#metamath

My last post was just me basically listing everything I need to do for myself this summer lol! BUT, with that, I also really wanted to talk about my thesis! While I don't really know what I want to do yet, I'm narrowing in a good way and I am getting REALLY excited to do this type of work! My initial topic was about interdisciplinarity and how we navigate contact points between scientists of different fields but it's slowly changed completely in ways that I'm getting ecstatic about writing about! I made that post about "Gödel, Escher, Bach" and I have been making my way further through that. I got to Hofstadter's first conversation on Isomorphic systems and I had just a complete moment of "clarity" (though, we'll see lol!). Me and my advisor had talked a lot last meeting about deconstruction and this idea that scientists use a lot of words that don't inherently mean anything (like, what does it mean to do something "urgently"?). So she wanted me to study God Terms and I myself found the term "floating signifiers" which both led me down some really great trains of thought. When I got to isomorphism with Hofstadter though I had this really great thought that I could almost use isomorphic systems as a framework of sorts to look at how different scientific organizations or disciplines discuss climate change. Obviously, if the science is accurate and up to date, the arguments that are built by independent scientists should be essentially 1:1 with the same information but with different signifiers and God Terms to better achieve their goal rhetorically. With that, if there is this perfect matching of concepts or data sets between two systems then I can map some sort of translation between their terms and then use God/Devil Terms and floating signifiers to explain why their systems appear different yet use the same data sets and information. I don't have all the kinks worked out and I really don't even know if something like this will work (especially as a thesis compared to like a dissertation), BUT conceptually I am SO fascinated about how I can iron this out and make these concepts work! ESPECIALLY since I'm interested in interdisciplinarity, I'm ECSTATIC that I might be able to use a metamath theory as a framework for rhetorical analysis!

So in honor of pi day (it’s always pi day somewhere probably), regarding Archimedes’ famous upper bound of pi of twenty-two sevenths and less famous lower bound of two hundred and twenty-three seventy-firsts.  This came from his use of two enneacontahexagons, one on the outside, one on the inside.  However, what if you don’t have access to geometry, what if you only know pi as the lowest strictly positive zero of the sine function (yes I have been on a certain website)?  Well, then, you could approximate Archimedes method by showing that that upper bound is less than ninety-six times the tangent of a ninety-sixth pi, and that that lower bound is greater than ninety-six times the sine of a ninety-sixth pi.  But how do you establish that these are on opposite sides of pi?

Well, first, how does Archimedes?  How does Archimedes even assert that a circle has an arclength?  Well, how would we define it now?  By the integral of the lengths between nearby points, essentially, treating this as a Riemann mesh.  That’s pretty much what the inner polygons represent, but also they can be seen as an appropriate lower bound by the notion that a straight line is the shortest between two points, which is as good a first principle as any.  It can then be shown that the outer polygons are a least upper bound since they get arbitrarily close and never less.  Consistency’s not shown, but it never can be.

This raises the question of what exactly the Euclidean distance is, since even this notion of arclength depends on it.  Of course, normally, the Pythagorean theorem is assumed as naturally as breathing.  Those of a certain persuasion will tell you the Pythagoreans “stole it from brown people,” meaning that Egyptians and Babylonians had the use of it, although no extant proof.  They can’t have cut it from whole cloth, though, so let’s ask ourselves - what exactly is the Pythagorean theorem?

Well, let’s look at what’s sometimes called the “Pythagorean proof,” one more visually obvious than Euclid’s - four equal right triangles, arranged first into two rectangles that leave two squares, then into a frame of a square on the hypotenuse.  Note that both of these form a square of side equal to the sum of the two legs, which can be broken down by algebra.  So in essence the proof is the proof of the existence of triangles; in the Euclidean distance, including in arclength, is hidden the abstract notion of the triangle.

But returning to Archimedes how to show from this alternate notion the boundaries on this zero of the sine?  What’s to be shown is that, just above zero, the sine is always less and the tangent always greater.  That the sine is always less can be seen from the series representation, at least for a sufficiently small parameter.  That the tangent will be greater can be seen by multiplying the series expansion of the cosine by the parameter and subtracting to see that the sine will be greater near zero.