Where do numbers come from?
Richard Dedekind and the nature of numbers
You probably learned about numbers when you were very small. You may have been subjected to worksheets in elementary school asking you to add, subtract, multiply, and divide. Later on, you learned you would replace those numbers with letters when you didn’t know what the numbers were and had to solve for x.
But I bet nobody ever explained to you where numbers come from or what they really are.
After all, when learning what cats, dogs, cows, and horses are as a small child, you can visit a farm and see all of them and point them out. They are real things. In fact, most words you know correspond to real things.
Likewise, if you learned that a cat is white, you could understand that.
A white cat makes sense. But what about white itself?
You can probably visualize white in your head. When I do it, I imagine a splotch of white paint against a black background.
I’m not sure if I can just imagine white. I can imagine white things. I can imagine a blank field of whiteness. But is that what white really is? Am I not simply imagining something white, perhaps something that takes up my whole field of vision, like I have my nose pressed against a white wall?
The same is true of numbers. Numbers used to just be adjectives. You couldn’t just have three. You had to have three apples or three cats.
You can visualize three somethings.
Try, however, to visualize just three by itself.
You can’t.
And you were never visualizing white by itself either.
You can’t.
The reason you can’t is that both concepts, threeness and whiteness, are abstractions. They are adjectives that have been stripped of their nouns and turned into nouns themselves.
This has a lot of advantages.
When I do math, I don’t need to worry about what the numbers refer to. I just want to know how numbers relate to other numbers or transform under certain transformations.
When I start to add units to numbers, which are the nouns these adjectives refer to, I am in the realm of science, not math.
Thus, numbers come from the same place that all adjectives come from: language.
They are a way of adding attributes or properties to nouns.
Indeed, all abstractions are really adjectives. You can have a beautiful vase, but Beauty is abstract. You can have a just person, but Justice is abstract.
This explanation for numbers may be satisfying to some, but it hardly goes far enough to equate them, generically, with abstract concepts. After all, numbers appear to have a reality all their own. They have a logic and relationship with reality that is so fundamental that it demands a new system of thought.
You also know, if you read me regularly, that things are about to get more complicated.
