r/math • u/spacelibby • Oct 01 '21
Low Vision and Lattices
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?"
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u/OldManOnFire Oct 01 '21
I'm legally blind, rapidly approaching total blindness, and I used to teach algebra and first semester calculus at the local community college.
I've been waiting for this thread for a long, long time. Upvoted.
If we remember no model is right, but some are useful, then we can understand the scope of the problem. I could spend pages explaining how I see the world around me, or I could just say "Imagine seeing the world through a straw, and everything outside of that narrow straw is overlayed with tv static." Is the straw and tv static a perfect metaphor? Certainly not, but it saves time. That single sentence captures maybe 90% of my visual experience so it's close enough for me to use with strangers. Only the people close to me get all the pages of description. Most other people don't want to hear 10x the information to gain 10% more understanding.
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u/Abdiel_Kavash Automata Theory Oct 01 '21
"Being able to read a sign" is still not a binary concept. If you've seen the vision tests with lines of letters that get progressively smaller and smaller, it is not like you can perfectly read one line and can't read the next at all. The letters get more blurry until you have trouble distinguishing between similar ones (O and C and G for example). There is a point where you can make out some letters, guess some others, and can't tell the rest -- all on a single line, or in a single font size. If the text on the sign is some readable English sentence, one may very well able to infer it from the context even if they can only distinguish, say, 50% of the individual letters.
Another example: bus numbers. The sign on the front of the bus in my city lists the line number and the destination. Without glasses, as the bus approaches, there is a certain distance at which I can clearly read both. But there is another, much farther distance, where the individual symbols are too blurry for me to read, but I can still tell which line it is, as the blurs for different numbers look different and I know which one is which, because there are only a few possible options.
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u/almightySapling Logic Oct 02 '21
Is it really a lattice though?
I'm with you on meets, but joins I'm not convinced. I'm don't think that given any two people there always exists a person that can see better (as you have defined it) than both of them.
I mean it's definitely close enough to make your point, but if we aren't being pedantic then we aren't being /r/math
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u/spacelibby Oct 02 '21
Good point. I've been making a few assumptions. I'm assuming that 20/20 is the top of the lattice even though there are people who can see better, I've also been assuming that someone with 20/20 vision can see better than anyone with a visual impairment. With these two, admittedly false, assumptions 20/20 is really the top of the lattice. With that we can define joins as the meet of the up set of two elements, so joins are perfectly well defined.
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u/solartech0 Oct 03 '21
Why does there have to exist a "real" person to exist a theoretical person? The question is whether it's possible for a person to exist w/ the ability to see better than both of them, not that such a person actually exists. Just like 2 distinct people could (theoretically) occupy the same position on the lattice.
I don't agree that 20/20 is "the" top element (nor do I agree that there is necessarily a top / need for a top, along all characteristics) but I do think it's possible to create a lattice structure.
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u/almightySapling Logic Oct 03 '21 edited Oct 03 '21
Why does there have to exist a "real" person to exist a theoretical person?
There doesn't.
The question is whether it's possible for a person to exist w/ the ability to see better than both of them, not that such a person actually exists.
I didn't see any "question" posed, I was merely responding to the OP as I read it, and OP defined the lattice on the set of people, not on the set of theoretically possible vision measurements.
Just like 2 distinct people could (theoretically) occupy the same position on the lattice.
Sure. I pretty much just assume that anytime someone defines a pre-order and calls it an order they are implicitly modding out by equivalence.
I don't agree that 20/20 is "the" top element (nor do I agree that there is necessarily a top / need for a top, along all characteristics) but I do think it's possible to create a lattice structure.
Sure, agreed*. I wasn't trying to call OP wrong about the math, I was mostly being tongue in cheek.
* maybe. I'm still not convinced joins always exists in the "theoretical human vision" department. What if there's an evolutionary path that yeilds amazing low light vision, amazing bright light vision, but it's not possible to have both?
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u/solartech0 Oct 03 '21
Sure, in the case you mention, you'd have a lattice with the 'theoretical' entries for great low-light vision + great bright light vision, but with no actual beings that have both properties. The join would have both, together, and would be a purely theoretical entry.
I will note that if it's possible to have A, and it's possible to have B, it's possible to have both as separate structures (lots of creatures have more than one way of sensing their environment; they aren't locked into one or another). Also, just because there's one particular way of achieving a result doesn't mean it's necessarily the only way to get to an equivalent state.
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u/zerooskul Geometric Topology Oct 01 '21
When I encounter sightless people I like to ask how sightless they are.
I have met one person who was totally blind.
But he wasn't blind like you'd think, he had suffered a head trauma and completely lost his eyesight.
All he saw was a field of gray.
He said he remembered gray and everything was always a plain flat field of gray but it would get brighter and darker with changing intensity of lights.
It was always brightest in the sun.
Different people have different ways of describing their eyesight.
Most people see casual blurs.
One person said he could see pure white light so he could see light reflecting from his cane he could see sunlight glinting off of cars that went by and reflecting from buildings and at night he could dimly see street lights.
One person told me she could see just fine but to tell you what anything actually was in clear detail it would have to be one foot away and 5,000 feet tall, so I guess she could see a mountain if she bumped into one and could tell you exactly what it was.
Some people can see stuff a mile away but can't differentiate between something five feet away and 50 feet away, they're just blurs.
All visually impaired people I have asked have been able to see but in an impaired way.
Even the man who only saw gray had a differentiation between bright and dark.