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This question is derived from just curiosity, but is there a specific reason for the situation I stated in the title? I am pretty sure that "positive integer" = {1, 2, 3, 4, ...}, which is just a set of natural number. But the majority of textbooks use the term "positive integer" instead of "natural number" when they explain some definition. (I can't find the proper tag for this post so I'm just using \calculus tag btw)

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    I use positive integer and non-negative integer so there is zero ambiguity. – Asinomás Apr 21 '21 at 15:25
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    "which is just a set of natural number" Be warned that some authors use $\Bbb N = {0,1,2,3,\dots}$ and other authors use $\Bbb N={1,2,3,\dots}$ – JMoravitz Apr 21 '21 at 15:27
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    I linked a thread above. The point is, different sources have different ways of defining terms like "natural number" and "whole number". It is better to be unambiguous. – Deepak Apr 21 '21 at 15:28
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    Mathematically, there is nothing Whole or Natural about numbers other than its literary history. So just call them positive integers, negative integers and zero and get over with this pointless confusion. – Nilotpal Sinha Apr 21 '21 at 15:51
  • This question is not a duplicate. The other question-asker is aware that there is a distinction between conventions, and this question-asker isn't. – hunter May 01 '21 at 23:38

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Unfortunately the English terms are not standard. Speaking anecdotally, in grade school in the US in the nineties, we learned that "natural numbers" mean "positive integers" and "whole numbers" mean "non-negative integers."

In academic math, however, more commonly "natural numbers" means "non-negative integers" (although sometimes it excludes zero), and "whole numbers" is not much used. Best to use the unambiguous phrase.

hunter
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    Speaking anecdotally myself, I tend to find that the algebraists in my life tend to start $\Bbb{N}$ at $0$ (due to the algebraic utility of an identity) and the analysts tend to start $\Bbb{N}$ at $1$ (since nobody wants to have to write $\frac{1}{n + 1}$ for their generic sequence tending to $0$). – Theo Bendit Apr 21 '21 at 15:33
  • With regard to the phrase "English terms", I don't think the language is relevant here. It's not about linguistic imprecision, there are actually conflicting mathematical conventions for defining these sets. But there is zero ambiguity in saying "positive integers". – Deepak Apr 21 '21 at 15:35
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    @Deepak Well, I think "English terms" is pretty accurate here. In french, $0$ is commonly taken as a natural number an $\mathbb{N} = {0,1,\ldots }$. Thus, linguistic translations can lead to two different mathematical objects. Also, in french, "positif" means "non-negative" and "strictement positif" (litteraly, "strictly positive") means positive! – Didier Apr 21 '21 at 15:41
  • @Didier Fair enough. When I talked about "linguistic imprecision" I was thinking about situations where natural language isn't a good fit for describing mathematical statements, like the subtly different implication of the word "or" in English vs mathematical logic. – Deepak Apr 21 '21 at 16:03
  • A side remark, for non mathematicians, men-in-the-street, "positive" is unambiguous, it means $>0$ (it would like strange to them to hear that 0°C is a positive temperature for example). – Jean Marie Apr 21 '21 at 16:15