I wonder where you get your idea of correct from. That everyone does it, doesn't make it correct, read my example above. You support mob rule? — Metaphysician Undercover

:grin: A trite reply, as was expected.

Clearly "three dozen" does not represent a quantity of three, just like "four score" does not represent a quantity of four, and "twenty six" does not represent a quantity of twenty. — Metaphysician Undercover

You missed the point of the example, as is your habit.

I've explained very thoroughly why 1/2 is not a number. — Metaphysician Undercover

Well, no; what you did was explain how you use the word "number" in a rather eccentric fashion. You told us nothing about numbers.

So that's an end to this discussion, I think. There's nothing here but Meta's queer usage.

I think I got it (incidentally, c=k here, right?) — InPitzotl

Yes, you got it. The point I'm making is that 1/k is positive and > 0. Even as you go to infinity 1/x can't be zero. So, you are summing an infinity of positive terms > 0 which is infinity, right? As the number of terms taken increases 1/x decreases but never becomes zero.

It might be interesting to revisit surreal numbers, as mentioned by @tim wood.

These are constructed by two limits, the left part describes some sequence with the surreal number as its lower limit, and the right number describes some sequence with the surreal number as its upper limit, and that further this mode of representation is powerful enough to produce all the other sorts of numbers... except imaginary numbers.

Now the Wolfram discussion on Limits makes the point that a limit is said to exist if the limit approached from below is the same as the limit approached from above; the same construction as the surreals.

An example fo the mathematics that disappears when one denies the argument in the OP.

There's something extraordinary in the creativity of mathematics. Consider the imaginaries; we all know you can't take the root of a negative number; but despite that if we call the root of -1 "i" we can have even more fun with numbers...

Theorem 1
Define 1/x such that 0 < 1/x < 1. If 1/x is summed to itself infinitely often, the sum is infinity. From this we conclude that any positive quantity added infinitely sums to infinity — EnPassant

Please, please, don't start calling this trash "theorems". And stay away from LaTeX, it's not like playing with a shovel and pail in a sandpile. What you don't know can hurt you. I beg you, leave this disaster zone and return to the relatively safe comforts of philosophical musing. :groan:

I said these do not represent any particular quantity, and ought not be considered as numbers. — Metaphysician Undercover

But I note that your OED definition talks about values referring to the same particular quantity. And I note that you've chosen of your own will in this post to not actually argue the relevant point... which was that .999...=1 is equivalence under equality, and under equality equivalence implies having the same value.

Until you do, there's nothing to argue against. You have no point to make, just a problematic claim. And by Hitchen's razor, I can dismiss that without argument.

It is the belief that they are numbers which is what I consider to be a problem.

Your problem is your problem though, not mine.

The difference between these two is a difference which does not make a difference, for them, so they say that it is the same value. — Metaphysician Undercover

No, they define .999... in such a way that it has the same value; it's not a different value that's close enough, it's the same value. But .999... having that value comes from the definition assigned to it. Like I said at first, this is a language barrier issue. You don't speak the same language.

But that doesn't prevent me from arguing that the claim that there is a difference which doesn't make a difference is a contradictory claim.

You can do whatever you wish, but I'm under no obligation to take you seriously, especially at your word.

My argument, if you've read what I posted, is that .999... does not represent a particular quantity. — Metaphysician Undercover

That's not an argument, it's a claim.

And I told you why that's inconsistent with the views you presented. It's still there in the post. To help you out, I repeated it at the top. But the barrier between us (and also you and many others) goes far deeper than this. You're trying to have a conversation without speaking the language. That's made even worse by your refusal to even consider speaking it, which is made even worse by your having unfalsifiable "opinions" on how the language should even work. All of this is a grand recipe for having pointless arguments, but nobody is interested in having pointless arguments with you. We have to clear this barrier before it's even possible to have a conversation with you.

I suggest that you come back when you've got an argument to make. — Metaphysician Undercover

Again, you didn't make an argument (it was just a claim) and, until you do, I can dismiss your claims with Hitchen's razor. Where we left off is your claim that .999... does not represent a "particular value" despite it being equal to 1, which does. I repeated the inconsistencies I pointed out last post in this post for you.

As the number of terms taken increases 1/x decreases but never becomes zero. — EnPassant

That's true for all finite x. But you need it to be true for an infinite x. To see the problem, here's a "troll proof" that infinity is finite. 1. 1 is finite. 2. For all x where x is finite, x+1 is finite. 3. By 1 and 2, and infinite recursion, infinity is finite, QED.

Think of this "troll proof" analogous to your conjecture. Your inequality holds for all finite x's, no matter how big the x is. But also, no matter how big the x is, you only have a finite number of terms. But to apply Theorem 1, you need two things: (a) a value x such that 0 < 1/x < 1, (b) an infinite number of those values. You can't have (b) with any finite number. You can't say (a) "at infinity". Since you need both, and never have both, you cannot apply Theorem 1.

Here's another "proof" (illustration) for why 0.999... = 1.

To write the difference between 0.999... and 1, write a zero, a decimal point, and then infinitely many zeroes, and then "when you finish", write the "final" 1.

Problem is, you will never finish, because the zeroes are infinite, so the difference between 0.999... and 1 is 0.000... forever. In other words, just 0.

And if the difference between two things is 0, there is no difference between them; they are the same.

You missed the point of the example, as is your habit. — Banno

There was no point to your example, as is your habit. You started from a false premise and tried to make something out of it.

Well, no; what you did was explain how you use the word "number" in a rather eccentric fashion. You told us nothing about numbers. — Banno

I took my definition of "number" straight from the first entry in my OED. I'm still waiting for an alternative definition, one which allows that 1/2 signifies a number. Your response to the definition was very lame: "Family Resemblance".

But I note that your OED definition talks about values referring to the same particular quantity. — InPitzotl

I don't deny that in some cases different symbols represent the same quantity. The op does not provide one of those instances.

And I note that you've chosen of your own will in this post to not actually argue the relevant point... which was that .999...=1 is equivalence under equality, and under equality equivalence implies having the same value. — InPitzotl

I deny that .999..., as presented in the op, represents a particular quantity, because there is no quantitative value given for 1/9. Therefore I deny that .999...which in this instance does not represent a particular quantity is equivalent to 1 which does represent a particular quantity.

Until you do, there's nothing to argue against. You have no point to make, just a problematic claim. And by Hitchen's razor, I can dismiss that without argument. — InPitzotl

I made my point, symbols such as 1/2, 1/3, 1/9, are representative of ratios between quantities, they do not represent any particular quantitative value. To represent a particular value they need to be qualified.

No, they define .999... in such a way that it has the same value; it's not a different value that's close enough, it's the same value. But .999... having that value comes from the definition assigned to it. Like I said at first, this is a language barrier issue. You don't speak the same language. — InPitzotl

I haven't seen that definition. care to provide it?

Again, you didn't make an argument (it was just a claim) and, until you do, I can dismiss your claims with Hitchen's razor. Where we left off is your claim that .999... does not represent a "particular value" despite it being equal to 1, which does. I repeated the inconsistencies I pointed out last post in this post for you. — InPitzotl

The argument is very clear in my discussion with Banno. You just cannot grasp the first premise, that 1/9 does not represent any particular quantity, and therefore it is not a number.

(a) a value x such that 0 < 1/x < 1, (b) an infinite number of those values. You can't have (b) with any finite number. You can't say (a) "at infinity". Since you need both, and never have both, you cannot apply Theorem 1. — InPitzotl

Ok, but can't this be also said for 0.999...? Adding terms and then saying 'at infinity'. You can't have (b) at any finite number of the terms 9/10, 9/100...but ya gotta say 'at infinity' sometime if you assert that the literal infinite sum is 1.

I do not argue against the fact that mathematicians believe that .999..., and 1 refer to the same value. The difference between these two is a difference which does not make a difference, for them, so they say that it is the same value. But that doesn't prevent me from arguing that the claim that there is a difference which doesn't make a difference is a contradictory claim. — Metaphysician Undercover

They don't just believe that 0.999... = 1. They've proved it.

There really is nothing to discuss here. The disagreements are flat Earther stuff.

I don't deny that in some cases different symbols represent the same quantity. The op does not provide one of those instances. — Metaphysician Undercover

MU, you're pretending here to be making an argument about .999... = 1:

As I've explained to fishfry already, that two things are equivalent does not mean that they are the same thing. Therefore what is on the left side of the "=" (which indicates equivalent) does not provide a definition of what is on the right side. — Metaphysician Undercover

A, therefore B, where A is .999... = 1, and B is some rambling about equivalence. But here we don't merely have equivalence, we have equality. Because we have equality, they do represent the same value. I've never heard of someone so far gone as to commit an amphiboly by changing the word. But in this post, and here?:

I deny that .999..., as presented in the op, represents a particular quantity, because there is no quantitative value given for 1/9. Therefore I deny that .999...which in this instance does not represent a particular quantity is equivalent to 1 which does represent a particular quantity. — Metaphysician Undercover

...you're still not talking about the relevant point. You're still not advancing any reason why you think equality represents different values, or why you think the same value representing a particular quality can actually wind up representing different qualities. Nor are you even trying to make this point; you're just, instead, playing hide-the-ball.

I haven't seen that definition. care to provide it? — Metaphysician Undercover

$\displaystyle\sum_{i=1}^{\infty}{a_i} = \lim_{n\to\infty}\sum_{i=1}^{n}{a_i}$
When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. — Wikipedia

That definition was already discussed in the thread. And that definition is used in the pdf provided by the op in section 1. By that definition, .999... = 1 exactly.

I deny that .999..., as presented in the op, represents a particular quantity, because there is no quantitative value given for 1/9. Therefore I deny that .999...which in this instance does not represent a particular quantity is equivalent to 1 which does represent a particular quantity. — Metaphysician Undercover

In other words, 65 pennies, a dime and a quarter is not worth a dollar because pennies are 1/100th of a dollar and that's not a particular quantity of money. I mean, sure, some pennies are smaller than other pennies slightly; but some dollar coins are also smaller than other dollar coins. But apparently the pennies being smaller implies that pennies aren't a particular value, whereas the dollar coins being smaller does not indicate such a thing. Such is the tomfoolery I've heard from you so far. That's a garbage argument that can be ignored just on its merits.

I made my point — Metaphysician Undercover

Yes, but it's all gibberish nonsense.

I made my point, symbols such as 1/2, 1/3, 1/9, are representative of ratios between quantities, they do not represent any particular quantitative value. — Metaphysician Undercover

Sure they do. 1/2 represents one half. As you said, one of anything represents a particular quantity. The quantity that half represents is very clear... that is the multiplicative inverse of two. It takes two halves to make the quantity one.

The argument is very clear in my discussion with Banno. — Metaphysician Undercover

What argument?

You just cannot grasp the first premise, that 1/9 does not represent any particular quantity, and therefore it is not a number.

Your "premise" isn't a premise... it's a pointless language game. It shows you cannot speak the language (or at the very least, refuse to). You invented some niche and uninteresting alternate meaning for "particular quantity" that mathematics speakers do not use. The way mathematics speakers use the term "particular quantity", 1/9 is indeed one of those things. So you're not really advancing a "view" of quantities, you're promoting a language that's uninteresting. Therefore, your real burden is to show what's wrong with the language of math; you can't just say, "I don't 'believe' 1/9 is a particular quantity"... you have to say, "saying '1/9' is a particular quantity leads to the following problem" and say what that problem is.

Ok, but can't this be also said for 0.999...? — EnPassant

.999... is an infinite string of 9's. There's no problem with that per se.

Adding terms and then saying 'at infinity'. — EnPassant

We don't have to say "at infinity"... it's an infinite string of 9's. We add one 9 for all finite numbers, there are an infinite number of finite numbers, therefore there are an infinite number of 9's. But I think you misunderstand what the problem is...

You can't have (b)

In the infinite string .999...., I can say that the first digit after the decimal is 9. That's how I construct that infinite string. It's done homogeneously... it's always there at all steps.

In contrast, you cannot say what finite value x is when you have your infinite case, or that 0 < 1/x < 1 is true in that case. Your x changes every time you increment even by 1. What x is is inhomogeneous; it's diferent every time.

9 is the n-th digit in my string at all steps in the construction of it starting at n, going on indefinitely. x is a different value at every step in your construction. When I extend my string to infinity, 9 is still the first and nth digit. When you extend your inequality to infinity, x isn't finite, and you can't say 0 < 1/x < 1 for an infinite x. You never have an infinite number of a finite 1/x where 0 < 1/x < 1.

There's nothing here but Meta's queer usage. — Banno

An extremely charitable remark.

We have something that at first makes no sense, then we MAKE it make sense with a clever definition. What exactly are you objecting to? — fishfry

And this, some time ago, s/h/b the last word in this thread by a participant qualified to make it.

When you extend your inequality to infinity, x isn't finite, and you can't say 0 < 1/x < 1 for an infinite x. You never have an infinite number of a finite 1/x where 0 < 1/x < 1. — InPitzotl

Ok, but isn't this what happens with 1/2^c in the sum 1/2 + 1/4...? If we're talking about an infinite sum the same applies: by that way of looking at it you would not have 0 < 1/2^c < 1

Or in terms of 0.999..you can't, by this criterion, say 0 < 1/10^c < 1

But the assertion being made is that 1/10 + 1/100+... can be taken to an infinity of terms and summed to 1. If we take a finite number of terms it won't be 1. That's what I mean by an actual or literal sum. You have to go the whole way.

Such as the vicissitudes of these things. Ramanujan summed the natural numbers and got -1/12.

Calculus is a way of reasoning but there are other ways of reasoning that arrive at different results. Calculus is fine as long as we are talking about finite sums tending towards a limit. But when you go to infinity things get sticky. This is one of the reasons calculus was formulated in terms of finite sums going to the limit, so that the paradoxes at infinity would not interfere.

But the assertion being made is that 1/10 + 1/100+... can be taken to an infinity of terms and summed to 1. — EnPassant

That doesn’t sum to 1, that sums to 1/9.

You do know how to calculate a limit, don’t you?

Ok, but isn't this what happens with 1/2^c in the sum 1/2 + 1/4...? — EnPassant

No.

Series 1
Step 1: 1/2
Step 2: 1/2+1/4
Step 3: 1/2+1/4+1/8
...
Step 10: 1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024
Step 11: 1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024+1/2048
...

Step with all finite numbers: 1/2+1/4+1/8+1/16+...

Series 2
Step 1: 1/2
Step 2: 1/4+1/4
Step 3: 1/8+1/8+1/8
...
Step 10: 1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024
...

Step with all finite numbers: ? + ? + ? + ? + ? + ...

In series 1, at the "step with all finite numbers", the first term is 1/2. 1/2 is finite, and 0 < 1/2 < 1. In series 2, what is the first term at the "step with all finite numbers"? It's just that "step with all finite numbers" that has an infinite number of terms. Can you name the finite number x such that ? is 1/x? Can you say that 0 < ? < 1? Can you even say what ? is?

Or in terms of 0.999..you can't, by this criterion, say 0 < 1/10^c < 1 — EnPassant

Sorry, what is c here and how does that relate to 0.999...?

Ramanujan summed the natural numbers and got -1/12. — EnPassant

Yes, by using a different definition for a divergent infinite sum. I toyed with that here:

So does this mean ...999.999... = 0? — InPitzotl

...since ...999=-1 in 10-adics.

Sorry, what is c here and how does that relate to 0.999...? — InPitzotl

It's the same idea 0.999.. = 9(1/10 + 1/100 + ...)

What seems to be happening here is that 1/x = 0 at infinity.

Then you have the absurd(???) conclusion that

1/infinity = 0

1 = 0 x infinity

So 0 + 0 + ... = 1 after an infinity of terms.

There was an Indian mathematician in the Middle Ages who asserted this (I forget his name)

But most mathematicians probably would not accept this.

But there is also a geometric way to "prove" this.

Take the x,y axis and mark off the unit length from 0 to 1.

This unit represents an infinite string of points all lined up in a straight line.

What is the width of each point? Zero. They are dimensionless.

But they add up to 1 unit width. How do da?

zero width + zero width + ... = extension???

Every time a mathematician draws a graph on the x,y axis they are implicitly accepting that 0 + 0 + ... = 1 because they are working under the assumption that an infinity of dimensionless points add up to extension; the unit. Go figure...

It's the same idea 0.999.. - 1/10 + 1/100 + ... — EnPassant

So you're talking about 0.111...? Then @Pfhorrest's post applies:

That doesn’t sum to 1, that sums to 1/9. — Pfhorrest

What seems to be happening here is that 1/x = 0 at infinity. — EnPassant

There is no "at infinity" here though. Every term here is a finite number; there's just an infinite number of finite numbers. Think of it intuitively this way... imagine the set you're trying to picture... it has "at infinity" in it, and maybe some other things. Remove every infinite-th step from this; we only care about finite steps. But we do want all of the finite steps. Now you still have an infinite set, but it only has finite terms in it. That is the thing we're describing.

So 0 + 0 + ... = 1 after an infinity of terms.
But most mathematicians probable would not accept this. — EnPassant

If we see 0 repeated an infinite number of times in a sum, we tend to say that the result is undefined.

But they add up to 1 unit. How do da? — EnPassant

But you could do the same thing with a segment of length 2, 50, 0, and -7. So that infinite sum could also add up to 2, 50, 0, or -7, or any other value. This is what undefined refers to.

Every time a mathematician draws a graph on the x,y axis they are implicitly accepting that 0 + 0 + ... = 1 because they are working under the assumption that an infinity of dimensionless points add up to extension; the unit. — EnPassant

No, they aren't. The unit position is not defined in terms of infinite additions of 0. That would be useless, since infinite additions of 0 is undefined.

That doesn’t sum to 1, that sums to 1/9. — Pfhorrest

That was a typo, I meant to say 9/10 + 9/100 + ...

If we see 0 repeated an infinite number of times in a sum, we tend to say that the result is undefined. — InPitzotl

Yes, because it can't be defined in terms of calculus but the question remains, what is it?

But you could do the same thing with a segment of length 2, 50, 0, and -7. — InPitzotl

Yes, but that is arbitrary as the unit can be taken as any width, as in geometry - the unit radius can be 1 inch or 1 light year.

Yes, because it can't be defined in terms of calculus but the question remains, what is it? — EnPassant

It's undefined! :wink:

Yes, but that is arbitrary as the unit can be taken as any width, as in geometry - the unit can be 1 inch or 1 light year. — EnPassant

No, it's not arbitrary. It's just infinitely non-specific. That sum genuinely is sometimes 1 inch, sometimes a light year, sometimes 0, sometimes negative. So it's undefined. You can't reduce the sum to 1 inch if it could be negative or a light year. So it's useless to ponder whether it "truly" is 1 inch or "truly" is a light here, because your infinite sum doesn't give you the information to distinguish any length from any other.

Sometimes you can get that information elsewhere. But from just this, you just can't say.

No, it's not arbitrary. It's just infinitely non-specific. — InPitzotl

Say 0 + 0 + 0 + ... = 50 units. Simplify-

(0 + 0 + 0 + ...)/50 = 1

But 0/50 = 0. It's just a matter of reducing to the lowest terms and the same logic obtains.

Say 0 + 0 + 0 + ... = 50 units. — EnPassant

Okay, so that sum is 50 units.

(0 + 0 + 0 + ...)/50 = 1 — EnPassant

You can only say that if you're literally talking about that 50 unit thing, because:

But 0/50 = 0. — EnPassant

Yes, and 0+0+0... can be equal to 1. And 50. And a billion. And negative 7. To recap, that sum is undefined.

Yes, and 0+0+0... can be equal to 1. And 50. And a billion. And negative 7. — InPitzotl

But you still have 0 + 0 + ... = something

But you still have 0 + 0 + ... = something — EnPassant

If you do. It can also be 0. It can also not be anything. It can also be anything.

If you do. It can also be nothing (0). It can also not be anything. — InPitzotl

But if you draw the x - axis and mark off one unit, there you have it. The sum of dimensionless points add up to a unit width.

But if you draw the x - axis and mark off one unit, there you have it. The sum of dimensionless points add up to a unit. — EnPassant

You're going in circles. 1 is one of the possible things that sum can be. Pause for a second and think about this; otherwise this could continue forever.

You're going in circles. 1 is one of the possible things that sum can be. Pause for a second and think about this; otherwise this could continue forever. — InPitzotl

The reason the unit circle is a standard in geometry is because what applies to the unit applies to a circle of radius 10 or 100 or 1000. It is the internal geometric logic of the thing that matters, not the arbitrary measurement. It is like the difference between inches and centimeters. 1 inch = 2.5 cm. This has more to do with convention than anything fundamental.

The reason the unit circle — EnPassant

Not talking about the unit circle... just unity on the number line, and the idiom "going in circles" which means to retrace your paths over and over.

There's nothing here but Meta's queer usage.
— Banno
An extremely charitable remark. — tim wood

Well, @Metaphysician Undercover hasn't addressed the two proofs from the OP. All he has done is to assert that 1/9, and other fractions, are not numbers. His argument is an appeal to the authority of the OED.

Incidentally, and to my great amusement, the OED definition of fraction is "...numerical quantity that is not a whole number...", contradicting Meta's assertion that fractions are nether numbers nor quantities.

The disagreements are flat Earther stuff. — Michael

Hang on... isn't the flat earth two dimensional?:

Space cannot be represented as distinct dimensions, as the irrationality of these two dimensional figures demonstrates. — Metaphysician Undercover