I think I’ve found my new favorite website: the Standford Encyclopedia of Philosophy. It’s much smaller than Wikipedia, but each article is very well written. I’m reading through the Category Theory article right now, and it’s quite tough. But slowly but surely I’m making progress. It may seem like abstract nonsense, but you’ve encountered useful examples of this already!
You learned about parity (even and odd numbers) in grade school. The parity of 1 is “odd” and the parity of 42 is “even”, and so on. We could write this like
Our intuition about adding even and odd numbers is
- Even plus even is even
- Even plus odd is odd
- Odd plus even is odd
- Odd plus Odd is even
We can just write this like
Now here’s something special about our function \(Parity\). Is the sum of 14 and 23 even or odd? One way to figure this out is to add 14 and 23, then find the parity.
But maybe you took a shortcut, and realized “Hey! 14 is even and 23 is odd, and I know that an even number plus an odd number is always odd.”
You don’t even need to know how to add numbers really. All you need to know is that 14 is even and 23 is odd.
It turns out that for any two numbers \(x\) and \(y\) (not just 14 and 23)
The above equation means “Adding two numbers and then checking if they are even or odd gives the same result as checking if the numbers are even or odd and adding them according to the even/odd rules.” Our function \(Parity\) is known as a Homomorphism, a way of preserving some structure.
It also turns out that a lot of things have homomorphisms. A lot of the results are mathematical, but it appears in everyday thought processes too.
You know the result of mixing paints by knowing how colors mix, because there is a homomorphism that associates a color with a can of paint. You could find out by mixing the jars of paint and then observing the resultant color, or you could observe the colors of the initial paint jars and then reference a color wheel to find out what color would be produced (which saves the energy of mixing the paints).
If you enjoyed learning about how thinking abstractly can benefit you, take a look at this article on Algebra. A word of warning: Algebra is not what you think it is.