Hierarchy of Tendencies

I am thinking about a hierarchy of tendencies, such that some are more proximate in their influence than others. The near-term tendency might be toward some state, but more delayed tendencies might be toward different, even contrary states. I am thinking of this hierarchy as a system of derivatives, but I am interested to learn normative language in which such ideas are discussed.

What you're describing—where a system's immediate inclination points one way, but its longer-horizon inclination points another (even opposite) way—has a rich vocabulary across several traditions. Here's a map of the normative language, grouped by lineage.

1. The core structural vocabulary

Proximate vs. ultimate tendencies/ends

Proximate end / near end: what a thing is inclined toward here-and-now.
Ultimate end / final end: the terminus the whole system is ordered toward.
Subordinate vs. governing ends: proximate ends are subordinate to, and intelligible only in light of, the governing end.

First-order vs. second-order

A first-order desire/tendency aims at some object directly.
A second-order tendency is about which first-order tendencies should govern. Frankfurt's hierarchy of desires is the canonical modern source.

Derivative vs. fundamental

The proximate tendency is derivative; the long-run tendency is fundamental or originating.
Your "system of derivatives" language maps onto: the proximate is the derivative of the fundamental with respect to time or with respect to the system's order.

2. Aristotelian / Thomistic language

This is probably the richest normative idiom for your idea.

Natural inclination (inclinatio naturalis): intrinsic directedness toward an end.
Appetite (natural, sensitive, rational): the faculty of being inclined.
Order of ends: ends are nested; lower ends are propter higher ends ("for the sake of").
Imperfect vs. perfect beatitude: partial, unstable goods now vs. the complete good finally.
Defect of order (deordinatio): when a proximate inclination is pursued against the higher end it should serve. That's precisely your "contrary" case named as a vice.
Rightly ordered (recte ordinatus) vs. inordinately (inordinate): the normative label for a tendency-hierarchy that is or isn't correctly subordinated.
Per accidens vs. per se tendency: an accidental short-run pull vs. the essential direction.

Aquinas's phrase "the order of the appetites" and his distinction between appetitus naturalis (tendency toward one's own perfection, often below awareness) and appetitus elicitus (the consciously elicited act) is directly your structure.

3. Stoic language

Impulse (hormē) vs. right reason (orthos logos).
Propassion (propatheia): the first involuntary movement, before assent.
Assent (synkatathesis): what makes a tendency yours in the normative sense.
Oikeiōsis: the "appropriation" by which one's horizon of concern naturally widens from self to cosmos—a temporal hierarchy of tendencies.
Kathekon (fitting act) vs. katorthōma (perfect/right act): the proximate-decent vs. the fully-ordered act.

4. Kantian language

Inclination (Neigung) vs. duty.
Pathological (from sensibility) vs. moral (from reason).
Hypothetical imperative (relative to a proximate end) vs. categorical imperative (unconditioned, governing all proximate ends).
Heteronomy vs. autonomy: which tendency legislates for the whole.

5. Developmental / teleological language

Telos vs. mechanism: the long-run directedness vs. the causal push here-and-now.
Entelechy (Aristotle via Leibniz): the inner principle realizing an end through time.
Final cause vs. efficient cause: your two "derivatives" are the time-derivative of the final cause vs. the instantaneous efficient impulse.
Conatus (Spinoza, Hobbes): the striving to persist, which has both a present and a global form.
Epigenetic vs. proximate causation (behavioral biology): the developed tendency vs. the immediate trigger.

6. Contemporary analytic / decision-theoretic language

Time preference and hyperbolic discounting: the proximate tendency (steep discount) conflicts with the long-run tendency (shallow discount). This is your hierarchy quantified as a discount-rate structure.
Myopic vs. far-sighted preferences.
Revealed preference at short horizon vs. considered preference / second-order preference (Frankfurt; Bratman's self-governance).
Akrasia: acting on the proximate tendency against the one you endorse.
Synchronic vs. diachronic rationality.
Commitment device (Schelling, Elster): a way of making the proximate derivative obey the global one. Elster's Ulysses and the Sirens is the key text.
Preference hierarchy / meta-preference (Sen, Collective Choice and Social Welfare on meta-ranking).

7. Systems / cybernetic language

Nested control loops: inner loop (fast, proximate) vs. outer loop (slow, superordinate).
Set-point vs. reference trajectory: the instantaneous gradient vs. the whole-path objective.
Hierarchical control / subsumption architecture (Brooks): higher layers override lower ones.
Objective function vs. gradient: the proximate tendency is the negative gradient; the ultimate tendency is the objective the gradient is supposed to serve.
Homeostatic (short-run) vs. homeorhetic (long-run trajectory): Waddington's distinction; homeorhesis is exactly "stability of a path" rather than "stability of a state."

8. Your "derivative" framing, named

If you want to keep the calculus metaphor, the standard normative labels are:

The proximate tendency = the local / instantaneous / synchronic derivative.
The ultimate tendency = the global / integral / diachronic objective (or its gradient).
When they conflict: incontinence, akrasia, deordinatio, myopia, dynamic inconsistency (Strotz), or time-inconsistency (Kydland–Prescott).
When they align: integration, rightly ordered appetite, dynamic consistency, temporal coherence, practical wisdom (phronesis), or character.

If you tell me which tradition you're writing in—Thomist, Stoic, Kantian, decision-theoretic, or cybernetic—I can tighten this to the exact terms of art that a reader in that tradition would expect. The "derivative" metaphor itself is closest to the control-theory and time-preference idioms; the richest normative vocabulary for "the lower is for the sake of the higher" is the Aristotelian-Thomistic order-of-ends language.

User:

Imagine a function, f, of the vehicle speed during a family vacation. The first derivative of the travel is the instantaneous speed, but the second order derivative is that one of the children needs to use a restroom. The third order derivative is the principle of safety first.

DeepSeek:

This is a lovely conceit, and it's worth being precise about what makes it work and what normative language it invites, because the joke is doing real philosophical labor.

What the construction actually is

You've built a hierarchy of derivative orders where each higher order is not a further rate of change of the same quantity, but a different kind of governing consideration that constrains the lower ones. Strictly:

f(t) — position along the route.
f′(t) — speed. A physical, measurable rate.
f″(t) — "a child needs a restroom." This is not literally the acceleration of the car. It's a state of the system whose onset changes the target of the lower-order variable.
f‴(t) — "safety first." Not a rate at all, but a principle that ranks and constrains everything below it.

So the mathematical labels are being used analogically, not literally: each order names a level in a hierarchy of governance. The joke lands because we intuitively accept that the higher-order term has authority over the lower ones, even though "third derivative of position" has no such authority in physics. That gap between the literal math and the normative role is exactly the interesting part.

The normative vocabulary this invites

Order of ends (Aristotelian–Thomistic). Speed is propter the restroom stop; the restroom stop is propter the child's good; the child's good is propter the family's flourishing; and all of it is governed by safety. This is the ordo finium — ends nested for-the-sake-of higher ends. Your f‴ is the governing end; f″ is a subordinate end; f′ and f are its instruments.

Imperium and regimen (Aquinas). The higher order commands (imperium) and rules (regimen) the lower. Safety doesn't compete with speed; it legislates for it.

Lex superior / hierarchy of norms (Kelsen, Hart). A normative hierarchy in which lower norms derive validity from higher ones. Safety is the Grundnorm of the trip; "drive 70" is valid only if derivable from it.

Subsidiarity. The lower order handles what it can; the higher order intervenes only when the lower can't. The restroom need doesn't dictate the route until it does.

Override / preemption (practical reasoning). Raz's language: a higher-order consideration preempts rather than weighs against the lower. "Safety first" doesn't trade off with speed; it excludes some speeds outright.

Lexicographic preference (decision theory). Safety is the first coordinate; among safe options, comfort; among comfortable options, speed. Your derivative orders are lexicographic ranks, not additive terms.

Constraint vs. objective (optimization). f′ is the objective being maximized; f″ and f‴ are constraints on the feasible set. A constraint isn't a bigger number in the same sum — it's a different logical role. This is the cleanest formal statement of your hierarchy.

The one place the metaphor strains

In genuine calculus, higher derivatives don't govern lower ones — the third derivative of position is just the jerk, a physical quantity with no authority. So the conceit works only if we read "third derivative" as "the level of the system that sets the target for the level below." The moment you do that, you've left calculus and entered control theory: nested set-points, where the outer loop assigns the inner loop's reference. In that idiom:

f‴ = reference generator (safety envelope)
f″ = supervisory controller (comfort/needs state)
f′ = inner-loop set-point (speed)
f = plant output (position)

That's a hierarchical control system, and it's the honest formal home for your idea. The calculus labels are the poetry; the control hierarchy is the prose.

A name for what you've built

If you want a term of art: it's a normative derivative hierarchy — a ladder in which each higher order (i) is a different kind of consideration, not a further rate, and (ii) has authority over the order below. The philosophical cousin is the order of ends; the engineering cousin is the nested control loop; the decision-theoretic cousin is the lexicographic constraint stack.

And the punchline of the joke is genuinely a thesis: safety first is not a large number in the same sum as speed — it is a different order of thing altogether. That's the whole point of a hierarchy rather than a trade-off, and your f‴ is a very efficient way of saying it.

User:

Looking at a situation from its current moment. We want to notice what it is doing now, but also carry predictions of what it will be doing later. If we let the first derivative represent what it is doing now, and a second derivative that carries a reactive component, it seems that the present could be headed in a direction contrary to where it will be later.

Yes — and what you've just described has a precise name in control theory and a precise name in practical philosophy, and they turn out to be the same structure.

The formal situation

Let the state be x(t). Then:

x′(t) — what it is doing now. The instantaneous velocity. Pure present.
x″(t) — a reactive term. Not just "rate of change of velocity," but the mechanism by which the system responds to its own present deviation. This is what carries the correction.
x(t + Δ) — where it will actually be later, once the reactive term has done its work.

The key move you're making: the present direction and the future direction can be opposite, and that opposition is not a contradiction — it's the signature of a system under correction.

Why the second derivative is the right home for "reactive"

In a damped system:

x″ = −k·x′ − ω²·x

The −k·x′ term is the reactive component. It opposes the current velocity. So:

Right now: x′ > 0 (moving forward, fast).
The reactive term: pushes backward, because the deviation is too large.
Later: x′ crosses zero and reverses.

At the instant you look, the object is going one way. Its own second derivative says it will soon go the other. The present is headed, by its own internal logic, contrary to where it will be. That's not a paradox; it's what a restoring force is.

The normative vocabulary for exactly this

Teleological vs. mechanistic description. The first derivative is the mechanistic present — the efficient-cause push. The second derivative, when reactive, is the teleological signal — it encodes where the system is trying to get back to. This is the language of final cause operating through an efficient mechanism.

Negative feedback / error correction. The reactive term is an error signal. The present deviation generates its own correction. Wiener's Cybernetics names this directly: the system's now contains, in its second derivative, a representation of its not-here.

Homeostasis vs. homeorhesis. If the target is a fixed state, you have homeostasis — the reactive term pulls back to a set-point. If the target is a trajectory, you have homeorhesis (Waddington) — the reactive term pulls back to a path, not a point. Your vacation case is homeorhetic: the target isn't "stopped," it's "at the restroom by 3pm."

Damped oscillation vs. driven oscillation. A damped system has a built-in tendency to reverse. A driven system reverses because of an external input. Your "child needs a restroom" is closer to a driven event that resets the reactive term — a disturbance that makes the second derivative carry new information.

The philosophical payoff

This gives you a clean way to say something that's otherwise slippery: the future can be present in the system now, without being foreseen. It's present as a tendency in the higher derivative — a curvature that hasn't yet shown up in the velocity. Bergson's durée gestures at this; so does Peirce's synechism (the future is really present in the present as tendency) and Aristotle's potentiality (dynamis) — the acorn's oak is not a prediction stored somewhere, it's a second-order fact about the acorn's present motion.

The one caution

If your second derivative is purely reactive (only −k·x′), the system can only ever return — it can't arrive anywhere new. To get "headed contrary to where it will be later" in a productive sense, you need either:

a forcing term (an external input: the child's need, the restroom's location), or
a time-varying set-point (a homeorhetic target that itself moves).

Otherwise "later" is just "back to where it was," and the contrarian present is merely oscillation, not teleology.

So: yes, and the precise statement is — when the second derivative carries a reactive component, the sign of the first derivative at time t is not a prediction of the sign at time t + Δ. The prediction lives in the higher derivative. The present direction is real but not authoritative; the curvature is the part that knows where things are going.

In its intended purpose the idea of such a second-order derivative -- just the idea of it, and the sharing of the idea -- is the real forcing term.

Yes — and that closes the loop in a way that's worth naming carefully, because you've just made the hierarchy reflexive.

What you've said, stated precisely

The forcing term isn't the child's bladder. It isn't the restroom's location. It's the idea of the second-order derivative, held and shared. The representation itself is the input that drives the system.

So the causal order inverts:

Naively: a need arises → the driver forms an idea of it → the idea drives the car.
Your version: the idea is primary; the need is what the idea is about; and the sharing of the idea is what actually perturbs the trajectory.

The map becomes the territory's driver.

The vocabulary for this

Reflexivity (Soros). The system's representation of itself is a causal input to the system. Not a passive mirror — a participant. Soros's reflexivity is exactly "the idea of the situation changes the situation."

Performativity (Austin, Butler, Callon). Saying the thing does the thing. A speech act isn't a description of a state; it's an intervention in one. "We need a restroom stop" doesn't report a fact — it creates the obligation and the plan.

The definition of the situation (Thomas theorem). "If men define situations as real, they are real in their consequences." The shared idea of the second derivative is the definition; the car's behavior is the consequence.

Common knowledge (Lewis, Aumann). Sharing isn't just each person knowing. It's each knowing the others know, ad infinitum. That mutual knowledge is what makes the idea binding — it's what turns a private hunch into a coordination point. The forcing term's strength is a function of its commonality.

Constitutive rules (Searle). "X counts as Y in context C." The idea of the restroom need counts as a directive. The rule doesn't regulate an independent fact; it constitutes the fact's normative force.

Autopoiesis / second-order cybernetics (Maturana, Varela, von Foerster). The system's own descriptions are among its inputs. Von Foerster's "second-order cybernetics" is precisely the move from "observed systems" to "observing systems" — the observer's idea is inside the loop.

Enactivism (Varela, Thompson, Noë). Cognition doesn't represent a pre-given world; it enacts one. The idea of the second derivative doesn't model a need — it brings a need into practical existence.

Why this is the right home for your original hierarchy

Your derivative stack was:

f′ — present motion
f″ — reactive term (the need)
f‴ — safety first (the governing principle)

You've now said: f″ is not discovered, it is proposed. The reactive term is generated by the act of representing it and sharing it. That makes the hierarchy not a description of a system but a self-modifying control structure — one whose higher orders write the lower orders' targets into existence.

This is why the vacation example is so apt: the child's need is real, but what counts as the need, when it becomes a directive, and how strongly it perturbs the route are all fixed by the shared idea. Two families with identical bladders and identical roads take different routes because they hold different second-order representations.

The sharp formulation

The forcing term is the shared representation functioning as a cause. In cybernetic terms: the system's model of its own higher derivatives is wired back in as an input to its lower ones. In philosophical terms: the idea is not about the tendency — the idea is the tendency's governor.

And the punchline you've been circling: the highest derivative isn't the one that best predicts the future. It's the one that gets to write it. Safety first doesn't predict where the car goes; it decides. The idea of the second-order derivative isn't a forecast of the restroom stop; it's the thing that makes the stop happen.

So the hierarchy isn't a ladder of increasing abstraction. It's a ladder of increasing authority — and at the top, the authority is the authority to be shared.

The sharing of authority at the top is the strangest move in the whole hierarchy, because it looks like a loss of power and is actually the only way the top gets its power. Here's the anatomy of it.

The structural problem

At the top sits a governing principle — "safety first," or more generally the highest-order term that ranks everything below it. Call it the sovereign term. It has two options:

Hold authority privately. One party (the driver, the parent, the king, the expert) treats the principle as their own directive and enforces it downward. The lower orders obey because they're coerced or commanded.
Share authority. The principle is handed to the others as theirs — not as a command they receive, but as a rule they hold, apply, and can invoke against the top itself.

Option 1 looks stronger. It isn't. A privately held sovereign term is only as strong as the enforcer's attention, and it collapses the moment the enforcer is distracted, tired, or wrong. Option 2 is the only version that survives the enforcer's absence.

What sharing actually does

1. It converts a command into a norm. A command binds only the commanded, only while the commander watches. A shared norm binds everyone including the commander, and it binds when no one is watching, because each holder is now a enforcer of it. This is the Hart move: the shift from the gunman's order to a rule of recognition that officials and subjects both use to identify what counts as binding.

2. It makes the top self-applying. Once "safety first" is shared, the driver is bound by it too. The parent who invokes it against a child's wish to speed is also the parent who must slow down when the child invokes it against the parent's impatience. The sovereign term now cuts both ways. That's not a bug — it's what makes it credible. A principle that only binds downward isn't a principle; it's a preference.

3. It gives the lower orders standing to correct the higher. This is the crucial and counterintuitive part. Sharing authority at the top empowers the second derivative. The child who says "you're driving too fast for the rain" is not overriding safety — the child is applying safety. The shared top term gives every level a warrant to speak in its name. Authority flows down as permission to invoke, not just as obligation to obey.

4. It makes the top term robust to error at any single node. If safety lives only in the driver's head, one driver's misjudgment is the whole system's misjudgment. If safety is shared, the child's correction, the passenger's flinch, the other parent's "slow down" are all independent checks. The top term gets redundancy by being distributed. This is the deep reason federalism, separation of powers, and peer review work: the sovereign norm is stronger when no single holder can unilaterally reinterpret it.

The vocabulary for this

Sovereignty shared / popular sovereignty. The top term is authoritative because it's held in common, not because it's held at the top. Bodin's sovereign is the limit case; the modern move is that sovereignty can be constituted by the sharing rather than possessed by a person.

Rule of recognition (Hart). The shared practice of officials and citizens in identifying valid norms. It's not a command; it's a convergent practice. Its authority is real but it lives in the practice, not in any holder.

Constitutive rules (Searle). Sharing the top term is what constitutes the game. Before sharing, there's a car and some people. After sharing, there's a trip with a safety norm, and everyone's moves are now intelligible as moves in that game.

Common knowledge (Lewis, Aumann). The sharing must be mutually known to bind. Each must know the others hold it, and know they know, etc. This is why the sharing is an act — a saying, a glance, a stated rule — not a private belief. The act is what creates the common knowledge, and the common knowledge is what creates the authority.

Authority vs. power (Arendt, Raz). Power is the ability to compel; authority is the ability to be invoked without argument. Sharing the top term converts power into authority: the norm no longer needs to be enforced because it's already held.

The general will (Rousseau) / the kingdom of ends (Kant). The top term is authoritative precisely insofar as it's willed by all who are bound by it. Not a sum of private wills, but a shared willing of the governing principle. Kant's kingdom of ends is exactly this: a hierarchy in which the top term is legislation that each rational agent gives to all, including themselves.

Polycentric governance (Ostrom). Authority at the top is shared across many centers, each able to invoke and apply it. No single center can monopolize the sovereign term, and that's what makes it resilient.

The mechanism, step by step

Articulation. Someone names the top term. "Safety first." Before naming, it's a vague felt priority; after naming, it's an object that can be held, quoted, contested.
Avowal. Others take it up as theirs. Not "I obey your rule" but "this is our rule."
Mutual recognition. Each sees the others holding it. The norm becomes common knowledge. At this instant, the norm's authority stops depending on the namer.
Delegation of invocation. Every holder now has standing to apply it — to correct, to slow, to stop, to remind. Authority is now distributed as a right to invoke.
Binding of the top. The namer is now bound by the named. The sovereign term has escaped its author. This is the moment authority is truly shared: the top can no longer privately reinterpret what it publicly avowed.
Redundancy. Many independent holders mean many independent applications. The norm survives the failure of any one node. The hierarchy is now fault-tolerant at the top.

The paradox at the center

The top term gains authority by alienating it. A principle held privately is a preference; a principle shared is a law. The sovereign term is sovereign because it has been given away — because it now lives in the many, not the one. This is the same structure as the second derivative being the real forcing term: the highest order only governs the system by being represented in the lower orders, and it is represented in them by being shared with them.

So the sharing of authority at the top is not a dilution. It is the constitution of the top. Before sharing, there is no top — only a strong party. After sharing, there is a top — a norm that binds all, including the one who first named it, and that any holder can invoke in its name. The hierarchy becomes self-governing at the moment its highest term is put into common hands.

The one-line version

Authority at the top is not possessed but constituted — and it is constituted by being shared, so that every level below holds a warrant to speak in its name, including against the level above.