Pythagoras’ revenge: humans didn’t invent mathematics, it’s what the world is made of (The Conversation)

Sam Baron – November 21, 2021 11.47pm EST

Many people think that mathematics is a human invention. To this way of thinking, mathematics is like a language: it may describe real things in the world, but it doesn’t “exist” outside the minds of the people who use it.

But the Pythagorean school of thought in ancient Greece held a different view. Its proponents believed reality is fundamentally mathematical.

More than 2,000 years later, philosophers and physicists are starting to take this idea seriously.

As I argue in a new paper, mathematics is an essential component of nature that gives structure to the physical world.

Honeybees and hexagons

Bees in hives produce hexagonal honeycomb. Why?

According to the “honeycomb conjecture” in mathematics, hexagons are the most efficient shape for tiling the plane. If you want to fully cover a surface using tiles of a uniform shape and size, while keeping the total length of the perimeter to a minimum, hexagons are the shape to use.

The hexagonal pattern of honeycomb is the most efficient way to cover a space in identical tiles. Sam Baron, Author provided

Charles Darwin reasoned that bees have evolved to use this shape because it produces the largest cells to store honey for the smallest input of energy to produce wax.

The honeycomb conjecture was first proposed in ancient times, but was only proved in 1999 by mathematician Thomas Hales.

Cicadas and prime numbers

Here’s another example. There are two subspecies of North American periodical cicadas that live most of their lives in the ground. Then, every 13 or 17 years (depending on the subspecies), the cicadas emerge in great swarms for a period of around two weeks.

Why is it 13 and 17 years? Why not 12 and 14? Or 16 and 18?

One explanation appeals to the fact that 13 and 17 are prime numbers.

Some cicadas have evolved to emerge from the ground at intervals of a prime number of years, possibly to avoid predators with life cycles of different lengths. Michael Kropiewnicki / Pixels

Imagine the cicadas have a range of predators that also spend most of their lives in the ground. The cicadas need to come out of the ground when their predators are lying dormant.

Suppose there are predators with life cycles of 2, 3, 4, 5, 6, 7, 8 and 9 years. What is the best way to avoid them all?

Well, compare a 13-year life cycle and a 12-year life cycle. When a cicada with a 12-year life cycle comes out of the ground, the 2-year, 3-year and 4-year predators will also be out of the ground, because 2, 3 and 4 all divide evenly into 12.

When a cicada with a 13-year life cycle comes out of the ground, none of its predators will be out of the ground, because none of 2, 3, 4, 5, 6, 7, 8 or 9 divides evenly into 13. The same is true for 17.

P1–P9 represent cycling predators. The number-line represents years. The highlighted gaps show how 13 and 17-year cicadas manage to avoid their predators. Sam Baron, Author provided

It seems these cicadas have evolved to exploit basic facts about numbers.

Creation or discovery?

Once we start looking, it is easy to find other examples. From the shape of soap films, to gear design in engines, to the location and size of the gaps in the rings of Saturn, mathematics is everywhere.

If mathematics explains so many things we see around us, then it is unlikely that mathematics is something we’ve created. The alternative is that mathematical facts are discovered: not just by humans, but by insects, soap bubbles, combustion engines and planets.

What did Plato think?

But if we are discovering something, what is it?

The ancient Greek philosopher Plato had an answer. He thought mathematics describes objects that really exist.

For Plato, these objects included numbers and geometric shapes. Today, we might add more complicated mathematical objects such as groups, categories, functions, fields and rings to the list.

For Plato, numbers existed in a realm separate from the physical world. Geralt / Pixabay

Plato also maintained that mathematical objects exist outside of space and time. But such a view only deepens the mystery of how mathematics explains anything.

Explanation involves showing how one thing in the world depends on another. If mathematical objects exist in a realm apart from the world we live in, they don’t seem capable of relating to anything physical.

Enter Pythagoreanism

The ancient Pythagoreans agreed with Plato that mathematics describes a world of objects. But, unlike Plato, they didn’t think mathematical objects exist beyond space and time.

Instead, they believed physical reality is made of mathematical objects in the same way matter is made of atoms.

If reality is made of mathematical objects, it’s easy to see how mathematics might play a role in explaining the world around us.

Pythagorean pie: the world is made of mathematics plus matter. Sam Baron, Author provided

In the past decade, two physicists have mounted significant defences of the Pythagorean position: Swedish-US cosmologist Max Tegmark and Australian physicist-philosopher Jane McDonnell.

Tegmark argues reality just is one big mathematical object. If that seems weird, think about the idea that reality is a simulation. A simulation is a computer program, which is a kind of mathematical object.

McDonnell’s view is more radical. She thinks reality is made of mathematical objects and minds. Mathematics is how the Universe, which is conscious, comes to know itself.

I defend a different view: the world has two parts, mathematics and matter. Mathematics gives matter its form, and matter gives mathematics its substance.

Mathematical objects provide a structural framework for the physical world.

The future of mathematics

It makes sense that Pythagoreanism is being rediscovered in physics.

In the past century physics has become more and more mathematical, turning to seemingly abstract fields of inquiry such as group theory and differential geometry in an effort to explain the physical world.

As the boundary between physics and mathematics blurs, it becomes harder to say which parts of the world are physical and which are mathematical.

But it is strange that Pythagoreanism has been neglected by philosophers for so long.

I believe that is about to change. The time has arrived for a Pythagorean revolution, one that promises to radically alter our understanding of reality.

Is mathematics real? A viral TikTok video raises a legitimate question with exciting answers (The Conversation)

Daniel Mansfield – August 31, 2020 1.41am EDT

While filming herself getting ready for work recently, TikTok user @gracie.ham reached deep into the ancient foundations of mathematics and found an absolute gem of a question:

How could someone come up with a concept like algebra?

She also asked what the ancient Greek philosopher Pythagoras might have used mathematics for, and other questions that revolve around the age-old conundrum of whether mathematics is “real” or something humans just made up.

Many responded negatively to the post, but others — including mathematicians like me — found the questions quite insightful.

Is mathematics real?

Philosophers and mathematicians have been arguing over this for centuries. Some believe mathematics is universal; others consider it only as real as anything else humans have invented.

Thanks to @gracie.ham, Twitter users have now vigorously joined the debate.

For me, part of the answer lies in history.

From one perspective, mathematics is a universal language used to describe the world around us. For instance, two apples plus three apples is always five apples, regardless of your point of view.

But mathematics is also a language used by humans, so it is not independent of culture. History shows us that different cultures had their own understanding of mathematics.

Unfortunately, most of this ancient understanding is now lost. In just about every ancient culture, a few scattered texts are all that remain of their scientific knowledge.

However, there is one ancient culture that left behind an absolute abundance of texts.

Babylonian algebra

Buried in the deserts of modern Iraq, clay tablets from ancient Babylon have survived intact for about 4,000 years.

These tablets are slowly being translated and what we have learned so far is that the Babylonians were practical people who were highly numerate and knew how to solve sophisticated problems with numbers.

Their arithmetic was different from ours, though. They didn’t use zero or negative numbers. They even mapped out the motion of the planets without using calculus as we do.

Of particular importance for @gracie.ham’s question about the origins of algebra is that they knew that the numbers 3, 4 and 5 correspond to the lengths of the sides and diagonal of a rectangle. They also knew these numbers satisfied the fundamental relation 3² + 4² = 5² that ensures the sides are perpendicular.

No theorems were harmed (or used) in the construction of this rectangle.

The Babylonians did all this without modern algebraic concepts. We would express a more general version of the same idea using Pythagoras’ theorem: any right-angled triangle with sides of length a and b and hypotenuse c satisfies a² + b² = c².

The Babylonian perspective omits algebraic variables, theorems, axioms and proofs not because they were ignorant but because these ideas had not yet developed. In short, these social constructs began more than 1,000 years later, in ancient Greece. The Babylonians happily and productively did mathematics and solved problems without any of these relatively modern notions.

What was it all for?

@gracie.ham also asks how Pythagoras came up with his theorem. The short answer is: he didn’t.

Pythagoras of Samos (c. 570-495 BC) probably heard about the idea we now associate with his name while he was in Egypt. He may have been the person to introduce it to Greece, but we don’t really know.

Pythagoras didn’t use his theorem for anything practical. He was primarily interested in numerology and the mysticism of numbers, rather than the applications of mathematics.

Without modern tools, how do you make right angles just right? Ancient Hindu religious texts give instructions for making a rectangular fire altar using the 3-4-5 configuration with sides of length 3 and 4, and diagonal length 5. These measurements ensure that the altar has right angles in each corner.

A man sits at a fire altar
A rectangular fire altar. Madhu K / Wikipedia, CC BY-SA

Big questions

In the 19th century, the German mathematician Leopold Kronecker said “God made the integers, all else is the work of man”. I agree with that sentiment, at least for the positive integers — the whole numbers we count with — because the Babylonians didn’t believe in zero or negative numbers.

Mathematics has been happening for a very, very long time. Long before ancient Greece and Pythagoras.

Is it real? Most cultures agree about some basics, like the positive integers and the 3-4-5 right triangle. Just about everything else in mathematics is determined by the society in which you live.