Mathematics as a cultural artifact.I have been thinking about Martin Gardner’s critique of Reuben Hersh’s ideas in an article he wrote in Eureka. I am not sure I understand Prof. Hersh’s ideas but Gardner’s comments were obviously wrong.
I am going to assume that Hersh thinks that mathematics does not exist independent of people. Gardner says that Hersh asserts that mathematics is a “cultural artifact”.
In the first place, to think that humanity has discovered timeless, universal truths is an amazing act of human hubris.
Since the right triangles of mathematics don’t exist the question arises: What does it mean to say there is something true about something that doesn’t exist? How do you say something true about unicorns?
Of course mathematics is a cultural artifact. How could one think otherwise?
In my view mathematics can be likened to a computer game. The symbols of mathematics are game pieces. The rules of mathematics are the rules of the game. The symbols don’t represent real objects. They are like the trolls, ogres and mages. The problem solutions and theorems are like the gold coins, better guns and health.
It turns out that humans can make game pieces that seem to model the approximate “real” world and then make up game rules that when followed seem to lead to a game solution of a “real” world problem. Then humans perform experiments to see how the game solution measures up to the humans’ approximation of what actually happened.
The game gives a useful answer often enough that mathematics game rules are taught in school.
Different cultures come up with different games with different game pieces. When someone says that the Pythagorean Theorem is true everywhere, they are wrong. It’s only true where the Earth mathematics game can be played and the Pythagorean Theorem is a secret place in the game.
In the “real” world there are no right triangles, there are no line segments to
use as sides for triangles or points to use as vertices. A board 2 feet long doesn’t exist. These are all game pieces but they were designed to model approximations to the “real” world as humans saw the “real” world.
The unit square is no more “real” than a dragon. Saying things about the unit square is like saying things about a dragon. What does it mean to say things about imaginary objects? Assuming that “true” and “exist” have meaning, can you make true statements about objects that don’t exist?
I feel like I’m swimming in molasses when trying to explain why mathematics isn’t real. I don’t see why the idea is so resisted. It doesn’t require believing in the impossible.
It is not hard to think of an environment where Earth mathematics can’t be played. I don’t think that the game can be played in the world of dolphins and whales.
What need would whales or dolphins have to count? They surely wouldn’t count plankton one by one. Humans tend to count because there are a lot of collections whose objects are stationary enough to allow counting.
I think that counting arose from a need. If there was no need to count, numbers would not arise. A culture that had no need of counting would have no need of numbers.
The purpose of numbers was to convey a certain kind of information. It would be remarkable if there were only one way to convey that information and that it required human physiology.
What does it mean to say that the Pythagorean Theorem is true everywhere? A culture may have an entirely different way to perceive the universe and have no way to think about geometry but if you, from Earth, were there would you see the truth of the theorem?
Well, how would you check the proof of it? You would set up your game and see if you could play it. So, what if there is no way to set up the geometry game? Is the Pythagorean Theorem still true?
On Earth there are no right triangles that satisfy the conditions of the theorem. There are no real triangles so the Pythagorean Theorem only makes sense in the world of the game.
I read somewhere that there are 50-100 billion neurons in the brain. (I have no idea why there is that big a spread.) 10 billion of these neurons pass signals around the brain over 100 trillion synaptic connections.
So to have a brain you need 10 billion “things” that pass messages among themselves and to, say, 100 billion other “things”. You probably need other stuff too but the 100 billion “things” is a good start.
In the human brain the “things” are called neurons and the connections are hardwired. I don’t see why a wireless connection wouldn’t work as well if not better. This means that the “things” wouldn’t have to be localized in a skull. The size of such a “brain” would be limited only by what speed of thought was satisfactory to the “brain”. And who knows how the “brain” looks at time. The “things” could be light years apart.
The “brain” could be 10 billion years old in which case a short interval of time would mean something quite different to the “brain” than it does to me.
How would such a “brain” look at truth?
Humans still think that they are the center of the universe. They think heaven is the next step in human development. Maybe the urge to give a mysterious reality to right triangles is the desire for God.
Why do humans insist on the reality of ghosts and goblins? Why do they insist that the universe was made in six days? What’s the point?
I can see where belief in God offers the possibility of salvation and good crops. What does ∞ do for you if you believe it is real and not just a game piece?
And that’s all it is…a game piece.
And that is what mathematics is…a cultural artifact.