Logic Meets Beauty Week 1

Logic Meets Beauty — Week 1

Patterns, Numbers, and a Larger Landscape

The more carefully we look, the more there is to see.

Beauty invites us to notice. Logic helps us understand why.

Welcome to Week 1 of Logic Meets Beauty: A Journey Through Numbers, Mystery, and Faith.

You do not need to be a “math person” for this journey. In fact, for the next few weeks, I would like us to approach mathematics less as a subject to master and more as a place to explore.

We are going to encounter numbers, fractions, geometry, infinity, and eventually a mathematical proof. But those are not our only destinations.

Along the way, I want us to think about larger questions:

• What do we notice when we slow down?
• What happens when we deliberately ignore some details?
• How can changing our perspective reveal something that was there all along?
• How do we know when something is true?
• And where do the patterns and structures we discover come from?

Think of this hour as a small retreat from the demands of everyday life.

Our posture will be simple:

Slow down. Notice. Wonder.

Beginning with Creation

Consider these familiar words from Genesis:

“And God said, ‘Let there be light,’ and there was light. God saw that the light was good, and he separated the light from the darkness. God called the light ‘day,’ and the darkness he called ‘night.’ And there was evening, and there was morning—the first day.”
— Genesis 1:3–5

It is easy to read quickly through a passage we have heard many times.

So let's slow down.

Look only at the ending:

“And there was evening, and there was morning—the first day.”

Slower still:

Evening. Morning. One day.

Already there is pattern.

There is distinction. Light and darkness are separated. There is naming. There is rhythm: evening and morning. There is number: one day.

And there is another phrase that occurs repeatedly throughout the creation account:

“God saw that it was good.”

The text doesn't portray creation merely as a sequence of things being produced. Again and again, it draws our attention to seeing.

God creates.

God sees.

And what God sees is good.

Perhaps that offers us an appropriate posture as we enter mathematics.

For a little while, let's stop producing and accomplishing.

Let's simply look at what is there.

What Is Mathematics?

Many of us learned mathematics as a collection of activities:

• arithmetic,
• equations,
• formulas,
• calculations,
• and perhaps a few proofs.

But mathematics can also be understood as the study of patterns, structures, and relationships.

That requires an unusual skill:

Mathematics intentionally ignores things.

That may sound strange. We usually think greater understanding comes from gathering more information.

Sometimes it does.

But sometimes we understand something precisely because we decide which information doesn't matter.

Consider three cats.

They might have different colors. Different weights. Different personalities. They need food. They shed. They may scratch the furniture.

Now consider three dollars.

Almost everything has changed.

But something has remained.

Now consider three stars.

Again, almost every detail has changed.

And yet we recognize the same pattern:

$[3]$

What exactly is 3?

It isn't a cat.

It isn't a dollar.

It isn't a star.

We discovered 3 by ignoring almost everything about the objects and preserving one thing they have in common.

That gives us one of our recurring questions:

What are we choosing to ignore—and what becomes visible when we do?

Three cats contain far more information than the number 3.

So something has certainly been lost.

But something remarkable has also been gained.

The pattern 3 can now apply to almost anything.

This is one of the great powers of abstraction.

But here is another question I want to leave open for now:

Is 3 “good”?

Genesis tells us that God saw creation—the light, the land, living things—and called it good.

What about the patterns we discover within creation?

Is a number such as 3 something we created, something we discovered, or something else?

I don't have a simple answer to that question.

For now, just carry it with you.

As we discover more patterns in numbers, we'll return to it.

Ignoring details can reveal powerful patterns—but it can also hide things that matter, especially when we apply labels to people. Abstraction is useful precisely because we choose carefully what to preserve and what to ignore.

Looking Outward: Telescopic Infinity

Let's begin our journey through numbers with the most familiar ones:

$[1,2,3,4,5,\ldots]$

These are the natural numbers, or counting numbers. We will denote them by

$[\mathbb{N}]$

Now imagine looking outward through a telescope.

Keep going:

$[10,\quad100,\quad1{,}000,\quad1{,}000{,}000,\ldots]$

Eventually the numbers become unimaginably large.

But here is the important question:

How do we know they never end?

We haven't counted forever.

We can't.

Yet the logic is remarkably simple.

Suppose someone claimed to have found the largest counting number. Call it (N).

We could simply form

$[N+1]$

And we would have another counting number.

So there can be no last one.

We have used a finite piece of reasoning to make a statement about something infinite.

Notice that.

It will become important later.

Looking Inward: Microscopic Infinity

Infinity isn't only found by looking toward unimaginably large numbers.

We can also look inward.

Consider the interval between 0 and 1.

There is a fraction between them:

$\frac12$

But now look between 0 and $(\frac12)$.

There is another: $\frac14.$ Then another: $\frac18.$ And another: $\frac1{16}.$

We could continue:

$\frac12,\frac14,\frac18,\frac1{16},\frac1{32},\ldots$

But this isn't special to 0 and 1.

Take any two different fractions. There is always another fraction between them.

For example, if $a<b$, then their average, $\frac{a+b}{2}$, lies between them.

So we encounter a second kind of endlessness.

With the counting numbers:

There is no last number.

With the fractions:

There is no next fraction.

I like to think of these as two different perspectives on infinity:

Telescopic infinity: keep looking farther outward.

Microscopic infinity: keep looking more closely between.

The rational numbers—the numbers that can be written as a ratio of integers—are denoted by

$\mathbb{Q}$

A rational number has the form, $\frac{p}{q}$, where ($p$) and ($q$) are integers and $q\neq0$.

Creating Space for Truth: A Mathematical Garden

Our number line has become very crowded.

Between any two fractions there are infinitely many more.

So let's try a different landscape.

Imagine a perfectly regular grid.

Before putting any numbers on it, simply look at it.

What does it remind you of?

A tiled floor?

A garden?

A city?

A Battleship board?

Notice the order.

The rows stay straight.

The columns stay straight.

Every intersection lines up perfectly.

Unlike a real garden, we don't have to weed it or maintain it. Nothing gets crooked. Nothing gets lost.

For a few minutes, let's simply enter this mathematical landscape and see what is already there.

First, notice the intersections.

Choose any three intersections and place a dot on each one.

You have just chosen three places in our garden.

Soon, each of those places will become a fraction.

### Giving the dots meaning

A fraction has two parts:

$\frac{p}{q}.$

We will use the denominator $q$ to tell us how far to move across, and the numerator $p$ to tell us how far to move up.

So

$\frac{2}{3}$

means:

- move 3 across,
- move 2 up.

Thus,

$\frac{2}{3}\longrightarrow(3,2).$

Now return to the three dots you chose.

Count how far across and how far up each one is.

### What fractions did you choose?

Nothing about the grid changed.

We simply learned a new way to see what was there.

## Something Appears

Let's choose three particularly simple fractions:

$\frac11,\qquad\frac22,\qquad\frac33.$

They correspond to the points

$(1,1),\qquad(2,2),\qquad(3,3)$

### What do you notice?

All three fractions equal 1.

And all three points lie on the same ray from the origin.

Draw the ray through them.

We have discovered that the number 1 can be represented not only as a point on a number line, but as a **direction** on our grid.

Now try another family:

$\frac12,\qquad\frac24,\qquad\frac36,\qquad\frac48$

Again, the fractions are equivalent.

And again, their points lie along one ray.

This is not an accident.

Equivalent fractions have the same slope:

$\text{slope}=\frac{\text{rise}}{\text{run}} =\frac{\text{numerator}}{\text{denominator}}.$

So a fraction can be thought of not merely as a location on a number line.

It can also represent a direction.

## One More Mystery Hidden in the Grid

Look again at the ray through

$(1,1),\quad(2,2),\quad(3,3),\ldots$

This is the line

$y=x$

It divides our garden into two beautifully symmetric regions.

### Below the line

The numerator is smaller than the denominator, so the fraction is less than 1.

For example,

$\frac23<1$

### On the line

The numerator and denominator are equal:

$\frac11=\frac22=\frac33=\cdots=1$

### Above the line

The numerator is larger than the denominator, so the fraction is greater than 1.

For example,

$\frac32>1$

Now imagine placing a mirror along the line $y=x$.

The point $(3,2)$ reflects to $(2,3).$ So $\frac23$ is reflected to $\frac32.$

The grid seems to pair fractions below 1 with fractions above 1.

That raises a strange question:

> ### **Could there somehow be just as many fractions between 0 and 1 as there are greater than 1?**

Our ordinary intuition may say no.

But the symmetry of the grid seems to be suggesting something else.

We will not answer that question yet.

In Week 3, we will develop a precise way to compare the sizes of infinite collections.

For now: Notice the mirror.

What Did We Gain? What Did We Lose?

This is worth thinking about.

Neither representation is “the correct one.”

The number line makes certain things easy to see:

• order,
• relative size,
• distance.

The grid makes other things easier to see:

• numerator and denominator separately,
• equivalent fractions,
• geometric relationships,
• direction and slope.

We gained something when we moved to the grid.

But we also lost something. On the grid, the ordinary ordering of fractions is no longer nearly as obvious.

This is a small example of a much larger principle:

A representation can reveal something true without revealing everything that is true.

Changing our point of view doesn't necessarily change reality.

Sometimes it changes what part of reality we are able to see.

And Now Something Strange.

So now reverse the question.

Instead of starting with a fraction and finding its direction, imagine starting with every possible direction.

Must every ray eventually pass through another grid point?

If such a ray exists, what would its slope represent?

Don't answer too quickly.

That is one of the questions I want you to carry into Week 2.

Logic Meets Beauty

We've covered surprisingly little complicated mathematics.

We counted.

We looked between numbers.

We separated the numerator and denominator of a fraction.

We plotted a few dots.

But notice where those simple moves have taken us.

We began with individual objects: $\text{cats}\rightarrow3.$

Then individual counting numbers became an infinite collection:

$1,2,3,\ldots\rightarrow\mathbb N.$

We looked between numbers and found another infinite collection:

$\text{fractions}\rightarrow\mathbb Q.$

Then we changed our perspective:

$\text{number line}\rightarrow\text{grid}.$

And points became directions:

$\text{fraction}\rightarrow\text{ray}.$

Each time, we ignored something.

Each time, we gained something.

And each time, a larger landscape became visible.

That is one sense in which logic meets beauty.

Beauty invites us to notice. Logic helps us understand why.

Questions to Carry With You

I want to leave Week 1 with two questions.

1. Can you think of anything in our world that is truly “forever”?

Not merely something very large or very old.

Something that really never ends.

And then ask the harder question:

How could you know?

We will discover that mathematics has an unusual ability to say things with certainty about processes that no human being could ever finish observing.

That will become increasingly important as our journey continues.

2. Can an arrow miss every dot?

Our grid contains infinitely many dots.

And every fraction gives us a direction that hits one.

But are the fractions all the possible directions?

Or is there something hiding between the rays we've already discovered?

Next week, we'll go looking.

A Final Thought

Genesis repeatedly gives us a simple phrase:

“And God saw that it was good.”

Perhaps seeing deserves more attention than we normally give it.

There are patterns we rush past.

Relationships we take for granted.

Things we've seen thousands of times without really noticing them.

Even something as ordinary as a fraction may contain a landscape we never knew was there.

So as you go through the coming week, practice the posture we've begun:

Slow down. Notice. Wonder.

The more carefully we look, the more there is to see.