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 the 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 the 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 the 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 the 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 the 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 the 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 the 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.