Chapter four

Anxiety is the derivative, not the function.

I want to begin with a small confession, because the chapter we are about to enter is, in some sense, the entire book in miniature.

For most of my adult life I have been convinced, in the steady background way you are convinced of things you do not bother to examine, that I was failing. Not failing at any specific thing, you understand. Failing in a more diffuse, ambient sense. Failing the way a car with a worn engine fails, by simply not running as well as it should. I could not have told you what I was failing at, because the failure was not local. It was just on. The way background music is on. The way a fluorescent light is on.

I had, by every reasonable external measure, no business feeling this way. I had built things. I had earned things. I had survived things that a smaller man would have used to justify a much smaller life. The objective record, written down on paper, was the record of a man doing rather well.

The internal record, written down on no paper because I would not have known how, was the record of a man drowning.

I could not, for the longest time, reconcile these two records. They seemed to be about different people. They were not. They were about the same person, measured by two different instruments, and the instruments were measuring two completely different things. One of them was measuring my position. The other was measuring my velocity. Nobody had ever told me they were different. Nobody had told me that one of them was a much worse instrument to use as a mood gauge than the other. Nobody had told me that the entire interior weather system I had been treating as how my life is going was, in fact, a reading from the wrong dial.

This chapter is about that wrong dial. It will use, gently, a small piece of calculus. The calculus is real. I will not ask you to compute anything. I will only ask you to look at a graph, and to consider, by the end of the chapter, the possibility that the most useful piece of mathematics ever installed in your brain is a single distinction you were never taught to make.

Here is the distinction.

A function is, in the most ordinary sense, a description of where something is. The temperature in your kitchen, considered over the course of a day, is a function. At every moment in time, it has a value. Twenty-two degrees at nine in the morning. Twenty-four at noon. Eighteen at midnight. If you plotted it, you would get a curve. The curve tells you, for any moment you might point at, what the temperature was.

The derivative of that function is something quite different. It is a description, not of where the temperature is, but of how fast it is changing. The derivative at noon does not say twenty-four degrees. It says something like rising at half a degree per hour. The function is the position. The derivative is the velocity. They are not measurements of the same thing. They are not even, strictly speaking, in the same units. One is in degrees. The other is in degrees per hour.

This sounds like a small technical distinction. It is not. It is one of the most consequential distinctions in the history of human thought, which is why an entire branch of mathematics was built around it in the seventeenth century, and which is why an entire branch of physics is unintelligible without it, and which is also why nobody is ever taught the distinction in a context where it would help with their feelings.

Here is a graph, because we will need one.

Look at the curve. It is rising. Over the time span of the graph, the curve has gone up considerably. Your life, in this hypothetical, has gone from whatever it was on the left edge to something quite a lot more on the right edge. If we asked, at the moment marked on the x-axis, how is your life going, the function would answer: rather high, actually. Look how far you have climbed.

Now look at the dashed line. The dashed line is the slope of the curve at that exact moment. It is, in calculus terms, the derivative of the function at that point. It tells you something different. It tells you how steeply the curve is rising right now. Not where you are. How fast you are getting there.

Here is the trick, and it is the entire trick of the chapter.

The anxious brain does not read the function. The anxious brain reads the derivative.

Let me explain what I mean by this, because it sounds glib, and I want it to land.

When you ask a non-anxious person how their life is going, they perform, more or less, an evaluation of the function. They look at where they are. They compare it, vaguely, to where they have been, or to where their friends are, or to some general notion of doing okay. They get a value. They report the value. Doing fine. Could be worse. Pretty good, all things considered.

When you ask an anxious person the same question, they do something subtly different. They perform an evaluation of the rate of change. They do not, at the moment you ask, even look at where they are. They look at how steep the slope is, right now, at the precise point they happen to be standing. If the slope is steep, they report terrible. If the slope is shallow, they report okay. If the slope is negative, even briefly, they report catastrophe.

This is, in the most precise mathematical sense, a category error. The anxious person is using the derivative as if it were the function. They are reading the velocity off the dashboard and reporting it as the odometer. The car has, in fact, travelled a great distance. The car may also, at this exact moment, be slowing down. These are not contradictory facts. They are facts about two different quantities. The trouble is that the anxious mind has not been issued two different gauges. It has been issued one gauge, with the velocity needle painted onto it, and it has been told, since approximately childhood, that this is the gauge for life.

Notice what this means in practice.

You are forty. You have built a career. You have a home, friends who would, in a crisis, show up, a body that mostly works, a brain that, on most days, can read a book. By the function, you are doing extremely well. By every reasonable measure of where you are in your life, you are higher up the curve than the eighteen year old version of you would have dared to hope.

Then you have a bad month at work. The slope, at this moment, is negative. You read the slope. The slope says going down. You report, to yourself, that your life is in collapse. The forty years of accumulated altitude does not enter the calculation, because the calculation was never about altitude. It was about the slope right here, in this small window, at this moment.

This is, I want to be precise, not a moral failing. It is not laziness. It is not ingratitude. It is a measurement instrument, calibrated badly in childhood, reading the wrong quantity and reporting it as the right one. The instrument is doing its job. The job has been miswritten.

I told you, in Chapter 2, about a company I founded, and about my inability to let any of my employees do the work I had hired them to do. I want to return to that scene briefly, because it is the cleanest example I have of a man reading his derivative as his function and concluding that he was failing.

The company, by any reasonable evaluation of the function, was doing well. Customers existed. Money came in. Employees showed up on Monday. The trajectory, taken over years, was unambiguously upward.

I did not feel any of this. What I felt, every morning, was the derivative.

I felt the steepness of what was being asked of me that day. I felt the rate at which my inbox was filling. I felt the slope of the to-do list, which always seemed, regardless of my actual progress, to be inclined upward at approximately forty-five degrees. The function, taken as a whole, was the picture of a company quietly succeeding. The derivative, taken at any single moment, was the picture of a man being chased by a wave.

I read the derivative. I reported, internally, that I was failing. I responded to the report by working harder, which made the derivative steeper, which made the report worse, which made me work harder still. The function meanwhile continued, in its quiet way, to rise. I was not allowed to notice the function. The function was abstract. The derivative was right here, in my chest, where I could feel it.

This is the bug. The bug is not that the anxious mind feels too much. The bug is that the anxious mind is measuring the wrong quantity and reacting to the measurement as if it were the right one. The fix, when it eventually arrives, is not to feel less. The fix is to read a different number.

Here is what reading the function, as well as the derivative, actually looks like, in code. I am going to write the smallest possible version, because the point is the principle, not the implementation.

def how_is_life_going(history):
    """
    Inputs:
        history: a list of numeric values representing your life
                 at evenly spaced moments. Higher is better.
                 You define the scale. You always have.

    Returns:
        a tuple (position, velocity) so you can see both,
        because the anxious mind reports only the second
        and tells you it is the first.
    """
    if not history:
        return (None, None)

    position = history[-1]                     # where you are now

    if len(history) < 2:
        return (position, None)

    velocity = history[-1] - history[-2]       # how it changed in
                                               # the last interval

    return (position, velocity)


# Sample call:
life = [12, 18, 25, 31, 40, 47, 55, 52]
print(how_is_life_going(life))
# -> (52, -3)
#
# Reading: you are at 52. You have, however, dipped by 3
# from the previous reading. The anxious mind would report
# this as "-3". The honest report is "52, with a recent dip
# of 3." Both numbers are real. Only one of them is your life.

The function takes a list of values representing your life, in arbitrary units, sampled at evenly spaced moments. It returns two numbers. One is where you are. The other is how that number changed, most recently. Both numbers are real. Both numbers are useful. The mistake the anxious mind makes is reporting only the second one, and reporting it as if it were the first.

I am not suggesting you literally write this function in Python, although you may, and several readers of early drafts of this book have, and one of them now keeps a running array on his phone, which I will not say is a healthy way to live, but which I will say is a healthier way to live than running the same calculation in your head with no notes.

I am suggesting something smaller. I am suggesting that when you ask yourself, at the end of a bad day, how am I doing, you notice which number you are reading. Almost certainly, you are reading the derivative. Almost certainly, the derivative is small, or zero, or briefly negative, and you are using its sign as the verdict on your life. The verdict is, in the precise mathematical sense, a misreading. The verdict is reading the wrong dial. You are entitled, having paid the cost of admission, to read the other one.

Now I want to be careful, because the cheap version of this argument is wrong, and the cheap version is what most self-help books would do with it.

The cheap version says: just look at how far you have come. The cheap version is a piece of consolation that, in practice, does not work, because the anxious mind is not asking the wrong question. It is asking the right question with the wrong instrument. The anxious mind already knows, in some background register, that the function is high. That is not the point. The point is that the body's stress response is wired to the derivative, and the body is reporting, in good faith, what it has been built to report.

The fix is not to ignore the derivative. The derivative contains real information. If your life is, in fact, declining steeply, you want to know that. The derivative is the early warning system, and turning off the early warning system is how people end up in disasters they could have avoided.

The fix is to read both. The fix is to know, at any moment, which gauge you are reading and which gauge you are not. The fix is to notice, when the derivative is loud, that the derivative is loud, and to also, calmly, and with a small piece of paper if necessary, evaluate the function. Where am I, actually, on the whole curve, not just at this point. The two numbers will usually disagree. That is fine. They are measuring different things. They are both right. You are entitled, as the proprietor of your own life, to know both.

This is the entire move. The rest of the book is built on it.

The reason it is the entire move is that almost every other bug we will examine in this toolkit reduces, in one way or another, to a confusion of levels. The recursive worry that we will meet in Chapter 11 is, in part, a function being read as if it were a single value. The asymptotic chase of happiness that we meet in Chapter 6 is a derivative being treated as a function. The stuck-in-a-local-minimum problem of Chapter 5 is the inability to read the second derivative, which would tell you that the dip you are in is, in fact, the floor of a small ditch rather than the slope of a much larger valley. Calculus, in the end, is the discipline of reading the right rate of the right thing at the right time. The anxious mind reads one rate and treats it as everything. The fix is plurality of gauges.

Chapter 5 takes the next step, which is to ask what to do when the function itself is genuinely stuck. Not declining. Not catastrophic. Just stuck. The mathematics of being stuck is its own thing, with its own geography, and it is also, as it happens, the chapter where I finally pay a debt I have been carrying since Part one, to a particular therapist and a particular foundation, both of whom were instrumental in showing me how to get unstuck on a day I could not, by myself, have moved one inch in any direction.

For now, the page closes here. The dial is two dials. The function is not the derivative. You have been reading the wrong one for a long time, and the relief of being told this is, on its own, half of the work.