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This is a very old consideration (cf https://www.jstor.org/stable/2183530 https://www.jstor.org/stable/2183530). If you want to frame a 'universal defining pro
by rhelz 2y ago
This is a very old consideration (cf https://www.jstor.org/stable/2183530 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.
- ysofunny 2y agowhence 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
- rhelz 2y agosigh 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.
- jhanschoo 2y agoMeet here is shorthand the comment author is using for: https://en.wikipedia.org/wiki/Join_and_meet 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."
- rhelz 2y agochuckle 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.
- yau8edq12i 2y agoHow 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)?
- rhelz 2y agoThe 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.