> if object X models object Y, then I’m going to say that X is Y

If you haven't read to the end of the post, you might be interested in the philosophical discussion it builds to. The idea there, which I ascribe to, is not quite the same as what you are saying, but related in a way, namely, that in the case that X models Y, the mathematician is only concerned with the structure that is isomorphic between them. But on the other hand, I think following "therefore X is Y" to its logical conclusion will lead you to commit to things you don't really believe.

> But on the other hand, I think following "therefore X is Y" to its logical conclusion will lead you to commit to things you don't really believe.

I would love to hear an example… but before you do, I’m going to clarify that my statement was expressing a notion of what “is” sometimes means to a mathematician, and caution that

1. This notion is contextual, that sometimes we use the word “is” differently, and

2. It requires an understanding of “forgetfulness”.

So if I say that “Cauchy sequences in Q is R” and “Dedekind cuts is R”, you have to forget the structure not implied by R. In a set-theoretic sense, the two constructions are unequal, because you use constructed different sets.

I think this weird notion of “is” is the only sane way to talk about math. YMMV.

I think the problem with insisting on using "is" that way is that you then can't distinguish between two things you might reasonably want to express, i.e. "is isomorphic to"/"has the same structure as" and "refers to the same object". I totally agree that math is all about forgetting about the features of your objects that are not relevant to your problem (and in particular as the post argues things like R and C do not refer to any concrete construction but rather to their common structure), but if you want to describe that position you have to be able to distinguish between equality and isomorphism.

(Of course using "is" that way in informal discussion among mathematicians is fine -- in that case everyone is on the same page about what you mean by it usually)

> I think the problem with insisting on using "is" that way is that you then can't distinguish between two things you might reasonably want to express, i.e. "is isomorphic to"/"has the same structure as" and "refers to the same object".

It’s reasonable to want to express that difference in specific circumstances, but it would be completely unreasonable to make this the default.

For example, I can say that Z is a subset of Q, and Q is a subset of R. I can do this, but maybe you cannot—you’ve expressed a preference for a more rigid and inflexible terminology, and I don’t think you’re prepared to deal with the consequences.

Most commenters are talking about the first part of the post, which lays out how you might construct the complex numbers if you're interested in different properties of them. I think the last bit is the real interesting substance, which is about how to think about things like this in general (namely through structuralism), and why the observations of the first half should not be taken as an argument against structuralism. Very interesting and well written.

It is very re-assuring to know, on a post where I can essentially not even speak the language (despite a masters in engineering) HN is still just discussing the first paragraph of the post.

Maybe the bottom ~1/3, starting at "The complex field as a problem for singular terms", would be helpful to you. It gives a philosophical view of what we mean when we talk about things like the complex numbers, grounded in mathematical practice.

Notably, the real numbers are not symmetrical in this way: there are two square roots of 1, but one of them is equal to it and the other is not. (positive) 1 is special because it's the multiplicative identity, whereas i (and -i) have no distinguishing features: it doesn't matter which one you call i and which one you call -i: if you define j = -i, you'll find that anything you can say about i can also be shown to be true about j. That doesn't mean they're equal, just that they don't have any mathematical properties that let you say which one is which.

Every father is a son to somebody...

Unfortunately tech journalists' judgement of source credibility don't have a very good track record

But a) that's the cost to the user -- we don't know how much loss they're taking on those and b) the number of tokens to serve a similar prompt has been going up, so that the total cost to serve a prompt has been going up in general. Any cost analysis that doesn't mention these is hugely misleading.

> people simply don't care

I don't think that's right, even for laypeople. It's just that the pain of things that take 5 seconds when they could take 50 ms is subtle and can be discounted or ignored until you are doing a hundred things in a row that take 5 seconds instead of 50 ms. And if you don't know that it should be doable in 50 ms then you don't necessarily know you should be complaining about that pain.

It's also that the people who pay the price for slowness aren't the people who can fix it. Optimizing a common function in popular code might collectively save centuries of time, but unless that converts to making more money for your employer, they probably don't want you to do it. https://www.folklore.org/Saving_Lives.html

> [...] three factors [...] Capital accumulation is one.

The obvious omission here is well developed in Imperialism, The Highest Stage of Capitalism: it's hard to accumulate capital when all of the productivity growth from foreign "investment" by the rich world is captured by the "investors".

There's always an excuse.

It's extremely naive to ignore the power dynamics that set the terms of who benefits from those investments, and the relationship between those power dynamics and the initial wealth of the participants. I'll withhold judgement on whether it's "cope".