Thanks for the suggestions! What you’re illustrating here with your comments actually gets at the heart of what I was trying to accomplish here.
The formal definition of a Ramanujan prime starts (and ends) with the most specific statement definition possible. In a math class or textbook, you might start with the definition, and then unpack the parts. In my opinion, this starts with confusion—a terse riddle that may be intimidating—and only gradually move into more familiar terrain. I wanted to try moving in the opposite direction, by talking about relatively familiar concepts and gradually making them more specific.
Clearly, for you at least, this reversal, or the way I executed it, made things more confusing rather than less. That’s an unfortunate outcome, but I still feel excited about continuing to try this approach with other math concepts.
You’re absolutely right about your nitpick, and I thought about that while writing, but ultimately decided to leave that bit of nuance out. It’s important, but my aim was to put a visual and intuitive way of thinking about Ramanujan primes into the reader’s head, from which they can more easily “recompute” the bits of nuance that the article doesn’t convey—or extract them from further conversations and reading in a more formal context. Just as we can’t expect a mathematical treatise that is optimizing for specificity and accuracy to also achieve a high level of intuitiveness and engagement, I decided to sacrifice some specificity in order to achieve greater intuitiveness. But I expect that for a mathematically sophisticated audience, this may not be ideal.
I would have found the explanation much clearer if you stated at the outset whether the length of the line is chosen by us or for us. As is, the explanation has a lot of parts that are allowed to move from the outset: the location where we are standing, the length of the line extending behind us, and the number of primes on that line. Since the goal is to impose a relationship between these three components, and ultimately our standing location is the only bit that is allowed to vary, it would have helped me make things more concrete if you started with something like “Wherever we plant our feet, the magic line extends backwards towards 0 and then whispers a number in our ear, hinting at the primes it contains.”
This sets up the three key questions for the theorem: how far does the line extend backwards, what number does it whisper, and what does that number have to do with primes?
“Wherever we plant our feet, the magic line extends backwards towards 0 and then whispers a number in our ear, hinting at the primes it contains.”
This is a beautiful sentence. I wish I’d thought of it :D I’ll see if I can find my own version, because it strikes a lovely balance between leaving the specific details for later in the article, while building anticipation for them.
Relatedly, I was horribly confused when I started reading the article because I didn’t understand that the “magic line” wasn’t actually a line in the mathematical sense (lines go on forever, line segments don’t). For me it would have been a lot better to call it a magic (piece of) rope.
Thanks for the suggestions! What you’re illustrating here with your comments actually gets at the heart of what I was trying to accomplish here.
The formal definition of a Ramanujan prime starts (and ends) with the most specific statement definition possible. In a math class or textbook, you might start with the definition, and then unpack the parts. In my opinion, this starts with confusion—a terse riddle that may be intimidating—and only gradually move into more familiar terrain. I wanted to try moving in the opposite direction, by talking about relatively familiar concepts and gradually making them more specific.
Clearly, for you at least, this reversal, or the way I executed it, made things more confusing rather than less. That’s an unfortunate outcome, but I still feel excited about continuing to try this approach with other math concepts.
You’re absolutely right about your nitpick, and I thought about that while writing, but ultimately decided to leave that bit of nuance out. It’s important, but my aim was to put a visual and intuitive way of thinking about Ramanujan primes into the reader’s head, from which they can more easily “recompute” the bits of nuance that the article doesn’t convey—or extract them from further conversations and reading in a more formal context. Just as we can’t expect a mathematical treatise that is optimizing for specificity and accuracy to also achieve a high level of intuitiveness and engagement, I decided to sacrifice some specificity in order to achieve greater intuitiveness. But I expect that for a mathematically sophisticated audience, this may not be ideal.
I would have found the explanation much clearer if you stated at the outset whether the length of the line is chosen by us or for us. As is, the explanation has a lot of parts that are allowed to move from the outset: the location where we are standing, the length of the line extending behind us, and the number of primes on that line. Since the goal is to impose a relationship between these three components, and ultimately our standing location is the only bit that is allowed to vary, it would have helped me make things more concrete if you started with something like “Wherever we plant our feet, the magic line extends backwards towards 0 and then whispers a number in our ear, hinting at the primes it contains.”
This sets up the three key questions for the theorem: how far does the line extend backwards, what number does it whisper, and what does that number have to do with primes?
This is a beautiful sentence. I wish I’d thought of it :D I’ll see if I can find my own version, because it strikes a lovely balance between leaving the specific details for later in the article, while building anticipation for them.
Feel free to use it wholesale.
Relatedly, I was horribly confused when I started reading the article because I didn’t understand that the “magic line” wasn’t actually a line in the mathematical sense (lines go on forever, line segments don’t). For me it would have been a lot better to call it a magic (piece of) rope.
That’s a great idea, might steal it and update the post! Thanks for the suggestion.