What does it mean for something to be "logically possible"?

[Speakpigeon]

I'm sure it is uncontroversial among mathematicians. And yet, the layman will look at the apparent and very visible formal difference in how "1" and "0.9999..." are written and squirm at the idea that there's no difference in value.

Can I tempt you into explaining why there's indeed no difference?

So, here is a reason for controversy:

0.9999... straightforwardly is 0.9 + 0.09 + 0.009 + 0.0009 + etc. You keep adding more terms, each one the tenth of the last term you just added. No big deal. Except, there's no end to it. It's an infinite sum. There's no end to it. So, to get both the whole sum and the whole number written, the one, then, which is said to be equal to 1, you have to assume that you actually finish writing this infinitely long sum. Wherein lies the problem. It would be one thing to say that you get closer and closer to 1 by adding more and more terms like 0.00...0009. And you can even say you could get as close as you might want to. But it's now a very different thing because you now have to say that you've effectively finished adding a sum which is supposed to have infinitely many terms: you would have finished adding all the terms of the sum even though this sum has no end. And this would be the equivalent of adding infinitely many 1s, the sum being ∞, which would make ∞ a number, an ordinary number like any finite, real number. Another way to say it is to observe that the limit of the summation is 1 but 1 is not only the limit, it is also the actual result of the summation.

Isn't that problematic?
EB

Do you have the same problem with 1/3=0.333.... ? Or 0 = 0.000... ?

I don't seem to have any problem with 1 = 0.9999... to begin with.

I offered a reason for a possible controversy. If no one finds anything of substance to say about it, it's fine by me.

Now, your examples are both different from the case of "1 = 0.9999..." in relation to the reason for controversy I provided.

"1/3 = 0.333..." is more similar to "0.9 + 0.09 + 0.009 + 0.0009 + etc. = 0.99999..." than to "1 = 0.9999...", and I wouldn't be able to articulate any reason for controversy there. I'm sure you've already thought about how 1/3 = 0.333... relate to 1 = 0.9999... but please take you're time before making any facile argument.

In the case of 0 = 0.000... I also couldn't possibly find anything interesting to say.

So, there you are, I replied. Now, surprise me.
EB

 

[Speakpigeon]

For the same reasons, this argument would also prohibit the sum of 1/2^n (from i = 1 to n as n goes to infinity) to equal 1.

Absolutely.

But it's not the same to say that the sum of 1 / 2 ^ n from n = 1 to infinity equals 1 and to say that the limit of the sum of 1 / 2 ^ n when n starts from 1 and then tends towards infinity equals 1.
EB

 

[Speakpigeon]

If the real number line is complete and constructed from rationals and irrationals, then infinitesimals exist as the number they represent on the real number line.

Curiouser and curiouser still.

It seems to be the first time I learn that infinitesimals are supposed to represent any particular, actual, Real!

Lest I somehow forgot.

My, and I had to wait till I'm nearly dead to learn this life-changing fact. I'm going to ask for a refund, and not in any infinitesimal money.


Nah, I'm pretty sure that if Newton, Leibnitz or any of the others had thought of infinitesimals in this way they would have specified this number, that you say infinitesimals are supposed to represent, straight away. So, why didn't they? Shyness? Lack of time? They didn't know of the zero? What?

It doesn't make sense to think that the infinitesimal segments that create the real number line change or cease to exist once constructed.

The real (R) value of an infinitesimal is 0. That is what should be expected since any other value wouldn't be infinitesimal.

Please explain again why infinitesimals should have any particular value at all?

The definition of a limit using delta and epsilon works without infinitesimals, but I don't think that necessarily means they don't exist.

You bet.
EB

- - - Updated - - -

I'm sure it is uncontroversial among mathematicians. And yet, the layman will look at the apparent and very visible formal difference in how "1" and "0.9999..." are written and squirm at the idea that there's no difference in value.

Can I tempt you into explaining why there's indeed no difference?

So, here is a reason for controversy:

0.9999... straightforwardly is 0.9 + 0.09 + 0.009 + 0.0009 + etc. You keep adding more terms, each one the tenth of the last term you just added. No big deal. Except, there's no end to it. It's an infinite sum. There's no end to it. So, to get both the whole sum and the whole number written, the one, then, which is said to be equal to 1, you have to assume that you actually finish writing this infinitely long sum. Wherein lies the problem. It would be one thing to say that you get closer and closer to 1 by adding more and more terms like 0.00...0009. And you can even say you could get as close as you might want to. But it's now a very different thing because you now have to say that you've effectively finished adding a sum which is supposed to have infinitely many terms: you would have finished adding all the terms of the sum even though this sum has no end. And this would be the equivalent of adding infinitely many 1s, the sum being ∞, which would make ∞ a number, an ordinary number like any finite, real number. Another way to say it is to observe that the limit of the summation is 1 but 1 is not only the limit, it is also the actual result of the summation.

Isn't that problematic?
EB

It's not. All mathematicians will make it clear what they mean by the notation "0.999..." if the question comes up, and they'll stipulate unambiguously that "0.999..." must denote 1.

There are subtleties when it comes to reasoning about the infinite, but this isn't one of them.

So apparently I failed to tempt you for real.
EB

 

[Kharakov]

There isn't a least amount of change. In 10^-100 seconds, the universe will have changed. In 10^-1,000,000 seconds (when the Earth has completed ~10^-999,993 orbits), the universe will have changed.

When something changes it has to change the least amount possible first.

So clearly it does exist.

There isn't a least amount possible in a continuous system. You can look at change over a decade, over a day, over a picosecond, over 10^-99999999999 seconds...

Unless you propose that the universe evolves like a clockwork orange?

 

[ryan]

For the same reasons, this argument would also prohibit the sum of 1/2^n (from i = 1 to n as n goes to infinity) to equal 1.

Absolutely.

But it's not the same to say that the sum of 1 / 2 ^ n from n = 1 to infinity equals 1 and to say that the limit of the sum of 1 / 2 ^ n when n starts from 1 and then tends towards infinity equals 1.
EB

From how I understand it, there are an aleph 0 infinite number of elements of n used for integrals, but n itself never actually equals infinity. So n going to infinity is sufficient to "fill" the area under a curve as there must be at least aleph 0 elements to do so, even though n only tends to infinity.

Nah, I'm pretty sure that if Newton, Leibnitz or any of the others had thought of infinitesimals in this way they would have specified this number, that you say infinitesimals are supposed to represent, straight away. So, why didn't they? Shyness? Lack of time? They didn't know of the zero? What?

What value is lim n-->infinity c/n

Think about it this way. If I start at 0 and don't gain any distance, I have moved 0 distance. I get a dot on 0, but at least I moved an infinitesimal. If I don't even start at 0, then I don't even get a dot. I gain no distance, not even an infinitesimal.

 

[Phil Scott]

It's not. All mathematicians will make it clear what they mean by the notation "0.999..." if the question comes up, and they'll stipulate unambiguously that "0.999..." must denote 1.

There are subtleties when it comes to reasoning about the infinite, but this isn't one of them.

So apparently I failed to tempt you for real.
EB

I'm afraid so. I'm happier to elaborate on the particular controversies I had in mind, the logics that have been put forward for dealing with them, and the cases where classical logic looks odd (and non classical besides, at least if we think of "classical" as referring to antiquity). This might involve talking about things like the Banach Tarski Theorem and the Continuum Problem, both of which are sufficiently odd to have got quite a few philosophers of mathematics worked up.

The discussion of 0.999... hasn't got philosophers of mathematics worked up, and near as I can tell, is mostly only a concern for people who haven't taken a real analysis course.

On the other hand, most people in this thread seem to be having quite enough fun in this thread as it is without my input.

 

[LordKiran]

OH GOD PLEASE NO NOT MATH

STOPPIT

 

[Speakpigeon]

Absolutely.

But it's not the same to say that the sum of 1 / 2 ^ n from n = 1 to infinity equals 1 and to say that the limit of the sum of 1 / 2 ^ n when n starts from 1 and then tends towards infinity equals 1.
EB

From how I understand it, there are an aleph 0 infinite number of elements of n used for integrals, but n itself never actually equals infinity. So n going to infinity is sufficient to "fill" the area under a curve as there must be at least aleph 0 elements to do so, even though n only tends to infinity.

Sorry, I think you're trying to dress your intuitive understanding of the issue as a properly logical or rational argument. No good. Either n somehow gets to equal infinity at some point and then your sum equals 1 or it doesn't (and you're now saying it doesn't) and then only the sum's limit equals 1.

Nah, I'm pretty sure that if Newton, Leibnitz or any of the others had thought of infinitesimals in this way they would have specified this number, that you say infinitesimals are supposed to represent, straight away. So, why didn't they? Shyness? Lack of time? They didn't know of the zero? What?

What value is lim n-->infinity c/n

Think about it this way. If I start at 0 and don't gain any distance, I have moved 0 distance. I get a dot on 0, but at least I moved an infinitesimal. If I don't even start at 0, then I don't even get a dot. I gain no distance, not even an infinitesimal.

That wasn't my question.

So, I can only assume you couldn't explain why all these great minds didn't get it right where you, on the other hand, did.
EB

 

[Speakpigeon]

OH GOD PLEASE NO NOT MATH

STOPPIT

God just had pity on you apparently.
EB

 

[Kharakov]

OH GOD PLEASE NO NOT MATH

STOPPIT

In base 11, 1 = .

 

[Kharakov]

An infinitesimal is a point on a line, a line on a plane (infinitesimal area), a plane in a volume (infinitesimal volume)....

They aren't necessarily without size. An infinitesimal hypervolume could still take up a lot of space.

A moment in spacetime is.....

 

[ryan]

From how I understand it, there are an aleph 0 infinite number of elements of n used for integrals, but n itself never actually equals infinity. So n going to infinity is sufficient to "fill" the area under a curve as there must be at least aleph 0 elements to do so, even though n only tends to infinity.

Sorry, I think you're trying to dress your intuitive understanding of the issue as a properly logical or rational argument. No good. Either n somehow gets to equal infinity at some point and then your sum equals 1 or it doesn't (and you're now saying it doesn't) and then only the sum's limit equals 1.

"As n goes to infinity" is the same as saying all n of the naturals. There are an aleph 0 infinity numbers in the set of the naturals, yet n never actually equals aleph 0 in the set.

Nah, I'm pretty sure that if Newton, Leibnitz or any of the others had thought of infinitesimals in this way they would have specified this number, that you say infinitesimals are supposed to represent, straight away. So, why didn't they? Shyness? Lack of time? They didn't know of the zero? What?

What value is lim n-->infinity c/n

Think about it this way. If I start at 0 and don't gain any distance, I have moved 0 distance. I get a dot on 0, but at least I moved an infinitesimal. If I don't even start at 0, then I don't even get a dot. I gain no distance, not even an infinitesimal.

That wasn't my question.

So, I can only assume you couldn't explain why all these great minds didn't get it right where you, on the other hand, did.
EB

No, it's because set theory hadn't been invented yet.

 

[untermensche]

When something changes it has to change the least amount possible first.

So clearly it does exist.

There isn't a least amount possible in a continuous system. You can look at change over a decade, over a day, over a picosecond, over 10^-99999999999 seconds...

Unless you propose that the universe evolves like a clockwork orange?

I know what a continuous system is in theory.

But I've never seen one.

How does something move without moving the shortest distance possible first?

I agree. In theory continuous systems have no shortest possible distance.

That's how we know movement in the real world is not continuous.

 

[Kharakov]

There isn't a least amount possible in a continuous system. You can look at change over a decade, over a day, over a picosecond, over 10^-99999999999 seconds...

Unless you propose that the universe evolves like a clockwork orange?

I know what a continuous system is in theory. But I've never seen one. How does something move without moving the shortest distance possible first? I agree. In theory continuous systems have no shortest possible distance.

That's how we know movement in the real world is not continuous.

Umm, how do "we" know that? The Planck length isn't a "shortest possible distance", rather it is the smallest distance that can be measured without producing a black hole.

There are distances that cannot be probed, with current physics, but this doesn't mean there is a "smallest possible distance" other than the length of your dick.

 

[untermensche]

I know what a continuous system is in theory. But I've never seen one. How does something move without moving the shortest distance possible first? I agree. In theory continuous systems have no shortest possible distance.

That's how we know movement in the real world is not continuous.

Umm, how do "we" know that? The Planck length isn't a "shortest possible distance", rather it is the smallest distance that can be measured without producing a black hole.

There are distances that cannot be probed, with current physics, but this doesn't mean there is a "smallest possible distance" other than the length of your dick.

You didn't answer the question.

How does something move without moving the shortest distance possible first?

 

[ryan]

There isn't a least amount possible in a continuous system. You can look at change over a decade, over a day, over a picosecond, over 10^-99999999999 seconds...

Unless you propose that the universe evolves like a clockwork orange?

I know what a continuous system is in theory.

But I've never seen one.

How does something move without moving the shortest distance possible first?

I agree. In theory continuous systems have no shortest possible distance.

That's how we know movement in the real world is not continuous.

The universe may be a static 4 dimensional object. In which case there is no movement.

 

[Kharakov]

Umm, how do "we" know that? The Planck length isn't a "shortest possible distance", rather it is the smallest distance that can be measured without producing a black hole.

There are distances that cannot be probed, with current physics, but this doesn't mean there is a "smallest possible distance" other than the length of your dick.

You didn't answer the question. How does something move without moving the shortest distance possible first?

Do you beg the question while you're beating your wife?

There isn't a shortest possible distance. There are transition points (where a gradient switches directions) in nature, but they have no length.

Otherwise, all things in nature are smooth/continuous (except for transition points).

 

[fast]

I'm not really so sure that the word "logical" can even be applied to the word possibility.

I wish you could see what I see. You can't treat an adjective as fully applicable when it's purpose is for distinction in name only. You must treat the two-worded term "logical possibility" as a name in its own right and not look to the individual meanings of the terms as if it calcuably gives rise to what "logical possibility" means. It's the same thing with "free will." The individual words are akin to a business name. It clues us in, but it doesn't tell the whole story.

You are doing the very thing you should be doing by questioning what is so logical about a purported possibility, but once you learn that a logical possibility (a term in its own right with its own specific meaning) is not a logical possibility (a kind of possibility that is logical) and recognize the ambiguity, you should drop what only appears to be intentionally conflating them.

You are not amazing me when you point out the absurdity of calling an impossibility as something possible. A physically impossible event surely is impossible, but it can nevertheless quite correctly be called possible (save for contradictions). It's easier to see when the ambiguity of the name is removed and replaced with a term that is less misleading. (Recall my previous post substituting jumbo peanuts with Strain 78).

Let's try another route:
If I refer to all women over 60 as red men, then no amount of logic is going to make it so that women are men, let alone red, but if I refer to all women over 60 with a proper name (oh say, Red Men), the meaning is stipulative. If it becomes apart of common usage, the meaning of red men (adjective and noun) will be vastly different than the proper name that gained notoriety red men (to not be treated as two one worded-terms but instead as one two-worded term.

See what it is you're doing when you ask about the applicability of the word "logical" to the word "possibility". You'd be doing the same thing as asking what does "free" to do with "will" or "red" to do with "men" or "jumbo" to do with "peanut."

A logical possibility is not some kind of in the same sense possibility that has some quality of being logical. After all, physical impossibilities are logically possible -- even if not a possibility of the logical variety--whatever that might mean.

 

[untermensche]

You didn't answer the question. How does something move without moving the shortest distance possible first?

Do you beg the question while you're beating your wife?

There isn't a shortest possible distance. There are transition points (where a gradient switches directions) in nature, but they have no length.

Otherwise, all things in nature are smooth/continuous (except for transition points).

You're refusal to answer the question is not my anything.

You have failed to address the question twice.

That is telling.

Nothing can move without making the smallest possible move first.

 

[untermensche]

I'm not really so sure that the word "logical" can even be applied to the word possibility.

I wish you could see what I see. You can't treat an adjective as fully applicable when it's purpose is for distinction in name only. You must treat the two-worded term "logical possibility" as a name in its own right and not look to the individual meanings of the terms as if it calcuably gives rise to what "logical possibility" means. It's the same thing with "free will." The individual words are akin to a business name. It clues us in, but it doesn't tell the whole story.

You are doing the very thing you should be doing by questioning what is so logical about a purported possibility, but once you learn that a logical possibility (a term in its own right with its own specific meaning) is not a logical possibility (a kind of possibility that is logical) and recognize the ambiguity, you should drop what only appears to be intentionally conflating them.

You are not amazing me when you point out the absurdity of calling an impossibility as something possible. A physically impossible event surely is impossible, but it can nevertheless quite correctly be called possible (save for contradictions). It's easier to see when the ambiguity of the name is removed and replaced with a term that is less misleading. (Recall my previous post substituting jumbo peanuts with Strain 78).

Let's try another route:
If I refer to all women over 60 as red men, then no amount of logic is going to make it so that women are men, let alone red, but if I refer to all women over 60 with a proper name (oh say, Red Men), the meaning is stipulative. If it becomes apart of common usage, the meaning of red men (adjective and noun) will be vastly different than the proper name that gained notoriety red men (to not be treated as two one worded-terms but instead as one two-worded term.

See what it is you're doing when you ask about the applicability of the word "logical" to the word "possibility". You'd be doing the same thing as asking what does "free" to do with "will" or "red" to do with "men" or "jumbo" to do with "peanut."

A logical possibility is not some kind of in the same sense possibility that has some quality of being logical. After all, physical impossibilities are logically possible -- even if not a possibility of the logical variety--whatever that might mean.

What is a possibility?

Is it something you just have to imagine?

If I imagine something how is the thing imagined a possibility? I already imagined it.

In what universe is it a possibility?