HS #106 2024.5.9
Is Mathematics Invented or Discovered?
Occasionally a reader suggests a topic for my column as with the above title. My thanks.
This question is important because it applies to more than mathematics. For example, did Michelangelo create (invent) David from a block of marble or discover David in the marble? We typically think of artists using their imagination to create their art, but Michelangelo thought differently. He claimed, “The sculpture is already complete within the marble block before I start my work. It is already there, I just have to chisel away the superfluous material.” So David was in the block – waiting to be discovered and released. This makes Michelangelo’s sculpture, “Slaves” especially poignant. The figures are only partially carved from the marble. The rest of the slaves’ bodies were not released – they stayed enslaved in the marble block.
It also applies to moral truth. Do we humans invent our own ethical systems through the creative and natural use of our empathy, or do we discover morals truths platonically existing outside our natural world – perhaps given to us by God?
Let’s reflect on the meaning of those two words. In my living room I have a favorite piece of petrified wood. It was discovered. In my closet is the board game Monopoly. That game was invented. Which of these two words best describes how we came upon mathematics?
Did not Pythagoras discover that the lengths of a right triangle obey “A squared plus B squared equals C squared”? Could he have chosen otherwise? Seems not. Seven is a prime number because this many things $-$-$-$-$-$-$ can’t be divided into equal piles. Wasn’t this true before humans existed – indeed, before the universe existed? Can 2 plus 3 equal anything but 5? Would a first-grade teacher allow a creative student to answer 7?
So it certainly seems that mathematical truths exist independent of humans and were there to be discovered.
But not so fast. On the other hand, some mathematicians claim that doing math is like playing a game. Once you invent the game of chess, for example, there are all sorts of questions to ask: Can a knight be manipulated to land on any square? What is the fewest number of moves needed to win? Moreover, if someone wishes, they can change the rules to invent a modified game.
Similarly, until the nineteenth century, Euclid’s geometry was considered the one true geometry discovered by ancient humans and organized by Euclid. Then Gauss and others started playing around with the Euclid’s five foundational axioms. By changing them, mathematicians invented new geometries to play with - very much like altering the rules of a board game. Some of these new geometries were fun to play but useless, while another was used by Einstein in formulating his Theory of Relativity, and yet another describes the geometry of the globe on which we live. Useful, but ostensibly invented by humans.
So is mathematics invented or discovered?
The 19th century mathematician, Leopold Kronecker, effectively answered “Both!” Referring to the simple integers {. . . -3, -2, -1, 0, 1,2, 3, . . . }, he famously claimed, “God made the integers, all the rest is the work of man.” That is, the foundational truths already exist for humans to discover. But once discovered, humans can use their imagination to create more. Much like baking bread – the recipe may be your own invention, but your choices are limited by the (discovered) inherent properties of the ingredients. Every yeast bread recipe must include time for it to rise.
The great 20th century mathematician Paul Erdos likely agreed. Every living mathematician has an “Erdos number”: 1 if they published a paper with Erdos, 2 if they published with someone who published with Erdos, etc. (My Erdos Number is 4.) Erdos traveled the globe asking mathematical questions and solving mathematical theorems (visiting Hope College a couple times). When an aspiring mathematician showed Erdos a new mathematical proof, Erdos sometimes gave his supreme compliment, “That one is from The Book.” That is, certain proofs were so beautiful – so much like Michelangelo’s David or Beethoven’s Ninth Symphony – that Erdos claimed they had been discovered from God’s book of proofs rather than just invented by human imagination.
Don’t we think the same? Who can listen to Beethoven’s Ode to Joy without sensing that this is a piece discovered from “God’s Book”. In contrast, the music written by the high school rock band in your neighbor’s garage - most likely invented.
