Our neural networks often generate these little "concepts". (See motivating example in addenda)
The powerful thing here, is that we can write such concepts down, or otherwise name or remember it. And once we've written it or remembered it, we can do things with it that we normally can't do when it is but a fleeting pattern of signals in our neural network.
As a still-quite-abstract example, sometimes we retrieve a mathematical idea from a very "fuzzy" intuitive idea, or some "hunch" that we have. We just know that some mathematical description of it ought to exist, and eventually we're able to write it down and "capture it" to some sufficient degree. This is done in the exact same way a painter captures a feeling or concept in a painting.. in the sense that the image of the painting in our minds can be processed in a way that evokes the concept or emotion in our deeper mind. The simplest way to do it is by naming, which is basically the bundling together of several intuitive thoughts, feelings, experiences into one without any additional structure. Of course we usually do something more than naming, using other pre-established names and relations to build something with structure, and in effect "reducing" the thing-in-our-minds into just these other, already labelled, "known" things arranged in some combinatorial way.
Anyways, once we've captured a concept as described, we can apply other little concepts to it, create more structures. In this way we build and build. Essentially we are "hacking" our own minds, conceiving of new ways to think about the same things in different ways, about different things in the same ways. Further, we often "extend" our neural network by using external memory (e.g. paper and pencil) and external processing (e.g. computers, grad students).
This idea of forcing our brains to do different things… this "hacking" thing…. Seems profound. It's like us trying to transcend our own minds, become greater than it by telling it how to think. But by definition… we can never transcend our own minds, right?
..Or can we? We might consider that indeed, we're not just using our minds, but other things: paper, pencil, computers, and… time.
The summative mathematical results of our mathematical work across all memory, external or otherwise, and time… may indeed surpass anything any mind by itself would have done, ever.
I mean, that's a nice fantasy. But what do you call math that you do entirely in your head, and learn via experience over time? What if, say, a man thinks up of an addition system all on his own, just over time?
Well… that man still made an effort to remember stuff in his head, effectively "writing". The point is that he still used your executive function to make sure his brain "burned in" certain things, certain "gymnastics". That's when you're doing math -- you're becoming aware, conscious, and basically manipulating your unconscious mind to make it do what you want. It often involves taking bits and pieces of the thoughts from the unconscious brain, naming them and storing the labels into the conscious brain, then using them to take control of what the unconscious brain computes for you. Usually, the unconscious brain is purely functional -- like an artificial neural network, it arranges itself in the optimal way in response to reinforcing stimuli. But when one does mathematics, one might say it becomes more than just a means to an end, but a means in itself. This idea, in fact, reflects in mathematics itself at several levels: On the actual doing of mathematics level, there's often theory that just is. It doesn't really "operate" on anything, and it's more like am abstract painting than anything else, just sort of describing this object. It's sort of the part of math where you build definitions, write axioms. And then there's mathematics that "says stuff". Theorems, lemmas, applications. And on the outside-of-math level, there's mathematics that's really "pure" and doesn't seem to be inspired by or say much about much of anything. And then there's mathematics that really came to be as an abstraction of real things and experiences.
You might say that the conscious brain is not as powerful, but with the help of the unconscious brain and by making it "its bitch", it can do some pretty impressive stuff, something the unconscious brain by itself wouldn't have really bothered to figure out on its own.
Of course, this is sometimes less true than other times. It's like native people who speak a language vs a foreigner who learned it "by the book". As foreigners, the native people, it might appear, "know" the grammar rules! But anyone who is even slightly introspective about how they speak their native tongue knows that that's just not how it works. We will often be surprised that certain rules even exist, despite the fact that we magically seem to always follow them (e.g. order of adjectives in English).
And indeed, there are structures in our deep unconscious minds that make it so (hence why we don't observe the rules consciously), although they may not actually be represented by literal graph-theoretic components strung together in some logical structure that the grammar books seem to imply.
In fact brain research hints at the contrary (brain surgery experiment), which is that logically modular "functions" seem to be more topologically distributed rather than actually topologically compartmentalized in the brain. And indeed, this may just be a better, faster way to do things, and the componentization is just a side effect of the deep structure of how the function really works.
So do we say that the native speaks "does grammar"? Not really, not in the same way that the foreigner "does grammar".
And it's true, it's quite beautiful in a way how native speakers can speak their language so well and so effortlessly. The foreigners might be slower, limited. But oftentimes we will be surprised at how eloquent the truly well-studied foreigners are. There's a running joke on social media sites about how people who say apologize saying "sorry, English is not my native tongue" often speak English better than half the native English speakers on the site!
And there is an advantage to knowing conscious, compartmentalized, formal rules of grammar. And that's provability, communicability, and extensibility. That is, with provability, we can extend, scale. And we can communicate and share proofs with other minds, because the elementary objects of the proofs are, purportedly, basic concepts we can all work with and understand.
With only our pure, unconscious neural networks, we can only do so much. It's true that we often overthink, theorize, and generally forget to get out of our heads and just simply do the task, but at the same time we're not going to be launching spaceships based on the advice of some seasoned veteran who "knows his rockets real well". Because first of all, such a veteran isn't going to exist…. I mean, how much "trial and error" are you gonna have when it comes to building rockets (which is what you need for the training phase of the unconscious mind)
I mean, raw experience and ingrained skill are great for most human tasks… maybe even all human endeavors and jobs/skills made up before some time period (hunting, farming, building). But even the slightly more complex stuff require some theory. And of course, once you're talking theory, you're talking about the conscious mind. Executive function. Rules. Logic. Legos.
And the two minds used together in this way, in what we call "mathematics", is truly powerful. And that's what mathematics, at its core, is really all about. Using your head real smart.
But I doubt mathematics is the only way to use your head real smart. It sure is an interesting way to do it, and it definitely seems to be the most executive-dominant method of the bunch.
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