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I came across https://books.google.com/books/about/Mathematical_Writing.html?id=wpQvBQAAQBAJ Where it talks abt the difference of non and not

Sounds like mathematician use non to mean “not necessarily” or “not definitely”

Where as “not” means “definitely not”

Does this mean mathematician always mean “non” as “not necessarily”?

Update

Sorry, I just realize the link above doesn't take you to the exact page.

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Kim Stacks
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  • It is hard to say with any certainty. But sometimes this is the convention. – rschwieb Jun 15 '20 at 21:06
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    I would say that "a non-zero real number $x$" means "$x\ne0$". Also "non-decreasing" is commonly used for "weakly increasing", so I guess that's more of an indication than it is a rule. –  Jun 15 '20 at 21:11
  • Is the set of cases where "non-whatever" means "not necessariily whatever" even non-empty? – Hagen von Eitzen Jun 15 '20 at 21:42
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    I think that this is more a question of English grammar than mathematics. – badjohn Jun 15 '20 at 21:54
  • How would a mathematician use it then? – Kim Stacks Jun 15 '20 at 21:55
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    Can you provide an actual example of your problem? "non" is not an English word, but rather a prefix used to form some English words but not according to any general rule: "non-decreasing" is a good word but "non-significant' just isn't a word (the right word is "insignificant"). – Rob Arthan Jun 15 '20 at 22:45
  • "non-negative" and "non-zero" seem to mean definitely not. So I'd say, no, mathematicians don't always do this. – fleablood Jun 15 '20 at 23:49
  • The more I think about it I'd say mathematicians never say "non" to mean "not nescessarily". At least, I can' think of any case where they do and every case I can think of they don't. Trouble is there doesn't seem to be any word for "not necessarily". – fleablood Jun 16 '20 at 00:06
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    Can you link to the passage in the book where they discuss this? – fleablood Jun 16 '20 at 00:07
  • reproduce the text in book as image when i realize the link doesn't really work – Kim Stacks Jun 16 '20 at 00:22
  • Thanks for including the extract. I think the author of the book was having an over-pedantic moment when he wrote the bit about a "not negative" function. I wouldn't expect anyone to know what I meant if I wrote "$f$ is negative" or negated that and wrote "$f$ is not negative". If you want to be clear, be explicit and say "$f$ is nowhere positive" or "$f$ is everywhere negative". Always take style guides with a pinch of salt! – Rob Arthan Jun 16 '20 at 00:33

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The prefix 'non' is not always used inclusively. For example 'let $x$ be nonzero' definitely does not allow the case that $x$ is $0$, and 'let $x$ be nonnegative' means that $x$ is either positive or $0$.

Oscar Cunningham
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    I don't understand your point; In both of your examples, the prefix "non" denotes a simple negation: "inclusivity" or (to be more relevant to the question") "necessity" is not meaningful in your examples. – Rob Arthan Jun 16 '20 at 00:23