> There’s a very deep problem with that—every time you invent a “superset”, do you then have to redefine the subset to be a subset of that “superset”?
I have thought about this too, and I'd initially agree with you. but I thought at some point how mathematical history is not extremely dissimilar from this. put in very rough terms:
at first humans discovered/invented numbers (i.e. the counting numbers); these started at number one — the first number. later on, at some point we had to go back and realize that there was a zero before number one which "silently" redefined the first number as zero and this created the natural numbers a the modern set-based N
edit: adding this alternative rendering of my intended comment triggered by a condescending reply: "mathematics silently redefines stuff all the time. deal with it"
There is a different answer here which is more satisfactory… which is to use notions of equality other than set theoretic equality. Which is what the article is talking about.
"Nothing" as a concept always existed of course. But it wasn't considered a number, generally. Certainly no one counted "nothing, one, two", and even today natural language doesn't include "nothing" or some equivalent as a numeral noun.
You need to be careful about the phrase “considered a number” since I believe one, or unity, was also not considered a number by some ancient civilisation - i.e. a number was only multiple copies of unity.
[I believe this YouTube video goes into more detail in its discussion of why 1 was not considered Prime in the ancient world: https://youtu.be/R33RoMO6xeA]
That's where I'm quite skeptical. Imagine you are in charge of trade or rationing important village resources in the winter. It just seems to me almost necessary that people would have a way to symbolically indicate that all the sheep are gone. As opposed to just not having any symbol for that at all.
Zero entered western writing systems through India, with limited usage in math before that. It seems like it was invented/borrowed as part of switching from additive numbers (such as Roman numerals) to positional numbers.
I have thought about this too, and I'd initially agree with you. but I thought at some point how mathematical history is not extremely dissimilar from this. put in very rough terms:
at first humans discovered/invented numbers (i.e. the counting numbers); these started at number one — the first number. later on, at some point we had to go back and realize that there was a zero before number one which "silently" redefined the first number as zero and this created the natural numbers a the modern set-based N
edit: adding this alternative rendering of my intended comment triggered by a condescending reply: "mathematics silently redefines stuff all the time. deal with it"