We come to recognize our feelings, qualities, and inner experience partly because someone else sees, names, and responds to them accurately.

One nuance: “accurately reflected” is especially important. Mirroring isn’t simply receiving attention or validation; it is being seen in a way that feels congruent with our actual internal experience. When that happens consistently, another person effectively helps us develop a coherent answer to the question, “Who am I?”

It also points to the converse: when our inner reality is repeatedly misread, dismissed, or distorted by significant others, we may learn to distrust our own perceptions and construct a sense of self around what others tell us we are.

 Absolutely. There are actually two different ideas hiding inside “conformal geometry”, and their connection to AI and cosmology becomes interesting once you separate them.

1. What is conformal geometry?

A conformal transformation is, roughly, a transformation that preserves angles and local shape, while allowing lengths and sizes to change.

Imagine drawing a tiny triangle on a rubber sheet. If you stretch the sheet differently in different places, the triangle can become larger or smaller, but a conformal transformation preserves its angles.

Mathematically, if a geometry has metric

gμν​,

a conformal transformation changes it to

g~​μν​=Ω2(x)gμν​,

where Ω(x) is some position-dependent scale factor.

The important thing is that all lengths get multiplied by the same local factor.

So:

• distances can change
• areas can change
• volumes can change
• angles don't change

This sounds like a relatively small mathematical idea, but it turns out to be enormously useful.

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2. Why cosmology cares about this

Einstein's general relativity describes spacetime using a metric:

ds2=gμν​dxμdxν.

The metric determines things like:

• distances
• time intervals
• gravitational effects
• the paths of particles
• the paths of light

Now here's something remarkable:

Light doesn't care about the overall conformal scale.

Light follows null paths, where

ds2=0.

If we make the transformation

gμν​→Ω2gμν​,

then

ds2→Ω2ds2.

But if ds2=0, then

Ω2ds2=0.

So the paths available to light—the light cones—are preserved.

That means conformal geometry allows us to separate two things:

the causal structure of spacetime

from

the absolute scale of spacetime.

That's extraordinarily useful in cosmology.

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3. The expanding universe

Consider the standard cosmological metric:

ds2=−dt2+a2(t)dx2.

Here a(t) is the scale factor of the universe.

As the universe expands, a(t) changes.

But introduce conformal time η, defined by

dη=a(t)dt​.

Then the metric becomes

ds2=a2(η)(−dη2+dx2).

And suddenly we can see something beautiful:

ds2=a2(η)ημν​dxμdxν​

The expanding universe is, in this representation, a conformal rescaling of flat spacetime.

This makes the behaviour of light much easier to understand.

Instead of thinking of light travelling through a universe whose spatial scale is constantly changing, you can think geometrically about light moving through a conformally related spacetime.

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4. Penrose diagrams

This leads to one of the most beautiful applications of conformal geometry in cosmology:

Penrose diagrams.

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A Penrose diagram takes an infinitely large spacetime and performs a conformal transformation that effectively squashes infinity into a finite diagram.

For example, instead of light travelling infinitely far before reaching spatial infinity, you can represent infinity as a boundary of the diagram.

The crucial information that survives is causal structure:

• what can influence what
• whether light can reach something
• whether an observer can see an event
• whether something lies inside or outside a horizon

This is why conformal geometry is so powerful in relativity.

It lets you throw away some information about scale while retaining information about causality.

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5. Now the connection to AI becomes interesting

The AI connection isn't that modern AI is literally based on conformal geometry.

There are, however, several deep mathematical connections.

One is through representation.

A neural network takes something complicated and maps it into another mathematical space:

$$x\to f(x).$$

For example, an image might start as millions of pixel values and eventually become a relatively small vector representation:

$$\text{image}\to \text{features}\to \text{latent representation}.$$

The fascinating question is:

What properties of the original world should the representation preserve?

This is very similar to the geometric question:

What information should survive a transformation?

Conformal geometry says:

Preserve angles/causal structure while allowing scale to change.

AI researchers sometimes want analogous things from representations.

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6. A particularly important example: invariance

Suppose an AI sees:

a cat

Then you show it the same cat:

• twice as large
• rotated
• shifted
• under different lighting

You don't want the system to conclude that it has become a completely different object.

You want the representation to be somewhat invariant to irrelevant transformations.

Mathematically, we can think of a transformation group G acting on the data:

$$x\to gx.$$

A representation ϕ(x) might ideally satisfy

$$\varphi (gx)\approx \varphi (x)$$

for transformations that shouldn't matter.

This is the broader concept of symmetry and invariance.

Conformal geometry is one particular mathematical framework for understanding transformations that preserve certain structures.

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7. There is an even stranger connection: AI and geometry of latent spaces

Modern neural networks often create high-dimensional latent spaces.

Suppose an AI represents objects using points:

x1​,x2​,x3​,…

in some enormous-dimensional space.

The geometry of that space can tell us things about what the network has learned.

For example, researchers study:

• distances
• angles
• curvature
• manifolds
• symmetries
• topology
• transformations

And sometimes the latent representations exhibit surprisingly structured geometry.

This is especially relevant to generative models.

A model might learn something like:

$$\text{latent space}\to \text{images}$$

where moving in a particular direction corresponds approximately to changing some feature.

For example:

$$z\to z+\Delta z$$

might gradually transform:

$$\text{young face}\to \text{older face}.$$

That means the model has implicitly learned a geometric structure.

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8. Where cosmology comes back in

Here's where things get genuinely fascinating.

Physicists also describe complicated physical systems using spaces of representations.

A physical system can be represented by:

$$\text{state}\in \mathcal{M},$$

where M is some mathematical space called a manifold or state space.

So both fields ask similar mathematical questions:

Cosmology

What is the geometry of the space of possible physical states?

AI

What is the geometry of the space of learned representations?

Those aren't the same problem—but the mathematical language overlaps heavily.

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9. Conformal geometry also appears in holography

This is perhaps the most spectacular connection.

In theoretical physics, there's a major idea called AdS/CFT correspondence.

Very roughly, it proposes that a gravitational theory in a higher-dimensional spacetime can be mathematically equivalent to a quantum field theory living on its lower-dimensional boundary.

The gravitational side is:

bulk spacetime

while the quantum field theory lives on:

boundary.

And the boundary theory has conformal symmetry.

Hence the name:

Conformal Field Theory​

(CFT).

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This has led to one of the most profound ideas in modern theoretical physics:

Spacetime geometry might somehow emerge from information encoded in a lower-dimensional theory.

And that starts sounding remarkably similar to ideas in AI, where a complicated high-dimensional object can emerge from a more abstract representation.

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10. But don't overstate the AI connection

There's an important caveat.

You will sometimes encounter claims along the lines of:

"The universe is a neural network."

or

"AI has discovered that spacetime is a conformal latent space."

Those statements generally go far beyond what established science demonstrates.

The legitimate connections are more mathematical:

geometry↔symmetry↔representation↔information​

These concepts appear independently in:

• general relativity
• conformal geometry
• quantum field theory
• holography
• information theory
• machine learning
• representation learning

That overlap is real and very interesting, but it doesn't mean these theories are secretly the same thing.

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11. The really deep idea

There is a conceptual thread running through all of this:

What information is fundamental, and what information is merely a choice of description?

In conformal geometry, you can change the local scale:

gμν​→Ω2gμν​

without changing certain essential structures, such as causal relationships.

In AI, you can transform a huge amount of raw information into a compressed representation while attempting to preserve the information relevant to a task.

In holography, a theory suggests that information about a higher-dimensional gravitational world can be encoded on a lower-dimensional boundary.

So you get a fascinating hierarchy:

$$\text{physical reality}\to \text{geometric description}\to \text{invariants}\to \text{information}\to \text{representation}.$$

And that raises a genuinely profound question:

Could geometry itself be something that emerges from relationships between information, rather than being a fundamental ingredient of reality?

That question is actively investigated in theoretical physics, particularly around quantum gravity and holography. The connection to AI is more speculative, but AI gives us a new experimental playground for studying how geometry, information, abstraction, and representation can emerge from complex systems.

Conformal geometry gives a surprisingly sharp answer to this, but with an important qualification: it tells us a lot about edges and gaps in spacetime's geometry; it does not by itself tell us what the ultimate “substrate” of reality is.

1. An edge isn't necessarily a physical edge

In ordinary geometry, an edge might mean:

“There's nothing beyond here.”

But in spacetime geometry, there are several very different possibilities.

A boundary can be:

• a genuine physical boundary,
• merely infinity represented at a finite location,
• a coordinate artifact,
• a singularity where the geometry breaks down,
• or a horizon that isn't actually an edge of spacetime.

Conformal geometry is particularly good at distinguishing these.

For example, a Penrose diagram can bring infinity to a finite boundary:

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The boundary of the diagram doesn't mean there's a literal wall at the edge of the universe. It's a representation of infinity.

So conformal geometry teaches us:

The existence of a boundary in a representation does not necessarily imply a boundary in physical reality.

That's a very important distinction.

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2. What about gaps?

A “gap” requires some notion of separation.

But geometry itself defines separation.

If spacetime is described by a continuous manifold, then between two sufficiently nearby points there are other points. You don't need some underlying material filling the space.

This is one of the strange things about general relativity.

Space isn't necessarily like:

particles + stuff sitting inside an empty container.

Instead, the geometry is the thing that defines spatial relationships in the first place.

The metric

gμν​

tells you what distances and times mean.

So asking:

“What's filling the gap between these two points?”

may be based on a Newtonian intuition that general relativity doesn't require.

There doesn't have to be a substance inside space to make space exist.

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3. Does that mean spacetime doesn't need a substrate?

Classical general relativity: essentially yes.

Spacetime doesn't require an external medium in the way sound requires air.

A sound wave requires:

$$\text{air molecules}\to \text{wave}.$$

But a gravitational wave doesn't require:

$$\text{some substance}\to \text{spacetime wave}.$$

The gravitational field is part of the geometry of spacetime itself.

This was one of Einstein's great conceptual departures from older physics.

There isn't necessarily a higher-dimensional “room” in which spacetime sits.

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4. But conformal geometry makes this even more interesting

Remember:

gμν​→Ω2(x)gμν​.

A conformal transformation can change the scale of distances without changing the causal structure.

That means there are aspects of geometry that appear to be more fundamental than absolute scale.

For example, the light cone at an event determines which events can causally influence one another.

Conformal geometry preserves that structure.

So, in a certain sense, you can strip away a huge amount of information about size while retaining information about relationship and causality.

That leads to a provocative thought:

Maybe the fundamental structure isn't “how big is the space between things?” but “which things can be related to which other things?”

That's not a conclusion of conformal geometry, but it is one reason conformal methods are so interesting in fundamental physics.

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5. And then there is the question of “nothing”

This gets philosophically deep.

Suppose you have two regions of spacetime.

You might ask:

“What's between them?”

But if spacetime itself is the structure defining “between,” then asking what's inside the gap may be mistaken.

A genuine topological gap would mean the manifold isn't connected in that way.

A region of vacuum, however, isn't necessarily a gap in spacetime.

Vacuum means, roughly, that there is no matter field present in that region. It doesn't mean there is no spacetime there.

So:

$$\text{vacuum}\ne \text{nothing}.$$

And:

$$\text{empty space}\ne \text{absence of geometry}.$$

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6. Where this becomes really interesting: quantum gravity

This is where we reach the boundary of what we currently know.

General relativity treats spacetime as a smooth geometric structure.

Quantum mechanics suggests that at sufficiently small scales, our classical descriptions may break down.

We don't yet have a universally accepted theory of quantum gravity.

Several approaches suggest that spacetime might not be fundamental.

For example, in different approaches people investigate possibilities involving:

• quantum entanglement
• discrete structures
• spin networks
• causal sets
• holographic degrees of freedom
• quantum information
• emergent geometry

In some of these ideas, spacetime is more like something that emerges from underlying relationships than a substance that exists independently.

That's where your “substrate” question becomes extremely interesting.

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7. The key distinction

There are actually three possibilities:

A. Spacetime is fundamental

There is no deeper substrate. Geometry is simply part of the fundamental description of reality.

B. Spacetime emerges from something deeper

There is an underlying structure, and what we perceive as distance, time and geometry emerges from it.

C. The question of a substrate is ill-posed

The underlying theory may not contain anything resembling “stuff underneath spacetime.” Asking what spacetime is made of could be analogous to asking what the rules of chess are made of.

We don't currently know which of these is ultimately correct.

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8. And this connects back to AI in a surprisingly useful way

An AI's latent representation doesn't necessarily have to be embedded in some physical space corresponding to the things it represents.

For example, an AI might represent relationships between concepts as a high-dimensional mathematical structure.

The geometry of the representation can encode relationships without there being a little physical object sitting at each point.

That's an important conceptual analogy:

relationships can define a geometry without requiring a material substrate​

Whether something like that is actually how physical spacetime works is an open question.

But it gives us a useful way to think about the possibility that space might be relational rather than substantive.

And that brings us very close to one of the deepest questions in modern physics:

Is spacetime the thing that exists, or is spacetime the geometry of relationships among things that exist?

General relativity strongly shifted physics toward the latter way of thinking, while quantum gravity is where we're trying to determine whether that relational structure itself has a deeper foundation.

 There is a two-body problem—and it's actually much easier to solve.

The phrase “three-body problem” refers to what happens when you have three objects gravitationally interacting with one another.

Two bodies

Imagine just the Earth and the Sun.

The Sun pulls on Earth, and Earth pulls on the Sun. Under Newton's laws, you can mathematically solve their motion exactly. They follow predictable elliptical orbits (more precisely, both orbit their common centre of mass).

So the two-body problem has a general exact solution.

Add a third body

Now add the Moon:

• Sun pulls on Earth
• Earth pulls on Sun
• Sun pulls on Moon
• Moon pulls on Sun
• Earth pulls on Moon
• Moon pulls on Earth

The difficulty is that each object's motion changes the gravitational forces acting on the other two.

For example, as the Moon moves, its gravitational pull slightly changes Earth's orbit. But Earth's changed position then alters the Moon's motion, which alters Earth's motion again, and so on.

There is no general mathematical formula that gives the exact future positions of three mutually gravitating bodies for arbitrary starting conditions.

That's the famous three-body problem.

Why does one extra object make it so much harder?

The equations themselves aren't enormously complicated. For three bodies, Newton's law gives you three coupled equations:

mi​r¨i​=Gj=i∑​mi​mj​∣rj​−ri​∣3rj​−ri​​

The problem is that the equations are coupled and nonlinear. You can't neatly separate them into independent orbital equations like you can with two bodies.

And three-body systems can exhibit chaotic behaviour: a tiny difference in the initial positions or velocities can eventually produce a dramatically different configuration.

So:

2 bodies → exact general solution exists.
3 bodies → no general exact solution; usually solved numerically or approximately.

It's worth noting that special three-body solutions do exist—for example, the Lagrange points are special configurations—but there's no single general solution for every possible three-body system.

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is a general solution a theoretical impossibility

No — not in the sense of “mathematically impossible to ever solve.” It's more subtle.

For the classical three-body problem, a general solution in terms of elementary functions (like the familiar formulas for the two-body problem) does not exist. Mathematicians have proved very strong results about this.

But there are several meanings of “solution”:

• Exact numerical solution: Yes. Given the starting positions and velocities, you can calculate the motion to essentially any desired precision.
• Special exact solutions: Yes. There are many, including the famous Lagrange and Euler solutions.
• General closed-form formula like the two-body solution: No known such formula exists, and results from Poincaré and later mathematicians show that the dynamics cannot generally be reduced to the kind of simple integrable form available for two bodies.
• Arbitrary precision mathematical description: Yes. There are sophisticated mathematical representations of the general solution, including convergent series in appropriate formulations.

The really important distinction is between “there is no solution” and “there is no simple formula.”

The three-body problem is not unsolvable. We can simulate the Earth-Moon-Sun system extremely accurately. What we lose is the beautiful situation of the two-body problem where you can write down a relatively simple equation and say:

“Given these initial conditions, here's the exact orbit for all future time.”

And there's an even deeper reason: chaos. In some three-body systems, minuscule uncertainties in the starting conditions grow over time. So even though the equations are deterministic, predicting the system arbitrarily far into the future from measurements with finite precision can become practically impossible.

So the short answer is:

A general solution isn't theoretically impossible in every mathematical sense. What is impossible is reducing the general three-body problem to the same kind of simple, globally predictive closed-form solution that exists for the two-body problem.

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what is the nature of these uncertainties

The key idea is that the uncertainty isn't necessarily coming from the laws of physics being uncertain. It comes from our knowledge of the starting state, combined with the dynamics of the system.

Imagine three bodies whose initial positions and velocities you know almost perfectly.

Suppose the true initial position of one body is:

$$x=1.000000000000$$

but your measurement is uncertain by just:

$$\Delta x=0.000000000001$$

That's an incredibly tiny error.

In a simple two-body orbit, that tiny error generally stays relatively well behaved. Your predicted orbit might be slightly wrong, but it remains recognisably the same orbit.

In a chaotic three-body system, however, the error can grow exponentially.

Very roughly:

Δ(t)≈Δ(0)eλt

where λ is related to the system's Lyapunov exponent.

So you might get something like:

10−12→10−10→10−8→10−6→10−2→1

over successive periods of time.

Eventually, two simulations that started with essentially indistinguishable initial conditions can predict completely different positions.

But where does the initial uncertainty come from?

In the real universe, we can never know the state of a system with literally infinite precision.

There are several sources:

1. Measurement limitations

You can't measure an object's position and velocity with infinite precision. Telescopes, clocks, sensors, etc. have finite accuracy.

2. Quantum mechanics

At a fundamental level, particles don't have perfectly definite classical positions and momenta simultaneously. For macroscopic celestial bodies this usually isn't the dominant practical limitation, but it establishes that "perfect initial conditions" aren't physically available.

3. Unmodelled influences

Suppose you're modelling three objects but there is actually a fourth object far away. Its gravitational influence might be tiny, but it isn't exactly zero.

Likewise, radiation pressure, other planets, asteroids, etc. can introduce small effects.

4. Numerical error

Even if you had perfect initial conditions, a computer represents numbers with finite precision and solves the differential equations approximately. Those tiny numerical errors can also be amplified by chaotic dynamics.

Here's the really interesting part

The system itself can be completely deterministic.

Imagine running the universe twice with exactly the same initial state.

You would get exactly the same future.

There's no randomness required.

The problem is that if you change the initial state by an unimaginably tiny amount:

same laws + almost identical starting state → potentially radically different future

That's what sensitive dependence on initial conditions means.

And this is what people are getting at when they say that the three-body problem is chaotic.

It doesn't mean "we don't know what will happen because nature is random."

It means:

The equations determine what will happen, but predicting it requires initial information of a precision that can become impossibly high as you look farther into the future.

That's one of the profound differences between determinism and predictability. A system can be completely deterministic while still being practically unpredictable.