 Part four of our discussion about geometry today. So this is going to be blending the past few weeks together I'm going to be talking about the purpose of geometry and the subject matter of geometry and how theories get constructed So simple way to think about it is this Do you build theories from the bottom up from? establishing the fundamentals and then seeing what logically follows from them to build more advanced structures of knowledge on the fundamentals or Do you go from the top down you have some theory that you like some conclusions that you like? And then if you're interested you can choose to try to found them on Fundamentals to try to find the fundamentals, but if the fundamentals are unclear, it's not a big deal Believe it or not this blew my mind when I discovered this just a few years ago but the approach of modern mathematicians for at least the last I'd say three centuries is Top-down it's not bottom-up Ironically enough Euclid had the bottom-up approach axiomatic deductive theory. I think that's the correct The correct way to go about building a theory. I just think you got the axioms wrong But the modern approach is to say look at what we can do with calculus look at what we can do with Geometry Euclidean geometry look how well it works Who cares about the underlying philosophy and fundamentals of it it works and that should be good enough I believe this is a completely wrong method for Constructing theories of any sort my favorite analogy is the Ptolemaic model of the solar system Now the Ptolemaic model of the solar system was where the earth was in the center of the solar system Everything revolved around the earth and if you look at a map of the Ptolemaic model It's beautiful got the earth in the center And you've got all of the heavenly bodies Orbiting around it and then on the orbits of those heavenly bodies You have these little little squiggles kind of reverse and then they go in their direction Those are called epicycles the reason for the epicycles is because the Predictions didn't quite work of the geocentric model And so they had to add little additions little epicycles here and there to try to get their predictions to fit the theory now I bring that up because the Ptolemaic model of the solar system worked and it worked Incredibly well in fact for a long period of time it worked better than a heliocentric model of the solar system a sun That the theory that everything revolves around the sun So those who were arguing methodologically speaking that the end conclusions are more important than the fundamentals are mistaken I would say in that circumstance. They're dogmatic and mistaken the person that sees little problems with that dominant Theory and can craft a new theory based on different completely different fundamentals. I think is a More careful thinker. I think that is the way that progress gets made in terms of our theoretical constructions now importing this back into geometry and mathematics Euclidean geometry works Extraordinarily well, what is the reason for that? Is it the case that because Euclidean geometry works it must be a true theory? No, that's not true as exemplified by the Ptolemaic model of the solar system predictions worked And yet it was completely fundamentally wrong So it's possible that Euclidean geometry works and calculus. Let's say works for the wrong reasons Not for the reasons that mathematicians and geometers think that they work my approach specifically to geometry is like this in Order to create a coherent Theory of the taking up of space, which is what I say geometry is about I am positing the existence of a three-dimensional unit if we're talking about physical space We're talking about two-day space a two-dimensional unit all of these The space that I'm describing can be reducible to those Fundamental units those base units that I call them. So in my theory space like this is Composite space. It's a bunch of little things put together This is like an atomic geometric way of thinking about how space works There's a bunch of base units that are indivisible you put them together and then you get extended space If you take that approach you get no rational numbers. You get no infinities zero infinities You get all of the predictive power of mathematics. You just understand things a little bit different So for example the Pythagorean theorem works if you have a huge amount of base units So what is a diet? What does it mean to say a diagonal line in terms of this theory? Well a diagonal line is a composite object made up a bunch of units And so you can approximate the truth about distance between two points by using the Pythagorean theorem a squared plus B squared equals C squared it's not going to be perfectly precise But that's because a perfectly precise triangle cannot be coherently constructed now I freely admit if my alternative theory didn't have the kind of Explanatory power of Euclidean geometry it would be a much bigger stretch to be proposing that this is a superior way of thinking about geometry But actually when you work through the logic of it It does give you the same predictive power and in fact it explains what you're looking at right now Which is a video on a computer screen that is a composite amount of space built up of a finite amount of pixels You think you see perfect curves you think you see circles you think you see squares on your computer screen But by the Euclidean standard you actually don't all you're seeing is a bunch of tiny little squares That are arranged in a particular way that gives your consciousness the illusion of some perfect smoothness of perfect continuity So my claim in approaching mathematics is as follows If your composite object Cannot be constructed out of a finite number of base units that object Does not and cannot exist because shapes are composite objects The Euclidean approach is to say we're gonna start with the concept of a perfect triangle And that perfect triangle what doesn't quite fit with the idea of being base unit therefore everything's infinitely divisible and it's made up of a finite number of points and lines, but the Points take up no space zero-dimensional lines take up one dimension And it's not exactly clear how the infinities work together and it's not exactly clear What irrational numbers mean unless we're saying that numbers exist separate of our mind But we like the idea of talking about perfect triangles and perfect circles So we're just gonna go with it two very fundamentally different approaches to thinking about the philosophy of mathematics and Unfortunately, there's not enough discussion about these ideas taking place people mathematicians in particular and intellectuals Put mathematics at on a pedestal They say there's no way that these mathematical ideas the Pythagorean theorem could possibly be wrong And therefore we're gonna use it as a metric to determine the intelligence of other people They disagree with this certainly true claim. They must not be intelligent and therefore we can't entertain their theories This I am claiming explicitly is Dogmatic it's no less dogmatic than a group of powerful theologians in the church getting together and saying the Litmus test for whether or not we're gonna treat your idea seriously is whether or not to accept the self-evident fact of the existence of God If you don't you're a heretic and we're not going to listen to you The same thing is going on in math mathematics and it ain't just geometry. It's shot throughout Especially modern mathematics arithmetic's probably fine But higher levels of math I think far too commonly are assumed being Immune from skepticism and so people don't even look if they were to look They would see what I've seen which is a bunch of dubious metaphysical claims that just aren't challenged