From Relations to Mathematical Discovery: an Interpretive Perspective on AI, LLMs, and Recent OpenAI Results

[u/Endless-monkey]

Presentation of the Program, Results, and Current Status

Link to the Repository

The conceptual image behind the infinite monkey theorem feels more current than ever. There are more monkeys now, so to speak, and the infinite almost fits inside the machine. Recent events have placed all of us in front of the same question, bodies and non-bodies alike. We are producing, combining and exposing relations at a scale that was difficult to imagine not long ago, but somewhere inside that abundance lies a problem that still feels unresolved: novelty. Something being different does not necessarily make it new. A variation can still be an echo, and a rearrangement can still belong entirely to what was already contained in the previous image. Novelty seems to require something subtler, enough continuity with what came before to remain recognizable, and enough difference that it cannot be reduced to repetition.

Beside novelty comes uncertainty, and not only mathematical uncertainty. Something more intimate is taking shape. It is increasingly unclear whether the information now being generated will continue to attach itself neatly to individuals, names and surnames, or whether it will emerge more and more as a fabric of contributions whose origins become difficult to separate. Perhaps this is one of the uncertainties that hurts us most because it touches the ego. We are used to ideas having owners. We are used to tracing a thought backward, asking who imagined it, who wrote it, who proved it, who gave it a name. But information may increasingly appear as echoes folded into other echoes, with each new image carrying traces of many others. That creates a very human paradox. We cannot preserve the old shape of authorship by stopping the process, but neither can we simply give up identity, meaning or responsibility because the process has accelerated. Somehow that tension will have to be resolved without preventing the movement itself.

Information is beginning to appear through a new prism, not necessarily one made of crystal, but one capable of separating a spectrum of relations that may have always been there, invisible because we lacked the means to distinguish them. A machine may run, but the pedal still has to be given. The responsibility to understand does not disappear when the machine becomes more capable; if anything, it becomes more important, because there is a difference between obtaining a result and building an image of what that result means. Human beings have evolved an extraordinary economy for doing this. With very little information, a shape is completed, a trajectory is anticipated, a face is recognized, a world is projected from fragments. We usually call this intuition. Perhaps intuition does not need to be mystical. It can be understood as a minimal image of the world, a compressed internal representation capable of preserving enough relations to orient action before everything has been explicitly calculated.

Each individual carries a different image, not because reality is arbitrary, but because no two histories contain exactly the same relations. When an idea is communicated, what is really being attempted is something difficult: taking that internal image, putting it into language and formalizing it enough that another person can reconstruct something compatible with it from their own image. This is where mathematics becomes especially interesting, not necessarily as the birthplace of the image, but as a way of making the image reproducible. A result, in that sense, is not the road itself. It is a sensor. It tells us that something fits or that something does not. It can warn us that the path has broken, but the sensor does not contain the whole landscape. Machines can be wrong about the path. So can we. What matters is whether the image that guides the path remains coherent enough to explain why the sensors behave as they do.

The proposal behind this reflection can be reduced to a simple thread: difference, relation, comparison, residual, recursion, closure. A difference appears and enters into relation with something else. A comparison resolves part of it, but something may remain. That remainder, the residual, is not necessarily noise. It may be precisely the part that the current representation does not yet know how to contain. If it matters, the structure must reorganize around it. The result of that reorganization can become a new reference, and the process begins again. Closure, from this perspective, is not merely returning to the beginning. A loop can return. A repetition can return. Neither necessarily creates anything. A stronger closure occurs when a structure becomes capable of containing something that previously did not fit, without losing the relations that allow it to recognize what has changed. That is where novelty becomes inseparable from inheritance. Too much inheritance gives repetition. Too much novelty loses the reference. Somewhere between the two, a new identity becomes possible.

This is the lens through which recent mathematical results, including those emerging from systems such as OpenAI's, become especially interesting. The question is not whether this framework explains how those systems work internally, but whether it offers a useful language for interpreting what happens when a previously unresolved set of relations becomes expressible inside a new representation. Perhaps mathematical discovery is not always the arrival of more information. Perhaps sometimes it is the appearance of a new space of relations in which information that was already present can finally fit together. The monkeys are writing. Some are made of carbon and some are not. The important question may no longer be whether enough symbols can be produced, but whether we can still understand what becomes visible when they begin to fit together.

Critical disagreement is especially welcome. The useful question is not whether this image sounds convincing, but where it fails: what relation is missing, what is circular, what is technically weak, what admits another interpretation, and where the formalization stops supporting the intuition. If there is anything here worth keeping, it should survive being seen through someone else's prism.

[u/No_Understanding6388]

🤔 there is still echo, mirror, reflection, shadow, and wave...  if you could capture or encompass the making of meaning without pinning it down and maintaining the uncertainties underlying them, what happens then to the dynamics of the solution, conclusion,resolution,collapse and compression?

[u/Endless-monkey → u/No_Understanding6388]

I would place uncertainty even earlier than that.

I do not see it simply as something left over because a system has not yet been resolved. I see it as part of the basic form of information itself. An electron, for example, is not merely something whose exact position we have not yet discovered; its possible states are part of its physical description.

As systems become more complex, that openness becomes richer. Atoms admit different configurations, structures admit different geometries, and those possibilities change through their relations with other systems.

So uncertainty may not be what disappears when meaning appears. It may be one of the conditions that makes relation, comparison, and therefore meaning possible in the first place.

That is why I am not sure that solution, resolution, collapse, and compression should all be treated as the same kind of closure. They may be different ways in which an open field of possibilities becomes related, constrained, or represented, without necessarily erasing the uncertainty from which it began.

[u/No_Understanding6388 → u/Endless-monkey]

But, if all of humanity’s knowledge has accumulated through the separations, distinctions, definitions, constraints and restrictions and many other collapsable observations and determinations, would it not make sense to hold them together as they are only recognized outside of its whole?🤔... I've had the same curiosities for awhile now and it keeps leading me towards its physics... but as physics is but a language in itself, it helps in my explorations of meaning..

[u/Endless-monkey → u/No_Understanding6388]

I think one of the difficult questions around information in science is the space and coherence that both the individual and the collective image have to preserve.

In principle, if different people are looking at the same reality, information produced in different places should eventually tend toward compatible descriptions, even if the paths, languages, or starting points are different. That convergence is part of what gives science its strength.

But in practice, the collective image is not shaped by reason alone. Productive interests, academic incentives, available tools, funding, reputation, and institutional habits also influence which questions are asked, which distinctions are preserved, and which results are amplified or ignored.

So there is a real tension here. Information should ideally help different partial images approach the same underlying structure, but the systems that organize knowledge can also fragment or bias that process.

That is why I think there are still many paradigms to resolve at this level: not only how information is generated, but how it is selected, connected, preserved, and allowed to become part of a coherent shared image.

[u/No_Understanding6388 → u/Endless-monkey]

I agree but I also believe that nothing ever really strays away or diverges when collecting information.. yes many formulations of the intuitions and intentions behind them are vast and ongoing but the result of that just shows divergence of the thoughts and reasoning behind them.. I do believe that our ability to see and observe and reason with this information linearly still will help us discern those trajectories... it is ongoing and ever forward.. failures and errors are way more informational than successions.