Good question. 1 is the product of zero primes: https://en.wikipedia.org/wiki/Empty_product — InPitzotl

First class example of deception.

The ontological fallacy is that unity and continuity is what is real. — Gary M Washburn

This is not necessarily a fallacy. It may be the case that the principles required to establish compatibility between the two, such that we can conceive of the two coexisting, as "what is real", have not been discovered.

Any system? And you wish to be taken seriously? And then there's the lie: what have you learned in this thread, for example? — tim wood

If you think that you can explain the meaning of "2+2=4" by saying that "2+2", and "4" represent the same mathematical object, and "=" represents "is the same as", then I would say that you sorely misunderstand mathematics, and you ought not be taken seriously.

Do you understand that for an "equation" to be at all useful in honest mathematical practice, the right side must necessarily represent something different from the left side? If not, the equation would be a useless tautology. But that is how some people might present "2+2=4" as an example of a useless tautology in which "2+2" represents the very same thing as "4". But clearly that is not how equations are used by scientists and mathematicians. It is only how sophists who are attempting to persuade someone that "2+2" represents the same object as "4" might use an equation, and this is surely not honest mathematical practice.

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? — dex

I partake in this forum to learn, and to help others learn. Learning is a communal process requiring sharing and consent. I apprehend a difference between understanding and being persuaded. One can very often persuade a person to act in a particular way, without the person understanding the need to act that way, or the reason why that action is being called for by the other. This difference is what allows for the existence of deception.

"Learning" is a broad term which is used to refer to both, understanding, and being persuaded to act without understanding. So for instance, as children we are highly susceptible to being persuaded, and our capacity for understanding is quite limited. We are taught principles, like arithmetic, and are persuaded to behave in a particular way, without understanding the reasoning, which is the theory behind those principles. "Learning rules" of arithmetic, and even the "rules" of higher mathematics is not actually a case of understanding principles, as "learning rules" seems to imply. Fundamentally it is a matter of being persuaded to act in particular ways in response to specific situations, and develop particular habits, like training a dog. We are persuaded to act in a particular way without understanding any actual rule, though the rule might be produced as a description of that behavior. Wittgenstein is relevant here. When we are older, and our capacity for understanding is increased, one might delve into theoretical mathematics, what fishfry called pure mathematics, in an attempt to understand these actions.

What is important to apprehend, is that in the general sense, understanding follows from acting, it doesn't precede it, as we learn from experience. So theory follows practice. We find a practice which works, and we employ it, then we develop the theories to account for why it works. In this theoretical process, which is the "understanding" of the practice, it is of the utmost importance to determine the faults of the practice, exceptions, places where the practice produces less than perfect results. This is where proper understanding, and formulation of theory in a way which accounts for these discrepancies can lead to a better practice in the future.

So for example, the ancient practice of astronomy was to map the orbits of the planets as circles. This practice worked very well, and provided very good prediction, as Thales apparently predicted an eclipse. But there were slight imperfections. What was required was theoretical analysis of the slight imperfections, to produce a true understanding of the real orbits of the planets. That new theory produced a whole new set of practices which today extend far beyond the solar system. But the new practices have demonstrated their own imperfections. Therefore we need to revisit all the theory from bottom up to understand and account for these imperfections.

He's yet to answer the question I posed most likely because it isolates the underlying hypocracy of his debate stance -- apparently truth has little to do with his posting motivation -- so it's only pointless to argue against that which his hypocracy is productive of. His whole intellectual orientation is faulty. But it's for some reason been useful enough for him to maintain it. — dex

To answer your question now, I believe it's a faulty goal to partake in this forum with the intent of persuading others. We are here as philosophers with the goal of understanding. We cannot treat each other as children to be persuaded, and even a minimal degree of participation will reveal that persuasion is never forthcoming. My goal in arguing the position I have argued in this thread is to bring to the attention of others, the slight imperfections which I've observed to exist within the practicing of mathematics. We can only move forward, collectively, by acknowledging, and accounting for these imperfections. To me, the imperfections are glaring, but every person perceives and apprehends things in one's own way. So some people cannot even see the imperfections, and others who see them dismiss them as being so minor that they're irrelevant (a difference which doesn't make a difference), so they end up denying that the imperfections are even imperfections. This is what I refer to as contradiction, to say that there is a difference which is not a difference, as the difference between ".999...", and "1".

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). — InPitzotl

And what about 1? Is it excluded as a positive integer, or natural number? Or have you made the fractions into integers? Where does 1 fit in this theorem?

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

That's simply an assertion. I have yet to see a definition of "mathematical object" which allows for the application of the law of identity. And the law of identity is what identifies an object as an object. To say that they refer to "the same mathematical object", says nothing more than that they are equal. And two distinct objects with the same value may be equal, and clearly not the same object according to the law of identity. So the phrase "they refer to the same mathematical object" is nothing but a deceptive use of jargon.

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. — InPitzotl

Sure, leaving the definition of "number" open is a "benefit"; to those who want to expand mathematical theory in any imaginable direction, like fishfry promotes, and also for those who argue by equivocation. For those who want to develop clear and consistent mathematical theory with universal applicability, it is detrimental.

Math is not subject to any standard of applicability. On the contrary, the only criterion for the worth of a piece of math is whether it's regarded as interesting and beautiful by mathematicians. — fishfry

This is the fantasy that aesthetics is valued over and above good. It is a fantasy because we can only passively enjoy beauty for an extremely short period of time before our bodily needs get in the way and we are urged to act. The natural human condition is to act, so even the purest forms of theory are influenced by the urge to act.

MU has a metaphysical theory of numbers, he's a believer in them in the full b-word sense (it's part of his identity... almost literally), and modern math is kind of a heresy wrt it. That's my take. I personally envision his theories as being roughly of both the form and value of Eric the half a bee. — InPitzotl

No. I don't seem to have a metaphysical theory of numbers, because I do not understand numbers well enough to create such a theory. What I do understand though, is that there is no metaphysical convention, and therefore no ontological coherency, in modern math. You might say that I believe in metaphysics, and modern math demonstrates a poverty of metaphysics.

I can't help being struck by the amount of mindshare Metaphysician Undercover holds here. — fishfry

I would blame Banno, for declaring that this thread is about me. I'm just doing whatever I can to live up to Banno's expectations of me. See my respect for you Banno?

The result will be to show in even greater relief that this is a thread about Metaphysician Undercover, not about maths. — Banno

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.

Obviously you haven't read those pages.

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.

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.

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.

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.

So your perspective is more psychological and related to our conscious experiences. Is this a Berkeley or Kantian strategy you are gleaning from in treating spacetime as a fundamental psychological process but nothing more? — substantivalism

Sorry, I don't think I can answer this question.

Technically physical objects in special relativity move relative to other frames of reference and are always going to happen to observe that the top casual speed is c. — substantivalism

Do you apprehend, as I do, that making the "top casual speed" (whatever you mean by "casual") as c, is to posit an absolute?

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.

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?

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.

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

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

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.

Space and time, as well as space-time are the concepts human beings have developed to understand their surroundings. We understand our environment as things which are changing relations to each other, and are also changing in themselves. Since these concepts are derived from the fundamental principles which describe our surroundings as things, it doesn't make any sense to talk about space and time as being independent from things.

There was a time when things were thought to move in space. Empty space was required in order that a thing could move, otherwise it would have to push on another thing which would push another an another, and nothing could move. But Einsteinian relativity conceives of things as moving relative to light. This allows that things might move through light without necessarily moving through space, and space and time as concepts, refer to the relations between things and light..

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.

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?

Can you show me a mathematician who has questioned rational numbers like 1919? — Michael

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.

The only thing I'm giving a shot at is for you to see how the math works. — InPitzotl

Well, if you think that I haven't already seen how math works, then you're wrong. And as I've already explained, the conclusion I've made from what I've seen is that a healthy dose of skepticism is needed in my approach to mathematics. That is why I've tried to take the discussion beyond rational numbers, to natural numbers, and number theory itself. It appears like you, and most in this forum believe this to be a pointless exercise. That doesn't really concern me. If that's what you think as well, then you're wasting your time here if your true intent is for me to see how math works. I've already seen it. That does not mean that I understand it. If you want me to see why math works, then drop your presuppositions and come to the bottom with me.

If it's just that I am providing entertainment for you and the others, at least it's of a healthier sort than that provided by the president of the USA.

There's nothing to make clear to me; this is illusory insight. — InPitzotl

Oh, so you do not see any difference of type between the object we call a pizza, and the object we call a number. That's revealing.

Try this... instead of 1/9, let's do 1/7. Now our description has to change, because we get 0.(142867). So yes, each "time" the machine is forced to "loop back" it's because there's a remainder. But what is the remainder to 0.(142867)? Is it 3, 2, 6, 4, 5, or 1? Note that "each time the machine is forced to 'loop back'" it is because there is exactly one of these left as a remainder. Is there exactly one of those left as a remainder to 0.(142867)? Can you even answer these questions... do they have an answer? I'll await your reply before commenting further. — InPitzotl

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. If one expression is more vague than the other, then it may or may not be a better way of hiding the fact that there is a remainder. To see what the remainder is at any given time, all we have to do is look to see at what point the machine is at when it loops back. Where's the problem?

But if we can't say which remainder this is, we can still talk about the same thing using an alternate view. Suppose we run our long division program and we're told that the result is 0.125. Then what can we say about the ratios it was dividing? I claim we can say it was dividing k/8k for some k. Now likewise suppose we run our long division program and we're told the output is 0.(142857) using the description given by a symmetric recursion and infinite loops. Now what can we say about the ratios it was dividing? I claim we can say it was dividing k/7k. — InPitzotl

I don't deny any of this, that's how math works, conventions are followed, and that's what convention has us call "dividing". The question is on what principles do we say that the conventions are right or wrong. 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 following the conventions works, there is quite possibly still a better way. We are inclined to say that the conventional way of doing things is "the right way" simply because it is the conventional way, but then what do we say when a better way is shown? One might follow a trail, between the residence and place of work, to and from, day after day, and following that trial always works to get the person where they are going. The person says it's the right way to go to get to and from my work. But that doesn't mean there's not a shortcut. How does a shortcut make the right way into the wrong way?

You're denying that we can divide at all.. — InPitzotl

This is an abysmal straw man.

But because you worship the idol of the integers, — InPitzotl

You're making the same mistake as fishfry. I do not worship any numbers. In the other thread I was using principles from the rational numbers to attack the real numbers. and for some reason fishfry got the idea that I strongly believed in the rational numbers, just like you think I strongly believe in the integers.

The real discussion then is whether we're doing integral division using decimals or rational division, and since decimals are driven by powers of tens (including powers of tenths), it's immediately apparent it's rational division. — InPitzotl

No, the discussion is whether rational division, as the inverse operation of multiplication, is a true form of division.

You've got it backwards. They're derived from the axioms of the system you're using. The axioms define various relationships between undefined terms. The application demands use of an appropriate axiomatic system whereby the mappings of the undefined terms have the relationships described by the axioms. — InPitzotl

This is the root of the difference between us. You seem to think that mathematics works because people dream up random axioms, then the axioms are applied, and voila, mathematics works. I think that mathematics works because people design the axioms so as to be applicable to the real world. So from my perspective, the real world puts limits on which axioms ought to be accepted. From your perspective, so long as the axioms are coherent and consistent, the mathematics ought to work in the world. Do you see how you are the one who has it backwards?

So I start with the fundamental principle of "pure mathematics", which states that a "unit", as a simple, cannot be divided. However, I qualify this by saying that whenever the "unit" is applied to the real world, in "applied mathematics", the nature of the object, which the unit represents in that application, determines how the unit might be divided, depending on the object's parts etc.. So the divisibility of the unit is dependent on the object it is applied to.

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. First, I will argue that this annihilates pure mathematics and number theory, making "one" signify a multitude. 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. So this backward conception of "unit", which you hold, misleads you in this way, actually deceiving you to the point that you will argue persistently that any object can be divided in an infinity of different ways.

Therefore, the approach which takes as fundamental, that a unit might be divisible in an infinity of different ways, and then might qualify this in application, tailoring divisibility to meet the specifics of the object, is the wrong approach. It is the wrong approach because it has misled you, and others of course, into thinking that mathematicians can produce axioms and the world will exist in the way that the axioms dictate. But when we hold in theory that the "unit" is fundamentally indivisible until its divisibility is proven through practice, we avoid this problem.

Because we define it. Incidentally in terms of application we can use this in arbitrarily complex ways. There are some 1080 atoms in the universe, but we can practically get far smaller than 10-80 by applying arithmetic coding to text. Note also that machines can far exceed what we can do, so the limits of what we can do are not bound by some smallest unit of some extant thing... they're bound by the furthest reaches of utility we can possibly get from machines. We can get much further not limiting our theories in silly inconsistent ways. But even without all of this, just for the math is all of the required justification. — InPitzotl

This demonstrates my point.


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. Of course it will do that or else it would not be an acceptable procedure. The question I thought, was whether there are doubts about the procedure. As I explained above, doubt arises if one believes that there might be a better way. To doubt in this way does not require that the skeptic produce the better way, only that the skeptic demonstrate issues with the accepted way, which might be improved upon.


Yes, that is what is at issue here, the validity of such inversions, when the inversion turns up something which is outside the rulebook of what it is supposed to be an inversion of.


I already answered this for you. Your request is outside the range of what I asserted, so not relevant.

Most folk can manipulate "one" in quite complicated ways. They learn to speak of one dozen, for example, understanding that they can treat twelve things as if they were an individual. They can have half a glass of water without having an existential fit about the non-existent other half. — Banno

This is applied mathematics. I was speaking about what fishfry called "pure mathematics".

Talking to you has similarities to talking to a pre-operational child. — Banno

In case you haven't noticed, that is my intent. I'm sure you've read Wittgenstein's "Philosophical Investigations". What he demonstrates is that to properly understand the nature of fundamental, basic concepts, upon which knowledge is built, an individual must get one's mind into that same condition which it is in when one learns those concepts naturally. This is the condition which you call "a pre-operational child". The time when a person learns such concepts naturally is the time when the "understanding" of the concepts occurs. Later, we take the concept for granted, and claim to understand it. The role of the skeptic is to analyze the actual "understanding" of the concept, which is performed by pre-operational children. The difference between the skeptic and the pre-operational child, is that when we revisit this condition, we can revisit it as an observer, and thereby learn something about the actual process which is called "understanding".

Multiple things. Playing a game. I'm trying to see how much perspective I can give you about your lack of competence in this area... that you're uncooperative makes it a bit challenging. But I'm being quite honest here; I don't take you seriously. — InPitzotl

In other words, you're trying to persuade me.

Well seeing as the pizzas themselves wouldn't be exactly equal either, why would we care? — InPitzotl

This is why the example, as proposed, is not useful. We are talking about what fishfry called pure math, not the application of principles to pizzas. We are dealing with numbers, not with pizzas, and discussing the basis (principles) upon which we divide quantitative values. I've tried to make this clear to you, but you keep going back to these examples. As soon as we come to a mutual agreement about the divisibility of quantitative values (abstract numbers), we can move on to examples of application. What I am trying to impress upon you, is the simple fact that some quantitative values cannot be divided in certain proposed ways. That's a fundamental feature of what a quantitative value is, being based in "the unit".

But you failed to prove there's still a remainder in an infinite string of 1's following a decimal point following a zero. — InPitzotl

I gave you an inductive proof and you refused it. I accused you of lying in denying the truth of my inductively derived premise. What else might I do?

Without an end, when do you have a remainder? — InPitzotl

Each time the machine is forced to "loop back" it is because there is a remainder which must still be divided. The machine does not stop looping back because there does not stop being a remainder. In learning long division, we are instructed to round off at some point, carry it to two decimals, three, whatever.

If we're talking about integers, sure. If we're talking about fields, no. It's intriguing to me that you take this sort of integral and/or whole and/or counting number realism to such extreme deepisms that you both transport the properties of such things into other number systems and trick yourself into thinking you've done something profound, but I have no actual interest in the broken theories that lead to this. I am however interested in the psychological aspects of why you're so committed to these deepisms... but not being a psychologist I'm content with just what I can piece together with reverse engineering. — InPitzotl

Good, we're making some progress toward principles of agreement. If you recognize that there are some restriction which may apply to the division of a unit, due to the nature of the unit, then you ought to understand that the conditions are derived from the real particulars of the application. So for example, one pizza might admit to certain equal divisions, and one octave might admit to other equal divisions. The divisibility of the unit, (the restrictions on how it may be divided), are dependent on the nature of the unit being divided. Why would you think that there is any type of thing, like a field or whatever, which would admit to any possible division imagined, whatsoever?

1. the arithmetic procedure gives 0 decimalpoint and endless 1s (provable by, say, mathematical induction, reductio, whatever) — jorndoe

This is what I disagree with. Instead, I think that one divided by nine is an impossible procedure, provable by induction.

This doesn't answer my question. Do you think that mathematicians are aware that "the axioms are full of inconsistencies and contradictions. A lot of these so-called 'proofs' are smoke and mirrors built on false premises and therefore unsound"? — Michael

Some are, some aren't.

If they are then why do they use them and not "fix" them? — Michael

Some are actively trying to fix them. There's not universal acceptance of all mathematical axiom because some mathematicians propose alternatives. They are trying to fix the problems.

If they're not then how are you, a mathematical layman, able to notice what the experts can't? — Michael

I'm a metaphysician, and some mathematical axioms are derived from metaphysical concepts such as the concepts of unity and continuity, which are features of "being", a subject of metaphysics. So I'm not exactly a layman on these issues.

We can prove things about switching 1/9 to decimal form without doing it (↑ stands on its own).
You understand...? — jorndoe

This is exactly what I've been arguing for the entire thread. What we can prove is that it can't be done.

If yes, then you're free to suggest a means to communicate this unambiguously, or you can follow typical conventions like 0.1¯=0.(1)=0.111…0.1¯=0.(1)=0.111…. — jorndoe

I've already communicated the point quite clearly throughout the thread. Some specific numbers cannot be divided by other specific numbers, that's a fundamental feature of "numbers" which is very evident, and we ought to respect it. However, the convention in mathematics is to use postulates such as every number is divisible by every other number (except zero perhaps), then dream up unsound axioms to support these postulates.

We've been through this MU. We're not debating... you're under the delusion that we're having a debate... that my goal is to persuade you, that I'm trying to do so, that something is riding on your agreement, and that it actually matters that I persuade you. We're not, I'm not, I'm not, it isn't, and it doesn't. But I have to say... it's all kinds of adorable that you think we're debating! — InPitzotl

What are you doing then? Do you not see that such discourse with a delusional person is pointless?

But I can. I can use a pizza cutter, I can slice a pizza into 3 equivalent parts, and then I have thirds of a slice. — InPitzotl

And what do you say when we weigh the slices and find out that they are not exactly equal?

There is a real thing I can do to distribute 24 pizzas among 9 people (above). Your objections fail to describe or affect that procedure; worse, they fail miserably to account for the fact that I wind up with no pizzas instead of six. — InPitzotl

I've never seen a pizza sliced in exactly equal pieces, and despite your minor in bs, I don't believe that it can be done. Sorry, but your example is what fails miserably.

The second run is qualitatively different, but we can still represent what it does. We know that there's no complete output of this program because we know it will never halt; but we know the program keeps generating 1's in perpetuity. We know we cannot write down the full output here, because we know it is infinite, but we know that its output will keep spitting 1's because it did so for a couple of steps and because the nature of the infinite loop is that of symmetric recursion. So we can represent the output as 0.111... meaning it never stops, and will always spit out 1's. "An infinity of 1's" is just a shortcut for saying the same thing. "One repeating" says the same thing as well; 0.(1) refers to the same thing. — InPitzotl

Sure, but as I said, there's still a remainder which hasn't been dealt with, even if you represent the situation as ".111..." Do you not comprehend that? There's something left which hasn't been divided. The machine keeps spitting out 1's forever, and the division problem is never solved. So representing 1/9 as .1... is the same as saying that this is an unresolvable, or impossible division to do. All ".111..." represents in your example, is that the machine could keep adding 1s forever, and the division problem would still not be completed.

Oh, thank you MU. It saves a lot of time when you make a claim but accidentally prove by contradiction that it's false (underlined). If you can describe this string as "an infinite string" and reason about what that implies, then I can refer to the same string as "0.111..." and reason about what that implies. As a bonus points, you've demonstrated that you yourself are just confused about this, which is something I keep saying. — InPitzotl

You seem to misunderstand. I'm not arguing that it's impossible to represent an infinite string of 1's, that's simple to do. What I'm arguing is that an infinite string of 1's. following a decimal point, following a zero, does not represent a solution to one divided by nine. There is no solution to one divided by nine, it is an impossible division. But instead of facing this very simple, and straight forward fact, which is nothing other than the way that numbers are, you and other mathematicians will argue to wits end, providing all sorts of smoke and mirrors illusions, claiming that you have actually resolved this impossible to resolve division.

There's no "end" to a program that never halts. — InPitzotl

Exactly! Without an end the problem is not resolved. The division has not been carried out. That's because it is impossible to do. The program never halts because the division is never completed, because it is impossible to do.

Hey InPitzotl, there doesn't seem to be anything new in your post. And, as you say we are not debating this, nor are you trying to persuade me of your point of view, so why continue? Are you learning anything yet? Would you consider the proposition that certain numbers just cannot be divided by each other? It's just something that's impossible to do.

The prime commandment is that One must be kept whole. At the alter he sacrifices all of mathematics beyond addition. — Banno

You know that the value of "one" is that of a whole, a single unit, do you not? If it is divided in half for example, then the two halves together can not have an equal value to the "one" which is a single, not a double. If there is such a thing as "pure mathematics", then the unit which is represented by "1", being simple, must be distinct from the unit represent by "2", or "3", being multiplicities. The need to divide the fundamental unit "1" is a feature of application. Only in reference to the particulars of the application can the divisibility of that which is represented by "1" be determined. In other words, the divisibility of "1" is dependent on, and determined by the divisibility of the object which it is applied to in application.

The fraction part of a mixed number specifies an exact portion of a unit. — InPitzotl

I disagree with this fundamental point, and reasserting it will not persuade me to agree. The fact that there is a repeating decimal when we attempt to divide one by three demonstrates that the unit cannot be divided in three exact portions. There is a remainder. Therefore it is impossible that 2/3 represents an exact portion of a unit. What you have argued is that you can define "one" or "unit" however you please, as consisting of three parts ( the same as "three"), or consisting of nine parts (the same as "nine"), or whatever number you want, and that's just contradiction plain and simple. In no way can "one" represent whatever number you want, without contradiction.

By contrast, the fraction specifies an exact quantity. It means a specific thing to give one person 2 2/3 pizzas. If I give each of 9 people 2 2/3 pizzas, then I have none remaining. — InPitzotl

You're intentionally avoiding the point, and I must say, lying, when you say 2/3 of a pizza is an "exact quantity". Sorry, no offence meant, but I feel it's necessary to point this out. You have no qualifications here to stipulate the size of the pizza and whether it might be divided in thirds, so it's impossible that this represents an exact quantity.

never" applies to all steps in the process. And all steps are finite. So this does not apply "after" we put an infinity of 1s. (I would argue there's no such thing as that after). — InPitzotl

OK, I agree with you here, so at least we agree on something. There can be no "after" we put an infinity of 1's, because it is impossible to put an infinity of 1's. If it were possible to do that, then someone might do it, and then there would be an "after:" it was done.

We can conclude that for all steps. But we cannot conclude that "after" we put an infinity of 1s, which is the very thing you're making a truth claim about. — InPitzotl

Actually my truth claim was that no matter how many 1's we put, there is still a remainder. So we can remove the needless qualification of "even after we put an infinity of 1's", since we both agree that this is impossible, and just adhere to the basic premise. No matter how many 1's we put, there is still a remainder.

I don't think that at some point we'll have enough 1s. — InPitzotl

OK, that's fine, I'll accept that as an honest answer. Now, can you give me an honest answer to how you think the remainder is dealt with then, such that we can end up with an "exact quantity".

The literal string .111... refers to an infinite string starting with .111 and followed by a 1 for every finite ordinal position; that is, if you count the first 1 as 1, the second as 2, and so on, there is no finite n such that the nth position does not have a 1 in it. — InPitzotl

Now, here you go and contradict the only thing we could agree on. We agree that one cannot put an infinity of 1's, and now you are claiming that ".1..." means that an infinity of 1's has been put there. Don't say I do not understand the language, because it's right there in English. Do you not apprehend a contradiction here? Or, are you saying that you're putting an infinity of 1's there, and insisting that there is no "after" this?

Your proof falters because it does not apply to the one thing you're making a claim about. .111... is an infinite string; that is the thing under discussion. — InPitzotl

No, .111... cannot refer to an infinite string, because we've agree that we cannot put an infinite string there. Now if you go and put an infinite string there you've reneged on our agreement, and I'll insist that there is still a remainder even after you've put your infinity of 1's there.

You are now claiming to do what we've agreed is impossible. Which do you accept as the truth, can we put an infinite string there or not? If you say that .111... refers to an infinite string that is somewhere else other than there, then how is it relevant?

What your proof being wrong means, instead, is that your reasoning that there is a remainder is invalid. — InPitzotl

Is that so? You've refuted my proof by proposing that there cannot be an "after" one puts an infinity of 1's there, and then going and putting an infinity of 1's there. Now we are at the point of after you put the infinity of 1's there, so all you have done is disproven the premise of your refutation.

It would seem so. Your comments are still off topic.

Back to the topic here:
We can prove that all the procedure does here is give us 0, decimal point, followed by endless 1s.
And we can prove that without writing down 0, decimal point, followed by endless 1s — it's an artefact of the procedure, and the proof involves mathematical induction and such.
Doesn't really matter much whatever anyone makes of it, that's how the arithmetic works.
That was the topic brough up, though we can prove more than just that (repetend length is 1).

But, proof or not, this should be intuitively clear. You understand? If yes, then you're free to suggest a means to communicate this unambiguously, or you can follow typical conventions like 0.1¯=0.(1)=0.111…0.1¯=0.(1)=0.111…. — jorndoe

You keep referring me back to the same post, so that I've read it numerous times now, and still don't see the point. You claim it ought to be "intuitively clear" but I'm sure my intuition is quite different from yours.

All I can say is that you seem to contradict yourself. First you say "Doesn't really matter much whatever anyone makes of it, that's how the arithmetic works.". Then you say "you're free to suggest a means to communicate this unambiguously, or you can follow typical conventions like 0.1¯=0.(1)=0.111…0.1¯=0.(1)=0.111…".[/quote]

Doesn't the second statement directly imply the falsity of the first?

You didn't really answer the question though. Do you believe that generations of mathematicians are aware of this, and yet for some reason continue to use them, or are they unaware, and you're just smarter than everyone else? — Michael

I don't think you can cast a net of generality on all mathematicians in that way. Some follow the discipline in such a way that they would apply the principles without being aware of the underlying issues. Fishfry might call this applied math. Some question the underlying principles, as indicated by jgill. Fishfry might call this pure math. If I understand fishfry's proposed divisions.

Furthermore, there are multitudes of complex problems involved with what might be called "pure math". If there is such a thing as "pure math" it would involve analyzing these problems. And those who are interested in addressing the problems direct their attentions toward the issues which interest them. For instance, jgill suggested I direct my attention toward the axiom of choice, but it's not my interest right now.

So there's no issue of anyone being smarter than anyone else, I don't know how you would even judge such a thing. It's a matter of where one's attention is directed. I happen to have an interest in music, and musicians work with a fundamental unit called an octave, along with divisions and multiplications, using frequencies to produce harmonies and dissonance. So the matter of what can and cannot be divided into equal parts is interesting to me. The issue of the acceptable divisions of a unit has never been resolved. And to claim as InPizotl seems to, that a unit can be divided in any way one pleases is totally unrealistic. However, notice that my interest in the problems of division is piqued by my interest in music, such that the pure side of my math interest is still guided by the applied side. And this is why I do not accept fishfry's proposed division.

I agree that just because you argue from certain premises doesn't mean you agree with them. But you are being disingenuous here. I could easily go back to our older discussions and show you where you accepted the rationals in order to deny 2–√2. I don't take this as a serious remark. Your prior posts don't support your claim that "I was only kidding about the rationals." You are retconning your posts and I'm not buying it. — fishfry

Sorry, but I have no idea what you're talking about fishfry. The stuff you claim here makes no sense to me at all. When did I say I was just kidding?

But that is fantastic! If you have discovered a specific inconsistency in the ZF axioms, you would be famous. Gödel showed that set theory can never prove its own consistency. To make progress we must either assume the consistency of ZF; or else, equivalently, posit the existence of a model of ZF. This by the way is what some readers may have heard of in passing as "large cardinals." For example there's a thing called an inaccessible cardinal. It can be defined by its properties, but it can't be shown to exist within ZF. If we assume that one exists, it would be a model for the axioms of ZF; showing that ZF is consistent. — fishfry

You know, ZF is only one part of mathematics. If axioms of ZF contradict other mathematical axioms, then there is contradiction within mathematics. In philosophy we're very accustomed to this situation, as philosophy is filled with contradictions, and we're trained to spot them. So we might reject one philosophy based on the principles of another, or reject a part of one philosophy, and so on. There is no reason for an all or nothing attitude. Likewise, one might reject ZF, or parts of it, based on other mathematical principles.

We do have exact definitions for natural numbers and integers, rationals, reals, complex numbers, quaternions, octonions, p-adic numbers, transfinite numbers, hypereal numbers, and probalby a lot more I don't even know about. But ironically, and confusing to many amateur philosophers, there is no general definition of number. A number is whatever mathematicians call a number. The history of math is an endles progressions of new things that at first we regard with suspicion, and then become accusotomed to calling numbers. — fishfry

So, mathematicians can call whatever they want, "numbers", but not philosophers? Whenever a philosopher uses the word "number", the mathematician has the right to say "your wrong, because you are a philosopher not mathematician", yet the mathematician can make "number" refer to whatever one wants, especially something different from whatever the philosopher wants it to refer to. Isn't "number" a weaselly little word? Whenever the philosopher comes close to nailing down a definition, the mathematician says no that doesn't suit me right now, I still want to be able to use the word in other ways.

I get the picture, the mathematician doesn't want "number" to be defined, in order to proceed in using the word however the mathematician pleases, in acts of deceptive equivocation. This is why philosophers are trained to recognize such inconsistencies, so that we can address such sophistry.

You're not only a mathematical nihilist. You're a mathematical Philistine. "One who has no appreciation for the arts." You deny the art of mathematics. You know nothing of mathematics. — fishfry

Actually I love the art of mathematics. You even said so yourself in this thread, that I obviously care very much about mathematics. Notice I didn't disagree with that #2. But people are always doing something with their artwork, and mathematicians like to "prove" things. And the nature of that art of mathematics is that when it is applied it is extraordinarily persuasive. So when mathematicians use their art for deception, I especially despise that, because it gives them an extraordinary power to succeed.

You're just confusing pure and applied math. And missing the lessons of history that what is abstract nonsense in one era may well and often does become the fundamental engineering technology of a future time.

You know when Hamilton discovered quaternions, nobody had any use for them at all. Today they're used by video game deveopers to do rotations in 3-space. Did you know that? Are you pretending to be ignorant of all of this? That when you run the world nobody will do any math that isn't useful today? — fishfry

Actually, until you demonstrate the validity of your supposed distinction between pure math and applied math, you have no argument here. The fact that someone discovers something which is useless to the person at the time, because they may have been doing something else at that time, does not mean that they were not involved in some application at that time. So, when a principle is discovered, and not put to use for hundreds or thousands of years, this does not mean that the person who discovered it wasn't involved in application at the time.

Thanks for the lesson in terminology InPitzotl. But I don't see how expressing the remainder as a fraction resolves the issue of the remainder. The fraction is just an expression of an unresolved division problem. So in expressing "24÷9=2 rem 6", as 24÷9=2 6/9" or "2 2/3", all you are doing is replacing the remainder with an unsolved division problem. It's not really any different than having a remainder because what you are doing is saying the division can be carried out to this point, but the rest remains not divided.

Prove it. — InPitzotl

It's been proven. It's called inductive reasoning. Every time someone adds another 1, there is still a remainder. And never ever is there not a remainder. And since the nature of the numbers stays the same, we can conclude that this will always occur. I don't see what makes you think that at some point we'll have enough 1s that there'll suddenly be no remainder.

You misunderstand — this is about the procedure, not about writing.
Go back, think about the procedure instead. — jorndoe

I don't see your point. The "procedure" demonstrates very clearly that there is a remainder in this division problem. So, it's quite obvious that 1 cannot be divided by nine. As I explained earlier in the thread, some numbers just cannot be divided by other numbers. It's impossible, and we ought to respect this simple brute fact which is inherent to the nature of numbers. I could go back over this again if you'd like, but you'd probably just deny the evidence like InPitzotl, and postulate like Michael, that any number is divisible by any other number. But why employ a false postulate?

You misunderstand — this is about the procedure, not about writing.
Go back, think about the procedure instead.

In fact, we can go much further, though it requires some abstract thinking, e.g.: Repeating decimal (Wikipedia)

Hm regarding abstract thinking, in analogy: suppose we want to prove p; then by some other means we find that we can prove that p can be proven; well, then we're done with our initial task (unless we're curious). — jorndoe

The op asks whether I think this "other means" is acceptable. My answer is no. The reason is that the so-called "other means" does not actually achieve what it is supposed to achieve. That is because the thing which it is supposed to do is actually impossible, by the very nature of numbers themselves, and mathemagicians like to use smoke and mirrors to create the illusion that they have figured out a way to do what is impossible.

We know that there isn't. I don't know why you think that you know more about maths than generations of professional mathematicians. With everything you're saying about numbers and division and the like, I honestly want to know what is going on in your head. Do you think that there's some grand conspiracy and they're lying to us? Do you think that you're an enlightened prodigy who is able to outsmart the people who have studied this stuff for years, despite probably having little to no formal training of your own? Seriously, I want to know. The psychology of this is fascinating. — Michael

I suppose we disagree then, on what "we know", and of course that's quite common here at TPF. I've seen some mathematical proofs, and as I've shown that the axioms are full of inconsistencies and contradictions. A lot of these so-called "proofs" are smoke and mirrors built on false premises and therefore unsound.

I think the most important thing here is, what is MU's criteria for truth? MU made an actual truth claim here to counter a proof. Can MU offer a proof in return, or does MU think he has a better truth criteria? Either way, I want to see the proof or this better criteria. — InPitzotl

The criteria for truth is honesty. I provided my argument, and you disagreed with the second premise, that no matter how many 1s you place after the decimal point there will still be a remainder. I think that this premise is true and I am honest in this claim. If you claim that you do not think that this premise is true, I think you are being dishonest. If that is the case, then your claim is false.

The thread might continue until someone produces an infinity of 1s, and you guys see that there is still a remainder. But then some smart ass will suggest that if we add another 1 the remainder could be resolved, and we'd start all over again and produce another infinity of 1s. And there'd still be a remainder.

The further point, which is not a matter of fact, but simply an opinion, is that the expression ".111..." creates the illusion that the remainder in the division problem has been resolved, and so there is no remainder. However, we know that no matter how many 1s we put, even after we put an infinity of 1s (whatever that means), there would still be a remainder. So this illusion, that the remainder has been resolved, is quite clearly a matter of deception.

The "unending 1s" indicates that there is a remainder. And, anytime we express the inverted division problem as multiplication, the remainder must be added. Example: seven divided by three equals two time three plus one. In this case, one is the remainder, and must be added into the multiplication expression.

In the case of the op "1/9=.111...", the "..." indicates that there is a remainder which has not been stated. So in the inversion, 9x(0.111...)the remainder is not indicated, and not accounted for. Therefore it is not an accurate representation.

Kind of dull I suppose, repetitive, something that most elementary schoolers catch on with quickly, but, anyway, the proof sure saves a bit of paper, so we'll then just write that as "0.111...". — jorndoe

Do you agree with my premises?
P1. Any time there is a remainder in division, that remainder must be added into the equivalent multiplication inversion, in order for there to be accuracy in the equivalence.
P2. No matter how many 1s you write, and however you express this multitude of 1's, there will always be a remainder when you divide one by nine.
P3. The multiplication expression of the op "9X(0.111...) does not add the remainder.

Tell me, what what you think is wrong with my conclusion that the expression of the op is not accurate.

You're only demonstrating your incompetence, over and over. You're just proving you don't speak the language. — InPitzotl

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. So I am very careful in learning language

I'm fine with defining division as the inversion of multiplication, if that's what you want, so long as you accept that any instance in which an operation of division would result in a remainder, this cannot be cannot be an act of division. As I've explained, multiplication has no place for the remainder, so under this definition of division, such cases cannot be called "division". Therefore under this definition, 1/9 cannot be a representation of division. Do you accept this. If not, then how do you represent in multiplication, the remainder which results from 1 divided by 9?

Remainders aren't fractions. But they do indicate the numerator of the fractional part of a mixed number. You have no real point here, though. No amount of confused gibberish you spew prevents me from sharing two pizzas evenly between three people, nor does it change the method by which I do so. All you're doing is inventing fake contradictions. — InPitzotl

It really looks like you're the one confused.

Wrong. I never mentioned foot long rulers... I mentioned foot long lengths. You could use a 50 foot tape measure to mark off these lengths starting from a point in the center of a 12 foot board. You don't even need to use that clumsy folding metal thing at the end of the tape... the distance from the 2 inch mark to the 14 inch mark is a foot. You can use foot rulers if you like, but all you need to measure a particular length is something that has that particular length, such as two marks on a tape measure. — InPitzotl

Whatever you use, sticks or markings on the ground, my criticism holds. You are not distinguishing between a unit of measure, "a foot", and a measured foot on the ground, or foot ruler. We were talking about the units of measurement, "a yard", "a foot", not a marked off measurement. Consider that the number "2" is a unit of measurement, rather than a collection of two things. then you will see the mistake you are making, like referring to the collection of two things (like the markings on the ground) as the unit of measure called "2" (like the unit of measure called "a foot"). Do you see the difference?

But I don't mind fiddling with the puzzle. — InPitzotl

OK then do you agree to what I stated above? If division is defined as the inversion of multiplication, then any proposed division in which there would be a remainder cannot be a division, because there is nothing in multiplication to account for the remainder.

Again, what's the problem? — Michael

The remainder is not identified or given a specific numerical value. It is hidden to create the illusion that it has been dealt with.

1313 , 655655, XIIIXIII, and 11121112 are the same number. You don't seem to understand that different symbols can be used to refer to the same thing. — Michael

You're right back to where InPitzotl first engaged me, and the discussion I had with fishfry in a previous thread. It is not true that "13" and "65/5" refer to the same thing. They are equivalent. Do you not understand the difference between being equivalent and being one and the same thing? "Equivalent" allows that two distinct things have the same value. "The same thing" does not allow for two distinct things. In the example above, "65/5" is clearly not the same thing as "13". It doesn't even have the same meaning. In order that they refer to the same thing, we need to reduce the meaning to a simple numerical value, and apply a principle which makes a value into an object. This way we can say that the value of "13" is the same as the value of "65/5" and since that value is an object, then they both refer to the same thing. But it's doubtful that there is any truth to the premise that a value is an object, or that there is an acceptable principle which turns a value into an object..

You're getting so lost in what the symbols look like that you're not paying attention to what they mean. — Michael

Oh, it's not me who is not paying attention to what the symbols mean. That's why I offered a definition of "number". It's people like you, who claim that "65/5" represents a number rather than what it really represents, a relation between two numbers, who are not paying attention to what the symbols mean.

You seem to be reifying. — Michael

Again, you've got this backward. It's the people like you, who claim that a numerical value is an object, and therefore "13" and "65/5" both refer to the same "number", who are reifying.

You misuse the word "correct" IMO. — jgill

"Correct" is a value judgement, and it needs to be grounded or based in some principles. I base "correct" in truth, but Banno clearly bases "correct" in what is conventional. I think that's Wittgenstein's influence, which gives this notion of "correct" which is unacceptable to me.

Instead of writing virtual tomes about the drivel on this thread you should apply your critical thinking skills to actual controversial items like the Axiom of Choice. — jgill

As I explained to Banno, I don't mind discussing trivial things, like the subject matter of this thread. But do you see how seriously some people take these trivial matters, hurling the insults at me as if I've just attacked the most sacred thing in the universe, instead of simply noticing that I have a difference of opinion? In philosophy we respect a difference of opinion. But for some reason in mathematics a difference of opinion is perceived as a threat, so the defenders must attack and belittle the person with a difference. It's as if the mathematicians know and accept that their principles are doubtful, so they are insecure, and therefore they must attack and keep the skeptic away. Can you imagine how offended they would be if I addressed something of more importance?

Maybe there is a mathematical universe, and somewhere, through all the "chess game rules" mathematicians study, a path to understanding it can be found. — jgill

If there is such a thing, then it is part of "our world". And so the mathematical axioms must be "true to it", in order to be correct. Then the chess game analogy fails.

1÷2=0.51÷2=0.5

0.5×2=10.5×2=1

What's the problem? — Michael

This is where the illusion is created, in incidences where there is no problem, just like my example of eight divided by two. The illusion takes the form of a general rule, that division is the inversion of multiplication. However, the cases of division in which there is a remainder demonstrate that the inductive reasoning which creates the general rule is faulty, if we allow that division can be carried out in these cases.

0.999... = 1, so you do get 1 by multiplying 0.111... by 9. — Michael

This depends on how one deals with "the remainder" in division. I was following InPitzotl's principles to demonstrate the inconsistency in what was argued. If one is divisible by nine, as InPitzotl claims, then division is not a direct inversion of multiplication because there is a remainder signified with "...".

One way to resolve that inconsistency, which I've been arguing for, would be to establish the true nature of a mathematical element signified by "1" as a unit which cannot be divided, as the common definition of "one" implies. It is not comprised of parts like two and three are, and therefore is not a multiplicity. If it's not a multiplicity it cannot be divided into constituent parts. But this principle would deny your other example, of one divided by two, as well, and in each case where there is a remainder, the proposed division would be denied as impossible . This would allow for the truth of the inductive principle that division is the inversion of multiplication..

.Furthermore, this does not mean that fractions are not valid mathematical representations. It just places them into a category other than numbers, so that they do not get conflated and confused with one another. It is to recognize, maintain, and uphold the real difference in meaning between symbols like this, "1", "2", "3", which represent a number (quantity), and symbols like this, "1/2", "1/3", "1/4", which represent a relation between numbers. There is a real difference between what is internal to an object, it's constituent parts, and what is external to an object, its relations to other objects. and this difference needs to be respected.

This distinction must be maintained because numbers are often conceived as Platonic objects, and when they are given such ontological status it is important to recognize that what exists between numbers is not of the same "material" (implying the same meaning) as the material (meaning) which comprises the object, a number. The Platonic object is an element of meaning, so different types must be separated categorically. So for example, "2" represents a number, but what is signified is that there are two parts, which are united by some principle of unity. But "1/2" represent a division by 2, a dissolution of that principle of unity which makes "2" signify one unity. So if "1" represents a fundamental number, with no such parts as a multiplicity has, there is no such unifying principle, and it cannot be divided. If a person wants to divide "1" into parts, this cannot be done by following the same rules which we would use to divide "2" into parts, because the principle of unity in the object "1" is completely different from the principle of unity in the object "2". What I propose is that the principle of unity in "1" implies that it cannot be divided into parts.

If we proceed to deny this distinction then there is no principle by which a number might be an object, and if it is insisted that numbers are objects, there is absolute lawless chaos as to what distinguishes one object from another because the features which separate one mathematical object from another, as the principles of divisibility, are completely ignored. .

By definition, division is the inverse of multiplying. — InPitzotl

I've never seen any such definition of "division". The usual definition involves dividing something and this has nothing to do with multiplying.

You turn a blind eye to the evidence, to insist on a falsity. Take the circumference of a circle, and divide it by the diameter, the result should be pi. But to start with the same diameter, and multiply it by pi, will give you a different number as the circumference, because you'll have to round off pi. This is the same situation in the op. Start with one, divide it by nine, and you get .111.... Start with .111... and multiply it by nine, and you do not get one, you get .999.... In these cases, when you take a number and divide it by another number, then take the quotient and multiply it by the divisor, the product is different from the original number. Therefore division is not a direct inversion of multiplication.

Of course you insist that .999.. is the same as 1, and therefore division is simply an inversion of multiplication. But this is just begging the question. Your false assumption that the two are the same thing, supports your conclusion that division is an inversion of multiplication, and the false assumption that division is an inversion of multiplication supports the claim that the two are the same.

You ignore the evidence of the fundamental difference between multiplication and division.. This evidence is that when you carry out an operation of division there is often a remainder. There is never a remainder in multiplication, nor do you start with a remainder, There is no place for a remainder in multiplication, yet there often is a remainder in division. Therefore division is not simply an inversion of multiplication.

Even if you provide examples where one is a direct inversion of the other, (eight divided by two equals four, and four times two equals eight, for example), this is not sufficient for the inductive conclusion that division is the inverse of multiplication. All it takes is an example or two, such as the ones I provided, to invalidate such a conclusion. Whenever there is a remainder, there is evidence that your conclusion is invalid. So you make the inductive rule (division is the inverse of multiplication), then when exceptions to the rule are shown to you, which ought to make you think twice about the validity of the rule, you simply deny that the exceptions are real exceptions, by claiming that .999... is the same thing as 1.

Obviously, you think that "the remainder" in an operation of division is not a real thing, that its existence can be denied and ignored, and so we can say that division is simply an inversion of multiplication. You turn away from, and ignore the overwhelming evidence that you are wrong.

Not true. 1 yard = 3 feet without your parts. There is a different sense of part that is in play here, though. The particular length that is 1 yard is length-equivalent to 3 feet in a specific way... there are two positions (particular points) along a yard-length section that separate a yard-length into 3 contiguous equivalent lengths. Each of these three contiguous length has the particular length of a foot. Conversely, if we take three foot-lengths so arranged such that they are laid out end to end meeting at these two points, then the total distance covered by these three foot-lengths is itself that same particular length we call a yard. So in this sense, a yard-length is composed of three foot-length partitions, each of which we can call a part. Note that you can slice the ruler at this point if you choose and make separable parts, but that does not in any way affect the invariant condition of being a particular length measured by these particular quantities (1, 3) of particular length-units (yard, feet). — InPitzotl

You seem to be conflating units of measurement, foot, yard, etc., with length, which is the determined measurement of something. So your argument here really makes no sense. You argue that three one foot long rulers makes up a length which is a yard, and you conclude therefore that a yard, as a unit of measurement consists of these three parts. But this is clearly false, because this is just one example of something which measures a yard, three one foot measuring sticks, and it in no way indicates that the unit of measurement "a yard" is actually composed of these parts.

So if there's no problem with the math, you're going to suffer. And that's exactly the situation you're in... there's no problem with the math, and you're suffering. Take another look at the reactions your getting and tell me I'm wrong. — InPitzotl

I'm not worried about that, because the problems in math are glaring. So if it takes "no problems with math" to make me suffer, I think it will be an extremely long time before I start to suffer.

What's with the appeal to others? Banno was in the same boat as you, implying that if others agree it must be correct. It's as if when someone comes up to you and pats you on the back saying "your right", this makes you right. Then you might have a whole group of people in a big circle jerk, patting each other on the back saying "you're right", and "I know I'm right, and so are you", onward and onward, blissfully unaware of the truth, when they're not really right. And if someone from the outside tries to point out your mistakes you shun them, saying you're not part of our circle, you don't understand our language, go away, we don't want to hear what you have to say, it interrupts our self-congratulations.

What's hurting you is the fact that by pitting yourself against the theory that defines division this way using your worthless theory, you're defacing your own image in the eyes of others who know better. There's a severe risk that people will equate your value to the value of your views, because your views are total garbage. — InPitzotl

Oh, poor me. Don't you just feel so sorry for a poor soul like myself? I'm standing up here in front of others, doing whatever I can to make a fool of myself. And you want to shelter me, and protect me. What kind of bullshit is this? You're even worse than Banno.

Math is a language that does what it says on the tin... this follows; that is consistent, and so on. — InPitzotl

What I am arguing is the lack in consistency in math. How many different "number" systems are there, natural, rational, real? How can you believe that there is any consistency within mathematics as to what "number" refers to?

I'm not exactly sure what it is you even think it means for aa to be divided by bb. — Michael

I think it's quite clear what division is, it's to divide something into parts. You think it's to do a certain type of calculation. I would go along with this, as a theoretical type of division, so long as there are some rules involved.

I hope you don't think that division is simply an inversion of multiplication. If you do though, then we ought to adhere to the rule that if there is going to be a remainder in any calculation of division, then this calculation cannot be carried out, because it cannot be inverted into multiplication. This would mean that some numbers cannot be divided by others. But if you insist that any number might be divided by any other number, then we need to accept that division is not a simple inversion of multiplication, because we can have remainders.

Point1: Ok a fair answer but still a deflection. The question is why you earlier believed in the rationals, but now do not believe in 1/9. Since 1/9 is a rational number, being the ratio of two integers, 1/9 is rational. — fishfry

I don't think that I said I believe in the rationals. I was arguing using principles consistent with the rationals, so you inferred that I believe in the rationals. But arguing using principles which are consistent with one theory doesn't necessarily mean that the person believes in that theory. So I don't see your point here, I think you just misunderstood.

Point 3: Do you regard the rules of chess as needing a "good dose of skepticism?" Why or why not? Perhaps you are putting more ontological certainly into math than math itself claims. I personally don't think that .999... = 1 is "true" in any meaningful sense. In the real world the notation isn't defined at all, since there are no infinite series because as far as we know, the axiom of infinity is false.

So YOU are the one setting up strawman claims on behalf of math, that math itself doesn't claim.

How can you complain about the rules of a formal game? How could one be "skeptical" about the rules of baseball? What does that even mean? — fishfry

What I argue against is inconsistency in the rules. And, if someone asked me to play chess, and I noticed inconsistencies in the rules, I would point them out.

I wonder what claim you think it being asserted by .999... = 1. It's a statement in the formal game of modern math. You can no more object to it than you can object to the rules of chess. — fishfry

As I've demonstrated, we can still object to a specific set of mathematical rules, using a different set of mathematical rules to make that objection. This is due to inconsistency in the rules of mathematics. Look at how many different systems of "numbers" there are. You, in this very post, have accused me of being inconsistent for switching from rational numbers to natural numbers. This is not my inconsistency, it's inconsistency within the rules of mathematics. Imagine if chess were like this, and every time you wanted to play a game with someone you had to discuss all these different and inconsistent conventions, deciding which ones to play by.

No. Math isn't true or false any more than chess is true or false. If you criticize math for having rules that are not technically true of the world, you must make the exact same criticism of chess. Do you? — fishfry

I don't agree with this analogy at all. We apply mathematics toward understanding the world, and working with physical materials in the world. This is completely different from the game of chess. If the principles of mathematics were not to some degree "true of the world", they would not be useful in the world. There is no such requirement in the game of chess. So it's completely acceptable to criticize the principles of mathematics when they are not "true of the world", because mathematics is used for purposes which require them to be true of the world. But the game of chess is not used in this way. So if I were to criticize the rules of the game of chess, it would be if I thought they were deficient for serving their purpose.

Suppose for sake of argument I say yes. The axioms of math are faulty by virtue of not being true of the world. Will you then grant me that the rules of chess are likewise faulty by virtue of not being true of the world? — fishfry

This is a nonsensical analogy. The rules of mathematics are used for a completely different purpose than the rules for chess. And the rules of math, to whatever degree they are not true of the world, lose there effectiveness at serving their purpose. The rules of chess are not used in that way.

I even provided quotes from Dr. Feynman, describing how when electrical energy moves, it moves through the field, rather than as electrons moving. But people here insisted that what was real was energy moving as electrons. We have the same problem here with regard to light energy. People here insist that electromagnetic energy moves as photons (particles) rather than as waves in the electro-magnetic field.

"Material" here is in the contemporary sense that if it is affected by and/or affects material things, it comes under the material world's purview (e.g. spacetime, electric fields, etc.) In short, if we can detect it, even indirectly, it gets classed as material. — Kenosha Kid

You're not going to get very far by limiting "immaterial" in this way. It's a straw man which doesn't in any way represent what a person who believes in the immaterial believes in. We believe that certain things which effect the material world, intention, soul, and God, for example, have an effect on the material world. In fact the existence of these immaterial things is commonly demonstrated by their effects on the material world.

he point was, is demonstration that it should exist sufficient to justify belief in it, even though we cannot demonstrate it itself. — Kenosha Kid

This is an ambiguous distinction. A logical demonstration is a demonstration of what "should" be. It is the only type of demonstration which can be used to justify the belief in anything. The necessity which supports belief is a necessity of what "should be".

The modern view of the material world is that everything, except maybe gravity, is quantum fields. If it exists, it exists as a collection of interacting excitations of those fields, fleeting or permanent. There are many fields, all with their own properties. These underpin the entire Standard Model. — Kenosha Kid

The idea that a "field" is something material, is what needs to be demonstrated. I once argued on this forum, that fields are believed by physicists to be real, active, causal, material things, and I was laughed at for this. But that's just an indication that people have the tendency to laugh at the things which I argue for. Perhaps you are better positioned to make this argument.

Then they define set "Q" as the set that contains the elements a/b where a and b are elements in set "Z" and b is not 0. — Michael

This is where I have a disagreement. There are many instance of a/b, which cannot be called an element. As I described already, in many cases a cannot be divided by b, it is impossible. One might express the ratio a/b, but the operation which is required to produce an element from this ratio cannot be carried out, therefore there is no element in these cases. So we have a faulty set here consisting of some necessarily non-existent elements.

A ninth is the multiplicative inverse of nine. A twenty fourth is the multiplicative inverse of twenty four. Dividing by nine is equivalent to multiplying by a ninth. "A ninth of" is multiplying by a ninth; just as "five ninths of" is multiplying by five ninths. There's no problem here. — InPitzotl

There is a problem, dividing is clearly not the inverse of multiplying. The evidence of this is the existence of irrational numbers, which are derived from dividing, but not derived from multiplying. For a mathematician to say that dividing is simply the inverse of multiplying is like a physicist who says that time can be modeled as going either way, future to past, or past to future, one is just the inverse of the other. There is ample evidence that this is not true, and those who overlook the evidence, like yourself, start making false claims.

A yardstick measures 1 yard. It has 3 feet in it. Each feet has 12 inches. Those 12 inches usually are marked in fractions of an inch; typically at least an eighth of an inch. Now don't get scared... an eighth of an inch is part of an inch which is part of a foot which is part of a yard. — InPitzotl

This is all wrong. These are measurements, and what you are describing is equivalencies. A
"yard" is equivalent to three feet, and a foot is equivalent to twelve inches. Each term refers to a particular length, and the length is one unit, without parts. If a yard, or a foot consisted of parts, there would have to be something within that unit to separate the individual parts, one from another. Clearly there are no such separations within a yard or a foot, and there are no such parts within these units. What would that separation be made of? And without the separation there are no parts. Do you know what a "part" is?

If the only problem with the language is that you have a problem with it, then you are the problem. — InPitzotl

I hope you realize that this is a very selfish expression. And I really hope you don't behave this way in your common interactions with people.

The reason I'm talking to you is that I care about you. — InPitzotl

Oh sure, the person who's trying to convince me that division is really just inverted multiplication is doing this because they care about me. I think you're like Plato's philosopher king, with the noble lie. You actually believe that your lying to me is for my own good. Or are you so naive to actually believe that there is no more to division than an inversion of multiplication?

I almost agree... your whining about something that works gets us nowhere. The only part where I disagree is that your whining about something that works has negative effects. — InPitzotl

Whether or not it "works" is not the issue. I have no doubt that it works. What is at issue is the truth. You know, until they're exposed, lies and deception work. Don't you?

Curious about your 1/9 concerns. A while back you told me you believe in rationals but not sqrt(2). But now you don't seem to believe in rationals. What's up? — fishfry

Different thread, different argument. What makes you think that I believe in any sort of mathematics? What I believe is that it's about time for a good dose of healthy skepticism to be directed at mathematical axioms.

Secondly, can you give me a yes or no response to this question? Do you agree, either by personal understanding or by taking my word for it, that regardless of whether .999... = 1 is "true" in any metaphysical sense, it is still the case that it's a formal consequence of the axioms of ZF set theory? — fishfry

Sure, why would I deny this? It's been shown to me in so many different ways. But if you have good reason to believe that the consequence is a falsity, then it's just evidence of the faults of those axioms. Do you agree, that if the the formal consequence of the axioms is to produce a falsity (whether or not you believe the present example is a falsity), then there is likely fault in the axioms?

Also it seems to me that what you call "numbers" mathematicians call "natural numbers" (or maybe "integers"; do you consider negative numbers as numbers?). There's more than just natural numbers in mathematics; there's rational numbers that include the commensurable fractions like 1919, real numbers that include irrational numbers like 2–√2, and more. — Michael

I can't answer this question without an acceptable definition of "number". It seems mine has been rejected. But as I just described, one cannot be divided by nine, because it means that one is a multiplicity, when it is defined as a single, or simple.

don't see what purpose there is in saying that non-natural numbers aren't numbers, and latching onto the OP saying "as a matter of representing numbers" completely misses the point of this discussion. — Michael

The point is that a ratio is not a number, it is a relation between two numbers. And some ratios cannot be expressed as a number, the relation expressed by pi for example. Which ratios can and cannot be expressed as a number is a matter which might be discussed. I am taking the extreme position to claim that none of the following 1/2, 1/3, 1/4, 1/9, etc., can be expressed as a number, because it contradicts the definition of "one" to say that one is composed of a multiplicity which can be divided.

If the others, in this thread, insist that all ratios can be expressed as a number, then I want to see that definition of "number" which allows for this. Or is "number" just some meaningless word which mathematicians can use however they please, in any random way?

For example, if the quantity expressed by "one" can be divided however a mathematician wants to divide it, then it must be a multiplicity composed of an infinity of parts. But that contradicts its definition of "single", and makes "one" into a meaningless term. Having no restrictions derived from what it means to be "a single object", or some such thing, a person might refer to any multiplicity whatsoever as "one". And this leaves "one" as an absolutely nonsensical term.

This rules out that you understand the language and refuse to speak it. You genuinely don't speak the language of math. — InPitzotl

Of course. If you're just now noticing, I refuse to use that deceptive language, loaded with contradiction in its axioms.

(a) 1/9 of nine is 19×9=119×9=1
(b) 1/9 of eighteen is 19×18=219×18=2
(c) 1/9 of 27 is 19×27=319×27=3
(d) 1/9 of thirty six is 19×36=419×36=4
Do you see the multiplication now? — InPitzotl

That is just deception. To divide nine into nine parts, or to divide eighteen into nine parts is very clearly division. To express this as multiplying nine by another number, "1/9", or 18 by "1/9", is an act of deception. Claiming that division is multiplication is deception.

When I slice one pizza into eight slices, it's still one pizza. — InPitzotl

No it isn't it has been divided. Either it is one object, or it is eight objects. To claim both is to claim contradiction.

What you fail to understand, MU, is that many things can be divided, even if you count one of them. Also, lots of things have whole-part relations; given a loaf of sliced bread with 24 (equal) slices per loaf, I can give you 3 loaves, or 3 slices... I'm still doing nothing but counting, but I'm giving you different "particular quantities" of bread. The slice quantity is much smaller than the loaf quantity. This is what's known as a unit. If I give you 3 slices, I'm giving you 3/24 loaves. We might also say 3/24 of one loaf = 3 slices. We can also apply units to continuous measurements, such as lengths along those dimensions you alone denied exist. — InPitzotl

If this is true, then we need to define how to distinguish a whole from a part, so that we are not referring to the part as "one", when it is really 1/8 of the whole, and we are not referring to the whole as "eight" parts when it is really one whole. I would enter a discussion of parts and wholes with you, so long as we have principles whereby we can distinguish one from the other, and not just randomly decide to call this a part, and that a whole, at will throughout the discussion, because that would get nowhere.

Also I think you're putting the cart before the horse. We don't start with some definition of "number" and then see which things satisfy that definition. Instead we have the mathematical terms 11, 1919, 2–√2, etc. which mathematicians place in sets that they decide to name "natural number", "rational number", "real number", etc. and then lexicographers try their best to come up with an adequate description of what the word "number" means when they write their dictionaries. — Michael

That's putting the cart before the horse. Before deciding which items go into which set, we need to define the conditions of the set. No one puts a whole bunch of random terms into one set, then names the set "numbers". if that were the case, why wouldn't we put "house" and "car" into that set called "numbers" as well?

A ninth is the specific particular quantity corresponding to dividing one into nine equal units. — InPitzotl

You don't seem to understand. "One" does not represent a quantity which can be divided. Any multitude such as two, three, or four, can be divided, because being a multitude means that it is composed of parts and therefore can be divided into those parts. If one could be divided, then you are saying that it is made up of parts and is therefore a multitude, and not one. If it could be divided in two, then you are saying that it is made of two parts, but that would mean that it's two, not one. If it could be divided in three, then that would mean that it consists of three parts, and is really a quantity of three.

If you think that the quantity represented by "one" can be divided in any way that you please, then you deny the meaning of "one" as a single thing, because you are saying that it's really a multitude of as many things as you want it to be, existing as a unity. But that's nonsense, because that's what the other numbers represent, multitudes which have a quantitative value. If you say that one can be divided any way you please, then you are saying that "one" represents a multitude with no particular quantitative value, it is however many things you want it to be. But that's nonsense, because we all know that "one" represents a single thing, not a multitude of however many things you want it to be.

That's quite interesting. What I was saying here is a direct analog of your points about fractions and pie applied to money according to my best assessment of what gibberish you're trying to push. So if you yourself don't understand this, maybe you should heed the advice you're trying to give me. — InPitzotl

It appears you just haven't taken the time to understand what I was saying.

What are you talking about? A whole pie is one pie, not nine pies, eighteen pies, or twenty seven pies. You mean groups. Taking a particular quantity of equal sized groups is just multiplication. If I were at a farmer's market and they had a carton of a dozen eggs, I might could barter getting one half of a dozen. He'll give me six eggs. Or maybe I need more... maybe I need two dozens. He'll give me 24 eggs. Even your precious one dozen is twelve eggs. You're choking on multiplication. — InPitzotl

Again, it appears you haven't taken the time to understand what I was saying. As a result, I have no idea what you're talking about. No one mentioned multiplication, the issue was division.


Concise Oxford,1990, p813.

That the mathematical definition of "number" changes like the weather is good evidence of what I've been arguing. We do not have any logically rigorous definition of "number", and mathemagicians just use the term however they please, referring to whatever they want as "a number".

Meta did not directly address this, or any other such proof. Instead he went to an irrelevance, his claim that 1/9 is not a number. — Banno

The issue, as I said, is the op's question: "As a matter of representing numbers, wouldn't most be fine with...". I'm not fine with it, because as I said 1/9 is not a number,.it is a ratio. The op doesn't ask for proofs or any such nonsense, it asks if you are satisfied with that way of representing numbers.

Even if 1/9 were not considered a number, the proof would stand. — Banno

I couldn't care less about the proof. The op asks, "as a matter of representing numbers, would most be fine with...". If 1/9 is not a number, then we ought not be fine with this, as it is presented as a representation of numbers. If 1/9 is not a number then the presentation, as a representation of numbers, is false.

Which premise is false? — Michael

That 1/9 is a representation of a number.

How you choose to define "number" has no bearing on whether or not 0.999... = 1. — Michael

The question of the op is are you fine with this, as a representation of numbers. It is not, do you believe that .999...=1. I really do not care whether .999...=1 or not, or how many proofs there are concerning this. I'm concerned about the question of the op, is this an acceptable way of representing numbers.

So the issue I've pointed to is whether 1/9 is a representation of a number, or not. I've argued that it is a representation of a ratio and therefore not necessarily a number. Some ratios are impossible to represent as a number. That is where we get the term "incommensurable".



My point was that ".999..." has a different meaning from "1". InPitzotl insisted that it is two names referring to the same thing. Clearly it is not, because .999... is derived from 1/9 in the op, and 1 has a simple meaning without any such baggage.

The result will be to show in even greater relief that this is a thread about Metaphysician Undercover, not about maths. — Banno

Flattery will get you nowhere.

Now you've struck the heart of the problem. Some quantities cannot be divided in certain ways. It is impossible. Three cannot be divided by nine, it is impossible. Nevertheless, mathemagicians are an odd sort, very crafty, wily like the fox, devising new illusions all the time. They like to demonstrate that they can do the impossible. Some people even believe that they actually do what is impossible. That is a problem. — Metaphysician Undercover

You didn't see this? Put up or shut up! — tim wood

I've already given my proof, based in my definition of "number". It's in my posts directed at Banno. Don't listen to Banno here because Banno's form of discussion is to pay no attention to what the other person says.

We can talk about making a theoretical division which is impossible to do, such as dividing one in nine equal parts, which is impossible to do. In reality it is contradictory to divide one into any parts, because then you are saying that it is not one, but however many parts you are dividing it into. Instead of recognizing that division of some quantities is impossible, some mathemagicians have proposed a new system of "numbers", which allows that impossible divisions can be represented as numbers.

The problem is that now they have so-called "numbers" which are outside the criteria of the definition of "number" (as presented by me), yet the mathemagicians provide no clear new definition of "number" which allows that these representations such as .111..., are actually representative of numbers. They give them a name "real numbers", so that they can refer to them, but the concept of "number" is just left vague, undefined, and full of inconsistencies.

Since I have provided a very clear definition of "number", and according to this definition many representations which are classed as real numbers are not actually numbers, so we can conclude that these real numbers are not actually numbers, based on that definition. If you want to demonstrate that my logic is unsound, I suggest you show me that my definition is false, by producing the true definition of "number", the one which allows that all real numbers are actually numbers. Otherwise we can look for a better name for things like 1/9, one which better represents what they are, such as "relations between quantities", or something like that .

perhaps in answer to Tim's question you might set out where the flaw is in this calculation - regardless of wether the items involved are numbers or not, where in your view does this go wrong? — Banno

I've already laid that out for you:

Now you've struck the heart of the problem. Some quantities cannot be divided in certain ways. It is impossible. Three cannot be divided by nine, it is impossible. Nevertheless, mathemagicians are an odd sort, very crafty, wily like the fox, devising new illusions all the time. They like to demonstrate that they can do the impossible. Some people even believe that they actually do what is impossible. That is a problem. — Metaphysician Undercover

The op uses an expression which represents an impossible division, 1/9, a ratio which cannot be expressed in numbers. There are different forms of divisions which cannot be expressed in numbers, some are called "irrational numbers". Since they are not numbers it was a mistake to start calling them numbers. But this is the bad influence which common vernacular has over logic, it inclines us to replace the rigorous logical definitions which are required for sound logic, with family resemblance. (It's similar to a number, so let's just called it a number, and then we can have a bunch of different types of numbers).

They've proved it. — Michael

They think they've proved it. Staring from a false premise does not make a sound proof. But if one doesn't recognize the falsity of the premise...

A, therefore B, where A is .999... = 1, and B is some rambling about equivalence. But here we don't merely have equivalence, we have equality. Because we have equality, they do represent the same value. I've never heard of someone so far gone as to commit an amphiboly by changing the word. But in this post, and here?: — InPitzotl

"1" represents a value which is a quantity. No one has demonstrated how 1/9 represents any particular quantity, because as I've explained, it does not represent a quantity. So how do you claim they are equal or equivalent?

That definition was already discussed in the thread. And that definition is used in the pdf provided by the op in section 1. By that definition, .999... = 1 exactly. — InPitzotl

I see no such definition. Perhaps you can produce that definition so that we can determine whether 1/9, and .111... represent numbers. According to the definition of "number" which I provided they do not represent numbers. Where's your definition of "number"?

n other words, 65 pennies, a dime and a quarter is not worth a dollar because pennies are 1/100th of a dollar and that's not a particular quantity of money. I mean, sure, some pennies are smaller than other pennies slightly; but some dollar coins are also smaller than other dollar coins. But apparently the pennies being smaller implies that pennies aren't a particular value, whereas the dollar coins being smaller does not indicate such a thing. Such is the tomfoolery I've heard from you so far. That's a garbage argument that can be ignored just on its merits. — InPitzotl

Sorry, I have no idea of what you're talking about again. I wish you could make a greater effort to make clear what you want to say.

Sure they do. 1/2 represents one half. As you said, one of anything represents a particular quantity. The quantity that half represents is very clear... that is the multiplicative inverse of two. It takes two halves to make the quantity one. — InPitzotl

I don't see how you can say that one half represents any particular quantity, when one half of two is different from one half of four which is different from one half of eight, etc.. The quantity represented by "one half" is clearly, and completely dependent on the context. So how could "one half" on its own, represent any particular quantity.

So, you claim that there is a quantity called one half. That's nonsense. There is no such thing as one half, unless it is a half of something, and that "something", which is required of necessity for the existence of the half, sets the value for the quantity which the half is a half of. If you really think that there is such a thing as a half which is not a half of something, and is an actual quantity all on its own, then show it to me, give me an example. Show me a half which is not a half of something. It's impossible, because "half" is by definition half of something. I'm sure you must really understand this though, that there is no such thing as a half which is not a half of something, and that one half cannot have any quantitative value whatsoever unless it is stated what it is a half of. Are you just playing dumb?

And no, it does not take two halves to make one, that's a falsity. Two halves are made by dividing one. Two equal things together, are two, not one. And one is a unity which is not necessarily made from two halves. I'm shocked that you are unfamiliar with these fundamental principles.

The way mathematics speakers use the term "particular quantity", 1/9 is indeed one of those things. — InPitzotl

Ha ha, that's ridiculous. 1/9 can be any quantity you want, depending on the size of the whole which is being divided nine ways. 1/9 can be one if the whole is nine, it can be two if the whole is eighteen, it can be three if the whole is 27, it can be four if the whole is thirty six, and so on and so forth. The idea that 1/9 itself, is a particular quantity is utter lunacy. And if mathematics speakers really use the term in this way, then I'd have to say that they really do not know what they're talking about. Don't you agree? How can something (1/9), which can be absolutely any quantity whatsoever, be said to be a particular quantity? And how can you not see the ridiculousness of the claim that it is a particular quantity?

Therefore, your real burden is to show what's wrong with the language of math; you can't just say, "I don't 'believe' 1/9 is a particular quantity"... you have to say, "saying '1/9' is a particular quantity leads to the following problem" and say what that problem — InPitzotl

I've already demonstrated this, numerous times already now. It's utter nonsense to insist that 1/9 is any particular quantity, when it's very clear that it can be any quantity whatsoever. And please don't suggest as Banno did, that 1/9 really means one divided in nine equal parts, because that's something impossible.

Well, Metaphysician Undercover hasn't addressed the two proofs from the OP. All he has done is to assert that 1/9, and other fractions, are not numbers. His argument is an appeal to the authority of the OED. — Banno

I'm still waiting for a better definition of "number". "Family Resemblance" doesn't suffice in logic.

ncidentally, and to my great amusement, the OED definition of fraction is "...numerical quantity that is not a whole number...", contradicting Meta's assertion that fractions are nether numbers nor quantities. — Banno

Actually, this doesn't contradict my claim at all. It just shows that the definitions of mathematicians contradict themselves. That's the problem with mathematics, which I will not cease to demonstrate, it's loaded with contradictions.

Perhaps there is something to be gained here, not by treating Meta's posts seriously, but by looking at how he avoids confronting the truth. — Banno

Let's just say that there is something to be gained from looking at how people, in general, avoid confronting the truth, and not single out any individuals here. Can we leave it at that? Or are you so absolutely certain that what you claim is the truth, and what I claim is false, that you would single out me as the one who is so certain?

For example he provides a restricted definition that suits his purposes, and when challenged he demands 'I invited anyone to provide a better definition of "number"'; a "have you stoped beating your wife" response. — Banno

The op clearly deals with "a matter of representing numbers". If it is the case that some of the symbols used in the op do not actually represent numbers, then we have a false representation. Therefore we require a definition of "number" to determine whether or not there is such a false representation. I provided a definition of "number". According to this definition of "number" we have a false representation in the op.

You are not willing to accept the truth of this, so you reject my definition. Now we have no definition of "number", and no way to resolve the question of whether the op gives us a false representation. Therefore I implore you to provide a better definition of "number", so that we can truly see whether there is a false representation or not, resolve this issue to everyone's satisfaction, and get on to something less trivial.

Again, there is the outstanding point that he fails to directly address the two arguments presented in the OP. I think this is in order to avoid rigour. — Banno

There is no point to addressing the arguments themselves until we determine whether or not there is a false premise. Is this a matter of representing numbers or not. I think it's very clearly not such a matter. You disagree. Where's your argument? What is your criteria for "a matter of representing numbers", which makes you so strongly believe that this is actually a matter of representing numbers?

I see what is expressed in the op as a matter of dividing magnitudes. And, it has been demonstrated numerous times, over and over again throughout history, that some theoretical divisions cannot be represented in number. Claiming to have a numerical representation of what cannot be represented in number, is clearly a false claim. Don't you agree?

You missed the point of the example, as is your habit. — Banno

There was no point to your example, as is your habit. You started from a false premise and tried to make something out of it.

Well, no; what you did was explain how you use the word "number" in a rather eccentric fashion. You told us nothing about numbers. — Banno

I took my definition of "number" straight from the first entry in my OED. I'm still waiting for an alternative definition, one which allows that 1/2 signifies a number. Your response to the definition was very lame: "Family Resemblance".

But I note that your OED definition talks about values referring to the same particular quantity. — InPitzotl

I don't deny that in some cases different symbols represent the same quantity. The op does not provide one of those instances.

And I note that you've chosen of your own will in this post to not actually argue the relevant point... which was that .999...=1 is equivalence under equality, and under equality equivalence implies having the same value. — InPitzotl

I deny that .999..., as presented in the op, represents a particular quantity, because there is no quantitative value given for 1/9. Therefore I deny that .999...which in this instance does not represent a particular quantity is equivalent to 1 which does represent a particular quantity.

Until you do, there's nothing to argue against. You have no point to make, just a problematic claim. And by Hitchen's razor, I can dismiss that without argument. — InPitzotl

I made my point, symbols such as 1/2, 1/3, 1/9, are representative of ratios between quantities, they do not represent any particular quantitative value. To represent a particular value they need to be qualified.

No, they define .999... in such a way that it has the same value; it's not a different value that's close enough, it's the same value. But .999... having that value comes from the definition assigned to it. Like I said at first, this is a language barrier issue. You don't speak the same language. — InPitzotl

I haven't seen that definition. care to provide it?

Again, you didn't make an argument (it was just a claim) and, until you do, I can dismiss your claims with Hitchen's razor. Where we left off is your claim that .999... does not represent a "particular value" despite it being equal to 1, which does. I repeated the inconsistencies I pointed out last post in this post for you. — InPitzotl

The argument is very clear in my discussion with Banno. You just cannot grasp the first premise, that 1/9 does not represent any particular quantity, and therefore it is not a number.

You're all over the place here. You have a definition of number that refers to a value (read the newer version of OED; cf to definition 1b of your revision). 1 and .999... being equivalent means they refer to the same value. And don't think I didn't catch that suddenly "refer to" changed to "are"; nevertheless, it's common language to use forms of "to be" to represent equivalence under equality.. — InPitzotl

I do not argue against the fact that mathematicians believe that .999..., and 1 refer to the same value. The difference between these two is a difference which does not make a difference, for them, so they say that it is the same value. But that doesn't prevent me from arguing that the claim that there is a difference which doesn't make a difference is a contradictory claim.

If .999... represents the same "particular quantity" that 1 does, they refer to the same value, which is what it means to say that they are the same thing. — InPitzotl

My argument, if you've read what I posted, is that .999... does not represent a particular quantity. I suggest that you come back when you've got an argument to make. The fact that mathematicians believe that .999... has the same value as 1 is just evidence that they are wrong, it's not an argument that .999... refers to a particular quantity.

What conversation pray tell are you even talking about? How can .999... have a second meaning if .9 means 9/10 and 9/10 is allegedly a problem? And how come you can't be honest about what you're inviting me to do? The problem isn't that you're missing that conversation about why there are numbers that have two representations in the decimal system... the problem is that you don't believe decimals are possible because you have a quixotic quest against fractions, and yet you present to claim that you believe .999... has a meaning at all. I'm not the problem here, MU; I can easily have that conversation with someone who isn't so wrapped up in your fictional world of fraction-denial. I just can't have this conversation with you because you can't face the fact that there's a thing to discuss. — InPitzotl

Sorry, I have no idea of what you're talking about here. I never said .9, or 9/10 is a problem. I said these do not represent any particular quantity, and ought not be considered as numbers. It is the belief that they are numbers which is what I consider to be a problem.

wonder if MU believes in negative numbers either, or just the naturals. Does zero count to him? — Pfhorrest

I believe that zero is a very complex idea with numerous different meanings, some inconsistent with each other, as exemplified by imaginary numbers.

I think your problem lies with the distinction between pure and applied maths rather than a distinction between 1 and 1/9. 1, 1/9, -1/9 , 0.9, 0.999i are all numbers in the realm of pure maths. — A Seagull

I agree that these are all considered by mathematicians, to be numbers. What I haven't seen yet is a definition of "number" which validates this belief. It is possible that a person, or even a whole group of people, believe that a certain thing is such and such a type of thing, but when a clear definition of that type of thing is made, it turns out that the thing is actually not that type of thing. Take Pluto for example. Everyone believed it was a planet, until a clear definition of "planet was made, then the people realized that Pluto actually wasn't a planet. The same might be the case with some of these things which people believe are numbers. Until a clear definition of "number" is produced we will not know if this is the case. According to the definition I proposed, some of these are not numbers.

Everyone here does that. No, what I'm curious about is the apparent absence of humility. Given that others have thought about these issues - many others, over centuries - and given that your way of thinking is so at odds with the way these others have approached the topic, I wonder at the absence of self-correction. — Banno

I wonder where you get your idea of correct from. That everyone does it, doesn't make it correct, read my example above. You support mob rule?

I guess your response will be to the effect that there are 36 pies, not three dozens, and hence that this is not an example of 3 divided by 9, but of 36 divided by 9. — Banno

If you knew your example was so bad, why present it? Clearly "three dozen" does not represent a quantity of three, just like "four score" does not represent a quantity of four, and "twenty six" does not represent a quantity of twenty. Sometimes I wonder Banno, how you can go so far as to conceive such bad arguments. It must take strenuous effort to make your arguments so bad.

Meta is of interest because of his inability to see that ⅓ is just another number, — Banno

I've explained very thoroughly why 1/2 is not a number. I have yet to see a counter argument, only your extremely bad example which premises that "three dozen" represents a quantity of three. That false premise disqualifies the argument as unsound.

Earlier you claimed this as some kind of problem. Your position seems absurd on its face. — fishfry

I didn't claim this is a problem, that was Pitzotl''s misinterpretation. I said that if the same thing has two distinct names, there is a reason for that.

I was shorter as a child than I am now. That child and the adult I am now are the same person. How can one person possibly be shorter or taller than themselves? The same way a mountain can be smaller at the top: we’re talking about an n-1 dimensional section of an n-dimensional whole. Some measure in the first n-1 dimensions changes over the last dimension. In the case of the mountain it’s diameter over altitude. In the case of me it’s height over time. — Pfhorrest

No, clearly that's not "the same way". You, as a growing human being, have grown taller over time. The mountain's circumference has not grown larger at the bottom. The fact that you call this "the same" baffles me.

You may think it’s a weird way of talking, but understanding that way of talking is necessary to understand what eternalists mean, and if you don’t, then you’re not talking about the same thing as them at all. — Pfhorrest

If understanding what eternalists mean requires accepting that two very different things are the same, then count me out. I can already see clearly that eternalists are wrong, by this statement.