 The most influential mathematical theory of all time is Euclidean geometry. Most people, when they think about geometric objects like squares or cubes and spheres and cylinders, they think in terms of Euclidean geometry. And in fact for thousands of years, learning mathematics began with, and was sometimes synonymous with, learning Euclidean geometry. So most people are content with accepting the premises of Euclidean geometry. They think that if the best minds for millennia have been examining this theory, well, the theory has got to be correct. Who am I to think that maybe the theory is wrong? And so they just accept that the fundamentals have been sorted out and work on higher levels of mathematics from there. Unfortunately, I think this is the way that dogma gets perpetuated. We can think of lots of other ideas which have been around for a few thousands of years that actually are incorrect or even fundamentally incorrect. So, though it might seem like at first glance, skeptical inquiry into Euclidean geometry is crazy, I think as will be revealed in a fairly short period of time, there is room for skepticism here. And maybe some of the principles of Euclidean geometry simply aren't true. So the purpose of this video and the next few videos on this topic is going to be to examine with a skeptical eye Euclidean geometry. The purpose is not to tell you about orthodox opinion so that you might repeat it and appear intelligent amongst civilized company. The purpose is to try to grasp the concepts of geometry ourselves. And if some ideas don't make sense to us, then we have to investigate until they do make sense, even if that means coming up with new theories in the process. So the first question that we need to examine is a very big picture. What is geometry? What is geometry about? What is it trying to describe? Geometers, I think, would generally agree that geometry is about the taking up of space, the nature of shape, position, maybe distance, length. And I think that geometry can actually be completely reduced to, fundamentally, the taking up of space. And then the objects which take up space, there are various properties. You get things like shape when you have objects which take up space and so on. Now if that's true, if that's what geometry is about, then it immediately presents us with a really big problem with Euclidean geometry for the following reason. Euclidean geometry is a theory that's based on an axiomatic deductive framework, meaning you start with premises, you start with self-evident premises in this case, or very intuitively appealing premises, and then you use logical deduction to see what follows from those premises. So the geometric objects in Euclidean geometry are defined by the axioms, and you get more and more advanced objects, all of which can be reduced to the fundamental objects defined by the axioms. So if it's the case that there's a problem with an axiom or a definition of an object, then it's the case that the conclusions which follow from that axiom contain errors and might be fundamentally completely wrong. And go figure, I think it's the case that in the defining of geometric objects, Euclidean geometry makes a rather large mistake. First object is that of the point, arguably the most important object in all of Euclidean geometry. What is a point? Well the English translation, modern English translation for what Euclid said a point was, is that which has no part. I really like this definition, but I like the modern English translation of it. What Euclid probably actually meant, which is coherent with his theory, would be something more analogous to that which has no dimension, has no length, has no width, has no depth, that I really don't like. But from the point we also get the next geometric object which is the line, which is a one dimensional object, that which only has breadth or length or what have you. Along that line you have an infinite number of points. So immediately we run into kind of a big problem with these definitions, or at the very least it's unclear. What exactly is a point? What is a geometric something which takes up no space and has no dimensions? Have you ever experienced or encountered a point? Something which has some spatial thing which has no length, breadth or depth. Have you ever encountered a line which only has length and no breadth? Have you ever encountered a circle? Something I'll talk about more, I would say no. According to the mathematicians nobody's encountered a circle, it's got really remarkable properties which I'll talk about in a minute. So the entire structure of Euclidean geometry is built on the axiomatic notion of the point in the line and the objects that are constructed by various assortments of points and lines. But it's not exactly clear what those are. This problem runs very deep and when you talk to mathematicians about it they give you some remarkable stories. They'll say things like, well, yeah, nobody's ever actually experienced points or lines or shapes or cubes or spheres. These are ideal platonic objects. They exist in another dimension, not the one that we directly experience. We don't see them, we don't visualize them, and we don't encounter them in the physical world. Well, is that an area that we can be skeptical? Can we say, hey, it's a nice theory you have, but I'm interested in a geometric theory about objects that take up space because, believe it or not, if you construct a geometric theory in which objects take up space, points take up space, things are composed of points which take up space, then you do not get the conclusions of Euclidean geometry. The entire theory falls apart. Take, for example, the geometry of the computer screen you're looking at right now. You might think that you've seen lines and points and circles on your computer screen, but according to the Euclidean geometry, no, in fact, you haven't. All of those objects and things that you see and experience are mere imperfect approximations of their ideal lines and circles and squares. Take a humble circle. It might be the most perfect circle you've ever seen in your life. This, they say, the perfect circle that you see is not actually a circle. In fact, this circle that you think you're talking about has a rational pie. It has a rational ratio. It's circumference to its diameter. It's made up of a finite number of pixels which take up space, and so it is imperfect. Again, are we allowed to be a little skeptical of this? Are we allowed to develop a theory of geometry which explains the phenomena that we experience, namely things that we see, things that we visualize, things that we can clearly conceptualize? I think so. One more problem in this video that's very related to this notion of zero dimensionality is the question, how can it be that zero-dimensional objects can compose objects which have dimensions? For example, along the line, there lay an infinite number of zero-dimensional points. How is that possible? What does that mean? It seems like it's saying if you add up a bunch of zeros, an infinite number of zeros, you will get a one, which of course doesn't make any sense. It doesn't matter how many zeros you add together, you're never going to get a one. I'd say it seems intuitive to say it doesn't matter how many zero-dimensional objects you have, you're never going to get an object that takes up space. Now, again, the remarkable thing is a lot of the mathematicians will agree and they'll say, aha, that just demonstrates that Euclidean geometry is not about all this regular stuff that we experience, no, it's a higher level of abstraction. Well, maybe that's true, but that sure presupposes a whole bunch of metaphysics about how we have access to this knowledge, whether or not this mathematical theory applies to the world, none of which is laid out in Euclid's axioms. So right off the bat, with definitions of Euclidean geometry, it gives us plenty of room for skepticism and I would say alternative theories of geometry, which we'll cover more in the future. I will say in all of my investigations in lots of different areas of thought and lots of different topics, I find mathematics to be really remarkable. The history of mathematics, the logic of mathematics, and the argument of mathematicians. Never have I encountered so many people who are so quick to say, ah, math doesn't apply to reality, aka, my theory doesn't apply to reality, as if it's a defense for the theory. They say, oh, well, you've never encountered any of the objects. You can't even clearly conceive of any of the objects in my theory. In any other area of thought, that would be synonymous with saying, oh, you got me, my theory is flawed. But no, not with the mathematicians. Much more to be said on that topic.