Comment by rhelz
1 year ago
This is a very old consideration (cf https://www.jstor.org/stable/2183530).
If you want to frame a 'universal defining property', you have to frame it in some language. The definition, therefore, is inevitably idiosyncratic to that language, and, if you specify the same property using a different language, the definition is going to change. What's more, the two definitions can be incommensurable.
And---all you have done is translate from the object language to the meet language--leaving the definitions of the terms of the meta language completely unexplicated.
The solution to this is as old too---like "point", "line", etc, these terms can--and arguably should--remain completely un-defined.
whence did you learn this idiom of "the meet language"???
i have working definitions of line, point, etc.. in terms of dimensions. but I have some issues with the precise definition of "dimension" due to having picked up the concept of "density". I cannot say what's the difference between density and dimension
Meet here is shorthand the comment author is using for: https://en.wikipedia.org/wiki/Join_and_meet
You should read it as: "the largest group of properties that both languages agree about said abstractly defined mathematical object."
chuckle thanks for this heroic effort to charitably interpret me as actually being able to type in a coherent sentence :-)
But, alas, I was just trying to type in "meta" and it was auto-completed to "meet". I really, really, hate autocomplete.
sigh I really hate autocorrect. It was supposed to be "meta", not "meet".
It's bad enough that most of the content I read these days is being generated by LLMs. Slowly and inexorably, even the content I write is being written and overwritten by LLMs.
How can you frame a paper from 1965 as "very old" when the linked article is about Grothendieck, who started his math PhD in 1950 (and presumably studied math before that)?
The very very old solution I'm adverting to is Euclid's axioms. They don't define what "point" or "line" means, they just specify some axioms about them.