What do you think of Infinity as a concept?

[warpus]

You can't reach i if you continue adding integers to each other either, but we consider i a number. It's not a "Real" number, and unlike any Real number it does not have express an ordering. Infinity is similar. It's not "Real" but unlike i it expresses an ordering. Unlike 1.3 it can express the number of objects in a group.

One of the differences between i and infinity is that i is a specific and very exact amount. It is exactly i, no more, no less. For this reason you can just plop it into any equation. You can't do that with infinity because it doesn't represent that very specific value. It doesn't even represent a group of values. Infinity + 5 doesn't make sense, nor does infinity / 7. Contrast that with i/5 -- makes perfect sense.

 

[Souron]

One of the differences between i and infinity is that i is a specific and very exact amount. It is exactly i, no more, no less. For this reason you can just plop it into any equation. You can't do that with infinity because it doesn't represent that very specific value. It doesn't even represent a group of values. Infinity + 5 doesn't make sense, nor does infinity / 7. Contrast that with i/5 -- makes perfect sense.

i can't be more or less than anything; complex number don't create relational ordering .

Infinity can be one or a group of numbers depending how you use it. And you can plop it in an equation. Infinity + 5 makes perfect sense. An immortal being is born 5 days earlier than expected. The observation that infinity + 5 is also a infinite quantity is a true, and it could imply that infinity + 5 is infinity, or that infinity + 5 is a different number that infinity. Either approach is reasonable. IEEE uses infinity as a single number, Hyperreals use infinity as a class of numbers.

What is 7 days to an immortal? 7 days / infinite days lifespan tells us that it's infinitesimal, or 0, and doesn't matter much at all. That's practical algebra with infinity.

 

[onejayhawk]

i can't be more or less than anything; complex number don't create relational ordering .

Infinity can be one or a group of numbers depending how you use it. And you can plop it in an equation. Infinity + 5 makes perfect sense. An immortal being is born 5 days earlier than expected. The observation that infinity + 5 is also a infinite quantity is a true, and it could imply that infinity + 5 is infinity, or that infinity + 5 is a different number that infinity. Either approach is reasonable. IEEE uses infinity as a single number, Hyperreals use infinity as a class of numbers.

What is 7 days to an immortal? 7 days / infinite days lifespan tells us that it's infinitesimal, or 0, and doesn't matter much at all. That's practical algebra with infinity.

It's a complex subject.

J

 

[warpus]

Infinity can be one or a group of numbers depending how you use it. And you can plop it in an equation. Infinity + 5 makes perfect sense. An immortal being is born 5 days earlier than expected. The observation that infinity + 5 is also a infinite quantity is a true, and it could imply that infinity + 5 is infinity, or that infinity + 5 is a different number that infinity. Either approach is reasonable. IEEE uses infinity as a single number, Hyperreals use infinity as a class of numbers.

The thing is that you can't add 5 to infinity, because.. well, because infinity is not a number, so you're not sure where exactly you're starting at and what you're adding the 5 to.

It doesn't make sense as a mathematical equation, but it can make sense in English (or whatever spoken language), in certain contexts. Engineers deal with applied math, usually not theoretical, so I can see why they'd cut corners and view it as a number, in contexts where it makes sense for whatever they're working on at the time.

The example with an immortal being is not a good one, because in that example you are present when he/she is born. How do you know this being will be immortal and will live forever? You don't. If you're always there to keep track of how old this being gets, you will always have a concrete number as the age, be it 15 years old, 20,000 years old, or 5 million years old. Infinity doesn't enter into it.

 

[Kyriakos]

The thing is that you can't add 5 to infinity, because.. well, because infinity is not a number, so you're not sure where exactly you're starting at and what you're adding the 5 to.

It doesn't make sense as a mathematical equation, but it can make sense in English (or whatever spoken language), in certain contexts. Engineers deal with applied math, usually not theoretical, so I can see why they'd cut corners and view it as a number, in contexts where it makes sense for whatever they're working on at the time.

The example with an immortal being is not a good one, because in that example you are present when he/she is born. How do you know this being will be immortal and will live forever? You don't. If you're always there to keep track of how old this being gets, you will always have a concrete number as the age, be it 15 years old, 20,000 years old, or 5 million years old. Infinity doesn't enter into it.

(Last bit was stressed by me)

While i can see your point, i don't think the example Souron gave was wrong, cause much like the age of an 'immortal' being, you can also account for set values of something which is still infinite. A basic example is any infinite series. The Fibonacci series has infinite positions. But you can account for what is in a finite position of it (eg after 0 there is 1, then 1, then 2, then 3, then 5, then 8 and so on).

There is no final Fibonacci number (and surely no final such number in the current math make-up of infinite series as a concept). But you can always be 'concurrent' (ie born at the same time as, in your counter-example) with any finite position of the series, and thus indeed know its number in that position

(the more boring, but parallel example, would be the series of 1+1+1+1+1+..., which would give you the specific year in each yearly position, and would be infinite too).

 

[classical_hero]

To be perfectly ho0nest, the concept of infinity is something we can't really understand since we are finite being, being constrained by time. I would say that we will never really be able to properly explain the concept in terms we are familiar with. For us to define this concept is impossible for us, since it is something we can never fully understand.

 

[Kyriakos]

To be perfectly ho0nest, the concept of infinity is something we can't really understand since we are finite being, being constrained by time. I would say that we will never really be able to properly explain the concept in terms we are familiar with. For us to define this concept is impossible for us, since it is something we can never fully understand.

Maybe. Or maybe if we define it, that would be a very critical breakthrough for all of humanity.

In my view humans sense infinity, which is why there is a concept of it and some theoretical work around (math primarily, and secondarily in art).
But it is one thing to sense something, and quite another to define it in a stable and concrete system of thought. I still think it will happen in our future.

 

[Souron]

The thing is that you can't add 5 to infinity, because.. well, because infinity is not a number, so you're not sure where exactly you're starting at and what you're adding the 5 to.

It doesn't make sense as a mathematical equation, but it can make sense in English (or whatever spoken language), in certain contexts. Engineers deal with applied math, usually not theoretical, so I can see why they'd cut corners and view it as a number, in contexts where it makes sense for whatever they're working on at the time.

Infinity is not a number except when it is...

If it acts like a number, why isn't it a number? If you can do algebra with it, use it in an equation, why isn't it a number? What definition of a number does it violate?

The example with an immortal being is not a good one, because in that example you are present when he/she is born. How do you know this being will be immortal and will live forever? You don't. If you're always there to keep track of how old this being gets, you will always have a concrete number as the age, be it 15 years old, 20,000 years old, or 5 million years old. Infinity doesn't enter into it.

Perhaps this is a better example:

In (some sects of) Christian belief, it is commonly argued that you should be good in life, so that you will not spend eternity in Hell. If you're good, you go to Hevean; if you're bad, to Hell. Life is short compared to eternity, so you should be good, for a life to be rewarded forever.

We can express that in math terms as [lifespan of immortal soul] >> [lifespan on earth]. In with few intermediate equations, we can get to [reward of being good] > [reward of being bad]. But [Life of immortal soul] is infinity. So if that expression is using useful mathematics, then it is treating infinity as a number. If you can't do mathematics using infinity in this way, then you can't claim it's conclusion, that the reward of being good is greater than the reward of being bad for an immortal soul.

I suppose you could try to formulate a similar answer with calculus, but why is that approach superior?

 

[uppi]

To be perfectly ho0nest, the concept of infinity is something we can't really understand since we are finite being, being constrained by time. I would say that we will never really be able to properly explain the concept in terms we are familiar with. For us to define this concept is impossible for us, since it is something we can never fully understand.

I disagree: Nothing in the universe is infinite, the concept of infinity is something that humans invented. If no humans understood it, it would be useless to humanity. That it remains in use as a concept implies that there are humans that can understand it. The conceptions humans have about infinity might differ, but as there is no "true" conception of infinity, that does not mater at all.

 

[Kyriakos]

I disagree: Nothing in the universe is infinite, the concept of infinity is something that humans invented. If no humans understood it, it would be useless to humanity. That it remains in use as a concept implies that there are humans that can understand it. The conceptions humans have about infinity might differ, but as there is no "true" conception of infinity, that does not mater at all.

That a concept of infinity is invented by humans does not mean that it has to be so, or it is clear that "nothing in the universe is infinite".
In fact it has little to do with whether it exists as a "reality" (no matter what that would be or mean). It only means we have a concept of it, and variations of that concept. It says nothing at all about the universe having it as a 'real' part of it. It only says that it can rise (in one manner or other) to be a concept.

I regard it at least as equally likely that the universe is more linked to "infinity" than being finite, and there is nothing which can establish proof of this either way. We have serious problems even establishing a proof on far less significant questions (riemann might ring a bell), let alone one of this enormous (or infinite ) magnitude.

 

[Souron]

Maybe. Or maybe if we define it, that would be a very critical breakthrough for all of humanity.

In my view humans sense infinity, which is why there is a concept of it and some theoretical work around (math primarily, and secondarily in art).
But it is one thing to sense something, and quite another to define it in a stable and concrete system of thought. I still think it will happen in our future.

I agree.

I think the difference here is also the difference between the number 1, and the property of being 1 in quantity. An apple can be red or green, and in a similar way there can be one or two or infinite apples. "One-ness" is a property of apples like redness. So is "infinite-ness". And I agree that in this sense, infinity is not at all hard to sense. We can understand what infinite apples means.

One is also a purely abstract number that you can manipulate without knowing what it represents. But when you do this, you define your algebra to be analogous to what it represents. Defining such an algebra is where you run into complexity in defining infinity. The usual resolution is calculus.

 

[nc-1701]

Infinity is not a number except when it is...

If it acts like a number, why isn't it a number? If you can do algebra with it, use it in an equation, why isn't it a number? What definition of a number does it violate?

You can't do algebra with it, that's the point

 

[warpus]

Infinity is not a number except when it is...

If it acts like a number, why isn't it a number? If you can do algebra with it, use it in an equation, why isn't it a number? What definition of a number does it violate?

It doesn't act like a number though. You can't use it in equations.. okay, sometimes you can, but as an approximation, I'm guessing. I'm no engineer, I have no idea how they use it. You can use it informally in equations as well, like I'm about to do below.

It's not a number. If it were a number, you would be able to add 1 to it and get a different number. But 1 + infinity is just infinity. 2 + infinity is infinity yet again. That's not how numbers behave.

We can express that in math terms as [lifespan of immortal soul] >> [lifespan on earth]. In with few intermediate equations, we can get to [reward of being good] > [reward of being bad]. But [Life of immortal soul] is infinity. So if that expression is using useful mathematics, then it is treating infinity as a number. If you can't do mathematics using infinity in this way, then you can't claim it's conclusion, that the reward of being good is greater than the reward of being bad for an immortal soul.

I suppose you could try to formulate a similar answer with calculus, but why is that approach superior?

You can sometimes use such expressions with infinity in them, because they're just comparisions. Of course infinity will always be greater than a finite number, so > is justified, in this case.

There is no final Fibonacci number (and surely no final such number in the current math make-up of infinite series as a concept). But you can always be 'concurrent' (ie born at the same time as, in your counter-example) with any finite position of the series, and thus indeed know its number in that position

I agree with everything in your post! It doesn't seem to contradict anything you quoted that I wrote

 

[Souron]

It doesn't act like a number though. You can't use it in equations.. okay, sometimes you can, but as an approximation, I'm guessing. I'm no engineer, I have no idea how they use it. You can use it informally in equations as well, like I'm about to do below.

It's not a number. If it were a number, you would be able to add 1 to it and get a different number. But 1 + infinity is just infinity. 2 + infinity is infinity yet again. That's not how numbers behave.

That's an arbitrary property to define numbers by. Why must any number + 1 always be a different number? Also, you could define a number system where infinity + 1 is a different infinite number.

Why does it matter if a use of an equation is "formal"? Even rough calculations can have a formal algebra described for them if one wanted to. Failure to utter such an algebra, does not change it's existence in the abstract sense.

You can sometimes use such expressions with infinity in them, because they're just comparisions. Of course infinity will always be greater than a finite number, so > is justified, in this case.

What's special about relational operators, that makes them operate on non-numbers?

 

[uppi]

That a concept of infinity is invented by humans does not mean that it has to be so, or it is clear that "nothing in the universe is infinite".
In fact it has little to do with whether it exists as a "reality" (no matter what that would be or mean). It only means we have a concept of it, and variations of that concept. It says nothing at all about the universe having it as a 'real' part of it. It only says that it can rise (in one manner or other) to be a concept.

I regard it at least as equally likely that the universe is more linked to "infinity" than being finite, and there is nothing which can establish proof of this either way. We have serious problems even establishing a proof on far less significant questions (riemann might ring a bell), let alone one of this enormous (or infinite ) magnitude.

The fact that nothing in the universe is infinite is not derived from humans inventing the concept, but (the current state of) physics. So humans have invented the concept without having a blueprint laid out by nature for them. Compare that to other concepts, where you might claim that.

It doesn't act like a number though. You can't use it in equations.. okay, sometimes you can, but as an approximation, I'm guessing. I'm no engineer, I have no idea how they use it. You can use it informally in equations as well, like I'm about to do below.

There are formal rules how to use infinity in equations. And these are not approximations but theorems written down in math books and taught by math professors.

 

[Kyriakos]

The fact that nothing in the universe is infinite is not derived from humans inventing the concept, but (the current state of) physics. So humans have invented the concept without having a blueprint laid out by nature for them. Compare that to other concepts, where you might claim that.

And you did fine (obviously) in adding that the current state of physics supports that. It does not mean it is true. It is not even certain whether our human science can ever decisively conclude on issues which are outside the formal system devised by us (and here i mostly mean math, not physics, which is - in my view - more of an amalgam and not a formal system by itself).

It is great to move forward, and even build up on previous work. However sometimes there is the serious risk of trapping oneself in the manner of:

He was happy and surprised that he moved so fast in the path of development. He was running it backwards.

 

[warpus]

That's an arbitrary property to define numbers by. Why must any number + 1 always be a different number? Also, you could define a number system where infinity + 1 is a different infinite number.

See, right off the bat that screams "nooooo" to me. There is no such thing as "an infinite number". There are numbers and there is the concept of infinity. The concept of infinity subsequently breaks down into countable and uncoutable concepts, as well as a couple other ones.

If you're describing the cardinality of something, it will either be a specific number (i.e. 17 apples), it will be an unknown, or it might fall into the "infinity camp". If you have an infinite amount of apples, and you add 1 apple, you remain with an infinite number of apples. That's because infinity is not a number, it's a concept. It's an abstract concept that attempts to explain the cardinality. The cardinality does not change if you add 1 or 2 or 17,251,125,158 apples. It would if you were talking about a number, but it does not with infinity.

I hope that explains the whole "infinity is not a number but rather a concept" thing a bit better.

As for the "definition" of number I used, it's not really a definition, just an example of how infinity just wouldn't work as a number.

Why does it matter if a use of an equation is "formal"? Even rough calculations can have a formal algebra described for them if one wanted to. Failure to utter such an algebra, does not change it's existence in the abstract sense.

By "formal" equations I mean properly written mathematical constructs on the level that would be appropriate to use in a formal proof.

Applied math often uses shortcuts and math "slang".

What's special about relational operators, that makes them operate on non-numbers?

They are able to compare sets of numbers and cardinalities - and while infinity is not a number, cardinality is something it does contain as a quality that you can at times analyze.

There are formal rules how to use infinity in equations. And these are not approximations but theorems written down in math books and taught by math professors.

[/quote]

Do you mean things like infinite sums?

I could very well be forgetting other examples of the infinity symbol appearing in formal proofs. Maybe throwing out this argument wasn't a good idea.

 

[uppi]

And you did fine (obviously) in adding that the current state of physics supports that. It does not mean it is true.

What we might discover in the future cannot be the basis of any concept right now. as we have not discovered anything truly infinite, we cannot link the human concept of infinity to anything in nature. We might or might not be able to do that in the future, but that does not have any impact on the concept today.

Do you mean things like infinite sums?

I could very well be forgetting other examples of the infinity symbol appearing in formal proofs. Maybe throwing out this argument wasn't a good idea.

I am not talking about infinite sums, but rules for addition and multiplication of infinity. If I remember correctly, these were then used in formal proofs (otherwise there would have been no reason to mention them).

 

[Daird]

Actually, I recall reading earlier this week that NASA's latest data suggests a universe that is infinite in space and time, but Google Fu has failed me miserably. Anyone got an opinion on this?

 

[warpus]

I am not talking about infinite sums, but rules for addition and multiplication of infinity. If I remember correctly, these were then used in formal proofs (otherwise there would have been no reason to mention them).

I would love to see some information on this!

(Well okay, not love, but I am curious)