Many people think 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 had a different point of view. Its defenders believed that reality is fundamentally mathematical. More than 2,000 years later, philosophers and physicists are beginning to take this idea seriously.
Professor Sam Baron of the Australian Catholic University below presents some arguments in favor of the fact that mathematics is an essential component of nature that gives structure to the physical world.
Why Are Bee Hives Hexagonal?
The bees in the hives produce a hexagonal honeycomb. Why?
According to the “honeycomb conjecture” in mathematics, hexagons are the most efficient way to tile the plane. If you want to completely cover a surface using tiles of uniform shape and size, keeping the total length of the perimeter to a minimum, hexagons are the shape to use.
Charles Darwin reasoned that bees have evolved to use this form because it produces the largest cells to store honey and the least energy input to produce wax.
The honeycomb conjecture was first proposed in ancient times, but it was only proven in 1999 by mathematician Thomas Hales.
Cicadas And Prime Numbers
Here is another example. There are two subspecies of North American periodical cicadas that live most of their life in the soil. Then every 13 to 17 years (depending on the subspecies), the cicadas emerge in large swarms over a period of around two weeks.
Why are they 13 and 17 years old? Why not 12 and 14? Or 16 and 18?
One explanation appeals to the fact that 13 and 17 are prime numbers.
Imagine that cicadas have a variety of predators that also spend most of their lives on the ground. Cicadas need to get out of the ground when their predators are inactive.
Suppose there are predators with life cycles of 2, 3, 4, 5, 6, 7, 8, and 9 years. What’s 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-, 3-, and 4-year-old predators will also be off the ground, because 2, 3, and 4 are all evenly divided into 12.
When a cicada with a 13-year life cycle leaves the ground, none of its predators will leave the ground, because none of 2, 3, 4, 5, 6, 7, 8, or 9 is evenly divided into 13. The same is true with 17.
It seems that these cicadas have evolved to exploit basic facts about numbers.
Mathematics: A Creation Or Discovery?
Once we start looking, it’s easy to find other examples. From the shape of soap films to the design of gears in motors to the location and size of the gaps in Saturn’s rings, the math is everywhere.
If mathematics explains so much that we see around us, then it is unlikely that mathematics is something we have created. The alternative is for the mathematical facts to be discovered: not just by humans, but also by insects, soap bubbles, combustion engines, and planets.
Plato’s View On Mathematics
But if we are discovering something, what is it?
The ancient Greek philosopher Plato had an answer. I thought that mathematics describes objects that actually exist – and included numbers and geometric shapes. Today, we could add more complicated math objects — such as groups, categories, functions, fields, and rings — to the list.
Plato also held that mathematical objects exist outside of space and time. However, such a point of view only deepens the mystery of how mathematics explains something since it 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.
The ancient Pythagoreans agreed with Plato that mathematics describes a world of objects. But, unlike Plato, they did not believe that mathematical objects existed beyond space and time. Instead, they believed that physical reality is made of mathematical objects in the same way that matter is made of atoms.
If reality is made of mathematical objects, it is easy to see how mathematics could play a role in explaining the world around us.
In the last decade, two physicists have mounted important defenses of the Pythagorean position: the Swedish-American cosmologist Max Tegmark and the Australian physicist and philosopher Jane McDonnell.
Tegmark argues that reality is just one big mathematical object. If that sounds strange to you, think about the idea that reality is a simulation. A simulation is a computer program, which is a kind of mathematical object.
McDonnell’s vision is more radical. She thinks that reality is made up of mathematical minds and objects. Mathematics is how the Universe, which is conscious, comes to know itself.
I defend a different vision: 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 increasingly mathematical, turning to seemingly abstract fields of research, 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 more difficult to say which parts of the world are physical and which are mathematics.
But it is strange that philosophers have neglected Pythagoreanism for so long.
I think that is about to change. The time has come for a Pythagorean revolution, one that promises to radically alter our understanding of reality.
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