The Subjectivity Of Everything: Music And Math Edition

Recently from a combination of world-weariness, pandemic-weariness, and me-weariness, I was seized by a desire to think about something abstract, useless, inert, and intellectually challenging. "I know what I'll do," I thought. "I'll buy a book about math."

I went to a small bookstore and asked, "Do you have a math section?" No. But there among the science books I came upon Music By the Numbers by math historian Eli Maor. I thought it would be about music theory and the mathematical underpinnings of classical music structures -- something I vaguely remember being interested in when I read about it in Gödel, Escher, Bach as a teenager forty years ago. But it was more about the fundamental mechanics of western music's organization of sound. Where did all these notes come from?

I knew that with a string, half as long means an octave higher. What I learned first is that the Pythagoreans and their followers based an entire scale on their idea of pleasing fractional intervals. The full story is a bit complicated, but taking fourths and fifths as a starting point, you can create a scale with a version of the "whole tone" interval we're familiar with (C to D, for example) based on a 9/8 ratio of a note to its predecessor (and a half-tone with ratio 256/243).

Maor says that while this way of creating a scale is mathematically elegant, it is out of step with the pitches produced in harmonic overtones -- and thus with the fundamentals of acoustics itself. From a philosophical point of view, Maor argues that their obsession with formal beauty led Pythagorean followers into a self-circular mathematical maze: by insisting on mathematical simplicity, pleasing ratios, and no irrational numbers, they "subject[ed] the laws of nature to their ideals of beauty."

But the more amazing moment for me came later, when I learned about tempering. As I kid, I knew about Bach's Well-Tempered Clavier, and I used to enjoy imagining that the well-tempered clavier had replaced a sour and irritable keyboard instrument known as "the ill-tempered clavier." But in all these years, I never learned what tempering is.

The background for tempering in the Western context happened around 1550, when a new scale was created bringing in intervals of thirds and their inversions. This "just-intonation" scale has intervals close to the ones we'd find on a modern piano. Maor argues that as it is based on the first six members of the natural harmonic series, and has pitches corresponding to the natural harmonics of musical instruments, it is acoustically and empirically superior to the Pythagorean scale. You could say that it has formalism that is not so much mathematical but rather musical.

However: a crucial feature of the just-intonation scale is that not all the tone ratios are the same. For example, in a C-major scale, the ratio of C to D would be 9/8 and from D to E, 10/9.

From a musical point of view, Maor argues, this is as it should be. It's from the practical and social point of view that these differences became a problem. For instruments with keyboards and fixed holes, a just-intonation tuning in one key will have notes at slightly different pitches from those in another key. Increasingly, people wanted to play together, with multiple instruments all at once. What if a group wants to play one piece in one key and another in another? The workarounds were complicated: early harpsichords had multiple keyboards, each tuned to a different key, for pieces written in different key signatures.

"Tempering," then, is creating a scale with even divisions, all the same. You just divide the octave into 12 equal semitones. Now all the instruments can play together, and playing in C is the same set of notes as playing in any other key.

Math people may already see that, unlike the just-intonation scale, the ratio of a note to its predecessor in the tempered scale is based on irrational numbers -- numbers that cannot be expressed as fractions. Having even divisions requires dividing the scale into equal parts -- in this case, 12 equal half-tones -- and to be equal, each ratio between a note and its predecessor in the sale would have to be the twelfth root of 2 to 1.

Maor: "This irrational number would have been regarded with horror by the Pythagoreans, as it cannot be written as a ratio of integers." !!!

Leaving aside Pythagorean worries, Maor describes tempering as "an acceptable compromise" between "the dictates of musical harmony" and "the practicality of playing a piece on the keyboard." The difference between the tempered semi-tone and the just-intonation semi-tone is just barely within what human ears can discern, a difference "most musicians were willing to live with." Maor alludes to a suggestion that Bach's Well-Tempered Clavier was partly PR: written to convince his fellow musicians of the benefits of this sociable way of organizing everything.

I found myself astonished by idea that the structural simplicity I associate with piano keys is the result of an "acceptable compromise" to solve practical problems of musicians playing together.

From a personal point of view, I guess I thought that the Western division of the octave into 12 equal semitones was somehow linked to the fundamental structure of music and sound. Not that it was the only way to exploit that structure -- I've always known that non-Western music was organized differently -- but that it was one way. Now this book says not only is that not so, but it's even weirder: there is structure of music and sound, and those equal piano divisions are really, deeply, not it.

From a philosophical point of view, the analysis provides an interesting reminder about the subjectivity of concepts like simplicity and elegance. For the Pythagoreans, these concepts translated to rationality. For the just-intonation fans, they relate to harmonics and ratios. For the temperers, it's like brute force to make it work -- hey, make 12 that are exactly the same, whether they are rational or not.

From a mathematical point of view, it's striking to see how slippery the difference between natural numbers and other numbers can be. The nineteenth-century mathematician Leopold Kroeneker is famous for having said "God made the integers; all else is the work of man." But when you can describe the tempered scale in terms of "12 equal divisions" or "12 irrational tone ratios" -- well, things start to seem less clear. Even in math, simplicity and elegance can mean one thing in one context and another in another.

From a sociological point of view, I found myself wondering: did books like this used to be more common, or more talked about, or something? I feel like Gödel, Escher Bach was a bit splash of a book, right around the time of other math books like the Gleick book Chaos. But now my bookstore doesn't even have a math section. Did the world change, or is it something about me that's different?

Statistics, Randomness, And My Misspent Youth

I've just been reading Nassim Taleb's Fooled by Randomness. It is a fun read, but at the risk of sounding intellectually immodest, I feel like it is full of things I already knew. Rare, unexpected, and often bad things can happen. It is easy to be fooled by statistics. It is easy to be fooled by life in general. You never really know what's going to happen. Plan accordingly.

Taleb's main target audience is traders and others who think they're succeeding and failing because of skill and insight or lack thereof. He says they are often just the victims of survivorship bias and other forms of bad reasoning. Survivorship bias is when you look at the winners and assume that whatever they did made them winners, while failing to look at the losers and what they were doing. Everyone might be throwing darts at the Wall Street Journal stock pages. Some might be winners, and tautologously, some will be winners relative to others. If you focus on the winners and blindly follow their strategy or their stock picks, you'll be SOL.

More broadly, the book is about unpredictability and how unexpected and surprising things can derail your life. This is something I knew from a young age, mostly because it features prominently in both fiction and reality. You might study for years then develop a brain tumor. You might train as an athlete then get hit by a car. You might raise your kids on organic food and still one of them might die -- from food poisoning, or a really bad flu, or ... anything, for god's sake. If I hadn't had the predictability of unpredictable events hammered home to me enough by the time I was fifteen, I definitely absorbed it when my father died suddenly -- and unexpectedly! -- from a heart attack at that time. He was forty-eight years old.

So I absorbed the epistemological lesson early. From a practical point of view, Taleb says that in the face of uncertainty, a reasonable response is to be risk-averse, and I completely agree. As I understand it, for a trader this means seeker smaller steadier gains, but to be honest I sort of zoned out on that part of the book because I am almost aggressively uninterested in thinking about investment strategy.

At this point you may be wondering: So, Patricia, did you become a risk-averse young person? Why yes -- yes I did! I skipped class, stayed up late drinking with friends, majored in two abstract and relatively useless majors (math and dance), drove around in my mom's ancient and poorly maintained VW Rabbit, and engaged in various activities we won't describe here but that are usually represented as paradigm cases of adolescent devil-may-care behavior.

To me, these were risk-averse activities. My reasoning -- which I stand by today -- was that the pleasures of impulsive behavior have an excellent chance of working out in the immediate moment, while the pleasures of planning and carefulness are distant in time and thus far less likely to actually work out.

The pleasures of hanging out with friends with you should be in class, eating cake for lunch, drinking and smoking, dancing at parties, spending a beautiful sunny day with a romantic interest instead of writing papers, etc. -- when you're a bright-eyed eighteen year-old, these are very dependable pleasures in the short-term run. Sure, they often lead to unhappiness in the long run. But whatever -- the long run is unpredictable. That is the whole point.

To forego these pleasures in the moment to get better grades, avoid lung cancer at middle-age, or even just try to live a longer period of time seemed to me like a crazy and extremely risky strategy. What if you followed all the rules and got hit by a bus at age 20? FFS. To me, my life strategy was based on getting while the getting was good -- a paradigm example of risk-averseness if ever there was one.

Over time, I stopped being so impulsive. Partly it was because the pleasures of short-term pleasures got less intense for me as I got older (as they do for so many people), and I started getting bored. Partly it was because I started to experience the negative effects of doing whatever I wanted to do in the moment: my health declined, I worked at shitty jobs like waitressing, and I had no health insurance. Partly it was because I formed close relationships with people who wanted me to flourish in the long term, which made me want to flourish in the long term as well.

For whatever it's worth, none of these reasons has to do directly with the kind of caring about the future me that is supposed to characterize rational risk-averse thinking in the standard model of human decision-making.

When people talk about risk-averseness and the rationality of overcoming impulse, I feel like there are assumptions in the background: that what it makes sense to do is plan for the future, store up your chestnuts for the winter, try to live to a ripe old age. But from the unpredictability-of-life perspective, those assumptions are peculiar and it's more the other way around. When you do whatever thing you want to do in the moment, that pleasure -- even if it's just the pleasure of a desire immediately satisfied -- is yours, and no one can take it away from you. It's a sure thing. Whereas the long term? You never really know what's going to happen.

Humanities Teaching Is Difficult And Time-Consuming

Before I studied philosophy, I studied math. I was working on a PhD in set theory when I became less interested in how-do-you-prove-the-theorem and more interested in questions like what-does-it-mean-if-you-can't-prove-or-disprove-the-theorem?

Among other things, this means that before I started studying philosophy I spent a certain amount of time teaching mathematics. I started as a teaching assistant for introductory courses like Calculus and Statistics, and then I was a teaching assistant for more advanced courses like Differential Equations, and then toward the end I taught a few classes myself, including one on how to do mathematical proofs.

Math is hard. But I found teaching mathematics to be mostly straightforward and rewarding. Students are usually externally motivated to learn: they want to do physics, or engineering, or more advanced math, or whatever, and to do it they have to learn some math. Except in the case of bogus requirements -- like baby calculus for no reason for majors like business to "weed out" students they didn't like  -- the importance and relevance of the subject was relatively obvious.

At the level of undergraduate teaching at least, math is also coherent and unchanging. Because of the nature of the subject, the same kinds of things confuse people, and similar kinds of questions arise again and again. Once I had explained concepts like limits, differentiation, and integration a few times, the ideas were cemented in my head in such a way that very little teaching preparation was required.

On top of everything else, because math is obviously difficult, a teacher's ability to break down difficult concepts to make them seem simple earns them great respect. And this was something I was relatively good at.

Several years into the process of studying for a PhD in math I switched to philosophy. I've now been teaching philosophy in one form or another for ... well, a lot of years. And my personal opinion is that teaching philosophy is way more difficult and way more time-consuming than teaching mathematics. I don't have a lot of experience with the other humanities, but it is my belief that the reasons apply to humanities teaching generally.


Those reasons are several. For one thing, math seems difficult and a teacher is there to make it seem simpler, but in the humanities, it's often necessary to start by taking something that seems simple and showing students how difficult it is. I teach ethics, and philosophy of sex and love, and contemporary moral problems, and philosophy of economics. In all of these areas there's a sense in which a student already knows what they think about things, and part of my job is to complicate that -- to raise questions about things that seem obvious, to showcase views that seem counter-intuitive, and to just generally show how many different factors and perspectives can come into play.

This is intellectually difficult, and it can also be emotionally draining. How do you frame the issues when students are coming into the room with very different background assumptions - and you don't even know what those background assumptions are? How do you encourage people to speak up when part of your job is to suggest they might be totally wrong? How, exactly, do you figure out the line between constructively challenging existing beliefs and just being a contrarian pain in the ass?

Some people love the way humanities thinking challenges them, but other people find it exhausting and annoying. Sometimes science students in my ethical thinking class tell me how frustrated they are by the lack of a "right answer" in philosophy. I sympathize! It can be frustrating as hell. Unfortunately, the problems we're talking about are the ones that don't have straightforward answers, so it's the best we can do to muddle through.

Another factor, of course, is the variation and unpredictability of what kinds of things are going to come up. The social and cultural world we're living in makes different things seem obvious in different times. Even just contingently there are classrooms where one thing seems really important that didn't seem important to some other group.

This variation and unpredictability is, of course, part of what makes humanities teaching so important, relevant, engaging, and fun. But it also means that while teaching an interactive mathematics class can feel like a going through a play you're performed a thousand times, teaching an interactive humanities class can feel like a high-wire act where the tricks are constantly changing.

And finally, of course, there's grading. While mathematics grading can be time-consuming (when I did it, we didn't just grade yes-or-no, we looked at student work for partial credit) it's not like grading a paper -- work that combines engaging with someone's novel ideas and helping them toward an amorphous goal like "writing well." As we've discussed before, it takes a lot of time and energy, and it's not something you can scale up.

A few times recently I happened to be in large university group settings, where people were coming from a range of disciplines. And in that context, I heard some remarks about how, from the point of view of the sciences, what we humanities might regard as a large-ish class -- like, 50 or 100 students -- is to them a very small class. No one said it, but I felt the suggestion that somehow we humanities people weren't pulling our weight, that what we were doing was some kind of niche thing, cute and nice if you can afford it, but not really where the action is.

And I can't really say, because I have no experience teaching science. I only taught math -- which to me is a completely different kettle of fish. But from my perspective, the time and energy to teach a philosophy class is way more than the time and energy of teaching a mathematics class. Even when the classes are a lot smaller.

None of this is meant as a complaint. I love university students, and I love being around them. I think the people who criticize the younger generation for being phone-obsessed and jobless are wrong and ill-informed, and that today's young people are the hope of the future. I regard helping these young people understand the complex world around them as one of the best things anyone can do.

I'm just saying: for me, anyway, teaching about utilitarianism is way harder than teaching what it means to take the limit as h goes to zero.

The "Attractor" Of Individualism And Its Basin Of Attraction

 Lately when I think about individualism in advanced capitalist societies, I find myself latching onto a certain mathematical picture -- something associated with what's called an "attractor."

With the refreshing literality characteristic of mathematics, "attractor" in this context means something like "attractor" -- a spot that attracts. More specifically: "an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system."

I'm thinking here of individualism in the sense sometimes associated with liberalism and neo-liberalism. In the basic sense: people are best understood as conceptually and practically independent from one another; they properly get what they need and want in life by negotiating and making deals; there is no such thing as society. In the more advanced sense: we should embrace the entrepreneurial self.

The reason I think of individualism as an attractor is that once you get going on the basic idea, the idea creates the conditions for its own flourishing. It's like an infection where the mechanism creates the environment where the infection can thrive. At the end it's like a cultural Roach Motel: people can get in, but they can't get out.

For example, consider poverty. It used to be possible to think of poverty as a structural problem: we live in a society that's not working for some people. Maybe there could be structural solutions?

But the attractor is close enough to exert its magnetic pull. Through some invisible process, the question gets reframed in terms of helping individuals by giving them a leg up. Characteristic of this phase is the bizarre idea of "education" as some kind of solution. Like: "Engineers make more than baristas. If we could train everyone to be an engineer, no one would have the problem that baristas are poor." Of course this is crazy: as long as we want coffeeshops, someone's going to have the problems of being a barista -- it doesn't matter what kind of education people have.

Now that we're closer to the attractor, though, the pull is even stronger. Having acknowledged that some people are going to be baristas and others are going to be engineers, individualism forces us into the possibility that it's OK that some people are baristas and poor and others are engineers and not. Like: Oh, baristas will be poor, but that's OK, because everyone has the chance to be an engineer. Life is what you make it, yada yada yada.

But of course, life isn't what you make it. People start from massively different starting points. If your parents are poor, or they don't speak English well or whatever, or you live in a crappy area with crappy schools, you are starting from way behind -- it's going to be massively more difficult for you to become an engineer.

Instead of taking this as a reductio of individualism's implications, the attractor moves people toward other ideas. Some of those ideas are things like charter schools, choice, teach character development to small children, whatever. In the end, the simplest way to avoid the cognitive dissonance is to go back to individualism itself, and here we find our way to, "Well, sure, some people are going to be baristas and not engineers, but what I can do about that? I mean, I'm just one person." 

Which -- given the creeping effects of individualism -- is actually more and more true. Because the closer you get to peak individualism, the stronger the magnetic pull toward individualism.

I don't know about you but I feel like I see this dynamic the time. Social problem identified. Solutions canvassed. Collective solutions rejected for being insufficiently individualistic. Possibility of collective power dismantled. Individualist solutions proposed. Individualist solutions rejected, on grounds that they won't make a difference anyway. Which at this point they probably won't. And so on and so on and so on.

In the elegance typical of pure mathematics, there's a concept called the "basin of attraction." Technically: "An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will eventually be iterated into the attractor..."

Which is a fancy way of saying: there's some range of starting places from which you can't help but fall into the attractor.

I don't know when, exactly, modern western society went from skirting around the edges to actually falling into the basin of attraction for individualism. Was it Reagan and Thatcher in the 80s? Was it back with colonialism? Did Locke have something to do with it? I have no idea. All I know is, I think we're in it now.

The Dream Of The Fundamental Law Of Human Behavior: WTF?

One of the trends in modern thinking that really mystifies and annoys me is the dream of the simple fundamental law of human behavior. What is up with that?

In fact I'm mystified and annoyed both by the credulity -- the belief that there is a simple fundamental law of human behavior -- and also by the desire -- the hope that there's a fundamental law of human behavior. What kind of person sees humanity this way?

I mean, it's one thing to think we're like overgrown mice -- the kind of animals whose behavior might be well understood through a massive data-oriented approach with mazes and observational studies and NHS funding. Though I've always had my doubts about the fruitfulness of this style of thinking, I don't find it alienating. I mean, in some ways we are like overgrown mice. I understand the appeal of an animalistic self-conception.


But the dream of the simple fundamental law seems to requires seeing humans not as overgrown mice but more like ... I don't know, planets or something. Large, inanimate bodies whose movement through space and time can be charted with trigonometry and laws like F=ma.

What kind of person wants to see humans -- wants to see themselves -- this way?

In a recent Facebook Q and A, the physicist Stephen Hawking asked Mark Zuckerberg which of the big questions in science he most wanted to know the answer to. Zuckerberg said, reasonably enough, that he was interested in questions having to do with people, like how they learn.

He then went on to say this:

"I’m also curious about whether there is a fundamental mathematical law underlying human social relationships that governs the balance of who and what we all care about. I bet there is."


He's surely not alone with his dream. But WTF?



For one thing, why think there's a law like this? While it's true we have some simple fundamental mathematical laws in physics, the fact that such simple laws work is widely regarded not as commonsensical but rather as a kind of miracle.

In 1960, Eugene Wigner wrote a classic article "The Unreasonable Effectiveness of Mathematics in Science." Toward the end, he says:

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve."

If it's a mystery and a "miracle" that mathematics works so well in describing things like planets and F=ma, why would you think it'll work in the same way for people, who seem so complicated?

But for another thing, why is this someone's fantasy or dream? I don't even get what's appealing about it. Suppose you did find some law like that. Now you see humans not just as part of the machine of the universe, but as a predictable part of the the machine -- a part you could use a pencil to work out where we're going from where we've been.

What's to like? Suppose you did find some basic law, so that what seems like the vast multiplicity of human feeling and culture and motivation, what looks like the incredible fabric of life, it really all comes down to the X factor. You go about your day, and instead of seeing kindness and anger and art and food and cooperation and conflict and music and flirting, you say smugly to yourself, "well, sure -- but it's really X, X, X, X, X, X, X, and X."

Who can see this as a dream come true?