 Continuing our conversation about geometry this week. So the question was asked what is geometry about? The central question. There's another related question which has equally important implications and it's what is the purpose of creating a geometric theory? And how you answer that question is going to determine what types of geometric theories you're satisfied with. Is the purpose of a geometric theory to explain the phenomena we experience? Is the purpose to describe the physical world or the world we think we inhabit? Is the purpose to describe another world, let's say the platonic perfect world where you have perfect objects? Or is the purpose of a geometric theory pure practicality? We're coming up with a system of ideas and methods that we can use if we're let's say carpenters and we want to use angles and lines and circles and shapes to make buildings and carve wood a particular way regardless of whether or not our theory is perfectly precise. So there are lots of different ways to answer this question. The Platonist would say well mathematics tells you certainly true things, geometry tells you certainly true things about the platonic universe. The Euclidean objects are out there, they're not in the physical world, they're not even really in the mental world because they can exist independent of our minds. They are in the platonic world of forms in which there are abstract objects that exist separate of us thinking about them. That's one approach. The opposite of that would be something like saying that geometry is just about pure practicality and whether or not it's perfectly logically precise is irrelevant. Concrete example. Take the Pythagorean theorem. Probably the most proven theory in all of mathematics ever. So most people think you can't doubt the Pythagorean theorem. Well Pythagorean theorem applies, presupposes a certain type of metaphysics is out a little straight. So in the Pythagorean theorem everybody is familiar A squared plus B squared equals C squared. Well in that formula it's supposed to tell you about the length of a hypotenuse of a triangle. You necessarily get square roots. Now in the dominant conception of how square roots work for the last few millennia is that square roots will yield irrational numbers or numbers with non-terminating decimal expansion. What does that mean practically speaking? That means if you're an engineer working with the hypotenuse of a triangle you are left with a non-terminating series of decimals. You cannot make a carve a piece of wood. You cannot create some machine. You cannot create a hypotenuse that includes an infinite number of decimals. You have to approximate. So is that okay? Is it okay that with square roots and irrational numbers we are actually when we use the formulas we are approximating? Does that mean that the theory is wrong? Does that mean that the world is just too imprecise for the perfect Euclidean space? There are a couple kind of standard answers to that question for the few people that actually have the guts to ask. One of the standard answers is to say no Euclidean space is perfect itself. It is into this other realm. Our physical world, our approximation is a distant shadow. This is a very platonic way of thinking. So we live in kind of a crude and imprecise world but it doesn't affect the perfection of that theory. The other approach is to say well it's good enough. Infinite decimal expansion means you can make the calculations as precise as you like and that should be good enough. Let me give you my own heretical approach. It's even more fundamental really than even Euclidean geometry which is if you are dealing with a supposed irrational number it is a demonstration that there is a flaw in your theory. And here's the reason why. I don't believe there is such a thing as actually existing infinite decimal expansion because I do not believe that numbers exist separate of our conceiving of them. What infinite decimal expansion presupposes is that numbers, the decimal expansion of a number somehow continues off in the distance after the dot dot dot without us thinking about them. For example, if you were to try to represent the quantity of one third in terms of decimal expansion as 0.333333 and so on and that's supposed to mean and there's a never ending amount of threes over there. I don't think numbers work that way. I think numbers have a metaphysical existence in our head. So to say 0.3333 dot dot dot and so on even when we're not thinking about it is nonsense. I would say it's a demonstration that the language of decimal expansion is not perfectly precise. You can represent the quantity of one third with a fraction you cannot with decimal expansion. That's the problem with decimal expansion. So how it applies to Euclidean geometry is if it's the case that you've come up with some hypotenuse and a triangle the only way you can try to grasp the length of the hypotenuse is by invoking irrational numbers, well it's not precise. It's demonstrably not precise. You have to approximate. In my theory of mathematics, I'm not okay with approximation. If we're talking about useful mathematics for engineers, approximation is fine. But I think there's a world of mathematics which is perfectly exactly logically certain and precise. We can build the geometric theory from it and it doesn't exist separate of our mind. Now in doing so, you don't get the conclusions of Euclidean geometry but I would say that's okay. And it's always remarkable to me to talk to mathematicians and here I would say just the incredibly aggressive dogma that they think there's literally no other way of conceiving of geometry or conceiving of mathematics in general to say that you could create a geometric theory that doesn't include irrational numbers is like saying to a theologian, God doesn't exist. It's just totally inconceivable and impossible. But in fact, when you play out the implications of a geometric theory built on base units which take up space actually it explains all the phenomena that we experience without irrational numbers without approximation whatsoever. So it's an ambitious project, I recognize that but I think it's completely doable and the first step on creating that theory is first recognizing some of the logical, metaphysical and philosophical problems with the standard approach to geometry and mathematics in general.