Math’s Path – Part 2

Calculus and the explosive growth of the rate of change in progress.

Continuing our exploration of the path of math through history, let’s explore calculus, the branch of mathematics that deals with things that change. Math is essentially the study of patterns measured in numbers, and calculus is of crucial importance to science because it allows the computation of things that change over time. Calculus is probably the most influential breakthrough in history, and its discovery by Isaac Newton is a mind-blowing story.

As described by Edward Dolnick in The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World:

No hero ever rose from less auspicious roots than Isaac Newton. His father was a farmer who could not sign his name, his mother scarcely more learned. Newton’s father died three months before his son was born. The baby was premature, so tiny and weak that no one expected him to survive; the mother was a widow, not yet thirty; the country was embroiled in civil war.

Newton went to Cambridge University, but then something happened that will ring familiar with today’s college students, but without the Zoom and even more social isolation:

When plague settled onto the town, the university shut down and sent its students and faculty away, to wait for a time when it would be safe to gather in groups again. In June 1665 plague struck Cambridge, and the university closed. A young student named Isaac Newton gathered up his books and retreated to his mother’s farm to think in solitude.

In a time without video games or other similar distractions, Newtown spent his free time at home … inventing calculus.  Newton started thinking about graphs like this, which show the position of an object (its height) as it travels through each point in time (its distance):

The question for Newton was: how can we figure out precisely the rate at which an object is changing at each and every point on this curve?  That is, how can we know an object’s velocity (the rate of change in its position) at each and every particular time, which would involve calculating the slope of the tangent line at each and every point of the graph, considering there are an infinite number of potential points on the graph?

As Dolnick explains (in a way worth quoting at length):

[The ancient Greek philosopher] Zeno cast his paradox in the form of a story about a journey across a room. In the 1500s and 1600s a few intrepid mathematicians reframed his tale as a statement about numbers. From that perspective, the question was whether or not 1 + ½ + ¼ + ⅛ + 1/ 16 + . . . added up to infinity. Zeno’s answer was “yes,” because the numbers go on forever and each contributes something to the sum. But when mathematicians turned from Zeno’s words to their numbers and began adding, they found something odd. They began with 1 + ½. That made 1 ½. Nothing dire there. How about 1 + ½ + ¼? That came to 1 ¾. Still okay. 1 + ½ + ¼ + ⅛? That was 1 7/ 8. They added more and more terms and never ran into trouble. The running total continued to grow, but it became ever clearer that the number 2 represented a kind of boundary. You could draw arbitrarily close to that boundary— within one- thousandth or one- billionth or even closer— but certainly you could never break through in the way that a runner breaks the tape at the finish line … [S]uppose you looked at [a] coach’s speed in briefer and briefer intervals and found that that sequence of speeds homed in on a limit? Then your troubles would be over. That limit would be a number— a definite, perfectly ordinary number. That was what “instantaneous speed” meant … [Newton] spent eighteen months between 1665 and 1667 at his mother’s farm, hiding from the plague that had shut Cambridge down. Newton was twenty- two when he returned home, undistinguished, unknown, and alone … Newton was indeed in his prime at twenty- three, for mathematics and physics are games for the young. Einstein was twenty-six when he came up with the special theory of relativity, Heisenberg twenty-five when he formulated the uncertainty principle, Niels Bohr twenty-eight when he proposed a revolutionary model of the atom. “If you haven’t done outstanding work in mathematics by 30, you never will,” says Ronald Graham, one of today’s best-regarded mathematicians.

Dolnick walks us through Newton’s thinking a bit more, referring to the graph below:

Pictorially, you would be drawing straight lines that passed through two dots on the curve. One dot was fixed in place at t = 1, and the other moved down the curve, like a bead on a wire, approaching ever nearer to the fixed dot … Those lines would approach ever nearer to one particular straight line. That “target” line was unique, in a natural way— it was the line that just grazed the curve at a single place, the point corresponding to t = 1. The target line— in mathematical jargon the tangent line— was the prize that all the fuss was about. (In the diagram below, the tangent line is the straight line made up of short dashes.) Until this moment, mathematicians had never managed to close their fingers around the notion of instantaneous speed … They had found a way to define a moving object’s speed at a given instant. Instantaneous speed was the number that average speeds approached, as you looked at shorter and shorter time intervals. The slope of the tangent line (short dashes) represents the speed of a falling rock at the instant t = 1 second.  The speed of a moving object at a given instant was just an ordinary number, the slope of the tangent line at that point. How did you compute that slope? By looking at the slopes of the straight lines that approached the tangent line and seeing if those numbers approached a limit. That limit was the number we were after, the grail in this long quest … Better yet, the same calculation that revealed the rock’s speed at a single instant also told its speed at every instant. Without bothering to lift a finger or draw another straight line (let alone an infinite sequence of straight lines homing in on a target line), this once-and-for-all calculation showed that the rock’s speed at any time t was precisely 32t. The speed was always changing, but a single formula captured all the changes. When the rock had been falling for 2 seconds, its speed was 64 feet per second (32 × 2). At 2 ½ seconds, its speed was 80 feet per second (32 × 2 ½); at three seconds, 96 feet per second, and so on … This new tool for describing the moving, changing world was called calculus.

As Dolnick describes the world-shaking importance of this discovery:

For the next two centuries, science would consist largely of finding ways to exploit the new power that calculus provided … How long will it take the world’s population to double? How many thousands of years ago was this mummy sealed in his tomb? How soon will the oyster harvest in the Chesapeake Bay fall to zero? … Of all the roller- coaster tracks that start at a peak here and end at a valley there, which is fastest? Of all the ways to fire a cannon at a fortress high above it on a mountain, which would inflict the heaviest damage? (This was [Edmond] Halley’s contribution, written almost as soon as he had heard of calculus. It turns out that he had also found the best angle to shoot a basketball so that it swishes through the hoop.) Of all the shapes of soap bubble one can imagine, which encloses the greatest volume with the least surface? (Nature chooses the ideal solution, a spherical bubble.) Of all the ticket prices a theater could charge, which would bring in the most money?

As Brooks describes the ubiquitous uses of calculus:

Whenever there is a continuously changing set of parameters, calculus steps up. Take a jumbo jet’s fuel requirements, for example: the amount of fuel needed to keep the plane in the air changes as it burns more fuel and reduces its own weight. Or how to calculate the annual income that can be derived from a savings account that has a variable interest rate. Or the market price of grain as supply and demand fluctuate. All these examples use calculus … If you drive or walk across a suspension bridge, a set of differential equations that describes the interplay between the mass, stiffness, and resistance to movement of the bridge’s materials … It’s a similar story in a skyscraper: issues from how the load on the foundation changes as the height of the building rises, to how much it will twist and sway in a storm, are calculated using differential equations in the design phase.

Now let’s step away from calculus and look at how even the simplest math concepts contribute to modern technological life.  Take prime numbers (numbers that are divisible only by the number 1 or itself).  Today they’re used to keep our information private when it’s sent over the internet:

Cryptography, the practice of creating and decoding secret communications, is perhaps the most under-appreciated branch of mathematics. Our freedoms and our prosperity are rooted in our ability to maintain privacy … It enables secure mobile banking, which enables Rwandan farmers to conduct business and create a livelihood. It’s central to Colombian anti-trafficking agencies co-ordinating a cocaine bust. Whistleblowers attempting to expose corruption need encrypted messaging services … [There are an] infinite number of prime numbers, which are only divisible by themselves and 1 … [In the 1970’s, Clifford] Cocks perform[ed] a mathematical operation that includes two large primes to generate his “public key” [that unlocks encrypted internet communications]. He can make this publicly available so that someone wanting to send him a secret can mathematically blend their secret with the public key. They then send the resulting string of data to Cocks. Because Cocks is the only person who knows the maths that used two prime numbers to create the public key, he alone can decrypt the message and reveal the secret … Reliable cryptography is so easy to implement now that these schemes protect our personal data, our credit card details, our communications, and anything else we choose to keep private.

Math was essential to winning World War II:

The calculus of fluid flow is at the heart of the way we design aeroplanes. Get it right, and you can use it to win crucial victories that change the course of history — the Battle of Britain, for instance [which was won by the aerodynamic design of the British Spitfire warplane.]

And math even helped save us from nuclear destruction:

Over the years, hordes of mathematicians have worked on the algebra of the arms race, but John Nash stands above them all. That’s because of his celebrated “Nash equilibrium”, an algebraic way to find the best resolution to a dilemma where two parties cannot trust each other. It uses a complex form of linear algebra, and describes a scenario where two opposed parties can find themselves in a situation where they cannot do better than their current standing. The equilibrium strategy might not be the optimal strategy for that player, but it is the only one that doesn’t make things worse … it’s a least-worst-option kind of thing … Nash gave both sides in the Cold War concrete proof that they should accept the frosty détente and not consider any further moves. Essentially he used algebra to make the world a safer place.

We started these essays on the role of math in civilizational advancement by noting how kids’ math scores are down, especially among lower-income kids, and then sought to explain how losing the power of math weakens not just life prospects, but blocks paths to social progress.  Yet today, there are perverse movements afoot dedicated to eradicating math in the name of “social justice.”  As Williamson Evers writes in the Wall Street Journal:

If California education officials have their way, generations of students may not know how to calculate an apartment’s square footage or the area of a farm field, but the “mathematics” of political agitation and organizing will be second nature to them … This will be the result if a proposed mathematics curriculum framework, which would guide K-12 instruction in the Golden State’s public schools, is approved by California’s Instructional Quality Commission in meetings this week and in August and ratified by the state board of education later this year. The framework recommends eight times that teachers use a troubling document, “A Pathway to Equitable Math Instruction: Dismantling Racism in Mathematics Instruction.” This manual claims that teachers addressing students’ mistakes forthrightly is a form of white supremacy. It sets forth indicators of “white supremacy culture in the mathematics classroom,” including a focus on “getting the right answer,” teaching math in a “linear fashion,” requiring students to “show their work” and grading them on demonstrated knowledge of the subject matter. “The concept of mathematics being purely objective is unequivocally false,” the manual explains. “Upholding the idea that there are always right and wrong answers perpetuates ‘objectivity.’ ” Apparently, that’s also racist.

Sadly, such movements will put math even further beyond the reach of kids who deserve to be exposed to its wonders, and to its potential for yet more progress we all might enjoy.  Such movements, intentionally or not, only threaten to keep the power of math in the hands of the elite by denying it to others.  As Brooks writes of the past:

It’s hard to know whether the gradual appropriation of mathematics by an elite minority was deliberate. The ancient Egyptians’ nilometers suggest it was: these gauges of the river’s depth were built within the boundaries of the temples to give the priests exclusive access. They alone knew when the floods were coming, and so possessed secrets that affected the lives of ordinary people, an important attribute when seeking power over the masses.

Math is for everyone, and its power works to the benefit of everyone.  To understand that, one need only count the things math has given the world.  But tragically, even that simple task seems beyond the abilities of some education officials today.

In the next essay, we’ll look at how math helps prevent the spread of disease, and also how panic spreads through poor estimations of risk.

[NOTE: A fantastic YouTube channel that visually explains math and calculus concepts in an easy-to-follow manner is 3Blue1Brown.]

Also, Leonard Mlodinow, in his book The Drunkard's Walk: How Randomness Rules Our Lives, provides a further discussion of Zeno’s paradox, as follows:

Though calculus represents a new sophistication in the understanding of sequences, that idea, like so many others, had already been familiar to the Greeks. In the fifth century B.C., in fact, the Greek philosopher Zeno employed a curious sequence to formulate a paradox that is still debated among college philosophy students today, especially after a few beers. Zeno’s paradox goes like this: Suppose a student wishes to step to the door, which is 1 meter away. (We choose a meter here for convenience, but the same argument holds for a mile or any other measure.) Before she arrives there, she first must arrive at the halfway point. But in order to reach the halfway point, she must first arrive halfway to the halfway point— that is, at the one- quarter- way point. And so on, ad infinitum. In other words, in order to reach her destination, she must travel this sequence of distances: 1/2 meter, 1/4 meter, 1/8 meter, 1/16 meter, and so on. Zeno argued that because the sequence goes on forever, she has to traverse an infinite number of finite distances. That, Zeno said, must take an infinite amount of time. Zeno’s conclusion: you can never get anywhere. Over the centuries, philosophers from Aristotle to Kant have debated this quandary. Diogenes the Cynic took the empirical approach: he simply walked a few steps, then pointed out that things in fact do move. To those of us who aren’t students of philosophy, that probably sounds like a pretty good answer. But it wouldn’t have impressed Zeno. Zeno was aware of the clash between his logical proof and the evidence of his senses; it’s just that, unlike Diogenes, what Zeno trusted was logic. And Zeno wasn’t just spinning his wheels. Even Diogenes would have had to admit that his response leaves us facing a puzzling (and, it turns out, deep) question: if our sensory evidence is correct, then what is wrong with Zeno’s logic? Consider the sequence of distances in Zeno’s paradox: 1/2 meter, 1/4 meter, 1/8 meter, 1/16 meter, and so on (the increments growing ever smaller). This sequence has an infinite number of terms, so we cannot compute its sum by simply adding them all up. But we can notice that although the number of terms is infinite, those terms get successively smaller. Might there be a finite balance between the endless stream of terms and their endlessly diminishing size? That is precisely the kind of question we can address by employing the concepts of sequence, series, and limit. To see how it works, instead of trying to calculate how far the student went after the entire infinity of Zeno’s intervals, let’s take one interval at a time. Here are the student’s distances after the first few intervals: After the first interval: 1/2 meter After the second interval: 1/2 meter + 1/4 meter = 3/4 meter After the third interval: 1/2 meter + 1/4 meter + 1/8 meter = 7/8 meter After the fourth interval: 1/2 meter + 1/4 meter + 1/8 meter + 1/16 meter = 15/16 meter There is a pattern in these numbers: 1/2 meter, 3/4 meter, 7/8 meter, 15/16 meter … The denominator is a power of two, and the numerator is one less than the denominator. We might guess from this pattern that after 10 intervals the student would have traveled 1,023/1,024 meter; after 20 intervals, 1,048,575/1,048,576 meter; and so on. The pattern makes it clear that Zeno is correct that the more intervals we include, the greater the sum of distances we obtain. But Zeno is not correct when he says that the sum is headed for infinity. Instead, the numbers seem to be approaching 1; or as a mathematician would say, 1 meter is the limit of this sequence of distances. That makes sense, because although Zeno chopped her trip into an infinite number of intervals, she had, after all, set out to travel just 1 meter. Zeno’s paradox concerns the amount of time it takes to make the journey, not the distance covered. If the student were forced to take individual steps to cover each of Zeno’s intervals, she would indeed be in some time trouble (not to mention her having to overcome the difficulty of taking submillimeter steps)! But if she is allowed to move at constant speed without pausing at Zeno’s imaginary checkpoints— and why not?— then the time it takes to travel each of Zeno’s intervals is proportional to the distance covered in that interval, and so since the total distance is finite, as is the total time— and fortunately for all of us— motion is possible after all.

Links to all essays in this series: Part 1; Part 2; Part 3; Part 4