thus, we use simplicity and beauty as tools to make understanding the world manageable. But the world itself does not conform to them. Hence, for example, Steiner and Howell cannot deduce anything from the premise “the universe is surprisingly simple and beautiful,” because the real world is actually neither. Only our idealizations are both. And idealizations are human inventions.
Nevertheless, when it comes to the actual laws of hydrostatics (not the human construct, but the actual way the universe behaves, which “physical laws” are only human descriptions of), those actual laws change as the physical facts change, just as with Archimedes’ Laws: if certain physical facts are true, Archimedes’ laws of hydrostatics are true; if other physical facts are true instead, some other mathematical law will be true instead. In fact, whatever the physical facts are, there will be a description of their resulting behavior, and as that behavior involves quantities, its description will always be mathematical, and we will call those mathematical descriptions physical laws. Always.
Therefore, laws of physics being mathematical can be no indication of there being a God. To the contrary, their being mathematical is simply an inevitable product of the universe being physical. Therefore, if we see something behaving mathematically, we don’t need to propose anything else is behind it but a physical world.
In a sense this all began in 1960 with a physicist, Eugene Wigner, who wrote on how mystified he was by “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” (Communications on Pure and Applied Mathematics 13.1). Which was most decisively answered in February of 2005 by Sundar Sarukkai in “Revisiting the ‘Unreasonable Effectiveness’ of Mathematics” (Current Science 88.3). Although it had plenty of refutations before that (Wikipedia has an entire article on it). As Sarukkai points out, that mathematics is effective in describing the world is no weirder than that English is effective in describing the world. It was invented to describe what was there. And quantities of stuff behaving in quantizable ways is what was there. Which is just what we should expect to be there in any godless world.
“The effectiveness of mathematization,” Sarukkai concludes, “significantly depends on the power of symbols to act like pictures of ideas, concepts and events. The role of mathematics in the sciences seems to be essentially dependent on the possibility of using mathematical symbols as ‘pictures’.” Meaning, pictures of the systems, their shape and quantifiable relations, that we want to describe and understand. Our minds can then explore that “picture,” like exploring a 3D sim, to discover things about it that weren’t obvious before. This is how we use language to discover things about the universe. But there is nothing remarkable about that. It’s exactly the same thing I was doing when I used sentences in English to discover what happened to my wallet.
Of course, anyone would figure all this out the moment they tried to describe a universe that wouldn’t be describable with mathematics. And this necessary step of reasoning (trying to imagine what would be the case, and thus what we’d observe, if the hypothesis you wish to defend were false) is routinely ignored by Christian theologians. Because they don’t understand how the logic of evidence works. That’s why they are Christian theologians. But if there is no way to describe a universe that isn’t describable mathematically, then the probability a universe will be describable mathematically is 100%, regardless of any hypothesis about God. And that means the mathematical describability of a universe can never be evidence for a god.
There is no immaterial “thing” that you can “remove” from existence, and somehow make it not the case that all mathematics is true of any and every universe that exists. Either actually or potentially, all quantities and ratios and shapes and structures necessarily exist in every possible universe (which means, any region of existence that isn’t absolutely nothing and thus the complete absence of a universe; although maybe still even then). Because you can’t show any such universe will lack quantities—and thus shapes, patterns, and ratios. That it will contain quantities is logically necessarily the case. And you can’t show that any potential rearrangement of the physical contents of any universe that would exist is logically impossible—and if you could, it would be mathematically impossible as well (and thus moot). Yet everything that isn’t logically impossible is by definition logically possible. And everything that’s logically possible, is a potential property of everything that exists.
Of course, defeated by this realization, religious tinfoil hatters will try to jump even further back from mathematical universes requiring a Mind (because clearly none ever would), to claiming logically possible universes require a mind, because “logic” is mental. Which is like trying to argue that paper money backed by gold is literally made of gold. A confusion of basic reasoning. I won’t go into that nonsense here, but I’ve addressed it extensively elsewhere, including in Sense and Goodness without God (index, “logic”) and in my extensive critique of the related Argument from Reason (cf. in particular the section on the Ontology of Logic, and then the related section on the AfPR).
We mustn’t confuse words with what they refer to. Numbers (and equations and operators and so on) are words invented by humans. They don’t exist without minds. But the only minds that need exist for them to exist, are our minds. God did not invent our language or teach us any mathematical words or symbols or equations. We invented all that ourselves, over tens of thousands of years of trial, error, experimentation, and testing. Quite the opposite of having any divine source for it. We had to figure this shit out wholly without guidance. Numbers and mathematics are all as invented as names and the entire English language. But we invented words in our languages to refer to actual things that exist, or could potentially exist—the things we can hypothesize or imagine.
And the real things our invented mathematics describes, are physical quantities and ratios, and physical shapes, patterns, and structures. Which all universes will contain, actually or potentially. In fact, which all godless universes will entirely consist of without remainder. Therefore all universes will be described by mathematics. Especially the godless ones. Like everything else, from thunderbolts to morals, gods have nothing to do with it.