In both of your examples equality holds for x in some specified domain. The first equation is not valid (outside of, say, analytical continuation) for x >= 1.
I thought the article was cool. From this comment, you are coming across as someone who is trivializing a potentially important piece of work. I didn’t really understand it, but I take mathematicians at IAS seriously.
Why the ad hominem? I have nothing against the mathematicians or their work. I don't even know the math; that's why I'm reading the article. And I'm just trying to wrap my head around the article's explanation and trying to figure out why its premise seems inconsistent with my understanding. The problem is more likely somewhere in my understanding, so I posted it here in case somebody could clarify whether/what I'm misunderstanding of the difference between what we've called "equality" and what they're calling "equivalent", given neither of them seems to necessarily mean "exactly the same".
> you are coming across as someone who is trivializing a potentially important piece of work.
I doubt it. The parent is confused and hence the question obviously. But I get the impression that the parent would benefit from reading an elementary textbook on discrete math as the post is gibberish (as are a few other posts in this thread). I know it sounds rude, but I don't know how else to put it. It's not only the fact that the parent doesn't have a conceptual idea about equivalence relations and in particular equality, isomorphism, let alone homotopy, but is also very confused about what it means for a series to converge which is also very elementary. Before discussing the merits of Dostoevsky in Russian, one must learn and know the Russian alphabet at the very least or something.
> I'm not sure how far you could take math if you treat equality as meaning "exactly the same"; it seems even middle-school math requires understanding that that's not the case.
I honestly didn't mean that in a rude way, more like a "hey dude, be careful how you're coming across here, as there may be things you don't understand." I read the above phrase as describing a mathematician's work as something a middle schooler should already know. But I take back what I said. I'm all good, everyone's good, I think there was just a misunderstanding here.
In the same vein, I don't see how you could read my message as needlessly insulting, as I thought it was pretty softly worded. But I guess that's the nature of written communication. I think you should have given me the benefit of the doubt, just as I should have done for the original poster.
No, it wasn't softly worded. Now you take back what you said, although it was fine and I'm the one in the wrong to think it was insulting?! Etc. I don't think those are reasonable interpretations. Your comment was good without the last paragraph. Can't help sinking the boot in, eh? That's not a very nice habit. But anyway, I have things to do today! See ya.
I thought the article was cool. From this comment, you are coming across as someone who is trivializing a potentially important piece of work. I didn’t really understand it, but I take mathematicians at IAS seriously.