Before we can speak about what exists between two things,
we must first establish that both remain real within the relation.
When I began to define health in formal terms, I recognized that any instrument we might eventually construct would inherit the underlying definition, embedding within each measurement our prior decisions regarding ontology, significance, and the physical relationships that enable a signal to become a meaningful claim. A thermometer is possible only after temperature has been defined; an electrocardiograph is possible only after electrical potential, time, conduction, and the body’s surface have been unified within a shared physical framework. By the time either device enters clinical practice, the number displayed encapsulates centuries of mathematics, experimentation, calibration, and consensus.
Health has emerged through a distinct historical trajectory, shaped by medicine’s growing capacity to identify disease, describe injury, classify impairment, and delineate thresholds beyond which the body enters danger. Tumors can be located, stenoses graded, infections identified, tissue destruction monitored, and probabilities of future events calculated. Within this context, health is often conceptualized as the interval preceding visible disease, the restoration of function following injury, or the segment of a population distribution in which individual measurements remain unremarkable.
I sought to determine what is affirmatively present when we describe a living system as healthy, what underlying structure the term seeks to denote, the conditions that enable this structure to persist, and what an instrument must observe to justify such a claim physically and mathematically. My aim was for the term health to be embedded within the measurement itself, allowing any claim to be traced through the mathematics that support it and back through each material event that made the claim possible.
This inquiry led us beneath the sensor, the signal, and the resulting numerical value, recognizing that an instrument may collect highly detailed data while the significance of that data remains contingent on the phenomenon it was designed to measure. Instruments can detect millisecond-level changes, identify patterns imperceptible to humans, compare individual bodies to large populations, and correlate outputs with diagnoses that may only become apparent years later. While these capabilities are valuable, measuring health demands a more foundational approach: a formal account of what health constitutes, which transformations preserve it, and which physical observations can substantiate its presence.
The pursuit of such an account begins with relationship, as the significance of a physiological quantity emerges through its interactions with other variables. For example, a heart rate of seventy-two beats per minute holds different implications for a sleeping adult, a frightened child, a trained athlete in recovery, or an individual experiencing circulatory insufficiency. Although the numerical value remains constant, its physiological meaning shifts according to factors such as respiration, blood pressure, temperature, posture, oxygen demand, recent activity, and the temporal context within the body.
Every familiar measurement reveals its complexity upon closer examination. A concentration is defined by the fluid, tissue, compartment, and time in which it is measured; a voltage is determined by the sites it spans and the path of charge movement; a pressure is characterized by the boundary containing it, the generating source, and the material’s capacity to yield. Each quantity acquires significance through the relationships that confer consequence, and health becomes formally accessible through the organization, responsiveness, and mutual availability of these relationships as the living system evolves.
A formal definition of health must therefore be constructed through variables of relatability—quantities whose significance arises in the transition between entities: from source to response, between distinct sites, from a current state to subsequent change, or from an internal process to the boundary that renders it observable. Temporal sequencing is essential because a response is contingent on its antecedent; support is necessary because each measured term must originate from a physically and mathematically substantiated source; distinction is required because two quantities can only be related if each maintains sufficient identity to contribute unique information; and continuity is vital because health pertains to what a living system can preserve as matter, energy, circumstance, and time progress.
I would like to share a recent example of what this looks like in formal methods, along with the reflection it prompted in me about our lived experience as human beings. Recently, I was working through a veritable tangle of mathematical nuance when I reached a section whose conclusion struck me as almost too small to justify the difficulty that had gathered around it.
The proof concerned a correction operation acting on formal words. I had already completed the zero-length case, in which the relevant word is empty, and was working through the next smallest case, where the original word contains a single occurrence. What I needed to establish was simple enough to state in one sentence: every correction output still contains at least two supported occurrences.
The construction introduces one distinguished occurrence while carrying another through the surrounding context. Because the word being acted upon is nonempty, both remain in the resulting expression. A derivative can remove one eligible occurrence, leaving at least one behind, so none of these outputs can contribute to the empty word. Once the presence of the two occurrences has been established, the conclusion follows as easily as elementary arithmetic:
2 supported occurrences − 1 removal = 1 surviving occurrence.
Yet the proof assistant cannot accept the two merely because I can see them or because their presence seems obvious from the way the construction was written. Each occurrence must be traced into the output; its support must remain intact; its identity must survive the surrounding transformations; and the two must remain distinguishable through every reduction permitted by the algebra. The conclusion becomes easy only after the proof has established that the world inside the expression really contains two valid things to count.
A relational measurement is never only a number. It is a claim about what happened between identifiable things, across an ordered route, under a particular interrogation, with a known boundary and a preserved history of composition.
As I worked through this, I began to think about how often our ordinary understanding rests on the same apparently childish certainty. We look at two people in a marriage and know there are two bodies, two names, two histories, and two lives gathered within the relationship, while the deeper health of that relationship depends on whether two living centers of care remain present within it. The mathematical problem had become unexpectedly recognizable: before we can speak about what exists between two things, we must first establish that both remain real within the relation.
Imagine, then, a marriage after a difficult year, when two people may still share a home, a history, and the practical structure of a life, while wondering what remains alive within the word we. They have passed through months of unfinished conversations and small withdrawals, through mornings when the day began before either found language for the night before, and through the accumulating fatigue of living beside an injury that keeps changing shape as each person carries it privately.
One evening, driving home after visiting friends, they reach the familiar turn toward their street and continue past it, following the road beyond the houses until the traffic thins and the lights of town begin to recede behind them. The person in the passenger seat has spent most of the drive watching fields disappear into the dark, while the driver holds the wheel with both hands and waits in the quiet. When the passenger finally says they want to find their way through this, the sentence enters the car carrying the strain of everything that made it difficult to speak.
That sentence echoes the newly inserted occurrence in the proof, an act brought into the shared space by one person, while the answer that follows another mile—an assurance that anger has not exhausted commitment—carries something older through the conflict. The second sentence has traveled from the life they shared before this year, through disappointment and self-protection, into the present moment, where it becomes available again as a continuing bond. One person has initiated a movement toward repair, and the other has transported their commitment across the rupture; together they seem to provide the two occurrences the relationship needs.
Their presence will be established as those sentences take shape beneath them as the road continues and as the days that follow gather around what was said. The invitation to repair gains support when the person who offered it remains available to hear what the year has felt like from the other seat, allowing memories to return with their different angles and permitting an answer to alter the story already assembled in solitude. The continuing commitment gains support when it enters the ordinary structure of their life, appearing in the conversation resumed the following morning, in a promise remembered later in the week, and in the willingness to approach again after the first effort produces more tenderness than resolution.
As each contribution develops its own history, the two must remain distinguishable, because one person’s invitation cannot supply the other’s willingness, and one person’s patience cannot serve as evidence that both have understood. An apology belongs to the person who offers it; forgiveness belongs to the person who may eventually give it; and the path between them is forged by the movement of two lives whose contributions meet without losing the origin from which each came.
A marriage may contain two voices, with one person providing the language, regulation, and forward movement carried by them both. The same person may recognize the distance, begin the conversation, soften the first answer, interpret the silence, propose the repair, and keep the passage open long enough for both people to move through it, until the visible exchange bears two faces while its active support continues to arise from one center.
The apparent count may still look like
1 + 1,
even while its interior structure has become something closer to
1 + echo(1).
The difference becomes visible whenever the person who usually carries the relationship grows tired enough to set the work down. Their silence creates an opening in which another source of care may begin to move: the other person may notice, approach, ask what has happened, and hold the continuity that had so often been held for them. What remains after one contribution is withdrawn reveals whether the second contribution had its own support all along.
Every lasting relationship passes through seasons when one person carries more, because illness, grief, work, children, fear, and exhaustion continually redistribute what each person can offer. Relational health lives within this changing asymmetry, the capacity to restore two living centers of care, allowing one person to hold the shared life while the other recovers and preserving a path for the one being carried to return with their own agency.
Over the years, the direction of care shifts back and forth between them. One person remembers the appointment, manages the finances, rises with the frightened child, or keeps conversation possible through a period of grief; then the structure turns, and the one who had been held becomes the one capable of holding. Each learns the weight of the other’s life by carrying it, and each learns the reality of their own place in the relationship by discovering that weariness does not make the shared world disappear.
The health of the marriage appears in this capacity to preserve their twoness through change, sustaining enough relationship for both people to act, receive, withdraw, return, and continue altering the life between them. Their joining gives rise to something larger than either life alone, while the vitality of that larger life depends on the continuing presence of the two distinct sources from which it is made.
The formal proof gives this familiar experience an unexpectedly precise shape. Before two occurrences can support a conclusion, each must be shown to exist; before their relationship can survive a transformation, each must remain present throughout; and before the removal of one can reveal what remains, the proof must establish that the second was more than a repetition generated by the first. The marriage allows us to feel why these obligations matter, while the algebra requires us to state them with enough precision that nothing passes through the argument merely because it looked obvious from a distance.
When the definition of health shifts from the algebra to an instrument, the same obligations enter physical matter, where the clarity of a symbol gives way to the dense simultaneity of the living world. The formal system knows where an occurrence came from because it was constructed there, whereas a sensor receives a mixture formed by the body, the applied source, the contact between instrument and skin, movement, environmental interference, amplifier behavior, clock timing, and every computational transformation through which the original event becomes a recorded value.
Two channels may produce two waveforms while carrying the same disturbance, just as two sentences in a marriage may draw their motion from one person. A response may seem to follow a source because both were shaped by the same reference, motion, or drift in timing; and a processing step may produce several visible outputs from one underlying event, multiplying the appearance of evidence while leaving its origin unchanged. The instrument must therefore establish the terms of the relationship before that relationship can support a statement about health.
This requires a physical lineage that extends through the entire measurement. The source must be delivered and independently witnessed; the response must arise from a known boundary; the timing between them must remain intact as the signal passes through amplification, conversion, and computation; and the identity of each term must carry forward into the final claim. When a witness disappears, the claim must weaken with it, preserving within the language of the result the exact limit that entered the physical record.
This is where the formal definition begins to shape the hardware itself, because a definition built on relatability asks the instrument to preserve relationships from the world in the measurement. It asks us to know which physical event produced a recorded value, which site received it, when it arrived, how it changed, what other event it remained related to, and whether that relationship survived the sequence by which matter became voltage, voltage became data, and data became a statement presented to another human being.
Sensitivity belongs within this chain, along with resolution, calibration, repeatability, and every other quality by which an instrument earns scientific trust. Yet the particular burden of measuring health extends into the relationships among these qualities, because a beautifully resolved signal still needs a valid origin, a precise timestamp still needs two events worth relating, and an elegant analysis still needs physical witnesses capable of supporting the terms that entered it. The instrument must carry the discipline of proof into the materials from which it is made.
This is why we undertook the formal work before asking the measurement to speak. We wanted health to have an object rigorous enough to govern the claims made in its name, a proof corpus capable of establishing what remains true as that object is transformed, and an engineering chain capable of preserving the required relationships from the living body into the record. The definition gives health a form that can be reasoned about; the proofs establish the conditions under which that form survives; and the instrument creates a physical passage through which those conditions can be observed.
What first appeared to be a problem of sensing therefore became a problem of relation all the way down, from the relations within a living system, through the relations represented in the mathematics, into the relations preserved by the instrument, and finally into the relationship between a measurement and the claim it authorizes. At every stage, the same care returns: establish what is present, preserve where it came from, keep distinct things distinct, and allow the final statement to carry only what the full chain can support.
The brief length-one proof contains this undertaking in miniature, beginning with two apparent occurrences and refusing to count them until each has been shown to be present, supported, distinguishable, and capable of surviving the permitted transformation. Its conclusion remains as simple as the couple driving beyond the familiar turn and discovering, somewhere along the dark road, that one person’s effort can cross into the shared life and meet another effort already moving toward it: there are still two, and the difficult work of measuring health begins with making that sentence true, from the abstraction, through the instrument, and into the living world.
The definition this reflection begins from is the subject of The boundary comes first; the proof corpus and the algebra whose closure laws force this discipline are described in A boundary-observable certification algebra; and the passage from the definition into a physical instrument is the subject of Reading the living boundary.