QS&BBQuarks, Spacetime, and the Big Bang

Lesson 3. Mathematics, The M Word

When I imagine a triangle, even though such a figure may exist nowhere in the world except in my thought, indeed may never have existed, there is nonetheless a certain nature or form, or particular essence, of this figure that is immutable and eternal, which I did not invent, and which in no way depends on my mind.
René Descartes (1596-1650), Meditations on First Philosophy (1641)

This lesson is constructed like most. We’ll start with some goals, move smoothly into a short biography of an important character in the story, and then to the content. This one’s a little different. I’m going to use some mathematics in QS&BB and it’s important for you to understand why and gain confidence that like Goldilocks, we’ll do just enough.

This lesson is a little more review and reference-like than the rest. Some of you may have seen much of the content of this lesson. I’d point you to the Descartes, M Word, and Functions, the QS&BB Way sections.

Goals of this lesson:
I’d like you to Understand:
Simple one-variable algebra.
Exponential notation.
Scientific notation.
Unit conversion.
I’d like you to Appreciate:
The approximation of complicated functions in an expansion.
I’d like you to become Familiar With:
Aspects of Descartes’ life.
The importance of Descartes’ merging of algebra and geometry.

A Little Bit of Descartes

The 16th and 17th centuries hosted a proliferation of pre-scientific and scientific “Fathers of” figures: Galileo Galilei, the Father of Physics; Nicholas Copernicus and Johannes Kepler, arguably the Fathers of Astrophysics; and Tycho Brahe, the Father of Astronomy. That leaves out some lesser-known, but influential dads-of, like Roger Bacon, Frances Bacon (no relation), and Walter Gilbert, all of whom share paternity as Fathers of Experimentation. But the Granddaddy…um…Father of them all was René Descartes (1596-1650), often referred to as the Father of Western Philosophy and a Father of Mathematics, if not a favorite Uncle of Physics. If you’ve ever plotted a point in a coordinate system, you’ve paid homage to Descartes.

Frankly, if you’ve ever plotted a function, you’ve paid homage to Descartes. If you’ve ever looked at a rainbow? Yes. Him again. If you ever felt that the mind and the body are perhaps two different things, then you’re paying homage to Descartes and if you were taught to be skeptical of authority and to work things out for yourself? Descartes. But above all—for us—René Descartes was the Father of analytic geometry.

He was born in 1596 in a little French village called, Descartes—what are the odds? (Okay. That came later.) By this time Galileo was a professor in Padua inventing physics and Caravaggio was in Rome inventing the Baroque. Across the Channel Shakespeare was in London inventing theater and Elizabeth had cracked the Royal Glass Ceiling and was reinventing moderate political rule. This was a time of discovery when intellectuals began to think for themselves. This is the beginning of the end of the suffocating domination of Aristotle.

René was sent to a prominent Jesuit school at the age of 10 and a decade later emerged with his mandated law degree. Apart from his success in school, his most remarkable learned skill was his lifelong manner of studying: often ill, he was allowed to spend his mornings in bed, a habit he retained until the last year of his life. There’s a story there.

His school required physical fitness and in spite of his health, he became a proficient swordsman and soldier—wearing a sword throughout his life as befitting a “gentleman.” For a while he was essentially a soldier of fortune, alternating between raucous partying in Paris with friends and combat assignments (a Catholic, fighting with the Dutch Protestants) in various of the innumerable Thirty Years War armies.

Somewhere in that period Descartes became serious and decided that he had important things to say. He wrote a handful of unpublished tracts and became well-known through a steady correspondence with European intellectuals.

By 1628 he began to suspect that his ideas were not going to sit well in Catholic France ( confirmed for him when Galileo was censured in 1633) and so he moved to Holland where he lived for more than 20 years. He’d been playing with mathematics during his playboy-soldier period and little did he know, he found he was a mathematical genius, solving problems that others couldn’t. He enrolled himself as a “mature student” in Leiden and devoted himself to mathematics. By 1637, he changed the landscape forever.

Descartes’ Algebra-fication of Geometry…

…or geometri-fication of algebra! Descartes brought geometry and algebra together for the first time.

The fledgling field of algebra (“al-jabr” from the Arabic, “reunion of broken parts” ) was slowly creeping into European circles…along with the decimal point (Galileo had neither) and solutions of some kinds of polynomial equations were appearing. The notation was clumsy.

So geometry held on as king of mathematics. What Descartes did was link the solutions of geometry problems—which would have been done with rule-obsessive construction of proofs—to solutions using symbols. He did this work in a small book called Le Géométrie (The Geometry), which he published in 1637, the same year he published his philosophical blockbuster, Discourse on Method. In it he instituted a number of conventions which we use today. For example, he reserved the letters of the beginning of the alphabet $a, b, c,…$ for things that are constants or which represent fixed lines. An important strategic approach was to assume that the solution of a mathematical problem may be unknown, but can still be found and he reserved the last letters of the alphabet $x, y, z…$ to stand for unknown quantities—variables. He further introduced the compact notation of exponents to describe how many times a constant or a variable is multiplied by itself, $x^2$ for example.

The early translators of al-jabr to, um, algebra considered equations in two unknown variables like $y = \text{some combination of } x$’s to be unsolvable. But Descartes linked one variable, say $y$ to the other as points on a curve that related them through an algebraic equation—what came to be called a function. He called one of those variable’s domain the abscissa and the other, the ordinate. The use of perpendicular axes, which we call $x$ and $y$, stems from Descartes’ inspiration which is why they’re called Cartesian Coordinates.

Mathematicians picked up on these ideas and extended them into the directions that we know and love. One of those was John Wallis (1616-1703), the most important Cambridge influence on Isaac Newton.

Descartes’ Philosophy: New Knowledge Just By Thinking?

The rigor of the mathematical deductive method stayed with him and became a new kind of philosophy that he called “analytic.” Famously, he convinced himself that he had deduced a method to truth: whatever cannot be logically doubted, is true. The clue was that when you mentally and relentlessly doubted something and can’t go any further, then that idea has become “clear and distinct.” True, for him. Using this method, he decided that this demonstrated that his mind exists and that he, a thinker, is thinking these things and therefore he exists.

So by using a mathematical-like deductive path, he believed that he had made an important discovery—a proof of his existence. This is his famous bumper sticker conclusion called forever “the cogito”: Cogito ergo sum, I think, therefore, I am. But that’s not what he wrote in Meditations on First Philosophy. This is closer: “So after considering everything very thoroughly, I must finally conclude that this proposition, I am, I exist, is necessarily true whenever it is put forward by me or conceived in my mind.” Big bumper. But you know how legends go.

This is the philosophy of Rationalism which he is the king—the discovery of knowledge through pure thought. Rationalism has been in direct philosophical conflict with the philosophy of Empiricism— and as you’ll see, often physics is caught in the middle. Rationalism is in the spirit of Plato, but unlike Descartes, the Greek gave up on the sensible world as simply a bad copy of the Real World, which is one of Ideas…”out there” somewhere. By contrast, by asserting that mind and matter were both existent realms, Descartes decided that one could understand the universe by blending thinking (mind) with observing (body).

We physicists take some inspiration through Descartes’ approach. Theoretical physicists are often motivated to gain knowledge through thought, always deploying mathematics—so maybe thought and paper. Experimental physicists sometimes claim that knowledge can only be obtained through observation (and in modern form, experiment). Most of us are of the latter devotion, but can sometimes be amazed at how often smart physicists by just thinking can lead to new knowledge of the world. We’ll meet many of these folks. It’s sometimes a strange way to make a living.

After a public dispute—even in the Netherlands—Descartes began to imagine that his time among the Dutch was coming to a close. Queen Christina of Sweden, was an admirer and an intellectual and she invited Descartes to Stockholm to work in her court and to instruct her. After multiple refusals, not being a monarch to whom “no” is an easy answer, she sent a ship to Amsterdam to pick him up. He eventually accepted the position which was the beginning of his end.

The Queen required his presence at 4 AM for lessons. This, from the fellow who had spent every morning of his life in bed until noon! He caught a serious respiratory infection and died on February 11th, 1650 at the age of only 53.

We moderns owe an enormous debt to this soldier-philosopher-mathematician. Both for what he said that was useful and for what he said that was nonsense, but which stimulated a productive reaction. I think that there is a direct line from every QS&BB lesson that goes right back to René Descartes.

The M Word

I promise that the math of QS&BB will not be hard and we’ll get through it together. In this lesson I’ll develop most of the tools that we’ll return to repeatedly: simple algebra, some familiar geometry, exponents, and powers of ten.

Wait. Why use mathematics in a book for non-science people? I’m not a math person!
Glad you asked. Two reasons. First, there is a direct connection between a mathematical description of a phenomenon and nature itself. As I said, we don’t know why that’s the case and the argument about whether mathematics is “discovered” or “invented” is endless.
Second, it’s much more economical than using words.
Finally, it’s a little deductive engine for many of our purposes. You can “discover” things by manipulating the symbols…things that will further explain the physics.
I guess I lied. That’s three reasons.
Oh. There’s no such thing as a “math person” at the level we’ll be using math!

I had a decision to make in designing a set of lessons about physics for non-experts: use no mathematics or use some. Let me show you what I decided, and why. But first, here’s my guide to the use of mathematics in QS&BB:

We’ll use mathematics as a language to be “actively read” and a part of the narrative. But you’ll not have to derive things on your own from scratch.

🖋 $\leftarrow$ (see it?) then the page will turn a color and you start “close-reading” the colored material by writing in your notebook…filling blanks, making notations, even copying what your eyes see---yes, **by all means copy what you are reading!** (I do when I learn something new.) Then when it’s time to stand down, the page will go back to white and you can go back to “just” reading,

You don’t have a notebook? Please get one for the full QS&BB experience ;)

I’ll wait.

A Tiny Bit Of Algebra

Our algebraic experience here will involve some simple solutions to simple equations. I’ll need the occasional square root and the occasional exponent, but no trigonometry or solving simultaneous equations and certainly no calculus. I’ll refer to vectors, but you’ll not need to do even two-dimensional vector-component calculations. What’s not to like?

If I’d chosen to avoid all mathematics in QS&BB then I think something important would be missing. To learn about QS&BB ideas would be like learning how to paint but ignoring a particular color…where “red” should be, you’d insert a tiny note saying that “red should be here.” I'm convinced that absorbing a simple equation, which stands for something in the world, is a cognitively different experience from reading its symbols in a sentence.

An example of the power in symbols

Later we’ll learn the most fantastic model of motion that Isaac Newton invented—his Universal law of Gravitation. It explained the moon’s orbit around the earth, the planets’ motions around the sun, and still guides spacecraft through the solar system today. I could just tell you about it, or I could write it as an equation…a model.

Let’s compare two extreme approaches: I’ll write out the content of the Gravitation rule in an English paragraph and in its algebraic form. Then we’ll compare.

In this corner: Newton’s Gravitational law as a paragraph

“The force of attraction experienced by two masses on one another is directly proportional to the product of those two masses and inversely proportional to the square of the distances that separate their centers. The constant of proportionality is called the Gravitational Constant which is $0.0000000000667408\; \text{m}^3 \text{kg}^{-1}\text{s}^{-2}.$”

There. A perfectly good, if not moving, literary description of Newton’s rule. Lots of words, but it’s complete and it’s accurate. But it’s also inefficient and worse, it’s… lifeless.

Let’s contrast this with the mathematical opponent:

And in this corner: Newton’s law of Gravitation in symbols:

$F$ stands for the force of gravitation, $m$ and $M$ stand for two masses, $R$ is the distance between them, and $G$ is a number…that tiny number in the paragraph.

That’s it.

I claim that in addition to the obvious efficiency of the symbolic, compact notation…there’s physics buried inside of an equation that’s not in an English sentence. For example, here’s a perfectly good interesting question about gravitation:

Sun and Earth and Moon

What is the approximate force of attraction that the moon feels from the Sun compared with the force of attraction that the moon feels from the earth?
The paragraph-representation is not immediately helpful—it just sits there. But the symbol-equation-representation is very easily manipulated to answer a question of it. Twist it around and it’s ready to tell you something new. We could answer the question by forming the ratio of the two situations. Here’s just the answer, postponing the actual solution to the lesson on gravity:

Putting in the values for masses and distances, you’d find that the moon feels the Sun almost twice as much as it feels the earth. That information was buried inside of the symbolic representation…but not in the paragraph.

Here’s another question that the paragraph can’t deal with.

construct an experiment

Suppose Myrtle wants to study Newton’s law of gravitation by measuring the force that one mass ($m$) has for another mass ($M$). She would set them up at some distance and measure the force between them. Simple. Let’s pretend that she owns a scale that measures force in Newton’s and the smallest force it can measure is 1 N. Her lab only has a 1 meter space in which to set up her apparatus. Finally, Myrtle has a single mass at her disposal and it’s 1 kg…that’s $m$. So she needs to choose the other, $M$, in order to carry out her experiment. What does it need to be?

With some even-sided manipulation she can isolate $M$ in the equation and calculate what she needs:

Putting in the values from the question and she finds that the mass required is 14,992,000,000 kg. About the mass of 200 aircraft carriers.

Myrtle needs to do a different experiment. But…she didn’t know that until she let the original equation tell the story its way, which she encouraged through some simply symbol manipulation.

The symbolic approach has the agility to tell us how to decide about the experiment. The paragraph just sat there. Watching.

Neither of these solutions came from difficult algebra, but look what they uncovered! They’re alive!

There’s physical insight to be gained by looking at a function that describes—or maybe is?—nature.

Functions, the QS&BB Way

The language of physics is mathematics, uttered Galileo a long time ago (although he said that the language of the universe is mathematics). Well, he was right. And we have no idea why that seems to reliably be the case!

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research…”

This is from the last paragraph of a notorious and actually, delightful essay written by the physicist Eugene Wigner in 1960. The title is The Unreasonable Effectiveness of Mathematics in the Natural Sciences where he tries to dig into the strange relationship between the physical world and mathematics. It’s famous: ask Mr Google to find “Unreasonable Effectiveness” and you’ll get 150,000 references to Wigner’s essay.

I hope you see my point: The English paragraph and the succinct mathematical function on the surface do the same job. However, that “miracle” of the connection between the universe and mathematics is really only apparent when we make full use of the manipulative features of symbols in an equation where we can uncover new things.

Functions rule

One of the remarkable consequences of the mathematization of physics that began with Descartes is that we’ve come to expect that our descriptions of the universe will be in the language of mathematical functions. Do you remember what a function is? The fancy definition of a function can be pretty involved, but you do know about function machines and I’ll remind you how.

What to Remember from Lesson 3?

This has been a whirlwind pass through lots of mathematics from you past. I’d like you to remember that functions are nothing more than little machines for taking a variable and turning it into a value. The world seems to be astonishingly well described by models made up of functions! Some of them are easy and some of them are complex. Only some very simple manipulations will be required. See the Fairness Doctrine of Algebra!

I’ll ask you to “read” functions sometimes, but I promise: only when they are simple and only when there’s physics insight to be gained from that. Otherwise, we’ll be content to read graphs to “evaluate” functions because, well, it’s the same thing.

Powers of 10.

That will be an important tool since we’re talking about the largest things in the universe and also the smallest things in the universe.

Geometry

There are some simple equations, areas, and circumferences of geometrical objects that I’ll need you to remember: line, circle, triangle.

The rest

The rest of the items in Lesson 3 are there for you to refer to when we touch on some science stories that need them.

1. This stands for meter-kilogram-second, as the basic units of length, mass, and time. It’s a dated designation as the real internationally regulated system is now the International System of Units (SI) which stands for Le Système International d’Unités. The French have always been good at this. ↩︎

2. Actually, the Declaration of Independence wasn’t fully signed until August 2, 1776—my birthday! The day, not the year. ↩︎