Mathematical definition of number?

[lolgarithms]

What is the mathematical definition of "number"?

It seems odd to me that one of the most commonly used mathematical objects has no clear definition. Should the definition of number be "a member of a mathematical structure"? The notion of "sturcture" isn't clearly defined, so I don't like it. Also it will include functions, vectors, tensors, etc.
Or should it be "a member of a magma [which includes group, rings, and fields] that can not be written as an n-tuple"? This will exclude complex numbers, which are isomorphic to R².

Both of these will exclude cardinals and ordinals, which are not sets, let alone magmas.

IMHO, no definition of "number" can be given such that it encompasses all of the inclusions and the exclusions. Shouldn't we just expel this term from mathematical parlance and just talk about "elements of [put name of set/class/space/vector space/structure here]"?


ps. admins, you deleted the wrong thread

 

[qntty]

What do you mean by a "number"? A real number? A complex number? Do p-adic numbers count too? How about infinity or transfinites?

 

[lolgarithms]

What do you mean by a "number"? A real number? A complex number? Do p-adic numbers count too? How about infinity?

That's what I'm talking about.

 

[HallsofIvy]

No, the question was which. Some of those are normally thought of as numbers, some of those are not.

In any case, "undefined terms" are at the heart of mathematics. We certainly don't want to get rid of them! The reason we can apply mathematics to many different fields is that we have undefined terms that can be given definitions from the specific field.

 

[lolgarithms]

In any case, "undefined terms" are at the heart of mathematics.

so the term "number" can be thought as simply a shorthand term to denote an element of a naturals, or an element of reals, or complexes, or whatever, depending on context.

 

[symbolipoint]

HallsOfIvy may wish to clarify what he said; his discussion suggests that "number" is accepted as undefined and is valid and justifiable as a fundamental concept in Mathematics. We just KNOW by ordinary and extraordinary human experience what "number" means and we do not need to define it; in fact "number" is undefined. Basic Notion?

 

[Hurkyl]

We just KNOW by ordinary and extraordinary human experience what "number" means and we do not need to define it;

I beg to differ -- the fact so many laypeople disagree so strongly about what 'number' means contradicts your assertion.

 

[lolgarithms]

This discussion is complicated by the fact that numbers aren't directly derivable from sets (of naturals, reals, etc); are numbers axiomatized from sets, or sets from numbers? Does structure come from elements, or elements from structures?

We just KNOW by ordinary and extraordinary human experience what "number" means and we do not need to define it;

Our intuitions are not always at the level of precision that ought to be in the definition of higher mathematical objects such as numbers.

 

[Dragonfall]

This discussion is complicated by the fact that numbers aren't directly derivable from sets (of naturals, reals, etc); are numbers axiomatized from sets, or sets from numbers? Does structure come from elements, or elements from structures?


Our intuitions are not always at the level of precision that ought to be in the definition of higher mathematical objects such as numbers.

What's wrong with set-theoretical definitionof any of those numbers?

 

[HallsofIvy]

I would consideder "set" a more fundamental than "number" but we then define operations on numbers that we do not have on sets.

 

[Cantab Morgan]

I would consideder "set" a more fundamental than "number" but we then define operations on numbers that we do not have on sets.

I agree, so I'm fascinated that numbers are far more graspable than sets. Almost all humans can count. But almost no humans know anything beyond naive set theory. It would seem that numbers are part of the physical universe, but sets are Platonic. It surprises me that fundamental doesn't imply simple.

 

[disregardthat]

Humans can count as a consequence of how the human brain individualize objects. When we percieve and think about a certain object we think of it as independent of its sorroundings. Therefore is quantification a natural way of ordering the objects we percieve. Numbers, however, are not a part of the physical universe as they are completely human constructs. Objects in themselves are no fundamentally different from their sorroundings. It is only because of us forcing order to nature numbers become a useful concept. In this perspective are sets no different. I would even argue that sets are more fundamental to us than numbers. In order to quantify something, for example count the amount of stones in a pile, we need a concept of what the objects need to have in common in order to be included. We form the 'set of stones' before we count them.

That's at least how i look at it.

 

[fleem]

"number" is an axiom. it doesn't have a definition. Its existence is assumed because of repeated experience that numbers of things are conserved, and thus deserves a name. Some cave man noticed this when he put three apples in a hole and noticed it still contained three apples a moment later. Although i will admit that there is some debate on exactly which concept we should call "axiom", since circular "definitions" can be stated, certainly there has to be axioms in the thought processes somewhere!

 

[disregardthat]

An axiom is a statement which is considered true, either for the sake of discovering its consequences, or because we consider it obviously true. "Number" is not a statement. "Numbers exist" is, however. But, by referring to numbers you automatically proves their existence. Numbers are not physical objects, but an abstract concept.

 

[junglebeast]

The definition of "number" is any member of several specific subsets; counting numbers, integers, reals, etc.

All of these other sets are defined in terms of "counting numbers" which are the most fundamental set. For example, rational numbers is defined by the set of fractions made of counting numbers, real numbers is defined as the set of digits that are counting numbers, integers are defined as the set of counting numbers including negatives, etc.

The set of counting numbers is a specific set of 10 definitions.

0 = none
1 = one countable unit
2 = two countable units
... etc

This representation maps the numerical symbols to observable meaning.

 

[Hootenanny]

For example, rational numbers is defined by the set of fractions made of counting numbers

According your definition, there are no negative rational numbers. Generally, one defines the set of rationals as the set of all numbers that can be expressed in the form p/q, where $p\in\mathbb{Z}$ and $q \in \mathbb{Z}\setminus 0$.

 

[junglebeast]

Alright, that wasn't intended as a precise formal definition...I was just trying to convey a point that all the number sets can be defined in terms of a small set of digits which have meaningful definitions

 

[Cantab Morgan]

The definition of "number" is any member of several specific subsets; counting numbers, integers, reals, etc.

All of these other sets are defined in terms of "counting numbers" which are the most fundamental set. For example, rational numbers is defined by the set of fractions made of counting numbers, real numbers is defined as the set of digits that are counting numbers,

Yes, it's possible to construct the reals out of the rationals (as equivalence classes of sequences, for example), but that's not the only way to make reals. One can simply postulate the existence of the set of reals and list its dozen or so axioms without referencing the rationals. I'm sorry if I'm being a bit pedantic, but I'm just suggesting that one can take take either a constructivist approach or an axiomatic approach to real numbers. And of course, people much smarter than me have shown that these sets are isomorphic.

In either case, we define numbers from sets (whether we start with the Peano postulates and construct from there or not). In this light, it's proper to say that sets are more fundamental than numbers. But I suggest that it's the usefulness of set theory to defining numbers that gives set theory its validity. It's not set theory that makes numbers valid, but the other way around. For example, if tomorrow we found that ZFC led to a contradiction when trying to define numbers, we would not throw away arithmetic, we would look for an alternative to ZFC.

 

[Cantab Morgan]

Humans can count as a consequence of how the human brain individualize objects. When we percieve and think about a certain object we think of it as independent of its sorroundings. Therefore is quantification a natural way of ordering the objects we percieve. Numbers, however, are not a part of the physical universe as they are completely human constructs. Objects in themselves are no fundamentally different from their sorroundings. It is only because of us forcing order to nature numbers become a useful concept. In this perspective are sets no different. I would even argue that sets are more fundamental to us than numbers. In order to quantify something, for example count the amount of stones in a pile, we need a concept of what the objects need to have in common in order to be included. We form the 'set of stones' before we count them.

That's at least how i look at it.

That is really interesting and thought provoking. However it is not obvious. Let me be clear that I am not necessarily refuting what you have written, just that I'm not on board yet. If there are three rocks on my lawn, there are three rocks on my lawn whether I perceive them or not. I don't see us as forcing order on nature as much as discovering it.

But I don't agree at all that sets are more fundamental to our human cognition than numbers. Maybe lists are, but not sets. Sets have almost no structure, and are extremely abstract. The lack of concreteness is tough on human brains, evinced by humans counting long before we were writing about sets.

 

[junglebeast]

In either case, we define numbers from sets (whether we start with the Peano postulates and construct from there or not). In this light, it's proper to say that sets are more fundamental than numbers.

Some specific sets of numbers (eg, rational numbers) are only defined in terms of sets...but the most basic numbers are simply digits, and each individual digit has a definition that does not require the notion of a set. Thus, the most basic type of number is more fundamental than a set...but most numbers that we use are defined with the help of sets.

 

[CRGreathouse]

Some specific sets of numbers (eg, rational numbers) are only defined in terms of sets...but the most basic numbers are simply digits, and each individual digit has a definition that does not require the notion of a set. Thus, the most basic type of number is more fundamental than a set...but most numbers that we use are defined with the help of sets.

I don't think that the notion of a digit is fundamental in any way.

 

[Phrak]

That is really interesting and thought provoking. However it is not obvious. Let me be clear that I am not necessarily refuting what you have written, just that I'm not on board yet. If there are three rocks on my lawn, there are three rocks on my lawn whether I perceive them or not. I don't see us as forcing order on nature as much as discovering it.

I think there is a natural inclination to force order, all the time. What's the difference beween a tree and a bush. Where do you draw the line?

How do you distinguish stones from pebbles from sand then silt?

 

[disregardthat]

If there are three rocks on my lawn, there are three rocks on my lawn whether I perceive them or not. I don't see us as forcing order on nature as much as discovering it.

It would not really make much sense to say that "three rocks are on the lawn" without a human observer defining what a rock is. There is nothing special about rocks which makes them different from other objects other that us naming objects which share certain similarities as rocks. In other words, we are imposing order in nature by naming and defining phenomenas and objects.

Don't think I by saying "imposing order" mean finding order where there really is none. 'Order' is a human concept so it is thus necessary for a human being to impose order for there to be order. If a person exposed to a hypothetical situation saw 'order' does not mean that 'order' is fundamental to the situation. Some people might even see order where others cannot. Thus it is also a relative concept.

But I don't agree at all that sets are more fundamental to our human cognition than numbers. Maybe lists are, but not sets. Sets have almost no structure, and are extremely abstract. The lack of concreteness is tough on human brains, evinced by humans counting long before we were writing about sets.

I didn't mean that humans was aware of the mathematical properties of sets, but rather that the notion of a set is necessary to impose order in a certain situation where it would be natural to quantify something. Before we, for example, count the "amount of clouds" we need a notion of what a "cloud" is. What differentiates a cloud from the rest of the sky? When we are confident in distinguishing a cloud from each other and the rest of the sky, i.e. defining the set of clouds, we can begin counting them. It would not make much sense to count the clouds you see without a clear sense of what a cloud is. This definition of a set may be more or less happening in our subconscience as it often is obvious what you are counting. Numbers, on the other hand, is considered more like a tool. Note that I am not saying that we necessarily call this way of thinking as "defining a set" the same way as we call counting for "counting".

(I hope my usage of the word "order" is not confused with the word "sequential order". By order I mean that there is some type of structure.)

 

[HallsofIvy]

The definition of "number" is any member of several specific subsets; counting numbers, integers, reals, etc.

All of these other sets are defined in terms of "counting numbers" which are the most fundamental set. For example, rational numbers is defined by the set of fractions made of counting numbers, real numbers is defined as the set of digits that are counting numbers, integers are defined as the set of counting numbers including negatives, etc.

The set of counting numbers is a specific set of 10 definitions.

0 = none
1 = one countable unit
2 = two countable units
... etc

This representation maps the numerical symbols to observable meaning.

Alright, that wasn't intended as a precise formal definition...I was just trying to convey a point that all the number sets can be defined in terms of a small set of digits which have meaningful definitions

Then it still not true. You are defining "numerals" not "numbers".

 

[lolgarithms]

Alright, that wasn't intended as a precise formal definition...I was just trying to convey a point that all the number sets can be defined in terms of a small set of digits which have meaningful definitions

I would agree with HoI's reply. Numbers, as mathematical expressions and statements in general, are independent of notation (be it symbols, numeral systems, or different human languages). The statement encapsulated by the statement in decimal numerals: 1+1=2 is still true in binary: 1+1=10.

(But then we cannot define our notations without using another notation. We can say that "the symbol 2 at n places left of the decimal is defined as the number two times ten to the (n-1)th power", but it is a definition in English, just another notation.)

But I don't agree at all that sets are more fundamental to our human cognition than numbers. Maybe lists are, but not sets. Sets have almost no structure, and are extremely abstract. The lack of concreteness is tough on human brains, evinced by humans counting long before we were writing about sets.

We can impose some kind of "order" on sets, but I don't believe the order that we naturally conceive of (our intuitive "greater than" or "less than" valuation (giving values) to members of sets that we call numbers) be derived from order from pure order theory, which merely consists of properties that the order has.

 

[junglebeast]

I would agree with HoI's reply. Numbers, as mathematical expressions and statements in general, are independent of notation (be it symbols, numeral systems, or different human languages). The statement encapsulated by the statement in decimal numerals: 1+1=2 is still true in binary: 1+1=10.

You can define numbers starting with any base, but individual definitions are needed for each digit that are more basic than the notion of a set. It doesn't matter which number system you use because they all equivalent notations of the same thing.

 

[lolgarithms]

You can define numbers starting with any base, but individual definitions are needed for each digit that are more basic than the notion of a set. It doesn't matter which number system you use because they all equivalent notations of the same thing.

Why do you believe that digits are more basic than sets?

 

[lolgarithms]

You can define numbers starting with any base, but individual definitions are needed for each digit that are more basic than the notion of a set. It doesn't matter which number system you use because they all equivalent notations of the same thing.

Why do you believe that digits are more basic than sets?

(But then we cannot define our notations without using another notation. We can say that "the symbol 2 at n places left of the decimal is defined as the number two times ten to the (n-1)th power", but it is a definition in English, just another notation.)

Plus, how do you define "two" using only sets and ordering? -> Then how do you define divition and rationals?
then,yeah, i would agree that numbers are fundamental in some respect.

 

[HallsofIvy]

Plus, how do you define "two" using only sets and ordering? -> Then how do you define divition and rationals?
then,yeah, i would agree that numbers are fundamental in some respect.

The standard method is this: The number "0" is the empty set: {}. The number "1" is the set whose only member is the empty set: { {} }. The number "2" is the set whose only members are 0 and 1. In general, the "successor" to a number, n, is the set containing the set "n" and all of its members.

Once you have done that, you can show that these "numbers" satisfy Peano's axioms and prove all the properties of the non-negative integers from that. After defining the non-negative integers, you look at the set of pairs of non-negative integers and define the integers to be equivalence classes of pairs of non-negative integers with equivalence relation "(a, b) is equivalent to (c, d) if and only if a+ d= b+ c". We define the addition of two integers by: if x and y are integers (equivalence classes of pairs of non-negative integers) choose a 'representative' from each class, that is (a,b) from x, (c,d) form y. x+ y is the equivalence class containing the pair (a+c, b+d). Of course, you would have to prove that choosing different pairs from the same two equivalence classes you would wind up with the same equivalence class. Similarly, xy would be defined as the equivalence class containing (ac+bd, ad+ bc). If, for some (a,b) in equivalence class x, $a\ge b$, you can show that every pair has first member larger than the second member, and, in fact, "first number minus second" number is always the samd so we can associate that equivalence class with the non-negative number a-b. If, for some (a,b), b> a, that is true for all pairs in the class and we associate that with the number -(b-a), the additive inverse of b- a.

To define rational numbers, consider the set of pairs (a, b) where a is an integer and b is a positive integer and use the equivalence relation (a, b) is equivalent to (c, d) if and only if ad= bc. The rational number "m/n" would be the equivalence class containing (m, n).

There are a number of different ways of defining the real numbers. For example, consider the set of all Cauchy sequences of rational numbers and use the equivalence class $\left{a_n\right}$ is equivalent to $\left{b_n\right}$ if and only if the sequence $\left{a_n- b_n\right}$ converges to 0. Another way is to use increasing sequences having an upper bound instead of Cauchy sequences and the same equivalence.
For example, the equivalence class containing the sequence {3, 3.1, 3.14, 3.141, 3.141, 3.1415, ...} corresponds to the number "$\pi$". A completely different way is to define the real numbers to be "Dedekind cuts", sets of rational numbers satifying
(1) A cut is non-empty: there exist at least one rational number in the set
(2) A cut is not all rational: there exist at least one ration number not in the set
(3) There is no largest number in the set
(4) If b is in the set and a< b, then b is also in the set.
For example, the set of all rational numbers, whose square is less than 2, together with all negative rational numbers, is a cut and corresponds to the irrational number "$\sqrt{2}$".

Does that satisfy you? These and proofs of all properties of numbers from them should be "five finger exercises" for any mathematician. But I still assert that the concept of "number" itself is an "undefined term". These are all definitions giving specific "instances" of "number".

 

[lolgarithms]

a number that is in the naturals is simply defined as the set (containing members from zero up to n-1) itself, and not as the set's cardinality?

 

[HallsofIvy]

Yes, because you can't define "cardinality" until after you have the natural numbers.