1=0.999... infinities and box of chocolates..Phliosophy of Math...

So, now you understand why a point can not be divided any farther.

 

Log in or Sign up to hide all adverts.

But we are not talking about a point as defined by math. We are talking about reducing a ball so that it is the smallest it can be and yet still be greater than zero. We may be attempting to explore how that definition may need to be changed.

We are not talking about the limitations of a mathematical system. We are talking about reducing a ball so that it is the smallest it can be yet still greater than zero.

"If we don't have the tools to deal with the situation then I suggest we invent some"

Isaac Newton developed/evolved calculus for just that reason. The infinitesimal was developed for just that reason. It appears that even these tools are insufficient.
The paradox can be expressed simply as:
If a ball has a diameter if 1 Planck Unit what is it's volume?

If a Planck unit has positive dimension than what makes a plank unit the dimension it is?
What is less than 1 Planck?
Other than zero...
Zero has both 3 dimensions and zero dimensions simultaneously. It is both immaterial and material simultaneously.
The proof of this is obvious both in Physics and Math. IMO

 

Log in or Sign up to hide all adverts.

So basically Mathematics has given up on Achilles and stated that the paradox is indeed a paradox as the only way they can resolve it is to state that there is a limit to x/2.
Zeno of Elea was a very smart man, and I wonder if he thought that his "paradox " would have been better revealed by someone else.

 

Log in or Sign up to hide all adverts.

Mr. Q, You've mentioned that example many times, and it's completely wrong.

If we are talking about a mathematical ball, there is no smallest positive radius. This is readily seen, since if the radius is r > 0 then r/2 is also positive and strictly smaller than r.

If you're talking about a physical ball, then you'll have to tell me what a "ball" is, since at tiny scales of measurement all you have is quarks or strings or branes or whatever the latest physical theory is. There are no exact balls or spheres in the physical world at all. Not the earth, not a basketball, not a marble, not a "perfect" steel ball bearing. If it's physical, it's not a perfect sphere.

So if we're back to the world of mathematics, there is no such thing as a ball that is the "smallest it can be." You can always make it smaller, since there is no smallest positive real number.

 

except that 0.999... = 1 because the difference between them is less than any real number.
[The reduction of the ball is the exact same paradoxical situation as 0.999... = 1]

The contradiction I am alluding to is that on one hand you are saying that x/2 is infinite reduction and that there is always a smaller real number and then on the other hand we see 0.999... = 1 because there is no smaller real number.

So math is saying from what I can read of it,

1. There is always a smaller real number and
2. there are occasions where there is no smaller real number.

An apparent contradiction!

Then why is the magnitude of the difference between 0.999... and 1 being set aside [ Is it because it is less than any real number? ]

Remember your answer previously:
Question: Why does 0.999... = 1?
Answer: Because there is no real number between them.

My response then was to suggest :

That based on the ration-al above, x = y because there is no letter between them. [ and if x= y because of it then a= z] [Mach's Principle starting to show through (if I am not mistaken) ie. 1+1 = 1]

 

What I am suggesting is that every possible co-ordinate location in 3 dimensional space has both 3 dimensional and zero dimensional attributes. The "zero point" both exists as a real 3 dimensional object and a zero dimensional object simultaneously. [ A zero point is not just a mere mental abstraction but is phenomenologically real and material because of it's 3 dimensional attributes]

So a volume of pure vacuumous space has 3 dimensions because it is a zero point that is both 3 and zero dimensional.

 

RP Still Sidestepping Common Sense for Invalid/Incorrect Mathematics-imho

RP, what label you want to a attach to the 0.999...is more mathematically illusionary, mental masturbation sidestepping of the primary issue, because, irrespective of any label you want to attach to the value 0.999..., if those dots are to represent an infinite processs/procedure, then, the only way to arrive a the next higher value 1, is as follows;

1) STOP the infinite process/procedure, i.e. remov/drop the dots, ergo 0.9--- that is what the wiki link does, and then,

2) round to the next higher value, and in this specific case, the next highe value is 1.

Common sense trumps invalid/incorrect mathematics, irrespective of extracurricular(?) lableing of stated finite and finite values

The rule of the game, is that to convert an infinite value to a finite value, then we must;

1) STOP the infinite process/procedure--- drop the dots ---, and then,

2) round to next higher value-- and that is is some manner ---what they actually do in the wiki link in the "infinite squence" section that your less complicated givens were almost word for word verbatim(?).

You refuse to answer my questions in regards to that specific wiki section of "infinite sequences" becuase you know that what I've stated, is for the most part, exactly what is happeing in that "infinite sequence" section where the dots "..." are mysterious just dropped, with no explanation, as to why their dropped.

I can tell you why there dropped, because that is the ONLY way to get to ifinite 0.999... = finite 1.0.. Common sense and no complex "mapping", or "infinite sequence" or "real" vs non-real, natural vs unatural, etc....processes and labeling will ever trump the common sense and valid explanations I've repeated given.

I am not ignoring finite 1.0 and infinite 0.999... and what those two represent. Your are ignoring the rational, logical, and common sense explanations I've repeated stated, and the questions I've asked you specifically in regards to this issue, except for the one place you made an effort to address my givens as stated, and that was you eventually, put the very complex "mapping" aside, and offered us a more simple alledged "proof" that I later found to be almost word for word, verbatim from the wiki link regarding 0.999...

Rp, please come back and talk to me, when your want to address my given rational, logic common sense explanations as stated and references to the wiki links "infinite sequences given formual equations, for starters.

If you cannot do that, then all I see you doing is a continual sidstepping of the majority of my givens.

r6

 

You're confusing smaller with smallest. What math is really saying is: given a real number r which is not zero

1. There is always a nonzero real number smaller (less than) than r
2. There is no smallest (least) nonzero real number

A least or smallest number is always less than any other number. If you confuse this lower bound with other numbers, you're fucked.

 

But they don't represent a process.
They represent numerals, an infinite string of 9s in this case.
The "..." does not mean "unless you write down an infinite string there will be trouble".

 

Why cannot you agree with a simple statement. In logic, mathematics or philosophy one seeks to lay out a convincing argument than a conclusion follows from the premises. If you never state what your premises are then the best you can hope for is a cadre of sycophants mindless repeating phrases they have no basis to understand.

You cannot argue with a definition and the real numbers have already been defined in a way that settles the question 0.999... = 1.000... so in order to have a real conversation, I am willing to contemplate a theory of number where 0.999... < 1.000... but, I assert such a theory of number is not generally useful in the same way that the real and complex numbers are useful.

So what concept of number exists where 0.5 is a number but 0.25 isn't? If you claim there is a smallest positive number smaller than 1, then what is it and what mathematical property makes it impossible to divide it by two. Does this also mean there is a largest positive number in your philosophy? If there is a largest positive number then what happens when you add 1 to it? Does it conform to all the properties that every other number has?

Links omitted, for if you had not shown interest in learning what was taught in one mathematical field, why would you care what was taught in another.

"..." cannot represent an infinite process unless, like Zeno, you are doing the process the wrong way. 0.999... is an abbreviated representation (just a posh word for name) for an infinite series (a sum over an infinite sequence) and while no one can add up a column of unrelated numbers, "0.999..." has a repeating pattern that means it has very low information content. And since that sequence converges "0.999..." is just a representation for a number.
To deny mathematicians can invent a self-consistent system of classifying and evaluating formally infinite series that only have finite information content is to commit the error of argument from personal incredulity and to fly in the face of over 2000 years of mathematical development. Indeed, the real numbers were invented for the very purpose of being about to talk about convergent sequences of rational numbers, be them continued fractions or decimals.

Incorrect use of the term of "mental masturbation" as the fields of logic, mathematics and philosophy exist as meaningful communications specifically contradicting the non-communicable virtues of "mental masturbation." That 1.000.. = 0.999.. is a consequence of defining the value of some infinite series in a consistent way. If you would prefer another way to do it, the burden is entirely on you to 1) point out that you subscribe to different axioms than those of the real numbers, 2) enumerate those axioms, 3) fairly and concisely explain just what the number 1.000... - 0.999... is and how your concept of number differs from the real numbers. To avoid your burden is to avoid differentiating your thoughts on the matter from "mental masturbation", got it?

Neither stopping nor rounding is in evidence in ANY of the proofs that 1.000... = 0.999... in the domain of real numbers. You simply didn't understand the demonstrations and probably did not understand what separates the concept of real numbers from rational numbers.

No it isn't. Even if what you said was true, which it is not, then at most you would have demonstrated that 1.000... = 0.999... has not yet been proven.

The process of education consists largely of replacing common sense ideas with better approximations of truth. To embrace your particular notion of "common sense" is to deny that conversation on the topic is useful, so all of your posts can be replaced with: "I disagree. Common sense!!!11!!" and lose nothing in translation. Naturally, I disagree and think the beginning of the conversation requires agreement on what axioms we are asserting.

Repetitive and untrue. The process of arithmetic you speak of is not the whole of mathematics.

Repetitive and untrue, for none of the demonstrations that 1 is not a higher value than 0.999... involve stopping at a finite number of 9's or rounding.

Repetitive and untrue, for common sense requires that one not deem a field incorrect until one has understood it. To prejudge that 0.999... < 1.000... when one has only tools that work with finite number of digits is patently unwise and thus an an affront to common sense. Why would common sense with no experience of infinity have anything to say about the infinite?

Super repetitive and untrue. Can you not see that you used this same baseless claim in this very same post? That's lazy even by the standards of professional bloviating know-nothings.

Grossly untrue.

You have given no explanations -- you simply have not had anything useful to say about infinite decimals, or the real numbers. You can't even point to specific illegitimate steps on Wikipedia or in my demonstrations. This speaks to me of your inability to understand the demonstrations. Your repeated invocations of your purported common sense speaks to me of your inability to articulate your thinking and the arrogance to believe that yours is the principled approach.

1.000... has the same number of digits as 0.999...

You need a theory of number to coherently communicate what you believe those expressions represent. The theory of real numbers say they represent the same number.

More like nakedly asserted.

None of which I recall because your misconceptions make your questions uninteresting and your lack of a clearly expressed theory of number leaves nothing of your viewpoint as a communicated message.

Citation requested. I certainly don't remember your post, my response or any wikipedia page that resembles closely what I wrote.

What givens?

 

Arfa--- "..."---dots---Infiinite---Process--Concetual---Metaphysical---Illusiinary

Arfa, I believe you are mistaken, on what appears to me, an obvious case of inherently infinite process/procedure. of creating and infinite set of 9'' if even if only[/B--- and the only way --- in an metaphysically conceptually, abstract sense.

Arfra, you yourself admits here and above--- if not briefly elsewhere -- admit/concede/acknowldege were dealing with an infinite set of 9's and infinite is can only be an eternally existent, aprori(?) process/procedure.

Ditto the above, Arfa because, from a rational, logical common sense and finite, thought process. the "..." not only, do represent an eternally existent, process/procedure of creating 9's--- if only if conceptually so ---, you, wiki and others cannot have infinite = finite, unless,;

1) STOP-- i.e remove ---the dots "..." from the equation--- as is done in the "infinite sequence" set on the wiki page, and then,

2) round 0.9 to next higher value of 1.

It is that simple, rational, logical and common sense.

And again, not only have none of you addressed those specicific comments above and elesewhere in many past posts by me, none of you in your camp, have addressed my given info regarding--- Origins given, wiki related info ---that I believe directly stems from my MS cal. caulator value 1 only;

1) after I divide 1 by 3--- MS cal. 1 / 3 = 0.333{n} * 3 = 1--- where {n} is meant to be the spaces on my cal. are filled

2) and not when I first I type in 0.333{n} and multiply,

3) and not when I first type in 0.333{n} and do the addition 3 times, as in #2, and # 3 process here above, I get value 0.999{n} and not value 1.

And none in your camp have addressed the other questions I have repeated asked about, why, on the cal. when we divide value 1, by numbers 9, 7, 6 and 3 all of the spaces on cal. are filled, and infinite process/procedure is inferred/implied,

and then why other numbers, of 10, 8, 5, 4 and 2 are divided into value 1 we get a finite number i.e. the spaces of cal. are not filled.

So, if those of you in your cam,p truly want to help others to understand where your coming from, then you could have the courtesy if not respect, to address these kinds of questions.

To be clear, I'm not here to learn mathematics or how to use a cal.. I'm here to discover and correlate our reality to cosmic truths in more grand theory of everything type way. And if along the way, I can learn some math or cal. skills as and aside that is ok also.

I nearly flunked general math in 9th grade some 30 or more years ago, so if you really think I'm going to truly grasp/understand/comprehend, continued repititions of natural vs real vs the many various terms posted here, by those who appear to have had months or years of math edcuation, is just silly on their part.

Address my comments/questions as specifically stated, as it a good and fair starting point(s) if you want me to understand/grasp/comprehend where you and those in your camp are coming from. Or as Wednesday Adams, states in the movie "Addams Family Values", "I want to Believe".

Please Register or Log in to view the hidden image!

r6

 

October 29: [post=3126046]"It's not about rounding it's about limits. Real numbers are about limits. Until you understand limits, you do not understand continuity. Until you understand continuity, you do not understand what separates the concept of real numbers from the rational numbers."[/post] Basically, I am saying that the proper field for understanding real numbers is analysis.
November 5: [post=3128828]"This invites the question, in what precise way do the Real Numbers differ from the Rationals?"[/post] I also answered that question.
November 12: [post=3131414]"0.999... = 1 because we are talking about the real numbers and"[/post] by definition 0.999... = 0.9 + 0.09 + 0.009 + 0.0009 + ... \(= \sum_{k=1}^{\infty} \frac{9}{10^k} = \sum_{k\in\mathbb{N}} \frac{9}{10^k}\). Just like 123 = 100 + 20 + 3 or 0.25 = 0.2 + 0.05, the decimal notation allows us to build up numbers with just a few symbols ( optional minus sign, optional decimal point, the digits 0 through 9 ). Thus decimal numbers are sums. And an infinite decimal number must be an infinite sum, which is hopeless if there is no shorter way to express it. However, if there is a repeating pattern, "..." means "and so on" or to continue the pattern forever. So a finite number of symbols represent an infinite decimal representation which describes an infinite sum whose value is a number.
November 12: [post=3131421]"The key point is finding out why people with no experience with formal proofs or knowledge of the definitions of the real numbers would assert an opinion here, and stick to that opinion with evidence from all authorities that they are in the wrong."[/post] Now that is an interesting question.
On [post=3135313]November 23[/post] I complained that rr6 has apparently given up on assigning any meaning at all to infinite decimal representations.
On [post=3136951]November 27[/post] I point out that rr6's own source claims 0.999... = 1 and I gave an example of what Jim Loy meant when I wrote "For example, if you wanted to find a solution for a decimal representation which solves "x = 0.9 + x/10" both 1.000... and 0.999... obviously solve it so there are two solutions in decimal representations and so the solution to a linear equation is not unique, but in real numbers there is only a unique solution, 1. So proves of existence aren't the proofs that are complicated, but proofs of uniqueness obviously are, as are proofs that count numbers of solutions." So rr6's would introduce his own authority and then whinge bitterly when I introduce elementary concepts from the set theory that authority mentions and that rr6 failed to understand.
On [post=3139674]December 2[/post] I comment on chinglu's endorsement of rr6's earlier post and point out that there is no argument being made.
On [post=3140417]December 4[/post] (also [post=3140482]here[/post]) I comment on the absurdity of claiming one's calculator allows exploration of the meaning of 0.999... because the calculator does not allow you to enter an unending succession of nines. (Although a clever programmer could construct one, it would cease to be a simple automation of arithmetic but would be a symbolic calculator handing some subset of algebra and analysis.)
One [post=3141009]December 5[/post] I remind rr6 that what he calls an infinite number is just a finite collection of symbols that represent an infinite decimal representation and does not remove the possibility that other finite representations for the same number exist. As long as rr6 concentrates on the representations and not the numbers, there is no way to argue about the numbers. rgb(0,0,1) and #0000ff and blue and koniro might all be names for the same color, but until you have a theory of color you cannot begin to make that argument. Likewise, rr6 still lacks a theory of number expansive enough to allow discussion of 0.999... because he stops thinking when he sees the digits go on forever.
On [post=3141992]December 7[/post] I support my claim that some people are dubious of the claim "0.999... - 0.0999... = 0.9".
On [post=3142050]that same day[/post] I point out that a mistake was made in using a calculator.
On [post=3143183]December 11[/post] I point out rr6's distortions, repetitiveness, and naked assertion without communicable principles. I asserted I did not know of the exchanges he described and going back over my posts to October I have not found them as he described. I'm guessing he's calling the reasoning in the December 7 post as similar to one he saw on Wikipedia, but then he hasn't been paying attention to these threads as it has been there in one form or another since at least [post=3125752]post #35 on October 28[/post].

I continue to see nothing attractive about rr6's embrace of ignorance and "common sense" over deliberate and formal arguments from an axiomatized theory of number, especially the real numbers.

 

I believe I am not mistaken, so which one of us is mistaken? Why does an infinite string have to be the result of a process? Does something happen if you assume there is no process?

In formal language theory, strings are just arbitrary compositions of characters or letters (i.e. abstract symbols), so there is a set containing all the finite length strings for some alphabet. But this set must contain an infinite number of strings if there is no restriction on their length, other than it must be finite.
This follows from the fact that, in the natural numbers, there is no largest number; there will always be strings with greater length than some n.

Strings can be considered as letters under concatenation, and that concatenation is a process. However, given a string composed from some alphabet, can you decide how it was composed? It seems that the process or operation of concatenation, if there is one, has no inverse. You can't "unconcatenate" an arbitrary length string because you don't know if the last character was the result of a concatenation--there is no information about the order the characters were added to the string.

 

беседка

Из вашего сообщения не очень понятно, чего вы в итоге хотите добиться… Просто повысить самооценку или найти цель вашей жизни или еще что-то? Хотя, соглашусь, семинар о мотивации – не самый худший вариант. Попробуйте, это точно не помешает. Расскажете и нам, что да как

Please Register or Log in to view the hidden image!

 

It's Greek to me!

 

Finite 1.0 vs Infinite 0.999... Process/Procedure

Once again, and as so many time with yourserlf or various others here, you have not and do not directly address my givens, of rational, logical and common sense that are relatively simple explanations to what is obvious to those who do not have their ego's invested in invalid mathematics and do not have grudge against r6.

You are mistaken and my last post, is all that you or anyone else's needs to the truth of this issue regarding finite 1.0 not equal to infinite 0.999... process/procedure. imho

Arfa, you obviously do not understand/grasp/comprehend what infinite means i.e. if and when can grasp that in regards to 0.999... were considering a mathematically infinite process/procedure, the perhaps you can begin to address my previous comments on this issue in the last reply--- that you and others in your camp for the most part, still do not directly address ---come to rational, logical and common sense conclusion, as I have from the very beginning of my posting in this thread.

0.999... = infinite process and it is the dots ".." that specifically represent an infinite process. RP likes to talk about knowing the rules and playing by the rules yet he and you, want to ignore the basic facts of infinite process/procedure.

There is only one other infinity-- associated with and refereneceable our finite Universe of ---and that is eternally existent, macro-micro infinite non-occupied space, that, exists beyond our finite occupied space we call Universe. My guess is that like Origin--- and some other alledged intellectual types around here ---you cannot grasp an eternally existent, non-occupied space, beyond or finite occupied space Universe.

Be clear here, the true, non-occupied space eternally existent and not a process/procedure, whereas an infinite process/procedure is also eterntally existent, in that process/procedure, whether having to do with an occupied space and metaphysical abstract concept is eternally existent or invokes a process/procedure that can only be be associated with an infinite process/procedure of creating 9's, 1'2, 3's zeroes or any symbol.

An infinite set, irrespective of physical somethingness type bits/info/data etc or a metaphysically abstract concept, inherently-- if not apriori ---are a process/procedure or, invoke an infinite process/procedure.

String, nonstring, real, non-real, natural un-natural etc.. type labeling, is aside to finite( terminal ) vs infinite process/procedure( non-terminating ). imho Ergo for most part irrelevant to the main/basic issue.

Your "string" approach is just more of the same attempts by you to avoid/ignore the facts of mathematically illusionary, mental masturbation, process/procedure, that you invoke to get around addressing the majority of my comments as stated for the factual truths they represent. imho

This following of yours below is more of the same, ergo ditto my last post to you and the comments here. I repeat again for you, RP, Origin etc...

finite value 1.0 does not--- and never will ---equal infinite 0.999... unless;

1) the dots "..." representing and infinite process/procedure are dropped/removed-- just as the do in the "infinite sequence" on the wiki link, and,
2) we round 0.9 to the next higher value 1.0.

This is not rocket science and certainly does not require years of mathematics to grasp/understand/comprehend.

What I see years of mathematics allowing for is many more ways of expressing mathematically illusionary, mental masturbation. I'm not opposed to that, kind of fun mental fun nor would I want to ever take that kinda of fun from you, others, or myself, however, please address my comments directly as I've stated them, if you want to continue with a rational, logical and common sense conversation.

Thank you for any sincere and earnest attempts to meet my given comments as stated.

r6

 

Does that mean it will take me forever to cut a pie into thirds?

 

Yet More Ego Death Need To address Majority of My Comments

GM, when you want address my comments as stated, then please come back and talk to me. Otherwise, no need to attempt conversation.

Finite 1.0 does not equal finite 0.999... process/procedure. Common sense not neccessarily relevant to a common pie.


r6:

 

Don't make it harder than it needs to be. It's an anomaly inherent in base 10 - nothing more.

 

Because you don't have any "givens" that I or so many others here can understand.

It seems that no matter how many explanations there are, you can't understand any of them, your brain can't handle it.

What you do seem to understand is that, given an infinite string of characters does exist, you can't write it down.
Instead, you write (something like) "0.999...", which represents an infinitely repeating decimal, a string of numerals, so you don't have to write down all the digits.

When are you going to start trying to understand what the word "representation" means? If you can't be bothered, just say so.
Or perhaps you can recognise that saying 1/3 is not the same as 0.333... because you can't write down an infinite number of 3s, is just stupid.