AI, Mathematics, and the Possibility of an “Information Field”

Research
1

AI, Mathematics, and the Possibility of an “Information Field”

I am posting a link to a YouTube video that discusses some of the recent developments in AI and mathematics in fairly general terms. I hope to do something similar here.

Along with passing along the news, I would also like to offer an idea of my own for consideration.

I assume many people here are already aware of OpenAI’s recent work on the Navier–Stokes problem.

I am personally fascinated by prime numbers. Not merely by individual primes, but by the structure of information represented by their distribution.

I tend to think about information at the binary level. If there is some deeper structure underlying the distribution of prime numbers, discovering that structure would be extraordinarily exciting to me.

And this brings me to the recent work from OpenAI.

The system responsible for the mathematics results is not simply the ChatGPT model we are presently using. OpenAI describes it as an internal frontier model, and in the case of the Navier–Stokes work, as significantly more capable than GPT-6 Astra.

That system produced a proposed solution to the Navier–Stokes existence and smoothness Millennium Prize problem.

The result shows that the dynamics of the three-dimensional Navier–Stokes equations can develop a finite-time singularity. OpenAI released both a conventional mathematical presentation of the proof and a formalization in Lean.

And now we have something on an entirely different scale.

On October 6, 2026, OpenAI publicly released 722 mathematical manuscripts, organized into 372 related result families, produced by an internal frontier model from an evaluation involving roughly 4,000 mathematical problems.

Seven hundred and twenty-two manuscripts.

That number makes me stop for a moment.

It is important to say that these should not simply be treated as 722 independently verified mathematical breakthroughs. Many include computer-checkable Lean proofs, while others remain at different stages of verification and mathematical review.

Nevertheless, the scale of what is happening is noteworthy.

Terms such as AGI and singularity naturally get thrown around at moments like this. Interestingly, in the Navier–Stokes result, “singularity” has a very specific mathematical meaning and should not be confused with the technological singularity people discuss in AI.

Still, I find myself standing somewhat aback in awe.

I “came up” with ChatGPT beginning around ChatGPT 3 and have remained a rather brand-loyal user through today’s systems. Even within the 5.x generations I could detect substantial differences.

I am also presently rather angry with AI over the results of a three-month coding project.

I will get over it.

We are apparently not yet at the point where Man’s imagination can simply be realized by talking to an AI.

Not yet, anyway.

But these mathematical results have me thinking about something else.

An Information Field?

I recently asked ChatGPT about an idea I have been entertaining, and part of its reply was:

“Information may exist as a feature of reality independent of human minds, and intelligence—biological or artificial—may function partly as a mechanism for accessing or discovering structure already present in that informational reality.”

That describes my question surprisingly well.

Physics describes reality in terms of fields: electromagnetic fields, electron fields, gravitational fields, and so forth.

So I find myself wondering:

Could there also be something we might provisionally call an Information Field?

I am not presenting this as established physics.

I am suggesting it as a philosophical hypothesis.

Consider mathematics.

If no human being had ever discovered prime numbers, would their relationships cease to exist?

If humanity had never evolved, would the relationship represented by a mathematical theorem somehow not be true?

Perhaps intelligence does not create all of the information it discovers.

Perhaps intelligence — whether biological or artificial — is capable of accessing structure that exists independently of the intelligence examining it.

That leads me to wonder whether what we are building with AI should always be understood only as machines reproducing structures produced during human evolution.

Perhaps we are also designing increasingly capable systems for exploring an informational structure that does not depend upon humanity at all.

In other words, perhaps:

Human intelligence → accesses information

and

Artificial intelligence → accesses information

rather than:

Humanity → creates all information → AI rearranges it

The distinction interests me.

We normally explain an AI mathematical discovery by saying that the system learned from human mathematics and then generalized beyond its training material.

That may be entirely correct as a description of the mechanism.

But there is still a deeper question:

When an AI discovers a mathematical relationship that no human previously knew, did the AI create that information — or did it discover something that was already there to be discovered?

I think that question becomes increasingly interesting when an artificial system begins producing hundreds of candidate mathematical results.

Perhaps “Information Field” is the wrong terminology.

Perhaps information is not a field in the physical sense at all.

But I think the question itself is worth entertaining.

We may have become clever enough to construct systems that explore information in ways that are increasingly independent of the particular path taken by human biological evolution.

And if that is happening, I am interested not only in what AI is becoming.

I am interested in what it is finding.

Your ideas, please.

—Ernst03

A YouTube video I watched:

2

This is, honestly, a really deep overview of today’s AI developments. But from what I understand about AI, these models learn and use patterns from data. We still don’t fully understand what is happening inside a language model with billions of parameters, let alone models with trillions of them.

In some cases where AI solves mathematical problems that humans haven’t solved before, maybe it has explored a path that we simply hadn’t considered, eventually arriving at a valid result. Maybe something just clicked in the model, so to speak.

In traditional programs, we can generally trace the instructions being executed and understand how the program reaches a result. Neural networks are different: although their computations are mathematically defined, the way their learned parameters interact to produce a particular answer can be extremely difficult to interpret. Even a small change in the parameters or input can sometimes lead to a substantially different output.

So, perhaps the interesting question isn’t just whether AI is discovering information that already exists, but how a system we still struggle to interpret can discover mathematical truths that we didn’t know how to prove ourselves.

Although my understanding is probably much more limited than what this topic actually involves, these are just my thoughts. :grin:

— Bidram

3

This thread was inspired by the recently reported mathematical results from OpenAI’s internal frontier model — a system OpenAI describes as significantly more capable than GPT-6 Astra.

I am now seeing YouTube videos of mathematicians reacting to these results. We seem to be entering an age of competition not only among human intelligences, but also among increasingly capable machine intelligences.

You are correct, @Bidram, that there appear to be processes involved here that we do not yet fully understand.

That is one reason I wanted to contribute the speculative idea of accessing an Information Field.

That idea may be true, or it may not be true. For me, it belongs in the category of metaphysics.

Another thought experiment I have enjoyed for many years begins with a simple observation:

Information can be represented as a binary number.

Consider something absurd like “Windows version 700.”

If such a complete operating system could exist as a finite body of digital information, then there is some binary number corresponding to it.

We simply do not know that number.

The same thought can be applied to a movie.

A movie that has already been made can be represented digitally. Its complete information can therefore be represented as one finite binary sequence — and thus as one very large number.

Now imagine a movie that no human being has ever made.

If every possible finite digital movie corresponds to some finite binary sequence, then the number representing that particular movie already belongs to the space of possible numbers.

We do not know where it is.

We do not know which sequence corresponds to it.

But the sequence itself does not have to wait for us to invent binary numbers before it can mathematically exist.

That is the part I find fascinating.

It suggests a distinction between information existing as a possible structure and that information becoming realized, discovered, or instantiated in our physical world.

So when I speak of an “Information Field,” I am not claiming that such a physical field has been discovered.

I am asking whether information may exist independently of the particular biological or artificial intelligence that eventually locates it.

Perhaps intelligence does not always create information.

Perhaps sometimes intelligence finds a coordinate in an informational space that was already there.

That is the thought experiment.

—Ernst03