The indicator function of the Cantor set is Riemann integrable. Like you said, though, the Dirichlet function (which is the indicator function of the rationals) is not Riemann integrable.
The reason is because the Dirchlet function is discontinuous everywhere on [0,1], so the set of discontinuities has measure 1. The Cantor function is discontinuous only on the Cantor set.
Plants "want" nitrogen, but dump fertilizer onto soil and you get algal blooms, dead zones, plants growing leggy and weak.
A responsible farmer is a steward of the local ecology, and there's an "ecology of friction" here. The fertilizer company doesn't say "well, the plants absorbed it."
But tech companies do.
There's something puritanical about pointing to "revealed preference" as absolution, I think. When clicking is consent then any downstream damage is a failure of self-control on the user's part. The ecological cost/responsibility is externalized to the organisms being disrupted.
Like Schopenhauer said: "Man kann tun, was er will, aber er kann nicht wollen, was er will." One can do what one wants, but one cannot will what one wants.
I wouldn't go as far as old Arthur, but I do think we should demand a level of "ecological stewardship". Our will is conditioned by our environment and tech companies overtly try to shape that environment.
Friction is an element of the environment like any other. There's an "ecology of friction" we should respect. Deciding friction is bad and should be eradicated is like deciding mosquitoes or spiders or wolves are bad and should be eradicated.
Sometimes friction is noise. Sometimes friction is signal. Sometimes the two can't be separated.
I learned much the same way you did. I also started a coding bootcamp, so I've thought a lot about what counts as "wasted" time.
I think of it like building a road through wilderness. The road gets you there faster, but careless construction disturbs the ecosystem. If you're building the road, you should at least understand its ecological impact.
Much of tech treats friction as an undifferentiated problem to be minimized or eliminated—rather than as part of a living system that plays an ecological role in how we learn and work.
Take Codecademy, which uses a virtual file system with HTML, CSS, and JavaScript files. Even after mastering the lessons, many learners try the same tasks on their own computers and ask, "Why do I need to put this CSS file in that directory? What does that have to do with my hard drive?"
If they'd learned directly on their own machines, they would have picked up the hard-drive concepts along the way. Instead, they learned a simplified version that, while seemingly more efficient for "learning to code," creates its own kind of waste.
But is that to say the student "should" spend a week struggling? Could they spend a day, say, and still learn what the friction was there to teach? Yes, usually.
I've worked in tech and lived in SF for ~20 years and there's always been something I couldn't quite put my finger on.
Tech has always had a culture of aiming for "frictionless" experiences, but friction is necessary if we want to maneuver and get feedback from the environment. A car can't drive if there's no friction between the tires and the road, despite being helped when there's no friction between the chassis and the air.
Friction isn't fungible.
John Dewey described this rationale in Human Nature and Conduct as thinking that "Because a thirsty man gets satisfaction in drinking water, bliss consists in being drowned." He concludes:
”It is forgotten that success is success of a specific effort, and satisfaction the fulfillment of a specific demand, so that success and satisfaction become meaningless when severed from the wants and struggles whose consummations they are, or when taken universally.”
In "Mind and World", McDowell criticizes this sort of thinking, too, saying:
> We need to conceive this expansive spontaneity as subject to control from outside our thinking, on pain off representing the operations of spontaneity as a frictionless spinning in a void.
And that's really what this is about, I think. Friction-free is the goal but friction-free "thought" isn't thought at all. It's frictionless spinning in a void.
I teach and see this all the time in EdTech. Imagine if students could just ask the robot XYZ and how much time it'd free up! That time could be spent on things like relationship-building with the teacher, new ways of motivating students, etc.
Except...those activities supply the "wants and struggles whose consummations" build the relationships! Maybe the robot could help the student, say, ask better questions to the teacher, or direct the student to peers who were similarly confused but figure it out.
But I think that strikes many tech-minded folks as "inefficient" and "friction-ful". If the robot knows the answer to my question, why slow me down by redirecting me to another person?
This is the same logic that says making dinner is a waste of time and we should all live off nutrient mush. The purposes of preparing dinner is to make something you can eat and the purpose of eating is nutrient acquisition, right? Just beam those nutrients into my bloodstream and skip the rest.
Not sure how to put this all together into something pithy, but I see it all as symptoms of the same cultural impulse. One that's been around for decades and decades, I think.
From John Dewey's Human Nature and Conduct, the fallacy that "Because a thirsty man gets satisfaction in drinking water, bliss consists in being drowned."
“The fallacy in these versions of the same idea is perhaps the most pervasive of all fallacies in philosophy. So common is it that one questions whether it might not be called the philosophical fallacy. It consists in the supposition that whatever is found true under certain conditions may forthwith be asserted universally or without limits and conditions. Because a thirsty man gets satisfaction in drinking water, bliss consists in being drowned. Because the success of any particular struggle is measured by reaching a point of frictionless action, therefore there is such a thing as an all-inclusive end of effortless smooth activity endlessly maintained.
It is forgotten that success is success of a specific effort, and satisfaction the fulfillment of a specific demand, so that success and satisfaction become meaningless when severed from the wants and struggles whose consummations they arc, or when taken universally.”
I co-founded Dev Bootcamp and while I was still there one of my not-so-secret missions was to make mathematics less alienating. I only say that because it was incredibly difficult, even in an environment where I had complete autonomy and authority to make whatever curricular and pedagogical decisions I wanted. The problem becomes combinatorially more complex in a public school where teachers have much less autonomy, have to teach to a common set of state-wide standards, and have students of varying levels of interest.
Here are my scattered thoughts, though. I'm going to try to not suggest a pie-in-the-sky solution like "new curriculum!"
First, I majored in mathematics at the University of Chicago, but I hate, hate, hated mathematics in high school. Take something you'd see in Algebra II like matrix multiplication, matrix inverses, and solving systems of linear equations. You're presented with these things called matrices and taught a bunch of rules. Where did these rules come from? Why are we calling this "multiplication" when it doesn't look or act anything like multiplication?
And sure, I see that when I go through the steps you tell me to go through like a monkey I get an answer that works, but how do we know there aren't more correct answers? How did anyone even come up with these steps in the first place? It's not like someone sat down and tried a trillion random combinations of symbols and steps until one of them happened to work.
Augh. In that world the only recourse for students is to memorize, usually just enough to do the homework or pass the test, and then promptly forget. The only experience they associate with math is the utterly humiliating feeling of being terrible at it.
So, I think that's one of the root problems. People remember what they feel and most people remember feeling stupid, humiliated, and possibly ashamed when it comes to mathematics. It's only a matter of time before that becomes part of their identity. "Oh, I'm terrible at math. Oh, I'm not smart enough to do math." and so on.
If I were a HS math teacher my top priority would be to watch out for when those counterproductive, self-defeating beliefs were forming and do whatever I could to preempt them.
Second, I think the way math is taught is overly symbolic. What most non-mathematicians don't realize is that when most mathematicians look at a set of abstract symbols they don't "see" the symbols per se, they see what those symbols are meant to represent. They freely move between a geometric and algebraic picture of the world, but the algebraic picture is usually incredibly compressed.
I think the key thing is not to pick a side -- algebra vs. geometry -- but to show the relationship between the two. Geometric objects admit a symbolic representation and vice versa.
Third, students have this idea that math is all about being "right" or "wrong", that it's "black" or "white", that there's some universe of Proper Math that is insisting on certain rules for no rhyme or reason
Here's a silly but illustrative example that I think students would cover in 6th or 7th grade: order of operations.
Hey class! Look at this expression: 45+6. What does it equal?
A bad teacher says "It's 26 and any other answer is wrong." An ok teacher says, "Remember the order of operations. If we apply those rules we get 26, so that's the right answer."
A great teacher shows their students that some things are necessarily true and other things are definitionally (or conventionally) true. This teacher would do something more like...
Who got 26? Who got 44? Students who said the answer was 26, how did the students who got 44 arrive at their answer? Students who said the answer was 44, how did the students who said 26 arrive at their answer? Neither of you are wrong per se. We could have chosen to live in either world, but we have to choose one consistent set of rules.
These rules lead us to 26. If we chose the other set of rules, we'd get at 44. We only do this because we don't want to have to write down parentheses all the time, but without them it's unclear what order we're supposed to apply + and . So we need to agree on a set of rules so that two people looking at the same expression both understand how to make sense of it.
It's like traffic laws. There's nothing stopping people from driving on the left side of the road. In fact, there are countries where everyone does drive on the left side of the road. The important thing is that everyone agrees on a convention -- left-side or right-side. It works as long as everyone agrees and breaks if people don't.
I could go on, but I'll stop here. Like I said, these are my scattered thoughts. :)
Hah, yes! I hated math in high school but wound up graduating with a BS in mathematics from the University of Chicago. It wasn't until I took my first math course at Chicago using Michael Spivak's Calculus that I thought, "Wait, if this is math, what was I studying all through high school?"
The indicator function of the Cantor set is Riemann integrable. Like you said, though, the Dirichlet function (which is the indicator function of the rationals) is not Riemann integrable.
The reason is because the Dirchlet function is discontinuous everywhere on [0,1], so the set of discontinuities has measure 1. The Cantor function is discontinuous only on the Cantor set.
Likewise, the indicator function of a "fat Cantor set" (a way of constructing a Cantor-like set w/ positive measure) is not Riemann integrable: https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%9...