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Did Humans Invent Mathematics, or Is It a Fundamental Part of Existence?

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Many people believe that mathematics is a human invention. For this way of thinking, mathematics is like a language: it can describe real things in the world, but it does not exist outside the minds of the people who use it.

But the Pythagorean school of thought in ancient Greece have a different perspective. Their proponents believed that reality is fundamentally mathematical.

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

As Sam Baron, an associate professor at the Australian Catholic University, argue in a new paper, mathematics is necessary component of nature that gives structure to the physical world.

Honeybees and hexagons

Bees in hives make hexagonal honeycomb. Why?

According to ‘honeycomb conjecture’ in mathematics, hexagons are the greatest efficient shape for tiling the plane. If you want to completely cover a surface using tiles of a uniform shape & size, while keeping the entire length of the perimeter to a minimum, hexagons are the shape to use.

Charles Darwin reasoned, bees 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 first-proposed in ancient times, but was only proved in 1999 by the mathematician Thomas Hales.

Cicadas and prime numbers

For example, there are two sub-species of North American periodical cicadas that live most of their lives in the ground. Then, every 13 or 17 years (depending on the sub-species), cicadas emerge in great swarms for a period of about two weeks.

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

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

Imagine, cicadas have a range of predators that even spend greatest of their lives in the ground. The cicadas require 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 & 9 years. What is the best way to avoid them all?

Compare a 13-year & a 12-year life cycle. When a cicada with a 12-year life cycle comes-out of the ground, 2-year, 3-year & 4-year predators will even be out of the ground, because 2, 3 & 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 correct for 17.

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

Creation or discovery?

It is easy to find other examples, once we start looking. From the shape of soap films, to gear design in engines, to location & size of 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 have created. The another is that mathematical facts are discovered: not just by the humans, but by insects, soap bubbles, combustion engines & planets.

What did Plato think?

But, if we are discovering something, what is it?

The ancient Greek philosopher, Plato had an answer. He believed that mathematics describes objects that actually exist.

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

Plato even maintained that mathematical objects exist outside of space & time. But this view only deepens the mystery about how mathematics explains anything.

Explanation showing how one thing in the world depends on another. If mathematical objects that exist in a realm apart from the world we live in, they do not appear to be 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 did not believe mathematical objects exist beyond space & time.

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

If reality is made of mathematical objects, it is simple to see how mathematics may play a role in explaining the world around us.

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

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

McDonnell’s view is greater radical. She believes that reality is made of mathematical objects & minds. Mathematics is how the Universe, which is conscious, comes-to-know itself.

Sam defended in a different view: world has two parts, mathematics & matter. Mathematics gives matter its form and matter give mathematics its substance.

Mathematical objects give a structural frame-work for the physical world.

The future of mathematics

It makes sense that Pythagoreanism is being re-discovered in physics.

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

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

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

Sam thought that is about to change. The time has reached for a Pythagorean revolution, one that promises to radically alter our understanding of reality.

Many people believe that mathematics is a human invention. For this way of thinking, mathematics is like language: it can describe real things in the world, but it does not exist outside the minds of the people who use it.

This article was originally published on The Conversation.

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