What is a natural number?

Question

It’s been on my mind lately. I do maths and work with them daily, but I’m not entirely sure of what they really are.

I understand they are symbols at a surface level, but there is obviously more to it. I’ve seen the set theoretic definition where they define the natural numbers in terms of empty sets. counting the elements of a set to be crafting a bijection from the set to the number n (matching the elements of the set with the empty sets).

Obviously the rest of the numbers (integers, rationals) are easily constructible after the naturals have been defined, so I feel like if I understand them better it will make me feel better.

I’ve seen the Peano axioms which generalise the properties, but where did these properties come from?

I am experiencing derealisation and it’s uncomfortable. I posted on maths exchange and they told me to post it here, or realise the properties of the natural numbers. But where do the properties come from?

Answer 1

As far as I can tell, Peano's Axioms arose from properties of the counting (natural) numbers that have been known for thousands of years:

1. Zero is a natural number
2. Every natural number has a unique successor that is itself a natural number
3. Different natural numbers have different successors
4. Zero is not the successor of any natural number
5. Every natural number but zero can be reached by a process of repeated succession starting at zero

These roughly correspond to Peano's Axiom's as expressed in the language of set theory as follows:

1. 0 ∈ N
2. ∀x ∈ N: S(x) ∈ N
3. ∀x,y ∈ N: [S(x)=S(y) ⟹ x=y]
4. ∀x ∈ N: S(x)≠0
5. ∀P ⊂ N: [[0 ∈ P ∧ ∀x ∈ P: S(x) ∈ P] ⟹ P=N]

Where N is the set of natural numbers, and S is the successor function on N.

You might say that a natural number is just an element of a set on which Peano's Axioms hold. The conventional labelling has 1=S(0), 2=S(1), 3=S(2), and so on. Other labelings are, of course, theoretically possible. The only number actually defined in Peano's Axioms is 0. There is nothing to stop the user from defining an unconventional symbol other than 1 for S(0), e.g. S(0)='2', S(2)='4', etc.

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Also, the notion of numbers being sets constructed from an empty set (as in ZFC theory), is not universally held in practice. None other than one of the most renowned mathematicians living today wrote:

"Among all the objects studied in mathematics, some of the objects happen to be sets; and if x is an object and A is a set, then either x ∈ A is true or x ∈ A is false. (If A is not a set, we leave the statement x ∈ A undefined; for instance, we consider the statement 3 ∈ 4 to neither be true or false, but simply meaningless, since 4 is not a set.)"

--Terence Tao, "Analysis I," p. 34-35

Answer 2

Begin with the idea of being. The being can be of an abstract mathematical object like Boolean TRUE, a finite instruction like "step forward", or an identifiable physical process like an outfit, a rock, or an army. What matters is that you can either have it, or not have it.

Introduce the idea of incrementation. If you can have rock, you can have rock and rock. If you can have rock and rock, you can have rock and rock and rock. Thence you can have rock and rock and rock and rock and so on.

Introduce decrementation, which is incrementation in reverse.

Create a symbol to represent the smallest amount of being, when you can't decrement any more without losing being, and a syntax that lets you describe objects with it. "Killer rabbit, when you can't have any less killer rabbit" becomes succinctly expressible as "One killer rabit."

Create a symbol to represent that smallest amount, incremented.

Create another symbol to represent the incrementation of that amount, and so on. Now you have the natural numbers.

However, you also need infinite symbols to represent all of the possible naturals, which is inconvenient for beings in a finite universe. You even need uncomfortably many symbols to represent enough naturals to count your family members or your steps.

Notice that you can iterate groups just like you can iterate individuals.

• Become amazed by this superb and numinous quality and spend the next ten thousand years trying to use numbers to do sorcery.

There will be one or more particularly salient numbers in your environment, like the number of digits on both hands (10), the number of knuckles you can touch one one hand with the thumb of the same hand (12) - which is also the roughly same number as lunar cycles for every cycle of the zodiac, the largest number of round sticks or stalks you can gather together without them shifting around (7), digits on one hand (5), the smallest number of round sticks or stalks you can gather together without them shifting around, which is also the ideal number of sticks to use to make a structure on uneven ground (3), number of ways that presence or absence can be (2).

• Have another digression in which you think that these numbers are magical and incorporate them into your culture and religion, while ascribing demoniac forces to selected incrementations or decrementations thereof.

Pick one of these salient numbers, or any two multiplied together, and express larger numbers as iterations of that number.

Express yet larger numbers as iterations of that number of iterations of that number.

Now you have a number base, which is a convenient way of representing numbers too big to get their own special symbol. Instead of needing 124 symbols to represent that you left home with 124 sheep and you should be going home again with 124 sheep, you can just know that you have two sixties and four; ten twelves and four; a ten of tens, two tens and four; or, if you're a robot, a two of twos of twos of twos of twos of twos, a two of twos of twos of twos of twos, a two of twos of twos of twos, a two of twos of twos, and a two of twos.

You can see why the last option didn't catch on for counting sheep.

Answer 3

This is an old and subtle question in philosophy of mathematics, but as a starting point, it’s hard to do better than the classic Benacerraf 1965 paper, What numbers could not be. It’s very readable — I definitely recommend clicking through! — and lays out several of the established answers, arguing persuasively for a structuralist position: that any choice of “the natural numbers” as specific objects is arbitrary, and the crucial thing is the structure of the (ordered) set of them. As Benacerraf summarises near the end:

Arithmetic is therefore the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions. It is not a science concerned with particular objects — the numbers. The search for which independently identifiable particular objects the numbers really are (sets? Julius Caesars?) is a misguided one.

An important followup paper is McLarty’s 1993 Numbers can be just what they have to, arguing that categorical theory gives a better foundation than ZF for articulating the mathematical structuralism that Benacerraf proposed.

Answer 4

Set theory such as ZF is a customary framework in which one expects all higher-level objects to be developed, but the particular construction such as von Neumann's shouldn't be taken as "the natural numbers" but merely one way of constructing them. Von Neumann's construction is convenient because the order relation among natural numbers becomes simply the inclusion at the set-theoretic level.

Thus, if 0,1,2, etc. are represented by {}, {{}}, { {}, {{}} }, etc. (hope there are no misprints here :-), then the inequality 1<2 corresponds to the membership {{}} ∈ { {}, {{}} }, etc.

Again you shouldn't be discouraged if this seems overly complicated (it is!). It is merely one of several possible constructions. The relevant properties of the natural numbers as far as your stage of learning is concerned are given by the Peano Axioms involving the successor function, induction, etc. (Copied from https://math.stackexchange.com/a/4827700/72694).

The only numbers that we really have a handle on are the metalanguage integers, which form a sorites-like subcollection (not subset!) of the object-language integers. The latter may not even be unique, contrary to a Platonist's dream.

Answer 5

I suggest you read Can Fish Count?: What Animals Reveal About Our Uniquely Mathematical Minds, by Brian Butterworth, which:

...shows that every creature shares a deep-seated Darwinian ability to understand the intrinsic language of our universe: mathematics.

One example of the utility of natural numbers is provided by lions. Some lions encounter some others, and one group is trespassing on the other's territory; do they attack or hide? According to Butterworth, 3 lions will attack a solitary intruder (an easy calculation), and 6 will attack 3 intruders (a bit more tricky).

The natural numbers, IMHO, are "out there" in the world. Lions, fish, and bees have no need to do calculations involving negative numbers, rationals, irrationals, or complex numbers, but they need the natural numbers.

Answer 6

We'll consider both syntax ("How do we describe the natural numbers?") and semantics ("How do we conceptualize the natural numbers?")

Syntax

In second-order logic, the natural numbers are the elements of the unique set, up to unique isomorphism, which has a distinguished element 0, a successor operation S, and which satisfies the following three axioms:

No successor of a natural number is 0:

∀x. ¬(S(x) = 0)

Equality of natural numbers is discrete and decidable:

∀x. ∀y. (S(x) = S(y) → x = y)

And the second-order induction axiom:

∀X. ([X(0) ∧ ∀y. (X(y) → X(S(y)))] → ∀y. X(y))

This phrasing comes from SEP's article on second-order logic and is standard in second-order presentations.

It is reasonable to wonder where these axioms originate. They were refined by many mathematicians over many decades in an attempt to reverse-engineer various structures; these axioms, just like Peano's axioms, were chosen both for their intuitive appeal and their utility in proofs. Note that second-order arithmetic Z₂ has addition, multiplication, and well-ordering; as such, it has more axioms defining those operations.

Semantics

External

The natural numbers are the initial semiring; that is, for any other semiring, there exists a unique mapping (up to unique isomorphism) from the natural numbers to that semiring. Thus, we may think of the natural numbers as operations in semirings.

Semirings naturally occur in computer science. For example, natural numbers correspond to certain elements of Kleene algebras and regular languages. Matrices over semirings are also semirings, as explored in Dolan 2013, "Fun with Semirings" (slides). Natural numbers also correspond to certain algebraic data types and arise in type-level arithmetic, as used in Blass 1994, "Seven Trees in One", which includes a characterization of the semiring of types.

Also, several answers have pointed out that the natural numbers are initial Church numerals, which is a way of formalizing the intuition that the natural numbers are: do nothing, do it once, do it twice, …

Internal

As mentioned by other answers, there are nonstandard models of the natural numbers, like polynomial semirings or tropical semirings.

The concept of Church numerals can itself be internalized, and yields natural numbers objects (NNOs) (WP, nLab), which generalize the familiar set-oriented construction that many mathematicians learn with e.g. ZFC. An NNO allows us to take an element of an object, along with a repeatable operation on that object, and not perform it; perform it once; perform it twice; …

NNOs are often found in Cartesian closed categories (WP, nLab) or topoi (WP, nLab), which are ubiquitous in computability theory and logic. Boolean algebras do not have NNOs in this sense, because they either don't prove something, or prove it once. But provability logics in general can fail to prove something, prove it exactly one way, prove it exactly two ways, …

Note that, for a particular choice of category, we speak of the NNO when it exists.

Answer 7

I'm not sure this is a direct answer to your question, but I think it helps answer 'where do natural numbers come from' in a very concrete way. According to a lecture on the history of numbers I saw, the first known use of a symbolic number system came out of a record keeping system for flocks of sheep in Mesopatamia.

Each year (maybe more) a jar would be filled with tiny clay balls, one for each sheep in a flock. Later, when someone wanted to know the what the number of sheep was at that time, they would break the jar open and see how many balls were in it. This one-to-one abstraction between the 'count' of the balls and the 'count' sheep is the essence of natural numbers.

IIRC, after this system was in place for thousands of years, someone came up with the idea of marking the outside of the jar so that the count of balls in the jar could be seen without cracking it open. The count of the sheep, the balls, and the marks are all independent physical objects, but the concept of the number is the same for all of them. For example, if I have 3 sheep, then there are 3 balls, and 3 hatch marks (perhaps). All physically distinct but they share a common abstract property of their number: three.

What fascinates me about this history is that after the idea of marking the jars came into common practice, they continued to put the balls into the jars for thousands of years before switching to just marking flat clay tablets. This suggests to me the abstract concept of 'natural number' wasn't as obvious or intuitive as it might seem today and that it was actually a technological development. This is also supported by reports that contemporary stone-age societies don't have a fully developed concepts of natural numbers. E.g., they might have words for 1, 2, and 3 but beyond that, it's just 'many'. In other words, I believe the idea that 'natural numbers' are an inherent property of nature is an illusion.

Answer 8

As you can see, there's quite a variation in themes about what natural numbers are, and part of the reason is that different philosophies of mathematics presume different things about mathematics as presuppositions to natural numbers. Your post reminds us of Kronecker's quotation: God made the integers; all else is the work of man. There is, therefore not a canonical answer (which means that not all of the great thinkers agree), so the question, as many in philosophy, is controversial. There are two broad ways from which you can approach this question in contemporary philosophy. The first is the realist position, and the second is the anti-realist position. All particular positions fall under the umbrella philosophy of mathematics (SEP), so it helps if you're familiar with the basics of structuralism, empiricism, constructivism, etc.

For realists, the hero and inspiration is Plato and his Platonic realism (SEP) though Gödel and others have more contemporary theories. In some way, natural numbers are somehow real things like rocks and birds. Modern mathematicians, however, are empirically more sophisticated than Plato, and so natural numbers are often case as abstract objects (SEP). From the SEP:

Frege famously distinguished two mutually exclusive ontological domains, functions and objects. According to his view, a function is an ‘incomplete’ entity that maps arguments to values, and is denoted by an incomplete expression, whereas an object is a ‘complete’ entity and can be denoted by a singular term.

In this way mathematical structure and entities are taken to be objective and real in some way or other.

Contrary to that are views that numbers are the construct of empirical experience or products of the mind, mere fictions that have a utility deriving from their origins and thought and language. These are positions like fictionalism in mathematics (SEP), intuitionism in mathematics (SEP), or nominalism in mathematics (SEP).

For instance, constructivism in mathematics (SEP) says that natural numbers are products of Peano's Arithmetic and the axioms that comprise it. Thus, what is a natural number? With the exception of the first natural number, it is merely a successor function likely recursively calling itself. N is the set comprised of some n where n might be s(s(s(s(s(0))))) which is the same thing as 5. And a natural number only exists once it is constructed.

Are the natural numbers degraded version of Platonic Forms? Are they a set of recursive successor functions? Are they idiosyncratic names for positions in a physical sequence? Are they natural language ontological artifacts (SEP) to compensate for the limits of subitization? Are they abstract objects our minds discover? Are they products of neural computations like Lakoff and Nuñez claim in Where Mathematics Comes From?

Ultimately, you're going to have decide for yourself what naturals are, because there is no one answer or position that is privileged.

Answer 9

Given their elementarity and the circularity of attempts to reduce them to something else, it stands to some reason that the natural numbers are, for better or worse, sui generis. Though he seems to disagree with this sentiment, I will quote the author of the SEP article on ill-founded set theory to illustrate what I mean:

[The foundation-theoretic] way of describing the iterative picture suggests that the ordinal numbers were somehow present “before” all the iteration takes place, or at least that they have a life apart from the rest of the sets.

(For the point-of-departure treatment of the iterative picture of sets, see the work of Gödel.) An actual proponent of the sui generis take on the natural numbers was, if I remember correctly, Skolem, although don't quote me on that until I find an actual citation!

Why might the natural numbers not be identified merely with iterations of sets? One issue is Benacerraf's identification problem. Consider that:

1. 0 = {}, 1 = {0}, 2 = {{0}}, ...
2. 0 = {}, 1 = {0}, 2 = {0, 1}, ...

... diverge as to whether 0 is an element of 2 or not. When the natural ordinals are formatted one way, we get the one result; in the other, the other; these are staple examples of what are called "junk theorems." (They seem to make no difference to other, more desired proofs; hence they get called "junk.") So which chain of equalities is correct, here?¹

Another option then is called "structuralism," where the natural numbers are reduced to positions in a structure, ℕ say. Or, when structuralism is reductive, it too faces a problem: how to identify a position as the zero position, when every slot in the structure is technically empty? There is still, it seems, a circularity, and a needless one, in the definition ("the first empty slot is the zero slot"), needless that is if we just take the natural numbers to be as fundamental as can be.

Yet another problem, though, is when we can have what are called "nonstandard" natural numbers. For example, if we had a well-founded set of all finite ordinals, this set would not be itself a finite ordinal: it would be infinite. So if we want to deny that there are infinitely many natural numbers, we seem compelled to imagine a self-containing finite ordinal, which will hardly be standard in nature (it will have no distinct successor, for example, but is either its own successor or has no successor at all). For more on more commonly studied nonstandard naturals, see about hypernatural numbers.

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¹Perhaps we shall have cause, someday, to evaluate each and every such reduction in such a way as makes some come out more aesthetically pleasing. For example, if we start with an ur-element u which is such that it is (A) the only element of the set that equals 1 and (B) is only an element of 1, then we would have that 2 = {1} and not {u, 1} without this being merely an effect of notational formatting.^u We would be reluctantly optimistic, then, that proceeding in a similar manner might turn our junk theorems into some sort of treasures, then (or we would look to the idea that one seeker's junk already is another's treasure...).

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^uThere is a sentence, established well enough modulo ZFC, according to which V = WF, i.e. the ZFC world "as a whole" is the class of well-founded sets. This is given from the exclusivistic version of the foundation axiom. Now, then, if one wishes to well-found a world by default, one has a choice to work from either pure ur-elements (elements that aren't anything besides elements) or pure sets, and the tradition is to start from zero as the empty set. The Fregean account of the empty set is that it is such as {x: x ≠ x}, which makes it the complement of the universe V = {x: x = x}. So note that for universal terms U and the complement operation C, U^C = 0 and 0^C = U. So identifying the base of the world as the empty set = 0 is sensible enough, and we might also take the well-founded set of all nonexistent sets, which being well-founded is not an element of itself and hence does not satisfy its inclusion parameter, i.e. it is not nonexistent, hence it exists (and is a set that is then empty, since none of its elements exist).

But another thing we could do would be to start with u again, and look for the simplest set that can be built using u, which is {u} = 1 on the aforementioned picture of foundations. Why would this be sensible, too? Because whereas the generic concept of a set is compatible with being/can be disambiguated as a set that is either well-founded or not well-founded, an ur-element, having no elements of its own, can enter into only the well-foundation relation with whichever set it relates to (when it does).

Alternatively, if we imagine a universal well-foundation operation WF(X) and set X = 0, we get the standard ZFC-theoretic class WF at once (see Hamkins, et. al.[15] for the details, as well as considerations involving meta-theoretic natural numbers, incidentally). Again, using 0 for a world-base can be eminently perspicuous since, for example, 0 can be identified as the trivial limit ordinal, which has the otherwise large-cardinal property of being weakly (resp. strongly) inaccessible, among others (if I remember correctly, 0 is also trivially measurable). Note also that 1 is weakly inaccessible (self-cofinal) but not strongly inaccessible (because 2⁰ = 1). I should think also that WF(u) = WF just about as much as WF(0) does.

Answer 10

Questions about what mathematical objects are, what mathematical sentences mean, and what it is for a mathematical sentence to be true are questions about the philosophy of mathematics. There are over a dozen different ways of understanding what mathematics is about and each will lead to different answers to those questions.

For mathematical platonists, mathematical objects such as numbers have an objective and independent existence that is not reducible to mental entities. Numbers exist outside of humans or other rational agents and we are capable of discovering their properties through rational thought. They may have properties that transcend our ability to prove them.

For intuitionists, mathematics is a constructive activity. Numbers are mental constructions and proofs about them are also mental constructions. Numbers do not exist outside our heads. This leads to the view that mathematical truths cannot outrun the ability of an ideal mathematician to prove them, or even that proofs create mathematical truths.

For logicists, mathematics is a specialised branch of logic. Frege built on the idea from David Hume that if you have a bunch of Fs and a bunch of Gs, the number of Fs equals the number of Gs if and only if the Fs and Gs can be placed in a one-to-one correspondence with each other. Some logicists treat this as a kind of logical law that allows us to boot up a theory of numbers from logic alone.

There are many varieties of formalism. Typically formalists tend to treat mathematical objects such as numbers as symbols that require interpretation in order to be applicable to real things. The formal systems of mathematics are more or less arbitrary but become real when their symbols are given a concrete interpretation.

There are several other positions as well, including predicativism, structuralism and fictionalism. Each will tend to yield different answers to the question about what numbers are. Some of these are consonant with traditional mathematics, others are revisionary. Intuitionism, for example, employs a different logic from classical logic and rejects Peano arithmetic and the use of non-constructive proofs..

The Stanford Encyclopedia has a useful brief introduction to the philosophy of mathematics and there are many books on the subject.

Answer 11

Disclaimer: I am neither a (professional) philosopher, nor a mathematician. Yet I will do my best to attempt to offer what for me constitutes a very logical and satisfactory approach to "finding" the natural numbers, and in fact not just numbers but other concepts.

You've seen both the Peano and the set-theoretic construction of natural numbers. Those are useful, but don't necessarily convey why someone would start with these constructions ‒ while you can show that they are equivalent, and to a mathematician that is enough, they don't look like natural numbers (one is an infinite one-ended chain, the other is a set of sets you've created in previous steps), they just behave the same way (they model the natural numbers).

Let's keep those constructions aside and start with a plain observation of the world around us. You know you can ascribe certain qualities to things you see around you, like size, colour, shape, etc. Many of those concepts are hard or impossible to define without getting a sensation or "sample" of them in the brain, they are qualia.

How could you know a red apple and a red rose have the same colour, without knowing what a colour actually is? While the language needs a property ("colour") to start with, the brain only needs to recognize the relation ("same colour") ‒ it knows the two items are alike in some way, and it can discern items based on that relation.

Switching to mathematics for a while, such relations are called equivalences (they are reflexive, symmetric, and transitive ‒ in the case of "samecolourness" in the real world, the transitivity may be fuzzy, but let's pretend for the moment that we can distinguish colour perfectly). When you have an equivalence on a particular set, you can partition such a set into equivalence classes ‒ these are disjoint subsets of the set with members equivalent to each other (under the given equivalence). When you have a member, you can immediately find its corresponding equivalence class, and when you have such a class, you can also find a representative in that class (thanks to the axiom of choice).

Applying this to the real world again, you can partition everything into equivalence classes based on the colour equivalence, and then you find that you get something which unites both a red rose and a red apple ‒ their shared equivalence class. This is colour.

No, really, you can define colour as the set of all things that have the same colour. This is not a self-referential definition, because "same colour" is given here as a pre-existing relation, something the brain provides, and you can use that to define colour itself. Finally, getting the equivalence class of a thing is abstraction (as a mental operation), getting a representative is the opposite (concretism).

Once we've moved up to the equivalence classes, it is important to realize that individual members of the original set are no longer relevant ‒ looking at everything through colour-tinted glasses makes you see nothing else than colour; even the sizes of the classes are irrelevant.

Now if you think treating equivalence classes as individual (and indivisible) entities is weird, this is actually what happens in many areas in mathematics ‒ take congruence (mod n) for example, giving rise to modular arithmetic where the equivalence classes under congruence are the new numbers that you can still perform operations with (since they all behave consistently within individual equivalence classes). These numbers are {0, n, 2n, ...}, {1, n+1, 2n+1, ...}, {2, n+2, 2n+2, ...}, and so forth. While you usually think of modular arithmetic as always using the remainder of division by n, that just picks the smallest representative, but it does not matter which one you pick since all are equivalent!

Finally, let's apply the same process to natural numbers!

1. Notice that there are collections around you.
2. Observe that you can discern when two collections have the same size. You can do this without knowing numbers ‒ our brains can naturally perceive small sizes, and we can still compare sizes without counting (take one thing from both collections until one of them is empty). You can see that this relation exists a priori without having to create numbers first.
3. We have an equivalence ("same size") ‒ take its equivalence classes. These are the partitions of all same-size collections.
4. Abstract away the fact that these are collections and consist of individual parts. You get the essence of what it means for two collections to have the same size.
5. You get the natural numbers.

In fact, if you apply this not only to finite (real-world) sets but even to all infinite ones, you get the cardinal numbers which are precisely the numbers used to describe the set-theoretic cardinality.

Unlike the other definitions where it feels you have to build something out of nothing that shares the properties of what you seek, all you do here is finding something that already existed, and just looking at it from a particular point of view which makes it clear that it is what you wanted.

It is also obvious that you can do the usual arithmetic operations with these equivalence classes ‒ incrementation (add one thing to every collection in one equivalence class and you get precisely another equivalence class), addition, multiplication and so forth; all of them work since they can be consistently mapped to operations on the individual collections.

Lastly, to address your conclusion ‒ natural numbers are not really symbols, they are not like letters. Letters form words and words denote concepts. You can write all natural numbers using digits in a particular base (and a deal of other methods), but that doesn't change their fundamental nature, which is conceptualizing how two collections have the same size, formed as an actual mathematical object thanks to the equivalence classes. In fact, it is equivalence classes all the way up ‒ integers are equivalence classes of (a, b) where the equivalence is "same difference between a and b" (again something you can discern without needing integers; simply compare which one is bigger, and their distance), fractions are equivalence classes of a/b in the exactly same way (same ratio), and the reals are equivalence classes of Cauchy sequences of rationals. You can go even further, for example the hyperreal numbers are equivalence classes of sequences of real numbers agreeing on indexes from a given free ultrafilter on ℕ (an ultraproduct), then you can make an ultraproduct of that and repeat indefinitely.

Answer 12

There is a nice interview titled: What Kind of Thing Is a Number? A Talk with Reuben Hersh

REUBEN HERSH: What is a number? Like, what is two? Or even three? This is sort of a kindergarten question, and of course a kindergarten kid would answer like this: (raising three fingers), or two (raising two fingers). That's a good answer and a bad answer.

It's good enough for most purposes, actually. But if you get way beyond kindergarten, far enough to risk asking really deep questions, it becomes: what kind of a thing is a number?

Now, when you ask "What kind of a thing is a number?" you can think of two basic answers—either it's out there some place, like a rock or a ghost, or it's inside, a thought in somebody's mind. Philosophers have defended one or the other of those two answers. It's really pathetic, because anybody who pays any attention can see right away that they're both completely wrong.

A number isn't a thing out there; there isn't any place that it is, or any thing that it is. Neither is it just a thought, because after all, two and two is four, whether you know it or not.

Then you realize that the question is not so easy, so trivial as it sounds at first. One of the great philosophers of mathematics Gottlob Frege made quite an issue of the fact that mathematicians didn't know the meaning of One. What is One? Nobody could answer coherently. Of course Frege answered, but his answer [link added by me] was no better, or even worse, than the previous ones. And so it has continued to this very day, strange and incredible as it is. We know all about so much mathematics, but we don't know what it really is.

Of course when I say, "What is a number?" it applies just as well to a triangle, or a circle, or a differentiable function, or a self-adjoint operator. You know a lot about it, but what is it? What kind of a thing is it? Anyhow, that's my question. A long answer to your short question.

When you say that a mathematical thing, object, entity, is either completely external, independent of human thought or action, or else internal, a thought in your mind—you're not just saying something about numbers, but about existence—that there are only two kinds of existence. Everything is either internal or external. And given that choice, that polarity or dichotomy, numbers don't fit—that's why it's a puzzle. The question is made difficult by a false presupposition, that there are only two kinds of things around.

But if you pretend you're not being philosophical, just being real, and ask what there is around—for instance, there's the traffic ticket you have to pay, there's the news on the TV, there's a wedding you have to go to, there's a bill you have to pay—none of these things are just thoughts in your mind, and none of them are external to human thought or activity. They are a different kind of reality. That's the trouble. This kind of reality has been excluded from metaphysics and ontology, even though it's well-known—the sciences of anthropology and sociology deal with it. But when you become philosophical, somehow this third answer is overlooked or rejected.

Answer 13

To answer “where do the properties come from”.

Non-mathematicians have a pretty good idea about natural numbers. Take a Movie Theater, nobody seated. One person enters, there is one person. Another person enters, there are two persons and so on. Zero, one, two and so on. We know how these numbers work.

Now came the mathematicians and formulated rules, meant to capture the real-life natural numbers and nothing else.

The number of people in the movie theatre equals the number of people in the movie Theater - peano axiom: For every natural number x, x = x. If the number of people equals the number of popcorns sold then the number of popcorns sold equals the number of people in the Theater. For every natural numbers x and y, if x= y then y = x. And so on.

There are some peano axioms that a non-mathematician might not think of: For every number, there is another number one larger. So the natural numbers don’t end say at 10,000. 0 is a natural number (yes, you need to state that there is at least one natural number). And for every natural number other than 1, there is a natural number one less. And some others.

Answer 14

Natural numbers are abstract items of recognition. Natural numbers are indicated by arbitrary symbols. Particular distinct items of recognition can be associated with the abstract concept of natural numbers. The recognition process is not arbitrary but we have the capacity to recognize and use arbitrary abstract symbols to map our recognition of particular distinct items into patterns of language and mathematics.

EDIT BELOW

The concept of one (1) is a generalization of the perception of any particular distinct item of recognition. One (1) is the smallest natural number. The symbols 1, 2, and 1/2 are THREE (3) particular distinct items of recognition! But now there are four symbols: 1, 2, 1/2, and 3! Symbols 1, 2, 3, 4 are designated natural numbers while 1/2 is not! Natural numbers arise as symbols and concepts when counting distinct items of recognition. Natural numbers begin with 1 and add 1 to derive the next natural number in an infinite sequence from 1, 2, 3, ... N.

Answer 15

For a natural number to be possible, the notion of thing is necessary.

Imagine things don't exist.

Part I: from empty space to things.

Imagine that the universe is just a foam, all bubbles in the foam being atoms. There are no rocks, rocks are part of us, which are the same of every imaginable thing. The most important fact in this universe is that there are no borders. The limit between your finger and the apple you're holding in your hand does not exist at all. You are part of the apple and it is part of the room and the sky and the stars.

In such universe, there are no limits between things. Everything is just part of the unique and unitary thing that the universe is. There are just differences in foam densities, but there are no limits. So, things don't exist. All would be part of the same foam.

Have you imagined that universe? Well, news for you: you don't need to imagine such universe: that universe is precisely the universe you live in.

Let me explain.

Everything is just atoms. Worst even, atoms are not things. They exist only in our minds. Out there, atoms are waves in universal fields that have no limits. Things don't really exist.

In other words, out of our minds, everything is just empty space.

What!!? But there are rocks! There are apples!

Yes, but they are like waves in the sea, or mountains: they exist just because our minds and our bodies determine it. Have you ever seen the boundaries of a wave in the sea? Never. Yet, you know waves. You have felt the power of a wave when swimming.

So, things are just subjective notions.

But things are not just metaphysically subjective ideas, as frontiers, streets, holes, football teams, rooms or quality apples: things are also physically subjective ideas.

If apples exist it is because your body is tuned to perceive them. If would have the size of an electron, apples would not exist. If your heart would beat once in a century, that is, if you would live in a different velocity you will not have time to perceive apples. If you would have the density of a black hole, you would be unable to get near apples before they completely vanish as stellar dust hundreds of kilometers before reaching your hand.

You see? You are tuned physically and educated metaphysically to perceive things. Things exist because you need them to survive. You exist and survive only because you perceive yourself as a thing. But that is just appearance: the universe is just a foam of atoms, which are just empty space.

Immanuel Kant experienced what is known as his Copernican Turn: his best contribution to philosophy is to understand that objects don't determine what the subject perceive: it is the opposite: the subject determines the object.

Part II: from things to numbers.

As soon as things exist in your brain, you can organize them to take profit and get the benefits of living in a universe of of things. Now, having things, you can take a sip of water (a thing which borders are defined by your mouth) and drink it.

But where really do things exist? Well, they are part of knowledge, which is just a model of the world.

Knowledge is a model. This means that knowledge is not reality. Knowledge is just a tale that exists in everyone's mind. The moon is not physically white, but in our knowledge, it is.

What is important about knowledge is that the map is not the terrain. Even the best map will hide almost the totality of the reality it depicts. Maps are just abstractions, ideals, representations of the world, they are not the world. So, you can't think that maps are true or false. A map is not the terrain, it is always different, it is always incomplete.

But knowledge, being a model that we use to survive, includes valuable information about the facts the map tries to describe: this can be a dangerous river, so, you can avoid it. Or the map can include abstractions of... quantity. The river lies after the three old trees.

So, natural numbers are just the simplest way of organizing things in such map (knowledge).

Consider that, from the atomic perspective defined along this discussion, all apples in the world are necessarily different. No pair of apples in the world have the same atomic configuration or the same atomic state. But in our minds, they have.

Such difference is expressed in Thermodynamics as micro and macrostates. Microstates represent the atoms in both apples, always different in nature, and macrostates represent their perceived appearance: two things can have the exact same temperature, although we know that microstatically their energies have different values.

1+1 would, strictly, require two identical apples for the result to be 2. But such is the microstatic reality, the terrain. Knowledge uses the macrostatic illusion, the dream in our heads, where two completely different rivers can be counted as if they were identical.

Natural numbers allows organizing such illusion in knowledge, even if it is strictly false.

Now, think what happen with positive real numbers... how many decimal figures are necessary to describe the exact and precise weight of an apple?

Answer 16

What is counting

It’s not hard (but a little Strange Planet) to characterize what counting actually is.

So, counting is fundamentally a song-and-dance game, kind of like musical chairs or so.

• The song is generally a chant where phrases, called “numerals”, are separated by rests. There are two key aspects of the chant, first that it in principle can go on forever, secondly that it never repeats a numeral.

• The dance involves pointing at the “things to be counted” in time with the music. The objects thus have to admit a structure, call it a “segmentation,” such that they can be pointed at or “counted.”

• The game rules state that the dancer loses if they do not perform the dance in time with the song, or do not sing the song properly, or point at the same object twice. They win if there are no more objects to point at without pointing at one twice.

If the dancer wins the game then the last numeral chanted is known as the size of the collection (according to that song). There are three exceptional cases,

• if the game ends before it begins because the collection is empty, we generally have a special numeral for that

• if objects appear or disappear in mid-game then the game ends without winning or losing; if the game cannot be won for this reason then we say that the count is “noisy” and has to be approximated

• if the player can never win because they will never run out of things to point at, we say that the count is “infinite.” We can distinguish between the “countably infinite” where a savvy dancer can guarantee that any given object will be pointed at in some finite time, and the uncountably infinite, where you can prove that every strategy for pointing at objects necessarily admits elements that it will never point at no matter how long the song continues.

Why do we care about it

This is kind of the bigger question. Most song and dance games are great for children but then we don't play them as adults, this one is almost exclusively played as part of the transition to adulthood. In the most extreme variant, called “paying cash,” objects that stand in for 100, 500, 1000, 2000, or even 5000 or 10000 pennies are used, you have to use these to count a number higher than a target number called a “price after tax” and the other player, the “cashier,” then has to count your “change” such that if you counted both the price-pennies and the change-pennies together it would match your number. Only strange people seem to enjoy this game.

So there are a couple themes about where counting works. The single most important theme is that the segmentation really matters.

The segmentation is the abstract pattern or structure that allows us to tell apart the things that we are counting. If I give you a basin of water and ask you to count the water, there is no way to do it directly because the water is not segmented into chunks that can be counted. But if I ask you to count the number of liters of water in that basin, suitable containers could be procured such that you could segment and count this water. In even more unusual cases there isn't even the hope of this segmentation: consider if three children are excited for Christmas Day and I ask you, rather bizarrely, to count their collective excitements. Or on the next cloudless day, perhaps we might count the sunshines. “What do those questions even mean...”—correct, these are segmentation failures. Without segmenting the whole, the entity is unquantifiable.

Most of the things we count do not have a direct material existence but rather occupy a social reality that overlays physical reality. We count minutes and seconds and months, which kind of have a half reality; they don't fit the pointing game all that well. Also we count points in a football/soccer game but that's for sure a social reality. Count the sides of a triangle; did the triangle need to be materially embodied to do so?

Two themes around segmentation:

• usually the segments are interchangeable for some purpose, in mathematics we would speak of an equivalence relation. This is why when segmenting apples it seems weird to specify one of the segments as containing two apples when all the rest have one; these don't seem like interchangable segments.

• usually the either-social-or-physical laws underlying the thing-we-are-counting, create some sort of conservation principle that helps to keep the quantity stationary. Points in basketball only increase when the ball passes through the net. I can count my calories in the food I eat due to conservation of energy.

So there is something about these sorts of circumstances where things are simultaneously mostly conserved and interchangeable, that makes it reasonable to measure them by counting.

I would also mention that we get some feeling of confidence in the abstract procedure by seeing other properties:

• when we count things twice, we usually get the same number or else we can find usually an explanation of the discrepancy

• different people agree on the count of things and whether the game was performed correctly

• we pick up a skill called subitizing where if you see six objects you can often count them without the whole song and dance just by looking at them.

If you combine those then we tend to develop some very strong intuitions for the importance and constancy of counting.

Answer 17

I guess your difficulty arise from the confusion of the different formulations (more precisely: explications) of the natural number concept. The set theoretical definition you mentioned may corresponds to the definitions that stem from Frege, Russell, Von Neumann or others while the Peano five axioms corresponds to the Peano's formulation of natural number.

Your difficulty seems to be clarified by questions like:

• "Why there are different definitions of the natural number?"
• "What are their differences?"
• "Why there arises different definitions/attempted formulations of natural numbers?"
• "Have we chosen to adopt one of them as the standard explanation of natural numbers?"

However, here I can only provide you about the source of information that discuss, or address the above questions, or even the more fundamental one (i.e. what is the meaning of (natural) number), as the comprehensive answer seems not suitable to briefly discuss it here. For the following cited paragraphs, they are all from Bertrand Russell's book - "Introduction to Mathematical Philosophy".

Bertrand Russell firstly explained the Peano's five axioms of natural number in previous pages and concluded that:

According to Russell (2022) chapter 1 - "The Series of Natural Numbers", (p.6-7):

It is time now to turn to the considerations which make it necessary to advance beyond the standpoint of Peano, who represents the last perfection of the " arithmetisation " of mathematics, to that of Frege, who first succeeded in "logicising" mathematics, i.e. in reducing to logic the arithmetical notions which his predecessors had shown to be sufficient for mathematics. We shall not, in this chapter, actually give Frege's definition of number and of particular numbers,

but we shall give some of the reasons why Peano's treatment is less final than it appears to be.

---

In (p.9):

In Peano's system there is nothing to enable us to distinguish between these difierent interpretations of his primitive ideas. It is assumed that we know what is meant by "0," and that we shall not suppose that this symbol means 1oo or Cleopatra's Needle or any of the other things that it might mean....

This point, that "0" and "number" and "successor" cannot be defined by means of Peano's five axioms, but must be independently understood, is important... We have already some knowledge (though not sufficiently articulate or analytic) of what we mean by "1" and "2" and so on, and our use of numbers in arithmetic must conform to this knowledge. We cannot secure that this shall be the case by Peano's method; all that we can do, if we adopt his method, is to say "we know what we mean by "0" and "number" and "successor," though we cannot explain what we mean in terms of other simpler concepts."

Russell illustrated why Peano's formulation is less final in this chapter with more clear examples than what I have just cited.

It seems Russell understood that: Peano's formulation of 0 did not capture the mathematical meaning of 0 in ordinary language. In Russell's understanding, the Peano's explication of the concept of (natural) number only described natural number as a progression but failed to capture the meaning of natural number we understand in the natural language. Hence, Russell concludes that Peano's formulation of natural number is a less final treatment.

The following chapter "Definition of Number" (p.18), Russell discussed his own view about the definition of number.

Accordingly we set up the following definition :- The number of a class is the class of all those classes that are similar to it. Thus the number of a couple will be the class of all couples. In fact, the class of all couples will be the number 2, according to our definition. At the expense of a little oddity, this definition secures definiteness and indubitableness; and it is not difficult to prove that numbers so defined have all the Properties that we expect numbers to have.

In (p.19),

In other words, a number (in general) is any collection which is the number of one of its members; or, more simply still : A number is anything which is the number of some class. Such a definition has a verbal appearance of being circular, but in fact it is not. We define " the number of a given class " without using the notion of number in general ; therefore we may define number in general in terms of " the number of a given class " without committing any logical error.

"What is a number?", this question has been the subject of much disputed debate in mathematical philosophy.

Russell seems to give an introduction and his answer to this matter in the book that I have cited. I kindly suggest you to take a look to this important book if you are interested in the massive issues related to this problem (including axiom of infinity, etc.).

To conclude, according to Russell's view, Peano's formulation merely talks about that there is an object served as starting point, each object is followed by one and only one different object, and so on. However, these objects only presented a progression and other different collections of objects that do not have the ordinary meaning of natural number can also satisfy Peano's axioms. Peano's formulation is a less final treatment and Russell gave his explication of natural number as the class of all classes that are isomorphic to each other (here I used a modern usage of expressing Russell's idea).

For most of Russell's ideas expressed with rigorous symbolic logic in this book, you may compare Prof. Rudolf Carnap's "Introduction to Symbolic Logic and its Applications" - the whole part of his Language C.

For Content page of Russell's book:

For Carnap's book, these may be useful for you:

34. Equivalence classes, structures, cardinal numbers:

a. Equivalence relations and equivalence classes

b. Structures

c. Cardinal numbers

d. Structural properties

35. Individual descriptions:

a. Descriptions

b. Relational descriptions

36. Heredity and ancestral relations:

a. Heredity

b. Ancestral relations

c. R-families

37. Finite and infinite:

a. Progressions

b. Sum and predecessor relation

c. Inductive cardinal numbers

d. Reflexive classes

e. Assumption of infinity

38. Continuity:

a. Well-ordered relations, dense relations, rational orders

b. Dedekind continuity and Cantor continuity

Reference

Carnap, R. (2012). Introduction to symbolic logic and its applications. Courier Corporation.

Russell, B., & Potter, M. (2022). Introduction to mathematical philosophy. Routledge.

Answer 18

Natural numbers and the basic rules of counting and addition are useful elementary abstractions from real world experience. Other similarly useful abstractions are the concepts of point and line as well as the basic rules of geometry. These kind of abstractions have been made independently in many societies.

But in some cultures the mathematical quantities have been detached from their everyday use and have been considered separate entities. In Europe this happened in the Greek culture since the 6th century BCE when mathematicians started to investigate number theory and geometry as two pure sciences. In the 3rd century BCE Euclid axiomatized the body of existing mathematical knowledge. Euclid’s The Elements of Geometry became the paradigm, also for the later Peano axioms and the different axiomatizations of set theory.

Today’s number theory is not so much concerned with the question what numbers are, than with discovering properties of the mathematical objects which derive from elementary number theory: Rings of residue classes, finite fields, p-adic numbers, number fields, arithmetic schemes, …

Less than the question What are numbers? the question is Which properties have the mathematical concepts derived from numbers?

Answer 19

There's an interesting story Piaget tells. Once he was out walking in the woods with daughter (she was young at the time, perhaps 6), and they saw a slug. They looked at it for a while and then walked on, and then a few yards later she said "Look, there's the slug!". Piaget was curious about the phrasing, so he asked her if it was the same slug: she looked at the slug carefully, walked back to the first slug and looked at, cane back to the second slug and looked at it again, as though it had never occurred to her that there might be two slugs in the world.

The point is that there's a cognitive development here. First we develop the concept of 'object' (something that is continually distinct and separate from the sensory background) and then we develop the concept of 'class' (objects that can be grouped together under sameness or functional interchangeability). The concept of 'class' entails at least proto-numeracy: qualitative quantifiers like less than, more than, and proximate to. The natural numbers are the first abstraction of these basic qualitative quantifiers.

The confusion here (generally speaking) is that we tend to treat numbers as though they are nouns when linguistically they are adjectives. I mean, colloquially we say things like "I have one" or "one plus one equals two" without acknowledging the unspoken object being invoked. In fact, the second sentence is not even true unless we specify (as we almost never do openly) that the objects referred to be each 'one' must be distinct members of the same class. One apple plus one orange doesn't work; the same apple counted twice doesn't work…

Answer 20

I’ve come to the conclusion, that numbers are symbols like letters are in the alphabet. In fact I would go so far to say as that they should be in the alphabet.

Reasoning: over centuries, humans have evolved to understand language, to the point where we can just read symbols and speak and our mind just knows how to interpret the sounds / symbols to a meaning.

Numbers are like this, over centuries humans developed the concept of counting, amount and quantity. They are in our world now and are used for so many things, like lengths, money, ordering, and in everyday life, that we just understand (I accept without question my understanding of language). You can’t conceptualise a number really by itself since I feel they are contextual.

But, we do know how they “ought” to behave, and the standard way of counting (built over centuries) objects in a collection is identifying each object in the collection and increasing the counter(symbol representing all the objects counted), where the final count is defined to be “how many” objects are in the collection. In this context, we can understand 3 as 3 items say a,b,c . We just understand there are 3 letters above like we just understand the definition of “philosophy” without really understanding how that string of symbols came about.

I understand numbers in many different contexts, time, measurement, labelling, counting, ordering, money etc. I just accept from now on like I just accept my understanding of language.

Asking what 2 really is is analogous to asking what “a” really is. With no context, they are both just symbols.

Anyway, I think I’m going to chill out a bit because I’m slowly losing my mind. Thanks for all the help

Answer 21

What is a natural number?

I can only give you a small sample:

0, 1, 2, 3, 5, 6, 7 etc.

Natural numbers have many properties, so I will only give you a small number of them:

1 + 1 = 2

501 - 56 = 445

Etc.

We can of course define natural numbers exhaustively using addition tables.

Natural numbers are also defined as the symbols we use to refer to the number of things:

We use 2 to refer to the number of things in the case where there are exactly two things.

Ultimately, we have to define the number 2 ostensively. For example, in the expression "AA", there are two occurrences of the letter 'A', which is to say that there are 2 'A' in "AA".

This applies to very many things as long as we can distinguish between them: eggs, cats, atoms, stars etc. Sometimes, it is a bit tricky, for example clouds and ideas.

Apparently, most people learn to recognise when there are two things, rather than three for example, when they are children, so we can suppose that this is an innate ability humans have.

We may be tempted to see numbers as just mathematical concepts, saying for example that the operation 2 = 1 + 1 defines what 2 is in terms of smaller numbers and the addition.

It is true that we can do this for any natural number we can think of, but we cannot do it for all natural numbers because there is a potential infinity of them, and for any system we use to express numbers, there are numbers which are too big to be expressed in this system.

This also does not say what numbers are. It just identifies one property that they have, and it is just as true that every number can be defined in terms of the subtraction and greater numbers than them. For example:

56 = 501 - 445.

What this sort of definition does not say is that by 'number' we mean an expression that we use to mean the quantity of objects we think there is in sets of objects that we can distinguish from each other. This is true just as much in the context of everyday life as it is in the context of mathematics. So we take 2 to refer to the number of things in case that there are exactly two things.