Part 2: August 28, 2022: Natural Generalization, Natural Borders, Representation, Tree Democracy and Ground Up Category Theory
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.
