, so neither convention nor honesty works...? Now what?

Meanwhile, we all understand that half a dozen is six, and what's meant by a third of the area of the lawn, so that works fine (presumably for you as well). But of course, we don't speak of a ninth of dislike for pizza with pineapple, at least not without some further clarification.

have to round off pi — Metaphysician Undercover

to round off at some point, carry it to two decimals, three, whatever — Metaphysician Undercover

if there are issues with similar division problems we simply round things off (like with pi, and some square roots, and other division problems) — Metaphysician Undercover

That is, there's something to round off. Seems you've already presupposed what you want to deny. (The division procedure isn't really the problem here.)

Maybe a professional will have better luck getting through. — Michael

:rofl:

If 1 is divisible, its divisibility is different in base nine from what it is in base ten. — Metaphysician Undercover

Call me crazy, but why isn't a base 9 and a base 10 representation of the same number a base 9 and base 10 representation of the same number?

If fractions bother you then we can use exponents instead. — Michael

It is the decimal equivalence which you are claiming that is what bothers me. That is the issue of the thread. You said 1/10 in base nine can be represented as .1 in base nine, and this is equivalent to .111... in base ten. I don't think you can represent a solution to a division problem in base nine, as a decimal, (.1), because decimals are proper to base ten, and that would be to conflate base nine and base ten representations. So your argument here, is nonsensical:

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

And you continue on with this nonsense, as if 1/10 in base nine could be represented as .1, but 1/10 is only .1 in base ten.

Meanwhile, we all understand that half a dozen is six, and what's meant by a third of the area of the lawn, so that works fine (presumably for you as well). But of course, we don't speak of a ninth of dislike for pizza with pineapple, at least not without some further clarification. — jorndoe

As I've already explained, I have no problem with these divisions in application. We know that the representation of "one" in the case of "one dozen" is a multitude of twelve, and therefore can be divided accordingly. We know that the area of a lawn is going to be represented in a multitude of square meters or some such thing, and therefore can be divided accordingly. And we know that "one octave" consists of a multitude of frequencies which can be divided The problem is when we talk about "one" in the abstract sense, as a "number", or "unit", in which case it is defined as a single, and not as a multitude which can be divided. "One" only submits to being a multitude when it is applied to a thing which can be divided.

That is, there's something to round off. Seems you've already presupposed what you want to deny. (The division procedure isn't really the problem here.) — jorndoe

As seems to be the case often, you don't seem to be able to express your point very well, and you leave me wondering what you're talking about.

Call me crazy, but why isn't a base 9 and a base 10 representation of the same number a base 9 and base 10 representation of the same number? — InPitzotl

It is only different if the fundamental unit "one" is considered to be divisible. Take a look at the divisibility of "1" in base two, and compare it with the divisibility of "1" in base ten, for a good example of how the divisbility of "1" changes from one base to another. If each unit in a base ten number is divisible in a particular set of ways, and each unit in a base nine number is divisible in a different set of ways, then we cannot say that the representation is of the same number. But If the premise is that the base unit. "one" is not divisible, then there is nothing different about the number being represented in the different base representations.

I don't think you can represent a solution to a division problem in base nine... — Metaphysician Undercover

Additional ad hoc hypothesis used to prevent falsification of the core assumption.

Take a look at the divisibility of "1" in base two, and compare it with the divisibility of "1" in base ten, for a good example — Metaphysician Undercover

Lousy example. The number's representation is no more the number than you are a white M in a pink rectangle.

You keep telling people to take a look at binary. Okay. 1/9 = 0.(000111)₂. And? That's just another name for 1/9. Do you have a real point or a confused one? What do you think 1/9 being 0.(000111)₂ proves? Five bucks it only proves you're confused.

ETA: If you really want to learn what the fuss is about, try this.

^^-- here's a picture using the number line abstraction.

It is the decimal equivalence which you are claiming that is what bothers me. That is the issue of the thread. You said 1/10 in base nine can be represented as .1 in base nine, and this is equivalent to .111... in base ten. I don't think you can represent a solution to a division problem in base nine, as a decimal, (.1), because decimals are proper to base ten, and that would be to conflate base nine and base ten representations. — Metaphysician Undercover

I'll just comment on this. You're right that decimal representation is exclusive to base 10 because that's what the word "decimal" means. Poor wording on my part. What I meant to say is that base 9 (and 10 and 2) fractions can be re-written using a radix point to separate the integer part from the fractional part. In base 10 we call it the decimal point, in base 2 we call it the binary point, and so on.

But nomenclature not withstanding, my point stands. In both base 10 and base 2, $\frac{1}{10}=0.1$, and $0.1_2=0.5_{10}$, and as the linked page shows, $1101.101_2=13.625_{10}$.

This isn't me making stuff up. These are mathematical facts. As I've said before, it's you against the entire world of mathematics. That you think that you're right and everyone else is wrong amazes me. You're not the next Newton or Einstein.

Lousy example. The number's representation is no more the number than you are a white M in a pink rectangle. — InPitzotl

Sorry, but we haven't resolved the question of whether "1" is the representation of a number or not. If you think it represents a number, then why is this number not a prime number?

You keep telling people to take a look at binary. Okay. 1/9 = 0.(000111)2. And? That's just another name for 1/9. Do you have a real point or a confused one? — InPitzotl

The point is that 1/9 is not a name for anything. It's a bunch of signs which have meaning in a conceptual scheme. Your notion that a mathematical expression names a thing, is the problem you need to deal with. This idea allows people like fishfry to argue that "2+2" refers to the same object as "4". But in this argument, fishfry neglects the meaning, or role, of the operator represented as "+". Thus we have the false premise that an expression with an operator has the same meaning (expressed as 'refers to the same thing') as an expression without an operator. I see the very same problem when it is assumed that "1/9" names an object, the meaning of "/" is not accounted for. Therefore it is false to say that 1/9 names the same thing as .111..., or any other numerical representation in another base.

But nomenclature not withstanding, my point stands. In both base 10 and base 2, 110=0.1110=0.1, and 0.12=0.5100.12=0.510. — Michael

It is the meaning of what is being represented which we are discussing, and the meaning of what is represented by .1 differs from one base to another. So your argument makes no point. If .1 in base nine has the same meaning as .111... in base ten, then you haven't resolved anything by changing the means of representation. You just show that "1/10X10/1=1" in base nine, represents the same thing as "9/1X1/9=1" in base ten. But that does not capture the issue expressed in the op.

The point I am arguing is "1/9" does not have the same meaning as ".111...", or ".1 in base nine", or whatever base you want to represent it. The reason is that in the expression "1/9", the symbol "/" has a role which is not represented in the other representation. By convention, we say that 1/9=.111..., just like the convention allows that 1/10=.1 in base nine. And, the convention allows that "=" expresses an equivalence of value, the two have the same value according to the convention. But if we desire to make the conclusion that because "1/9" and ".111..." are expressions of equal value, they are therefore referring to the same thing, we need a further premise. This further premise, that two things of equal value are the same thing, is what I dispute.

This is the same argument which I had with fishfry on the other thread. Fishfry insisted that "2+2" refers to the same mathematical object as "4". But this assertion neglects the role of "+", just like the assertion that "1/9" refers to the same mathematical object as ".111..." neglects the role of "/".

What the op demonstrates is that by the conventions of modern mathematics, division is not an exact inversion of multiplication. If we start with one, and divide it by nine, then take the solution and multiply it by nine, we come up with something different from one. Further conventions implore us to accept that division is an exact inversion of multiplication, therefore the two are equivalent, ignore the difference. Thus we are inclined to ignore the difference. We assume 'a difference which doesn't make a difference', and get on with the calculations.

But whether 'a difference which doesn't make a difference' is an acceptable principle in mathematics, which strives for exactitude, is another question. And, ignoring the difference does not make it go away. To argue that there is no difference, like participants in this thread do, as if this argument could make the difference go away, is not the same as ignoring the difference. So if you choose this option, you'll have to discourse with people like me who will look for whatever ways possible to bring attention to the difference, trying to refute the false assumption that you can make a difference go away through argumentation. In reality such argumentation only brings attention to the difference.

Sorry, but we haven't resolved the question — Metaphysician Undercover

Sorry, but we haven't resolved that there's an actual problem here (not to me, or to anyone else here that I've seen).

If you think it represents a number, then why is this number not a prime number? — Metaphysician Undercover

If turtles are animals, why do they lay eggs? Since when does being called a prime have anything to do with being a number? The very fact that you even asked this question and actually think it's relevant shows that something is majorly wrong with your "problems".

The point is that 1/9 is not a name for anything. — Metaphysician Undercover

the question of whether "1" is the representation of a number or not. — Metaphysician Undercover

1/9 and 1 sure look like they name points on the number line to me. So where's the actual problem again?

Your notion that a mathematical expression names a thing, is the problem you need to deal with. — Metaphysician Undercover

We already know what numbers are and what expressions mean. The only problem here is you, and we don't even have to deal with that problem. But you've diverted 11 pages on this thread so far on your ego tripping delusions of having a problem. That's the problem.

This idea allows people like fishfry to argue that "2+2" refers to the same object as "4". — Metaphysician Undercover

Looks like the same point on the number line to me. So where's the actual problem again?

In the eleven pages of your rantings, I have yet to see an actual problem.

As seems to be the case often, you don't seem to be able to express your point very well, and you leave me wondering what you're talking about. — Metaphysician Undercover

Not much to it.

have to round off pi — Metaphysician Undercover

to round off at some point, carry it to two decimals, three, whatever — Metaphysician Undercover

if there are issues with similar division problems we simply round things off (like with pi, and some square roots, and other division problems) — Metaphysician Undercover

What exactly are you rounding off to decimal notation...? 1/9 π √2 ... You already acknowledge those numbers that you round off, only to go ahead and deny them. Inconsistent.

Numbers in the abstract are quantities of whatever we may want to examine, where the rules of mathematics are invariant (e.g. division) or otherwise set out. Whatever that "One" you mention is, it's apparently not among them, perhaps like distaste for pizza with pineapple. That's something you've added here.

So, I ended up thinking that you're no longer talking mathematics.

Sorry, but we haven't resolved that there's an actual problem here (not to me, or to anyone else here that I've seen). — InPitzotl

Denial is one of many possible responses.

If turtles are animals, why do they lay eggs? — InPitzotl

What kind of nonsense is this? Birds are animals too. What does laying eggs have to do with this?

Since when does being called a prime have anything to do with being a number? — InPitzotl

They are called "prime numbers". And "one" fulfills all the conditions of "being called a prime", except that it is not a number. Therefore the only reason why "one" is not a prime number is that it is not a number.

The very fact that you even asked this question and actually think it's relevant shows that something is majorly wrong with your "problems". — InPitzotl

You simply assume that one is a number, and class it with the other numbers. But it's not a number otherwise it would be one of the prime numbers, not divisible by two other numbers. Once you've made your faulty assumption that one is a number, you proceed to call fractions numbers too. Clearly you haven't got a clue what a number is, yet you keep insisting that such figures represent numbers.

We already know what numbers are and what expressions mean. — InPitzotl

This is very clearly not true, as I think everyone else on this thread has admitted, except you. There is no clear definition of what a number is, and there are supposed to be different sorts, natural numbers, rational numbers, real numbers. What the other participants in this thread have indicated is that "number" is just a vague term with no real defining features. That's why they rejected the definition I proposed at the beginning.

Looks like the same point on the number line to me. So where's the actual problem again? — InPitzotl

Sorry, but I have no idea what your little diagram is supposed to be showing. It's obviously not providing a definition, or any sort of indication as to what a number is. So how is that diagram supposed to argue your case?

What exactly are you rounding off to decimal notation...? 1/9 π √2 ... You already acknowledge those numbers that you round off, only to go ahead and deny them. Inconsistent. — jorndoe

We've been through this already, application is different from theory. There is no inconsistency in using a theory which one recognizes as less than exact (eg. having to round off), and still arguing that the theory is less than ideal. One can use a theory, and also at the same time, recognize and argue that the theory needs to be improved on. The problem is when someone like me recognizes that the theory needs to be improved upon, but others argue that it is already ideal.

Numbers in the abstract are quantities of whatever we may want to examine, where the rules of mathematics are invariant (e.g. division) or otherwise set out. — jorndoe

There is an abundance of evidence which demonstrates that the rules of mathematics are not invariant. First, you can look at the history of mathematics and see how the rules have changed. Then you can look at the rules which exist today and see variance and inconsistency between one branch of mathematics and another. Clearly "the rules of mathematics" are not invariant.

So, I ended up thinking that you're no longer talking mathematics. — jorndoe

If mathematics to you, is a subject where the rules are invariant, I never was talking mathematics.

Clearly you haven't got a clue what a number is, yet you keep insisting that such figures represent numbers. — Metaphysician Undercover

Still attacking those windmills with your insightful lance, eh? I have to admit, you've got gumption! :nerd:

Denial is one of many possible responses. — Metaphysician Undercover

And you're immune to it?

What kind of nonsense is this? — Metaphysician Undercover

It was meant to be an analogy... primes are numbers, but not all numbers are prime, was the point. But apparently you're even more messed up than this:

They are called "prime numbers". And "one" fulfills all the conditions of "being called a prime", except that it is not a number. — Metaphysician Undercover

Utterly wrong. There is a history to the concept of prime numbers... after some time in the development of number theory, it was quite apparent that it would be more useful to exclude one from the definition of primes in particular to avoid having to keep making exceptions for it, especially in the fundamental theory of arithmetic which is heralded as being an especially important theorem. That has nothing to do with considering one as a number though... that ship has long since sailed:

1 (one, also called unit, and unity) is a number, and a numerical digit used to represent that number in numerals. It represents a single entity, the unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. — one

...but ultimately it's just a loss of religion. There's no actual deep reason to not consider 1 (and 0) a number, except a bunch of meaningless mumbo jumbo.

TL;DR version: That one is not considered prime has nothing to do with the consideration of one being a number. It's just yet another confusion of yours.

This is very clearly not true, as I think everyone else on this thread has admitted, except you. — Metaphysician Undercover

You still have no idea what you're talking about... consistent with everything I've been saying for 11 pages, this is a language barrier and you're still confused.

There is no clear definition of what a number is, and there are supposed to be different sorts, natural numbers, rational numbers, real numbers.

You're referring to the fact that @Michael listed some categories of numbers here; namely, N (the whole numbers), Z (the integers), Q (the rationals), R (the reals), and C (complex numbers). Those are indeed categories, but there are more; beyond C, there are quaternions and octonions. In contrast to R, there are surreals and hyperreals. Take just Z into the complex plan and you get Gaussian Integers. This is not an exhaustive inventory. All of these things have their own kinds of numbers, and we can even make up new kinds of numbers on the fly.

I'm far from unaware of this MU... in fact, we've both gone through this. Here is the post where you said you were "trying to learn the language". And here is the reply I gave you seven days ago. Numbers defined differently is not a problem for math; it's just homonyms... just a feature of languages. To avoid the issues a language speaker just applies context.

What the other participants in this thread have indicated is that "number" is just a vague term with no real defining features. — Metaphysician Undercover

Vagueness is not transitive. An animal can be anything. My pet is an animal. But my pet cannot be anything; my pet is a cat. A number in general likewise could be just about anything. But 1/9 is a fraction, and 0.(1) is a repeated decimal. Generally discussions of such things are in R, though Q suffices.

Sorry, but I have no idea what your little diagram is supposed to be showing. — Metaphysician Undercover

That doesn't surprise me, but I gave you a link to it. So I guess a bit more spoon feeding you is in order:

In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by $\displaystyle \mathbb {R}$. Every point of a number line is assumed to correspond to a real number, and every real number to a point. — number line (wikipedia)

On the same page:

Two numbers can be added by "picking up" the length from 0 to one of the numbers, and putting it down again with the end that was 0 placed on top of the other number. — number line (wikipedia)

Compare to here.

It's obviously not providing a definition, or any sort of indication as to what a number is. — Metaphysician Undercover

The diagram tells you how you're supposed to play the language game with real numbers. 0.(1) is a real number.

If you actually knew what your nonsense babblings were trying to whine about, then you should recognize that in this picture:

...we have a number line (top), whereby we add 2 (purple) to 2 (blue) by applying the addition rules (see links) to get 2+2 (green), and that refers to the same number (point on a number line, black, circled) as 4 (same point, literally). This counters your idea that 2+2 and 4 don't refer to the same number.

So how is that diagram supposed to argue your case?

Silly MU, there is no case. You have no jurisdiction, the defendant is a non-entity (language), there is no standing, and there is nothing actionable.

You're objecting to the rules of a language game by playing different language games, and pretending you've said something meaningful. That's all there is, except for the fact that there are 11 pages of it.

Utterly wrong. There is a history to the concept of prime numbers... after some time in the development of number theory, it was quite apparent that it would be more useful to exclude one from the definition of primes in particular to avoid having to keep making exceptions for it, especially in the fundamental theory of arithmetic which is heralded as being an especially important theorem. That has nothing to do with considering one as a number though... that ship has long since sailed: — InPitzotl

Your so-called history of prime numbers is backward compared to what Wikipedia has to say:

Most early Greeks did not even consider 1 to be a number,[34][35] so they could not consider its primality. A few mathematicians from this time also considered the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.[34] By the Middle Ages and Renaissance mathematicians began treating 1 as a number, and some of them included it as the first prime number.[36] In the mid-18th century Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be prime.[37] In the 19th century many mathematicians still considered 1 to be prime,[38] and lists of primes that included 1 continued to be published as recently as 1956.[39][40]

So, according to Wikipedia, and contrary to your claims, 1 was first considered as other than a number, therefore not a prime number. Then, in more modern times mathematicians wanted to treat 1 as a number, so they had to include it in the prime numbers and this created a problem. Now they've excluded 1 from the prime numbers, by definition.

...but ultimately it's just a loss of religion. There's no actual deep reason to not consider 1 (and 0) a number, except a bunch of meaningless mumbo jumbo.

TL;DR version: That one is not considered prime has nothing to do with the consideration of one being a number. It's just yet another confusion of yours. — InPitzotl

On the one hand you say mathematicians "keep having to make exceptions" if one is a prime number, and one the other hand you say that there is "no actual deep reason" not to consider one a number. It's starting to become crystal clear which one of us is actually the confused one.

Let's see what's the case here. We apply a rule, the rule of primality, to the whole infinity of "numbers", and find that there is one exception to the rule, the exception is "1". The rule applies to all the numbers, allowing mathematicians to create theories based in that rule, therefore we can say that it is a defining feature of "numbers". However, the rule does not apply to 1, as 1 needs to be excluded from these number theories. In your mind, what's the logical thing to do, make an exception to the rule, to allow that 1 is still a number despite being an exception to this defining feature of numbers, or conclude that 1 is something other than a number?

I'm far from unaware of this MU... in fact, we've both gone through this. Here is the post where you said you were "trying to learn the language". And here is the reply I gave you seven days ago. Numbers defined differently is not a problem for math; it's just homonyms... just a feature of languages. To avoid the issues a language speaker just applies context. — InPitzotl

I know that "numbers defined differently is not a problem for math". What is a problem is conceited people making the universal, uncategorized statements like "we already know what numbers are", when it's very evident that they haven't the foggiest idea of what a number is.

Vagueness is not transitive. An animal can be anything. My pet is an animal. But my pet cannot be anything; my pet is a cat. A number in general likewise could be just about anything. But 1/9 is a fraction, and 0.(1) is a repeated decimal. Generally discussions of such things are in R, though Q suffices. — InPitzotl

No, an animal cannot be anything, a rock is not an animal, a plant is not an animal. Likewise, a number cannot be anything.

That doesn't surprise me, but I gave you a link to it. So I guess a bit more spoon feeding you is in order:
In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by RR. Every point of a number line is assumed to correspond to a real number, and every real number to a point.
— number line (wikipedia) — InPitzotl

It is not logical to refer to a property of a special type of number (real number) to demonstrate what a number is in general. This is like referring to your cat's meow to say what an animal is. In philosophy we call this the difference between an essential property and an accidental property, and being able to make the distinction is fundamental to proceeding with deductive logic. That 1 can be represented on a number line as a feature of real numbers, is an accidental property, specific to one type of number, real, and not an essential property, describing, or defining numbers as a whole.

The diagram tells you how you're supposed to play the language game with real numbers. 0.(1) is a real number. — InPitzotl

I apologize for not joining your little game, but I see no reason to restrict our discussion of "numbers" to real numbers.

That's all there is, except for the fact that there are 11 pages of it. — InPitzotl

That's 11 by your convention, not by mine.

Page after page of nonsense beyond eccentricity, MU. What is your point? Do you have one?

Obviously you haven't read those pages.

nonsense — tim wood

Don’t use that word! It’s @Metaphysician Undercover’s and it has a technical meaning with unique connotations.

Obviously you haven't read those pages. — Metaphysician Undercover

You mean obviously I have, which would be generous of you. Because I have not read all, but many, and enough. But as absolutely standard MU m.o., you ignored the question.

I'll ask it again. What is your point? Do you have one? I'm not asking about the dance, but why you're dancing.

If you're pointing out that between the English language and the language of mathematics there can be slippage, I believe that's acknowledged - they're two very different languages. But you carry the notion of slippage into one of the languages and that's a false step, and you make a spectacle of yourself in the process. Hence the question: what''s your point? Do you have one?

Apparently, he doesn't have "1"! One over infinity must be taken as the mathematical equivalent of zero, or the calculus, the basis of physics, is founded in a fallacy. Rationalizing the irrational is the essence of science. But what if that tiny value, excluded because of its irrationality, is what reality is? How much of a diversion from the causal nexus is needed to force the admission of freedom, consciousness, and moral value? How many is one? And, does the certitude which 'one' is which, so necessary to do logic, the same enumeration as the count of however many nines after the decimal makes 'one'? If not, don't we lose both? But if only the irrational quantity excluded by reason is that resolution between which and how many is 'one', then the excluded term is not a quantifier at all, but a qualifier. The qualifier is the heart of reason.

Your so-called history of prime numbers is backward compared to what Wikipedia has to say: — Metaphysician Undercover

...

So, according to Wikipedia — Metaphysician Undercover

...while we're on the subject, what does the very next paragraph say?

Then, in more modern times mathematicians wanted to treat 1 as a number, so they had to include it in the prime numbers and this created a problem. Now they've excluded 1 from the prime numbers, by definition. — Metaphysician Undercover

Welcome to the year 2020. So what's the problem?

What is a problem is conceited people making the universal, uncategorized statements like "we already know what numbers are", when it's very evident that they haven't the foggiest idea of what a number is. — Metaphysician Undercover

...so where does that leave you? Do you have the foggiest idea what a number is? Do you make universal, uncategorized statements about numbers?

It is not logical to refer to a property of a special type of number (real number) to demonstrate what a number is in general. — Metaphysician Undercover

I think you're lost, MU. This is supposed to be a thread about 0.(9)=1.

I apologize for not joining your little game, but I see no reason to restrict our discussion of "numbers" to real numbers. — Metaphysician Undercover

Well that's really easy MU. Here's the primary motivation, in your words:

I'm trying to learn the language, and I don't like inconsistency or contradictions within the language I use. Such things lead to misunderstanding and even deception. — Metaphysician Undercover

The way to avoid inconsistencies and contradictions that lead to misunderstandings and deceptions (aka, amphibolies/equivocations) where languages have homonyms is to restrict the conversation to applicable shades of meaning. When in a pool hall and someone talks about how to sink the 7 without sinking the 8, English should be regarded as a pool-technique, so it simply means to invoke a spin on the ball... countering a discussion invoking the use of English with debates about how some hypothetical guy from England might sink the 7 is a meaningless distraction. In this context, we're supposed to be talking about 0.(9)=1. 0.(9) is a repeated decimal. Repeated decimals are special cases of fractions, suggesting a treatment of at a minimal Q, though decimals just commonly invoke R. So to avoid misunderstandings and deceptions, to meaningfully talk about Q and R, we should be employing the context of one of these two things.

Incidentally MU, even if we don't restrict our discussions to the reals, 2+2 and 4 refer to the same object in the reals, and you claim they don't refer to the same object (again, in case you missed it, "Do you make universal, uncategorized statements about numbers?"). This implies you're flat wrong in at least one context. According to your pretended concerns about the inconsistency or contradictions with language leading to misunderstandings or deceptions... according to your pretense of avoiding smoke and mirrors, just having this single context in which you are wrong is challenge enough to warrant a clarification of your claims anyway... you know... to... avoid misunderstandings and deceptions?

TL;DR, we should restrict our discussion to the reals because that's the context within which 0.(9)=1 and 0.(1)=1/9 are meant to be discussed; i.e., it is this context from which the meaning of such things derives. Ranting and raving about what some guy in 300BCE would have called 1 is a meaningless distraction.

We've been through this already, application is different from theory — Metaphysician Undercover

Your previous side-track doesn't really matter much here; it's about the numbers, 1/9 π √2 ... By rounding them off, you've already admitted them. Denying them is hence inconsistent; you wouldn't have anything to round off in the first place.

"the rules of mathematics" are not invariant — Metaphysician Undercover

Saw the word "invariant" and took it for a ride? Having five fingers on each of your two hands means having ten fingers on them, not none, not a dozen. 5 + 5 = 10 = 2 × 5 (and 5 < 10 by the way). Notice how that goes for toes and claws as well? Whether yours or mine or the Pope's? You don't mysteriously get a dozen fingers in that case. That's what's meant by invariance here, + - × /, and what you tried to dismiss with a casual handwave. Oh, also, √2 × √2 = 2 (and 1 < √2 < 2), irrespective of your rounding, so there. ;)

"One" only submits to being a multitude when it is applied to a thing which can be divided — Metaphysician Undercover

As mentioned, whatever your "One" is, this is something you've added here, much like I added distaste for pizza with pineapple. Your "One" apparently does not figure as the number 1 does in arithmetic.

Stick to the topic.

Has anyone ever persuaded a change of an opinion or belief you've held?

Are you solipsistic, by any chance?

But what gets "rounded" out? Is it nothing? What if it's everything? There may be nothing unique in the mathematician's mind, but there is a great deal of evidence that every particle in the universe is, that reality itself is uniqueness. Can this be "rounded out"? If 1 is unique there is no arithmetic at all. Add this one apple up all you want and you won't get an orange. The question is, philosophically, what is number for? If not, that is, to sustain the logician's conceit that the excluded middle is law?

The question is, philosophically, what is number for? — Gary M Washburn

The question is, philosophically, what is a grapefruit spoon for? Do you see the problem?

As a matter of representing numbers, wouldn't most be fine with 9/9 = 9 × (1/9) = 9 × (0.111...) ? — jorndoe

I'll ask it again. What is your point? — tim wood

I've already made the point numerous times. The op asks: "As a matter of representing numbers, wouldn't most be fine with 9/9 = 9 × (1/9) = 9 × (0.111...) ?" I agree that most would be fine with that, but I am not. If you are interested in the reasons why, you can read the thread. I started with the need for a definition of "number", as necessary in order to determine the acceptability of a matter of representing numbers.

So what's the problem? — InPitzotl

Your reference to the history of the prime numbers neglected the fact that for millennia 1 was not considered to be a number. It was only in relatively modern times that mathematicians wanted 1 to be a number, and this created the problem which required an exception to be added into the rule of primality.

Do you have the foggiest idea what a number is? — InPitzotl

I provided a definition at the beginning of the thread, this was my idea of what a number is, an arithmetical value representing a particular quantity. It was rejected, and then it was explained to me that "number" is not a defined term in mathematics. So I concluded that no one really has the foggiest idea of what a number is. Then you contradicted this, claiming that we know what numbers are.

The way to avoid inconsistencies and contradictions that lead to misunderstandings and deceptions (aka, amphibolies/equivocations) where languages have homonyms is to restrict the conversation to applicable shades of meaning. — InPitzotl

This is the point I've argued from the beginning of the thread. To know whether the op offers an acceptable representation of numbers, we need a working definition of "number", and restrict the conversation so as to use "number" only in that way, and thereby discuss whether the op offers an acceptable way of representing numbers or not. As I stated earlier in this thread, I don't think that .111... is acceptable as "a number" because it does not represent a particular quantity. But it was claimed that my definition of "number" was unacceptable.

Incidentally MU, even if we don't restrict our discussions to the reals, 2+2 and 4 refer to the same object in the reals, and you claim they don't refer to the same object (again, in case you missed it, "Do you make universal, uncategorized statements about numbers?"). — InPitzotl

Clearly "2+2", and "4" do not refer to the same "object" by any conventional definition of "object". So I think it's time for you to start learning the language.

TL;DR, we should restrict our discussion to the reals because that's the context within which 0.(9)=1 and 0.(1)=1/9 are meant to be discussed; i.e., it is this context from which the meaning of such things derives. Ranting and raving about what some guy in 300BCE would have called 1 is a meaningless distraction. — InPitzotl

It ought to be clear to you by now, that I do not accept "the reals" as a representation of numbers. Any system of interpretation which ignores the role of "+" within an equation, to claim that "2+2" says the same thing as "4", cannot really be taken seriously.

Your previous side-track doesn't really matter much here; it's about the numbers, 1/9 π √2 ... By rounding them off, you've already admitted them. Denying them is hence inconsistent; you wouldn't have anything to round off in the first place. — jorndoe

That's not true. Due to the nature of representation, using symbols which represent quantities does not necessitate that the person believes in the existence of numbers. I can ask for two coffees for example, using the term "two" to get what I want, without believing that "two" represents some sort of mathematical object called a number. So I can do all sorts of arithmetical operations, using those symbols in the way that I am taught to, including the rounding off of quotients, without believing that there is any such thing as numbers. There's no inconsistency between using those symbols and denying the existence of numbers.

Saw the word "invariant" and took it for a ride? Having five fingers on each of your two hands means having ten fingers on them, not none, not a dozen. 5 + 5 = 10 = 2 × 5 (and 5 < 10 by the way). Notice how that goes for toes and claws as well? Whether yours or mine or the Pope's? You don't mysteriously get a dozen fingers in that case. That's what's meant by invariance here, + - × /, and what you tried to dismiss with a casual handwave. Oh, also, √2 × √2 = 2 (and 1 < √2 < 2), irrespective of your rounding, so there. — jorndoe

But your claim was "the rules of mathematics are invariant", not the number of fingers on my hand is invariant. I gave you a clear explanation of how the rules of mathematics are not invariant. Your logic is appallingly bad similar to InPitzotl's. You give me one example of an invariant rule and conclude therefore all the rules of mathematics are invariant. It's as if you are arguing that "5+5" is equal to "10", in all instances, therefore all the rules of mathematics are invariant. Look at the conventions for multiplying negative integers, and imaginary numbers as an example of how mathematical rules are not invariant.

As mentioned, whatever your "One" is, this is something you've added here, much like I added distaste for pizza with pineapple. Your "One" apparently does not figure as the number 1 does in arithmetic. — jorndoe

I'm still waiting for someone to explain to me how the so-called "object", or "number" which is represent by "1" and is by definition not a multitude, and therefore not composed of parts, can be divided into nine parts. Care to explain how the division might take place? I'm not asking for a demonstration in symbols, because it's easy to represent something with symbols, which is actually impossible to do, just like we can talk about doing things which are impossible to do. I'm asking what makes it possible to divide a unit which is not composed of parts?

Has anyone ever persuaded a change of an opinion or belief you've held? — dex

My beliefs change like the weather. But to be honest, I wouldn't say that it's others who persuade me to change.

My beliefs change like the weather. But to be honest, I wouldn't say that it's others who persuade me to change. — Metaphysician Undercover

In that case, may I ask why you're arguing your position here? If you yourself can't be persuaded by others, what makes you think others will be persuaded by you?

It ought to be clear to you by now, that I do not accept "the reals" as a representation of numbers. Any system of interpretation which ignores the role of "+" within an equation, to claim that "2+2" says the same thing as "4", cannot really be taken seriously. — Metaphysician Undercover

Why is this thread still going on?

It was only in relatively modern times that mathematicians wanted 1 to be a number, and this created the problem which required an exception to be added into the rule of primality. — Metaphysician Undercover

What are you talking about, "problem" and "required"? The fundamental theorem of arithmetic states, in the modern reading, that all positive integers can be represented as a unique product of primes (barring order). That's perfectly phraseable with prime including 1, it's just clumsy: "All positive integers can be represented as a unique products of primes, barring order, excluding from said product the number 1". Both phrases describe the same fact. One is just clumsier.

I provided a... — Metaphysician Undercover

Pretty much.

This is the point I've argued from the beginning of the thread. To know whether the op offers an acceptable representation of numbers, we need a working definition of "number", — Metaphysician Undercover

Sort of, but not really. "Number" applies to a lot of things. But that's not a problem; it's actually a benefit. The definition of number should not merely not be nailed down; it should be open. But part of the point of categorizing these numbers is so that we can give particular kinds of numbers names.

Clearly "2+2", and "4" do not refer to the same "object" by any conventional definition of "object". — Metaphysician Undercover

This is jargon... they refer to the same mathematical object.

Any system of interpretation which ignores the role of "+" within an equation, to claim that "2+2" says the same thing as "4", cannot really be taken seriously. — Metaphysician Undercover

Anyone who uses the decimal system to count above 9 shouldn't take your pronouncement seriously.

What was Einstein's supposed definition of insanity again?

Among the reasons why I think that while being courteous and a gentleman has its place, there comes a time when those behaviours and the spirit that animates them is a mistake, a failure to understand the situation. MU has made it completely clear he's a nut-case, and not even an honest nut-case, yet some people don't get it. And a parallel with Trump and similar people. The right approach to them is to treat them appropriately. And in my experience, bad and crazy and evil people soon enough tell you clearly and exactly what they are.

Of course Einstein did say it with more succinct elegance.