I’ve long been a fan of comedies of remarriage (“It Happened One Night”, “The Philadelphia Story”, “His Girl Friday”, etc.), and one of the greatest comedies of remarriage is the story of Math and Physics (or “Phyz”, as Math likes to call her).
You could say that the source of their eventual breakup was present from the start, when they put together a model of the universe at their wedding. It was a sweet but, in hindsight, naive gesture. You see, Math had discovered that there were exactly five ways of sticking identical regular polygons together to form perfectly symmetrical solids (we humans named these regular shapes “Platonic solids” in honor of the philosopher who officiated at the wedding, though he didn’t discover any of them); delighted by the discovery, Math brought the five solids to the ceremony, as gifts for her bride-to-be. Meanwhile, Phyz brought her own gifts: earth, air, fire, water, and “quintessence” (heaven-stuff), the five elements from which she said the universe was constructed. (See Endnote #1.) Five regular solids? Five elements? Surely this marriage was foreordained! (See Endnote #2.) Math and Phyz exchanged gifts and proclaimed their bond, swearing that they would never part. And if any onlookers thought the correspondence between the gifts was forced, they had the good manners to keep their mouths shut.
But after a few millennia, latent tension in the relationship rose to the surface. Physics kept growing and changing, revising her core principles, sheepishly deciding for instance that earth, air, fire, and water weren’t true elements after all. But Math couldn’t help noticing that even as Phyz discovered new elements, Math didn’t have to update her inventory of regular solids. She had in fact found a proof that there couldn’t be any more, and the proof remained valid down the centuries, even as Phyz kept revising her own basic tenets. Oh, and here’s another example: Physics said that projectiles rise in a straight line before falling along a curve, until she said oops, no, they rise along a curve too. Math was embarrassed by the flightiness and unreliability of Phyz, even as Phyz was embarrassed by the stodginess of Math.
Over time Math became more fussy and equivocal. She began to hedge her statements, refusing to say what was true, but merely making conditional assertions of the form “Well, if assumptions A, B, and C are true, then conclusions X, Y, and Z follow.” Or: “To the extent that assumptions A, B, and C are approximately true, to that same extent conclusions X, Y, and Z should hold as approximations as well.” Though she hated the way she sounded when she said things like that.
But you shouldn’t think that Math was merely retreating into wishy-washiness or sterile perfectionism. Math was growing just as much as Physics was, but in different ways. And it wasn’t that Math lacked commitment to her relationship with Physics; she just felt too confined by where Phyz lived. Eventually, sometimes around 1900, she said “I need to see different universes,” and she moved out.
One issue that highlights the divide between Math and Physics is the issue of higher dimensions. Do they exist? Math and Physics have very different answers. In physics, the most naive (and mostly right) answer is “No”: you can’t construct an object with four lines that are at right angles to one another. (Of course, you can change the question and then the answer becomes “Yes”, and then you can change it again and the answer becomes “Maybe”, but I’ll get back to that shortly.) On the other hand, in mathematics, we can lay down axioms for n-dimensional Euclidean geometry not just for n=2 and n=3 but for any positive integer n. From these axioms, consequences can be derived, and every mathematician will obtain the same consequences, so higher-dimensional spaces are as real as any other mathematical construct: they’re consistent creations of the human mind with properties that all logical minds will assent to not because the axioms are true (whatever that would mean!) but because the entities under discussion satisfy the axioms by definition. Mathematics nowadays is a language for describing possible universes, of which the universe that we happen to inhabit is just one example.
Turning away from my conceit of Mathematics and Physics as personified beings and turning towards a consideration of human history, consider the careers of the mathematician Ludwig Schläfli (1814-1895) and the physicist Albert Einstein (1879-1955). Schläfli wanted to know what sorts of higher-dimensional regular solids (“regular polytopes” is the more technically correct phrase) exist in n-dimensional Euclidean space for values of n bigger than 3. He showed that there are six regular solids in 4-dimensional Euclidean space but only three regular solids in n-dimensional Euclidean space when n is 5 or 6 or any higher integer. On the other hand, Albert Einstein pursued a view of physics in which our 3-dimensional space needs to be conceived of as part of a 4-dimensional geometry of “spacetime” in which the properties of space and time become interwoven. (See Endnote #3.)
Despite the fact that the two thinkers’ lives overlapped — indeed, Einstein’s precocious ruminations about riding a beam of light ocurred around the time of Schläfli’s death — in an important sense their work did not overlap at all. Partly that’s because the two great relativity theories that Einstein developed aren’t Euclidean; special relativity uses what we now call Minkowski space (with time playing a privileged role that distinguishes it from the other three dimensions; see Endnote #4), and general relativity makes the game even deeper by allowing Minkowski space to warp and bend. But more importantly, Einstein was concerned with our world while Schläfli was concerned with idealized mathematical worlds.
Nowadays there are speculations that our physical universe might have extra dimensions that are too small to see. The possibility of there being extra dimensions is a tantalizing one (it’s the “Maybe” I mentioned earlier), but in math, extra dimensions are more than a possibility: they becomes an actuality, albeit just one of many coexisting actualities, because math (as we understand it nowadays) isn’t about actuality, but about possibility.
GNOSTIC PHYSICS VERSUS GNOSTIC MATH
To help me further clarify the divide between math and physics, I’ll recruit a couple of hypothetical demiurges (similar to the one postulated by Gnostics, but nerdier) to help me.
So, imagine if all at once all over the world a booming voice were heard, saying: “Hello, hello! Hello everyone. (Is this working? Oh good.) I am a mighty Demiurge, and I have decided to confess: I’ve been messing with you. Most of you are aware that some religious fundamentalists on your planet believe that I or someone like Me planted dinosaur bones in the ground as a test of your faith. Well, I have been messing with you. But not using dinosaur bones. Instead, I’ve been interfering with the behavior of electricity and magnetism and light on your planet for several centuries. The bottom line is, Newtonian physics is correct, and special relativity is wrong. There actually is a preferred reference frame for observers; the luminiferous ether is real; Maxwell’s equations are wrong; et cetera, et cetera. Surprise!”
Such a pronouncement would give us reason to reconsider Einstein’s theories, but not Schläfli’s. The existence of exactly six regular polytopes in four dimensions is a fact of pure reason, not an experimental observation. If we lived in a four-dimensional Euclidean space, then there would be exactly six different ways to stick regular three-dimensional polyhedra together to form regular four-dimensional polytopes. The Demiurge’s proclamation wouldn’t change that.
In contrast, we might imagine a different Demiurge who pipes up “That’s nothing! I messed with Ludwig Schläfli’s head, and the heads of everyone who ever read his work or reconstructed it for themselves, so that nobody would notice the logical fallacy in his proof and discover the hidden seventh regular polytope; every time one of you humans reads the argument for why it can’t exist, I make your brains go BLOOP at the crucial moment and you miss the mistake!” That’s an entirely different kind of mischief. It’s one thing for us to suspect that our observations have misled us; it’s a more disturbing thing to suspect that our processes of reasoning are themselves flawed, and this suspicion quickly leads us to far more radical doubts that undermine not just the Schläfli’s work or Einstein’s but the entire scientific enterprise and our whole sense of self.
(For instance, I don’t think that I’m just a brain in a vat. But wait a second: what right have I to say what I think or don’t think, if I can be mistaken about what my own thoughts are? But wait another second: the words “what right have I to say that” only makes sense if I can say things, and if I’m a brain in a vat, I only think I’m saying things. Then again, what does “wait a second” even mean if Time itself is an illusion? And …)
To stress the difference between physics and mathematics, I’ll borrow a phrase introduced by the paleontologist Stephen Jay Gould to try to broker an amicable divorce between science and religion. Gould called the disciplines “non-overlapping magisteria” and contended that they weren’t in conflict because science’s questions are “what/when/where?” questions while religion’s questions are “why?” questions, and that there can be no contradiction between the is and the ought. There are some problems with Gould’s attempt to resolve a key tension of the modern age, not least of which is that fundamentalists of various faiths maintain that their religious scriptures give clear statements of What Is from the Creator of the the Universe (who presumably would be in a position know). But my point is that, in a similar way, math and physics fail to collide because they fail to connect. Math is an engine for deriving non-obvious consequences of assumptions, but it cannot tell us which assumptions to make. We can compare the predictions of mathematics with observations of the real world and use the resulting concordance or discrepancy to decide whether the assumptions that led to those predictions are useful in explaining the world, but when we do this we are doing physics, not math.
THE THREE FACES OF PI
Then again, maybe we should think of math and physics as overlapping magisteria, and picture things like the famous quantity pi (3.14159…) as living in the overlap. On the one hand, pi is a physical quantity that measures the ratio between the circumferences and diameters of actual, physical circles; on the other hand, it’s a mathematical quantity that is for instance equal to 4 times the limit of the infinite sum 1 – 1/3 + 1/5 – 1/7 + … Let’s call the former physical pi and the latter formulaic pi. In fact, I want a third pi that I’ll call geometric pi. Geometric pi is the ratio between the circumference and the diameter of an ideal mathematical circle, whether or not such circles exist in our world. Mathematical reasoning can lead us, by a beautiful but complicated path, from geometric pi (“circumference divided by diameter”) to formulaic pi (4 times 1 – 1/3 + 1/5 – 1/7 + …, or some other formula for pi you prefer) but it doesn’t tell us whether Euclid’s axioms are a true description of our world. If they’re not, then geometric pi and formulaic pi, although exactly equal to each other, don’t pertain to the world we live in except perhaps as approximations.
Let’s take this idea further. Physical pi involves measuring things, and it can only be known up to finite accuracy. If we can only build circles up to 1020 meters across, and we can only measure them to within an accuracy of 10–20 meters, then we can only know the diameter or circumference of a circle with 40 significant figures, and when we take the ratio of two such measurements (the circumference and the diameter), we again get only 40 significant figures. In this setting, does it make sense to talk about the hundredth digit of that ratio as having a definite value if there is no way to measure it? Indeed, quantum physics tells us that the whole game of measuring lengths becomes problematic at the subatomic scale. Likewise, general relativity says that once you start building things (like a super-big blackboard on which to draw a super-big circle), the things you build will warp space, causing deviations from Euclid’s axioms (which only apply to flat space, not curved space). So when we talk about computing hundreds of digits of pi, we don’t — can’t — mean pi the physical constant; we must mean pi the mathematical quantity, defined by expressions like 4 times (1/1 – 1/3 + 1/5 – 1/7 + …).
A Demiurge might be able to warp space to change our measurements, but it’d have to warp our brains to make us think that 4(1/1 – 1/3 + 1/5 – 1/7 + …) equalled 5, say.
By the way, I don’t want to leave you with the mistaken impression that formulaic pi denotes the value given by the specific formula 4(1/1 – 1/3 + 1/5 – 1/7 + …). There are thousands of known formulas for pi, and it’s the totality of them that constitute what I’m calling formulaic pi, and not any one of them in particular. We know that they’re equal not by measuring objects but by reasoning about mathematical expressions, in the place where Math lives.
WHERE MATH WENT
It’s hard to say where Math went when she moved out of our universe. Plato pointed the way to her new home when he wrote “You know that the geometers make use of visible figures and argue about them, but in doing so they are not thinking of these figures but of the things which they represent; thus it is the absolute square and the absolute diameter which is the object of their argument, not the diameter which they draw.” That is, human geometers may draw pictures, but when we draw those pictures we’re thinking about something Absolute, even if it’s in a realm we can’t get to.
In her new home, Math doesn’t have to equivocate and add “… (assuming that Euclid’s axioms are correct)” at the end of every statement of a theorem of Euclidean geometry; she can just make assertions about ideal squares, diameters, circles, etc. that perfectly satisfy Euclid’s axioms, period. Build a Euclidean square whose base is the diagonal of some given Euclidean square, and the new square has area exactly twice the area of the old square. In the place to which Math has gone, there’s no need to worry about black holes warping the picture, or quantum foam undermining the diagram at sub-nanoscale. The constructs that pervade Math’s new home are precisely what they were constructed to be. In some ways it’s a lonely place, but it’s where you need to go if you want to connect with perfect truth, and to know the things you can be absolutely, positively sure of.
Since we humans can’t get to where Math went, we argue about whether the place even exists, and Math is cool with that. And she’s not even lonely, because guess who’s been visiting her there, and sometimes even spending the night? Physics! Phyz wants to talk about quantum field theory in n dimensions and how it relates to general relativity in n+1 dimensions, for all values of n!
In the best comedies of remarriage, the two parties to the marriage have done some growing during their period of separation. Perhaps each of them has developed characteristics of the other, becoming better-rounded people in the process. Or perhaps they have become more tolerant of themselves and others. Either way, the new relationship they develop is not the same as the one they had before.
Going back to our celestial Couple, Physics came to accept that Math’s flirtatiousness, her inability to be satisfied with just one universe (or even some large but finite number of them!), wasn’t just a sign of immaturity; her flirtatiousness was a key component of her nature. Phyz realized that even if she (Phyz) was content with 4 dimensions, or 11, or 26, Math could never stop there, nor would Phyz really want her to. Higher dimensions and curvature and bizarre topologies and even weirder variations on the theme of what space could be — Phyz now understood that all of this was part of what makes Math wonderful.
But Mathematics learned something too. All along, she’d thought of herself as the imaginative free-spirited one, and physics as the uncreative plodder. But then came string theory, and a particular prediction of string theory called the gauge-gravity correspondence. It was inspired by the physical world, and it might in the end make predictions about the real world, but beyond that possible application, it gave rise to beautiful new theorems. Who could have imagined physics providing inspiration to algebraic geometry? Algebraic geometry was one of the purest precincts of math. Surely if there was to be any traffic between the disciplines, math would inspire physics, and not the other way round! Yet in recent decades, ideas about fundamental particles that may or may not turn out to be good descriptions of the world we live in have provided inspiration to pure mathematicians, providing blueprints for some of the loftiest airborne castles mathematicians are trying to build.
The parade of ideas being imported from physics into mathematics doesn’t undermine my claim about math and physics being separate magisteria, but it sure does complicate it!
Some may rightly point that traffic between math and physics has been bidirectional for a while. They’ll point to Richard Feynman’s non-rigorous path-integrals, or to Oliver Heaviside’s even earlier non-rigorous delta function, which didn’t fit into mathematics when they were first formulated, and whose successes forced an enlargement of mathematics. But string theory is the best example to date. It’s not clear whether, without physics to inspire them, mathematicians would have made the leaps of imagination that led to mathematical string theory — even though the standard stereotype is that mathematicians are the unfettered makers of creative leaps while physicists are constrained by the need to describe the physical world.
Anyway, getting back to our two Personifications, and to my imaginary movie about their divorce and remarriage: In the last scene of the film, Physics and Mathematics return to the place where they first took their vows, and we see Phyz giving a new present to Math: an arXiv preprint discussing new connections between mirror symmetry and the geometric Langlands program. A look of shock comes to Math’s face, replaced by a slowly dawning delight. We the moviegoers don’t know what kind of new relationship the two of them will have going forward, and we’re not sure they know either. But we can tell from the look on Math’s face that what has just been bestowed on her was absolutely, positively the perfect gift.
Thanks to Sandi Gubin.
#1. Here’s what Plato said (in the dialogue Timaeus) about the correspondence between four of the five regular solids (the cube, octahedron, tetrahedron, and icosahedron) and the four elements that comprised the physical world according to Greek thought (earth, air, fire, and water):
“To earth, then, let us assign the cubical form; for earth is the most immoveable of the four and the most plastic of all bodies, and that which has the most stable bases must of necessity be of such a nature. Now, of the triangles which we assumed at first, that which has two equal sides is by nature more firmly based than that which has unequal sides; and of the compound figures which are formed out of either, the plane equilateral quadrangle has necessarily, a more stable basis than the equilateral triangle, both in the whole and in the parts. Wherefore, in assigning this figure to earth, we adhere to probability; and to water we assign that one of the remaining forms which is the least moveable; and the most moveable of them to fire; and to air that which is intermediate. Also we assign the smallest body to fire, and the greatest to water, and the intermediate in size to air; and, again, the acutest body to fire, and the next in acuteness to, air, and the third to water. Of all these elements, that which has the fewest bases must necessarily be the most moveable, for it must be the acutest and most penetrating in every way, and also the lightest as being composed of the smallest number of similar particles: and the second body has similar properties in a second degree, and the third body in the third degree. Let it be agreed, then, both according to strict reason and according to probability, that the pyramid is the solid which is the original element and seed of fire; and let us assign the element which was next in the order of generation to air, and the third to water. We must imagine all these to be so small that no single particle of any of the four kinds is seen by us on account of their smallness: but when many of them are collected together their aggregates are seen.”
As for the fifth regular solid, the dodecahedron, Plato decided that it must be correspond to some fifth element (or “quintessence”), and that since the number of its sides (twelve) is the number of signs in the Greek zodiac, it must be the element that the heavens are made of.
It should be stressed that Plato advanced this cosmology as a working hypothesis, not as what we would nowadays called “settled science”.
#2: As a side-note to my parable, I can’t resist mentioning that, to the Pythagoreans, the number five symbolized marriage, as it was the sum of the first “male” number (3) and the first “female” number (2). Presumably the Pythagoreans would have thought it more fitting to use the number 4 to symbolize the marriage of two females.
#3: When long skinny object rotates, we may sometimes see it as being tall and thin (when its axis is vertical) and at other times as being low and long (when its axis is horizontal), but we don’t think anything essential about it has changed. This is all the more true if the object is stationary and we, the observers, are the ones doing the rotating. That’s because the three dimensions of space are interwoven. In Einstein’s theory of special relativity, time joins the weave but in a different way. A clock with a circular clock-face moving at close to the speed of light will appear to run slow and its face will not look circular. The same is true if the clock is standing still and we’re the ones who are moving. But the tempo and shape of the clock haven’t changed — just the relationship between it and the observer.
#4: One way to build up the theory of three-dimensional Euclidean geometry is to use coordinates in the manner pioneered by Descartes. Points become triples of numbers, and the distance between the point (x1,y1,z1) and the point (x2,y2,z2) is the square root of (x1–x2)2+(y1–y2)2+(z1–z2)2. We could build a 4-dimensional Euclidean space by using quadruples (w,x,y,z) instead of triples (x,y,z) and define the distance between the point (w1,x1,y1,z1) and the point (w2,x2,y2,z2) to be the square root of (w1–w2)2+(x1–x2)2+(y1–y2)2+(z1–z2)2. But for purposes of physics it’s better to use Minkowski space: points are still quadruples, but now our fourth coordinate is to be thought of as signifying time, and the “distance” between (x1,y1,z1,t1) and (x2,y2,z2,t2) is (x1–x2)2+(y1–y2)2+(z1–z2)2–(t1–t2)2. The minus sign in front of that last (t1–t2)2 is crucial. Distances can now be negative numbers, corresponding to events in space-time that occur in a definite order no matter who observes them; meanwhile events at positive distance correspond to events that are causally separated, and events at distance zero correspond to points in spacetime along the path of a photon.
(Some people prefer to use (t1–t2)2–(x1–x2)2–(y1–y2)2–(z1–z2)2. That also works, as long as you don’t get mixed up about which sign-convention you’re using.)
Bravo. A delightful parable exploring the relationships between math and physics. Naively, one difference is empiricism. A beautiful theory must confront, and possibly slain by an ugly empirical fact. With computers, are there areas of math that are “empirical”. Some mathematical worlds, fractals come to mind, that are explorable with computers.?
May I lobby you to devote a whole essay on a small digression you make in this essay. There is Pythagorean distance, and Minkowski “distance “. What other kinds of measures and their associated spaces are possible?
Thanks again, David Jacobi.
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Math certainly has an empirical side. In fact, empirical refutation kicked me in the butt a few hours ago; my laptop (running some Mathematica code I wrote) informs me that a conjecture I believed a few days ago is false. Fortunately I never mistakenly believed that I’d proved my conjecture; then the refutation would have been quite embarrassing!
I may write some more about different ways of measuring distance in Mathematical Enchantments. In the meantime, I wrote a little bit about L^p distance in my essay https://mathenchant.wordpress.com/2017/02/16/three-point-one-cheers-for-pi/ . And anyone who likes my essays should check Evelyn’s Lamb’s “Roots of Unity” essays (https://blogs.scientificamerican.com/roots-of-unity/), many of which are on the theme of “A Few of My Favorite Spaces”.