MATHEMATICAL REALISM
Numbers are real. They exist. They have a use in the real world. I love starting out with statements that everyone agrees with. But my first academic love was philosophy and philosophers, as Bertrand Russell observed, start with premises that no one in his right mind could deny and end with conclusions that no one in his right mind could believe. Let’s see what happens with the proposition, “Numbers are real.”
Have you ever written a story and thought it was horrible? Sure. We all have. So you change it. This happens in fiction all the time since we’ve been hit with prequels. You don’t like Captain Kirk being thirty-five? You change the timeline and he’s twenty-two. Starbuck, from Battlestar Galactica, becomes a woman, Kara Zor-El finds her mother still alive, and Shaft becomes a white guy.
Uh, wait a minute. Shaft becomes a white guy? That can’t happen. Why not? Because he’s inherently a black guy, has the experiences of a black guy, wouldn’t make sense as a white guy. (Same thing goes– dare I say it? –in spades for Easy Rawlins. James West can be a black guy, but not Easy Rawlins.}
Numbers are like that. Even more so. You can switch the racial context in a book. Ever see “White Man’s Burden,” where America was first occupied by Africans and the Europeans were the underclass? In that society, Shaft and Easy Rawlins could be white guys. But can it be the case that 2 + 2 ≠ 4? No.
This is important. Numbers are mental, just like fictional characters. They are not things, but ideas. But numbers have inherent, objective properties as if they were things. Just like gold has 79 protons, a^2 +b+2=c^2. And there ain’t nothing you can do about it.
To name some of the objective properties of numbers:
- Trichotomy. Exactly one of the following is true: y = x; y < x; y > x.
- Transitivity. If x < y, then x + z < y +z
- Order of Addition: If x < y, then x + z < y +z
- Order of Multiplication. If z is positive and x < y, then xz < yz. If z is negative and x < y, then xz > yz
These are called the order properties of numbers. They are immutable. Even in a world where black is white, where impossibly young people command starships, and your dead mother is alive as she, per impossible, survived your planet’s destruction, these properties of numbers will continue.
As will these:
Communitive laws: x + y = y + x; xy =yx
Associative Laws: x + (y+z) = (x+y) + z and x(yz) =(xy)z
Distributive law: x(y+z) =xy +xz
Identity: x +0 =x and x times 1 =x
Inverses x-x=0 and x times 1/x =1
But wait. There’s more.
These properties are true for all places, all times; they are non-local, not situated in time. They are as true in AD 5000 on the planet Mongo as they are on Earth in AD 2019. We know this even though we haven’t observed it.
Numerical relationships, such as those in algebra and geometry, though, are never perfect in the material world. In calculus, the limit for a series of fractions is not 1, it’s .99999…In the material world, nothing is perfect. As Jesus observed, it’s all moths and rust. Hannibal Smith can make a plan come together perfectly, but you can’t. In the world of numbers, we may screw the numbers up, but the relationships, the real calculations are always precise.
This perfection of numbers led Plato to think that they existed apart from the physical world; that the physical world instantiated them. Numbers were forms; and forms were perfect, i.e., eternal (non-local). This led later philosophers to think that they existed in the mind of God. They were ideas, all right. But their objectivity meant they were not in the mind of beings that were fallible, could change their minds, could screw up and need an eraser. They must be in the mind of someone infallible and who doesn’t make mistakes. That explains their objectivity. Mathematics becomes the divine language, the code by which the universe runs. The world is imperfect, being composed of matter, which is corruptible. The forms are the really real.
So there you are. We have gone from a premise that no one can deny to a bizarre notion that numbers are eternal, they are more genuine, more real than the physical world, that there can be entities that are pure form without matter. It’s just a short leap from that to a belief in angels, folks. St. Thomas Aquinas thought angels were also pure form but limited by their properties. And God was a pure form, but unlimited.
In a later blog, we will address what skeptics think about mathematical realism.
Dr. Fred Young
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