The value of abstraction

So I finished reading Anathem, by Neal Stephenson, which my sister recommended. If weaponized epistemology sounds like a mashup you’d be interested in exploring, grab a copy. Otherwise, just know that it is really about knowing what we know and quantum mechanics. Mix that up with two (differing) blogs I’ve stumbled over – the rationalist Less Wrong and the mystical Interesting Ideas That Miss the Point – put the whole thing in a blender set to purée, let it stew for a week, and out comes the central idea of blog post: chaos gives rise to order.

Our understanding has to start somewhere. There are axioms you must accept without question, whether in arithmetic or in philosophy. Interesting Ideas uses the concept of a belief-space and a mind-space. The former contains everything you accept and therefore do not think about consciously; the latter contains what follows from those base beliefs. Belief-spaces are internally, tautologically consistent. But you can work with multiple belief-spaces, which is essentially doublethink. The most common example is a scientific belief-space (science has beliefs, which we’ll get to momentarily) and a religious one. Ideas are evaluated within the context of their belief-space, which vary from religion to religion and from person to person. This is why two intelligent people can disagree, whether on religion or politics or economics – they evaluate the same idea in different belief-spaces.

The beliefs of Science (science and philosophy and reason and so on) validate the tools used in that enterprise. These beliefs are as simple as possible (think Occam’s razor). For example: we accept the conclusions of sound and valid logical arguments. But we have to believe that such acceptance is a legitimate (trustworthy) way of knowing, because using a logical argument to prove that we should accept logical arguments is circular reasoning. Similarly, we believe that empirical observations are also legitimate. These beliefs can never be fully justified, but they can be supported by seeing a great deal of internally-consistent, largely unique knowledge is found accepting them. That is, they are useful beliefs. Utility, though only a rule of thumb, is the best tool available for evaluating beliefs when constructing a belief-space.

Sometimes that construction is subconscious. Much of what we learn, or are told, from early childhood becomes an embedded part of our belief-space(s). Much of formal education is about consciously standardizing belief-spaces. This is the other benefit of the Science: its simplicity makes it easy for everyone to adopt a similar copy of the same belief-space, facilitating communication. This is both because simple beliefs are easier to teach, and the lack of unwarranted assumptions leaves little room to dispute to legitimacy of the belief-space. This communication is the third belief of Science: that the work of others, using logic, empiricism, and the work of still others using the same, is legitimate. The acceptance of the work of others is critical to any belief-space, because otherwise you’ll end up reinventing the wheel, rather than standing on the shoulders of giants.

We normally think of belief as chaotic, and thought as orderly. But we see that all ordered thought rests on unproven beliefs – but all human knowledge doesn’t come crashing down. We can agree on physics while disagreeing on metaphysics.

One of the recurring metaphysical discussions in Anathem surrounds the nature of Scientific laws, especially mathematical objects. The example used is an isosceles triangle. Formalism states that the triangle is just part of a game played with meaningless symbols, syntax, without intrinsic meaning. Platonic realism states that there must be some perfect triangle out there with semantic meaning, possibly even existing physically, and that all triangles we see are imperfect copies.

I reject that a perfect triangle exists somewhere in Platonic heaven, but I also reject that an isosceles triangle is an unimportant figment. But there is a third option.
Our models do not arise directly from the way the universe works (i.e. quarks), nor do they flow from a Platonic heaven. They exist in the collective consciousness of humanity. They exist in the belief-space of Science, as standardized and communicated by the education and scholarship systems of humanity.


We all carry around models of the universe in our heads. They need to be similar enough to someone else’s model to communicate. They also must accurately describe and predict the physical world. How do they do that?

Nature doesn’t have physical laws as we know them, like F=ma. They are merely abstractions and approximations that allows us to work with what is interesting on the macroscopic level, i.e. a map. The universe operates on the scale of quarks and electrons, the territory. I can’t run a quark-by-quark analysis of a 747, like Nature does, because quantum mechanics is strange. Its simple rules beget complex phenomenaI have to fall back on my aerodynamics approximations. And therefore, I can predict with good-enough precision how that 747 will behave, using a lot less math than Nature uses. I have a multi-leveled map to a mono-leveled territory, and I’m trained to use the right level to get the most relevant information. (Well, not me specifically, but a properly trained engineer. Working on that one.)

Let me take a moment to make a crucial distinction, because it is how layers of abstraction (the map) are implemented (in the territory). I’m going to define my terms, in this case, complicated and complex.

Complicated – consisting of components that are individually very sophisticated, and different from one another, and therefore difficult to understand
Complex – consisting of components that are individually very simple, and similar to one other, but difficult to understand because of the number of components and the connections between them

There’s a term in philosophy called the mind-body problem, which is how your simple brain can give rise to your complex, even complicated, consciousness. Answer: through abstraction. If anything, we should talk about the body-quark problem, which is how your simple elementary particles can give rise to your complex, even complicated, body.

If you try to go from atoms to Aristotle directly, it is indeed complicated. But breaking it down into steps reduces it to merely complex. That quark is is part of a proton which is part of an carbon atom which is part of a lipid molecule which is part of the cell membrane structure which is part of a neuron cell which is part of grey matter tissue  which is part of your brain, an organ, in your body. Repeat, mutatis mutandis, for every quark that’s part of you.

You can reverse the process and peel away models. This isn’t models all the way down; rather our models (maps) stop at quarks. That’s where the map meets the territory, and is therefore completely useless. Sometimes, we need to dig down to electrons to explain something. Usually we don’t.

So go the other way. It’s like a powers of ten exhibit, except we’re more concerned with the layers of knowledge we have, not the relative size of things. We’ve abstracted our way up to the brain, so is it so hard to say that the same process abstracts the brain into the mind? It’s the most complex system we know of, but the abstraction process is fundamentally the same.

Keep zooming out though, and something interesting happens. Around human-size, complexity peaks. Once you leave the planet, you’re back to almost nothingness. The empty space between stars bears an peculiar similarity to the empty space between atoms.

So far, I’ve spoken of our maps being drawn to the right scale (level of complexity/abstraction). (As complexity increases, so does abstraction, since we can only handle so much information at once.) However, evolution takes advantage of this peak in complexity – where things appear solid rather than empty space – and has designed our bodies to be adapted to the particular scale they are. Our eyes don’t see quarks. They don’t need to. So clearly there is something special about this level of the map, if only that we are most comfortable with it because our observations are of things people-sized. We’re hard wired to work in abstractions. That’s not a complicated mess of quarks – that’s a person.

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