Can You Prove a Theory True?

The question is partly rhetorical. You’re supposed to get affronted and say, “No, of course you can’t. Everyone should know that. You can only prove a theory false.”

But to dig into the question properly, you have to define “theory” and consider the difference between closed and open systems. The word “theory” is used in many different senses; as Marcelo Gleiser points out, you must observe its context to understand its meaning.

A theory is more than a hypothesis, which in turn is more than a guess or one-time prediction. A hypothesis is a tentative explanation of a general phenomenon: for instance, “Rain occurs more frequently where land is cultivated.” This can later be developed into a theory (e.g., “Rain follows the plow“). Sometimes there is disagreement over whether something should be called a hypothesis or a theory–but the distinction remains.

A theory must be well substantiated, have broad application, and explain a general phenomenon or class of phenomena. When a theory takes the form of an apparently unwavering principle, it may be called a law.

If I say, “There is oil on Mars,” I am not positing a theory; I am just making an untested assertion. If I were to say instead, “Where rocks on a planet show chemical composition X, there is oil beneath the planet’s crust,” that would be a hypothesis; with strong basis and explanation, it could be a theory. If I were to find a unifying principle predicting the presence or absence of oil, I might call it a law. Any of these can be refuted: assertion, hypothesis, theory, and law. Only the assertion can be proven true, in the case that oil is found on Mars. Even then, there are caveats.

Now, growing up in a family of mathematicians, I assumed in childhood that you could prove a “theory” true (provided that your axioms were true). Because mathematics works within a closed system, you can work logically from axioms to conclusions and thereby demonstrate that the latter proceed from the former. (In mathematics, it is a “theorem” that you prove, not a “theory.” The word “theory” in mathematics usually refers to a body of knowledge. The usage is not entirely consistent, though; one hears of Ramanujan’s theory of primes, for instance.)

In the natural sciences, you never have a completely closed system, except in the theoretical fields. This is where things get tricky. In theoretical physics, for instance, you can determine from your axioms and laws how  your model will behave. Models do not completely match the natural world, though. In the natural world, there are mitigating factors (and the truth of the axioms matters a great deal).

The social sciences may be the farthest from any closed system–because so many factors, past and present, can influence human behavior. You usually work with high degrees of uncertainty. What do you do? Do you just give up? There are those who believe the social sciences are pure nonsense, but I am not among them. I favor efforts to make sense of our lives from the standpoints of many different fields–including philosophy, literature, mathematics, theology, languages, statistics, physics, psychology, and more.  Each field contributes in some way to our understanding.

But how does one work with so much uncertainty? First of all, enjoy it now and then. It would be a dreary world if we could figure everything out. One doesn’t have to be perpetually cheery about it, but one can take courage from it. Second, find ways of working with degrees of uncertainty–not treating all uncertainty as alike, but determining which are greater than others. Models in the social sciences can bring much insight; one must just take care to observe their divergence from actual phenomena.

It’s important, when doing this, to avoid the null hypothesis fallacy. Some might say, “Well, I can’t prove a theory true, but I can prove its negation false, and that’s essentially evidence for the theory.” No, it isn’t. The two are not the same. When proving the negation false, one does not win evidence for the theory; one is still firmly fixed in the wobbles of doubt.  One must figure out how to view the doubt clearly.

In any case, the answer to the initial question is double. If you are working within a closed system, you can prove a theory (or theorem) true; within an open system, you cannot. However, in the latter case there is still much you can learn.

A mini-glossary:

Hypothesis: A proposed explanation for the way something works. (It is more than a “guess”; it must have a basis in evidence and reasoning, and it must be testable.)

Theory: A hypothesis that has been tested, substantiated, and extended, and that applies broadly to a natural phenomenon or class of phenomena.

Law: Like a theory, but unified into a general principle that unerringly explains a phenomenon (to the best of our knowledge and understanding).

Model: A representation of a real-world phenomenon, designed to assist with observation, testing, and explanation.

Note: I revised this piece substantially after posting it. In particular, I clarified the terms, changed the examples, and added some links. I cut the part about literature, since it needs a post of its own.

One Nation Not on Speaking Terms with Itself

As I woke up intermittently throughout the night and disbelieved myself into the news,  I realized that I knew of only one person (among friends, acquaintances, family, relatives, and colleagues) who may have voted for Trump. There may be more; I’m just not aware of them.

How much do we talk to each other in this country? I don’t just refer here to people from Vermont talking with people from Kansas. I refer to the people in our  neighborhoods, families, professions, and so on. I am not that talkative overall, so there are many with whom I never talk daily. Why would those happen to include just about everyone who voted for Trump?

The “sorting” phenomenon in the U.S. is well known, documented, and discussed; people tend to associate with those who generally agree with them. Sure, some disagreements come up, but often the big ones stay out of the picture, to the point where we simply don’t know anyone who views things in a markedly different way.

I have generally thought of myself as a semi-outsider, as someone who stays out of gossip and chatter and holds somewhat unconventional views. But when I look from a distance, I see that I “fit in” with my environment more than not. I do not have daily encounters with people who believe that I am going to hell because I read things other than the Bible (or because I read the “wrong” parts of the Bible). I have never (to my knowledge) hobnobbed with a creationist, at least not about creationism. Only a few times have I discussed immigration with people who feel that the U.S. should close its gates. Except for a few artists, writers, and entrepreneurs, I know no one who sees college as a waste of time. In addition, I know few U.S.-born poor and working-class Caucasians. Nearly all my friends, family, colleagues, and acquaintances are middle-class, immigrant, non-Caucasian, or some combination of the three.*

But the insulation goes a bit deeper. I suspect that we know as little about ourselves and those close to us as we know about those from the “other side” of the political divide. If it’s true that we know ourselves at least partly through dialogue, then our self-knowledge depends on the quality and extent of such dialogue. If we are limited to certain topics, certain people, certain moods, and certain assumptions, then a lot will remain unexamined.

My response to the election is to look at my conversations. With whom am I talking, and with whom not? Which ideas and topics come up, and which do not? It is worthwhile to live in the uncertainty of this: to listen beyond the familiar sounds.

Note: I made a few edits and additions  to this piece after posting it. Also, see today’s excellent pieces by Jesse Singal and Drake Baer.

*Added after the initial additions and revisions.

Beyond the Introvert-Extravert Divide

Over at New York Magazine, Drake Baer has been challenging the introvert-extravert dichotomy with vigor. “‘Introvert or Extrovert’ Is the Wrong Way to Define Your Identity,” declares one October article; an article from July has a similarly bold title (“Why Declaring ‘I’m an Introvert!’ Limits Your Life“). In both articles, and in some earlier pieces, Baer emphasizes the complexity of personality and the influence of occupation and context. I would go even farther than he does—for instance, I am skeptical of the Big Five theory of personality—but I applaud his combination of boldness and subtlety.

The introvert issue has been so overhyped that it swept other discussions into its hot air. It created a “groupthink” of its own. In 2012, a few months after Republic of Noise came out, I was interviewed for an Education Week article on introverts in the classroom (as was Susan Cain). When speaking with Sarah Sparks, I emphasized the distinction between solitude and introversion. Solitude is essential to education (in some way and in some form) no matter what your personality type. Instead of trying to make the classroom amenable to introverts (who are a highly diverse bunch, with a wide range of preferences and needs), pay attention to the subject matter. It just isn’t true that “introverts” prefer online discussion to class discussion. If you are approaching the subject keenly, your class discussion will not be dominated by table-thumping loudmouths anyway. People will have to think, because there will be something to think about. Of course you should pay attention to the students—but for their ideas and unique qualities, not their type.

But these points were left out of the article;  Sparks and other reporters continued to present issues in terms of introverts and extraverts. I have wondered why. It seems part of our country’s tendency toward polarization. It isn’t so far removed, in other words, from the climate of the election. It is all too easy to identify yourself with an oppressed group (in this case the introverts) and let someone else tell you who  you are and what you need. Someone shows up who seems to tell your story, explains how you and your kind have been mistreated, and promises a revolution.

But maybe this isn’t quite your story; maybe your personal oppression (to the extent that it exists) comes from many places, including the self; maybe liberation lies not in an uprising of your personality type but in good independent thought. I don’t mean that one should reject all alliances, but no alliance should demand a reduction of the mind or soul. There should be room to challenge not only the dominant train of thought but its underlying suppositions. There should be room to say, “this isn’t quite right.”

I see Baer’s articles as a promising step in that direction. A shout-out to Melissa Dahl too.

Note: I originally mistitled the first Baer article; the error is now fixed.

Formal and Informal Research

I have been thinking a lot about formal and informal research: how both have a place, but how they shouldn’t be confused with each other. One of my longstanding objections to “action research” is that it confuses the informal and formal.

Andrew Gelman discusses this problem (from a statistician’s perspective) in an illuminating interview with Maryna Raskin on her blog Life After Baby. It’s well worth reading; Gelman explains, among other things, the concept of “forking paths,” and acknowledges the place of informal experimentation in daily life (for instance, when trying to help one’s children get to sleep). Here’s what I commented:

[Beginning of comment]

Yes, good interview. This part is important too [regarding formal and informal experimentation]:

So, sure, if the two alternatives are: (a) Try nothing until you have definitive proof, or (b) Try lots of things and see what works for you, then I’d go with option b. But, again, be open about your evidence, or lack thereof. If power pose is worth a shot, then I think people might just as well try contractive anti-power-poses as well. And then if the recommendation is to just try different things and see what works for you, that’s fine but then don’t claim you have scientific evidence one particular intervention when you don’t.

One of the biggest problems is that people take intuitive/experiential findings and then try to present them as “science.” This is especially prevalent in “action research” (in education, for instance), where, with the sanction of education departments, school districts, etc., teachers try new things in the classroom and then write up the results as “research” (which often gets published.

It’s great to try new things in the classroom. It’s often good (and possibly great) to write up your findings for the benefit of others. But there’s no need to call it science or “action research” (or the preferred phrase in education, “data-driven inquiry,” which really just means that you’re looking into what you see before you, but which sounds official and definitive). Good education research exists, but it’s rather rare; in the meantime, there’s plenty of room for informal investigation, as long as it’s presented as such.

[End of comment]

Not everything has to be research. There’s plenty of wisdom derived from experience, insight, and good thinking. But because research is glamorized and deputized in the press and numerous professions, because the phrase “research has shown” can put an end to conversation, it’s important to distinguish clearly between formal and informal (and good and bad). There are also different kinds of research for different fields; each one has its rigors and rules. Granted, research norms can also change; but overall, good research delineates clearly between the known and unknown and articulates appropriate uncertainty.

Update: See Dan Kahan’s paper on a related topic. I will write about this paper in a future post. Thanks to Andrew Gelman for bringing it up on his blog.

“That boatman am I”

floydsrowFor the past four days, at the ALSCW Conference, I have been in my element: presenting poems and papers, listening to others, leading seminars, participating in other seminars, and conversing seriously and jovially about literature. The talks, poems, fiction, music (including Floyds Row, pictured here), and keynote speech woke me beyond the usual waking and dreams.

It wasn’t just a matter of intellectual thrill. Here was a chance to go back to past readings and memories, learn about works I had not yet read, take in cadences and inflections, and participate thoroughly. I heard people read their own and others’ work; speak on Homer, Euripides, Pindar, Ovid, Augustine, Chaucer, Dante, Rilke, Woolf, James, Milton, Orwell, Hardy, Housman, Shakespeare, Stickney (from whom the title of this blog post comes), and others; and point out details, meanings, structures, and gestures. I read two poems (“Afternoon Visit” and “The Nose’s Arrest“); presented on Gogol’s “The Nose,” my translation of Venclova’s “Pestel Street,” and Cortázar’s “Final del juego“; and led a seminar that burgeoned into discussion (during the session and afterward).

cua

Now back to the book and the books. I start the tenth chapter this week.

Let Daydreaming Daydream

11

Painting: “11” by Karen Kaapcke, an entry in the 2016 Atlas Art Contest.

I have written about daydreaming numerous times (see here, here, here, here, and here in the blog, see here in Republic of Noise, and see my story “The Diagnosis“). I have daydreamed all my life; since infancy I was able to absorb myself in something simple for hours. I was kicked out of ballet class at age six because I would dance around the room instead of following directions (and was completely unaware that I wasn’t following directions). I was terrible at sports involving quick reactions, because my mind was on other things.

Generally I like being this way. It slows me down but also allows me to play with ideas, words, sounds, images. I am usually working on a story in my head over a period of months. It may not be anything I write down; I simply enjoy working out the details and carrying it in my mind. At other times, I work on projects or just let the thoughts wander.

All of this goes to say that I have some experience with daydreaming. Usually, when I read discussions of it, I find that they are slightly on the wrong track. They seem to focus on how daydreaming helps or hinders productivity (or so-called “creativity,” which is usually meant as corporate creativity). This carries two questionable assumptions: (a) that mental processes are valuable only insofar as they serve productivity (and so-called “creativity”), and that if we just found that key to productivity and creativity, people would be ever so much more productive and creative.

So it was somewhat refreshing to see Emily Reynolds’s New York Magazine piece “Everyone Should Make More Time for Daydreaming.” After that iffy title, the piece hit some good subtleties. Challenging the assumption that daydreaming is “a waste of time,” Reynolds cites some research and commentary suggesting otherwise, and goes on to say that daydreaming takes different forms, some helpful, some not. But not all daydreaming has to boost your output, she notes:

But this isn’t to say that you should reframe daydreaming as a “productive” activity, one aimed at particular or favorable outcomes. “Positive constructive daydreaming need not have a goal,” Kaufman agrees. Whether you do it mindfully or mindlessly, it’s worth spending a little time each day imagining the world beyond the present moment.

All fine and well, except for two things. First, there was really no need to cite Kaufman here; is the idea to give her statement a kind of scientific glow? Something from Dante or Emerson (for instance) might have worked better.

Second, I am not sure that daydreaming should be practiced deliberately. That seems to turn it into something else. Reynolds advocates some kind of “mindful daydreaming”–a combination of whimsy and awareness–but isn’t that already second nature to some people? If people set out to do this for the sake of becoming more creative, wouldn’t that corrupt the endeavor?

There is something wrong with the search for a “key” to creativity (or productivity). The people clamoring for it are not typically yearning for more poetry; no, they want more creativity on the job, in the service of profit. It is creativity on someone else’s terms. Also, they neglect the interaction of subject matter and creativity. Creativity exists only in relation to something. The best way to increase your creativity is to immerse yourself in that subject. You will start thinking about it, playing with it, imagining its possibilities, daydreaming about it. You won’t get there by trying to become more creative.

In his scathing (and brilliant) article “Ted Talks Are Lying to You,” Thomas Frank writes that “the literature of creativity [is] a genre of surpassing banality” in that it exemplifies conformity, not creativity, and is directed not at artists, musicians, actors, and writers, but at the professional-managerial class. Reynolds’ piece certainly doesn’t fall in this category, but it could step more boldly outside the trend.

In short: It’s good to recognize that daydream is not just a waste of time–that it is essential to some natures and endeavors. But there’s no need for daydreamer-chic, daydreamer mindfulness training,  or Amazon (Inc.) treehouse daydreaming sessions. Let daydreaming do what it does best: take its own way.

The Secret to Education

rainydayThe Secret to Education … that One Thing that will Change Everything … the Great and Shocking Truth … one by one, I reject these titles, until I finally pick the first, just for fun.

It is a dim and rainy day (photo taken just now); before I take off for New Haven, where I will be spending the afternoon and evening, I thought I would put together some thoughts on teaching.

I taught for approximately nine years in New York City public schools: first at a middle school in Boro Park Brooklyn (for three years), then at an elementary school in East New York, Brooklyn (for one year), and then, for the last five years, at Columbia Secondary School, where I served first as curriculum adviser, then as philosophy teacher and coordinator.

In addition, I taught for several years in other contexts. I taught first-year Russian at Yale for a year (as a graduate student), second- and third-year Russian at Trinity College in Hartford for a year (as a Mellon Fellow), and literature for six consecutive summers at the Sue Rose Summer Institute for Teachers at the Dallas Institute of Humanities and Culture. (This is ongoing.) Finally, I taught English in Kyrgyzstan for a summer and taught elementary enrichment summer school on the Crow Reservation in Montana.

So, after all this time (which pales in comparison to many teachers’ experience), what would I say that our schools need? I say emphatically that there is no one answer. None! I have no secret, no great solution.

Or rather, if there is one thing schools need, it’s good judgment: the ability to recognize good curricula and practices and apply them discerningly.

One truth presents itself again and again: teaching requires focused, quiet thought, which the school systems do not emphasize or honor. Yes, teachers need to collaborate, but to do so well, they also need to think about the subject on their own. This has little room in the school day; if you want time for quiet thought or focused study, you have to find it on your own.

Nor is “more time” the answer; there has to be a strong understanding of what that time is for. A teacher’s work must be perceived as intellectual. For that to happen, there must be more time for intellectual life overall. That will not come overnight, nor will any one reform bring it closer.

With all my skepticism, I do have a few ideas. They are not mass solutions, but they could set an example for many.

I would start with a good curriculum: that is, not a script, not a pacing calendar, but an outline of the concepts, works, and problems to be studied, along with the major assignments and projects. I would find schools willing to adopt the curriculum and education schools willing to base their program on it. This curriculum is not meant to be constricting; rather, it builds flexibility, as it gives everyone a working base.

Prospective teachers would begin by studying the actual subject matter of the curriculum (before thinking about how to teach it). They would learn it backwards and forwards, pose questions about it,  give presentations about it, and attend lectures and seminars. They would study their own subject matter and another subject (and possibly a third). Those already familiar with the subject matter would study it at a higher level.

The following year, they would translate the curriculum into lesson plans, practice giving lessons, and serve as student teachers at participating schools. They would not have to reinvent the wheel year after year; if lesson plans already exist, they might review them and modify them for their own teaching. They would develop more than one way to teach a given topic and would anticipate student questions and errors.

Then, when they entered a school, they would be well prepared to teach not only the subject but the actual curriculum itself. They could put their efforts into their new responsibilities.

Of course there are problems: what  if there aren’t enough education programs or schools? What if some district mandate comes along and topples  the curriculum that was constructed with such care?

Any number of things can go wrong; this is no magic solution. Still, I see promise in (a) having prospective teachers focus first on subject matter, then on curriculum and pedagogy and (b) having schools and education programs work with a shared curriculum. To some extent, this is the approach of the Dallas Institute’s Cowan Center and (in a different way) the Yale-New Haven Teachers Institute. Such an approach takes time, but this is precisely the right kind of taking of time: going far into subject matter and figuring out how to bring it to students.

In This Grand Primordial Mess

notmessy

Messy people (including me) may be on the up-and-up. Behold, to the left, a desk, my desk. This is about as unmessy as it gets. At least once a week, the piles at least triple. They flow onto each other. They threaten to converge and topple. So I bring them down a little and start again. That has been my life since adulthood. In childhood and adolescence, it was much worse; my mess didn’t even organize itself into piles. But I enjoyed it in some way and did not want to become neat. Others tried to get me to organize myself; although I did, a little, over time, I also kept a good deal of messiness, since it allowed me to focus on other things.

So I was delighted to see Jesse Singal’s article on mess. Apparently there are more mess-defenders in the world than I thought. I learned about a new book, Messy:  The Power of Disorder to Transform Our Lives, by Tim Harford. Unfortunately, though, the title gave me IS (Instant Skepticism). It sounds like another “Great Secret to Creativity” book. I hope it’s not that. There’s lots to be said for a degree of messiness, but I don’t for a messy second believe that becoming messy will make you more creative or successful. (It may be that the title only flops askew over the book’s actual contents; I will wait to see.)

When and how can messiness be good? Well, first of all, it’s just the way some of us are. My students have described me as organized, but that’s probably because I have learned over time how to handle my mess. Even so, I don’t organize myself more than I have to. It takes too much time, and I have my mind on other things. I work better if I don’t have to worry all the time about putting things in their  proper places. As long as I know where to find them, and as long as I keep them in good condition, I’m fine.

I need some messiness; I need the freedom to pile book on top of book while I am looking into an idea and writing out an argument. Also, I like the look and feel of mess (up to a point); it reminds me of things I and others have been doing, and it keeps an array of materials at hand. This cannot and should not be pre-engineered; it’s just the way I work.

It may well be true that all creativity involves some messiness. This does not mean that you arrive at creativity by generating mess. Mess comes in different forms; there are people who maintain an impeccably neat exterior but allow themselves a pile of loose ends in the mind. There are those whose mess occurs in blogging, or in speaking, or in musical tastes. It’s unlikely that any “messy regime” will help anyone produce a work of brilliance.

On the other hand, it is nice to see some people questioning the despotism of neatness. Talk about hegemony. Some of us (including me) have had points taken off, throughout our lives, because we didn’t write as neatly as others, organize our notebooks clearly, take legible notes in class, or put everything away immediately after using it. For the sake of justice alone, I am happy to join in praise of limited mess.

Speaking of mess: I was delighted to come upon some videos of a 1978 concert by the Roches. I first heard them in 1982 (thanks to a friend who insisted I come hear them). I had forgotten just how beautifully messy (yet in time and in tune and inspired) they were. Here they are performing the wonderful “We.”

Oh, the title of this blog: Once upon a time, in 1989, someone’s beautiful mess, and the occasion of a tornado, inspired a sonnet from me. Here it is.

Tornado, July 10, 1989

The winds began to imitate your prance,
a rolling soda can became the lyre,
the sirens sang the lyrics, mixing fire
with something like your name. The dance grew dense,
a cat shot an accusatory glance,
and time was canceled. Wood, debris, and wire
were pulled like windowshades to curb desire,
since pagan hail had trampled down the fence.

Thinking survival hardly worth the cost,
I risked electrocution or success,
clambering over what was once a street,
with hopes that in this grand primordial mess
finding you in your element, I’d greet
what never had been had, and still was lost.

Lectures, Teams, and the Pursuit of Truth

One of these days, soon, I’ll post something about teaching. Since I’m not teaching this year, I have had a chance to pull together some thoughts about it.

In the meantime, here are a few comments I posted elsewhere. First, I discovered, to my great surprise, that Andrew Gelman seeks to “change everything at once” about statistics instruction—that is, make the instruction student-centered (with as little lecturing as possible), have interactive software that tests and matches students’ levels, measure students’ progress, and redesign the syllabus. While each of these ideas has merit and a proper place, the “change everything” approach seems unnecessary. Why not look for a good combination of old and new? Why abandon the lecture (and Gelman’s wonderful lectures in particular)?

But I listened to the keynote address (that the blog post announced) and heard a much subtler story. Instead of trumpeting the “change everything” mantra into our poor buzzword-ringing heads, Gelman asked questions and examined complexities and difficulties. Only in the area of syllabus did he seem sure of an approach. In the other areas, he was uncertain but looking for answers. I found the uncertainty refreshing but kept on wondering, “why assume that you need to change everything? Isn’t there something worth keeping right here, in this very keynote address about uncertainties?”

Actually, the comment I posted says less than what I have said here, so I won’t repeat it. I have made similar points elsewhere (about the value of lectures, for instance).

Next, I responded to Drake Baer’s piece (in New York Magazine’s Science of Us section), “Feeling Like You’re on a Team at Work Is So Deeply Good for You.” Apparently a research team (ironic, eh?) lead by Niklas Steffens at University of Queensland found that, in Baer’s words, “the more you connect with the group you work with—regardless of the industry you’re in—the better off you’ll be.”

In my comment, I pointed out that such associations do not have to take the form of a team—that there are other structures and collegial relations. The differences do matter; they affect the relation of the individual to the group. Not everything is a team. Again, no need to repeat. I haven’t yet read the meta-study, but I intend to do so.

Finally, I responded to Jesse Singal’s superb analysis of psychology’s “methodological terrorism” debate. Singal points to an underlying conflict between Susan Fiske’s wish to protect certain individuals and others’ call for frank, unbureaucratic discussion and criticism. To pursue truth, one must at times disregard etiquette. (Tal Yarkoni, whom Singal quotes, puts it vividly.) There’s much more to Singal’s article; it’s one of the most enlightening new pieces I have read online all year. (In this case, by “year” I  mean 2016, not the past twelve days since Rosh Hashanah.)

That’s all for now. Next up: a piece on teaching (probably in a week or so). If my TEDx talk gets uploaded in the meantime (it should be up any day now), I’ll post a link to it.

Gradus ad Parnassum

gradusadparnassumI took this picture yesterday in Fort Tryon Park; it is one of my favorites. It made me think of a book I loved in childhood: The Study of Counterpoint, from Johann Joseph Fux’s Gradus ad Parnassum. The book teaches you counterpoint through a dialogue between teacher and student. Step by step (with some leaps and longer views), you learn the principles and practices.

I am not especially systematic when it comes to learning new things or advancing my knowledge. I like to plunge in at a much-too-difficult level and figure things out. But even that requires a sequence; I find myself going as far back as necessary to basic concepts and then working toward the problem at hand. I enjoy finding out again and again that it can be done—with languages, music, mathematics, and even human conundrums.

Here is the beginning of the dialogue in The Study of Counterpoint:

       Josephus.— I come to you, venerable master, in order to be introduced to the rules and principles of music.
       Aloysius.— You want, then, to learn the art of composition?
       Joseph.— Yes.
       Aloys.— But are you not aware that this study is like an immense ocean, not to be exhausted even in the lifetime of a Nestor? You are indeed taking on yourself a heavy task, a burden greater than Aetna. If it is in any case most difficult to choose a life work—since upon the choice, whether it be right or wrong, will depend the good or bad fortune of the rest of one’s life—how much care and foresight must he who would enter upon this art employ before he dares to decide. For musicians and poets are born such. You must try to remember whether even in childhood you felt a strong natural inclination to this art and whether you were deeply moved by the beauty of concords.

Once Josephus convinces Aloysius, the instruction begins.

Today the idea of inborn talent is unpopular—but Aloysius’s point is not that talent rules over all, but rather that the hard work of music requires great and strong desire. It can’t be a passing whim or a light interest.

On the other hand, once you have committed to the ascent, all you have to do is ascend, step by step, over many years. It doesn’t matter if sometimes you rush ahead and then backtrack, or pause for a long time at a given level; even then, you lead your life on the stairs.