jamespropp

I agree (see endnote #5).

Thanks for catching this! I fixed the error.

I added a sentence: “(Warning: This is sort of a trick question, so don’t expect a textbook-style answer!)” I hope that this will prevent others from being similarly frustrated.

I plan to continue using AI tools, but to do so in a mindful way, to avoid becoming addicted to them, and to be prepared to back off if I don’t like the direction they take me in.

Perhaps an additional Moral is, if a calculation can be done in an afternoon, it's probably been done before. (I say this because it's been pointed out in the Comments section of my essay that Jon Lu asked and answered the same question back in 2016.)

I'm not sure I understand your question. Given that it takes an infinite amount of time to write down the decimal expansion of pi, what might it mean to compute pi in finite time?

Thanks for catching this! What was in my mind when I wrote the first of those two sentences was the entirety of my past experience using ChatGPT as a research tool, and all the times it made mistakes. (For a funny example, see my January 17, 2023 essay “Denominators and Doppelgängers” in which I describe ChatGPT's proof that .999... is less than 1.) But you're 100% right that the version of my essay that I posted last week didn't make this clear; I'll update it appropriately. I'll also add a brief description of ChatGPT's bibliographic blunder.

As usual, I'm most interested in comments that will help me improve the article, but feedback of every kind is welcome!

I’ve updated this passage. Let me know if the new version is clearer.

Thanks for pointing this out. I’ll work on this passage tomorrow.

Yes!

Do you disagree with my take or think I’m missing Witt’s point? I’d be happy to hear from people who disagree with me.

Richard Dedekind didn't just invent Dedekind cuts (and rings and ideals); he gave us a new way to think about (or avoid thinking about) the ultimate nature of mathematical reality.

What makes the real number system different from all the smaller number systems it contains? Richard Dedekind found a simple answer: a geometric axiom that Euclid missed.

This Mathematical Enchantments essay describes three contexts in which a sensible way to count how many times you've performed an operation is "zero, one, two, one, two, one, two, ..."

What can the counterintuitive nature of some mathematical results teach us about the pursuit of virtue?

This spoof of popular science journalism envisions a parallel world in which mathematicians make their discoveries in a radically different order yet science journalists invent the exact same cliches (and a few famous folks have suspiciously familiar names).

This essay explores the boundary between two aspects of math: its unchanging truths on the one hand, and the ever-changing ways we try to capture those truths in symbols on the other. Modern symbolical algebra is relatively new on the scene (only five hundred years old); this essay tells how it came to be and discusses the benefits and pitfalls of its extreme concision.

This essay describes operations on finite sets that mimic the operations of adding, multiplying, and exponentiating counting numbers and uses these operations to explain why discrete mathematicians and computer scientists take 0^0 to equal the number of functions from the empty set to itself, which is 1.

How mathematicians learned to stop worrying and to just get on with the business of creating new universes.

I don’t have classroom experience with this kind of teaching. I may write a companion to “When 1+1=0” called “When 1+1=1”, at which point I’ll have to think harder about these issues. (Are we using a different 0 and a different 1 here? Or a different +? Or a different =?)

I’ll look into this. Thanks!

This essay tells the story of how the modern theory of vectors was gradually discovered, and covers the colorful yet often forgotten late-19th century quarrel between quaternionists and vectorialists.

This essay discusses finite fields and the role they played in the early years of the computer revolution (remember paper-tape drives?) and in the ongoing Voyager mission to explore the outer solar system.

In this month’s Mathematical Enchantment, I wrote about finite fields and the role they played in the early years of the computer revolution (remember paper-tape drives?) and in the ongoing Voyager mission to explore the outer solar system.

Yeah, I admit the title is a bit click-baity. But I got the word “marvelous” from the Audre Lourde poem, so it’s not completely gratuitous.

Thanks, fwlr! Your words of appreciation made my day.