JAN24

SAT2026

The Ledger of Selves

Personal identity as a stochastic process:

\theta_t = \theta_{t-1} + \text{drift} + \text{jumps}

philosophyidentitystochastic-processespsychologychange

Are you the same person you were ten years ago?

I don't mean in the trivial biological sense — most of your cells have turned over, your neural connections have rewired, your microbiome has shuffled. I mean something more slippery: is there a you that persists through time, some thread that connects the twenty-year-old and the thirty-year-old and the fifty-year-old into a single coherent entity?

The question has haunted philosophers for millennia. I want to offer a different framing — not to resolve it, but to make it more useful. What if "you" isn't a fixed point, but a stochastic process? A distribution over possible selves that evolves, drifts, occasionally jumps? The question then becomes not "are you the same person?" but "what's the transition kernel?"

The Ship of Theseus, Revisited

The ancient paradox: if you replace every plank of a ship over time, is it still the same ship? And if you keep the old planks and build a new ship from them, which is the original?

Applied to persons, the paradox sharpens. Derek Parfit — the philosopher who thought most rigorously about this — concluded that personal identity is not what matters. What matters is psychological continuity: the chain of memories, personality traits, intentions, and beliefs that link your past selves to your present self. The chain can be thin, can have gaps, can even branch (in thought experiments about teleportation or brain splitting). But as long as some thread persists, you have enough.

This is liberating. It means you don't have to defend some mystical essence called "you." You just need continuity of the right sort — and continuity admits of degrees.

But I think we can push further. Parfit's account is qualitative. What if we made it quantitative?

The Self as State Vector

Model a person at time $t$ as a high-dimensional vector $\theta_t \in \mathbb{R}^d$ : beliefs, values, personality traits, memories, skills, habits, attachments. The dimensions are fuzzy — you can't literally measure "conscientiousness" to twelve decimal places — but the metaphor is useful.

Your trajectory through life is a path through this space:

\theta_0 \to \theta_1 \to \theta_2 \to \cdots \to \theta_T

The question "are you the same person?" becomes: how do we relate $\theta_0$ to $\theta_T$ ? Is there a conserved quantity? A persistent mode?

Drift: The Slow Changes

Most change is gradual. You read books, have conversations, make mistakes, adjust. Each experience nudges you slightly in some direction:

\theta_{t+1} = \theta_t + \mu(\theta_t, e_t) + \varepsilon_t

where $\mu(\cdot)$ is a drift function that depends on your current state and your environment $e_t$ , and $\varepsilon_t$ is noise — the unpredictable flutter of daily experience.

This is just a random walk with state-dependent drift. The same mathematics that describes stock prices, particle motion, and epidemiological spread. Your "self" is diffusing through possibility space, pulled by forces you only partly understand.

The drift term is where most of life happens:

Learning. You encounter new ideas and your belief-vector updates. Bayesian inference, if you're doing it right.
Habituation. Repeated exposure dampens novelty. What once felt thrilling becomes routine. Your hedonic set-point drifts back.
Aging. The biological substrate shifts. Fluid intelligence declines; crystallized intelligence accumulates. Risk tolerance often drops.
Social shaping. You become like the people you spend time with. The environment pulls on the drift term.

Drift is gentle. It respects continuity. The you of tomorrow is recognizably descended from the you of today.

Jumps: Discontinuous Change

But not all change is smooth. Sometimes the process jumps:

\theta_{t+1} = \theta_t + J_t \cdot \mathbf{1}[\text{event}_t]

where $J_t$ is a large perturbation that occurs when some trigger fires. This is a jump-diffusion process — the same model used in finance to capture sudden crashes, in physics to model quantum transitions, in biology to describe punctuated equilibria.

Psychological jumps include:

Conversion experiences. Religious awakening, ideological shift, falling in love. The vector doesn't drift — it teleports.
Trauma. PTSD rewrites the fear circuitry. Grief reconfigures attachment. The person after is discontinuously different from the person before.
Revelation. A sudden insight that reorganizes your worldview. Reading Parfit on personal identity, for instance, might itself be a jump.
Psychedelics. The literature suggests they can catalyze rapid, persistent changes in personality — openness especially. A chemical jump.

Jumps are where Parfit's "thin psychological continuity" gets tested. After a sufficiently large jump, how much of the old person remains? The memory chain might persist, but if values, personality, and attachments have all shifted, in what sense is it the same person?

I don't think there's a clean answer. The question admits of degrees. And maybe that's fine.

The Narrative Self

Here's where it gets recursive: the story you tell about yourself shapes who you become.

Psychologists call this the "narrative identity" — the internalized, evolving story of the self that provides a sense of unity and purpose. You select from the flux of experience, weave a coherent plot, cast yourself as protagonist.

Mathematically, the narrative acts as a filter on the state space. Not all trajectories through $\mathbb{R}^d$ are equally available to you, because some would contradict the story you're telling. If you identify as "a person who doesn't quit," that constrains your transitions. If you identify as "fundamentally broken," that constrains them differently.

P(\theta_{t+1} \mid \theta_t, \text{narrative}) \neq P(\theta_{t+1} \mid \theta_t)

The narrative creates path dependence. Two people with identical state vectors but different self-stories will evolve differently. The story is part of the transition kernel.

This cuts both ways:

Generative narratives expand your option space. "I'm someone who's always learning" makes it easier to try new things.
Constraining narratives shrink it. "I'm not a math person" makes it harder to develop quantitative skills, even if you could.

Therapists spend a lot of time helping people rewrite their narratives. The new story doesn't change the past — the events are fixed — but it changes the meaning of the past, and therefore the future.

When to Let a Self Die

If identity is a process rather than an essence, then maintaining a particular identity has costs. The sunk cost of self-concept.

I've been thinking about this in the context of professional identity — the topic of a future post on careers. Your "self" includes "financial analyst" or "academic" or "startup founder." These labels constrain your transitions. They're useful for coordination and legibility, but they also trap you.

When should you let a self die? Some considerations:

When the drift has already carried you away. If you look at your current state vector and realize you're far from where the identity places you, maybe the identity is already false. You're just maintaining a fiction.
When the environment no longer supports it. Identities are ecological. If you built your identity around a skill that's become obsolete, or a community that's dispersed, the identity may be a drag. It's like carrying the planks of the old ship long after you've built a new one.
When it forecloses better futures. The narrative constrains the transition kernel. If "I'm not the kind of person who X" is blocking a transition you genuinely want, the narrative has become a cage.
When you've earned a jump. Conversion experiences can be manufactured — not reliably, but sometimes. Changing environments, trying psychedelics (legally, safely), starting therapy, making a dramatic life change. These are ways of inducing jumps when drift alone won't get you where you want to be.

The hard part is distinguishing "letting a self die" from "abandoning commitments you should keep." Parfit would say: don't privilege the person-stage that made the commitment over the person-stage evaluating it. Both are real. The question is what the chain of selves, taken together, reflectively endorses.

Implications

If I take this framing seriously, a few things follow:

Identity claims become probabilistic. "I am X" becomes "my distribution over possible selves is currently centered near X." This is more honest. It acknowledges that you might be wrong about yourself, that introspection is noisy, that the boundary is fuzzy.

Continuity is a matter of degree. There's no bright line between "same person" and "different person." There's just more or less overlap between $\theta_t$ and $\theta_{t'}$ , more or less connection between the narrative at $t$ and the narrative at $t'$ .

Change is not betrayal. The person you'll be in ten years is allowed to have different values, different beliefs, different priorities than you do now. That's not a failure of integrity — it's the natural evolution of a stochastic process. What matters is whether the transition kernel was chosen or merely suffered.

Strategic identity change becomes possible. If you understand your own drift term and jump triggers, you can steer. Not perfectly — you're not outside the process — but somewhat. Choose environments that pull in directions you endorse. Create conditions for beneficial jumps. Edit the narrative to expand option space.

Connections

This framing echoes ideas from several fields:

Behavioral economics. Thaler's "multiple selves" model treats the present self and future selves as different agents with conflicting interests. Commitment devices are ways of constraining the transition kernel.
Buddhist philosophy. The doctrine of anatta (non-self) denies a permanent, unchanging self. What exists is a stream of causally connected mental states — not so different from a stochastic process.
Process philosophy. Whitehead's ontology replaces static substances with dynamic processes. Reality is "becoming," not "being."
Control theory. If you model $\theta_t$ as a state vector and the narrative as a control input, the problem becomes: design a controller that steers the system toward desired regions of state space. Life as an optimal control problem, with noisy dynamics and uncertain objectives.
Life extension. In my post on hazard shaping, I framed longevity as steering a survival process. The parallel is exact: instead of steering $S(t)$ , you're steering $\theta_t$ . Both are about reshaping a process under constraints.

Coda: The Ledger

A ledger is a record of transactions. Each entry is a change — debit or credit — that transforms the balance.

Your self is a ledger. Each experience is an entry, modifying the running total. Some entries are small (a conversation, a book, a night's sleep). Some are large (a marriage, a death, a revelation). The ledger is the self — there's no separate account holder.

And here's the strange part: the ledger is also the accountant. The process is self-referential. The story you tell about the ledger becomes part of the ledger. The interpretation of past entries affects future entries.

This isn't a paradox to resolve. It's a structure to inhabit. You are the process, and you are also — partially, imperfectly — the hand that writes the process.

What kind of ledger do you want to keep?