When we say $1,2,3…$ are natural numbers, why don't we include rational and irrational numbers?
Isn't $\pi$ something natural?
Shouldn't we say all real numbers the Natural numbers?
Shouldn't we, just, say them "Counting Numbers"?
Maybe I don't know the correct reason for which the word "natural" is prefixed. So, please tell me IYK.
Best Answer
In the formative days of modern mathematics, there was some debate as to what the natural numbers should be considered. Grassman even suggested that the natural numbers are a result of a recursive definition, and as such should not be considered natural. Later, the Bourbakis decided that zero should be included in the naturals; differing conventions exist to this day.
The decision of the nomenclature "natural" largely became one of the convention settling to an almost-stable state -- if any numbers are to be called naturals, it is either $\{0,1,2,\ldots\}$ or $\{1,2,\ldots\}$.
As this set of numbers can be used to wholly-construct the reals, one might say that the naturals are the bottom-most foundation of the real numbers.
Another example is to consider anthropological evidence. It is known that many civilizations in antiquity separately came upon the concept of counting systems. Some included zero, some excluded zero as a placeholder digit, some even excluded the number one. Because these conclusions were established more or less in parallel between societies that had no contact, one might think that the naturals are the most fundamental natural consequence of human intellect.
Coming full circle, however, to support Grassman's standpoint on the issue is the fact that other societies don't have a concept of numbers beyond "a few." This is evidenced by many languages, some surviving today in tribal cultures, that don't have counting words, instead having words for "one," "two," and "many." Indeed, as it is known that language affects neural plasticity, perhaps the natural numbers are not natural at all, as there are groups of living humans who, in the entire history of their lineage, have never known the concept of 87.392.