Please reply in the comments with things you understood recently. The only condition is that they have to be useless in your daily life. For example, "I found this idea that defeats procrastination" doesn't count, because it sounds useful and you might be deluded about its truth. Whereas "I figured out how construction cranes are constructed" qualifies, because you aren't likely to use it and it will stay true tomorrow.
I'll start. Today I understood how Heyting algebras work as a model for intuitionistic logic. The main idea is that you represent sentences as shapes. So you might have two sentences A and B shown as two circles, then "A and B" is their intersection, "A or B" is their union, etc. But "A implies B" doesn't mean one circle lies inside the other, as you might think! Instead it's a shape too, consisting of all points that lie outside A or inside B (or both). There were some other details about closed and open sets, but these didn't cause a problem for me, while "A implies B" made me stumble for some reason. I probably won't use Heyting algebras for anything ever, but it was pretty fun to figure out.
Your turn!
PS: please don't feel pressured to post something super advanced. It's really, honestly okay to post basic things, like why a stream of tap water narrows as it falls, or why the sky is blue (though I don't claim to understand that one :-))
Yeah I know. I'm only looking at it now because intuitionistic logic can't be reduced to finite truth tables like classical logic, it really needs these pictures. That's kind of weird in itself, but hard to explain in a short post.
I guess today I'm learning about Heyting algebras too.
I don't think that circle method works. "Not Not A" isn't necessarily the same thing as "A" in a Heyting algebra, though your method suggests that they are the same. You can try to fix this by adding or removing the circle borders through negation operations, but even that yields inconsistent results. For example, if you add the border on each negation, "A or Not A" yields 1 under your method, though it should not in a Heyting algebra. If you remove the border on each negat... (read more)