Archive forMathematics

Math’matics: the clashes, with tildes and dashes

Note the pleasant rhyme and rhythm of the title (:

Anyways, what I say today is “wtf is with all the different ways to write various forms of equivalence in math, and how come everyone uses different ones?”

Don’t worry about what “equivalence” means (usually just something almost like equality, but for things other than numbers), or what any technical terms mean, the gripe persists through the smokescreen. Worst comes to worst, smile and nod.

Consider the following symbols (way easier said than done, huh?):
≃ ≂ ≅ ≈ ≡

Now first keep in mind these aren’t even all of the possible tilde/dash combinations, just the 5 most common, aside from - (which, yes, is almost always minus), = (which is always equals) and ~ which is almost always “is equivalent to” in general.

But the problem starts when ≃,≅, and ≈ are used interchangeably to mean either “is isomorphic to” or “is homeomorphic to” (in algebra, and topology respectively). But thats not all! sometimes “≃” or ≈ might mean “is homotopic to”. Or ≂ might mean isomorphic or homotopic (but rarely homeomorphic). Then you have the problem where ≅ and ≡ are used semi-interchangably in specific instances in set theory and number theory, although the former is more common in set theory and the latter in number theory, except that you’ll find just enough ‘incorrect’ uses to confuse you.

Edit: I forgot to say, that in Teχ, ≡ is called “\equiv”, and “≅” is calld “\cong”, when really ≡ is used in certain cases of congruence (specific type of equivalence), and the general equivalence symbol ~ is called “\sim”, and similarity has hell of different meanings, many of which are specific types of equivalence. Rarely, if ever, will you see “≅” meaning congruence.

No doubt, enough context will always clarify which meaning is, but often not after confusing the brbs out of you.

Maybe its just because as an undergraduate I read more textbooks (and hence read more old textbooks), which are intra-consistent but not inter-consistent with regard to usage. Hopefully modern journals and textbooks have adopted more standardized notation, and I will one day be able to appreciate a world in which mathematics actually is the universal language, at least between mathematicians.

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