Register | Sign In


Understanding through Discussion


EvC Forum active members: 65 (9162 total)
4 online now:
Newest Member: popoi
Post Volume: Total: 915,808 Year: 3,065/9,624 Month: 910/1,588 Week: 93/223 Day: 4/17 Hour: 1/1


Thread  Details

Email This Thread
Newer Topic | Older Topic
  
Author Topic:   0.99999~ = 1 ?
New Cat's Eye
Inactive Member


Message 196 of 237 (544614)
01-27-2010 2:10 PM
Reply to: Message 195 by Jon
01-27-2010 1:05 PM


Please, correct me where I have erred.
OK.
an infinitely long string of 0.9999| represents what would be infinite divisibility
Right there.
It represents the asymptote, itself, not the approach to it.

This message is a reply to:
 Message 195 by Jon, posted 01-27-2010 1:05 PM Jon has replied

Replies to this message:
 Message 197 by Jon, posted 01-27-2010 2:38 PM New Cat's Eye has replied

  
Jon
Inactive Member


Message 197 of 237 (544617)
01-27-2010 2:38 PM
Reply to: Message 196 by New Cat's Eye
01-27-2010 2:10 PM


an infinitely long string of 0.9999| represents what would be infinite divisibility
It represents the asymptote, itself, not the approach to it.
Right; I should have said it represents the process of infinitely dividing, with the infinitely divided being the space between. The point remains, do these concepts of Math have Real-world counterparts or are they merely conceptual?
quote:
Wikipedia (emphases added):
The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem counter to everyday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a computer screen have a positive width. So if the they were to be extended far enough they would seem to merge together, at least as far as the eye could discern. But these are physical representations of the corresponding mathematical entities; the line and the curve are idealized concepts whose width is 0 (see Line). Therefore the understanding the idea of an asymptote requires an effort of reason rather than experience.
Jon
Edited by Jon, : No reason given.
Edited by Jon, : No reason given.

[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

This message is a reply to:
 Message 196 by New Cat's Eye, posted 01-27-2010 2:10 PM New Cat's Eye has replied

Replies to this message:
 Message 198 by New Cat's Eye, posted 01-27-2010 2:45 PM Jon has not replied

  
New Cat's Eye
Inactive Member


Message 198 of 237 (544618)
01-27-2010 2:45 PM
Reply to: Message 197 by Jon
01-27-2010 2:38 PM


with the infinitely divided being the space between.
Huh?
0.9R is 1, there's no space between.
The point remains, do these concepts of Math have Real-world counterparts or are they merely conceptual?
There might be some math concepts that are merely conceptual, but I don't think 0.9R is one of them.
quote:
Wikipedia (emphases added):
The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem counter to everyday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a computer screen have a positive width. So if the they were to be extended far enough they would seem to merge together, at least as far as the eye could discern. But these are physical representations of the corresponding mathematical entities; the line and the curve are idealized concepts whose width is 0 (see Line). Therefore the understanding the idea of an asymptote requires an effort of reason rather than experience.
A line that represents my path from here to there does have a width of zero, regardless of my inability to draw the line on a piece of paper that way.

This message is a reply to:
 Message 197 by Jon, posted 01-27-2010 2:38 PM Jon has not replied

  
Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 199 of 237 (545037)
01-31-2010 6:51 AM
Reply to: Message 192 by caffeine
01-27-2010 9:05 AM


caffeine responds to me:
quote:
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?
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.

Rrhain

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.

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

This message is a reply to:
 Message 192 by caffeine, posted 01-27-2010 9:05 AM caffeine has not replied

  
Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 200 of 237 (545038)
01-31-2010 7:05 AM
Reply to: Message 193 by Dr Adequate
01-27-2010 11:29 AM


Dr Adequate responds to me:
quote:
quote:
Infinity - infinity is undefined.
Not at all.
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.
quote:
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.
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.

Rrhain

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.

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

This message is a reply to:
 Message 193 by Dr Adequate, posted 01-27-2010 11:29 AM Dr Adequate has replied

Replies to this message:
 Message 203 by Dr Adequate, posted 02-02-2010 12:45 AM Rrhain has replied

  
Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 201 of 237 (545042)
01-31-2010 7:21 AM
Reply to: Message 195 by Jon
01-27-2010 1:05 PM


Jon responds to me:
quote:
at some point Achilles must move one whole unit
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.
quote:
an infinitely long string of 0.9999| represents what would be infinite divisibility (Message 173), which Reality does not permit.
And yet, 0.999... is identical to 1. Not merely equal, but identical.
How are you going to reconcile your claim with reality?
quote:
To check this, MATHSYSTEM introduces further rules to which Reality is not subject in order to mend the disconnect.
Incorrect. You are confusing notation with reality. "Ceci n'est pas un pipe." Do not confuse the symbol for what it represents.
quote:
Please, correct me where I have erred.
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.

Rrhain

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.

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

This message is a reply to:
 Message 195 by Jon, posted 01-27-2010 1:05 PM Jon has replied

Replies to this message:
 Message 202 by Jon, posted 02-01-2010 2:02 PM Rrhain has replied

  
Jon
Inactive Member


Message 202 of 237 (545126)
02-01-2010 2:02 PM
Reply to: Message 201 by Rrhain
01-31-2010 7:21 AM


Rrhain writes:
quote:
To check this, MATHSYSTEM introduces further rules to which Reality is not subject in order to mend the disconnect.
Incorrect. You are confusing notation with reality. "Ceci n'est pas un pipe." Do not confuse the symbol for what it represents.
Wrong, sir. I have maintained an extremely sharp contrast between the Reality and the Math. My entire babbling since about my third post into this thread has been in regards the fact that the Math permits distinctions in symbols which the Reality does not. A single Real existence has separate encodings within the Math system:
R
     /|\
    / | \
   /  |  \
  1  3/3 0.9999|
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.
And yet, 0.999... is identical to 1. Not merely equal, but identical.
That is misleading. They are not identical; they represent identical Realities according to the Math rules of transcription.

[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

This message is a reply to:
 Message 201 by Rrhain, posted 01-31-2010 7:21 AM Rrhain has replied

Replies to this message:
 Message 204 by Rrhain, posted 02-02-2010 3:23 AM Jon has replied

  
Dr Adequate
Member (Idle past 284 days)
Posts: 16113
Joined: 07-20-2006


Message 203 of 237 (545173)
02-02-2010 12:45 AM
Reply to: Message 200 by Rrhain
01-31-2010 7:05 AM


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

This message is a reply to:
 Message 200 by Rrhain, posted 01-31-2010 7:05 AM Rrhain has replied

Replies to this message:
 Message 205 by Rrhain, posted 02-02-2010 3:34 AM Dr Adequate has not replied

  
Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 204 of 237 (545181)
02-02-2010 3:23 AM
Reply to: Message 202 by Jon
02-01-2010 2:02 PM


Jon responds to me:
quote:
A single Real existence has separate encodings within the Math system
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.
quote:
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.
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.
quote:
That is misleading. They are not identical
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.
quote:
they represent identical Realities according to the Math rules of transcription.
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.

Rrhain

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.

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

This message is a reply to:
 Message 202 by Jon, posted 02-01-2010 2:02 PM Jon has replied

Replies to this message:
 Message 206 by Jon, posted 02-02-2010 9:19 AM Rrhain has replied

  
Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 205 of 237 (545182)
02-02-2010 3:34 AM
Reply to: Message 203 by Dr Adequate
02-02-2010 12:45 AM


Dr Adequate responds to me:
quote:
I read what you intended to be a minus sign as a dash.
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.

Rrhain

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.

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

This message is a reply to:
 Message 203 by Dr Adequate, posted 02-02-2010 12:45 AM Dr Adequate has not replied

  
Jon
Inactive Member


Message 206 of 237 (545216)
02-02-2010 9:19 AM
Reply to: Message 204 by Rrhain
02-02-2010 3:23 AM


Rrhain writes:
...that is what makes them identical...
...
...different terms to refer to the same object...
Ever bother reading your own post? Two things that are different cannot be identical. Tell me, are the following two figures, (a) and (b), identical?
a) OO
b) U

[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

This message is a reply to:
 Message 204 by Rrhain, posted 02-02-2010 3:23 AM Rrhain has replied

Replies to this message:
 Message 207 by xongsmith, posted 02-02-2010 6:02 PM Jon has replied
 Message 208 by Rrhain, posted 02-03-2010 7:22 AM Jon has replied

  
xongsmith
Member
Posts: 2578
From: massachusetts US
Joined: 01-01-2009
Member Rating: 6.8


Message 207 of 237 (545269)
02-02-2010 6:02 PM
Reply to: Message 206 by Jon
02-02-2010 9:19 AM


Jon objects:
Ever bother reading your own post? Two things that are different cannot be identical.
Again, don't confuse the ascii letter representation with the number itself.
0.999| is a perfectly good equivalent of 1.0 or 1. It's just not as efficiently represented by our ascii character set.
3.14159 is an approximation of pi and not identical to pi. We have no way of representing the series of digits continuing out ad-infinitum for pi.
We might try pi = 4*arctan(+1.0) or 4*atan(1) in some computer languages which is 9 ascii characters.
or
ln( -1)
----------- could be ln(-1)/i, which is only 8 characters.
i
or arccos(-1) is another way.
Some computer languages will even allow you to write "pi", only 2 characters, but inside the machine there is a value of the 3.14159 kind, with as many digits as their long double may hold. So it's really, again, not identical. And this is also true of those other constructs above, because ultimately the computer has a limited storage ability for calculating things like 4*atan(1).
The notation we use to represent the number is never meant to be the same as the number itself.

- xongsmith, 5.7d

This message is a reply to:
 Message 206 by Jon, posted 02-02-2010 9:19 AM Jon has replied

Replies to this message:
 Message 209 by Jon, posted 02-03-2010 1:09 PM xongsmith has not replied

  
Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 208 of 237 (545331)
02-03-2010 7:22 AM
Reply to: Message 206 by Jon
02-02-2010 9:19 AM


Jon responds to me:
quote:
Ever bother reading your own post?
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.
quote:
Two things that are different cannot be identical.
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.
quote:
Tell me, are the following two figures, (a) and (b), identical?
a) OO
b) U
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?

Rrhain

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.

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

This message is a reply to:
 Message 206 by Jon, posted 02-02-2010 9:19 AM Jon has replied

Replies to this message:
 Message 210 by Jon, posted 02-03-2010 1:10 PM Rrhain has replied

  
Jon
Inactive Member


Message 209 of 237 (545370)
02-03-2010 1:09 PM
Reply to: Message 207 by xongsmith
02-02-2010 6:02 PM


Symbols are Things Too...
xongsmith writes:
Again, don't confuse the ascii letter representation with the number itself.
Huh? Can you show me where I did this? Can you show anywhere in any of my posts mention of ASCII character representation schemes?
0.999| is a perfectly good equivalent of 1.0 or 1.
Yup. What does that have to do with them being identical though? The property identical can be determined applicable or non to a collective description of two things or more based solely on their existence and availability to the process of comparison. In the case of 0.9999| and 1.0, it is true that they are not identical - I mean, just look at them. Equality (or, as you say equivalence), however, is a function of a Symbology, that is, a system of symbols and the conventional ways in which they are said to represent what they represent. Within the generally accepted Math system 0.9999| and 1.0 are equal because within the Symbology of the Math system they are agreed to represent identical realities. Identicalness is a property of things which can be determined irrespective of any system of interpretation which derives meaning from those things. Equality is a property of symbols within a Symbology in which those symbols are agreed to represent identical realities.
Let me give a non-math example, since trying to explain it with math has made the waters more muddy, I think.
The following two things are identical:
SON
SON
We can determine they are identical without any reference to their meaning. However, if we are to decide whether they are equal or not, we must understand the Symbologies to which they may each belong. If we are told that they each belong to the same Symbology, then we may conclude they are equal without question and without having to know what it is they represent within that Symbology. To state it as a rule that can be used in understanding symbols, we have the following:
If two or more identical symbols are members of the same Symbology, then they are equal.
or (where I = "two or more symbols are identical", S = "the same two or more symbols are from the same Symbology", and E = "the same two ore more symbols are equal"):
1 (IS)→E
So, if I tell you that our original examples (SON and SON) are part of the same Symbology, then you can conclude they are equal, because they are identical. For sake of argument, let us just name that Symbology, which is English. Being identical members of the same Symbology allows us to conclude they are equal, and indeed, this is what we feel as being accurate as speakers of English (i.e., users/understanders of the Symbology). But now, let me tell you these are not part of the same Symbology; what can we make of them now? They may still be equal, of course, even though they are members of separate Symbologies, but we can no longer make the assumption that they are based solely on the fact that they are themselves identical. Therefore, we need more information on the nature of the Symbologies. So, I tell you that one Symbology is Spanish and the other English. Now, with this information, we may interpret the meaning behind each to decide if they are equal or not, and indeed they are not. So, we have another rule:
If two or more identical symbols have the same meaning within their respective Symbologies, then they are equal.
or (where M = "the two or more symbols have the same meaning within their respective Symbologies", with the other variables already defined):
2 M→E
So if we do not have both I and S, we can use M to determine the status of E. This is important, because we now turn to two different examples:
SOL
SUN
Are these identical? No, they are not, which we can determine merely by looking at them. Thus, our first rule cannot be used. So, now we must use our second rule, which requires that we determine their meaning, which requires us to know a little something about their respective Symbologies. I will tell you that the first is Spanish and the second English, like before. Now, we must find what they represent within those Symbologies. Do they mean the same thing? Indeed, thus they, despite being not identical, are equal, because they satisfy the antecedent of our second rule. Of course, the try-this-then-that approach used should make it clear that our rules can be simplified into one super rule:
If two or more symbols are identical and members of the same symbology, or have if they are either identical or not but have the same meaning; they are equal.
or:
3 ((IS)vM)→E
The notation we use to represent the number is never meant to be the same as the number itself.
Well I would certainly hope not.
Jon
Edited by Jon, : quote attribution

[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

This message is a reply to:
 Message 207 by xongsmith, posted 02-02-2010 6:02 PM xongsmith has not replied

Replies to this message:
 Message 211 by nwr, posted 02-03-2010 4:20 PM Jon has replied
 Message 212 by lyx2no, posted 02-03-2010 5:41 PM Jon has replied

  
Jon
Inactive Member


Message 210 of 237 (545371)
02-03-2010 1:10 PM
Reply to: Message 208 by Rrhain
02-03-2010 7:22 AM


Where Does Identicalness Lie?
Rrhain writes:
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.
Are the following two figures, A and B, identical?
A)
B)
A simple one-word answer will suffice.
Jon
Edited by Jon, : quote attribution

[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

This message is a reply to:
 Message 208 by Rrhain, posted 02-03-2010 7:22 AM Rrhain has replied

Replies to this message:
 Message 217 by Rrhain, posted 02-04-2010 2:11 AM Jon has replied

  
Newer Topic | Older Topic
Jump to:


Copyright 2001-2023 by EvC Forum, All Rights Reserved

™ Version 4.2
Innovative software from Qwixotic © 2024