Number

[Resha Caner]

I don't know if this fits better under "philosophy" or "science", but the science forum seems to get more traffic, so I thought my chances might be better here.

"Number" is one of those things where it's like, "I think I've got it. Yeah, I've got it ... no I don't."

So, if anyone would share their thoughts about "number" (enlightening, sarcastic, condescending, or otherwise), I'd appreciate it.

 

[Tinker Grey]

I'll speak out of my ignorance. It'll give somebody something to correct.

A reasonable starting point, I think, is this: A number is a concept that expresses either 1) how much of something there is, or 2) a relationship between two numbers of definition 1.

 

[Resha Caner]

I'll speak out of my ignorance. It'll give somebody something to correct.

A reasonable starting point, I think, is this: A number is a concept that expresses either 1) how much of something there is, or 2) a relationship between two numbers of definition 1.

Yes, so #1 equates to the cardinal numbers and #2 equates to the ordinal numbers. I don't see that there is anything to correct, but I could focus on your use of "concept." So are numbers only a concept?

 

[leftrightleftrightleft]

So are numbers only a concept?

I would say numbers are only a concept. I don't think they actually exist in the physical world.

I'm one to ascribe to the view of forms and ideas expressed by Plato. Numbers are ideas and are, as such, idealized types of the physical world (forms).

 

[AV1611VET]

I don't know if this fits better under "philosophy" or "science", but the science forum seems to get more traffic, so I thought my chances might be better here.

"Number" is one of those things where it's like, "I think I've got it. Yeah, I've got it ... no I don't."

So, if anyone would share their thoughts about "number" (enlightening, sarcastic, condescending, or otherwise), I'd appreciate it.

Numbers speak of quantity and order.

 

[quatona]

So are numbers only a concept?

Why "only"? A useful concept is quite something.
"Only a concept" - as opposed to what?

 

[Resha Caner]

I would say numbers are only a concept. I don't think they actually exist in the physical world.

OK. Not that I subscribe to Plato's Forms, but I'm OK with taking number as only a concept.

So, let me extend that a bit to clarify that number, then, only exists in the mind. Again, not a problem. But there are still some interesting aspects of number - especially if it only exists in the mind.

First, I can't think of anything that contrasts with number ... if that's the right word. Or would it be better to say it has no duality. For example, "love" has an opposite: "hate." Or "existence" has "nonexistance." Or the "Parallel Postulate" has an alternative: rather than only one parallel through a point to a line there may be none or there may be an infinite amount.

So, is there a concept for:
* opposite-number
* non-number
* alternative-number

If there is, I can't think of it. Are there other concepts like that?

 

[quatona]

OK. Not that I subscribe to Plato's Forms, but I'm OK with taking number as only a concept.

So, let me extend that a bit to clarify that number, then, only exists in the mind. Again, not a problem. But there are still some interesting aspects of number - especially if it only exists in the mind.

First, I can't think of anything that contrasts with number ... if that's the right word. Or would it be better to say it has no duality. For example, "love" has an opposite: "hate." Or "existence" has "nonexistance." Or the "Parallel Postulate" has an alternative: rather than only one parallel through a point to a line there may be none or there may be an infinite amount.

So, is there a concept for:
* opposite-number
* non-number
* alternative-number

If there is, I can't think of it. Are there other concepts like that?

If it´s permitted to create an opposite concept by simply defining and denoting it ex negativo (like you did with existence-nonexistence) the opposite concept of "number" would be "non-number".
Personally, though, I don´t consider non-existence a concept. Rather it´s the epitome of a non-concept.

 

[Tinker Grey]

Countable and uncountable.

 

[Resha Caner]

Countable and uncountable.

Hmm. I can see how "uncountable" is a possibility, and maybe that's good enough. But I can't actually think of something that is uncountable.

I tried to do the same with the ordinals. Is it possible that there is something that can't be ordered? I can't think of it. Suppose we take color. Even though my sensation of color attaches no value to color, it could be ordered by its frequency, i.e. a rainbow.

I was going to say that it seems odd that there is a concept that we are basically compelled to accept. We have no workable alternative to "number."

But, you might be on to something Tinker, so I'll need to withhold that conclusion.

 

[quatona]

Hmm. I can see how "uncountable" is a possibility, and maybe that's good enough. But I can't actually think of something that is uncountable.

Water?

 

[Jazer]

So, if anyone would share their thoughts about "number" (enlightening, sarcastic, condescending, or otherwise), I'd appreciate it.

I do not know if this is a Urban Myth or not. But they say that numbers are something you can not make up. If you cheat on your taxes by making up numbers the computer will spit your return out for investigation. We have been living in this house for about 7 years now and the government is still sending bills for taxes that the women who lived here before owes them. She was running a cleaning business with 15 employees and got into trouble on her taxes. So if they owe you money chances are you will not hear anything from them. If you owe them money you may never see the end of it.

 

[Resha Caner]

Water?

1 mole, 2 moles, 3 moles, ...

or

1 cup, 2 cups, 3 cups, ...

 

[quatona]

1 mole, 2 moles, 3 moles, ...

or

1 cup, 2 cups, 3 cups, ...

These are certain numbers of moles and cups, not numbers of water.
Maybe your question is not about numbers but about quantities in general?

 

[metherion]

Well, you see, you had to attach a unit to the water before you counted, and you weren't counting the water, you were counting the units of water.

And numbers can also be used to do things that aren't counting. Take pi. Pi, among other things, is a ratio between two numbers, the radius of a circle and the diameter of a circle. The units would cancel out, so pi isn't counting anything or measuring any units of anything.

And something that is uncountable might be something whose exact number we cannot know due to technology. Like, we can count how many moles of water there are, but not exactly how many molecules there are in, say, the oceans of the world. We can find a range, but I don't think we can find the exact number. So it would be uncountable.

Metherion

 

[KCfromNC]

Yes, so #1 equates to the cardinal numbers and #2 equates to the ordinal numbers. I don't see that there is anything to correct, but I could focus on your use of "concept." So are numbers only a concept?

Math is a formal language. Like natural language, it can be used to communicate ideas about things which exist in the real world and can also be used to communicate ideas about things which don't. Numbers are one of the constructs of this language, so they are things which may or may not relate to something outside of abstract concepts depending on the context in which they are used.

 

[Resha Caner]

Well, you see, you had to attach a unit to the water before you counted, and you weren't counting the water, you were counting the units of water.

I see that as a false dichotomy. You've merely defined water in a way that makes it uncountable. If I choose to define water differently, it becomes countable ... and that is what scientists have done (rather clever of those chemists methinks).

The same thing follows for all your examples. My ability to count individual water molecules should have no bearing on whether something is countable. If I can conceive of a way to do it (given that numbers are only concepts anyway), the rest is mere execution.

 

[Resha Caner]

Hmm. OK. I expected someone to bring up Frege. I'm no expert on his work, so I hoped someone might know more of him than I.

Frege is one of those guys whose importance is demonstrated by the fact that everybody likes to point out why he was wrong. But I have to say, he noted some very cool things.

For example, prior to Frege many considered "number" to be a property. Frege showed how "number" does not meet the criteria of a property. For example, if I have 5 green houses, I can note that "green" is a property of each and every house: "the house is green."

But I can't say, "The house is five." I can't even say, "the houses are five," in the same way as "the houses are green." So, Frege insisted that "number" is not an adjective. It is a noun. And, I think you'll find that most dictionaries list "one", "two", "three", etc. as nouns. That is very curious.

Why?

Well, consider the noun "tree." It can be used as a noun or an adjective. I can walk up a mountain and define a "tree line." The line dividing the place where trees grow and where trees don't grow is of type "tree."

I can't do the same with number. I can't find a "number line" that divides the place where numbers exist from the place where numbers don't exist. So, "number" is not the same as "tree" in that way. Nor is it the same as "tree" in my ability to say, "There is a tree." I can't point out, "There is a five." Five must always be associated with something else, be it five objects, the word (five), or the symbol (5).

There are numerous other examples like that with other nouns.

Yet, Frege seems to be very successful with his definition of number as an object. For example, his definition of "zero" is "the count of things which contradict themself" (or something like that). Pretty cool. Most of all because it embeds the law of non-contradiction in number theory. In other words, the fundamental law of logic also becomes the fundamental law of "number." Just so cool.

 

[metherion]

I'm sorry, I truly did not see it as a false dichotomy. This was my line of thought:

Me: (walks up to an ocean). I'm going to count the water! But how do I count water? Water is H2O. Well, that's one molecule of water. So to count water, I need to find a unit of water. Water is not something with inherent units to count, so it is not countable. But units of water are countable, therefore, I cannot count water, I can count units of water. I can say there is one ocean here. I can say there are billions of gallons, or millions of tons, or bajillions of moles, or ten to a very large power of molecules, but those are units of water. So I am not counting water, I am counting units. I don't have 10 waters, I have 10 moles of water, or 10 gallons of water, or 10 molecules of water, but I don't have 10 waters. I suppose I could have ten TYPES of water (salty, potable, mixed with things, and so on), so that might be 10 waters, maybe.

I also appear to have been making the wrong distinction in countability. I was referring to practical countability (such as the exact amount molecules in the oceans) versus something impossible to count.

Metherion

 

[Resha Caner]

metherion,

I understand what you're saying. I just disagree. It's not really something I think is worth arguing about, so I'll just explain how I see it.

Every object that is to be counted is assigned a "unit." Since some units are obvious, I think you're trying to make it "self-evident", and then excluding as uncountable those objects that don't have obvious units.

If I count pennies, I have implicity assumed a unit of "penny." But I don't have to. I could count 1/2 pennies (that used to be legal tender). It would not be proper to claim that because one has adopted the 1/2 penny unit that pennies are uncountable.