Are we inventors or discoverers?
With mathematics, the best answer may be that it is both discovered and invented
PremiumPythagoras is known for his famous theorem that describes right-angled triangles. You know, the one about the squares on two sides adding up to the square on the hypotenuse. But if that’s a fundamental part of mathematics, Pythagoras also had some ideas about mathematics more generally and its place in the world.
Over 2,500 years ago, he suggested that mathematics is one of just two languages - yes, it is a language - that can describe the, well, nature of nature. The architecture of nature, if you like. Whatever you make of that, you probably want to know what the other language was that he thought of in this way. Not Greek, not Latin, no languages of that kind. He meant ... music.
More of that, another time.
For now, let me run past you a few phenomena we know and understand well. Right-angled triangles follow a strictly mathematical rule we’ve named for Pythagoras. A circle is effectively defined by a mathematical constant we call “pi". Sunflowers and seashells grow in distinctly mathematical ways. Bees use near-perfect geometric shapes called hexagons to build their honeycombs. There’s more.
Now I’d like to think Pythagoras was an admirer of the sunflower, not least since its botanical name, Helianthus, comes from the Greek words “helios" (sun) and “anthus" (flower). But of course, I have no evidence that he noted the mathematical essence of sunflowers or any of those particular phenomena above apart from the triangle. Still, they allude to that mathematical spirit that clearly moved him.
Pythagoras believed numbers were fundamental to everything around us. In fact, he believed that in some sense, the universe itself is built on mathematics. The reality, he was certain, is fundamentally mathematical. So in this way of looking at the world, mathematics is what lends definition and structure to the universe we live in.
That’s an interesting way of considering our surroundings, no doubt. And once you subscribe to it, you start noticing mathematics all around.
There are the long-hibernating cicadas I once wrote about here.
Why are they underground for 13 and 17 years, meaning why those numbers in particular? There’s a persuasive theory that it’s because 13 and 17 are primes. Who would have thought prime numbers mean something to cicadas - and in what sense is that true?
Or take the planet Saturn’s famous and beautiful rings. They are made up largely of chunks of ice, with the occasional rock thrown in. Scientists have long known that the chunks that are larger are less common. If you think that’s only natural, think of this too: the sizes follow a mathematical precept called the “inverse cube law." It’s not just that the larger ones are less common; it’s that this law governs how many chunks of each size there are, compared to ones of another size. That is, chunks twice as large as others are eight (the cube of 2) times less frequently found; three times as large, and they are 27 (cube of 3) times less frequently found.
Or think of bubbles of soap. Their very appearance is already mathematical. To enclose the volume of air inside the bubble, a sphere is a shape with the smallest surface area. Put another way, how can a given quantity of soap film enclose the largest possible volume? Answer: Form a sphere. Its volume is nearly 40% greater than a cube formed with the same quantity of soap film. This is why soap bubbles look like they do, not cubes.
But there’s more to them too. Joseph Plateau, a Belgian physicist, spent plenty of time observing soap bubbles in the 19th Century. He postulated two rules - Plateau’s Laws - that apply when bubbles touch. First, soap films invariably meet in threes, forming angles of 120 degrees and an edge called the Plateau Border. Second, four such Plateau Borders form a vertex, meeting at an angle of about 109 degrees. There are trigonometric reasons for those two specific angle measures.
Finally, consider this quite different scenario. In the 1990s, economics and social science researchers started using an experimental game called Dictator to test how people respond to certain situations. There are two “players", anonymous to each other. One, the “dictator", is given a sum of money. She is asked to divide it between herself and the other player, but how she chooses to divide is left entirely to her. She can leave nothing. She can leave the whole amount or some fraction. You’d think that most people would decamp with the whole amount, leaving nothing for the other player. But the researchers were struck by how many people chose to leave something. One study (Gary E Bolton et al., “Dictator game giving: Rules of fairness versus acts of kindness", International Journal of Game Theory, 1998, suggests that the most frequent amount the dictator leaves is 30% of the original sum.
Question: where did this 30% come from? Is this a mathematical phenomenon in the same sense as Plateau’s Laws, or the inverse cube law? Whatever your answer, the point here is about a number that crops up, seemingly out of nowhere.
Not for nothing did Pythagoras come to believe that mathematics explains the architecture of nature. Still, not everyone in mathematics agrees with him. He’s on one side of a debate in the field that has gone on forever: is mathematics discovered, as Pythagoras believed, whether by humans or bees or inanimate objects like soap bubbles?
Or is mathematics a human invention? Is it because we invented right-angled triangles, the notion of prime numbers, and the idea of averages that we see them in different ways all around us? Did we invent them at all? Do they, too, just exist so that they can be discovered?
Not easy questions to answer definitively, at least for me. But mathematicians have indisputably invented are proofs - of the Pythagoras Theorem, the infinity of primes, and even of Plateau’s Laws. It’s such proofs that tell us about truth in these phenomena, and sometimes the lack of it.
For example, the Pythagoras Theorem is true for all right-angled triangles - but only on flat surfaces. Mark out such a triangle on the Earth’s surface - use the lines of latitude and longitude - and not only will the theorem not hold, but the triangle will also have some unexpected characteristics.
What’s more, there are mathematical ideas for which we don’t (yet) have proofs: the Twin Prime Conjecture and Goldbach’s Conjecture, to name just two, that involve primes. So in what sense can we say they exist and were discovered?
I realize some of this seems philosophical, and maybe it is. But, that’s the nature of this debate. It’s reminiscent of the nature vs nurture question about the growth and development of human beings. The best answer may be that we are all a mixture of nature and nurture.
With mathematics, the best answer may be that it is both discovered and invented. There is certainly mathematics out there that we have discovered, that we are still to discover. But there are also mathematical ideas and principles that humans have invented.
Seen that way, it’s both a pat on our collective backs and a reminder of the need for a certain humility. I like it that way—music to my ears, at any rate. You know, that other language Pythagoras spoke of.
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