Err, and err, and err again; but less, and less, and less.

As a child, I learned that atoms are tiny, indivisible particles.

This was useful, but wrong.

In middle school, I learned atoms are made of smaller parts: electrons zipping around a nucleus of protons and neutrons, like planets around a sun.

This was still wrong, but less wrong.

In high school, I learned electrons don’t move in tidy orbits, but buzz in probabilistic clouds, because something mumble mumble quantum states.

This was even less wrong (and if I actually understood it, that’d be even better).

Each step of the way, I held in mind a slightly false view, a useful fib that served the needs of the moment. As a math teacher, the process reminds me of a Taylor series, each successive term offering a better approximation of a transcendental function. But this kind of advancement is perhaps better captured in this verse by architect/mathematician/poet Piet Hien:

The road to wisdom? —Well, it’s plain
and simple to express:
and err
and err again
but less
and less
and less.

Sometimes I wonder: does the same go for moral progress? Do we get better and better and better, passing through various stages of wrongness on the road to moral truth?

We all know the MLK quote: “The arc of the moral universe is long, but it bends towards justice.” Isn’t there a whiff of calculus in the geometry of this image? Euphemia Lofton Haynes, the first African-American woman to earn a PhD in mathematics, made the connection explicit:

The concept of a limit is merely an expression [in] mathematical form of an ever receding goal of perfection for which man yearns and for which he strives, yet never attains. With each new approximation, he is merely closer…

I find the vision beautiful. I’m less sure about “true.”

It seems to me that history walks like one of those strange birds or lizards, zig-zagging back and forth. I can never tell which way it’s headed until it’s already gone there. And so, while the last few centuries have seen an unprecedented improvement of humanity’s material conditions, I’m wary of generalizing the pattern.

Still, even if a civilization does not progress this way, perhaps a single person can.

As William James argues in The Principles of Psychology (1890), humans are “gregarious animals,” always seeking the approval of others. Alas, this need for approval can lead us astray: “many a man truly great, many a woman truly fastidious… will take a deal of trouble to dazzle some insignificant cad whose personality they heartily despise.”

We struggle to tolerate censure, even the censure of idiots. Our social instrument is strung so tight, the least disturbance leaves us resonating for days.

But each of us also possesses a spiritual self: “a sort of innermost center within the circle, of sanctuary within the citadel.” These two selves present a sort of puzzle, a contradiction to reconcile. How can the gregarious animal escape its lust for status, stop dazzling the cads, and start cultivating its soul?

James presents a simple formula:

When for motives of honor and conscience I brave the condemnation of my own family, club, and ‘set’… I am always inwardly strengthened in my course… by the thought of other and better possible social judges than those whose verdict goes against me now… [T]he emotion that beckons me on is indisputably the pursuit of an ideal social self, of a self that is at least worthy of approving recognition by the highest possible judging companion…. This self is the true, the intimate, the ultimate, the permanent Me which I seek.

To rise above the petty judgments of our neighbors, we retreat into imagination. We envision better neighbors.

And then even better ones.

As our imaginary judges grow more just and discerning, our morality grows by corresponding steps.

“All progress in the social Self,” James says, “is the substitution of higher tribunals for lower.”

We grow by successive approximations.

Is moral progress like a mathematical limit?

17 thoughts on “Err, and err, and err again; but less, and less, and less.

  1. I don’t know if this is where Ben got it, but I found the quote in the Notices of the AMS, which in turn cited the Haynes-Lofton Family Pages held at the Catholic University of America. The quote can be found in the very final paragraph (on page 1002).

    Kelly, S., Shinner, C., Zoroufy, K. (2017) Euphemia Lofton Haynes: Bringing Education Closer to the “Goal of Perfection.” Notices of the AMS, October 2017, 995-1003.

  2. I’ve personally found Orthodox Christianity helpful in the kind of moral ascent Ben is illustrating. The Christian ideal is to not only share your french fries, but to forgive your enemies even when they are beating and torturing you. However, the interesting thing is that the higher you go, the lower you feel…The moment you think you’ve arrived at a superior moral point, you’re judging someone else for being lower than you are. Those who have truly arrived think that everyone else is better than them. “Most people are probably going to heaven, but I might not if I don’t repent”. It’s also interesting that it’s not possible to achieve this kind of morality without suffering. While from a utilitarian perspective, people in past centuries might have been more violent, from a virtue theory perspective, the average person was probably exerting a lot more self-control than people today. It’s a moral achievement to not beat your wife when you are starving, there are enemy marauders threatening to burn your village, you’ve lost three kids to childness illness, and you could die any moment of a fever. I’m not sure it’s much of a moral achievement to not beat your wife when you are well-fed, surrounded by comforts, likely to live long, and have access to fancy cars and toys. Similarly, you can’t know whether you would forgive your enemy until someone is treating you unfairly. Perhaps this is why many world religions recommend ascetic practices like fasting and vows to poverty for cultivating patience and morality. We should also keep this in mind when passing judgment on those in an out-group that is considerably less well-off than we are.

    1. “The moment you think you’ve arrived at a superior moral point, you’re judging someone else for being lower than you are.”

      You should be proud of your humility.

      1. If you’re interesting in thinking deeply about worldviews, my husband really likes Astralcodexten He says the people who write and comment there are all really thoughtful and the comments are remarkably civil for an internet forum. Have a great day!

        1. Whatever my other feelings on Astral Codex Ten, I think Scott Alexander (1) is a stupendous writer, and (2) has done great work cultivating a community in his comments section, which is hard, hard work (wrote the blogger in the midst of his own oft-neglected comments section).

  3. I like the sentiment and will surely ponder a few of the quotes and comments for a while. Thank you!

    But isn’t \sum_{n=1}^{\infty} 10^{-n} just “zero-point-one-recurring” and thus equal to 1/9?

    Am I missing a punchline here?

      1. Yes, the first number ever proved transcendental!

        As I recall the proof, it hinges on a funny question: when does an irrational number lend itself to a suspiciously good rational approximation?

        E.g., one nice approximation to sqrt(2) is 99/70. It’s almost suspiciously close; we’re only slicing the number line into 1/70ths, and yet we’re accurate to within 1/10,000th. It’s like we’re getting extra accuracy for free.

        But with algebraic (i.e., non-transcendental) numbers, there’s some bound on how often you get these suspiciously good approximations (and how close they can be). [if I had William Dunham’s “Calculus Gallery” I’d look up the details here…]

        But with Liouville’s number, you keep getting more and more suspiciously good approximations. The number takes the form 0.1100010000000000000000001… with longer and longer strings of zeros between each one. This means that, if you truncate the number after one of the 1’s, you’ve got a “suspiciously good” approximation, accurate to many additional orders of magnitude beyond what you’ve specified.

        (Just glanced at Wikipedia and it says all this more concisely, but what is my blog if not a rambling series of paraphrases from Wikipedia?)

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