Pythagoras thought that the ultimate truth behind what we see around us lay in number. Galileo believed that the universe was written in the language of mathematics. Eugene Wigner famously described the effectiveness of mathematics in describing nature as unreasonable, and likened it to a miracle. Finally, Max Tegmark, in his book Our Mathematical Universe, completes the circle by claiming that reality is mathematics. Is there really some deep truth uniting mathematics and reality that demands an explanation? Is God a mathematician?
(Note: This article follows on from an earlier related article, What Do Physics Theories Describe)
It is often remarked that mathematics and numbers feature everywhere in the natural world. Throw a ball, and the trajectory you get describes an upside-down parabola. Some numbers seem imbued with a certain significance; the proton is 1,836.15267 times heavier than an electron, for example, and the speed of light clocks in at just under 300,000,000 metres per second. Fine-tuning arguments make the case that certain indispensable parameters, such as the value of the cosmological constant and the decay rate of protons, are so unlikely to have arisen by chance that there must be some mysterious force operating in the background (typically God) tweaking the dials. We also hear of certain numbers or mathematical functions which seem to crop up with unnatural frequency in the natural world, like the Fibonacci sequence, which has been invoked in reference to the shape of pinecones, the number of petals on flowers, and even to body proportions in animals.
In addition to these claims, many physicists seem convinced that there is something special about mathematics because it is so effective, unreasonably so, at describing the universe. This belief rests on the tremendous success of physics theories, which don’t just allow physicists to describe the motion and interactions of balls, planets, and particles, but also allow them to make predictions based on these mathematical models and abstruse equations. In one memorable instance of the power of mathematical modelling, physicist Murray Gell-Mann successfully predicted the existence of a new type of fundamental particle (the quark), purely from the construction of a mathematical theory. What exactly is going on here?
What is Mathematics?
In Our Mathematical Universe,Max Tegmark defines mathematics as a set of abstract entities and the relations that hold between them. There are two important features evident in this definition. First, mathematics is abstract, meaning that it ignores the details of the particular things it describes. The four interior angles of a rectangle are each 90°, but the rectangle for which this is true is an ideal, perfect one, not the rectangular business card on your desk which you trimmed with your scissors to make it fit in your wallet. Second, it is principally concerned with relations between these abstract entities. Perhaps the best way to envisage this is with the equation, which is literally a proposition showing how a number of terms (constants or variables) are related to each other. E = mc2, for example, precisely describes how energy, mass, and the speed of light are all related to each other. This means that mathematics is a framework.
As important as it is to ask what mathematics is, we also have to clarify what it isn’t. In light of the above definition, it is clear that the shapes, angles, and numbers you see represented in things all around you aren’t mathematics. The circular coffee cup lid, the parallel lines of the picture frame, the angle the corner of my desk makes; none of these are mathematics, or even mathematical. What are they? Real-world things; things which appear as particular shapes, and which are certainly describable in the precise language of mathematics, but nevertheless real-world things with real-world significances, not empty mathematical abstractions, like squares, circles, and right angles. The ideal, mathematical circle in which the ratio of its circumference to its diameter is 3.1415… is about as far removed from the lid of my coffee cup as you can get. Mathematics is an abstract, relational framework that we apply to a world full of real objects; it isn’t the objects themselves.
Is Mathematics Discovered or Constructed?
Interestingly enough, this is perhaps the only question which has the power to leave our ordinarily uncompromising, determinist, materialist physicist at a loss for words, and gesturing vaguely in the direction of some Platonic realm where mathematical truths exist independent of the humans who think them. It is certainly tempting to think of mathematical truths in this way. The ratio of a circle’s circumference to its diameter (pi) has the value that it does irrespective of whether we approve of it or not. Moreover, it has had that value since the beginning of time, and perhaps even further back than that. Pi has always been, and always will be, 3.1415…. If anything qualifies as an eternal truth; that is, as something we discover, surely it is this.
—– Henri Bergson —–
20th century French philosopher, Henri Bergson, weighs in on this topic when he argues, contrary to popular opinion, that our minds have a natural tendency towards the mathematical. The primary animal motivation, he asserts, lies in action. Before there were thinking animals, there were acting ones, and, more importantly, the former capacity only arose out of a desire to improve the latter. Considered from an evolutionary perspective, one might say that we think in order to act. Now, to act well, we must have a goal, an end, in mind. The fulfilment of this goal requires a plan, which, in turn, requires a detailed description of the mechanism by which that plan can be effected. Such a description is only possible if we have identified constant relations between events, such that we can anticipate future outcomes based on present states. This is the mechanical rendering of the natural world, which becomes increasingly mathematical the more rigorously and systematically these relations are expounded and formalised.
There are two things that stand out regarding Bergson’s account. The first is the idea that we are naturally predisposed to a mathematical way of thinking. Mathematics is highly abstract and unrelated to practical, everyday concerns; one doesn’t need calculus or geometry to catch a woolly mammoth for dinner. However, the tendency to mathematical thought is something quite different. This emerges naturally from the evolutionary mandate that we act in the world. The second point supports what I said earlier; namely that this mathematical way of thinking is just that, a way of thinking. Mathematical relations, functions, formulae, and even numbers, don’t exist out there in the world of things; rather, they only exist in the way we think about, describe, and make sense of that world. If I throw a spear and single-handedly bring down a woolly mammoth, at no point in that daring escapade did an upside-down parabola manifest in the world. Yes, I can describe the arc the spear follows as an upside-down parabola, I can reduce its trajectory to differential equations, I can calculate the force (F) with which it will impact the mammoth, but none of this establishes anything about the world as it is in itself. There are no parabolas, equations, or mathematical constants out there. What is there? A spear and an unlucky mammoth.
What does this mean for the question of whether we discover or construct mathematical truths? Well, there’s no doubt, I think, that we do discover mathematical truths. We certainly don’t create them in the way that a writer creates a story, nor can we change them to suit our preferences. However, the specific values we give to mathematical truths are wholly arbitrary. A degree (as in, a right angle is equal to 90 degrees) isn’t something we found in nature handily labelled as such; rather, it is the unit of angle measure which, when multiplied 360 times around a point, completes a single rotation. There is nothing which mandates that exactly, and only, 360 of these ‘degrees’ describes a complete rotation; it’s just that the Babylonians had a base 60 number system, and a year of 360 days. So, while there is a certain inventiveness and arbitrariness that goes into establishing the foundations of calculation, the relations that follow from those initial definitions will remain things we discover.
But this article isn’t just about abstract, idealised mathematical truths; it’s about the way these supposedly relate to reality in an unreasonably effective way. It is precisely here that we find construction. Mathematics is an abstract, relational framework that lives in the pristine realm of pure thought. The world, on the other hand, is a messy place (figuratively speaking) populated by actual things. If we apply the former to the latter, as Bergson suggests is our evolutionary predisposition, then, given that we are deliberately choosing the framework within which we will interpret, or understand, the world, this is very much a matter of construction as opposed to discovery.
Consider a page full of printed text. One way of seeing this is to read it, in which case it means something, perhaps an exciting discourse on the nature of perception. Another way to see it is to look at the patterns all of those squiggly black lines make on the white page, in which case it means something totally different. Now, no matter which way I go here, I am discovering something, information about perception in the first case, and visual patterns in the second, but the choice about which way to take up the page in the first place isn’t a discovery. It isn’t a discovery for the simple fact that there isn’t anything being discovered here. Instead, what’s happening is we’re making a decision about how the page of writing will appear for us; we are, in a very real sense, constructing the page.
Mathematics and the World
—– Heidegger and the Mathemata —–
In 1936, German philosopher Martin Heidegger gave a series of lectures in which he discussed the mathematical in a way that bears no little similarity to the way we are using it here. He begins by asking a deceptively simple question; are mathematics and number so closely connected because the mathematical is numerical in character, or is it the case, rather, that the numerical is, in fact, mathematical? He finds in favour of the latter.
So, what is the mathematical then? Exercising his etymological muscle in a way that goes more than a little beyond our present concerns, Heidegger concludes that the mathemata is a learning which is about things we already know. In other words, it is not an original learning. Rather than investigating the things themselves, it presupposes them, and merely applies knowledge we already have to things which have already been revealed to us. One example Heidegger gives of this is what happens when we see three chairs. These three objects tell us absolutely nothing about the number “three.” On the contrary, we can only count these chairs as ‘three’ things because we already know what “three” is; that is to say, we brought this knowledge with us before we interrogated the chairs (which already appeared before us as chairs in some way that has gone unquestioned).
This example also handily reveals why number is mathematical; i.e. because it is that which we most easily recognise in things without actually deriving it from them. In counting or calculating things, we make use of them without actually having to pay them any regard. The physicist’s equations and mathematical models reveal nothing more about the things they pertain to than what they already bring with them.
Heidegger’s conception of the mathemata as simply applying what one already knows to things already present to one (as opposed to a genuine revealing of the things themselves) parallels quite closely the idea I have been arguing for here of the mathematical as an abstract framework that reduces real things to plotted points in a Cartesian plane, identified by an array of numbers that describe the strength of various fields at that point, which therefore don’t reveal anything essential about them. In other words, the world isn’t mathematical, so mathematics doesn’t actually describe it, let alone in some mysteriously effective way that demands an explanation. The mathematical is a framework we bring with us when we want to quantify the world and the things in it. It certainly appears as if there is a mystery surrounding the effectiveness of mathematics at describing the world, but this is only because we are looking at the world through mathematically-tinted glasses. There is no small element of absurdity in this. We deliberately construct abstractions of real-world things, deliberately removing anything about the object that isn’t conducive to a mathematical interpretation, and then stare in amazement at the fact that we are left with something amenable to a mathematical description.
Imagine you’re playing cricket. You’re out fielding and the batter hits the ball. As it bounces along the turf towards you, how might you, as a fielder, describe this projectile? Well, the first thing you might notice is its colour; red. You haven’t touched it yet, but you know that it also has a texture. It is hard and smooth, although this smoothness is broken by a raised seam winding around its surface. It also appears with a location that has significance for you in relation to other things. It is between you and the batter, above the ground, and a little off to your right. It’s also moving; a feature you intuitively understand as a warning that it will pass you by unless you move quickly to intercept it. Reflecting on your situation, you realise that the ball also has a wider significance in the game you are playing. It calls to your body, inviting you to move towards it, to grab it and throw it to the wicket keeper. It acts as a pole for your action, orienting your movements in ways you don’t intellectually determine, as your body, with a ‘feel’ that is instinctive as much as it is calculated, ‘fits’ itself to the situation. And we still haven’t even come close to exhausting what the ball means for you.
Imagine you’re playing cricket. You’re out fielding and the batter hits the ball. As it bounces along the turf towards you, how might you, as a physicist, describe this projectile. Well, first you would discount all ‘secondary’ qualities, which are subjective, and therefore irrelevant. That means it has no colour and no texture, although it does reflect light of certain wavelengths, and it does have a specific topography. Its location manifests as a set of x, y, and z co-ordinates you centred on the middle of the projectile, and plotted in an imaginary 3-dimensional container. Likewise, this central point reveals a motion which you calculate as a velocity. The ball has no wider significance in a game, or for our body because… well, now that I think about it, there isn’t actually a ball at all. This is what Tegmark calls “baggage;” fuzzy, imprecise terms we humans have invented to describe mathematically precise events that we just lack the faculties to perceive.
Of course, I’m not suggesting the physicist’s view of things is ‘wrong’ in some way. It’s a perfectly legitimate, and powerful, way of viewing the world. It’s just a world that has been completely stripped of meaning, a world no one lives in (the physicist included), a world no one could live in (the physicist included). And what do I mean when I say the physicist’s world has been stripped of meaning? Nothing more than that it has removed the subjective, that is to say, the non-quantifiable aspects of the thing. What kind of madness is it that declares anything non-quantifiable persona non grata in an account of reality, and then, in apparent sincerity, wonders how amazing it is that reality is so amenable to being quantified?
A Couple of Objections
Fine. The mathematical is a framework we fit real-world things into (not a description of the things themselves), but isn’t there still a mystery as to how and why those things fit into that framework so neatly?
This objection has forgotten what we learned from Bergson above. It implies that mathematics has somehow appeared in the world (through the minds of human beings) independently of it. The source of the physicist’s amazement at this coincidence relies on the assumption that the two have entirely different sources. We have the physical world over here, mathematical propositions over there, and inexplicably we can map the former (with a little subtraction) onto the latter.
This is not only wrong, but a violation of basic materialist principles. If the physical is all there is, by definition, everything, including human thought, must be reducible to it. In such a world, how can there be an entire realm of truth that stands outside the physical and is therefore capable of comparison (and subsequent amazement at the correlation)? As Bergson pointed out, our mathematical way of thinking emerged from the demand placed on all living beings; namely, that they act in the world. This makes it clear that the physicist has it all backwards. It isn’t the case that when we look around us, we see shapes that conform to our geometry, lines that admit of being divided into parts, and objects that can be counted. Rather, it is our geometry and arithmetic which imitate the real world. How could it be otherwise? Mathematics emerged from our interactions with, and subsequent analyses of, real-world, 3-dimensional objects, which, being physical objects in space, are (as we saw above with the cricket ball) able to be conceived in terms of lines, angles, curves, and number.
I have talked of mathematics as an abstract framework we apply to the world. This is true. But before it became this universal body of knowledge dealing with ideal, perfect objects, it was a simplified description of parts of real things in the real world. This is the arbitrariness I spoke of earlier when I acknowledged that mathematical truths are discovered. They are indeed discovered (no one created pi), but only after they have received an initial grounding in real-world objects. Aren’t I contradicting myself? What happened to Heidegger and his claim that the number three is an example of knowledge we bring with us to the thing; i.e. that specifically isn’t found in the real world? What isn’t found in the real world is the mathematical concept of the number three; the third thing in a series of ‘whole’ numbers which extend to infinity. Of course there are ‘three’ chairs over there, this is what led us to imagine the concept of number in the first place (who would have, (no, who could have) thought of numbers if there were no things to count in the first place?). Heidegger’s point is that the number three, applicable to chairs just as readily as to tables and desks, isn’t derived from the chairs themselves; rather, it comes from something we notice about the chairs. It is a concept that couldn’t have been formulated without the chairs, but at the same time, one that completely passes over the chairs themselves.
It is true that the physicist describes the cricket ball by stripping away the non-quantifiable qualities, but that is precisely what makes it objective, and therefore, true. The fielder’s description was wholly confined to their individual perspective, and exists only for them.
There is a presupposition in this objection that lies underneath the entire scientific project; namely, truth is objective. The reason science insists on objectivity is that it just wouldn’t work without it. I’ve argued that objectivity is secured through the rigorous whittling away of anything related to perspective. Fine. But what makes this true? Why do we think the objective description is somehow truer, or more real, than the description that includes a perspective? Where do we get this bias from?
What is the objective truth of a book about physics? What is the book in reality? Well, it can’t have anything to do with the ideas contained in the words because they can only be deciphered by a reader who speaks the particular language in which the book is written; i.e. a person approaching the book from a particular perspective. In fact, once we get rid of all of Tegmark’s “baggage,” the objective truth of the book will have reduced it to nothing more than a meaningless jumble of subatomic particles. It’s true that every observer now has equal epistemological access to the book, but the price we pay for this equality is the replacement of a thing infused with detail and significance with a sterile collection of numbers that don’t mean anything to anyone.
The assumption that the objective is also the truth is actually a metaphysical claim that has gone entirely unquestioned. Why can the truth of the thing not be different for different perspectives? The book is a rich source of knowledge for the reader who speaks the right language, but it is also an intriguing collection of black shapes on a white background for speakers of other languages. Again, we obviously need objectivity in order to make scientific and technological progress, but this is a very different claim from the claim that there must only be one single truth for everything; a truth that objectivity, as the lowest common denominator, satisfies.
We started this article with a world full of mathematics. Everywhere we turned we saw mathematics, from numbers in the constants of nature and shapes in the objects around us, to Fibonacci sequences in… well, just about everything if you believe the proponents of this idea. More seriously, we also encountered the claim that the fundamental structure of the universe was itself mathematical. Somehow, the idealised, perfect world of mathematics and the real, messy world of the physical were connected.
Once we dug a little deeper into exactly what mathematics was, we discovered that it was a purely abstract, relational framework. This meant that, rather than actually appearing in the world, it was a way of thinking about the world, a way of thinking that we bring to, and impose, upon the real things we encounter. Pursuing this thought led to the insight that, not only is mathematics not in the world, it can’t even describe it as it really is; at least not without pulverising the thing into pieces, and selectively retaining only the quantifiable ones.
But then we saw our determined (and imaginary) interlocutor insist that there was still some strange coincidence to be explained in the mere fact that reality, albeit after some specific alterations, was ultimately able to be successfully described by an abstract, pristine mathematics. Even if the connection wasn’t as direct, or as thorough, as it appeared to be at first, we still had an overlap that required an explanation.
We traced this intuition to the underlying assumption that mathematics and reality began in completely different places, and were somehow later discovered to have a mysterious correspondence. It turned out, however, that mathematics had a dirty little secret in its past. While it claimed to be of a wholly different pedigree from the riff-raff in the real world, a quick genealogy revealed that one of its parents was actually from the wrong side of the metaphysical tracks. The abstract realm of mathematics had a lowly birth in the world of the real. The irony then is that even though mathematics is incapable of describing reality as it really is, it nevertheless has its roots in it.
So, what are the take home points from all of this? First, the physicist is wrong in demanding an explanation for how unreasonably effective mathematics is at describing reality, simply because mathematics doesn’t describe reality. On the contrary, it describes a pale, sterile version of reality that has deliberately been rendered pale and sterile precisely so that mathematics can describe it. Secondly, the physicist is quite right in insisting that even if it is only a watered down depiction of reality that can be described mathematically, there must still be an explanation for this correspondence. The answer, however, rather than elevating the underlying structure of the universe to some higher-order realm of universal, mathematical truth, turned out to be the exact opposite; namely, the discovery of mathematics’ humble beginnings in the world of the real.
Coda: Subjective Experience
This whole article has concerned itself with only one portion of reality; the physical. There is a whole different aspect of the world we haven’t discussed though; subjective experience. There is, of course, a certain overlap with this and the qualities of the thing that reveal themselves to a specific perspective, but first-person experience is much more than this thin slice of subjectivity. It is also that feeling all of us conscious beings have, a feeling that one might even say defines what it is to be conscious; namely, that feeling that it is like something to see red, to hear music, to just be us. This feeling that lies at the base of subjective experience is at least as much a part of our universe as the physical furniture we share it with. The physicist’s equations and formulae are deafeningly silent on this.
I don’t want to harp on about this too much here because when the physicist says mathematics is unreasonably effective at describing the world, they are pretty obviously talking about the physical world. However, it is important to note, particularly these days when our intellectual elite are almost unanimous in their belief that everything can be reduced to mechanical, third-person processes, that any description of the world that even pretends to completeness, must be able to include subjective experience. Given that the mathematical theories of physics are essentially, and intentionally, third-person, there is no way they can possibly extend to the realm of first-person experience, and no way they can clear this high bar.