“Imagine you had to take an art class where you were only taught how to draw a fence … but were never shown the paintings of the great masters. Would you love art?”
That is professor of mathematics Edward Frenkel arguing that one of the reasons why people dislike math is because they were only taught to “draw the fence” and are not aware of the mathematical masterpieces: Euler’s Identity, Euclid’s Elements, Fermat’s Last Theorem, etc.
But we can’t blame them. At the end of the day, if you want to experience Picasso, DaVinci or Wadinsky you can go to a museum but the same isn’t true for mathematics—people don’t know where to find it.
But why bother ourselves so much with the exposition and appreciation of maths? I mean, is maths even real? Is it a feature of the Universe or is it a product of the human brain made to better understand the world?
Well, here are some of the most popular interpretations of the subject:
Mathematical Fictionalism considers numbers and mathematical concepts to be a product of our minds to describe patterns in the Universe that aren’t inherently mathematic.
The problem with this philosophy is that it doesn’t seem to justify that the study of the behavior of the natural world—science—has math embedded deep within it which has only proven favorable. This that defining math as false would imply that the way that nature behaves isn’t true.
Some also argue that since fictionalism requires a human mind to create the math, their thesis falls apart if we go back in time. In other words, if two Australopithecus get together for “grooming and chill” and make another Australopithecus there’s a total of three Australopithecines (2+1=3) whether we see it or not. According fictionalists this wouldn’t be true since no human was there to carry out the math.
Nominalism argues that numbers don’t exist but instead they describe real physical things.
Think of it as how we are taught math as children. For example 3×3=9 if taught as “if you have three boxes of three sweets each, you have a total of nine sweets”.
One of the challenges of Nominalism comes when we throw complex numbers into the mix. It’s easy to imagine two slices of pizza to represent the number 6 or half a latte macchiato to represent 1/2, but what about √-1?
This is known as an imaginary number (i) since we can’t actually show you a certain amount of pizza equivalent to √-1. Ironically, imagining it is pretty hard too.
Pi (an irrational number) doesn’t like to play along with Nominalism either and that is a big problem.
Pi is a naturally occurring number—it’s everywhere there’s a curve or circle such as our DNA double helix or the Sun—and if it’s in nature it surely must exist… right? Therefor we should be able to show you a concrete object to represent π but we can’t.
So it seems that to Nominalists one of the most important numbers in nature doesn’t exist. But how can that be? I mean, nature is real and if pi occurs throughout it, then does nominalism reject the existence of a part of the Universe?
Not quite. A common counter argument is that when we calculate with π we’re actually using an approximation to the actual number which, once fully calculated, could be physically represented by an object.
Mathematical Realism’s stand on the debate is that mathematics and all its concepts exist independently of us and they are disembodied in the Universe as part of its DNA. All we do is uncover and find uses for them.
In a way what Mathematical Fictionalism tells us is that the Universe is an immense painting and mathematical concepts are the techniques used to design it.
When you look at the Mona Lisa you don’t see the names of the techniques used by DaVinci written all over it as if they were objects, you see the result of them interacting together and the same applies to the cosmos.
Like all the previous interpretations, Realism isn’t perfect. It is sometimes criticized by people who deem it to be less about scientific views and more about faith since we can’t directly observe it happening.
I don’t know who’s on the right side of the debate. No one does. But I do know that mathematics is the language of nature and as a consequence of its omnipresence it’s a part of our culture and we should accept it as such.