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The definition of natural numbers from Wikipedia:

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

And the cardinal numbers:

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set.

Based on those definitions, I think that finite cardinals are equivalent to natural numbers ("The cardinality of a finite set is a natural number: the number of elements in the set."). But I noticed that philosophers tend to make a distinction between those two concepts (see, for example, Giaquinto's paper Knowing Numbers). Note that I'm not refering to cardinals in general, but only to finite cardinals which are just sizes of finite sets. Aren't the natural numbers supposed to describe sizes, too? If not, what differentiates these two ideas? Is one part of the other? Are they completely different?

I'm aware that answer to my question will depend on the person answering it. For example, mathematicians, philosophers and "regular" people could understand these concepts differently. Obviously, since I'm asking here, I'm most interested in the philosophers' view, but all points of view are welcome.

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    They're the same; the interest in cardinals comes from the fact the definition works for infinite sets and then you have 'infinite numbers' or the infinite cardinals. – Mozibur Ullah Dec 17 '17 at 20:45
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    @MoziburUllah: Thank you for your answer, but I think that at least in the case of an article I linked those two concepts aren't treated the same. For example, a quote from an article: "Even if my case for the possibility of a naturalistic account of knowing cardinals is accepted, one might protest that we still lack a parallel case for the objects of number theory. Those objects are not set sizes; they are, if anything, positions in the natural number structure." I can be wrong, but doesn't the author speak about natural numbers as something different from cardinals considered earlier? – bg5 Dec 17 '17 at 22:44
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    I agree that in the article he doesn't treat these the same - I was merely answering your question; he appears to be centering his discussion largely in terms what it means to know a number; for example he asks 'Do we have acquaintance with cardinal numbers greater than 3?' and 'what about knowledge of much larger numbers?'; I'm not sure what he means by 'set sizes' but the following sentence is illuminating ' Pure mathematicians at work in number theory are not concerned with finite cardinals as opposed to finite ordinals and the myriad other systems that exemplify this structure...' – Mozibur Ullah Dec 18 '17 at 0:22
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    ... they are concerned with the structure itself'; that sounds about right; professional number-theorists don't generally concern them-selves with set theory - they leave that to the professional set-theorist, and in the books they write they often offer a standard disclaimer saying that they won't concern themselves with foundations and take sets to be understood naively. – Mozibur Ullah Dec 18 '17 at 0:24
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    See the Wiki's entry again: "the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country")." Modern set theory generalize the concept of the "counting" aspect of number to cardinal numbers and the "ordering" aspect to ordinal numbers. – Mauro ALLEGRANZA Dec 18 '17 at 14:04
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It is fair to say that the concepts are "equivalent" in the modern mathematical practice. However, they have different histories and different overtones of meaning. The notion of natural or counting numbers goes back (officially) to Pythagoreans and unofficially to prehistoric times, Ishango bone, a counting artifact, is 20,000 years old. We are dealing with a loose concept imported to mathematics from natural language after centuries of practical and then more abstract use. While in one sense cardinals generalize natural numbers, in another sense they are a much more narrow, technical, and context bound notion.

The notion of "cardinal number" or "cardinality" (even finite) is much more recent, it is one way to make the everyday concept more precise for mathematical use, the one favored by the current formalization of mathematics. The notion was introduced by grammarians in 1590-s to distinguish between counting numerals (one, two, three) and order numerals (first, second, third), so-called ordinals. But their modern mathematical life begins at the end of 19th century in Cantor's set theory, which imported the distinction and made it even more technical. Cantor replaced the elementary concept of a number with a derived and abstract one based on sets (for what it is worth, the informal counting conception is closer to his ordinals than to his cardinals, the notions are equivalent in scope in the finite case but diverge dramatically for infinities). Two sets are equipollent if they can be put into a bijective correspondence (this is now called Hume's principle), and the cardinal number of a set is something like the equivalence class of all sets equipollent to it, or so Cantor wanted to say. This turned out to have technical problems, and now more "concrete" definitions are preferred, ones that identify it with one of very specially generated sets like ∅,{∅},{{∅}},... or ∅,{∅},{∅,{∅}},... , see Set-theoretic definition of natural numbers.

This multiple realizability is already a problem for philosophers, we would like to think that there is a unique concept of "two", expressed as 2 or {{∅}} or {∅,{∅}}, but set theory can only provide us with an arbitrarily chosen token for it. This leads to popularity of mathematical structuralism, the philosophical position that it is not what it is that makes 2 a 2 but only its place in a structure, in this case the structure of the counting number series described by the Peano axioms. Peano arithmetic already shows that natural numbers, unlike cardinals, can be lifted off of set theory since it does not depend on it. Neither does category theory, which has its own versions of natural numbers. Even within set theory the notion of "finite cardinal" can come apart from that of a natural number if we play around with the axioms. For instance, Dedekind defined "finite cardinals" as equipollents of those sets that can not be put into bijective correspondence with their proper subsets. Well, Russell and Whitehead showed in 1912 that Dedekind finite cardinals can be infinite (on the usual conception) in set theory without the axiom of choice, so-called ZF, or other alternative set theories. Intuitively, this is because those models of set theory do not have enough bijections to "detect" infinity Dedekind's way.

Cantor's set theory, and even Hume's principle it is based on, were historically controversial and not without alternatives, see Measuring the size of infinite collections of natural numbers: Was Cantor's theory of infinite number inevitable? by Mancosu. If mathematicians decide one day that they prefer categorical or some other formalization of mathematics to the set-theoretic one the notion of cardinal number may go away, but we can be sure that the natural numbers will still be around.

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If two different names are used to describe something that seems similar, it is useful to maintain the distinctions since they may lead to interesting conclusions and also to ways to justify any disagreement with the conclusions.

Take the natural number 3. We can add 3 to some other natural numbers and get a result that is again a natural number. The addition does not depend on anything but those natural numbers.

Take the cardinal number 3 as the property of, say, the number of chairs around a particular table. Call this the HowMany property. Although this 3 could be viewed as an object in a set of cardinal numbers, it is also, in this particular example, the value of the HowMany property of another object, the chairs around a table. Can we add or multiply this 3 with any other cardinal number like we did with the natural numbers above? No. To change the value of the HowMany property of chairs around the table we have to change the number of chairs around the table.

In Marcus Giaquinto’s “Knowing Numbers” he wants to show that we don’t need a supernatural mode of apprehending numbers by showing that we can know them naturally. I can imagine that this is not easy to do, but a simple place to start would be with cardinal numbers viewed as the values of HowMany properties of natural objects.

Is Giaquinto right? Paying attention to the distinctions he makes is helpful to reach a conclusion one way or the other.

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