Infinity is a number greater than every integer.

And infinity + 1 = infinity is true only in the sense that if you take an infinite number and add one to it, you get an infinite number. Basically, only if infinity does not refer to a specific infinity (i.e. only if infinity - infinity =/= 0.) Otherwise, if you're working with specific infinity, infinity + 1 > infinity.

Thank you very much.

As you say, an infinite sum may not have a limit. If you say that the concept of infinite sum and the concept of limit are one and the same concept then how is it possible for an infinite sum to have no limit?

Furthermore, the difference between the two concepts is bigger than that. For example, two infinite sums that approach the same value can represent different quantities. For example, $0.9 + 0.99 + 0.999 + \dots$ represents a number greater than $0.5 + 0.25 + 0.125 + \dots$ even though they both aproach but never reach $1$. Indeed, there are numbers greater than $0.999\dots$ but lower than $1$. Hexadecimal $0.FFF\dots$, for example, lies somewhere between the two numbers.

↪Magnus Anderson I can't even hazard a guess as to how you think "most people" define "0.333~" (I am more accustomed to the ... notation, but I assume you mean the same thing). — SophistiCat

I avoid the "..." notation because it looks ugly when used in forums without LaTeX support. But yes, that's what I mean.

"0.333~" represents the infinite sum 3 x 1 / 10^1 + 3 x 1 / 10^2 + 3 x 1 / 10^3 + ... + 3 x 1 / 10^inf. It does not represent its limit.

Some perfectly sensible and familiar rational numbers, such as 1/3 = .3333333..., have infinitely-long decimal representations. — fishfry

This depends on the meaning of the symbol "0.333~". According to the way most people define "0.333~", it is not true that "1/3 = 0.333~". By standard definition, 0.333~ is a number smaller than 1/3 (in the same way that 0.999~ is a number smaller than 1.) It's not a decimal representation of it. "0.333~" does not mean "the limit of 0.333~".

You said points have no size. I do not see how any part of time could have no size. If it has no size, then no time is passing at that "point", therefore it is not part of time. The same principle holds for space. If it has no size, then it cannot be part of spatial existence, because there is no space there. It is very clear to me, that if points have no size, then they are excluded from space and time, because things existing in space and time have size. Having size is what makes them spatial-temporal. Do you not understand this? — Metaphysician Undercover

I do not understand why you think that things that exist in space and/or time must have size. Why is it impossible for something to exist (in space and/or time) and not have size?

Points don't exist in physical space. According to the description they are non-spatial. — Metaphysician Undercover

I am not sure why you think so, Points do exist (both in time and space.) Consider that at any point in time, you occupy certain point in space. So there exist at least some of the points that we can imagine. Points that do not exist cannot be occupied by anything under any set of circumstances.

Physical space is made out of points. The fact that physical space is made out of things that have no size (points) does not mean that it has no size itself. Not sure why you think so.

We've already placed the no-length point as right out of the category of things to be measured, so how can a point appear on a line to be measured? — Metaphysician Undercover

Though you cannot measure how long a point is (since it has no length, as per definition) you can identify a point. And we do so through a complex process that involves the movement of our bodies (if we're talking about identifying points in physical space, that is.)

There are many things that have no size but that nonetheless exist (and are not logically contradictory, illogical or otherwise irrational.) The word "existence" does not imply size. For example, colors and feelings exist, and yet, they have no size. A typical counter-argument is that colors are light waves and that light waves have size (their wavelength.) But light waves are not colors. Rather, light waves are things that cause colors. (This is evident in the fact that light waves can exist without conscious beings whereas colors can't.)

If they had the same positions relative to all other objects, they would have the same identity, which means they would be one apple, not two. — litewave

My point is, not necessarily.

In general, history is very important. Have you heard of Richard Gregory's top-down theory of perception?

When it comes to identity, matters are different and space-time properties are critical to its meaning. One object cannot occupy two locations in space at the same time and it is this impossibility that gives objects their identity. So, two/more objects can be identical because they share all properties except space-time properties but they all have different identities because one object can't occupy two locations in space at the same time. — TheMadFool

Identity can also be (and it mostly is) established by history. So two objects can occupy the same position in space at the same time and still be identified as two different objects simply because they have different histories.

In general, two things are identical if and only if everything that makes up the first thing also makes up the second thing and vice versa.

The more important question, perhaps, is what constitutes a thing. Who decides what constitutes a thing? Us, of course.

Let's say we have two apples both of which are completely identical except in one regard: they occupy different positions. Are they identical or not? The answer depends on whether an apple's position constitutes its identity. In general, the answer is no, since we all operate with the concept of apple according to which an apple remains the same apple regardless of its position in space. Therefore, we can conclude, the two apples, though occupying two different positions, are identical.

If two things appear to be identical they would still be different if they are in different locations for example. — believenothing

In most cases, that amounts to sophistry, since position is rarely part of an object's identity.

If you check the OP, I did consider the possibility that points have no size. That leads to the size of a point being UNDEFINED and all line segments having an UNDEFINED length. — Devans99

But that's not what it leads to. If points have undefined length it does not follow that line segments have undefined length.

It is when we use non-sensical definitions like ‘points have no size’ that we find the maths always leads to contradictions. — Devans99

What's non-sensical about the statement that points have no size?

You cannot simultaneously hold that line segments (which have length) are made out of points and points have no length - that’s a plain contradiction. — Devans99

How's that a logical contradiction? How does "Line segments have length" contradict "Line segments are made out of points that have no defined length"?

Colours have a wavelength, so do sounds so they can be said to have existence. — Devans99

Colors do not have wavelength. Rather, it is light waves that have wavelength and light waves are not colors, they are the cause of colors.

To have no length is by definition to not exist in the realm of geometry — Gregory

It depends on how you define the word "existence". You can define it any way you like. You can define it in a way that implies length. That which exists has length. If we accept that definition then it follows that points (among many other things, such as sounds, colors and other sensations) do not exist. And since definitions have no truth-value, you can't argue against such a definition by saying "That's not true definition of existence!" Unless, of course, you're arguing against it on the basis of use-value. And that would be my response. Such a definition of existence has a limited (even questionable) use-value. It's certainly not how most people define existence. Most people define existence (not necessarily verbally but certainly intuitively) in such a way that even things that have no length (such as points, colors, sounds, etc) can be said to exist.

Continuum is a set of points where for every two points in the set there exists a point in the set that is in between the two points.

The fact that points have no size (which, by the way, does NOT mean that points have zero-size), whereas line segments (whether continuous or discrete) do, does not mean that line segments are not made out of points. It merely reflects the fact that distance is something that exists between points.

You don't measure the length of a line segment by counting how many points it has, you measure it by counting how many pairs of points-at-certain-distance it has. "This line is 10cm long" means "This line is made out of 10 pairs of points-at-1cm-from-each-other".

There is no need to consider that the line is made up of points. — A Seagull

But it is made out of points. It's just that the length of a line is not measured the way @Devans99 thinks it is measured. You don't measure the length of a line by summing the lengths of its smallest parts (which are points.) Points have no length. They do not have such a property. Length is something that exists between two points. In order to measure the length of a line you must count the number of pairs of points-at-a-certain-distance that constitute it.

resulting in a mental model of the segment as containing an infinite number of zero-length points (something than does not make physical sense - but we can do it purely in our minds). — Devans99

Points have no zero-length. They have no length at all. I think this is part of the problem. A lot of people do not understand that the statement "Points have no length" does not mean "Points have zero-length".

What does Schopenhauer mean when he says "the problem of existence"? Can someone help me understand the problem he's speaking of?

I might be blind or stupid but I never came across an argument for theism (:

There are people who are not willing to sacrifice themselves and others for the benefit of the collective. That's why they reject what you call objective morality.

The correct meaning of the word "chair" is far too complex to be a part of a short forum post. This is why I had to make it simpler than it really is by disregarding all correct meanings except for the one most widely in use. I could afford to do this because my point wasn't to define the word "chair" in the best possible way. My point was to illustrate that there are correct and incorrect definitions of words.

↪ChrisH Yes, the implied circumstance. Did you think he was just saying what he said in absolute terms, context free? :brow: — S

When I said that the correct meaning of the word "chair" is "a separate seat for one person, typically with a back and four legs" I had no specific context in mind. I was talking about what the word "chair" means in general. So yes, I do think that @ChrisH is right. However, it's an insignificant mistake that was made on purpose + it's not true that I was being defensive (I merely failed to understand what he was trying to say the first time I responded to him.)

Just try not to be like Kant, Hegel, C. S. Peirce, Heidegger, Derrida, Lacan, Baudrillard and the like.

That has nothing to do with what I asked you. I said, "If S is not trying to match the convention, then telling S that they're not matching the convention is irrelevant."

You're positing S not matching the convention and S telling U that U is wrong. — Terrapin Station

The way I understand you, what you're saying is that it makes no sense to criticize someone for not using words the way most people do if that's not what they are trying to do.

Is that what you're trying to say? If so, my response is an adequate one.

You stated the following:

If they're not trying to match the convention, then telling them that they're not matching the convention is irrelevant, right? — Terrapin Station

And I responded by saying that I disagree.

There are times when it is relevant to remind them that they are using words in an unconventional way -- even though they themselves have no interest in using words the way most people do. For example, if they are misunderstanding others because they forgot or simply never realized that other people are using words in a different way, then you have to remind them of this fact.

If I say something like "Cats cannot fly" and you tell me I am wrong merely because you fail to realize that I don't define the word "cat" the way you do -- you define it to mean "dragon" -- then it would be more than relevant to remind you that your use of words is unconventional.

Don't you agree?

What you just said.

Your initial response contained no correction and repeated your error. That looks defensive to me. — ChrisH

It was your second post that made your point clear.

I am telling you that it is relevant to criticize their lack of regard for conventions because it makes them blind to reality. Have you ever come across people who claim that the brain is inside consciousness and not the other way around?

If you're not trying to match the convention, because of a lack of regard for it, how could you make a mistake in word definition/usage? — Terrapin Station

You can form mistaken beliefs.

Sure you were. (Being defensive) — DingoJones

Maybe you'd like to go back and read my initial response to him:

I think you're mistaken.

Any good dictionary (essentially a record of existing usages) will give at least 6 different meanings. — ChrisH

That's true. Maybe I can correct myself by saying that's one of several correct meanings of the word? — Magnus Anderson

I do, however, think that he's nitpicking and missing the point.

Defensiveness? Not really.

You're saying that even if they're not trying to match the convention, they could be making a mistake? — Terrapin Station

I am saying that their lack of regard for conventions can lead them to making mistakes of all sorts.

If they're not trying to match the convention, then telling them that they're not matching the convention is irrelevant, right? And they're certainly not saying something not true, because they weren't trying to match the convention. — Terrapin Station

The mistakes they make might stem from their lack of regard for conventions. Such cases are numerous. But you're right in the sense that it's not always relevant. Sometimes. it simply does not matter.

And yes, if you speak your own language, that does not necessarily mean what you're saying is wrong.

I thought that might be the case but I wasn't sure. My initial response to you gave you the opportunity to correct your mistake but for some reason you decided to go defensive. — ChrisH

I don't think I was being defensive in the slightest.

Sorry, I don't see the relevance of that question.

"Correct" has a prescriptive connotation. Because people would rather be surrounded by folks whose beliefs are true as you say. — Terrapin Station

In most cases, yes.

You said your definition was "the correct meaning of the word chair". You repeated this in your follow up reply to me.

As I'm sure you're aware, there's an important distinction to be made between "the correct meaning" and "a correct meaning". — ChrisH

Well, that's not what I meant. Let's say I expressed myself in a way that wasn't the best. What I wanted to say is that the word "chair" has a number of correct meanings (one or more) and a number of incorrect meanings (again, one or more.) I don't know the exact numbers, I just know that the number of correct meanings is >= 1 and the number of incorrect meanings is also >= 1.

Can we make statements about something other than language in your view? — Terrapin Station

Absolutely. You can make statements about any portion of the universe -- not just language. For example, you can say that Donald Trump's face is orange. Nothing to do with language.

And would you say that you never use "correct" prescriptively? — Terrapin Station

Saying that something is correct or incorrect is not a prescriptive statement, it is a descriptive one. If you say that "The sky is red" and I say "That's not correct" that is not the same kind of statement as "You should adopt the view that sky is blue" or "People should have true beliefs". Of course, I'd rather be surrounded by people whose beliefs are true . . . if that's one of the things you're asking me.

"2 + 2 = 4" is a statement that claims that the symbol "2 + 2" is equivalent to the symbol "4". It's a statement about language. So yes, it has to do with conventions.

<---doesn't for a second believe that anyone here is actually using "correct" to simply descriptively refer to what's conventional, with no hint of a prescriptive connotation to it. — Terrapin Station

The correct answer to the question "Is 2 + 2 = 4?" is "Yes". What's prescriptive about that?

You're right. He assumed that I'm claiming the word "chair" has only one correct definition. He somehow ignored my main point which is that the word "chair" has a number of correct meanings (one or more) and a number of incorrect meanings (again, one or more.) The exact numbers are irrelevant. Even if I did make a claim that there is only one correct definition of the word "chair" it's irrelevant.

I think you're mistaken.

Any good dictionary (essentially a record of existing usages) will give at least 6 different meanings. — ChrisH

That's true. Maybe I can correct myself by saying that's one of several correct meanings of the word?

I'd take issue with your claim that you've given "the correct meaning". It's 'a' meaning but not the only one in current use.

New usages may even emerge in the future. These new usages, in my view, wouldn't be incorrect. — ChrisH

It's important to understand what the other is saying before one proclaims that what they are saying is true or false.

Specifically, it's important to understand what it means to say that a word is being using in a way that is incorrect. What it means is that the word is being used in a way that is not used or that was not used by some group of people at some point in time.

"Word W is used incorrectly" simply means "Word W is used in a way that is different from the way that it is used or was used by some group of people G during some period of time P."

Nowadays, when people use the word "chair" what they mean is "a separate seat for one person, typically with a back and four legs". In the future, the definition might change, but when I say that this is the correct definition of the word, what I mean is that this is how people use the word nowadays.

I agree that most of everything Terrapin says is absurd, but I think there's a confusion between understanding chair as a nominatum (the thing named) and chair as nominans (the name 'chair'). Qua nominans, yes, to understand what a 'chair' is requires a community of users who use the word in that way, etc etc. Qua nominatum, you need a great deal more than that, including all the stuff I mentioned regarding the grammar of chair (used for sitting, moveable, etc). I only insist that we can't treat the two nomen separately, and its only at the 'shallow' level of the nominans that one can argue about individuals vs groups and so on. — StreetlightX

In other words, it's one thing to study the word "tree" and another to study physical objects that can be represented by the word "tree". I am pretty sure most people here are aware of this distinction, so I would claim that no confusion is taking place. (Notice also how I didn't use a single Latin word to express myself.)

I believe this thread is about words and not about things that can be represented by words.