#math journal

Imagine a space where you can see the form of time from some standpoint, and imagine if you can only see if it you can tell it apart from a similar form that does not result in a second mirror world eating you up when you weren't noticing the details. This is how my continued foray into the following paper feels.

We begin with a quick clip by one of favorite mathematicians, Cohl Furey. At the time when I found out about her, 2018, I didn't have the spoons, the space, the time or the safety to further pursue my interest in what she taught. It is true that apolitical positions are privileged positions, and that some people are successfully perched on a safety that allows them to pretend like evading the boiling lava of politicization is a choice. That, or they are just completely blind to their own political positions and consequences, or simply in denial. Only now I have I finally ended the relationship that took me from several opportunities in analyticity. I revisit her work.

Her teaching is void of unnecessary distraction, organized, preplanned, clear, and distinctly moving forward with an insight for the student in mind. She shows high comprehensive skill therefore.

Still curious about division algebras, I found out that they are fields that are associative, aka, more or less coupled with their cancelling function with the exception of the Cayley numbers. Oddly, these are another name for the octonions, though apparently they were discovered independently.

In addition, Cayley numbers are non-associative. That means that changes in groupings have real effects, even if the contents are the same. It sounds like what I am learning about in complexity science and emergence. In addition to being non-associative, we also have to remember that as a division algebra the multiplication operation doesn't necessarily commute, meaning that direction of multiplication may have an effect as well. So, so far we know there are real effects on direction and group. When I hear that, I immediately think time--direction going to physics, group going to finance, and the link between the two going to complexity theory.

Moving on, we touch on involution--where I take to mean the shape of something coupled with its functional instantiation results in the identity. My hypotheses are that this identity can be either considered the (a) category or (b) some kind of pivotal location necessitated by the system constraints for this form to both exist and operate. It is dark, but I think of a gothic cathedral, and a gothic cathedral when it is on fire, operating, made of stuff. Genes, and genes when they are in the burning, mortal version of a human life.

Learning about involution and inversion, I couldn't help but to think how a shift in direction of an operation related to the inverting function. It seems to me inversion is just multiplicative commutation but on a different dimension. I often love to teach inversion to my kids using the fraction's bar as the mirror, and the fractions jump across the mirror world to find themselves in the identity function 1. Interestingly, the identity function is, as we see in the following video, also the first generator of a potential Cayley graph...this is me looping back to how Cayley category theory ties into graph theory.

This piece by Visualmath is amazing not only because it is visual, and most mathematics is best taught with the end in mind (the final picture; I always view the computational approach as an artist who sketches a lot of lines while drawing each individual piece of a picture, which is more than a little energetically expensive, and I view a mathematician as one who draws in a preplanned, certain way) but because I love that he laughs and says intimidating elements of math are actually really simple and easy to understand. I wish more math was taught this way. In fact, I wonder why it isn't?

Some things to note: I will be noting one after another the additions I have to make to my original reading to show why autodidactic, thorough mathematical endeavor has a serious complexity issue that I don't think the university system solves.

Initial inquiry http://www.csun.edu/~asethura/papers/DegreeDetVars.pdf

Resulting supporting inquiries while trying to solve the initial inquiry to be read as supplement while reading 1, added August 26, 2022 (https://inspirehep.net/files/1e02ad81af7a278f19b217e99715a973)

https://www.researchgate.net/publication/308943414_Commutativity_Theorems_in_Rings_with_Involution

Supplementary reading: Applications of category theory, to be looped into each entry as an extra challenge.

Other items to note; subvariety is a smaller item in a larger field. The question of codimension is how many dimensions are not shared...I find this an interesting use of the word co. So essentially, "what is the distance between that I am, and that which contains me, dimensionally speaking?" What's even more interesting is that, if I understand correctly, which I probably don't, the smallest possible difference between a subvariety and a field is the field's dimension, and the larger the subvariety becomes, the "greater" the codimension terminologically speaking, while it is actually numerically smaller. All I can think to explain this is that the term is referencing how much space they mutually compare. To me this seems like an inverse function. Anyone else irked by this?

Finally, I was also interested in the transposition here and how it relates to commutativity, associativity, and the real effect of direction and group--perhaps the group formed by the traces themselves.

Well, we're one small paragraph into a six page paper, and already in this deep. I've always been one for a steep learning curve though...I've been celebrating completing my financial analysis specialization! My favorite quote or something like it being finance is, in the end, the study of time. Forever the metaphysician.

As anyone who knows me may know, I have been working on reading Alain Badiou's Mathematics of the Transcendental for a few years now. And as you may not know, I am currently studying the evolution of networks and graph theory, specifically the Greatest Strongly Connected Component.

I am still baffled and relatively starved of a good resource to contextualize category theory, though I sense its power in the Badiou. I attempted to read a paper about applications of category theory, but there was still some context lacking. I am still very much a very OCD individual, and need the context for why things arise as much as I need the how of how they arise.

One of the common themes I see here is that references in category theory that denote an option most referred to by other categories would be an equivalent to the GSCC. The difference being the interrelations between the other terms have not yet been mapped, nor necessarily have to be in the strictly categorical system. Only when they become a living system, inter-referring, do they begin to be embedded on a graph.

As I continue with my exploration of the paper here, I found Michael Penn's video relatively helpful (we'll go into that later). I am now working through the fourth paragraph in this blog post. The proof states that the degree of the polynomial can be as large as necessary, so things can get very weird. It's easy to imagine x^3, for example, but it is not in any way easy to imagine x^20th. Nevertheless, the nature of the matter is such that these powers, regardless of how weird and wily they get, still have to finagle with some specific laws in math space. First of all, points that create shapes on algebraic sub-varieties, or sets/systems of solutions usually of the polynomial type (in fact, perhaps always of the polynomial type, though my understanding is not yet sufficient to understand the necessity there) must obviously have to be rational to exist. That means that any nature we derive theoretically must map actually somewhere on the intersections of all the component parts (see weird shape that resulted from such an operation above), and that even if we get it wrong, those points we observe do have a natural explanation. Yay. The creatures are not weirder than that, fortunately. Also, all fields operated on have to have an algebraic closure. More or less, this is the anti-flat earther statement of mathematics. Things have to fold in on each other, and there's a certain gravity to the possibilities supported by any given field. Finally, an irreducible subvariety means that it is within a given constraint, however, within that constraint, it cannot be further "gardened down" into seperate varieties. That's the math carrot, the math radish, or the math flower, whether you like it or not. It's not salad time over here, dude.

So far so good. We're getting there. Things I notice as I learn: I, again, don't understand why powers up to insanely complex magnitudes can be so reduced, other than reducing the whole function by a common denominator. But say you couldn't do it. It just strikes me as mind-boggling, but, it is mathematics.

Other things to note; there weren't any exceptionally strong videos on rational points. What didn't I like about the videos I found?

They don't explain the strategy behind the moves in a proof. They just show you how to do it. That doesn't relay a deeper understanding.

As usual, the why of solving the issue at hand isn't addressed either.

Both of these issues are problematic, as our mind needs relevancy and cause to learn best. Yes, we can show the mind how to do something, but if it doesn't know what for, as teachers we're going to get weak retention scores. And that's on us.

As I continue to attempt my autonomous maneuver of the following in my mathemagic space travels, the next paragraph down talks about strict bounds.

Strict bounds according to this are essentially precisely stated inequalities that show acceptability of a series of models that fit the constraints of a function.

This wasn't particularly interesting. What was interesting however, was towards the end (oh how the mathematicians like to test who's a ride or die). The shapes with the least spikes and lower data points generally have the tightest formulas for strict bounds. The more information that fits a given model however, the more difficult it is to create a strict bound. I must admit, the bit at the end was very curious to me. Essentially, the more information fits but the higher the variety, the more you should vouch for a supersolution (??). They tend to rely on second order variables as well as calculus based rate of changes to describe the change in models themselves.

Which one our garden of hypersurfaces fits is yet to be revealed...much less even what, really, a hypersurface is most applicable to...I guess that's on my outside venturing.

Also, I have never played the eternal cylinder, so I cannot vouch for contents, but this is a cool sci-fi title to suffice for a cylinder with a lot of signals going off in every direction while a radar wheel does a loopdy loop.

Yesterday, I spoke on the fact that higher level representation carries all of the properties of spatial reasoning–especially for abstract algebra–and that it therefore sees a drop in ability to follow because spatial reasoning is not a normative type of learning. It is nonverbal, and the nonverbal is viewed as “antisocial” as there is an erroneous claim that what is not verbal is not social. That’s a whole other debate. Needless to say, we establish that language is not antisocial or social, as we see antisocial and social uses of language all the time. The same applies to spatial reasoning. These are just means and methods of establishing connections between things, and one carries more social information (tones, pitch, body motion) but that doesn't inherently mean the results lead to more of that type of interaction. In fact, they can often signal that these interactions must be terminated. 

One species’ poison is another species’ treasure: These plants flock and frolick to all those terrible things you kept pent up inside in your journal. Ah yes, CO2. And finally somebody listens! Makes sense (plant cents) to these guys too. It’s just a win-win all around. 

Moving on from that, we ask the question–what then is this representation that causes the drop off? In mathematical notation, a great deal of information is compacted. The specification or analysis of these notations, as their authority is diffused into natural language, then becomes a hot-spot politically for the descriptions thus. For instance, as I read the resource I found on Representation Learning for Natural Language Processing by Liu, Lin, and Sun I found a phrase I consider to be better described as “organized” described thus; 

“distributed representation is able to represent data in a more compact and smoothing way.”

I thought that this instance was very funny for several reasons. First, it illustrated my beef with notation. Here we were in a book on representation in relation to natural language, and the function of organization had its authority diffused into the natural language description of “compact and smoothing”. Now, in this situation it may see somewhat comical. And of course, political. “Oh, I see what she’s saying.” “No I don’t. I think they meant something else.” In the end, it doesn’t matter. Why? Because the point’s already been made. We had it easy, as “organization” is relatively easy for all of us to understand as a function. Though we may have different mental constructions, it seems that most of us arrive at the same point about what is and isn’t organized. But, this isn’t the case with other mental notations. These functions are not something easily identifiable as the difference between your daughter’s room and your son’s room. In different cultures, there might be an understanding that one is more likely to be more organized than the other one. Again, it doesn’t matter. The politicization is stripped away for the general, trans-national function, organization. Clean and apolitical. Supposedly. We’ll get there later, don’t worry. 

The Organizing System Concept

So we start by saying that notation is something different than a word. Instead of “to run”, we have a name of a function, running. However, here, “running” is used descriptively, while notation supposedly contains everything required to instantiate, right then and there, running itself. We see this in our application of involution–that when I apply the symbol of the cathedral to the function of the cathedral, the human glory that coheres the two is the result. Or when I apply the function of complexity to its possible genetic set, a human being bafflingly arrives out of the womb. 

So not only is representation spatial, but it is the organization of functions, in a form that is both kinetic and potential depending on the circumstances–just the syntax of a mathematical sentence can decide if it has kinetic or potential power in the given sentence. So is that language as we know it? Or is it dancing? Or dancing as language? When I think of the forms described by the late Maryam Mirzakhani or the “shape of proofs” this is what I mean.

The final comedy is that what is often called “intuitive” is a way to create rigor–an unpacking of the implicit politicizations in a contained moving or non-moving function, in order to tease out the errors or accuracies in analysis, and send it back up to the sort as a specified function. In fact, that is most of the work of many researches…simply teasing out inefficiencies in the original description, releasing powers to the function that were previously unknown due to the carelessness of tongue. And that these things that are called “intuitions” actually stand greater chances when naturally described than those who may see a representation’s inherent organization and endearingly call it a “smooth and robust appearance”, indeed these big guys may be intuiting most of the language with which they speak.

As with all magics, one must speak in a careful tongue. Only then do the most encrypted of all creatures, the functions of nature, feel safe enough to speak back. These trees don’t have brains, but they are the products of moving and dancing with each other and the light, and doing what is needed to finance the endeavor. What looks like chaos for a leaf is another denser network (oh I’m sorry, human)’s tree. And maybe this is the vision that belongs to the greatest mathematicians among us, if their understandings are true, and their pursuit of closeness to the subject sincere. One doesn’t need money to see the forest for the trees. It will just lead you back here. A sharp tongue, a sharp spatial sense, and a little bit of magic.