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Since numbers are infinite, does any one number express any real value? Let's take the number n. Since there are an infinite number of integers that are both greater and less than n - is n really expressing a value that is different from n + 1 or n - 1?


There is no passion to be found playing small, in settling for a life
that is less than the one you are capable of living. - Mandela

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quote:
Originally posted by MBM:
Since numbers are infinite, does any one number express any real value? Let's take the number _n_. Since there are an infinite number of integers that are both greater and less than _n_ - is _n_ really expressing a value that is different from _n_ + 1 or _n_ - 1?



When I get my checking account statement in the mail, I still have a distinct preference for positive numbers over negative numbers... winkgrin

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The Nuclear Resister
School of the Americas Watch
miserable failure



Cauca, Colombia

"Since numbers are infinite, does any one number express any real value? Let's take the number n. Since there are an infinite number of integers that are both greater and less than n - is n really expressing a value that is different from n + 1 or n - 1?"

Yes. The number line is infinite, but any individual number n is not; it is finite, concrete. If n = 2, for example, or even 2^1231231231, it is a discrete quantifiable number, fundamentally distinguishable from every other number. If it were otherwise, we would have to consider each number a conceptual representation of all numbers which had infinite simultaneous values. In other words, any one number could represent every number, which would get a bit confusing. It could make for a hell of an argument on your calc final, though.
But I guess what I'm asking though is in the "universal" scheme of things, where there are infinite numbers greater than and less than any single number, does the value of one number have any real meaning?

quote:
Originally posted by djonmaila:

If n = 2, for example, or even 2^1231231231, it is a discrete quantifiable number, fundamentally distinguishable from every other number.


But, respectfully, "so what"? In a universe of infinite numbers what is the relevance of any one number?

quote:
If it were otherwise, we would have to consider each number a conceptual representation of all numbers which had infinite simultaneous values. In other words, any one number could represent every number, which would get a bit confusing.


No, I think what I'm asking is the inverse of this. I'm not suggesting that numbers have no value, I'm just questioning the relevance opf that value within an infinite context? Of course humans create contexts for the numbers to have meaning (for example in temperature). In a universal or theoretical context though, our scale is wholly arbitrary and meaningless though.



There is no passion to be found playing small, in settling for a life
that is less than the one you are capable of living. - Mandela


[This message was edited by MBM on December 15, 2003 at 10:31 AM.]
Individual integers still have properties that are shared with no others.

For example 0 has the property that 0+0=0. No other integer has that property.



It's all a question of what operations and relations that you consider fundamental to the integers.

If you consider the integers with only the order relation, that is, the structure (Z,<), then indeed there is nothing special about 0 or any other number. On the other hand, if you consider the integers with addition, that is, the structure [Z,+), then 0 is special. However, there is nothing to distinguish 1 from -1.

In the structures (Z,+,<) or [Z,+,*), you can distinguish every integer from any other.

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The Nuclear Resister
School of the Americas Watch
miserable failure



Cauca, Colombia

"But, respectfully, "so what"? In a universe of infinite numbers what is the relevance of any one number?"

So, if every number is distinguishable, no number is irrelevant in the "universal scheme of things", and n should not be conflated with n + 1. No number is diminished by its company of fellows.

"I'm not suggesting that numbers have no value, I'm just questioning the relevance opf that value within an infinite context?"

I didn't say they would have no value either, I said they'd have infinite value, every value. And the concept of "relevance" only makes sense in the context of its interaction with an organism (generally ourselves). That we don't conduct our daily lives on the scale infinite quantities is a fuction of our comprehension of the finite. Our number systems are arranged to be relevant to us, consumable by us. So to say that any number is rendered irrelevant implies the presence of an actor (us) for whom a number loses meaning. But what I am saying is that in the world of numbers, our great "universal scheme", the actor's understanding of a number is irrelevant to the number system, so her perspective is not a factor. Numbers exist outside the sphere of our comprehension of them, and therefore cannot be made irrelevant to the universe based on our human understanding of them. So it seems to me that any number which cannot be represented by any other number is not only universally relevant, but uniquely so.

"Of course humans create contexts for the numbers to have meaning (for example in temperature). In a universal or theoretical context though, our scale is wholly arbitrary and meaningless though."

Of course. It has to be in order to make any sense out of it. We cannot approach the world from the perspective of the universe, because that perspective is not relevant to us and that's where the arbitrariness of convention enters.
In our world, numbers are only valuable because they express a value. One dollar is greater than two dollars because we have created a context where that is the case. That context seems arbitrary and, in fact, out of context with the universal existence of numbers. I'm trying to understand the value of numbers relative to all the other numbers.

Can you answer whether the value of any one number is at all different from the value of another? In a world of infinitely large and small numbers, does the number 1 have a value that is different from 1, 234, 567, 890, for example. Aren't there infinite numbers less than and greater than each? Doesn't that indicate that their values are therefore equal?


There is no passion to be found playing small, in settling for a life
that is less than the one you are capable of living. - Mandela
There are two types of infinite numbers, ordinal numbers, which have to do with well orderings (counting), and cardinal numbers, which have to do with size comparrisons.

Every cardinal number is an ordinal number, but not vis versa. They agree on the finites, however, which is why the distinction is seldom mentioned in elementary mathematics.

In either case, there are different degrees of infinity.

In both cases, there are no negatives, and 0 is as small as things get.

If two sets can be put in one-to-one correspondence, they are said to have the same cardinality, and their size is represented by the same cardinal number.

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The Nuclear Resister
School of the Americas Watch
miserable failure



Cauca, Colombia

"Can you answer whether the value of any one number is at all different from the value of another? In a world of infinitely large and small numbers, does the number 1 have a value that is different from 1, 234, 567, 890, for example. Aren't there infinite numbers less than and greater than each? Doesn't that indicate that their values are therefore equal?"

No, but this answer is the same as the previous one. No two numbers can be equal if they are distinguishable. Also, if we are considering numbers in an infinite context, we should also be aware that there is infinity between any two numbers. If you take the numbers 0 and 1 as the outer bounds of your universe, there is still an infinite number of numbers between them.

So if the number of numbers greater than or less than n is infinity and the distance between any numbers n and m is infinity, then the same concept that makes any number or scale look infinitesimal also renders each number infinitely distant from any other number. So the distance between 1, 234, 567, and 890 is therefore just as significant as the difference between n and infinity.
I agree. So then how is there a difference in the value between two numbers? You say that they are "distinguishable? In what way? They may be distinguishable as two discrete points on a number line, but I'm focusing on the value of the numbers themselves.


There is no passion to be found playing small, in settling for a life
that is less than the one you are capable of living. - Mandela
I'm thinking about the real value of the number. For example, 2 is 100% greater than 1. In just considering those two numbers, that is a concept that I understand. But in a universal scheme of things where there are an infinite amount of numbers - does the fact that 2 is 100% greater than 1 have any meaning? (Even the concept of percentages is made irrelevant on an infinite scale.)

On a closed number line from 0 - 10. The number 5, for example, is meaningful because it is 50% between 1 and 10. There are an equal number of integers less than and greater than 5. That provides needed context to understand the value of that number.

With an infinite amount of numbers less than or greater than any number - I'm not sure I understand what information there is to differentiate any one number from any other. Do you see what I'm saying? It would be like if we believed our universe was infinitely big, then would the concept of a mile really hold any meaning? (You've flown a mile in your space ship, but still have an infinite number of miles to go.) Meaning would seem to only come into play if we created some artificial subset to consider it.


There is no passion to be found playing small, in settling for a life
that is less than the one you are capable of living. - Mandela
"I'm thinking about the real value of the number. For example, 2 is 100% greater than 1. In just considering those two numbers, that is a concept that I understand. But in a universal scheme of things where there are an infinite amount of numbers - does the fact that 2 is 100% greater than 1 have any meaning? (Even the concept of percentages is made irrelevant on an infinite scale.)"

I guess it depends on who you are. For us, yes, because we deal with units as small as 2 and 1 regularly, but for some other being somewhere, it may have no meaning at all. But the meaning itself is really just a relationship between the values and a mind that contemplates them. If you don't consider the mind, the question of meaning is an empty one. I understand what you're saying, and I agree that our perceptions and perspectives are dwarfed by the totality of all possible scales, but who's to say that it's not the minutia that's meaningless, but the infinite? You could just as easily argue that because of the vastness of all values, any value that doesn't have a use to us is meaningless, irrelevant. Likewise, percentages in the inverse scale are meaningless (to us). What need to we have of conceptualizing a value that is some infinite times the size of a kilometer, for example, even though the percentage may be incomprehensibly large?

And if you consider that the difference between 2 and 1 is next to nothing compared to the difference between 1 and infinity, it's still greater than precisely 0, and that's enough to make a difference on some scale.


"With an infinite amount of numbers less than or greater than any number - I'm not sure I understand what information there is to differentiate any one number from any other. Do you see what I'm saying? It would be like if we believed our universe was infinitely big, then would the concept of a mile really hold any meaning? "

But what differentiates any number from the next is its value. So if there are infinite miles in the universe, the relationship between the concept of a mile and a linear unit a space is inflexible (in classical physics, anyway), so even if there are uncountable miles, the distance of a single mile is still concrete. On a line of infinite length you run into trouble because there is no beginning or end. You can have perfect knowledge of where you began and still never know how much farther you have to go, so counting progress in miles, in fact counting at all, becomes rather pointless because you'll never stop. But the miles are still there beneath you and the distance is still real.

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