I didn't say any did. I imagine some have, or maybe not. That's not relevant because it doesn't mean that it's wrong for me to. — Metaphysician Undercover

So it's just you against the world of mathematics. If that can't convince you that your views are the problem, not mathematics, then I don't think anything will.

Well, if you think that I haven't already seen how math works, then you're wrong. — Metaphysician Undercover

What I mean is something like this:

I haven't seen that definition. — Metaphysician Undercover

I have no idea what you're talking about — Metaphysician Undercover

I've never seen any such definition of "division — Metaphysician Undercover

...

I've made from what I've seen is that a healthy dose of skepticism — Metaphysician Undercover

I can't see how this makes any relevant point. You've just demonstrated another smoke and mirrors method to hide the fact that there is a remainder. — Metaphysician Undercover

...no; blaming the mathematicians for your not finding the remainder is not healthy skepticism.

Where's the problem? — Metaphysician Undercover

You didn't answer the question, and I think the reason you didn't is because the question doesn't make sense. That carries over to your previous claim that 0.111... has a remainder.

I don't deny any of this, that's how math works, conventions are followed, and that's what convention has us call "dividing". — Metaphysician Undercover

...so... 0.(1)=1/9?

The question is on what principles do we say that the conventions are right or wrong. — Metaphysician Undercover

Logical deduction based on the axioms.

Do you agree that for any particular way that an action is carried out (an action being the means to an end), in this case a mathematical operation, it is possible that there might be a better way? So even if — Metaphysician Undercover

Sure, why not? Something like: 97*104=97+(100-3)(100+3)=97+10000-9=10088?

You seem to think that mathematics works because people dream up random axioms, then the axioms are applied, and voila, mathematics works. — Metaphysician Undercover

That's quite a fair description of Finite Geometry.

I start with the fundamental principle of "pure mathematics", which states that — Metaphysician Undercover

No it doesn't:

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. — Pure mathematics (wikipedia)

Part of the reason I post this definition (and rearrange this) is context for the response below:

I think that mathematics works because people design the axioms so as to be applicable to the real world. — Metaphysician Undercover

Examples of pure mathematics becoming useful (exact opposite of what you just said) here.

You start with the opposite (and what I claim backwards) position, that the fundamental "unit" is divisible any way one can imagine, an infinity of different ways. — Metaphysician Undercover

I start (frontwards) with the axioms.

First, I will argue that this annihilates pure mathematics — Metaphysician Undercover

I wouldn't go that far... it's just useful.

Second, I will argue that it leads you to believe, as you've demonstrated in this thread, that any object is divisible in any way imaginable, i.e. an infinity of different ways. — Metaphysician Undercover

Interesting... you claim that I've demonstrated that I believe any object can be divided an infinity of ways, and yet, in the same post, you quote me as saying there's about 10⁸⁰ atoms in the universe. Ladies and gentlemen... MU's healthy skepticism!

This demonstrates my point. — Metaphysician Undercover

No clue what you mean by that, so I'll just generically offer that arithmetic encoding works based on the same concepts we're discussing here... converting fractions (representing ranges of relative symbol frequencies) into placement systems (representing the coding).

Here's the problem. — Banno

Having an idea which is inconsistent with the conventional demonstrates "a problem". I agree.

So it's just you against the world of mathematics. If that can't convince you that your views are the problem, not mathematics, then I don't think anything will. — Michael

I told Banno already in this thread, I do not believe in mob rule. I'm an individual, and what makes an individual an individual, is to not be identified as a part of a group. Therefore it's natural that I be different from the others. This is what makes "one" fundamentally different from "half of two". The former identifies the thing being spoken about as an individual, the latter identifies the thing being spoken about as a member of a group. The mistake which many people partaking in this thread make, is that they think that because "one" is equal to "half of two", in mathematical applications, they both mean the same thing. But I believe myself to be "one", an individual, and my identity is not based in being a member of that group. Therefore I need not partake in their mistake.

..no; blaming the mathematicians for your not finding the remainder is not healthy skepticism.
Where's the problem?
— Metaphysician Undercover
You didn't answer the question, and I think the reason you didn't is because the question doesn't make sense. That carries over to your previous claim that 0.111... has a remainder. — InPitzotl

You seem to always misunderstand, or misrepresent what I say. I didn't ever blame the mathematicians for "not finding the remainder", I blamed then for hiding the fact that there is a remainder. And this is clearly evident from what is expressed in the op, when it is asserted that the follow is an accurate representation: 1/9=.111..., and .111...X9=1. And, I suggested that mathematicians ought to respect the fact that it is impossible to divide one by nine equally, instead of using smoke and mirrors tactics to make it appear like this impossible thing is possible. To me, it just makes the mathematicians look bad, more like mathemagicians.

No it doesn't:
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications.
— Pure mathematics (wikipedia)
Part of the reason I post this definition (and rearrange this) is context for the response below:
I think that mathematics works because people design the axioms so as to be applicable to the real world.
— Metaphysician Undercover
Examples of pure mathematics becoming useful (exact opposite of what you just said) here. — InPitzotl

You're ignoring a key part of your Wikipedia definition, "These concepts may originate in real world concerns...".

No it doesn't: — InPitzotl

If your desire is to dispute what I have stated as a fundamental principle of "pure mathematics", which you have defined through your wiki quote as "the study of mathematical concepts independently of any application outside mathematics", then to simply assert "no it's doesn't" is completely insufficient. But I don't see any point to trying to dispute any stated fundamental principle of pure mathematics, without having a good reason. That such and such a principle is not supported by such and such application as is your demonstrated mode of arguing, is not a good reason. Your reason must be based in logic, as the reason for my stated principle (which you dismissed for no reason) is.

Interesting... you claim that I've demonstrated that I believe any object can be divided an infinity of ways, and yet, in the same post, you quote me as saying there's about 1080 atoms in the universe. Ladies and gentlemen... MU's healthy skepticism! — InPitzotl

Your logic is way off bud. This is what you've argued. An object, represented as "1", can be divided in an infinity of different ways. Here's one of that infinity of different ways that an object might be divided. Now you seem to be claiming that by providing one possible way out of an infinity of possible ways, you have demonstrated that you do not really believe that an object might be divided in an infinity of different ways.


Hey, that looks like me, trying to get these guys to see their mistakes. At least the exercise is good. Now, you need a picture of two, one going each way, the other will be InPitzotl trying to get me to see my mistakes. Isn't philosophy fun?

I told Banno already in this thread, I do not believe in mob rule. I'm an individual, and what makes an individual an individual, is to not be identified as a part of a group. Therefore it's natural that I be different from the others. This is what makes "one" fundamentally different from "half of two". The former identifies the thing being spoken about as an individual, the latter identifies the thing being spoken about as a member of a group. The mistake which many people partaking in this thread make, is that they think that because "one" is equal to "half of two", in mathematical applications, they both mean the same thing. But I believe myself to be "one", an individual, and my identity is not based in being a member of that group. Therefore I need not partake in their mistake. — Metaphysician Undercover

Mathematicians aren't making mistakes. $\frac{1}{9}$ is a number and $0.999...=1$. If you don't understand this then you don't understand mathematics. I suggest you study more before wildly claim that mathematics is contradictory and derived from false premises.

You seem to always misunderstand, or misrepresent what I say. I didn't ever blame the mathematicians for "not finding the remainder", I blamed then for hiding the fact that there is a remainder. And this is clearly evident from what is expressed in the op, when it is asserted that the follow is an accurate representation: 1/9=.111..., and .111...X9=1. And, I suggested that mathematicians ought to respect the fact that it is impossible to divide one by nine equally, instead of using smoke and mirrors tactics to make it appear like this impossible thing is possible. To me, it just makes the mathematicians look bad, more like mathemagicians. — Metaphysician Undercover

It's been mentioned before but it's worth mentioning again.

$\frac{1}{9}$ in base 10 is equal to $\frac{1}{10}$ in base 9, so $0.111...$ in base 10 is equal to $0.1$ in base 9. It's divided equally.

Explain to me what you disagree with.

$9_{10}$, $10_9$, and Roman numeral $\mathrm{IX}$ all refer to the same number. You're getting lost in what the symbols look like.

I told Banno already in this thread, I do not believe in mob rule. I'm an individual, and what makes an individual an individual, is to not be identified as a part of a group. Therefore it's natural that I be different from the others. This is what makes "one" fundamentally different from "half of two". The former identifies the thing being spoken about as an individual, the latter identifies the thing being spoken about as a member of a group. The mistake which many people partaking in this thread make, is that they think that because "one" is equal to "half of two", in mathematical applications, they both mean the same thing. But I believe myself to be "one", an individual, and my identity is not based in being a member of that group. Therefore I need not partake in their mistake. — Metaphysician Undercover

This is madness.

Mathematicians aren't making mistakes. 1/9 is a number and 0.999...=1. If you don't understand this then you don't understand mathematics. I suggest you study more before wildly claiming that mathematics is contradictory and derived from false premises. — Michael

And this should be the last word.

I didn't ever blame the mathematicians for "not finding the remainder", I blamed then for hiding the fact that there is a remainder. — Metaphysician Undercover

So, the answer to the question of what the remainder of 0.(142857) is, is that there is in fact a remainder, it's clearly evident, mathematicians ought to respect that it is impossible to divide one by seven, and mathematicians are using smoke and mirrors to hide the fact that there is a remainder?

You're ignoring a key part of your Wikipedia definition, "These concepts may originate in real world concerns...". — Metaphysician Undercover

You're ignoring a key part of your Wikipedia definition: "may", not to mention the bolded part.

... then to simply assert "no it's doesn't" is completely insufficient. — Metaphysician Undercover

Multiple examples provided in link of utility following math.

An object, represented as "1", can be divided in an infinity of different ways. — Metaphysician Undercover

Of course. But the object represented as 1 is a mathematical object, not an onion. Nobody is claiming you can chop an onion into infinite pieces. But we can subdivide 1 indefinitely; there's no "math-atom" we run into. We can apply arithmetic coding for example to encode 10G of text, which means we can generate an arbitrary precision number with on the order of ten's of G's of symbols. So it turns out, you're the one confused, not me; you think 1 is a pizza or an onion. It's not. It's a number.

Regardless, I think I have found the core issue here. Your theories of where math originates suggests you think math is about just physical objects and, since it's not, you find counterexamples. But rather than taking this as being proven wrong, you double down, positing that mustachioed mathematicians conspire to lie about the nature of physical objects to themselves and others... something like that?

Mathematicians aren't making mistakes. 1919 is a number and 0.999...=10.999...=1. If you don't understand this then you don't understand mathematics. I suggest you study more before wildly claim that mathematics is contradictory and derived from false premises. — Michael

It's not the case that I don't understand, it's the case that I understand but disagree. You've been at tpf long enough to know that this is common, people understand but disagree. Why would you think that principles of mathematics have special status such that if you understand them you'll necessarily agree with them?

1919 in base 10 is equal to 110110 in base 9, so 0.111...0.111... in base 10 is equal to 0.10.1 in base 9. It's divided equally. — Michael

I don't see that your making a point. Base 9 is going to have its own numbers which are impossible to divide into each other. So this just emphasizes my point, what can and cannot be divided is dependent on the application.

Of course. But the object represented as 1 is a mathematical object, not an onion. Nobody is claiming you can chop an onion into infinite pieces. — InPitzotl

Great, now we're making some progress. You see that your pizza analogy is completely irrelevant, and we are talking about dividing the number represented by "1", not some physical object. Is the number represent by "1" a single unit or a multiplicity of units? Since it is not a multiplicity, as it is defined as a single, then how do you propose that it might be divided. You cannot take a knife or a pizza roller to it. What do you think, that you can imagine that it's really made of parts, a multiplicity, and you can divide it according to those parts? Of course that image would contradict the definition. So I really want to know what principles you are applying to divide 1, because you seem so insistent that you can divide it however you please.

Great, now we're making some progress — Metaphysician Undercover

Oh boy, made it to 400 posts!! That's progress! :nerd:

we are talking about dividing the number represented by "1", not some physical object... how do you propose that it might be divided. You cannot take a knife or a pizza roller to it. — Metaphysician Undercover

With the Platonic form of a pizza roller??

It's not the case that I don't understand — Metaphysician Undercover

It is the case that you do not understand.

how do you propose that it might be divided. — Metaphysician Undercover

By just doing so. I gave you an example, which is quite relevant, to help you understand. You ignored it. But it's still there. If you're going to ignore what I say, I'm not going to pretend we're having a conversation.

If you're genuinely interested in this:

So I really want to know what principles you are applying to divide 1 — Metaphysician Undercover

...then you need to understand that example.

I don't see that your making a point. Base 9 is going to have its own numbers which are impossible to divide into each other. So this just emphasizes my point, what can and cannot be divided is dependent on the application. — Metaphysician Undercover

It doesn't have its own numbers. It has its own numerals.

1111 in binary, F in hexadecimal, and XV in Roman numerals aren't different numbers to 15 in decimal. They're the same number.

$1111_2 = F_{16} = \mathrm{XV} = 15_{10}$

Great, now we're making some progress. You see that your pizza analogy is completely irrelevant, and we are talking about dividing the number represented by "1", not some physical object. Is the number represent by "1" a single unit or a multiplicity of units? Since it is not a multiplicity, as it is defined as a single, then how do you propose that it might be divided. You cannot take a knife or a pizza roller to it. What do you think, that you can imagine that it's really made of parts, a multiplicity, and you can divide it according to those parts? Of course that image would contradict the definition. So I really want to know what principles you are applying to divide 1, because you seem so insistent that you can divide it however you please. — Metaphysician Undercover

If we can talk about dividing a single cake into nine equal slices then we can talk about each slice being one-ninth of a cake, and if we can talk about each slice being one-ninth of a cake then we can talk about $\frac{1}{9}$.

Here's a cake, it's 90cm x 10cm x 10cm. I cut it into nine equal slices of 10cm x 10cm x 10cm and share it between my friends.

Does it make a difference if I describe the measurements in cm rather than in michaelmetres, where 1 michaelmetre = 90cm, and so the cake is 1michaemetre x $\frac{1}{9}$michaelmetre x $\frac{1}{9}$michaelmetre? Why?

And does it make a difference if I describe it as 1 cake and $\frac{1}{9}$ of a cake rather than just 1 and $\frac{1}{9}$? Why?

By just doing so. I gave you an example, which is quite relevant, to help you understand. You ignored it. But it's still there. If you're going to ignore what I say, I'm not going to pretend we're having a conversation. — InPitzotl

I ignored your example for two reasons. It doesn't answer my question, and it's false. First, my question concerns the principle by which you divide a number, not the act by which you represent this, which is what your example describes. Second, your example is false and invalid because "ten's of G's of symbols" is not the same as infinite.

I have no doubt about your capacity to represent "1" as being divided, we do this with 1/2, 1/3, 1/4, etc., and with .5, .3, .25, etc.. And this is what your example is, an example of a machine making a representation. What I am doubtful of is the "principle", the rule, which says that "1" is a number which can be divided. We can say, and represent whatever we want, but what I want to know about is the rule which makes the representation a valid representation. What rule makes the mathematical object represented by "1" divisible?

Here's another related question. Why is 1 not a prime number? I would say that 1 is excluded from the list of prime numbers by designating that it is something other than a number. If this is the case, then what is the relationship between 1 and all the numbers, 2,3,4, etc.? They are distinctly different types of mathematical objects. And back to my original question, if 1 is something other than a number, let's suppose it's called a "unit", on what basis can the unit be divided? That's the rule I'm asking for.

If we can talk about dividing a single cake into nine equal slices then we can talk about each slice being one-ninth of a cake, and if we can talk about each slice being one-ninth of a cake then we can talk about 19 — Michael

I have no problem talking about 1/9 in that application. In applications, if there are issues with similar division problems we simply round things off (like with pi, and some square roots, and other division problems), or we say "I can't do the task I'm being asked to do" (like if you asked me to cut the cake into three million equal pieces).

Where the problem is, is in what fishfry called pure math, which is when we are working solely with abstract concepts. In abstract math the thing being divided into nine parts is the "number" one, or the "unit" one, and this division is said to give a "number" with the value of "0.111...". This is where I see a problem , as I've tried to explain.

Where the problem is, is in what fishfry called pure math, which is when we are working solely with abstract concepts. In abstract math the thing being divided into nine parts is the "number" one, or the "unit" one, and this division is said to give a "number" with the value of "0.111...". This is where I see a problem , as I've tried to explain. — Metaphysician Undercover

And as we've tried to explain, it isn't a problem.

$\frac{1}{10_9} = 0.1_9$

Is there a problem with the number $0.1_9$?

I can't say that I completely understand your representation so I can't give an honest answer here. Perhaps you could explain better.

I can't say that I completely understand your representation so I can't give an honest answer here. Perhaps you could explain better. — Metaphysician Undercover

It's $\frac{1}{10} = 0.1$. I'm just specifying that I'm using base 9 numerals instead of the traditional base 10. Although in this case it isn't strictly necessary as $\frac{1}{10} = 0.1$ in every base.

Yeah sure, that's fine, I addressed that issue already.

Yeah sure, that's fine — Metaphysician Undercover

And do you understand what bases are? Do you understand that $9$ in base 10 and $10$ in base 9 are the very same number? To explain that, let's count the number of apples in this picture:



If I were to count the number of apples in base 10 I would count "1", "2", "3", "4", "5", "6", "7", "8", and "9".

If I were to count the number of apples in base 9 I would count "1", "2", "3", "4", "5", "6", "7", "8", and "10".

$9$ in base 10 and $10$ in base 9 are the very same number: the number of apples in this picture. And this is true even if we're not counting apples; it's true when we're doing "pure" maths.

It doesn't answer my question, and it's false. First, my question concerns the principle by which you divide a number, not the act by which you represent this, which is what your example describes. Second, your example is false and invalid because "ten's of G's of symbols" is not the same as infinite. — Metaphysician Undercover

So we can't divide a pizza into 9 slices because the slices don't weigh the same, and we can't divide 1 into 9 because 9 isn't infinity.

Sorry MU, but I'm not interested in playing Calvinball with you.

The problem is in dividing "1". In different representations the problem will appear in different ways, as I explained before. The manner of representation is a matter of application, and to show that the problem takes a different form when we change from this application to that, does not make the problem go away.

The problem is in dividing "1". — Metaphysician Undercover

There is no problem. If you accept that $9$ in base 10 and $10$ in base 9 are the very same number (which they are), then you must accept that $\frac{1}{9}$ in base 10 and $\frac{1}{10}$ in base 9 are the very same number (which they are).

I don't see how this discussion of pizzas or apples is relevant. You're just distracting from the topic.

Of course. But the object represented as 1 is a mathematical object, not an onion. — InPitzotl

This is the subject, 1 as a mathematical object, not pizzas.


The problem is in the supposed equivalence between the fraction and decimal representation. Do you understand, that by moving to base nine, you are actually removing the possibility of dividing one by nine, because nine has been excluded as a number? So all you are doing is obliging me, giving me what I asked for, making one divided by nine impossible. But that's the point of my argument in the first place.

The real problem though, is that one divided by numerous other numbers is also impossible. To demonstrate that you have actually dealt with this problem, show me the decimal representation of 1/7 and 1/8 in base nine. If there is no problem, then we can proceed to the other fractions in base nine as well, just to confirm that there are no such problems in base nine.

If there is still a similar problem in base nine, we might try base eight, and if a problem presents itself we could move to base seven etc.. Or, we could skip all that and just start at base two. Can you show me how to divide 1 in binary?

Or, we could skip all that and just start at base two. Can you show me how to divide 1 in binary? — Metaphysician Undercover

$\frac{1}{10} = 0.1$.

This is equivalent to

$\frac{1}{2} = 0.5$ in base 10.

The problem is in the supposed equivalence between the fraction and decimal representation. Do you understand, that by moving to base nine, you are actually removing the possibility of dividing one by nine, because nine has been excluded as a number? — Metaphysician Undercover

Nine hasn't been excluded as a number. There are nine apples in the picture above regardless of what base you use to count them. This is exactly what I mean by saying that you don't understand maths.

You're just getting confused by what the numerals look like. Whether I use $0.111...$ or $0.\overline{1}$ or $\begin{align}\sum_{n=1}^{\infty}\frac{1}{10^n}\end{align}$ or $0.1_9$ or $\mathrm{MichaelI}$, I'm talking about the same number.

Despite what you seem to be a saying, a number doesn't have to be representable as a terminating base 10 decimal. There are an infinite number of numbers that can't be represented this way. Some can be represented as terminating decimals in other bases. Others can't be represented as a terminating decimal in any base. And they're all still numbers.

You're just distracting from the topic. — Metaphysician Undercover

Oh that's rich.

my question concerns the principle by which you divide a number — Metaphysician Undercover

^^ 1/9 is that thing.

is not the same as infinite. — Metaphysician Undercover

^^ distraction.

FYI, under the rationals, there's no such thing as 1/infinity.

Meta has revealed that one cannot subtract from a whole. Subtraction only works if you have more than one individual. And division leads to the heresy of fractions. — Banno

:D We're no longer talking mathematics. (An acute case of ∞-phobia?) Maybe we could call it metamathonomy or something.

There is no such thing as one half, unless it is a half of something whatever/anything (hence an abstract quantity) — Metaphysician Undercover

As an aside,

The criteria for truth is honesty. — Metaphysician Undercover

... doesn't seem right. You can be both honest and wrong.

So, @Metaphysician Undercover,

• A ninth is a contradiction concealed by smoke and mirrors (in metamathonomy)? (A ninth kilometer, a ninth of three dozen, ...?)
• And there's a largest natural number? And a smallest positive rational number? (The division procedure somehow becomes invalid for "too small" numbers?)
• And ...?

Your replies are vague and hard for me to understand. That the procedure proves what the procedure is supposed to prove is not the issue. — Metaphysician Undercover

Mentioned procedure just writes 1/9 as 0.111... (in the common decimals). You have to understand what you're objecting to first.

:D We're no longer talking mathematics. — jorndoe

I don't think you will succeed in showing an inconsistency in @Metaphysician Undercover's mathematics that he will recognise. Rather, we have a choice between two mathematics.

In one, we can divide 1 into fractions, and hence 0.999...=1, and infinities can have differing cardinalities, and i² = -1, and we can use maths to do velocity, navigation, electronics, engineering and so on.

In the other, 1 cannot be divided. And thats all.

In the end it is the poverty of Meta's mathematics that we leave behind.

This is equivalent to

12=0.512=0.5 in base 10. — Michael

This is not true at all. If 1 is divisible, its divisibility is different in base nine from what it is in base ten. That's why I asked you to look at base two as an example, because it becomes very clear there, that if one is divisible, changing the base changes the divisibility of one. Therefore, if fractions are numbers we cannot transpose these numerical values from one base to another in the way that you propose.

Nine hasn't been excluded as a number. There are nine apples in the picture above regardless of what base you use to count them. This is exactly what I mean by saying that you don't understand maths. — Michael

Again, examples of objects only confuse the issue, because we a talking about the numbers themselves. And "nine" has a different meaning in base nine from what it has in base ten, especially if we allow that one is divisible, so your example is just an example of equivocation.

Despite what you seem to be a saying, a number doesn't have to be representable as a terminating base 10 decimal. There are an infinite number of numbers that can't be represented this way. Some can be represented as terminating decimals in other bases. Others can't be represented as a terminating decimal in any base. And they're all still numbers. — Michael

Due to the fact that what can and cannot be represented is dependent on the mode of representation, this claim employs equivocation in the term "numbers". This is the reason why we use different numbering systems, natural, rational, real, for example, and that "number" has a different meaning in each of these systems, just like the base unit "one" has a different meaning in each base system, if "one" is divisible. To claim that "they're all still numbers" is just a matter of equivocation, similar to saying that all uses of "right" refer to the same type of thing a right, unless you can demonstrate a definition, or category of "number", which encompasses all the different numerical systems. Under this definition of "number", we could say that they are all numbers without equivocation. But what I've been trying to demonstrate, is that if we allow that one is divisible, such a definition will prove to be impossible. Your example of using different bases should make this very clear to you, especially if you consider base two.

You can be both honest and wrong. — jorndoe

I touched briefly on the lack of an acceptable criteria for right and wrong on this thread, in my discussion with Banno and Michael. They seem to think that to act according to the convention is to be right. If this were the case, there would be no sense in discussing the op, because it expresses the convention, and asks if this is right. If we define right and wrong as consistent with the convention, there is nothing to discuss here.

So in order to have anything to discuss on this topic we need to get beyond the idea that the convention is necessarily right. Therefore we must define "right" in relation to something else. I proposed that we define it in relation to what one truly believes. This allows not only that the conventions might be wrong, but also that it would be wrong to use the conventions deceptively. One of the problems with defining right and wrong in relation to conventions is that it makes it extremely difficult to demonstrate that a person using conventions deceptively is actually wrong.

Mentioned procedure just writes 1/9 as 0.111... (in the common decimals). — jorndoe

Right, this is the convention which I object to as a convention which facilitates dishonesty. That dishonesty is demonstrated when people who know that ".999..." does not means the same thing as "1" insist that it does.

This is not true at all. If 1 is divisible, its divisibility is different in base nine from what it is in base ten. That's why I asked you to look at base two as an example, because it becomes very clear there, that if one is divisible, changing the base changes the divisibility of one. Therefore, if fractions are numbers we cannot transpose these numerical values from one base to another in the way that you propose. — Metaphysician Undercover

It is true. You just don't understand maths.

If $9_{10}=10_{9}$ then $\frac{1}{9_{10}}=\frac{1}{10_9}$

If fractions bother you then we can use exponents instead.

If $9_{10}=10_{9}$ then $9_{10}^{-1}=10_{9}^{-1}$

Due to the fact that what can and cannot be represented is dependent on the mode of representation, this claim employs equivocation in the term "numbers". This is the reason why we use different numbering systems, natural, rational, real, for example, and that "number" has a different meaning in each of these systems, just like the base unit "one" has a different meaning in each base system, if "one" is divisible. To claim that "they're all still numbers" is just a matter of equivocation, similar to saying that all uses of "right" refer to the same type of thing a right, unless you can demonstrate a definition, or category of "number", which encompasses all the different numerical systems. Under this definition of "number", we could say that they are all numbers without equivocation. But what I've been trying to demonstrate, is that if we allow that one is divisible, such a definition will prove to be impossible. Your example of using different bases should make this very clear to you, especially if you consider base two.

The symbols don't matter. How many apples are in this picture?



$9$ if I'm using base 10. $10$ if I'm using base 9. $\mathrm{IX}$ if I'm using Roman numerals. $\frac{90}{10}$ if I'm using base 10 fractions. $٩$ if I'm using Arabic. $九$ if I'm using Chinese. Different symbols, same number.

just like the base unit "one" has a different meaning in each base system

It doesn't. 1 is the same in every base.

I'm giving up now. Clearly nothing I can say is going to teach you. Go take a math class. Maybe a professional will have better luck getting through.