In the latest issue of the Bulletin of the American Mathematical Society, Eugenia Cheng has written about the ways in which technology can make the mathematics discipline more "congressive", meaning "bringing people with you and bringing people together and unifying and making connections between things."
Her essay made me think about the ways in which we talk about developments in large language models (LLMs) and how our "ingressive" interpretation of these tools may hinder our ability to make good use of them.
Aside: My writing progress has stalled while I work out some bugs in my reconstruction for the error function coefficients. You can view the minimal code here, which does a decent job generating a polynomial approximation, but not at the level of efficiency I want for my essay project.
LLMs really are "just" information retrieval systems, where the information emerges from the data encoded by the model weights. Regarded as a tensor product, prompt vector inputs compute response vector outputs. From a purely mathematical viewpoint, it's effectively matrices all throughout. (See Geordie Williamson's summary of deep learning models.)
Now, rather than discuss if an LLM isn't "just" an information retrieval system, let's grant that premise. Let's also assume the simplified representation of an LLM as a tensor product. (Simplified, because we're not admitting recurrent loops.) Isn't human intelligence "just" an information retrieval system?
Don't answer that question. Contemplate the question for a few moments, feel your body as you think, watch your emotions rise and fall, allow your senses to perceive the world around you. More likely than not, your answer turns between "yes" and "no" many times. You have directly taken on the prompt vector input -- "Isn't human intelligence 'just' an information retrieval system?" At the same time, you've been indirectly prompted by your emotional, physical, and social contexts. Some context's present in real time, some context's latent in your memory from lived experiences.
In the Bulletin and elsewhere, Eugenia Cheng's discusses congressive mathematics, which I take as compatible with a more broad intellectual activity. This involves working on open-ended problems that last years or generations before reaching a solution, if at all. Like all intellectual activity, congressive efforts involve teaching and learning, asking questions, collaborating, disseminating, and reasearching. To make these efforts congressive often allows allowing technology to take on tasks that are purely information retrieval -- not just "just" -- and orienting human activity to make the indirect prompting more direct.
Cheng writes, "Knowledge used to be power, but it isn't any more; the ability to search the internet and distinguish between the hits is much more important... More critical for me is an awareness of which sorts of [people] work on which sorts of things."
In other words, attention turns to people and what they're doing rather than exactly how they're doing it. Cheng continues
Has technology changed the intellectual parts of mathematics? Can it do so in the future, and should we embrace that?
I don't think technology has really encroached on the deepest, most creative, and therefore most human aspects of mathematics research. The deep creative parts involve coming up with ideas in the first place -- ideas for definitions, ideas for proofs, ideas for making connections between different parts of mathematics, ideas for new ways to express things, ideas for notation and terminology, ideas for diagrammatic reasoning, and ideas for visual representation.
I read this as highlighting the human experience in doing mathematics, which is far more expansive than recording and retrieving facts. Let's also be more mindful of the more expansive view on doing work, such that human roles aren't reduced to "just" mechanical efforts that may be replaced by machines.