“I Want” Is the Derivative of “I Am”
“I Am” is a position. “I want” is a velocity. Those are two sentences of analogy, not theorem — but hold them still and the calculus starts doing philosophy: desire becomes the rate of change of identity, every want splits into a part that grows you and a part that turns you, and the reason you cannot feel a want until something resists it falls out of a lesson this series already taught.
Contents
01 Two verbs
The oldest argument in Western philosophy is an argument between two verbs. Parmenides put everything on is: what is real is one, ungenerated, unchanging — motion a confusion of the senses. The tradition puts flows in the mouth of Heraclitus — the exact words panta rhei appear in no surviving fragment, but the river sayings are his, and Plato already paraphrases the doctrine: everything gives way, nothing stays. Being versus becoming. Position versus velocity. Twenty-five centuries later the argument is still running, mostly because both sides keep being right.
Here is the shape of a truce, in two sentences. “I Am” is a position. “I want” is a velocity.
Let me say plainly what kind of claim that is, because the numbered lessons in this series earn their keep by being literally true, and this essay does not get that privilege. The circle in Part One is really a circle; the momentum operator in Part Four really carries that i. A self is not literally a point in a state space, and nothing about persons will be proved below by i² = −1. This is an analogy — offered without apology, and held to the series’ standard for analogies: it has to predict something, it has to have a failure condition, and the people whose field is being borrowed from have to recognise it. The ledger near the end names exactly where it breaks.
One piece of housekeeping the philosophers will want up front. By “I Am” I mean qualitative character — who you are like — not the numerical persistence that occupies the personal-identity literature. Whether you are the same person who went to sleep last night, Locke’s question and Parfit’s question, is untouched here. The moving point models what you are like and how that changes. It says nothing about whether the point persists.
02 The derivative
Now take the two sentences seriously, and notice what the mathematics immediately does to them. A velocity is not a second object standing next to a position, the way a hat stands next to a coat. Velocity is defined from position: it is what the position is doing, extracted by a limit. Write the self as a moving point and the slogan sharpens itself into something stronger than it meant to be:
On this reading, wanting is not something a self has, the way it has memories or a coat. Wanting is the derivative of the self: who-you-are, differentiated. A self with no wants is not a self that happens to be missing an accessory; it is a self whose story has a horizontal tangent. And a want with no self to differentiate is not anything at all.
Philosophy got to the vector before I did, as it usually does. Anscombe’s Intention has the famous pair of lists — the shopper’s list says what the world should become, the detective’s record says what the world is — and when the shopping goes wrong the fault is in the world, not the list. The phrase for this, direction of fit, is Austin’s, from 1953; Mark Platts gave it its modern belief-and-desire form. Belief points mind-to-world: the mind is to be corrected until it matches what is. Desire points world-to-mind: the world is to be moved toward where the mind aims. A thing whose entire nature is to point from where you stand toward where you would go — philosophy’s standard picture of desire was already a vector wearing prose. The scholastics even named the zero end of the scale: velleitas — Aquinas describes it as the incomplete act of will toward what one takes to be impossible — a wanting of magnitude too small to move anything. English kept the word. A velleity is a velocity that rounds to zero.
A self with no wants is not missing an accessory. It is a curve whose tangent has gone flat.
03 Growth and turning
Here is where the analogy starts paying rent, because the mathematics of a moving point is not a blank screen you can project anything onto. It has structure, and the structure makes distinctions the slogan did not know it was committed to.
Take any differentiable path x(t) away from the origin. How fast is its size changing? One line of calculus, the same line Part Six built its whole argument on:
Read it slowly. The rate of change of your magnitude depends only on the component of your velocity along your position. Whatever part of the want lies perpendicular to who you are contributes nothing to growth — and everything to turning. Every want, every single one, splits cleanly into exactly these two pieces:
Every want decomposes: a part that grows you, a part that turns you
- who you are
- what you want
- parallel part — growth
- perpendicular part — turning
The two pure cases are both lessons this series has already taught. A want aimed entirely along who you already are is the equation x′ = kx with k real, and Part Two showed what that is: the exponential, motion along a ray, e as where you land if you never turn. More of the same self, compounding. Never other — only more. (And k < 0 is decay along the same ray; the mathematics is indifferent to whether you call that humility or erosion.) A want held exactly perpendicular to who you are is Part Six’s theorem run in reverse: perpendicular at every moment means |x| never changes. Pure transformation — becoming someone else at constant size, never more, never less, only other. And the generic case, k = σ + iω, is neither and both:
Three destinies of the equation x′ = kx
- only growing
- only turning
- both at once — the spiral
This is the vocabulary the slogan bought without reading the fine print, and it is better than what it replaces. We usually sort wants by their objects — money, love, standing, rest. The decomposition sorts them by their geometry: how much of this want is amplification, and how much is transformation? The want for a second house points almost entirely along the ray. The want to finally learn to listen is nearly pure arc. Most real wants are spirals, and the useful question about a desire is not “what is it for?” but “what is its angle to who I already am?”
04 You cannot feel a velocity
Part Three is the keystone of this series, and its claim was physical, not poetic: the body’s one instrument reads force, force is a real multiple of position, and so you have never once felt your velocity — not on the smoothest flight at five hundred knots — only its changes. In Part Three that assignment of felt and unfelt was forced, because there was an instrument to consult. Here there is no instrument, so what follows is offered as a fit, not a proof. But watch how well it fits.
Carry the claim across the analogy: you do not feel your wanting. You feel your wanting being bent.
You have been up for an hour. Kettle, inbox, the ordinary liturgy. You reach for the coffee machine’s handle and it comes off in your hand, and the machine is dead — and now, for the first time all morning, you want coffee. Vividly. Almost angrily.
The want is an hour old. It steered every step you took around that kitchen. It became visible the moment the world pushed back.
Phenomenology noticed this long before I wrote it as a derivative. Heidegger built a method on it in Being and Time: equipment in smooth use disappears — the hammer is not an object of experience but a transparency you act through — and only when it breaks does it announce itself as a thing. Csikszentmihalyi’s flow research found that the signature of absorbed, unobstructed activity is precisely the disappearance of self-consciousness: desire running at full magnitude, felt not at all. Medicine had the aphorism first — the surgeon René Leriche defined health as life lived in the silence of the organs, and Canguilhem made the line famous. The philosophy of desire already distinguishes standing desires (you want your children to flourish even while asleep) from occurrent ones; the claim here is one notch stronger, and I will state it with its hedge attached: even an occurrent want, running unresisted, tends toward transparency. Tends. There are unresisted wants that stay vivid — savouring, appetite at a good table, the pleasure of anticipation — and they are this claim’s standing counterexamples, named here so that nobody thinks I have not met them.
But the tendency is real enough to invert how we usually talk. We say desire is what we feel most immediately. The mapping says nearly the opposite: what you feel is force — the want thwarted, the want redirected, the want colliding with another want. Desire in free flight is like velocity on the night flight: enormous, and silent. Which is why the people who know exactly what they want are so often the ones who have recently been refused it.
05 “I know,” a conjecture
Here is the rung I am least sure of, and I will mark it as conjecture all the way down. If “I Am” is the position and “I want” is the velocity, the ladder does not stop. What is the acceleration — the thing that bends wanting itself? A candidate: “I know.” Coming to know something is what redirects desire. Learning what the job actually is kills the want for it; learning what the person is actually like creates one. On this reading, knowledge is not a warehouse. It is the second derivative of the self.
That is not a new idea; it is one of the oldest on record, and honesty requires naming it. It is Socrates. Knowledge, he argues in the Protagoras, is not a slave to be dragged about by passion; no one does wrong willingly; what looks like weakness of will is mismeasurement — ignorance wearing desire’s clothes. Real knowing, for Socrates, necessarily moves the knower. The conjecture is Socratic intellectualism written as a second derivative. And its failure condition has been named for twenty-four centuries: akrasia — knowing the better and wanting the worse anyway. Aristotle spends Book VII of the Ethics resisting the Socratic denial of it — though it is worth knowing he ends closer to Socrates than his opening suggests, conceding by VII.3 that what gets overridden in the akratic moment is not knowledge in full possession of itself.
The deep opponent is Hume, and the fault line is the oldest in moral psychology: “Reason is, and ought only to be the slave of the passions, and can never pretend to any other office than to serve and obey them.” On the Humean theory of motivation — the modern statement is Michael Smith’s — belief alone moves nothing; cognition can steer existing desires toward their objects but can never originate motion. Be precise about how strong the conjecture is: the moderate anti-Humean claims that cognition can motivate. Socrates — and this conjecture — claim that real knowledge must. That is the far end of the anti-Humean spectrum, and depression is its standing counterexample: knowing everything you knew last month, and wanting none of it. The mapping does not settle this fight. What it does — and this is the honest whole of the claim — is locate it: whether “I know” can be the acceleration of “I Am” is precisely the Humean question, restated in one derivative.
The ladder does offer the conjecture one strange gift. In Part Three the derivatives alternate axes: position on the felt rung, velocity on the transparent one, acceleration back on the felt rung. Carry that pattern here and it predicts: being yourself is an experience, wanting is transparent, and being changed is an experience again. That matches something real. Realization has a phenomenology — the moment of coming-to-know lands with a felt jolt that mere wanting never has. And the acceleration slot has a second claimant with a better legal team: Frankfurt located personhood not in what we want but in our second-order volitions — wanting which wants should move us. That is a structural rhyme with the second derivative, not an identification: Frankfurt’s hierarchy is a relation of aboutness, and a derivative is not about anything — a wanton whose desires swing wildly has a large second derivative and no second-order volitions at all. But notice where both claimants live. The knower and the chooser — the two things philosophy has most wanted the self to be — are both bidding for the same rung, and it is the rung the ladder hands back to the felt axis.
The ladder, carried across
06 The burning bush
A word for the theologians, who saw the title coming. The most famous “I Am” in history is the answer from the burning bush, and in the Hebrew it is not grammatically a position. Ehyeh asher ehyeh — the verb is first-person imperfect, the conjugation of incomplete action, open toward the future. “I will be what I will be” stands in the margins of the major translations; the NRSV footnotes it, and the JPS declines to translate the phrase at all. It was the Greek translators of the Septuagint who rendered it with the participle of pure being — egō eimi ho ōn, the One Who Is — and the later Jewish revisers Aquila and Theodotion who pushed back with esomai hos esomai: I will be who I will be.
Be careful with the inference, because a famous mistake lives here. Reading Hebrew verb morphology as a metaphysics of becoming — dynamic Hebrew versus static Greek — is exactly the move James Barr spent The Semantics of Biblical Language demolishing: the imperfect is aspect and mood, not ontology, and no grammar was ever a theology. What can be said with a straight face is smaller and still remarkable: the sentence’s grammar is open toward the future, and the translation tradition has been negotiating between position and velocity — between ho ōn and esomai — for twenty-two centuries. The oldest name in the book sits exactly on this essay’s axis.
Classical theism took the Greek fork to its limit. Aquinas’s God is actus purus — pure actuality, nothing potential, nothing left to actualize — and since Plato’s Symposium had already taught that desire is of what one lacks, a perfect being has nothing left to want. In this essay’s vocabulary that sounds like a self with zero derivative, and the Fifth Exercise asks you to argue it. Fair warning before you start: the Thomist’s best reply is not that God’s velocity is zero. It is that God is not in the state space at all — not a stationary point but not a point, with Boethius’ definition of eternity ready to hand: the whole, simultaneous, and perfect possession of interminable life. Life, note. Not stillness. The doctrine’s claim is that pure act out-lives motion, and you have not argued the other side until you have argued that.
Where does this leave the two sentences we started with? Better than they started. Parmenides and Heraclitus each kept half the calculus: a position with no derivative, a flux with no thing flowing. The derivative was the truce all along — x and x′ are not rivals but one object read twice. The claim of this series is that mathematical structure is why the world is as it is. The claim of this essay is smaller, and I want it exactly as small as it is: the structure is, at minimum, a language precise enough that the old questions about selves stop being gestures and become addresses. Desire is the derivative of identity. Growth and transformation are its orthogonal components. The felt life alternates rungs. Where the language stops working, the ledger below says so out loud.
07 Where this stops being true
- What is load-bearing
- Every mathematical statement here is a theorem and computed: d|x|/dt = (x·x′)/|x|; perpendicularity at every moment is equivalent to constant magnitude (Part Six); x′ = kx yields the ray, the circle, or the spiral as k is real, imaginary, or complex. The figures’ numbers are calculated, not drawn. And the philosophical scaffolding is real scholarship, cited: direction of fit, the Humean theory of motivation, Frankfurt’s hierarchy, the grammar of Exodus 3:14.
- What is convention
- Modelling a self as a point in a state space at all; reading |x| as “how much self”; and the assignment of felt and transparent rungs. In Part Three that assignment was forced by an instrument — the body reads force. No instrument forces it here. I borrowed the assignment because it fits the phenomenology; a critic who assigns differently has not made an error, only a different analogy.
- Where the shorthand breaks
- Wanting to remain who you are maps to zero velocity — so the mapping calls contentment and fidelity “not wanting,” which is false. To repair it you must promote conservative desire from a velocity to a restoring force, station-keeping against perturbation — and then the slogan is gone. Also: unresisted wants that stay vivid (savouring, appetite) bound the transparency claim; and “I know” as acceleration is the strong Socratic thesis, not the moderate anti-Humean one — akrasia and the listlessness of depression stand against it, unanswered here.
- Where I would push back on myself
- The self is not a point in a metric space. There is no addition of selves, no scalar multiplication, and no topology in which the limit defining x′ could converge — so “derivative” is here a disciplined metaphor, not an operation. Nothing about persons is proved by i² = −1. What the calculus supplies is a vocabulary precise enough to locate questions — growth versus turning, felt versus transparent, can-knowledge-move versus must-it — and locating a question is a real contribution and a much smaller one than solving it. If this essay is remembered as claiming more, I mis-wrote it.
08 Exercises
- Run the decomposition. You are at x = (3, 4) and you want x′ = (2, −1). Compute d|x|/dt and both components. (You should find growth of exactly 0.4 and turning of exactly 2.2 — a want that barely grows you and mostly makes you someone else. Name a real want with this shape.)
- Solve the no-turning life. Solve x′ = kx for real k > 0 and show the path never leaves the ray through x₀. Connect this to Part Two, then answer: what is the standard English word for a person whose every want points exactly along who they already are?
- Prove pure transformation. Show in one line, via d(x·x)/dt, that a want perpendicular to the self at every moment leaves |x| exactly constant. Then name one real change of life that this models: becoming different without becoming more.
- Stress the Frankfurt rhyme. Write down one of your own second-order desires — a want about your wants. The essay claims this rhymes with x″. Say precisely where the rhyme fails, using the wanton: wildly swinging first-order desires give a large second derivative with no second-order volition anywhere in sight.
- Argue the other side. Steelman actus purus: a perfect being has nothing left to actualize, hence — by the Symposium’s own logic — nothing left to want, and this is perfection, not poverty. Then meet the Thomist’s real move: God is not a stationary point in the state space but not in the state space, and Boethian eternity is the possession of life, not the absence of motion. Say what this costs the essay’s mapping — and what accepting it costs the Thomist who also wants to say God loves.
Sources
- Plato, Symposium 199c–201c; Protagoras 345d, 352b–358d; Meno 77b–78b.
- Aristotle, Nicomachean Ethics VII.2–3 (1145b21–27; 1147b14–17).
- Parmenides DK B8; Heraclitus DK B12, B49a, B91; Plato, Cratylus 402a.
- D. Hume, A Treatise of Human Nature (1739), 2.3.3, “Of the influencing motives of the will.”
- M. Smith, “The Humean Theory of Motivation,” Mind 96 (1987), 36–61; The Moral Problem (Blackwell, 1994).
- H. Frankfurt, “Freedom of the Will and the Concept of a Person,” Journal of Philosophy 68(1) (1971), 5–20.
- G. E. M. Anscombe, Intention (1957), §32; J. L. Austin, “How to Talk,” Proc. Aristotelian Soc. 53 (1953); M. Platts, Ways of Meaning (1979); I. L. Humberstone, “Direction of Fit,” Mind 101 (1992).
- M. Stocker, “Desiring the Bad,” Journal of Philosophy 76 (1979); T. Schroeder, Three Faces of Desire (Oxford, 2004).
- M. Heidegger, Being and Time (1927), §16.
- M. Csikszentmihalyi, Flow: The Psychology of Optimal Experience (Harper & Row, 1990).
- Aquinas, Summa Theologiae I q.3, q.9, q.10; I-II q.13 a.5 ad 1 (velleitas).
- Exodus 3:14 (MT; LXX; Aquila and Theodotion); J. Barr, The Semantics of Biblical Language (Oxford, 1961).
- Boethius, Consolation of Philosophy V.6.
- M. Tegmark, “The Mathematical Universe,” Found. Phys. 38, 101 (2008).
3 thoughts on ““I Want” Is the Derivative of “I Am””