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If we are talking just about '5', without it being with respect to any 'amount', does the idea of the number itself as a point on a line imply that it is itself some kind of abstract 'quantity'. It can be used to talk about quantity, and as an idea or point on a line represent the idea of a certain either whole or continuous quantity, but is it, in it's very existence a quantity itself?

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    If it was, it would have to be an amount of something. So the question appears, "5 is what amount of what?" The close-knitting of these numbers, and elements/sets, supplies one answer: the number 5 is 5 sets (or, the {}-notation iterated 5 times over the empty element, or whatever). Commented Apr 3, 2022 at 18:59
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    No. Numbers are symbols, which not only signal quantities, but also other properties, like identity, priority, enumeration, order, etc. Prisoner 671176 is not necessarily member of a jail having 671176 inmates. In this case, 671176 is not a quantity, and does not represent the idea of a quantity.
    – RodolfoAP
    Commented Apr 3, 2022 at 21:04
  • That is a good point. Perhaps though, these notions are closely related to the idea of quantity, to have priority, enumeration there is a concept of a quantity, as there is only 'three' people above you etc, If we consider a 'number' to be a quantity of abstract units, could we then define order, or member ship as being the one who makes the quantity have 3 abstract units.
    – Confused
    Commented Apr 3, 2022 at 22:02
  • If you define numbers to be the set of all logic objects obtainable by finite or infinite series of arithmetic operations on instances of the number 1, numbers are points on a 2-dimensional plane of complex numbers, not a 1-dimensional line of reals. e.g. 1+(1+1)(1-1-1)^(1/(1+1)) = 2i+1.
    – g s
    Commented Apr 4, 2022 at 19:19
  • I think if we are imagining that there is a 'oneness' or 'threeness' which is universal, there's some merit. But is there a universality to 'minus fourness' or '6i-ness'. It seems less plausible, and so what is special about positive real numbers?
    – Richard
    Commented Apr 5, 2022 at 0:17

5 Answers 5

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They are abstractions, ways of grouping and pointing at what things can have in common. When we use mathematics on the world, we make choices about formal equivalence, eg of one orange with another, and ignore a lot of detail like shelf life.

Number lines, are an abstraction of continuous symmetries, ie the idea of the mathematics of moving things along a line, regarding the positions as formally equivalent but with different labels. Discussed here, with background about concept-forming: As humans, do we require a total understanding of information to fully embody it as knowledge?

People get in knots over whether imaginary-numbers, infinities, or zero are 'real'. They are labels, they help link domains of mathematics, that helps our minds impact the world - so in that sense they are 'real'. Our world has 'real patterns', like continuous symmetries, and objects alike enough to consider them formally equivalent (eg, the physics of bosons explicitly depends on this). Reflecting such fundamental patterns in abstractions could be described as the maths being 'out there', but I'd call that misleading. We thought geometry was 'out there', but realised Euclidean geometry is just a local simplification.

The picture of truth from mathematics is now understood to be about the relation between definitions and abstractions, and we have learned a better picture of truth comes from paying close attention to which abstractions are valid, by observations. See: Philosophical assumptions underlying science

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Depends on precisely how you want to define "quantity". Since one abstract property of numbers (integers, rationals, reals) is a total order 1<2<3<etc, then, yeah, m<n means the (abstract) "quantity" of n is greater than that of m. But if you strictly mean concretely, then, as mentioned earlier, you need physical units, e.g., 2pounds<3pounds. And note that forall units x : 2x<3x, which gives some additional credence to the abstract notion 2<3. And Webster https://www.merriam-webster.com/dictionary/quantity seems to have a bit of an abstract bias, too.

But physics, on the other hand, has a very definite penchant for physical units. And although there are various dimensionless constants, e.g., the fine structure constant ~1/137, and the proton-to-electron mass ratio ~1836, it would be physically meaningless to suggest one is "greater" than the other.

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I agree with you that numbers like 1,2,3, … originated as abstraction from quantities of physical objects. But that’s not the last word in mathematics about numbers:

A first big step was the introduction of the number „zero = 0“: Here are no quantities at all to abstract from. And similarly the negative numbers, the rationals, the irrationals, the reals, the complex numbers ... . And even more: Different infinite cardinals as introduced by Cantor.

Along this way the numbers obtained an existence of their own. They are no longer linked to quantities. Contemporary number theory studies the algebraic properties of numbers, e.g., the operation of addition and multiplication and the inverse operations if they exist. The first topic of number theory is the factorization of numbers into primes and studying the many properties of primes.

Hence numbers in number theory are no longer considered from the viewpoint of quantities, but from the viewpoint of abstract structures with new formal properties.

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  • Even if we think of them for representing quantities, are they themselves quantities? Can we think of numbers as representing a quantity of abstract 'units'?
    – Confused
    Commented Apr 3, 2022 at 20:49
  • Even in cases where numbers represent quantities i do not consider them quantities themselves but abstract symbols: Numbers like words or names belong to a different category than the objects which they refer to.
    – Jo Wehler
    Commented Apr 3, 2022 at 20:55
  • Perhaps, it's all a question of choice, perhaps though the idea of the number, and the amount of abstract 'units' that define it are different?
    – Confused
    Commented Apr 3, 2022 at 22:03
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Short Answer

Are numbers, given just as mathematical objects, quantities in themselves?

Yes, however numbers can be interpreted as more than quantities, as they are often numerals, descriptions, structures, ordinals, and identifiers.

Long Answer

First, the term 'number' is a broad term, and is related to other terms such as numerals. It's best to think of 'number' as a linguistic entity in the same category as 'ethical', 'truth', 'real', or 'love'. What it means to be a number is at the heart of the debates in the philosophy of mathematics. To start off, let's start with a quotation by Leopold Kronecker. From cantorsparadise.com:

Natural numbers were created by God, everything else is the work of men — Kronecker (1823–1891).

This broadly captures a sentiment that is amenable to a philosophical position known as mathematical constructivism. To an intuitionist such as myself, subitization and intuitive pairing such as those used in Cantor's diagonal proof are the psychological and intuitive basis of all numeracy. Something like a theory of pairing is a complex language constructed to capture intuitional ideas long after the fact. In fact, the brilliant Charles Sanders Pierce held empirical ideas of mathematics, and quasi-empiricism in mathematics, though a minority position holds much appeal to those who rely heavily on empirical evidence.

To a mathematical intuitionist, everything after intuition is essentially a construction of language to make rigorous intuition using formal systems. For instance, in many paleolithic cultures, there are no numbers other than none, one, and many, and mathematical praxis consists of pairing things until there's leftovers. Even in English, 'eleven' and 'twelve' comes from Old English and means one and two left (over), because if you pair 11 and 12 items with ten, there is a remainder of 1 or 2, respectively.

So, this counting from 1 to n, as Kronecker sees it is the basis of what we have when we have numbers, we have labels for ordering and counting. Thus, a number may be quantity or a quality, in the case of indicating where in a sequence something is. English, of course, still retains a difference in syntax to reflect a quantity 'five', and an ordinal 'fifth'. There's a semantic difference between five pebbles and the fifth pebble. In modern mathematics, a sequence is usually defined as a mapping between the naturals and another set of numbers.

Also, since ordinals are useful, they can also be used as identifiers. In many organizations such as prisons, people are given unique identifiers that allow quick and easy access to information related to the person. In the language of modern computation, these are called primary keys. Numbers in this regard can actually become symbolic of evil, such as the tattoo of a number on the forearm of a prisoner of Auschwitz. Such images of numbers are frequently used in art to evoke feelings or provide context. Here, is a great example of how a number is more than just a quantity.

Let's raise another issue of how a number is more than a quantity, per se. Imaginary numbers are defined as that which is historically taken to be undefined, that is the root of a negative number. If there is no real that can be squared to get a negative real, what does it mean to count to 5i? In fact, George Lakoff and Raphael in their Where Mathematics Comes From makes a compelling argument that one can in fact count to 5i, if i's are construed as rotations around the Argand plane. The notion of i was once very controversial, but you won't find any educated electrical engineer decry their existence: they're just too darned useful not to be real! (In philosophy, what is real is a question for ontology.)

So, are numbers a quantity, in one sense, yes, but in many senses no. But this is an excellent question, because it leads into the philosophical about the ontological status of mathematical objects, and helps to characterize a philosophical position. For someone like Plato, for instance, numbers were not just quantities, they were Forms, and as such, numbers were proof that our existence was illusion!

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No they are not. Numbers in and of themselves are closer to an alphabet than they are an abstract measure of quantity. What makes us think of them naturally as quantities is the fact that we tend to associate some basic concepts of algebra to them (https://en.wikipedia.org/wiki/Group_theory). This is why you will hear people make differing arguments (all reasonable) such as numbers indicate order, abstract quantity, or innate comparison. These arguments stem from the conflation that numbers are defined with inherent algebraic properties - which they are not.

We can pretty easily see how our numbers exist outside of algebra by considering the English alphabet. Is "a" greater than "b"? How many "c"s would it take to have a "g"? Is "s" + "w" = "q"? It is all nonsense until we rigidly define some operations onto this alphabet (perhaps stringing letters together?). What numbers represent more than anything is uniqueness/distinction/difference (whatever you want to call it). Math is a language just like any other. It's definitions are typically more rigid than others - but it is a language nonetheless.

Finally, a point that I find important here is that numbers are defined using sets. This means that order is unimportant and there is only one number in the set that is itself (no duplicates!). So, from the mathematician's point of view, the only difference between any two numbers inherently is that they are not each other. Personally, I find this way of thinking about numbers to be sufficiently boiled-down and abstract.

Also, I just read the part about a number on a number line implying quantity. In that case, yes absolutely. Any number line is specifically defined with an operation in mind - or else there would be no way to order the elements. So no - numbers are not quantities, but yes - numbers on a number line inherently belong to a sequence, group, etc and thus have some quantitative implications.

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