The Deeper Paradigm

I went down a large number rabbit hole a couple months ago — actually I’ve gone down that hole a few times — and came across a piece by Scott Aaronson discussing them. He argues:

Even within mathematics, big numbers are often considered trivialities, their study an idle amusement with no broader implications. I want to argue a contrary view: that understanding big numbers is a key to understanding the world…

Who can name the bigger number? Whoever has the deeper paradigm.

I’ve thought about this concept of a deeper paradigm often since reading the piece. As I understand it, a shallow paradigm might be addition. If we want to generate more of something, we can keep adding to it — 3 + 3 + 3 = 9. A slightly deeper paradigm would be multiplication — 3 * 3 * 3 = 27. Multiplication is just repeated addition, now we’re printing three threes at a time rather than individually. It’s the difference between making things by hand one at a time, and having a machine churn them out en masse.

But multiplication is linear growth. To model something like a viral video or population growth, you need exponentiation — 3 ^ 3 ^ 3. That’s either 3 ^ 9 = 19,683, or 3 ^ 27 = 7.63 trillion, give or take, depending on your order of operations, something I discovered is far from settled. Instead of printing threes three at a time, you’re now increasing your ability to print three fold with each iteration. Exponentiation is repeated multiplication. It’s like making a machine that makes machines, one that churns out the three-at-time printers three at a time, and then prints three of those machines at a time and so on. But the exponentiation paradigm, while bigger than multiplication, is not remotely the biggest.

The next step is tetration, symbolized by arrows — 3↑↑3. While 3 ^ 3 = 27, 3↑↑3 is 3 ^ 27 = 7.63 trillion. After that is pentation, symbolized by three arrows — 3↑↑↑3 — which means 3 ^ 3 ^ 3… 7.6 trillion times, i.e., a power-tower of threes that would reach to the sun if I we stacked them upwards from this page. You can imagine what’s next 3↑↑↑↑3, which I can’t properly describe, but Tim Urban does a pretty good job of it here. With each arrow, the size of the paradigm grows. (Of course, we can put in any number of arrows, and you could as easily start with 9 (or any other number) instead of three. But this paradigm is not remotely the biggest, and you can dive into this much further in the links here.

The point is there are ever bigger paradigms describing increasingly mind-blowing number growth, and while we know the shallowest ones (addition, multiplication) must be deepened to describe many physical or biological phenomena which are exponential, the uses of the even deeper paradigms (tetration/pentation) are unclear — at least to me. But just as someone who doesn’t understand exponential growth would lack the tools to comprehend the spread of a virus, perhaps there are processes we don’t understand because they are tetrational or pentational.

The idea for this post occurred to me because I started (maybe 60 pages in) reading The Beginning of Infinity by David Deutsch. In it he distinguishes the process of biological evolution, i.e., natural selection over time, transmitted through changes in DNA, from cultural evolution wherein knowledge is passed down generationally outside the genome, something that’s faster than natural-selection-based evolution. The next step, though — the one that ushered in the scientific enlightenment after which technological progress not seen for millennia takes place during individual human lifespans — is seeking good (scientific) explanations for phenomena and passing not only the knowledge down, but also the tradition of conjecture and criticism. The good explanation-seeking species grows knowledge and technology at a much faster rate than animals (natural selection) or pre-scientific humans (cultural knowledge, rules of thumb, e.g., how to build a fire without necessarily understanding why rubbing two sticks together works.)

So if we liken natural-selection-based evolution to addition, cultural hand-me-downs as multiplication, scientific knowledge is like exponentiation. But in this model, what might tetration or pentation be? What is the bigger paradigm, if one exists, than our current practice of science? It might be AI, wherein not only are we seeking good explanations for phenomena, but building machines that seek good explanations. It would be like The Matrix where Keenu Reeves downloads martial arts skills or an entire language to his brain, but after instantly downloading all the scientific knowledge, we would then have the creative capacity of someone with all of that knowledge, spurring an explosion in scientific understanding, which would then itself be absorbed and spur an even bigger explosion, and this process would iterate ever more rapidly, like tetration, pentation, etc.

I suppose this is what people have described as The Singularity — which is also, not coincidentally, the same term used for the state of the universe prior to its birth via the Big Bang, and also the infinitely dense center of black holes which break spacetime. Maybe that’s where this all leads eventually — a pulling back of reality’s curtain to reveal something more fundamental wherein the laws of nature, on which even the deepest possible paradigms are based, themselves are shown to be provisional rather than axiomatic. Whatever mastery we achieve over energy and matter, time and space, in that case, would be tangential to understanding the deepest truth, the way in which no large number generating function can ever touch infinity.