What is a Number? - Math Philosophers' Views

[dobry_den]

Is there any consensus among mathematicians (or rather philosophers of mathematics) on the concept of a number - what is really a number?

This question has quite bothered me recently. Do we left the concept of a number undefined and take it as an intuitive one with no need of definition?

Leopold Kronecker wrote "God made the integers; all else is the work of man." (http://en.wikipedia.org/wiki/Kronecker" about Kronecker at Wikipedia). Is this idea prevalent?

Are there any recent books that deal with the concept of a number in modern mathematics?

 

[arildno]

From what I know, a "number" is just the term we use for an ELEMENT of a SET called a "number system". In addition, certain OPERATIONS (like addition or multiplication) must have been defined in order to call our set a "number system".

So, what you have fractured the number concept into, are the concepts of
1. A set
2. Elements of a set
3.Operations we can do on such elements.

The concepts of a "set" and "element" are so primitive that they are not defined in terms of other concepts; rather, it is specified what is true (or not true) of elements and sets in a particular set THEORY (for example Zermelo-Franckl).

 

[octelcogopod]

A number is an element in a language, much like the english language, that we created to describe what we see in the world.
The human brain quantifies everything, so it makes sense that we made something to quantify and do other mathematical operations with.

So imo a number is the way we interpret things.

 

[neurocomp2003]

a number(Natural number N) is built from the succ() and pred() operators from the set {1}. An integer(Z) is built much teh same way...from there you can build the set(Q-Rational,R-Real,C-Complex,Pn-Polynomial,Vectors etc).
Like Arilno said, the terms "element" and "set" are rather primitive though i think element comes from set. If you want to go further than that you jump into psychology on why as intelligent beings we begin to label things...once you label things, you call them objects(hence 1)..from there a group of objects is called the set and and element is one of these objects in the set. From there you define and build your operators like succ() and pred()

 

[Jimmy Snyder]

Are there any recent books that deal with the concept of a number in modern mathematics?

Roger Penrose's book "The Road to Reality" has a discussion on this very topic. He seeks a definition of number that is divorced from the physical world, a strictly mathematical definition. He gets one by using sets as follows:

0 \equiv \emptyset
1 \equiv \{\emptyset\}
2 \equiv \{\emptyset, \{\emptyset\}\}
3 \equiv \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}
...

to define the natural numbers and uses these to define negative numbers, rational numbers, real numbers, and complex numbers.

If you are looking for a definition of number that IS rooted in the physical world, then I expect that you run up against certain problems such as extremely large numbers having no physical meaning. That is, there are only so many particles in the universe and numbers larger than that may fail to mean anything.

 

[neurocomp2003]

jimmysnyder...given a finite large number of objects..the permutation/combination will be larger and given states or time the sequence of permutation will grow.

 

[Jimmy Snyder]

given a finite large number of objects..the permutation/combination will be larger and given states or time the sequence of permutation will grow.

Even then, there is a limit. The number of permutations of a finite number of particles in a finite space for a finite period of time is still finite. If you bring in continuity arguments to get an infinite number of spatial and temporal states, then you lose the ability to count altogether.

I'm not sure that you can't base number on some physical phenomenon. I just am pointing out a problem that you may need to overcome.

 

[matt grime]

Just to clarify, the notion of number as cardinals of sets is not an idea of Penrose. It predates him by many decades (as a formal concept, I mean: it predates him as an intuitive idea by millenia).

 

[Jimmy Snyder]

Just to clarify, the notion of number as cardinals of sets is not an idea of Penrose. It predates him by many decades (as a formal concept, I mean: it predates him as an intuitive idea by millenia).

Yes. I wrote that from memory, here is what Penrose says (page 64):

Let us consider one way (developed by [Georg] Cantor from ideas of Guiseppe Peano, and promoted by the distinguished mathematician John von Neumann) that natural numbers can be introduced merely using the abstract notion of set.

He raises the issue of finiteness and a second issue about basing the definition of numbers on physical phenomena:

Suppose ... our universe were such that numbers of objects had a tendency to keep changing. Would natural numbers actually be 'natural' concepts in such a universe?

I don't know if he was referring to the idea in quantum mechanics that numbers of objects do have such a tendency in this universe.

And a third issue:

We can even envisage a universe which consists only of an amorphous featureless substance, for which the very notion of numerical quantification might seem intrinsically inappropriate.

My daughter's bedroom?

 

[Mickey]

The thing about Cantor's approach is that numbers describe sets that are progressively inclusive of one another. The 3 set includes the 2 set, 1 set, and empty set as its elements. This isn't how other languages work. I'd very much like to see how any language can be constructed from nothingness.

The human brain quantifies everything...

What neurological literature did you scoop that from?

 

[neurocomp2003]

Mickey: I don't think he needs a neurological reference. Considering As humans we do like to give things labels, doesn't matter the language. But I'm sure you can find one about how we take a retinal image and store it in memory and with language label it. Therefore as the undamaged brain "sees" an image the string of sounds we use to describe objects in that image begin to fire. I'm sure you can also find literature on the brain quantifying size...large/small more/less etc.

as for teh cantor set...interesting enough he didnt' defin it as
1={0}
2={{0},{0,0}}
3={{0},{0,0},{0,0,0}}

N=N-1 U {N-1 U 0}

 

[Mickey]

But I'm sure you can find one about how we take a retinal image and store it in memory and with language label it. Therefore as the undamaged brain "sees" an image the string of sounds we use to describe objects in that image begin to fire. I'm sure you can also find literature on the brain quantifying size...large/small more/less etc.

I'm sorry, I was under the impression that memory storage has been a persistently elusive problem in neuroscience.

If you can help point me to some literature that shows how the brain "quantifies everything," or stores quantities, I'd be very thankful. :shy:

 

[Pythagorean]

Conjecture:

I've always assumed that the origin of numbers was more of a philosophical debate, divided mostly between people who think numbers are an intrinsic part of the universe (like pythagoras) and people who think numbers a human construct used to interpret the universe. To me, these can coexist easily, because we are an intrinsic part of the universe, but I am both pluralist and agnostic towards most ambiguous matters.

I would think the scientific method somewhat avoids confronting that, since it's such an intangible question. As far as most physical scientists are concerned, it's there, and it's the language that developed from the natural philosophers (old school physicists) as answers to the questions they asked.

I wouldn't be suprised if other life on Earth could count, either. The questions to me, is whether or not they're conscious about it.

 

[neurocomp2003]

Mickey: it is a localization problem. To my knowledge much of what was taught to me states that its in the areas know as the HC,ERC,PRC,SB. Then again the textbooks that i read could be wrong.

 

[Jimmy Snyder]

as for teh cantor set...interesting enough he didnt' defin it as
1={0}
2={{0},{0,0}}
3={{0},{0,0},{0,0,0}}

N=N-1 U {N-1 U 0}

But 0 is a number. Cantor defined numbers in terms of sets alone. What's more, the set {0, 0} is not considered to be anything different from the set {0}.

 

[Jimmy Snyder]

I wouldn't be suprised if other life on Earth could count, either.

I don't remember where I read this, so I can't provide a citation.

Apparently, baboons have a behavior that indicates that they may be able to count to 3. That is, there was a farm field where baboons were feeding. Three farmers walked into the field and the baboons ran away. The farmers hid themselves in the field, but the baboons did not return. Two farmers then got up and walked away from the field, but the baboons still did not return. The third farmer got up and walked away and then the baboons returned to the field. When four farmers walked onto the field, hid themselves and then three got up and walked away, the baboons 'lost count' and returned to the field with the one hidden farmer in it.

 

[neurocomp2003]

my bad i donnt' have the EmptySet symbol

 

[Jimmy Snyder]

my bad i donnt' have the EmptySet symbol

Quote this message and you will have the empty set symbol.

The problem with the definition you propose is that the set \{\emptyset\} is not considered to be anything different from the set \{\emptyset, \emptyset\}

 

[neurocomp2003]

yah i realized that...i think iwas thinking of ordered sets ()...but typed {} then again i shouldn't stay up past 5am

 

[Jimmy Snyder]

yah i realized that...i think iwas thinking of ordered sets ()...but typed {} then again i shouldn't stay up past 5am

I'm not 100% sure, but I think your ordered sets would work. Now the problem is that your definition which uses ordered sets is by no means simpler than Cantor's which only uses sets.

 

[neurocomp2003]

wouldn't the simplist be using the succ operator? (if we are dealing in sets something like
N_k=N_(k-1)U{forall(x in N_(k-1)|succ(x)}

 

[Jimmy Snyder]

wouldn't the simplist be using the succ operator? (if we are dealing in sets something like
N_k=N_(k-1)U{forall(x in N_(k-1)|succ(x)}

I don't know what the succ operator is, but I do know what the empty set is, so for me personally, using succ would not be simpler. Perhaps someone familiar with it can tell you whether it is a simpler definition than Cantor's. Or you could define it and I could express an opinion.

I hope I have not left you with a false impression. I do not know why Cantor chose the definition he did. My answers to your question evolved as your notation evolved. I think your question meant that you felt you had a simpler definition than his and I was unable to find any criterion for determining that it was simpler. That does not mean that I think Cantor found the simplest possible definition or even that he was looking for it. (But if I was a betting man ...)

 

[neurocomp2003]

succ() is +1...succ(x)=x+1;

 

[Jimmy Snyder]

succ() is +1...succ(x)=x+1;

Then it is taboo for the purpose of defining number.

 

[neurocomp2003]

but without teh actual meaning you could still define a number by succ()
succ(succ (x)); succ( succ( succ(x))) etc.
I just find it confusing that someone would begin counting with "empty set".
Unless of course in your example its not really the empty set but a symbol like {1,{1,{1}},{1,{1},{1,{1}}}}..

 

[Mickey]

but without teh actual meaning you could still define a number by succ()
succ(succ (x)); succ( succ( succ(x))) etc.
I just find it confusing that someone would begin counting with "empty set".

Counting with the empty set, a set with no elements, allows one to count without counting anything in particular.

The existence of an empty set is an axiom of set theory, and is sometimes written as {}, without any symbol between the brackets. We could rewrite Cantor's approach this way:

{} = 0
{{}} = 1
{{}, {{}}} = 2
{{}, {{}}, {{{}}}} = 3

But that doesn't look very easy on the eyes. To be clear, we need the axiom of extensionality to know that there is only one empty set. "The" empty set is derived. So, even the notion that emptiness is a unique idea has to be derived from other axioms. Pretty clever, huh?

 

[Jimmy Snyder]

but without teh actual meaning you could still define a number by succ()
succ(succ (x)); succ( succ( succ(x))) etc. }}}}.

Funny, I thought of this on my drive home from work. I am suspicious of your idea of using succ at all if it is to be denuded of any meaning. However, at its heart, your idea seems equivalent to what I came up with:

0 = {}
1 = {{}}
2 = {{{}}}
3 = {{{{}}}}
...

I don't know if this can hold up as well as Cantor's definition nor if it can be considered simpler or better in any context.

I just find it confusing that someone would begin counting with "empty set".
Unless of course in your example its not really the empty set but a symbol like {1,{1,{1}},{1,{1},{1,{1}}}}..

No it's really the empty set. Consider this quote from the book:

This may not be how we usually think of natural numbers, as a matter of definition, but is is one of the ways that mathematicians can come to the concept. Moreover, it shows us, at least, that things like the natural numbers can be conjured literally out of nothing, merely by employing the abstract notion of set.

"literally out of nothing" means out of the empty set.

 

[neurocomp2003]

gotta go back to my set theory text...can't remember all the axioms...I think there's one for identity "exists"x=x, ...i really should learn to use LaTeX. The axiom that there exists an object.

I just find it odd to use the empty set to count because you can get 0 from 1 but how do you get 1 from 0. However If it was symbols without meaning i can understand. It would be like putting strings together in language theory.

 

[matt grime]

but without teh actual meaning you could still define a number by succ()
succ(succ (x)); succ( succ( succ(x))) etc.
I just find it confusing that someone would begin counting with "empty set".
Unless of course in your example its not really the empty set but a symbol like {1,{1,{1}},{1,{1},{1,{1}}}}..

And 1 is what? How do you know it is a valid element in the model of your set theory?

 

[matt grime]

I just find it odd to use the empty set to count because you can get 0 from 1

eh? what?

but how do you get 1 from 0.

It is the cardinal defined by the set that contains the empty set as its unique element.

 

[neurocomp2003]

yeah that's the part of set theory i never got...because I didn't understand why {{}}!={}

Always thought that set theory started with one element/singleton {x}

 

[Jimmy Snyder]

To be clear, we need the axiom of extensionality to know that there is only one empty set.

I never studied axiomatic set theory, so when I first read this, it seemed very weird to me to worry about how many empty sets there are. However, after thinking about it, I decided that it was probably something like the proof in group theory that there is only one identity element. That was the first proof that I ever saw done strictly from the axioms and when I did, I immediately fell in love with mathematics. Here is my pseudo-proof based on a pseudo-axiom (modeled on that group theory proof).

pseudo-axiom: There is a set {} called an empty set. with the property that for any set A, {} union A = A union {} = A.

theorem:There is only one empty set. Pseudo-proof. Let {} and {}' be two empty sets. Then
{} = {} union {}' = {}'
qed

Is that how it's done?

 

[Jimmy Snyder]

yeah that's the part of set theory i never got...because I didn't understand why {{}}!={}

The empty set is not "nothing", it is a set. It is empty, but it is a thing. Like an empty bag is not "nothing", it is a bag. {{}} is not empty. it has a thing in it. {} IS empty, so {{}} and {} are not the same thing. Like a bag that has a bag inside of it is not the same thing as a bag with nothing inside of it.

 

[neurocomp2003]

but a bag is an object.

 

[Jimmy Snyder]

but a bag is an object.

Yes, and a set is an object too. Even the empty set is an object. And just as a bag with an empty bag in it is not itself empty, neither is {{}} empty.

 

[matt grime]

but a bag is an object.

It is also an analogy.

 

[matt grime]

yeah that's the part of set theory i never got...because I didn't understand why {{}}!={}

because one is a set that contains no elements and the other is a set that contains one element, and that element is itself a set, the empty set. It is important you distinguish between a subset of a set and an element of a set.

Always thought that set theory started with one element/singleton {x}

Nope, set theories do not say what the sets or the elements of the sets are, it just tells you the rules that the sets obey. The empty set, and the sets whose existence are implied by the existence of the empty set and the other axioms (such as constructibility) are the only ones that must be in the model.

 

[Mickey]

Jimmy, I don't think so!

We need the axiom of extensionality to tell us what it means for things to be equal. Then we are able to look at empty sets and see that, since they all have the same property, they are one and the same.

In your pseudo-axiom, you seem to already define your empty set as identity, taking for granted the set theory steps.

Also, you only have that one pseudo-axiom, so your pseudo-proof is more like an interpretation of that pseudo-axiom. It's like it's attempting to be an extension of that pseudo-axiom, but you forgot to have another axiom for what it means to "extend" something.

 

[neurocomp2003]

matt grime: so your saying a set that contains a set of no elements? wouldn't that be no elements? What is a set of no elements? just a pair of brace brackets?

Micky: doesn't teh axiom of extensionality states there exists something?

 

[honestrosewater]

matt grime: so your saying a set that contains a set of no elements? wouldn't that be no elements? What is a set of no elements? just a pair of brace brackets?

A pair of brackets is something. As you guys are using them, a pair of brackets denotes a set. The empty set is a set. It seems like you're looking through the brackets, as if you can just delete them if there is nothing inside of them. Is that how you're looking at things? That isn't how it works.

Micky: doesn't teh axiom of extensionality states there exists something?

No, it defines (or extends) equality on sets in terms of the membership relation: two sets are equal iff they contain the same members. Perhaps you are thinking of the Axiom of Infinity. That and the Axiom of the Empty Set are the only two axioms that I have ever seen included in any of the ZF axiomatizations that actually give you a set. At most, the other axioms give you sets if you already have other sets.

Also, ZF is not the only axiomatization of set theory. You don't have to take membership as primitive and define the subset relation, union operation, intersection operation, difference operation, etc. in terms of membership. You could simply take them all as primitive, or, for example, the equivalence that defines subsets in terms of membership

(1) \forall x, y, z \in D \ [(y \subseteq z) \Leftrightarrow (x \in y \rightarrow x \in z)]

where D is your domain of sets, can just as well be used to define membership in terms of susbets, making any necessary changes of course.As for what a number is, if you think that a number is defined by its internal structure, you can let a number be any structure that satisfies, or models, some theory of numbers, a theory of numbers being some set of formulas that you think anything called a number should satisfy. Or if you think that numbers are defined by their relationships with other things, you can let a number be a member of the domain of any structure that satisfies your theory of numbers. That is how I would answer similar questions:

Q: What is a set?
A: A set is a member of the domain of a structure that satisfies the theory of sets.

Q: What is a group?
A: A group is a structure that satisfies the theory of groups.

Q: What is an equality relation?
A: An equality relation is a structure that satisfies the theory of equality relations.

 

[matt grime]

matt grime: so your saying a set that contains a set of no elements? wouldn't that be no elements?

You have just asserted that the empty set does not exist. In the notation used {{}} is a set that contains a set. It therefore has an element in it, namely the set {}, but it would not matter if it were some other set: you're still making the same fallacious step*. You have also asserted that the empty set is a set that contains itself (as an element), so that is two times where you contradict the axioms of ZF (at least), three if we throw in the axiom of extension discussion you're having independently of me.

* the fallacious leap is that you think that the following two things are equal:

A set S, and the set {S} which is a set that contains S.

{S} has one element irrespectiveof what S is, be it the empty set (you agree the empty set is something) or the integers, or some large cardinal.

 

[3trQN]

yeah that's the part of set theory i never got...because I didn't understand why {{}}!={}

Was this adopted in axiomatic set theory after such problems as Russell's Paradox? Was naive set theory approach such that {{}} = {}?

 

[neurocomp2003]

ok i sort of get what you all are saying now...thanks for explaining itto me.

Is there a "Set Existence Axiom" ( thereexists x = x)

matt grimes: As pertaining to the 2 things are equal S= {S}...only for the empty set because i never really understood the empty set.

 

[honestrosewater]

ok i sort of get what you all are saying now...thanks for explaining itto me.

Is there a "Set Existence Axiom" ( thereexists x = x)

There certainly could be. As things are usually done, I think you're bordering on the logical matters, which are usually left in the background in math. Assume your background theory includes first-order logic, which I think is the most used logic in math. For any theory that includes an equality symbol, that symbol gets interpreted, by convention, as the identity relation on the domain of your structure. So for any such theory, the formula

(2) \forall x \in D \ [x = x]

will be satisfied by every structure, i.e., it will be true in every structure. If you also include as part of your background theory the assumption that all structure domains are non-empty or that

(3) \forall x \ [\phi] \rightarrow \exists x \ [\phi]

is satisfied by every structure for any formula \phi, it follows that

(4) \exists x \in D \ [x = x]

will also be satisifed by every structure that satisfies (2). So your background, logical theory and interpretation conventions might give you one or more sets, but you don't know anything else about them except that each is equal to itself. I think it's more of a technicality anyway, as I think you can change this stuff without it having any non-boring effects.

i never really understood the empty set.

The empty set arises naturally from several places. One rather intuitive place, I guess, is the connection between properties and sets. A property that no object has, or that is satisfied by no object, corresponds to the empty set. If no object has the property of being a square circle (or not being equal to itself or being a penguin on my lap), the set of all objects that have the property of being square circles (or not being equal to themselves or being penguins on my lap) would be empty. What objects would such a set contain?

Relating back to your other question, given that (2) is satisfied by your structure, you could define the empty set as the set of all members of your domain that are not equal to themselves:

\emptyset =_{\mbox{def}} \{x : x \not= x\}.

 

[honestrosewater]

Was this adopted in axiomatic set theory after such problems as Russell's Paradox? Was naive set theory approach such that {{}} = {}?

What does that equation mean, exactly?

Russell's Paradox showed that you can't just make up any property whatsoever and turn it into a set (or that not every property defines a set, or that the extension of every property is not a set, or however you want to think of it). You have to put some restrictions on what kinds of things are allowed to be sets. The axioms from which the antinomy is derived are too loose, too inclusive -- they allow you to derive both some formula and its negation ((x \in y) and \neg(x \in y)), so you need to change them to rule out one or both of those formulas. I think that's the basic idea, at least.

 

[neurocomp2003]

honestrosewater: so then the empty set arises because one assumes that there exists objects and these objects are not members of this set [\phi]. Is that correct?

 

[matt grime]

The empty set 'arises' because it is the set of all real roots of x^2+1=0, because it is the intersection of the set of even integers with the set of odd integers, because we find it so much easier to have an empty set so we can talk about aribtrary interesections and sets that possibly might have no element satisfying its defining properties. To do without it would be like trying to do addition without 0, or multiplication without 1 (and that is more than just an unfounded analogy: the empty set is the identity element for the operation of symmetric difference).

 

[honestrosewater]

honestrosewater: so then the empty set arises because one assumes that there exists objects and these objects are not members of this set [\phi]. Is that correct?

No, not really. I meant phi to be a formula, a well-formed string of the language in which your theory is expressed, but I don't think explaining that is going to help anyway. The empty set can arise in many ways. The most straightforward is to just say that it exists.

(1) There exists a set that contains no members.

Why does (1) bother you while

(2) There exists a set that contains one member.

doesn't? People have already said everything else I can think of to say. Can you try to explain why it bothers you? That might help us figure out where the problem is.

 

[loseyourname]

Russell's Paradox showed that you can't just make up any property whatsoever and turn it into a set (or that not every property defines a set, or that the extension of every property is not a set, or however you want to think of it). You have to put some restrictions on what kinds of things are allowed to be sets. The axioms from which the antinomy is derived are too loose, too inclusive -- they allow you to derive both some formula and its negation ((x \in y) and \neg(x \in y)), so you need to change them to rule out one or both of those formulas. I think that's the basic idea, at least.

Russell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.

Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves "R." If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself.

...

The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. For example, assuming both P and ~P, any arbitrary proposition, Q, can be proved as follows: from P we obtain P or Q by the rule of Addition; then from P or Q and ~P we obtain Q by the rule of Disjunctive Syllogism. Because of this, and because set theory underlies all branches of mathematics, many people began to worry that, if set theory was inconsistent, no mathematical proof could be trusted completely.

Russell's paradox ultimately stems from the idea that any coherent condition may be used to determine a set. As a result, most attempts at resolving the paradox have concentrated on various ways of restricting the principles governing set existence found within naive set theory, particularly the so-called Comprehension (or Abstraction) axiom. This axiom in effect states that any propositional function, P(x), containing x as a free variable can be used to determine a set. In other words, corresponding to every propositional function, P(x), there will exist a set whose members are exactly those things, x, that have property P. It is now generally, although not universally, agreed that such an axiom must either be abandoned or modified.

link

Don't you just love the Stanford Encyclopedia? I was getting ready to pull out my copy of The Principles of Mathematics, then I realized I didn't even have to. Notably, in that volume, Russell does define the empty set as the set of all objects that are not equal to themselves, which I believe is one of the definitions Matt gave above, the set of all x's such that x does not equal x.

 

[matt grime]

That wasn't one of the definitions I gave, at least not knowingly. I believe the best definition to give is that the empty set is the (unique) set X for which the statement x in X is always false. In particular this is the kind of thing we have to bear in mind when proving "x in X implies something": this is always true if X is the empty set.