Imagine, if you will, a triangle. I’ll make it a very specific one. It has a base of 18 centimetres and a perpendicular height of 9 centimetres. It even has a specific orientation with its base horizontal on a grid. Here it is (or at least a version of it)
Am I being wholly honest? The actual picture I’m showing you is a .png file and you are viewing it on a computer screen. The image is made of tiny squares (visible with a magnifying glass) and the file itself, the more abstract description of the image, is a .png i.e. in principle a set of coordinates of colour information for a grid of data. Does that matter? The answer is “It depends”.
Here’s the image again and superimposed a set of largeish squares. They are there to represent a low-resolution version of the pixelated triangle.
Make the squares a bit smaller and our shpae made from squares begins to look a lot more akin to a triangle:
You get the picture. Smaller and smaller squares make for a smoother image. The first set of stacked squares really doesn’t look much like a triangle but with smaller and more squares the image gets more and more like a triangle. How small do the squares have to be for it to be REALLY a triangle? Trick question.
Here’s a different way of looking at it. As I make the squares smaller the area of the stacked squares gets closer and closer to the are of the triangle we would find from the classic half-the-base-times-the-perpendicular-height formula. Our stacked squares shape literally gets more triangular as far as area goes as we improve the resolution.
But what about the perimeter? Well that’s a mess. It’s not even clear whether perimeter is meaningful. We could define it taking into encounter the edges of the nominal squares. If we do that then (I think, correct me if I’m wrong) the perimeter would tend towards the perimeter of a 9 by 18 rectangle. Ooops. My pixelated triangle is getting more rectangular. However, if I treat the perimeter as meaning “number of pixels on the outside of the shape” then I get a different limit.
Yeah but make them small enough and it is basically a triangle right? Sure, unless you really care about counting pixels, in which case not so much. Every real, physical manifestation of a geometric entity is a messy, not entirely correct compromise. Doing actual useful maths with real things requires understanding the extent to which a thing is and isn’t the “pure” mathematical entity it resembles.
This is a basic fact about the universe. It’s true whether you take a strong Platonic realist view of mathematical entities (i.e. they really are really real, maybe even more real than other things) or not (i.e. they are essentially fictional abstractions that are useful but less real than physical stuff that you can bump your head on).
Too cut a long story slightly shorter I sort of maybe tried to engage our old pal and the inaugural Dragon Award Winner for Best Horror Story That Is Actually A Space Opera Brian Niemeier about this. My excuse is that he said that “necessary being is what theologians mean by God.” and logically necessary things are to me like a flame is to a moth. Brian’s going to be doing the ontological argument for the existence of god and by golly if there’s one think I have opinions on it’s that. Then he used the necessary properties of a triangle as an example…
Anyway, the guy thinks I’m a demon anyway, so if he does the equivalent of drawing a summoning circle for me, I really have no choice but to manifest in a puff of sulphur. https://www.brianniemeier.com/2019/04/what-we-talk-about-when-we-talk-about.html
The discussion ended with this:
“You have two choices.
1) Confess that Jesus is the Christ, and God has raised Him from the dead. I shall forgive your violation of my hospitality, and you may continue commenting here.
2) You decline to make this just and logically irrefutable profession, and you cease commenting here. Persist in commenting without meeting my terms, and you get spammed.”
Point 1 apparently is how he expels demonically controlled beings from his blog. Which worked because I then vanished back to my demonic lair without even once twisting my head around 360 degrees. Brian then followed it up with an additional rant about the left being controlled by demons, in particular Doris: https://www.brianniemeier.com/2019/04/im-not-saying-its-demons.html
Anyway, how was your day?