The Platonic space of ideals is real. And we’re but humble explorers.
For much of history, we humans have believed that we have the capacity to create. To construct new ideas, new theories, new art. We prided in our monopoly on creativity, which raised us above the baser animals. We frolick in the beautiful garden of
LLMs have shattered this innocent dream.
The theory of the Platonic space of ideals (Platonic space) denies humans the ability to create things from thin air. Instead, the products of our creativity already have existence in this alternate plane which contains all possible ideas and concepts. When we “create”, we simply pluck objects from Platonic space into our physical reality. It doesn’t deny that creative work is hard work, it simply says the fruits already existed somewhere. You can build a chair, but the concept of chairs wasn’t created by you as a result. You can build a mathematical framework, but it merely describes something already in existence. Description, not creation.
Before LLMs, Platonic space was just a philosophical framework, an alternative and rather quaint way to view the creative process. LLMs in many ways demonstrate that Platonic space is real, as real as the electromagnetic field.
LLMs are next-token prediction machines, but I think there is a slightly better way to look at them: they are conditional probability machines. At each token, it tells you the conditional probability of each possible next token, given the previous sequence of tokens. To give a more visual picture, you’re basically standing at a location on a landscape, and the LLM’s logits tell you what the landscape looks like from your perspective and where you should take your next step to make your trek the easiest. An LLM starts at a point on this landscape and simply rolls down the hill, taking the path of least resistance. Yet despite this mechanical process, LLM agents are able to do things like code up a website or prove advanced mathematical theorems.
LLMs are essentially maps of Platonic space.
The maps were charted by humanity. Every data point, every random blog post, every word from a book written 2,000 years ago, they all contribute to painting this map during the LLM’s pre-training process. Each makes the map a little bit clearer, a little bit more detailed. And then, AI agents simply follow the map, walking the trails that humans have created.
But agents are not limited by what humans have created in the past. They have the capability to extrapolate. Conditional probabilities are composable: you simply multiply them. In this way, agents are able to arrive at new possibilities, new results, new theories that no human has ever created.
Humanity is no longer alone as explorers of Platonic space. Now we have machines.
This fact has cause much dismay. Artists claim machines can’t produce art. Mathematicians ask who gave people permission to let AI do math and claim it’d be the death of mathematical research. Understandable sentiments, but all futile.
If we find value in the exploration of Platonic space for its own sake, which our history of creativity suggests to be the case, then the better attitude is to embrace LLMs and agents as tools to help us better chart this final, if not only, frontier.