Because a number isn’t just a representation of a size or amount - that’s called a scalar. A number can also represent a point in a space, the label of a vertex on a graph and probably some other things too.
When you count upwards you start from 1, and go up. However, when you count down you usually end on 0. Surely this means 0 satisfies the definition.
The natural numbers are derived, according to Brouwer, from our intuition of time of time by the way. From this notion, 0 is no strange idea since it marks the moment our intuition first begins ^_^
On the contrary - to be countabley infinite is generally assumed to mean there exists a 1-1 correspondence with N. Though, I freely admit that another set could be used if you assumed it more primitive.
I’m arguing from the standpoint that we establish the idea of counting using the naturals - it’s countable if it maps to the naturals, thus the link. Apologies for the lack of clarity.
I don’t personally know many programming languages that provide natural number type in their prelude or standard library.
In fact, I can only think of proof assistants, like Lean, Coq, and Agda. Obviously the designer of these languages know a reasonable amount of mathematics to make the correct choice.
There can’t really be an argument either way. It’s just a matter of convention. “Natural” is just a name, it’s not meant to imply that 1 is somehow more fundamental than -1, so arguing that 0 is “natural” is beside the point
If we add it as natural number, half of number theory, starting from fundamental theorem of arithmetics, would have to replace “all natural numbers” with “all natural numbers, except zero”.
I’m not too good at math but i think it’s because the set of integers is defined as the set that contains all natural numbers and their opposites, while the set of natural numbers is defined using the successor function - 0 (or 1) is a natural number; if a number n natural, then S(n) is natural where S(n) = n+1.
Counterpoint: if you say you have a number of things, you have at least two things, so maybe 1 is not a number either. (I’m going to run away and hide now)
I just found out about this debate and it’s patently absurd. The ISO 80000-2 standard defines ℕ as including 0 and it’s foundational in basically all of mathematics and computer science. Excluding 0 is a fringe position and shouldn’t be taken seriously.
Ehh, among American academic mathematicians, including 0 is the fringe position. It’s not a “debate,” it’s just a different convention. There are numerous ISO standards which would be highly unusual in American academia.
FWIW I was taught that the inclusion of 0 is a French tradition.
The US is one of 3 countries on the planet that still stubbornly primarily uses imperial units. “The US doesn’t do it that way” isn’t a great argument for not adopting a standard.
This isn’t strictly true. I went to school for math in America, and I don’t think I’ve ever encountered a zero-exclusive definition of the natural numbers.
I’m an American mathematician, and I’ve never experienced a situation where 0 being an element of the Naturals was called out. It’s less ubiquitous than I’d like it to be, but at worst they’re considered equally viable conventions of notation or else undecided.
I’ve always used N to indicate the naturals including 0, and that’s what was taught to me in my foundations class.
I could be completely wrong, but I doubt any of my (US) professors would reference an ISO definition, and may not even know it exists. Mathematicians in my experience are far less concerned about the terminology or symbols used to describe something as long as they’re clearly defined. In fact, they’ll probably make up their own symbology just because it’s slightly more convenient for their proof.
I hate those guys. I had that one prof at uni and he reinvented every possible symbol and everything was so different. It was a pita to learn from external material.
My experience (bachelor’s in math and physics, but I went into physics) is that if you want to be clear about including zero or not you add a subscript or superscript to specify. For non-negative integers you add a subscript zero (ℕ_0). For strictly positive natural numbers you can either do ℕ_1 or ℕ^+.
An English dictionary is not really going to tell you what mathematicians are doing. Like, its goal is to describe what the word “integer” means (in various contexts), it won’t tell you what the “integer series” is.
The gist I see is that it’s kind of ambiguous whether the whole number series includes negatives or not, and in higher math you won’t see the term without a strict definition. It’s much more likely you’d see “non-negative integers” or the like.
I have seen arguments for zero being countable because of some transitive property with not counting still being an option in an arbitrary set of numbers you have the ability to count to intuitively.
But there are an infinite number of things that you don’t have any of, so if you count them all together the number is actually not zero (because zero times infinity is undefined).
Other fun arguments in the same vein: Is atheism a religion? Is not playing golf a sport? For extra fun, try explaining the answers to both in a non-contradictory way.
How are those the same? You need to define “religion” and “sport” rigorously first.
Since you haven’t provided one, I’ll just use the first sentence on the wiki page:
Religion is a range of social-cultural systems, including designated behaviors and practices, morals, beliefs, worldviews, texts, sanctified places, prophecies, ethics, or organizations, that generally relate humanity to supernatural, transcendental, and spiritual elements.
“Atheism,” without being more specific, is simply the absence of a belief in a deity. It does not prescribe any required behaviors, practices, morals, worldviews, texts, sanctity of places or people, ethics, or organizations. The only tenuous angle is “belief,” but atheism doesn’t require a positive belief in no gods, simply the absence of a belief in any deities. Even if you are talking about strong atheism (“I believe there are no deities”), that belief is by definition not relating humanity to any supernatural, transcendental, or spiritual element. It is no more religious a belief than “avocado tastes bad.” If atheism broadly counts as a religion, then your definition of “religion” may as well be “an opinion about anything” and it loses all meaning.
If you want to talk about specific organizations such as The Satanic Temple, then those organizations do prescribe ethics, morals, worldviews, behaviors, and have “sanctified” places. Even though they still are specifically not supernatural, enough other boxes are checked that I would agree TST is a religion.
I have no idea what you’re on about with not golfing being a sport.
How are those the same? You need to define “religion” and “sport” rigorously first.
This is really the crux of the argument. There are no absolute authorities on religion, sport, or in the case of the original post, mathematics. We can have definitions by general consensus, but they are rarely universal and thus it’s easy to cherry pick a definition that supports any particular argument with no ability to appeal to authority.
I have no idea what you’re on about with not golfing being a sport.
It’s mostly a troll argument, but you can easily trip up people with interchanging the definition of “sport” as a thing (“golf is a sport”) or an activity (“playing golf is a sport”). Then after trying to hammer down the definition more exactly, you can often poke holes in it with more questions like is chess a sport? Is playing Counter Strike a sport? Is competitive crocheting a sport? All of these ambiguities are possible because of the lack of a universal authority in the realm of sports, though some people try to pick an authority such as the Olympics to prove their point.
I’d argue that atheism is a feature of a belief system and that the system may or may not be a religion. There are religions that don’t feature a belief in any gods. Similarly, your personal belief system may not be a full blown religion, even if you did happen to be theistic.
Oh no, are we calling non-negative integers “whole numbers” now? There are proposals to change bad naming in mathematics, but I hope this is not one of them.
On the other hand, changing integer to whole number makes perfect sense.
I’d learned somewhere along the line that Natural numbers (that is, the set ℕ) are all the positive integers and zero. Without zero, I was told this were the Whole numbers. I see on wikipedia (as I was digging up that Unicode symbol) that this is contested now. Seems very silly.
I wouldn’t be surprised. I also went to school in MS and LA so being taught math poorly is the least of my educational issues. At least the Natural numbers (probably) never enslaved anyone and then claimed it was really about heritage and tradition.
Actually “whole numbers” (at least if translated literally into German) exist outside America! However, they most absolutely (aka are defined to) contain 0. Because in Germany “whole numbers” are all negative, positive and neutral (aka 0) numbers with only an integer part (aka -N u {0} u N [no that extra 0 is not because N doesn’t contain it but just because this definition works regardless of wether you yourself count it as part of N or not]).
Natural numbers are used commonly in mathematics across the world. Sequences are fundamental to the field of analysis, and a sequence is a function whose domain is the natural numbers.
You also need to index sets and those indices are usually natural numbers. Whether you index starting at 0 or 1 is pretty inconsistent, and you end up needing to specify whether or not you include 0 when you talk about the natural numbers.
Edit: I misread and didn’t see you were talking about whole numbers. I’m going to leave the comment anyway because it’s still kind of relevant.
In some countries, zero is neither positive nor negative. But in others, it is both positive and negative. So saying the set of natural number is the same as non-negative [integers] doesn’t really help. (Also, obviously not everyone would even agree that with that definition regardless of whether zero is negative.)
the standard (set theoretic) construction of the natural numbers starts with 0 (the empty set) and then builds up the other numbers from there. so to me it seems “natural” to include it in the set of natural numbers.
On top of that, I don’t think it’s particularly useful to have 2 different easy shorthands for the positive integers, when it means that referring to the union of the positive integers and the singleton of 0 becomes cumbersome as a result.
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