 So we're talking about geometry today. I'm going to focus again on points. Just a short video about different ways to think about points. As I said in the last video, I think a point makes sense in a geometric theory if the point is something that takes up space. Geometry, I think, is fundamentally about the taking up of space and therefore the objects in your geometric theory must take up some space. There are different ways to think about points. Like I said, Euclid said that they don't take up any space whatsoever. But there's another way of thinking about points, a more modern way, maybe post-day cart, which is to treat a point as a pair of numbers, as you might think of coordinates in a Cartesian plane. This is a really interesting approach I disagree with. It's something that a mathematician who's worked on a fan of Norman Weilberger, who's a work I'd recommend on YouTube if you hadn't seen it. He's doing very important work. I talked to him about it, and he seems to take that approach, that geometric objects, lines, points, circles are defined by numbers and pairs of numbers, sets of numbers. If you take that approach, it allows you to do some really nice and fancy mathematics. In fact, the way that Norman Weilberger does it, he does it in a very careful way where you can avoid even irrational numbers, which I think is a benefit of the theory. But regardless, I don't think that gets the metaphysics right. It might be the case that you can have a language which describes points in terms of numbers. So the numbers are supposed to correspond to the object. I'm fine with that. But to treat geometric objects themselves as numbers, I do not think is about geometry. That would be a theory about numbers and a nice theory about the relationship between numbers doesn't tell you about the taking up of space. It'll get you close, but it won't get you perfectly precise. I just discovered a Greek philosopher that actually agrees with some of my criticisms of Euclidean geometry, and to my surprise actually has somewhat similar theory. It's a good old Epicurus. Apparently, the Epicurans disagreed with the foundational axioms of Euclidean geometry. They thought that space and geometry must fundamentally have units which take up space. You have to have fundamental objects that take up space in order to build a coherent theory of geometry. I agree with them. But if you do that, you immediately lose a lot of the conclusions of Euclidean geometry, including the Pythagorean theorem, including the supposed transcendentalness of pi, all kinds of ideas that you lose out on. That simply logically follow from treating points as objects which take up space. Now, like Epicurus, I also disagree with one of the premises of Euclidean geometry and Greek geometry in general, of the infinite divisibility of space. I don't think that makes any sense. If it's the case that there is any space at all, that is, composite space that has parts you can reference, it must be the case, I think for logical reasons, that you have fundamental base units. Now, this idea has gigantic implications in multiple areas of thought that I can't go into right here, but suffice to say that people don't realize that the claims of Euclidean geometry explicitly presuppose the infinite divisibility of space. For reasons that the Greek philosopher Zeno pointed out, although it kind of accidentally didn't intend to point out, I don't think that it makes sense to say that space is infinitely divisible, that between any two points is another point. I think that's logically and empirically false. Think again about the geometry of your computer screen. Your computer screen has the ability to create composite geometric objects which appear like they're perfect. Circles that you see, for example, I used to be a motion graphics designer and I would create beautiful looking circles in motion graphics, all of which were fundamentally reducible to pixels. And if you examine these pixels, you'll note that the pixels are side by side. It's pixel here, pixel here, pixel here, pixel here. And between two touching pixels, there's no middle pixel, it's not infinitely divisible. It's one side by side to another. And yet it allows you to construct the objects that we actually experience. So why would we think that we need infinite divisibility, which I think is incoherent fundamentally, when we have a great analog of non-infinite divisibility, namely pixels on a computer screen, which create objects that we experience. Put on a VR headset, go into the world of virtual reality and look around and tell me that it doesn't look just like three-dimensional space. VR headsets fundamentally have pixels that are reducible to finite units. Between the two pixels, there is no in-between pixel. Not only that, but the math that actually comes out of the theory of geometry in which there are base units is way more beautiful and it's perfectly precise. There are no irrational numbers. There are no incommesurable quantities. There are no transcendental numbers. It's all perfectly precise. Every unit of space has its own distinct, discrete location in space. And in fact, I think this correlates to physics really nicely too. But that's a video for another time. So again, at the very bottom of Euclidean geometry are these premises and ideas which we can coherently doubt. We don't have to just accept the idea that space must be infinitely divisible and points must have zero dimensions, just because that's the way that people have been thinking about it for a few thousands of years. Not to mention who knows how many thinkers throughout history didn't make it into the textbooks, didn't make it into the history books and disagreed. I disagree. Fortunately, I have the tools available with modern technology to get my ideas out there, but it's very possible that people who were atomists, geometric atomists, as they might be called, their indivisible geometric units, were simply excluded from intellectual conversation. If I remember correctly, there was a quote over the School of Athens that said, let no man ignorant of geometry enter. Well, you might listen to that and think, oh, wow, must be that Euclidean geometry is beautiful and perfect in an intellectual discipline. Or you might listen to that and say, oh, wow, that's very dogmatic. If I were to say, let no man ignorant of the Holy Bible enter, you might go, oh, that's a red flag. But yet because it's geometry, we think, no, well, we can't possibly coherently doubt any of the premises. I suggest maybe that's dogmatic. Don't take my word for it. Keep examining these ideas for yourself. And as we go along, I think hopefully a little bit of doubt will be seeded in your head as well.