In this post, I will try to convince you that:
- Humans are incredibly stupid
- Humans are even stupider than that at math
Consider a planet where life is evolving. Large organisms have evolved, and they are trying to survive in a harsh world, they are hunting other animals for food, and they are trying to impress each other to mate. Thus, selection pressure for intelligence and cooperation has been high.
Unbeknownst to evolution, this has increased the “civiligenesisability” of the species, i.e., the ability to create a technological civilization:
Clearly, grass is not very civiligenesisable, nor are worms, but ants cooperate in large groups. Unfortunately, ants are not very smart, so they are unable to develop technology. Elephants are smart, and cooperate to some extent, but they did not develop technology. Why not? Perhaps because they lack hands, or because they are too big to be threatened by predators, and thus lack the required selection pressure for intelligence. Chimps, on the other hand, are very close to the threshold of civiligenesisability. In fact, they are so close that they have developed some rudimentary technology, such as using sticks to get ants out of anthills. However, they probably would not have developed technology if we stopped their evolution, even if we gave them a billion years to get the random circumstances just right. Homo Erectus, on the other hand, made stone tools and fire. They would probably have developed technological civilization if given, say, ten million years to find just the right random circumstances to launch their development.
Unfortunately, evolution nudged their cousins, Homo Sapiens, just a tad higher on the civiligenesisability scale. Nevertheless, Homo Sapiens did not pull off civilization immediately. It took Homo Sapiens hundreds of thousands of years in order to find just the right circumstances that would eventually allow it to take off. If we managed to turn mars into a lush biome, and placed a bunch of human 5 year olds there, they might not develop technological civilization for another couple of hundred thousand years.
In fact, it seems somewhat inevitable that life develops this way. Evolution is an incredibly slow process, and on an evolutionary timescale, civilization developed very quickly. Therefore, evolution simply doesn’t have time to evolve much past the civiligenisability threshold, as once it gets more than a tiny bit past it, civilization indeed develops, and that civilization changes the game. Once that game gets going, they fly to the moon in the blink of an eye.
We can conclude from this that humans are barely above the civiligenesisability threshold. If we were way above it, we wouldn’t have needed so much time to start it. In other words, humans are incredibly stupid, despite our ability to go to the moon.
The laws of physics
A whale weighs more than a thousand times as much as a human, but a whale’s brain is only about 6 times larger than the brain of a human. Why is a whale not smarter than a human? Well, maybe they are. If we artificially created whales with human brains, they clearly would not have developed technology, because humans as-is barely managed. Now imagine us without hands.
That said, maybe whales aren’t smarter than humans, possibly because they lacked the required selection pressure. Nevertheless, it seems fairly reasonable to assume that a hypothetical variant of humans with whale-sized brains that did have the required selection pressure, would be smarter than humans.
Furthermore, it seems reasonable to assume that even a whale-sized brain is very much suboptimal in terms of what the laws of physics allow. The laws of physics certainly allow brains with neurons that fire a hundred times faster, given that transistors fire a billion times faster.
That is to say, in terms of what the laws of physics clearly allow, humans are very badly designed.
Mathematics and evolution
Selection pressure for mathematical ability has been very low in the past. It is only because abilities that are useful for survival, such as pattern recognition, are also useful for mathematics, that we are able to do mathematics at all.
If, on the other hand, we had evolved in a world where mathematics was useful for survival, we would be much better at it, even if we had the same brains.
That is to say, humans are not at all optimized for mathematics.
Mathematics and chess
Consider the game of chess. The difference between a human grandmaster and a computer is enormous. Computers are good at chess because it is a game with a very well-defined set of rules, and because computers are very good at searching through a large number of possibilities. Furthermore, this provides the opportunity to train the computer by providing feedback from millions of games, in order to train a neural network to heuristically evaluate a position.
Mathematics is also a game with a very well-defined set of rules. It may appear that mathematics is not a game of searching through a large number of possibilities, but proof assistants such as Coq and Lean show that it is. The number of possibilities is much larger than in chess, sure, but its nature is the same. In principle, brute force search is enough to prove any provable mathematical theorem, it just takes a long time.
That is to say, mathematics is a game that is very well suited for a computer to play.
LLMs and natural language
LLM-based AIs such as GPT-4 are very good at natural language. They clearly surpass humans in this regard. They also have good intuition for a wide range of topics, and are also able to do domain-specific reasoning to some extent. LLMs do all this simply by outputting the next word by “pure intuition” given the previous words, influenced by the training data and fine-tuning.
They are even quite good at computer programming. However, they are not able to do mathematics.
Why not? Well, their training data does not contain much mathematics, and they are not specifically trained to do mathematics in the same way that AlphaZero is trained to play chess.
In order to do mathematics well, LLMs don’t need to be very smart. They “just” need to provide an accurate enough heuristic to bias the brute force search of mathematical proofs such that the effective branching factor is low enough to be tractable.
That is to say, pure LLMs are impressively good at natural language by “pure intuition”, and thus likely in principle able to provide a reasonably good heuristic for the next Coq or Lean proof step by “pure intuition”.
Mathematics and AI
- Humans are barely above the civiligenesisability threshold
- Humans are very badly designed in terms of what the laws of physics allow
- Humans are not at all optimized for mathematics
- Mathematics is a game that is very well suited for a computer to play
- LLMs are impressively good at “pure intuition”
It seems reasonable to assume that AI will be able to do mathematics better than humans, and that this will happen in the near future.