0.99999~ = 1 ?

Jon writes:

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I see no reason why Math could not permit such infinite divisions in contradiction to Reality, which clearly does not.

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You're still adding woo-woo where none exists.Let's try something simple: Repeat after me:0.999... is identical to 1.Can you say that with a straight face?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Jon writes:

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Care to explain?

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Let's see if I can provide an example of how infinite process still manage to complete.For this thought experiment, we need to make a few assumptions:1) You have an infinite number of coconuts in a pile, all numbered: 1, 2, 3, ....
2) You have a pit big enough to hold all of the coconuts (but they aren't in there.)
3) Superman and Captain Marvel both exist and can travel any distance in any length of time.Superman and Captain Marvel decide to play a game. At precisely 12:00 noon, Superman throws coconuts numbered 1 and 2 into the pit. Captain Marvel flies into the pit, grabs coconut #1, and throws it out of the pit.They sit around, having some coffee, talking about their adventures, and at 12:30 precisely, Superman throws coconuts numbered 3 and 4 into the pit. Captain Marvel flies into the pit, grabs coconut #2, and throws it out of the pit.Again, they sit around and socialize. At 12:45, in go coconuts 5 and 6, out comes 3. This continues, each time halving the amount of time they wait.Question: When 1:00 comes around, and 1:00 always comes around, how many coconuts are in the pit?The answer is: None. They all got thrown out. For every single coconut you care to name, I can give you an exact time when it was tossed out of the pit: Coconut #1 was tossed out at 12:00. Coconut #2 was tossed out at 12:30. #3 at 12:45, #4 at 12:52:30.So even though an infinite number of coconuts are involved, every single one of them was involved in the process and thus, not one is left in the pit when 1:00 comes around.This is why 0.999... is identical to 1: You ask how many 9s?All of them. Because we have all of them, 0.999... is not merely equal to 1, it is identical to it.And that's an even stronger statement.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Phat responds to me:

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No matter how many coconuts were involved in the doings of the fractions of time 1:00, at exactly 1:00, Coconuts X and Y are taken to the pit by superman and Coconut Y is dutifully removed, at that exact fraction in time.

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Nope.This isn't about how many coconuts are in the pit before 1:00. This is about how many are in the pit at 1:00. Yes, you can calculate which coconuts are in the pit at any given time, but the ultimate question is what is the state of the system at 1:00, not before.1:00 always comes around. The universe does not sit and wait for us to make up our minds about what has happened.No matter what number you give me, I can give you a precise time that is before 1:00 when it comes out.If there were a coconut in the pit at 1:00, then it would have a number. But for every number you give me, I can tell you exactly when it left the pit, which was before 1:00.Therefore, the pit is empty.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Phat writes:

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Captain Marvel and Superman operate at the exact same speed.

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Perhaps, but that isn't what was stated. Instead, the given is that Superman and Captain Marvel can move any distance in any time.

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Superman always deposits 2X as many coconuts in the pit as does Marvel take out.

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Indeed, but when you examine the problem, that isn't really that relevant. The heart of the question is: When do the coconuts come out? The problem is completely identical to this scenario:You have a pit filled with an infinite number of coconuts, all numbered. Assume Superman exists and can move any distance in any time. At 12:00, coconut #1 comes out. At 12:30, coconut #2 comes out. Coconut #3 comes out at 12:45, #4 at 12:52:30, etc.When 1:00 comes around, and 1:00 always comes around, how many coconuts are there in the pit?See, we've removed the entire "two go in" process and yet, the result is exactly the same and for the same reason: For every coconut you can name, I can tell you when it was removed from the pit, which was before 1:00. If there is a coconut in the pit at 1:00, then it has a number, but for every number, there is a time before 1:00 when it was removed.Therefore, the pit is empty.

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Time can be divided into infinitely small units of measure, and Superman and Marvel both can keep up the pace.

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That's the assumption mentioned above: They can travel any distance in any time.

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Does it not then follow that for any given unit of time (T) there will always be 2 coconuts in and one coconut out?

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Only in the sense that the scenario indicates that at each successive halving, Superman throws in two coconuts and Captain Marvel removes one. But that isn't the number of coconuts in the pit.At 12:00, there is one coconut in the pit: #2.At 12:30, there are two coconuts in the pit: #3 and #4.At 12:45 there are three coconuts in the pit: #4, #5, and #6.But this isn't about how many are in the pit before 1:00. It's about how many are in the pit at 1:00.And for every coconut, I can tell you exact when it was removed, which was before 1:00:#1 was removed at 12:00.#2 was removed at 12:30.#3 was removed at 12:45.It doesn't matter when the coconuts entered the pit. We can start with all the coconuts in the pit. The only thing that matters is when they leave.And they all leave before 1:00.Therefore, the pit is empty.

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Thus, even if every single coconut can be accounted for by a number, there is always at that same moment a number twice as great going into the pit...thus at any given moment of time, X+X (or X+Y?) Coconuts are going into the pit, while either X or Y are coming out of the pit...leaving the contents of the pit always at some variable of X (or Y) be it 1X, 2X, google-X, or Infinity X.

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Irrelevant. Start with all the coconuts in the pit.The only thing that matters is when the coconuts left. And they all left before 1:00.Therefore, the pit is empty.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Huntard writes:

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(superman's infinite coconuts) - (captain marvel's infinite coconuts) = 0
Sounds logical to me, though it's probably completely wrong.

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No, you're right, but it isn't that simple. When I was first introduced to this question in my Advanced Calculus class, it was to point out part of the nature of infinity.The next day, Superman and Captain Marvel decide to alter the game:At 12:00, Captain Marvel throws in coconuts #1 and #2 and Superman flies in, grabs coconut #1, and tosses it out.After half an hour, at 12:30, Captain Marvel throws in #3 and #4, Superman flies in, grabs coconut #3, and tosses it out.After fifteen minutes, at 12:45, in go #5 and #6, out comes #5.When 1:00 comes around, and 1:00 always comes around, how many coconuts are in the pit?Easy: An infinite number. In this case, all the even ones.The moral of this story? Infinity doesn't play like normal numbers. Infinity - infinity is undefined. You need to explain what the infinity is in order to determine what the result is. The physical process of the first game and the second game are the same: Two go in and one comes out, but the result is different because what is coming out is not the same in both scenarios.As I mentioned to Phat, the description of the two going in is really a red herring. Start with all the coconuts in the pit. Then it becomes more clear what is going on: In the first case, all the coconuts are processed and thus, there are none left over.In the second case, only every other coconut is processed. And since the size of the odd numbers is the same as the size of the Natural numbers, even though you're processing the same number of coconuts, not all of them are touched and thus, there are some left over. An infinite number of them.We can adjust the process so that any arbitrary number of coconuts are left. Suppose that in the first scenario, Captain Marvel first throws out coconut #2, then #3 at 12:30, #4 at 12:45, and so on?When 1:00 comes around, there is exactly one coconut in the pit: #1.Infinity - infinity is undefined.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

caffeine responds to me:

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Could it be that the apparent contradictions in this sort of hypothetical arise simply because it's something that couldn't happen except in mathematical abstraction?

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That's why it's called a "thought experiment." It cannot happen as a typical physical process.But that doesn't mean it isn't true. And there is no contradiction.Infinity - infinity is undefined. When you can define the process, what the value is becomes defined.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Dr Adequate responds to me:

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Infinity - infinity is undefined.

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Not at all.

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Incorrect. I've already shown you how you can create a process such that infinity - infinity can result in any Whole number you care to name.

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A set is infinite if and only if it is the same size as (i.e. can be put into a one-to-one correspondence with) a proper subset of itself.

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Irrelevant. This isn't about the size of a set. It's about the arithmetic of infinity.Infinity - infinity is undefined. In the equation x - x, as x goes to infinity, the result is 0. But in the equation x - x², the result goes to negative infinity.This is basic Real Analysis. I'm surprised you don't recall it. Were you not given the homework assignments of finding what a function's limit is? It all depends upon how you approach it. Sometimes, the value a function takes depends upon which way you approach it, from the positive or from the negative.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Jon responds to me:

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at some point Achilles must move one whole unit

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Irrelevant. Even in a universe that is infinitely divisible, he still catches the tortoise. Weren't you paying attention? 1:00 always comes around and thus, he eventually crosses the entire distance.

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an infinitely long string of 0.9999| represents what would be infinite divisibility (Message 173), which Reality does not permit.

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And yet, 0.999... is identical to 1. Not merely equal, but identical.How are you going to reconcile your claim with reality?

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To check this, MATHSYSTEM introduces further rules to which Reality is not subject in order to mend the disconnect.

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Incorrect. You are confusing notation with reality. "Ceci n'est pas un pipe." Do not confuse the symbol for what it represents.

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Please, correct me where I have erred.

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That would be the point where you tried to claim that the way we write something down has something to do with what it is referring to. "Blue" is a word that refers to a color, but it is not the color itself.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Jon responds to me:

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A single Real existence has separate encodings within the Math system

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But we do that with all language. We have lots of terms to describe the color of the sky, but there is no confusion about what we're talking about and the reason we use any particular term depends upon the context in which it happens.The reason why we choose "1" over "0.999..." has to do with the properties of the notation. In proving that the set of Reals is uncountable, we need to make sure that each number in the Reals is represented uniquely. We are trying to achieve a one-to-one correspondance and we have to ensure that a number doesn't show up twice.By leveraging the notation used, we can ensure that and thus achieve what we were trying to prove.The part you are having trouble is that 1 and 3/3 and 0.999... aren't merely equal...they are identical, which is a stronger mathematical relationship.

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This multi-encoding scheme used by Math can give the false impression that there are three separate realities each represented by three distinct symbologies, which, of course, is not the case.

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It can only "give the false impression" if someone is naive regarding the nature of mathematics. Just as we don't claim the sky is a different color simply because we call it "blue" in one context and "azul" in the other, we don't claim that 1 is something different from 0.999... simply because we use one in one context and the other in another.

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That is misleading. They are not identical

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Incorrect. They are, indeed, identical. Two dollar bills are "equal," but they are not "identical." But 1 and 0.999... are more than just equivalent. They are identical.

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they represent identical Realities according to the Math rules of transcription.

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Incorrect. You are again confusing notation with reality. Because they represent the same thing, that is what makes them identical despite the notational peculiarities required to talk about the objects of mathematics.The fact that we can use different terms to refer to the same object is not indicative of any sort of distinction in the object. It simply means that some contexts do better with one method while other contexts to better with another.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Dr Adequate responds to me:

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I read what you intended to be a minus sign as a dash.

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Ah! That's different.One of the things I was taught in Real Analysis is not only does infinity not play well, zero doesn't, either.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Jon responds to me:

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Ever bother reading your own post?

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Yep. As I have repeated asked you, quite politely:Do not confuse the symbol for what it is referring to.Indeed, "1" and "0.999..." are not written out the same way.However, that is irrelevant. They refer to the exact same thing. That is why they are identical. That one person calls the sky "blue" while another calls it "azul" doesn't mean the sky has actually changed color.

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Two things that are different cannot be identical.

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But that's just it: They're not different. You are confusing notation with substance. The numerical value of "one" is the same whether we write it down symbolically as "1" or if we write it down as "0.999...."Just as the color of the sky is the same whether we use the word "blue" or "azul."Do not confuse the notation for the object.

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Tell me, are the following two figures, (a) and (b), identical?

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a) OO
b) U

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Again, you confuse notation for the object it represents. Attempting to equate notational quirks of human language with notational aspects of mathematics is comparing apples and oranges. Language, being a living thing that is generated by the millions of people who speak it. It's why "bank" meaning the edges of a river and "bank" meaning a place to store money are different words with different histories and different roots even though they are spelled exactly the same way.Mathematics, on the other hand, is more rigidly controlled. The symbols have very specific meanings that exist for specific reasons.Here's a linguistic example: Suppose you wanted to count the number of words in the language. But there's a problem: Some words are spelled the same but are different. So if we're going to do a counting method by letter-by-letter comparisons, we're going to have to figure out a notational method to distinguish "bank" meaning the sides of a river from "bank" meaning a place to store money.One possibility is to extend the length of all words so that they are all the same length. Any words that were shorter than that will have extra, silent letters attached to the ends that indicate their uniqueness but do not change the pronunciation or meaning in any way. It's just a matter of notation so that we can do the count and ensure that we have accounted for every word once and only once.Do not confuse the notation for what it is referring to.Do you have any justification that 1 and 0.999... are not identical beyong your hangup over spelling?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Jon responds to me:

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Are the following two figures, A and B, identical?

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Once again, you confuse the symbol for what it represents.Here's an example of your failure: The notation used by engineers when referring to various aspects of differential equations and analysis is not the same as that used by mathematicians. In mathematics, the symbol for the square root of -1 is i. In engineering, it's j. In math, derivatives are marked by a tic. In engineering, it's a dot. There's no difference at all between the actual mathematical processes between the two, it's just different conventions.Even the discovery of the calculus shows how you must be careful not to confuse the notation for what it represents: Newton and Liebniz ostensibly discovered the calculus independently. And yet, the calculus we have today is a combination of the two. The methodology that we use is mostly Newton's but the notation that we use is mostly Liebniz's. Newton had the stronger process but Liebniz had the symbology to make it understandable. It is because the calculus is independent of notation that allows us to do this.

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A simple one-word answer will suffice.

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No, it won't.I don't know what those symbols represent and since I know not to confuse the symbol with the object it represents, I cannot answer the question.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Jon writes:

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Identicalness has nothing to do with meaning or interpretation or representation.

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And this is your fundamental failure. It is exactly the other way around. Identicalness has everything to do with meaning. That's precisely how you determine if things are identical: If they mean exactly the same thing.This is why things like infinity - infinity are undefined: Not all infinities are the same.Tell us what your symbols mean and then we'll know if they're identical.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

nwr writes:

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In mathematics, "equal" means "identical".

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Incorrect. Two things being identical is a stronger relationship than equality.Things that are identical are the same no matter what. Things that are equal are only so in certain circumstances. That's why the identity operators are called such: When they operate, they return the exact same object you started off with no matter what that object is. With an additive identity of 0, then x + 0 = x no matter what x is. With a multiplicative identity of 1, then x * 1 = x no matter what x is.^*a * x/x is only equivalent to a when when x <> 0. Therefore, a * x/x is not identical to a. It at best is equal to a.0.999... is not merely equal to 1, it is identical.^*As an example of why we must not confuse the symbol for what it refers to: The additive identity 0 differs depending upon the object it works on. For a number, 0 = 0. For a matrix, it is a matrix filled with 0s. But which matrix 0 is depends upon the original matrix. If you have an m x n matrix, then 0 is also m x n. But if you have an n x m matrix, then 0 is n x m. Similarly, 1 for a number is 1 but for a matrix, it is a square matrix that is as wide as the original matrix. This is if you have an m x n matrix, then 1 is an n x n matrix with 1s down the diagonal and 0s everywhere else. But if your original matrix is n x m, then 1 is an m x m matrix with 1's down the diagonal and 0s everywhere else.And yet, we use the symbols 0 and 1 to refer to all of these different things. The have equivalent concepts, but are not equivalent themselves.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Jon writes:

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Identical meaning does not make the figures 0.9999| and 1 identical; it does, however, make them equal.

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Incorrect. The two have the same meaning makes them not merely equal but identical. That's the point behind identity: They are the same object.Do not confuse notation with the object.

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So you are incapable of telling me whether two things are identical without knowing their symbolic meaning?

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Precisely. Notation is an artificial construct we place upon meaning. It is not the object itself. The reference to an object does not change the object in any way. Two objects are identical not because they are symbolized the same way but rather because they function the same way.That's why the sky doesn't change color just because one person calls it "blue" while another calls it "azul."It's why calculus use's Newton's methods but not his notation. The calculus did not change simply because Liebniz had a better way of writing it down.It's why "one" doesn't change just because sometimes we write it as "1" and other times we write it as "0.999...."Do not confuse the notation for the object it represents.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.