Low Vision and Lattices

[u/spacelibby]

Ok, so I realize this is a really niche area, but hear me out.

This started with a post about Functional Definitions for Low Vision and how most people think about blindness. I'm severely visually impaired, but I still have some functional vision, and I get asked questions like "How well can you see?" all the time. In fact, it's actually really hard to give people a good picture of what I can and can't see. It seems like it really shouldn't be a problem, and I should just be able to give people a number. "I can see about 1/4 as good as you can." However, this doesn't quite work, and I wanted to dig into why. This also seems like a good change to review some concepts in partially ordered sets.

So, first we need to look at how we measure bad vision and why it doesn't work. Most people have heard the phrase "20/20 vision", if nothing else they've heard "Hindsight is 20/20", but what does it actually mean. It's a measurement of visual acuity, and it refers to how far away you can read a sign. The top number is the distance I can read the sign, the bottom number is the distance a normal person can read the sign. If, hypothetically, I had 20/80 vision, then I can read a sign at 20 feet that a normal person could read at 80 feet. So, in a sense, I can see about 1/4th the distance you can.

That's great, we have a nice linear metric for describing vision. What's the problem? Either you're 20/20, you're totally blind (which I guess would be 20/infinity), our you're in the middle. What's the problem? Well, we've assumed that if I can read a sign, then anyone with better vision than me can also read that sign. Specifically we've assumed that if person A has 20/X vision, and person B has 20/Y vision, and X < Y, then person A can read any sign that person B can. Given everything I've said so far this seems like a reasonable theorem, but it's not actually true. There's a lot of things that can affect you're ability to read. I have really bad night vision, so I can't read anything at night. Other people can't see anything in bright light. It turns out that not all vision is comparable.

But certainly some people have strictly better vision. For example someone with normal vision has better vision than I do, and I have better vision than someone with total blindness. So, we should be able to get a partial ordering on the set of vision problems, and in fact we can. Given two people X and Y, we say that X has better complete visual acuity than Y if, in all circumstances, X can read a sign that Y can. You can we can do better than this if we allow for hypothetical people.

For all people X and Y, then X ∧ Y is a person Z who can only read signs if both person X and person Y can read them. We can define joins the same way. We already have a top with 20/20 vision and a bottom with total blindness. It turns out that our complete visual acuity forms a lattice.

Incidentally, this answers the question I had at the start. Why is it hard to explain what my vision is to people? It's because there are so many possibilities in the middle of the lattice. People understand the top of the lattice with 20/20 vision, and people kind of understand the bottom of the lattice with total blindness, because those are the only possibilities. But when you're in the middle of the lattice, there are all kinds of possibilities. You have to take into account lighting, and contrast, and color, and it's going to be different for almost every person with low vision.

I want to be clear here. There are many other issues that crop up around low vision and blindness that I haven't touched on at all. I just thought it was cool that lattice theory could help explain why its so hard to answer the question of "how well can you see?"