0.99999~ = 1 ?

I think there is a fine difference between growing to infinity and infinitely growing. I would say that the latter is possible in respects 0.9, etc., while still maintaining it has a value less than 1.0. Afterall, the number we are working with is infinite, is it not? With no proof eliminating infiniteness (not quite the same as infinity, I don't think), starting with 0.0 and adding the 0.9 cuts the remaining distance between 0.0 & 1.0 to 1/10 its previous size, no?1.0 - 0.0 = 1.0
1.0 - 0.9 = 0.1
(1/10)(1.0) = 0.10.1 is 1/10 of 1.0Now, if we add another 9, so 0.99, the remaining 1/10 distance is cut itself to 1/10 of its former size. I won't do the proof for that, but it seems more intuitive to me that as we added 9s, the distance between 1.0 and 0.0 would simply become infinitely smaller, but never be a gap bridgeable through the adding of infinite 9s or without rounding. (I hear the word 'Zeno' being whispered in my ear, but I am not sure why.) It seems that left as it is, Math can end up describing a paradoxical nature... or, a reality that is not Reality.I see no reason why Math could not permit such infinite divisions in contradiction to Reality, which clearly does not. (Damn, there's that 'Zeno' whisper sound again.) Now, I realize that it doesn't work this way, and so long as the MATHSYSTEM maintains the DISTINCTness criterion it too will not permit such infiniteness. But that it needs a check-sub-system to eliminate contradictions with Reality should be a good sign that the MATHSYSTEM and its parts do not follow necessarily from Reality (REALWORLDSYSTEM), just that we have found a way to make the system both internally consistent and eliminate its contradictions to Reality. Anyway, more on this relationship when my dern thread gets (if ever) promoted. JonEdited by Jon, : zeros and such

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

Hello my fellow Minnesotan Always good to be able to add some more cold, frigid sanity around here Jon

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

It's not? How many decimal places does 0.9999| go out to?Jon

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

Care to explain?

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

Indeed. It is well understood by both y'all and myself that Reality is not infinitely divisible, which is why Zeno's paradoxes do not workat some point Achilles must move one whole unit, be it however small, because moving only 1/2 of that distance is not an option as the space it represents is not further divisible, and so he overtakes the tortoise.¹And herein lies the whole catch: an infinitely long string of 0.9999| represents what would be infinite divisibility (Message 173), which Reality does not permit. To check this, MATHSYSTEM introduces further rules to which Reality is not subject in order to mend the disconnect. Math having a rule which reality does not is part of the conventionalism that I discuss in the other thread.Please, correct me where I have erred. Jon
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[size=1]¹ In fact, Zeno starts his paradox arguments with the premise that space and time are infinitely divisible, notices his answers from such a premise are not in accord with Reality, and then should have concluded his assumptive premise on the infinite divisibility of space and time to be incorrect, which would mean space and time are not infinitely divisible but are made up of however small non-divisible wholes, which would have proven the Atomists had it right all along. BUT, I digress...[/size=1]Edited by Jon, : Link broken

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

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:

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Wikipedia (emphases added):

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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.

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JonEdited by Jon, : No reason given.Edited by Jon, : No reason given.

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

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. That is misleading. They are not identical; they represent identical Realities according to the Math rules of transcription.

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

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

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

Huh? Can you show me where I did this? Can you show anywhere in any of my posts mention of ASCII character representation schemes? 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
SONWe 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)→ESo, 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→ESo 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
SUNAre 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)→EWell I would certainly hope not.JonEdited by Jon, : quote attribution

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

Are the following two figures, A and B, identical?A) B) A simple one-word answer will suffice.JonEdited by Jon, : quote attribution

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

Are the following two figures, A and B, identical?A) B) A simple one-word answer will suffice.Jon

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

Yup. The NAME 0.9999| and 1.0 are DIFFERENT, though the NUMBER they REPRESENT is the SAME. Are the following two figures, A and B, identical?A) B) The miscommunication appears to be here. When I write 0.9999| and 1, and refer to them, I am not using them as representations of some number and thus indirectly speaking of the number; but when I say 0.9999| and 1 are different, I am saying so in reference to them as strings, as purely a cluster of shape and symbol. In fact, were 0.9999| and 1 themselves identical, this thread would not exist, because no one would have ever questioned whether they stood for the same thing within the same Symbology.1 is identical to 1
0.9999| is identical to 0.9999|
3/3 is identical to 3/3
1 is not identical to 0.9999|
0.9999| is not identical to 3/3
3/3 is not identical to 1Identicalness has nothing to do with meaning or interpretation or representation. It is a purely physical and objective, unless, of course, you find the following two figures, A and B, to be something other than 'not identical':A) B) Jon

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

Which is why they may be equal, just not identical.

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

This is the position I have maintained the whole time. Go back and reread my posts without the presumption that I am referring to the represented values of the figures and just read it as it is, a figure, and you will see. Identical meaning does not make the figures 0.9999| and 1 identical; it does, however, make them equal. So you are incapable of telling me whether two things are identical without knowing their symbolic meaning?

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

Indeed; I agree with you fully. And it is because we have the different conventions that we end up with non-identical modes of representation. I have a feeling your answer is artificial, which means you are lying. Or, your answer is honest, in which case you cannot definitively tell me that a big green blocky blob is not identical to a smaller more fluid white blob. In either case, it ain't much worth continuing on with you, but I will finish with saying the following, a final attempt to show your understanding of identicalness to be flawed:

quote:

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Wiktionary:
....
3. (not comparable, mathematics) Exactly equivalent.
....
In mathematics, this adjective can be used in phrases like "A and B are identical", "A is identical to B", and, less commonly, "A is identical with B".

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Only in mathematics can you refer to two things that are equal as being identical, with the understanding that the definition used is especially tuned for the field of mathematics and that the word, when used, refers to the things represented and not the representations themselves. (In other words, Math does not set 0.9999| and 1 as identical, but sets their meanings as such).When referencing things outside of mathematics, your definition is non-applicable, in which case you should have concluded the two blobs to be non-identical. Your application of specialized definitions to generalized reasoning is creating a miscommunication problem.Unless you aren't just using the term in a specialized way but actually do believe identicalness to be dependent on meaning, in which case:Are we to assume that creatures without the mental capacity for symbolic representation cannot distinguish things? Does a cat not know a mouse from a dog? The problem with this understanding is that it rests identicalness on the existence of Symbologies. This reasoning is human-centered: Identicalness is a property dependent on Meaning.
Meaning is a property dependent on Symbologies.
Symbologies are dependent on humans.¹

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I→(M→(S→H))To maintain your logic as true, making H false (getting rid of humans) would have to render, ultimately, identicalness a non-property. But this is not what we see in the Real world. When two identical things are in identical environments, identical things do happen to them. The Natural World (Reality) does recognize and function subject to identicalness, and would continue to do so without the grace of our bipedal selves. Being identical is a property of the Real world, and is not dependent on whether or not we agree to interpret something as identical or not. The laws of Nature will not change if you call the blobs identicalthey will still be treated as different.This definition of identical as being ultimately dependent on Symbologies is just far far removed from what would be given to the word by any average speaker of the English language.Jon
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¹ Though I use humans, we could replace them with 'any entity capable of Symbologies', which does not change the logic, it just means that to prove the point we would have to exterminate many more critters

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