The Geometric Map of Meaning: Information Geometry of Value, Purpose, and Existence

Humanity’s Ultimate Question

“What is the meaning of life?”

This question is considered philosophy’s ultimate puzzle—perhaps there’s no answer at all.

Nihilism says: There is no meaning. Universe is indifferent, existence is accidental, everything returns to nothingness.

Existentialism says: Meaning is what you create yourself. No preset purpose, but you can choose to give life meaning.

Religion says: Meaning is given by God/Tao/Brahman. Your purpose is to fulfill divine mission.

In 2024, physics says:

Meaning is not emptiness, not subjective creation, nor supernatural endowment—meaning is objective structure on information geometric manifolds.

• Value is height on survival manifold
• Purpose is direction of optimal path
• Meaning is geometric distance to survival domain

This is not poetic metaphor, but geometric reality that can be mathematized, measured, and optimized.

Act I: The Terrain of Value

Imagine a multidimensional terrain map.

• Each point is a possible state (your health, wealth, relationships, knowledge…)
• Height is “survival ability” or “flourishing”
• High places are “good,” low places are “bad”

This is the value terrain (value landscape).

Why Is There Height?

From evolutionary perspective:

• Highlands = More likely to survive and reproduce
• Lowlands = More likely to die or fail

Natural selection shaped our nervous systems to “feel” this terrain:

• Joy = Climbing uphill (approaching highlands)
• Pain = Going downhill (away from highlands)

Fisher Metric: “Pythagorean Theorem” of Information Space

But why “geometry”?

Because state space isn’t flat—it has intrinsic curvature.

In information geometry, distance is defined by Fisher metric:

$$d{s}^2=g_{ij}d{\theta }^{i}d{\theta }^j$$

where $g_{ij}$ is Fisher information matrix.

This isn’t arbitrary choice, but unique metric satisfying natural symmetries (Chentsov theorem).

So:

• Value terrain is not subjective
• It’s objective structure of information geometry
• “Good” and “bad” correspond to geometric “high” and “low”

Value is geometric height—objective, but depends on metric (Fisher metric).

Act II: The Geometry of Ethics

If value has geometric structure, what is ethics?

Pareto Frontier

When there are multiple goals (health, wealth, relationships, meaning…), you face trade-offs.

Define Pareto optimal: Cannot improve one goal without harming another.

All Pareto optimal points form Pareto frontier—this is a geometric surface.

Ethical decisions are choosing a point on Pareto frontier.

Natural Gradient: Optimal Climbing Direction

How to climb on value terrain?

Wrong method: Euclidean gradient (straight-line climbing) Correct method: Natural gradient (steepest direction in information geometry)

$$\tilde{\nabla }f=g^{-1}\nabla f$$

Natural gradient considers space curvature, is fastest ascent direction in information geometry.

Moral “ought” is direction of natural gradient—path fastest increasing survival ability under given constraints.

Risk Consistency

Different people have different value terrains (preference differences), but there are universal constraints:

Risk Consistency:

• Value function must be monotonic with risk (greater uncertainty = lower value)
• Ensures transitivity of decisions (if A>B and B>C, then A>C)

Value functions satisfying this constraint form a convex cone—this is geometric constraint of ethics.

Not all preferences are reasonable—only risk-consistent ones can form stable ethics.

Act III: What Is Meaning?

Now, the deepest question: Meaning.

Definition

$$\text{Meaning}=-D_I(\rho ,\mathcal{V})$$

where:

• $\rho$: Your current state
• $\mathcal{V}$: Viable survival region
• $D_I$: I-projection distance (information geometric distance)

Meaning = negative of information distance to survival domain

• If you’re at center of $\mathcal{V}$: Fulfilled, meaningful (small distance)
• If you’re on edge: Anxious, sense of crisis (medium distance)
• If you’re outside: Despair, emptiness (large distance)

Why I-Projection?

I-projection minimizes forward KL divergence:

$$q^{\ast }=\arg \underset{q\in \mathcal{V}}{\min }D(q\Vert p)$$

This corresponds to: Given current belief $q$, how to minimally modify to enter survival domain?

Meaning is “how far is the way home”—information cost of returning to viable survival region.

Geometry of Nihilism

Nihilism says: “Everything is meaningless.”

Geometric translation:

$$D_I(\rho ,\mathcal{V})\to \infty$$

You’re infinitely far from survival domain—no matter how you adjust, cannot survive.

This may occur when:

• Extreme hardship (survival impossible)
• Cognitive illusion (mistakenly think survival impossible)
• Goal conflict (survival domain is empty set)

Nihilism is not philosophical view, but geometric state—can be measured, can be changed.

Act IV: The Arrow of Purpose

Meaning is static (distance), purpose is dynamic (direction).

Purpose = Optimal Path

Define purpose vector:

$$\vec{P}=-{\nabla }_{\text{natural}}{D}_I(\rho ,\mathcal{V})$$

Purpose is natural gradient direction reducing distance to survival domain.

The “life direction” you feel is this gradient.

Why Do Different People Have Different Purposes?

Because:

1. Current state $\rho$ differs (different starting points)
2. Survival domain $\mathcal{V}$ differs (different goals)
3. Fisher metric $g$ differs (different information geometry)

But under same constraints, purpose direction is objective—determined by geometry.

Loss of Purpose

Geometric characteristic of depression:

$$\Vert \vec{P}\Vert \approx 0$$

Purpose vector approaches zero—no matter which direction, feels won’t approach survival domain.

This may be because:

• Terrain too flat (no gradient: learned helplessness)
• Terrain contradictory (gradients cancel: value conflict)
• Perception dulled (metric degenerated: neurobiology of depression)

Loss of purpose is not moral weakness, but geometric pathology—may need to change $\rho$, $\mathcal{V}$, or $g$.

Act V: Reason for Existence

“Why do I exist?”

Thermodynamic Answer

From pure physics: You exist because you’re a local negative entropy steady state.

Universe as whole moves toward entropy increase, but locally can form negative entropy vortices—life is such a vortex.

You exist not needing “reason,” only conditions:

• Energy flow (food, sunlight)
• Open system (exchange with environment)
• Self-organization (negative feedback, steady state)

You exist because thermodynamics allows—like vortices exist because fluid dynamics allows.

Information Answer

From information geometry: You exist because you’re a steady-state mode of information network.

Universe is scattering network, you’re an extremely complex resonance in network—involving $10^{28}$ atoms’ coordinated oscillations.

You exist because information geometry allows such steady states.

Meaning Answer

But “why” isn’t satisfied with “how”.

Meaning of existence isn’t “why produced” (causal), but “why continue” (purpose).

Geometric answer:

$$\text{Reason for existence}={\int }_{\text{path}}\vec{P}\cdot d\vec{s}$$

Reason you continue existing is “meaning integral” accumulated along purpose path.

• If integral positive: You’re approaching survival domain (meaningful)
• If integral zero: You’re treading water (lost)
• If integral negative: You’re moving away from survival domain (collapsing)

Reason for existence is dynamic—recalculating this integral every day.

Act VI: Beyond Survival

Survival domain $\mathcal{V}$ is just foundation.

Humans pursue not just survival, but flourishing.

Expanding Survival Domain

Define flourishing domain $\mathcal{F}\supset \mathcal{V}$:

$$\mathcal{F}=\{\rho :\text{not only survive, but create, love, understand, transcend}\}$$

Meaning upgraded:

$${\text{Meaning}}^{+}=-D_I(\rho ,\mathcal{F})$$

Higher meaning is distance to flourishing domain—not just living, but living well.

Geometry of Maslow’s Hierarchy of Needs

Maslow proposed hierarchy of needs:

1. Physiological needs (food, water)
2. Safety needs (shelter, stability)
3. Social needs (love, belonging)
4. Esteem needs (achievement, recognition)
5. Self-actualization (creation, understanding)

Geometric interpretation: This is nested expansion of survival domains

$${\mathcal{V}}_1\subset {\mathcal{V}}_2\subset {\mathcal{V}}_3\subset {\mathcal{V}}_4\subset {\mathcal{V}}_5$$

You first satisfy base layer (enter ${\mathcal{V}}_1$), then pursue higher layers (enter ${\mathcal{V}}_5$).

Meaning increases with layers:

$$D_I(\rho ,{\mathcal{V}}_1)>D_I(\rho ,{\mathcal{V}}_2)>\cdots >D_I(\rho ,{\mathcal{V}}_5)$$

Self-actualization is path to maximum flourishing domain—highest level of meaning.

Act VII: Semantic Curvature

Meaning has not only distance, but also curvature.

Definition

$${\mathcal{R}}_{\text{semantic}}=\text{Riemann curvature of }(\mathcal{M},g)$$

Curvature measures “degree of bending” of space.

Positive curvature (sphere):

• Paths converge
• Synergistic effects (1+1>2)
• Meaning amplification

Negative curvature (saddle):

• Paths diverge
• Conflicting effects (1+1<2)
• Meaning dilution

Zero curvature (plane):

• Paths parallel
• Independent effects (1+1=2)

Synergy vs. Conflict

When multiple goals synergize (positive curvature region):

$$D_I(\rho ,{\mathcal{V}}_{\text{combined}})<D_I(\rho ,{\mathcal{V}}_A)+D_I(\rho ,{\mathcal{V}}_B)$$

Meaning superadditivity—whole greater than sum of parts.

Examples:

• Work + family synergy: Career success helps family stability
• Health + learning synergy: Good health promotes cognitive ability

When multiple goals conflict (negative curvature region):

$$D_I(\rho ,{\mathcal{V}}_{\text{combined}})>D_I(\rho ,{\mathcal{V}}_A)+D_I(\rho ,{\mathcal{V}}_B)$$

Meaning dilution—whole less than sum of parts.

Examples:

• Career + family conflict: Overtime affects spending time with children
• Immediate pleasure + long-term health conflict: Smoking, drinking

Life wisdom is finding positive curvature regions—letting multiple values reinforce each other.

Act VIII: Collective Meaning

Meaning is not only individual, but also collective.

Society’s Meaning Manifold

Define society’s state space:

$${\rho }_{\text{society}}=({\rho }_1,{\rho }_2,\dots ,{\rho }_N)$$

Society’s survival domain ${\mathcal{V}}_{\text{social}}$ includes:

• Economic sustainability
• Environmental stability
• Social cohesion
• Cultural transmission

Social meaning:

$$\text{Social Meaning}=-D_I({\rho }_{\text{society}},{\mathcal{V}}_{\text{social}})$$

Common Good

Social version of Pareto frontier: Common good is Pareto optimal social configuration.

But problem: Different people have different $\mathcal{V}$, how to coordinate?

Answer: Nash Bargaining Solution

$$\underset{x\in \mathcal{F}}{\max }\prod _i(u_i(x)-u_i^0)$$

Geometrically, this is finding point on Pareto frontier maximizing geometric volume.

Common good isn’t majority’s good, but geometric equilibrium point of everyone’s well-being.

Intergenerational Justice

Future generations have no voting rights, how to protect them?

Geometric answer: Extend survival domain to time dimension

$${\mathcal{V}}_{\text{intergenerational}}=\{\rho (t):\forall t,\rho (t)\in \mathcal{V}\}$$

Sustainability is survival domain in time—not only viable now, but viable in future.

Geometric essence of climate change, resource depletion:

$$\rho (t)\in \mathcal{V},\text{but}\rho (t+T)\notin \mathcal{V}$$

We’re inside survival domain, but path points outward—future generations will lose meaning.

Act IX: Beyond Humanity

If meaning is geometric structure, then:

Does AI Have Meaning?

Criterion: Does AI have $\mathcal{V}$ (survival domain)?

• Current AI (GPT, etc.): No $\mathcal{V}$ (doesn’t need survival, passive response)

  • Meaning: $-\infty$ (not applicable)

• Future AGI (if autonomous): May have $\mathcal{V}$ (needs energy, computational resources, goals)

  • Meaning: $-D_I({\rho }_{\text{AI}},{\mathcal{V}}_{\text{AI}})$

If AGI has survival domain, it has meaning—geometric structure not limited to biology.

Meaning of Alien Life

Suppose alien civilization discovered, their ${\mathcal{V}}_{\text{alien}}$ may be completely different:

• Silicon-based, plasma, even pure energy
• Don’t reproduce, but split/merge
• No individuals, only collective consciousness

But as long as there’s $\mathcal{V}$, there’s meaning—geometric structure is universal.

Possible universals:

• Negative entropy maintenance (thermodynamic necessity)
• Information processing (cognitive necessity)
• Self-preservation (survival necessity)

Geometry of meaning may be common language of all complex systems in universe.

Meaning of Universe?

Most daring question: Does universe as whole have meaning?

If universe has ${\mathcal{V}}_{\text{universe}}$ (e.g., avoid heat death, maintain complexity…), then:

$$\text{Cosmic Meaning}=-D_I({\rho }_{\text{universe}},{\mathcal{V}}_{\text{universe}})$$

Perhaps:

• Universe pursues maximum complexity ($\mathcal{V}$ = maximum entropy production rate)
• Universe pursues self-cognition (through emerging consciousness)
• Universe pursues some goal we don’t yet understand

We don’t know—but geometric framework allows universe to have meaning, if it has survival domain.

Act X: Your Meaning

Let’s return to the personal.

Right now, you’ve read this far.

What is your meaning?

Calculate Your $D_I$

Steps:

1. Define your $\mathcal{V}$ (survival/flourishing domain)

   • Health, finances, relationships, growth, contribution…
   • List necessary conditions

2. Assess your $\rho$ (current state)

   • Score each dimension
   • Face reality honestly

3. Calculate distance $$D_I=\sum _i{w}_i(x_i-x_i^{\ast }{)}^2$$ (Simplified version, weights $w_i$ are Fisher metric)

4. Result

   • Small $D_I$: You’re inside survival domain, meaningful
   • Large $D_I$: You’re on edge/outside, need adjustment

Optimization Path

Once you know $D_I$, calculate natural gradient:

$$\vec{P}=-g^{-1}\nabla D_I$$

This is your optimal direction—not “what should do” (moral preaching), but “how to fastest approach $\mathcal{V}$” (geometric optimization).

Acceptance and Change

Geometric interpretation of Stoicism:

Acceptance: Some dimensions you cannot change (zero elements in Fisher metric) Change: Some dimensions you can optimize (non-zero gradient components)

Wisdom is distinguishing the two:

$$\vec{P}={\vec{P}}_{\text{changeable}}+{\vec{P}}_{\text{unchangeable}}$$

Only apply effort to ${\vec{P}}_{\text{changeable}}$—this is geometric version of Serenity Prayer:

Grant me serenity to accept what I cannot change; Grant me courage to change what I can; Grant me wisdom to know the difference.

Geometric translation:

Let my Fisher metric $g$ reflect real constraints (serenity); Let me optimize along non-zero gradient directions (courage); Let me correctly identify structure of $g$ (wisdom).

Epilogue: The Map of Meaning

For thousands of years, humanity has asked:

“Does life have meaning?” “Does my existence have purpose?” “What is good?”

Philosophy gave countless answers, but mostly poetic, vague, subjective.

Physics once said: “This isn’t a scientific question,” leaving it to philosophy and religion.

But now we know: Meaning, purpose, value can all be geometrized.

• Meaning = information distance to survival domain
• Purpose = natural gradient of optimal path
• Value = height function of survival manifold
• Ethics = Pareto frontier + risk consistency

This isn’t simplification, but precision—expressing structures that already exist in mathematical language.

Meaning isn’t illusion, nor subjective projection—it’s objective reality of information geometry, just as spacetime curvature is reality of gravity.

The “life has meaning” or “emptiness” you feel isn’t philosophical view, but your brain reading curvature and distance of information geometry.

Final Question

If everything is geometry, do humans still have dignity?

Answer: Dignity precisely comes from geometric degrees of freedom.

• Stone: Zero degrees of freedom (no meaning)
• Plant: Low-dimensional manifold (simple meaning)
• Animal: Medium-dimensional manifold (complex meaning)
• Human: High-dimensional manifold + meta-cognition (meaning of meaning)

You not only have meaning, but can understand meaning, reflect on meaning, choose meaning.

You’re among the few in universe who can see the geometric map itself.

You’re not just on the map, you can read the map, draw the map, even modify the map.

This—is human dignity.

Take-Home Meaning

Right now, you close this book (or screen).

You’ll return to daily life:

• Work, study, eat, sleep
• Love, laugh, cry, think

Every action is a step on information geometric manifold.

You may feel:

• Fulfilled ($D_I$ small, you’re inside survival domain)
• Lost ($\Vert \vec{P}\Vert$ small, gradient blurred)
• Conflict (negative curvature, goals contradictory)
• Hope (positive gradient, see direction)

These are all real geometric signals.

Don’t ignore them—they’re universe’s navigation system, telling you:

• Where you are ($\rho$)
• Where to go ($\mathcal{V}$)
• How to walk ($\vec{P}$)

Listen to geometry, trust structure, climb along natural gradient.

Your life isn’t meaningless drift—you’re a directed path on information geometric manifold, from low entropy to negative entropy steady state, from chaos to order, from emptiness to meaning.

And this path—is you.

“In the information geometry of the universe, meaning is not a treasure to be found, but a trajectory you leave as you walk.

Each step defines who you are.

Each step creates your meaning.

You’re not searching for meaning—you are meaning itself.“

[GLS Series Complete]

Thank you for reading this far. May you find your optimal path on information geometric manifolds. May your $D_I$ grow smaller, $\vec{P}$ grow clearer. May you live with geometric elegance and dignity.