4 ms·

I like how this video focuses on the question of how to model the world. People often hold mathematical results up as truth when they're really just the conseq

by panic 7y ago

I like how this video focuses on the question of how to model the world. People often hold mathematical results up as truth when they're really just the consequences of taking particular definitions. It's possible to reject any mathematical conclusion -- a straw obviously doesn't have one hole, it has two! -- and then work backward to find new ways of modeling that make your conclusion true. Of course, you'll probably find strange new consequences, like the fact that a plate now has a hole. But then you can go back and tweak your definitions some more, or even the entire premise. Maybe "being a hole" isn't a binary yes-or-no thing, for example, but a continuum from cup-like (very holey) to plate-like (barely holey). This process of finding strange consequences, fixing the model to avoid them (or include them!), finding new strange consequences, and so on, is really what math is about, I think.

PopeDotNinja 7y ago
I think it's valuable to understand that most of what we're taught is an abstraction for describing something. Using negative integers as an example, it wasn't until I was introduced to unsigned integers in C that it even occurred to me to wonder about what the heck a negative number is. It was then I realized how cool it was to know that I could make up my own version of numbers where there are no negatives. From now on, numbers all have a length and direction! Instead of +3 and -3, now we was 3 in-that-direction, and 3 in-the-exact-opposite-direction! I decided it was easier to keep using regular negative integers, but it was cool to realize someone just made them up many moons ago. Then I felt smarter and/or less dumb, and I started having fewer "I could never do..." thoughts after that.
dwaltrip 7y ago
Great comment. I'll share a potential practical "definition" that popped into my head that is clearly different from the topological approach. If one starts with the idea of a hole as: "something that an entity can fall into or through", then it becomes very clear that a cup has a hole and a plate doesn't. An ant can fall into a cup from one side, but there is no way for an ant to fall into or through a plate. For falling "into" something, clearly some conception of "height" is important for this definition. Perhaps its related to being able to draw certain lines -- ones that are nearly parallel and of a relatively large length -- from one boundary to another surface of the object. For falling "through" something, I'm drawing a bit of blank right now, but I think there is some fairly simple definition that can clearly express this idea. I would argue the above is pretty close to what humans naturally consider as holes. --- EDIT: Thinking about this a bit more... I think we can simplify the above by viewing holes as "openings". This covers both cases of "falling into" or "through", and removes the distracting reference to gravity. It should be possible to come up with some sort of fairly practical definition of "opening". I think it has do with relatively sizing and geometry of the "cavern" that the hole leads to. However... inherently, there is no sharp line we can draw here, as the difference between a small scratch or dent and an actual cavernous hole is just a matter of degree. It seems, like many things in life, the distinction will remain a bit fuzzy and is not perfectly resolvable.

by panic 7y ago

PopeDotNinja 7y ago

I think it's valuable to understand that most of what we're taught is an abstraction for describing something. Using negative integers as an example, it wasn't until I was introduced to unsigned integers in C that it even occurred to me to wonder about what the heck a negative number is. It was then I realized how cool it was to know that I could make up my own version of numbers where there are no negatives. From now on, numbers all have a length and direction! Instead of +3 and -3, now we was 3 in-that-direction, and 3 in-the-exact-opposite-direction! I decided it was easier to keep using regular negative integers, but it was cool to realize someone just made them up many moons ago. Then I felt smarter and/or less dumb, and I started having fewer "I could never do..." thoughts after that.

dwaltrip 7y ago

Great comment. I'll share a potential practical "definition" that popped into my head that is clearly different from the topological approach. If one starts with the idea of a hole as: "something that an entity can fall into or through", then it becomes very clear that a cup has a hole and a plate doesn't. An ant can fall into a cup from one side, but there is no way for an ant to fall into or through a plate. For falling "into" something, clearly some conception of "height" is important for this definition. Perhaps its related to being able to draw certain lines -- ones that are nearly parallel and of a relatively large length -- from one boundary to another surface of the object. For falling "through" something, I'm drawing a bit of blank right now, but I think there is some fairly simple definition that can clearly express this idea. I would argue the above is pretty close to what humans naturally consider as holes. --- EDIT: Thinking about this a bit more... I think we can simplify the above by viewing holes as "openings". This covers both cases of "falling into" or "through", and removes the distracting reference to gravity. It should be possible to come up with some sort of fairly practical definition of "opening". I think it has do with relatively sizing and geometry of the "cavern" that the hole leads to. However... inherently, there is no sharp line we can draw here, as the difference between a small scratch or dent and an actual cavernous hole is just a matter of degree. It seems, like many things in life, the distinction will remain a bit fuzzy and is not perfectly resolvable.