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I’ve been obsessed with large numbers for a few years now, trying in vain to get others to care. But the “why” eluded me. I made one partially satisfying attempt to explain last year. Yes, he who has “the deepest paradigm can name the biggest number,” but to what end?
Now I think I’ve found the end, the reason I can’t quit this line of thinking, even if it’s driven me half mad: that the number TREE(3) is quite possibly a miracle. Despite arising from a simple game, TREE(3) can be hard to understand, so maybe it’s best to start with one of its “competitors”, Graham’s Number, to grasp what it is not.
Now Graham’s Number is only a competitor because the two are often compared, not because there is really any kind of competition. TREE(3) dwarfs Graham’s Number the way the breadth of the observable universe dwarfs the dimensions of an ant. (Actually, as you will see, that comparison vastly understates the disparity between the two numbers.)
But Graham’s Number, unfathomably vast in its own right, is both easier to understand and is generated via a different process. To get to Graham’s Number, we have to start with the most basic math that exists: counting.
I’ll excerpt from my post on growth that covers this:
First you have counting 1, 2, 3, 4…
If you want to speed up counting, you can add.
_Instead of counting from three to six, you can just add 3 + 3. Addition therefore is _ just repeated (iterated) counting.
But instead of adding 3 + 3 + 3 + 3 + 3 + 3, you can just do 3 * 6 because multiplication is iterated addition.
But instead of multiplying 3 * 3 * 3 * 3, you can just do 3 ^ 4 because exponentiation is iterated multiplication.
That’s where most people leave off in their education, and they feel perfectly content to live their lives only because they don’t know what they’re missing. Beyond exponentiation lies tetration, or iterated exponentiation.
Instead of 3 ^ 3 ^ 3 ^ 3, you can just do 3 ↑↑ 4 (a power-tower of threes, four high.)
You would say it “three to the three to the three to the three” in exponentiation terms, or “three arrow arrow three” in tetration terms. The number before the arrows determines the base and the number after them how high the tower goes. Let’s calculate some easy ones.
2 ↑↑ 2 is a power tower of twos, two high. That is 2 ^ 2 = 4.
3 ↑↑ 2 is a power tower of threes two high. That is 3 ^ 3 = 27. Easy.
2 ↑↑ 3 is a power tower of twos three high. That is 2 ^ 2 ^ 2 which is 2 ^ 4 = 16.
_3 ↑↑ 3 is a power tower of threes three high. That is 3 ^ 3 ^ 3 which is 3 ^ 27 = 7,625,597,484,987.
_Wait, what happened? You just encountered a fast-growing function.
Ok, tetration is cute, but if you want to generate Graham’s number, you’ll need to add more up arrows. Iterated tetration, symbolized by three up-arrows, is called pentation and would look like this: 3 ↑↑↑ 3. If 3 ↑↑ 3 is 7.6 trillion, what would 3 ↑↑↑ 3 be?
Well, it’s just iterated tetration, which means it’s a series of double-arrow operations with a base of three, three long, i.e., 3 ↑↑ 3 ↑↑ 3.
And since we know the second half, (3 ↑↑ 3) = 7.6 trillion, we can simplify it to 3 ↑↑ 7.6 trillion. What does that mean? It means a power tower of threes, 7.6 trillion high.
Okay, that sounds big. How big?
Consider a power tower of threes five high, i.e., 3^3^3^3^3 or 3 ↑↑ 5, is bigger than a googolplex.
To get the scale of a googolplex (one with a googol zeroes), consider you could not fit the zeroes it would take to write it out in the universe, even if you put one trillion zeroes on every atom. Again, we are not talking about the number itself, merely the number of digits required to write it out.
Consider a number with 200 digits is so massive, it’s far more than the number of Planck volumes (smallest known unit of measure) in the universe, but it’s trivial to write out. But you do not have space to write out a googolplex even while using a trillion digits per atom, let alone what those digits, if you could even write them, represent.
Your odds of entering every lottery on earth for the rest of your life, from the local bake sale to the mega millions, and winning all of them are far, far, far greater than 1 in a googolplex.
Your odds of guessing all the private bitcoin keys on earth without making an error are greater than one in a googolplex. A googolplex is an unfathomably large number. And yet it is smaller than 3 ↑↑ 5, or 3^3^3^3^3.
But 3 ↑↑↑ 3 is a tower of threes not five high, but 7.6 trillion high! When you get even to 10 high, you’ve exceeded a googolplex to the googolplexth power. The human mind cannot fathom the number you arrive at even at 100 or 1000 high, but we have to get to 7.6 trillion.
Okay, now that we’ve multiplied out the entire power tower to 7.6 trillion, guess what, we have to add another arrow. Not 3 ↑↑↑ 3 but 3 ↑↑↑↑ 3.
That’s hexation which is iterated pentation, in this case with a base of three and three terms, i.e., 3 ↑↑↑ 3 ↑↑↑ 3. We already know the second half is, whatever the incomprehensible result of the multiplied-out 7.6 trillion-high power tower was, call it X. So it’s 3 ↑↑↑ X.
And that means iterated tetration with a base of three, X times, i.e., _3 ↑↑ 3 ↑↑3 ↑↑ 3… X times.
To solve this, we go term by term. The first one is 7.6 trillion, which feeds into the second, the multiplied-out power tower 7.6 trillion high, i.e. X, the third is a power tower of threes, X high, multiplied out, and so on, and there are X of these entire towers, each one unfathomably, astronomically taller than the last.
Once we get through all X (remember itself an unfathomably large number) of the terms we’re at 3↑↑↑↑3.
That number is G1.
To get to G2, we just take 3 ↑↑↑↑↑↑↑↑↑↑…G1 arrows… 3.
Wait, what?
Remember each individual move up the scale from counting to addition to multiplication to exponentiation turbo-charged the growth of the function, and now in this function, they’re telling us to add G1 (3↑↑↑↑3) moves up the scale all at once!
Put differently, from counting by ones to the insanity of hexation, there are only six steps. To get G2, there are 3↑↑↑↑3 steps!
To get G3, it’s 3 G2 arrows 3. To get to G4, it’s 3 G3 arrows 3.
And so on until we hit G64 which is Graham’s Number.
It’s an indescribably massive number, not relatable to anything in the universe, not even in terms of the possible ways the atoms could be arranged taken to the power of the number of ways history’s chess games could have been played.
There is no way to visualize or imagine it except by walking vaguely through the steps to get there and straining your brain to grasp the process.
But as I said, Graham’s Number is trivial, basically zero compared to TREE(3), and that is so much the case that if instead of going to G64 via the steps, you went to G(Googolplex), or even G(Graham’s Number), i.e., G(G64), you would still be at zero relative to TREE(3).
But here’s where it gets fascinating. While Graham’s Number is generated via ramping up increasingly powerful iterative operations (as we did in the beginning) TREE(3) comes from a simple game.
There is a good article in Popular Mechanics that lays it out, building off this excellent Numberphile video with Tony Padilla:
You can click on the article and video for the specific (and relatively basic rules), but essentially, the TREE function has to do with “seeds” (dots) and “trees” (combinations of dots and lines), such that you make the maximum amount of unique “trees” (dot-line combos) per the types of seeds available.
If you have only one kind of seed, say a green one, there is only one unique tree that can be made.
So:
TREE(1) = 1.
If you have two seeds, say a green and a red, there are three different kinds of unique trees you could make.
TREE(2) = 3.
If you have three seeds, say a green, a red and a black, there are TREE(3) different kinds of trees you could make. As it turns out, that number (which is not infinite) is so much bigger than Graham’s number the two are not even in the same universe:
Here’s Padilla comparing TREE(3) to Graham’s Number if you want to see the difference:
Okay, so what does all this mean? It means that Graham’s Number, which is generated by successively more powerful iterations of mathematical operations, cannot compete with TREE(3) which comes from a game with simple rules.
Graham’s Number is built the way a machine would do it, the way an AI would go about making a huge number — mechanically increasing the rate of construction.
Consider if you had a machine that made products one at a time, that’s like counting. And if you had a machine that made products three at a time, that’s like adding.
And a machine that made machines that made products three at a time, that’s like multiplication. And a machine that made those three at a time would be exponentiation, etc., etc.
Each successive machine-making machine would take you into a deeper paradigm of growth. But you can see this is a mechanical process, no matter how deep you go.
By contrast, the tree series is what happens when you design a game with simple rules and let it play out. The growth (once you realize it to the extent the human brain can even grasp it) is not just faster than the mechanistic model, it’s on another plane.
The takeaway then is bottom-up complex systems (games) with a few simple rules can spawn a paradigm so much bigger than top-down mechanistic growth models.
The human brain (neocortex in McKenna’s terms) is just such a system, and yet we train ourselves to be like machines! Practice, routine, iteration, follow-these-10-steps to happiness, learn these five keys to investing, etc. Yes, you can get somewhere with these recipes, but nowhere near the destination of which you are inherently capable.
The key is a few simple inputs — good nutrition, enough sleep, a decent environment — and to let the mind have its space to play out the sequence in full.
In modern society the conditions needed for greatness, since the basics are relatively easy to come by, are achieved more by getting rid of negatives. Don’t be a drug addict or alcoholic, ditch the porn, the video games, the excessive social media use, etc. Then let the game play out.
Of course, this is easier said than done, as we’ve been deeply conditioned by the mechanistic paradigm, and remember TREE(1) is only 1, while G(1) is 3↑↑↑↑3, i.e., the more powerful growth function doesn’t necessarily reveal itself at the outset. But that changes in short order, and once it does, the mechanistic growth is no match for the “most densely ramified complexified structure in the known universe.”
I’ll end on a speculative thought, one that occurred to me while recording a podcast on this topic: Might what we think of as good vs evil actually just be a battle between the mechanistic and the complex, the difference between top-down compulsion and bottom-up free choice?
BONUS VIDEOS For those who want to dive deeper into this, there are some good videos by Carbrickscity on You Tube here, here and here. And Tim Urban’s article on Graham’s Number is worthwhile too.
TUBii on Nostr: Your brain will not get this. Much fun!! ...
Your brain will not get this. Much fun!!
