A number of numbers

[Speakpigeon]

I'm starting something without much of an idea as to where it might go. So, we'll have to see.

The target is a very precise mathematical point I'd like to clarify but I need to understand first exactly what concept mathematicians have of numbers. I plan to go into the question of the opposition between rational and irrational numbers, and then transcendental numbers, and then computable and non-computable numbers. And I'm no specialist! Sooo, if you fear it's just going to be too metaphysical, or perhaps too physical, feel free to move it accordingly.Just remember that there may not be anybody that will have anything to say on the subject over there.

So, I tend to make the distinction between three things. First, quantities. Physical quantities essentially but I'm not opposed to the idea of counting angels. Quantities, we suppose metaphysically, are things, perhaps properties, that exist somehow out there, in the physical world, like perhaps the always changing number of children in the courtyard or the varying quantity of potential energy of two bodies orbiting each other.

Second, there is our intuitive notion of quantities. I look at physical things and inevitably see them as quantities of other things. I see water as a quantity of molecules, humanity as a quantity of human beings etc.

Third, we have numbers. Numbers, I suspect, are thought of by most mathematicians essentially as symbolic expressions humans use to represent physical quantities. Mathematicians don't care much, I think, what they are used for. They are interested in their formal properties. Then, perhaps, physicists take a look at those numbers and try to figure out whether they might represent real physical quantities or what.

I'm going to stop here to see first if there is any interest in the subject. The question for now really is just this: What exactly is a number?
EB

 

[Jimmy Higgins]

A number is a step of a range.

 

[Speakpigeon]

A number is a step of a range.

Sorry, that's bit too cryptic. Doesn't make sense to me. What kind of step and what kind of range?!
EB

 

[fast]

Number! I'd say it's the referent to which numeral's refer!

Let me tell you what a number isn't. It isn't a product of human intellect. They did not spring forth from our mental concepts of them. We didn't invent them. Through intelligence, we discovered them. Hey were here all along.

Numerals, like words or symbols, sure, we invented those, but numbers, well, they've existed for all time. There were eight planets orbiting our sun prior to our human arrival. Sorry Pluto, we still love you.

You know I can go on, but I'm just throwing in a bite; after all, the thread might meander a different direction in need of a different answer for a different question.

 

[fast]

A number is a step of a range.

Sorry, that's bit too cryptic. Doesn't make sense to me. What kind of step and what kind of range?!
EB

I think he's eluding to a spot on a number line. Think of it like a ladder: might help if we tilt it vertically. A number then, would be a step on the ladder. So, there's where "step" comes in--a spot or point--a place between two ends (or nonendings); hence the range.

 

[Jimmy Higgins]

A number is a step of a range.

Sorry, that's bit too cryptic. Doesn't make sense to me. What kind of step and what kind of range?!
EB

It is a scale. A number has no universal meaning, only the relative ranges between different numbers on that particular scale.

It is really simple, take a metric and english thermometer. What are the numbers on them? Why can they have different numbers for the same temperature?

 

[Speakpigeon]

Number! I'd say it's the referent to which numeral's refer!

Let me tell you what a number isn't. It isn't a product of human intellect. They did not spring forth from our mental concepts of them. We didn't invent them. Through intelligence, we discovered them. Hey were here all along.

Numerals, like words or symbols, sure, we invented those, but numbers, well, they've existed for all time. There were eight planets orbiting our sun prior to our human arrival. Sorry Pluto, we still love you.

I'd be minded to agree here. There's a number of children in the courtyard and that number shouldn't be thought of as somehow dependent on whatever symbolic system we happen to have invented and want to use to represent physical quantities.

Still, is that at all acceptable to mathematicians?

Number
1 (Mathematics)
a. a member of the set of positive integers; one of a series of symbols of unique meaning in a fixed order that can be derived by counting.
b. a member of any of the following sets of mathematical objects: integers, rational numbers, real numbers, and complex numbers. These sets can be derived from the positive integers through various algebraic and analytic constructions.

See?

There's apparently no non-ambiguous definition of "number" that would match what you seem to mean here...

Although there's something good enough for particular numbers, like for example "five" here:

five
3. the amount or quantity that is one greater than four

Is that it?

But it doesn't actually say here that "five" in this case is a number. It's a quantity. Is it a number as well?
EB

 

[Speakpigeon]

A number is a step of a range.

Sorry, that's bit too cryptic. Doesn't make sense to me. What kind of step and what kind of range?!
EB

It is a scale. A number has no universal meaning, only the relative ranges between different numbers on that particular scale.

It is really simple, take a metric and english thermometer. What are the numbers on them? Why can they have different numbers for the same temperature?

Oh, I think I got it! My fingers on my hand is a scale and number one is my little finger, and like this up to five for my thumb.

Is that it? Is this all there is to it, do you think?
EB

 

[Jimmy Higgins]

It is a scale. A number has no universal meaning, only the relative ranges between different numbers on that particular scale.

It is really simple, take a metric and english thermometer. What are the numbers on them? Why can they have different numbers for the same temperature?

Oh, I think I got it! My fingers on my hand is a scale and number one is my little finger, and like this up to five for my thumb.

No, not like that. That is counting, not a scale.

 

[Speakpigeon]

OK, go on, I'm listening.
EB

 

[Jimmy Higgins]

OK, go on, I'm listening.
EB

No, you’re not.

 

[Juma]

Number! I'd say it's the referent to which numeral's refer!

Let me tell you what a number isn't. It isn't a product of human intellect. They did not spring forth from our mental concepts of them. We didn't invent them. Through intelligence, we discovered them. Hey were here all along.

Numerals, like words or symbols, sure, we invented those, but numbers, well, they've existed for all time. There were eight planets orbiting our sun prior to our human arrival. Sorry Pluto, we still love you.

You know I can go on, but I'm just throwing in a bite; after all, the thread might meander a different direction in need of a different answer for a different question.

On the contary: a number is nothing but a product of our intellect. Its not an invention since its a result of how our brain is wired. Numbers arise from our ability to construct groups of features in our perception.

 

[steve_bank]

Number! I'd say it's the referent to which numeral's refer!

Let me tell you what a number isn't. It isn't a product of human intellect. They did not spring forth from our mental concepts of them. We didn't invent them. Through intelligence, we discovered them. Hey were here all along.

Numerals, like words or symbols, sure, we invented those, but numbers, well, they've existed for all time. There were eight planets orbiting our sun prior to our human arrival. Sorry Pluto, we still love you.

You know I can go on, but I'm just throwing in a bite; after all, the thread might meander a different direction in need of a different answer for a different question.

So you are from the math as a separate reality crowd? They are 'out there' somewhere? Math like language evolved. Human creative invention, controlled fire, arrows and spin stabilization with feathers without sciince and math. Sharpened stone spear tips. Chimps, monkeys, and birds demonstrate creative invention.

Numbers used to count objects do not exist without someone to count things. Counting likely arose out of meccesity, the need to communicate about objects and animals.bCounting enhanced survival of the group.

 

[bilby]

A number is a step of a range.

Sorry, that's bit too cryptic. Doesn't make sense to me. What kind of step and what kind of range?!
EB

It is a scale. A number has no universal meaning, only the relative ranges between different numbers on that particular scale.

It is really simple, take a metric and english thermometer. What are the numbers on them? Why can they have different numbers for the same temperature?

English thermometers are metric. If you want a non-metric thermometer, you need to ask an American.

 

[steve_bank]

A number is a step of a range.

Sorry, that's bit too cryptic. Doesn't make sense to me. What kind of step and what kind of range?!
EB

It is a scale. A number has no universal meaning, only the relative ranges between different numbers on that particular scale.

It is really simple, take a metric and english thermometer. What are the numbers on them? Why can they have different numbers for the same temperature?

I agree. Numbers are abstractions no different than words. Meaning is whatever we give both numbers and words.

 

[fast]

The number three is the class of all triples; that, which is itself a class, belongs (as a member) to the even higher order class, numbers. So yes, numbers are 'out there', but spooky they are not, just actual truths of the immaterial and intangible variety.

Also, we do not need an intelligent being to count how many planets we are from the nearest star, but even in the absence of someone to count, there are a specific number of them. The system we have created and decided to use to count them most certainly does depend on us, but truth of what is there for our discovery predates the system and thus demonstrates that the facts observed are independent of us, even should it so be that ... .

 

[fast]

It is a scale. A number has no universal meaning, only the relative ranges between different numbers on that particular scale.

It is really simple, take a metric and english thermometer. What are the numbers on them? Why can they have different numbers for the same temperature?

I agree. Numbers are abstractions no different than words. Meaning is whatever we give both numbers and words.

Don't agree. Accept the ambiguity of "abstract." There are two different meanings. One pertains to the realm of the mental whereas the other does not. An abstract concept is substantively different than an abstract object.

 

[Speakpigeon]

It is a scale. A number has no universal meaning, only the relative ranges between different numbers on that particular scale.

It is really simple, take a metric and english thermometer. What are the numbers on them? Why can they have different numbers for the same temperature?

English thermometers are metric. If you want a non-metric thermometer, you need to ask an American.

Thanks, bilby. Now, I guess, I understand what JH said.

Still, I don't think I understand the point he's trying to make and it sure doesn't look like he's ready yet to explain anything much.
EB

 

[Speakpigeon]

Number! I'd say it's the referent to which numeral's refer!

Let me tell you what a number isn't. It isn't a product of human intellect. They did not spring forth from our mental concepts of them. We didn't invent them. Through intelligence, we discovered them. Hey were here all along.

Numerals, like words or symbols, sure, we invented those, but numbers, well, they've existed for all time. There were eight planets orbiting our sun prior to our human arrival. Sorry Pluto, we still love you.

You know I can go on, but I'm just throwing in a bite; after all, the thread might meander a different direction in need of a different answer for a different question.

On the contary: a number is nothing but a product of our intellect. Its not an invention since its a result of how our brain is wired. Numbers arise from our ability to construct groups of features in our perception.

I can see your point but one doesn't preclude the other.

I'm really interested in how mathematicians conceive of numbers, not whether their conception is correct. Still, the idea here is to go through the main conception people have of numbers and see what could come out of that.

My personal view is that numbers as we routinely conceive of them are indeed somehow human inventions, like the colour red or even pain, but that there likely are things out there that somehow correspond to our invented numbers and I don't see the point of not also calling these things numbers, just like the word "tree" is no doubt a human invention to which nonetheless correspond actual physical things that it would be needlessly fussy to try and not call "trees". We can always clarify by specifying what it is we mean, either the human-made concept or the thing that's supposed to exist out there. My point is that we certainly don't know that something much like numbers don't actually exist in the physical world although they may not be much like our concept of numbers.

Perhaps the crucial point is that our interpretation of the world should remain consistent through our use of the concept of number. So, the question is whether there is any problem in our use of the concept of numbers. Is there anything contradictory with other things we think we observe of the world?
EB

 

[Speakpigeon]

e ambiguity of "abstract." There are two different meanings. One pertains to the realm of the mental whereas the other does not. An abstract concept is substantively different than an abstract object.

"Abstract object" sounds to me like a contradiction in terms. Can't we just say that a number is a physical quality for example, or a physical property? If there are five children in the courtyard, then it seems to me that the number 5 here can be thought of as a property of the physical world. And in this case a very concrete property, obviously. The word "abstract" just seems to bring confusion.
EB