My day job finds me explaining math at various levels of rigor. At one end of the spectrum, math manifests as a game, not to be played but rather hacked. “What is the path of least resistance to the correct multiple choice answer on the SAT?” Honestly, not so fun, as all the beauty and structure and glory of math is sucked out by the greedy and lazy test-taking mindset. When I am fortunate enough to engage with students at the advanced high school or college level, warm coals of that glory start to radiate through, especially to receptive students. All math-lovers wish that love of math were more contagious, that teenagers could learn to truly dig humanity’s greatest accomplishment rather than just transact with it. Realistically, it’s probably impossible to devise a worthwhile “romantic” math curriculum, and in any case, it’s more important for students, young ones especially, to earn basic skills in arithmetic and reasoning. That said, I gripe about the rather ahistorical treatment of math in high school and college. Math biographies and timelines are not so interesting, but the history and development of the concepts themselves reveals a side of math that students rarely see: its fluidity, its improvisational and searching nature, its fundamental arbitrariness despite its miraculous and unreasonable effectiveness.
Take geometry, which is in many ways the black sheep of the high school curriculum. Geometry is typically the first and indeed only subject where a student is exposed to proofs. Proofs dismay and discourage most students too; they require strategic and formal yet open-ended thinking, unlike algebra, which is mostly about symbolic manipulation and “tricks of the trade.” One might rightfully ask, “Why should geometry be the home turf of abstract reasoning?” Its domain of objects -- shapes, pictures -- is less obviously ‘mathematical’ than the domain of algebra -- equations, variables. But such a question betrays a lack of historical perspective. Geometry predates algebra by an order of magnitude; symbolic manipulation is a far more recent mathematical style than geometrical (i.e. visual) argument is. In fact, most algebraic ideas are, in one way or another, abstractions/crystallizations of geometrical ideas. Also, algebra, at least at the high school level, is relatively devoid of theorems, while geometry is all about its theorems. There is lots to do in algebra, but not all that much to prove. But the value and legacy of mathematics lies in its magnificent, vast collection of true statements derived from (nearly-)self-evident axioms: its theorems. Theorems undergird all the practical techniques and applications as well as the aesthetic internal theorizing. But that doesn’t answer the question. Why learn logical arguments in a pictorial medium? Or, equivalently, why do the foundations of mathematical reasoning go back to Euclid, a geometer (rather than, say, Diophantus or al-Khwarizmi, the OG algebraists)?
My own answer hints at the hidden, fluid nature of math I brought up earlier. The supreme power, the miracle of math, is to lift situations and patterns from the real world into the transcendent abstract realm where mathematical technology can perform its work. True equality does not exist in the physical world, but an equation can capture and generalize the notion of a balanced scale, or two things of the same type, or a fair transaction. Likewise, circles and triangles are fictions of the mind, austere and perfect yet rigid models we try to squeeze real shapes into. Real shapes, i.e. any visual thing at all, are infinite and totally ubiquitous, even more ubiquitous than the phenomena captured by equations/algebra, which makes them the ideal starting ground for this mathematical procedure of abstraction and generalization that unlocks the potential of reasoning. The genius of Euclid and other Ancient Greek geometers was not just the treasure chest of theorems and propositions they proved, but the messy, playful compromising between real and abstract that enabled those theorems to 1) exist, 2) cohere, and 3) link together in an astounding beautiful and smooth way. It’s almost impossible to imagine, but there was a time before such a concept as “square” existed, and obviously yet amazingly, some human mind had to be the first to come up with that concept. But it’s not like once one person thought of “square,” it was immediately understood by everyone else. This is the battle frontier of math! A word must be chosen. What does that word refer to? Is a “square” merely a four-sided figure? Does it need more structure? Equal sides, equal angles? (Sides and angles are already abstract terms, relatively easy to intuit but not so easy to pin down with enough precision for mathematics.) If something is like a square but does not have equal sides, does that need a new word? Where does that conceptual bubble begin and end? (Think of rectangles and rhombi, distinct yet overlapping and both under the umbrella of parallelograms.) There is simply no way there was instant consensus. Early mathematicians must have wrung out these ideas and fought endlessly, internally and externally, consciously and subconsciously. And it’s not like this is some effete, silly, cushy-armchair philosophical game; the stakes are staggeringly high, for mathematics cannot recover from even a single successful attack; it is both the strongest and most delicate human creation.
Once the dust settles and a concept takes form in a rigorous and complete way, the usual mathematical muscles can get involved. Cold, logical reasoning stacks statement upon statement, never worrying about foundations. Despite the existence of innumerable beautiful and powerful theorems and techniques, this mode of mathematics impresses me less. I am instead in love with the mathematician whose reach initially extends beyond his grasp, who wrestles with dirty and mysterious half-formed ideas until he can confidently display an austere, coherent conceptual gem that plays nice with others. The domestication of the beasts of the mind.
The histories of mathematical ideas are not taught in school, so students don’t get a sense of that process. We enjoy peace earned through the battles of the past. But as concepts stack up, often precariously in the adolescent mind, so grows the implicit weight of abstraction. Teenagers, laden with thousands of years of sophisticated scholarship boiled down into extremely efficient mathematical/linguistic machinery, inevitably hit some point where a new idea simply does not fit into their model. The notion of, say, graphing a quadratic equation is both 1) needlessly complicated and 2) seemingly arbitrary.
“Why do we care what it looks like?”
“Why should the solutions, which sometimes you call ‘roots’ and sometimes ‘zeroes’ and sometimes ‘factors,’ have anything to do with crossing the x-axis?!”
“Why do these so-called parabolas only smile and frown but never tilt to the side?”
Most teachers dismiss these questions because they cannot answer them satisfactorily.
“You’ll get used to it.”
“It’s not about the math per se; it just makes you smarter.”
“You’ll need this next year.”
The subtle tragedy in all this is that the students are right: it is totally arbitrary, at least at the most foundational level. Math is not real. It could have been something else. But, the greatest minds of the past fought hard, not for the flawless definitions, but for the best compromises. The most comfortable perches on the tightrope between abstract generality and practical specificity. The concepts were designed to solve problems, as much as homeworks would have one believe that problems are designed to explain concepts. To chime in on an admittedly inconsequential debate, mathematics is certainly invented and not discovered. Being right on this question brings me no joy, but I would like to see this truth acknowledged by the education system. It can be acknowledged gingerly at first: take 25 minutes in a geometry class to dissolve Euclid’s fifth (parallel) postulate and peel back the curtain to non-Euclidean geometry. Show 7th graders what kind of nonsense you can come up with if addition is no longer commutative. Convince a college freshman that the additive identity (0) and the multiplicative identity (1) cannot be the same number. An informal discussion of paradoxical statements like “this statement is false” could open the door to Gödel’s mind-bending work. A similar project could be undertaken to demystify computation, which to most people might as well be sacred yet unapproachable magic we cannot live without. Music theory is often viewed as a frustrating game of puzzles devised by nerds in lieu of making real music, but one could instead view music theory, and indeed music itself in many instances, as a solution to puzzles that arise naturally. Tonality is a compromise between tuning, modality, and symmetry. Why should parallel fifths and octaves be forbidden in counterpoint? (What “proof by contradiction” can you come up with if they are allowed?) Chord-scale jazz thinking is a somewhat-fast algorithm for finding the right notes -- what are its weaknesses that offset its speed, and why? When is it worth it? (To be fair, I think music theory actually does a better job, on average, of presenting itself as a series of historical developments compared to math.)
I bet that acknowledging this intuitive, historical, imperfection-embracing mindset wouldn’t just tantalize the especially curious students but also bridge the gap to abstract/theoretical/technical topics for struggling students. Pull back the veil, demystify, show the cracks in the armor -- if for no other reason than to make math (or whatever subject) less intimidating. Wouldn’t it be empowering if a teacher acknowledged a student’s complaints about mathematics as valid, but then worked with her to see why things are the way they are? The answer would not merely be “by decree.” On a related note, Common Core math has lots of issues, especially at the elementary level, where students need to build up fluency in math through repetition and memorization. CC’s goal of making math more intellectual and less mechanical is misguided for young kids, but by the late middle school/early high school level, forays into history a la the examples I mentioned above would accomplish these goals better than the silly renamings and fragmentations of ideas that currently comprise CC. If nothing else, give kids a little buzz of excitement now and then by telling some of the rare tales of math being exciting. Archimedes running nude through the street shouting “Eureka!” upon discovering his principle of water displacement; Turing cracking the Enigma code; Eratosthenes calculating the circumference of the entire freaking Earth using nothing but wells and shadows. My own middle school math teacher spun some great tales and we as a class were transfixed. It is far more engrossing to work hard on problems when you feel you are tapping into some great ancient legacy, gliding along the swift river of ideas and accomplishment. That’s part of the joy of music transcription or even just reading sheet music, or playing chess, or celebrating an old holiday. Zoom out and appreciate the magnificent edifices upon which we stand!
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Long Coda: the Example of Continuity
To put my money where my mouth is, I’ll do my best to build the formal notion of continuity from the ground up. It’s one of my favorite examples of mathematicians pinning down a slippery concept precisely without sacrificing flexibility. Also, many teachers never define continuity satisfactorily because the somewhat vague intuitive notion of its meaning is sufficient for most purposes, even through calculus. If anything, students are merely given examples of non-continuity and expected to intuit the definition from there, as if swimming were the act of not drowning. But it is rewarding to wrestle directly with an evasive concept that seems so obvious at first! Without further ado:
“Continuous” in everyday speech means something like “smooth; unbroken; happening always.” In a mathematical setting, this notion naturally applies to the idea of a smooth, unbroken curve in the plane or in space, but also the idea of a shape or quantity smoothly morphing over time. Think of a candle melting, a tree growing, or water swirling around in a cup. This is already tricky for a mathematically conservative and uncreative mind: firstly, how does one reconcile the paradox of a smooth curve made of individual infinitesimal breadthless points? and secondly, how would one extend that notion to 2-, 3-, or N-dimensional, potentially physical objects like candles, trees, or water?
To the uninitiated, the answer is complicated, baffling, likely meaningless:
Let f : [a,b] → R and x0 ∈ [a,b]. f is continuous at x0 if for
every ε > 0 there exists δ > 0 such that | x − x0 | < δ implies | f(x) − f(x0) | < ε.
Not helpful. Let’s step back, way back, and find the path there ourselves.
First things first -- let’s focus on the idea of a smooth curve with no gaps. A curve that can be drawn without lifting your pencil. An unbroken string of pearls. Recall that most often, plane curves are generated from equations relating X and Y, as in the X- and Y-axes of the coordinate plane. More specifically, one typically considers functions relating X to Y. A function is a rule (an equation) that allows one to plug in an X-coordinate and generate a Y-coordinate so that a point on the plane, having an X-coordinate and a Y-coordinate, represents, potentially, a solution to an equation. The simplest possible example is the function Y = X, which tells us to draw a line through all the points where the Y- and X-coordinates are the same. An infinite diagonal line from Southwest to Northeast. Y = 2X + 3 is a slightly steeper and higher line including points such as (0, 3), (10, 23), and (-1.5, 0), as well as everything in between and beyond. More exotic functions generate curves rather than straight lines, but the principle remains the same: plug in X, return Y, plot the pair. Already we can see the roots of continuity: we are assuming X and Y are real numbers, which is to say we are allowed to plug in any value from the continuum of all possible values. Not just whole numbers, but fractions and irrational numbers too. An infinitely fine grain, the smoothest sand on the beach of the mind. The fact that there exists so many damn numbers, an infinitude between even the smallest gaps, gives us a clue as to how we might capture the notion of drawing a curve without lifting the pencil. Maybe there’s a way to do it “one point at a time” but leverage the infinitude of points into a suitable definition for curves, which is to say functions.
However, the problem with that approach is obvious. A single point “knows nothing” of the curve it’s on; it doesn't “know its neighbors.” A point is very slim on data: just two individual numbers, bundled together. Darn. But we do like how “zoomed in” a single point is; a continuous curve should be unbroken even at atomic scales and beyond (it would be a shame to discover gaps between the pixels). Can we have our cake and eat it too -- compromise between the global shapeliness of a curve and the local specificity and numerical properties of an individual point? Herein lies our first big strategic move: we will take the indirect approach of calling a function continuous at a point rather than over its whole domain. An approach that seems backwards at first, but the convenient fact is that if we do this successfully, we can call a whole function continuous if we know it is continuous at every point individually. In other words, we employ the smoothness of the numbers themselves to do work for us. We, as mathematicians, can zoom in to a specific location and analyze behavior there rather than having to juggle a whole snake of points all at once.
But we aren’t there yet. How do we imbue a point, a naïve and minimal chunk of data, with more structure and awareness of its surroundings? The key is to zoom out from the point, in the slightest way possible, to a “neighborhood” around the point. One can imagine a little blob surrounding a point on the plane. But we must be more precise, and we must appeal to the curve itself, not merely the space on the plane. We also must be extremely careful not to zoom out too much, for even a tiny segment of a curve is as infinite and unwieldy as the whole thing. Hence the foundational innovation undergirding limits, continuity, calculus, and the whole subject of mathematical analysis: ε. This measly Greek letter (epsilon) represents an arbitrarily small yet finite value. Think of the game you can play with any child:
“What’s the smallest positive number?”
“1.”
“What about ½?”
“Oh...well then what about ½ of ½?”
“Now you’re getting it.”
The notion of ε scales to whatever resolution of tininess we need at the present moment. Let’s see it in action.
To draw a curve without lifting the pencil, each point on the curve must be close enough to its neighbors that no pencil-lift is necessary to get there. On the pencil-scale, a distance of, say, .01 millimeters will surely do: that is, if there are no gulfs in the theoretical curve longer than .01 mm, we will be safe dragging our pencil across any smaller gaps. In this case, ε stands in for .01 mm. On the extra-thick Sharpie scale, a full millimeter might be sufficient for ε. On the electron microscope scale, ε would be on the order of nanometers. The magic is in the flexibility of the idea of ε. Let’s break it down further and get specific. Call whatever point we’re zoomed into, on whatever curve, p. p has an X-coordinate and a Y-coordinate determined by its X-coordinate. Call its X-coordinate x0 and thus its Y-coordinate f(x0). (Formally, “f of x-zero;” i.e. the result of applying the function f to the number x0.) We want to encode the idea of minuscule intervals around p in both the X- and Y-directions. Mathematically, all points on the curve a distance less than ε away, where ε can be any small positive (greater than 0) number. In order to disambiguate the X- and Y-directions, we introduce δ (delta) as another arbitrarily small positive value, to be used shortly. We are ready to put it all together.
We can now make precise the notion of sliding an infinitesimal amount along our curve, starting from the point p with coordinates (x0, f(x0)). The neighborhood around x0 consists of all x-values less than δ away, which is calculated through subtraction. Mathematically, all values of x such that | x − x0 | < δ. (The absolute value just turns all distances positive in order to defend against the semi-nonsensical notion of negative distance.) Likewise, the neighborhood around f(x0) (now we are oriented on the vertical Y-axis) is all those values of f(x) such that | f(x) − f(x0) | < ε. δ and ε can be freely specified at any point; we imagine in the meantime numbers like .000001. To finally define a continuous function (and thereby a continuous curve), we play a game. Imagine your adversary encloses point p within a small vertical box of size ε, and challenges you as follows:
“Your curve must not change more than ε. If you slide left or right along the curve and the y-value jumps more than ε, the curve is discontinuous and I win.”
Let’s imagine the two scenarios. If the function f is indeed continuous as we imagine it, that is smooth and unbroken, we should be fine. Imagine p is attached to a smooth line on both sides, and say ε is .001. Zoom in super-close to p and mark a ceiling and floor .001 above and below it. The smoothness of f implies that the ceiling and floor will cut off a box such that a minuscule segment of f with lies totally within. Call the width of the box δ. Here is a picture:
The adversary is defeated for now. He is free to shrink ε even further, but we can counter with a smaller δ that does the job. We believe in the smoothness of the real numbers to win this battle for us. Because ε can shrink to any microscopic scale at all and we can still find such a δ that encloses our smooth curve, we confidently declare that f is continuous at x0.
How do we know this definition is working? We check the other scenario, where f has some gap or jump, and hope that the adversary wins the game. So pick p somewhere where f jumps; let’s say that to the right, it is attached to a smooth curve, but has a gap on the left:
The adversary sets a small ε and challenges us to ensure that f changes less than ε vertically within a horizontal interval of δ. On the right side, we are OK. Tiny rightward nudges produce only tiny upward nudges, so our quest for δ begins swimmingly. However, on the left side, we run into a major problem. Even the smallest leftward nudge pops us downward far more than ε. We cannot find a small yet positive δ that closes the box; we fail! f is discontinuous at x0. Exactly as planned -- the definition “feels right.”
Note that we would have won our continuity-testing game anywhere else on f, even in the second scenario. It thus makes sense to say that f would be continuous on intervals that do not include that pathological point p. Remember, f being continuous as a whole curve/function amounts to all its individual points having the continuity property individually.
All that is left to do is summarize the procedure of this game in mathematical notation. One step at a time:
Let f : [a,b] → R and x0 ∈ [a,b].
All this is saying is that our function, our curve, is defined as mapping x-values on an interval of the real numbers ([a,b] means every number from a to b) to a real-numerical output. It’s a fancy way of placing us on the X-Y plane. Our special point has to have its X-coordinate, x0, in the domain of f. Read ∈ as “belongs to.”
f is continuous at x0 if:
for every ε > 0 there exists δ > 0 such that | x − x0 | < δ implies | f(x) − f(x0) | < ε.
Now the meat. “For every ε > 0” means that the adversary is free to challenge us with an arbitrary small positive number to box us in. Our job is to find, i.e. show that “there exists δ > 0 such that” the horizontal neighborhood around x0, i.e. the x-values fulfilling | x − x0 | < δ, don’t pop us out of the vertical neighborhood, i.e. the y-values fulfilling | f(x) − f(x0) | < ε.
In summary: the adversary assigns ε, and if we can always counter with a δ that keeps f comfortably in the adversarial box, we establish continuity at that point.
Deep breath. We have successfully defined an intuitively clear but mathematically slippery notion: continuity of a curve (represented by a function). The point here was to show that these definitions don’t just appear from nowhere; they are earned through head-scratching, compromise, and creative workarounds to paradoxes and imprecisions in intuition. While the definition may at first seem needlessly technical, abstract, or indirect, it turns out to be instead an ingenious balance of specificity and flexibility. We, counterintuitively, zoom into single points to define a global property of a curve; we play a recursive game with simple individual steps involving finite, definite numerical values that generalizes to infinitesimal scales so that we are satisfied without having to appeal to the paradox of zero-length nudges or zero-area boxes. Better yet, this definition is totally agnostic to the properties of f: it tells us whether points or intervals are continuous or not no matter how wild our curve gets. It also easily generalizes to higher dimensions with a rather minimal amount of mathematical massaging. In fact, it generalizes to domains beyond numbers in the case of general topology. An even more abstract definition, presented without comment or explanation:
Let (X, TX ) and (Y, TY) be topological spaces. A function f : X → Y is said to be
continuous if the inverse image of every open subset of Y is open in X.
(The real puzzle here is the meaning of open.)
I would have loved to grapple with this stuff in high school. Sure, it’s difficult. Merely presenting the definition and expecting a student to understand it would have been an obviously stupid idea. But with an hour of mental wrestling, led by a teacher, working through examples and trying out some hypotheses, this becomes a rewarding exercise that reveals the glorious, dirty, living, breathing, miraculous nuts and bolts of mathematics, the hard-earned truths that unlock our simple computational methods. Not exactly “light afternoon reading,” but instead of the ten-thousandth arcane, soulless symbolic manipulation of the sort that pervades high school math? Yes please!
If you read this far, good for you and please accept my gratitude. I hope you don’t feel I wasted your time. Some music recommendations to make up for it:
An arrestingly beautiful Horowitz performance of Bach
Be well and keep it real.