Artificial intelligence is here, it’s improving and iterating rapidly. Some prominent and seemingly well-informed people are worried about its impact on humans, going so far as to question our purpose once AI “becomes better than us at every mental task.”

I write to tell you fret not. The method by which AI learns is not as powerful as the one by which humans learn. AI will seem more impressive at lower level tasks — it will certainly multiply and divide faster, play chess better and pass college exams more easily. But at higher level functions, it will get destroyed.

On what basis do I claim this?

Let’s take a step back and consider how inputs (information) get converted to outputs (results). Like humans, AI takes in information and outputs results.

A simple way to model this process is via mathematical functions.

If f(x) = 2x, then the input is x, and the output f(x). In this case, let’s say the information (x) = 2. Then the output of f(x) = 4.

If instead f(x) = x^3, then if x = 2, f(x) = 8.

The second function is more powerful.

(Note the first function (2x) would have a larger output than the second (x ^ 3) when x = 1, i.e., it is possible for the more powerful function to seem less powerful initially.)

The question in which we’re interested is which function AI(x) or Mind(x) is more powerful.

Let’s consider how AI works. It’s a machine that can iterate rapidly. It can learn what it learns and then learn how to learn and how to learn how to learn, etc. How can a human compete with a machine that can absorb massive amounts of data and not only process that data but learn to process it ever more efficiently as it goes? Surely those who fear AI displacing humans at almost every task must be correct.

I will propose as a rough model for AI, a very fast growing function that accounts for its ability to improve itself rapidly.

And then I’ll propose a model for the human mind that grows much more slowly at first before leaving it in the dust.

Choose Your Fighter:

**AI = Graham’s Number.**

Graham’s Number was discovered in the 1970s by mathematician Arthur Graham as the solution to problem in the field of Ramsey Theory (something of which I had never heard until I searched for it two minutes ago.) What’s important, though, is how Graham’s Number is generated — the function G(x), so to speak.

To understand how it works, we have to begin at the beginning, with counting:

Let’s say you wanted an AI to generate the most massive finite number it could muster.

It might start by counting: 1, 2, 3, 4…

But it would soon discover you can speed up counting, by adding:

Instead of counting from three to six, it could just add 3 + 3 because addition is just repeated (iterated) counting.

But instead of adding 3 + 3 + 3 + 3 + 3 + 3, it could just do 3 * 6 because multiplication is iterated addition.

But instead of multiplying 3 * 3 * 3 * 3, it could just do 3 ^ 4 because exponentiation is iterated multiplication.

Beyond exponentiation lies tetration, or iterated exponentiation.

Instead of 3 ^ 3 ^ 3 ^ 3, it could 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.)

Before we go further, let’s understand tetration by calculating 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. (You go from right to left with power towers.)

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? We 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 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 (see above), 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.

If AI(x) is akin to Graham’s Number, it is a powerful information processing function indeed.

**MIND(x) = The TREE function**

While Graham’s Number is generated via ramping up increasingly powerful iterative operations, TREE(x) 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 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.

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. Graham’s Number is trivial, basically zero compared to TREE(3), and that is so much the case that if instead of going to G(64) 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). In other words, even if AI could instantly scan all the information in the universe (the inputs), it can’t even compete with TREE when it’s input is only 3! Imagine by what margin TREE(4) or TREE(100) would destroy even G(G(G(G(64)))))!

The input — the amount of information — is not nearly as important as the processing power of the function. And the functions with the most processing power are from complex systems with simple axioms, not machines that iterate. Consider that humans evolved from the simplest of organisms, and even we ourselves began with a few simple instinctual directives for basic survival.

Also recall that when the input (x) is 1 or 2, G(x) destroys TREE(x). So don’t be alarmed when AI seems like it has magic powers. It will be powerful indeed. Just not nearly as powerful or capable as a human mind sufficiently opened.

I’ll let Naval Ravikant have the last word — as he succinctly sums it up: