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
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
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
If the only problem with the language is that you have a problem with it, then you are the problem. — InPitzotl
The reason I'm talking to you is that I care about you. — InPitzotl
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
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
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
:up:What makes you think that I believe in any sort of mathematics? — Metaphysician Undercover
What I believe is that it's about time for a good dose of healthy skepticism to be directed at mathematical axioms. — Metaphysician Undercover
By definition, division is the inverse of multiplying.There is a problem, dividing is clearly not the inverse of multiplying. — Metaphysician Undercover
Silly MU. Given any integer a; and any nonzero integers b, c, d:The evidence of this is the existence of irrational numbers, which are derived from dividing, but not derived from multiplying. — Metaphysician Undercover
This being a false analogy, we can ignore your conclusions, except insofar as they reveal your state of mind. But for that, I'll just let your post speak for itself.For a mathematician to say that dividing is simply the inverse of multiplying is like a physicist — Metaphysician Undercover
Yes, they are measurements.These are measurements, and what you are describing is equivalencies. — Metaphysician Undercover
Yes! Let's math this using the equivalence symbol.A "yard" is equivalent to three feet, and a foot is equivalent to twelve inches. — Metaphysician Undercover
Thank you! So let's math that:Each term refers to a particular length, — Metaphysician Undercover
Without parts you say? Interesting:and the length is one unit, without parts. — Metaphysician Undercover
But by your own words, we have an equivalence relation without your parts. So we have something that works already.If a yard, or a foot consisted of parts, — Metaphysician Undercover
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).there would have to be something within that unit to separate the individual parts, one from another. — Metaphysician Undercover
Nope. The above suffices to make 1 yard equivalent to 3 feet without needing your parts. Given it works without your separable parts, your parts are superfluous.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. — Metaphysician Undercover
You misunderstand MU. You are the problem, and you are suffering because of it. You have chosen to pit your views against math. But you've handcuffed your own personal identity to your views; and, you're here in this thread sharing them. Because of the nature of the battle you yourself picked, it's you versus math. 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.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. — Metaphysician Undercover
You continue to misunderstand. I don't care if you believe division is inverted multiplication or not; that's not what's hurting you. 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. But you're not. My goal here is simply to give you some perspective so that you can see what I see... that you're just hurting yourself.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. — Metaphysician Undercover
Dysphemisms and appeals to my alleged gullibility isn't an argument.Or are you so naive to actually believe that there is no more to division than an inversion of multiplication? — Metaphysician Undercover
There's no sense of math being "true" other than that it works. You're basically trying to sell us a belief. Math is a language that does what it says on the tin... this follows; that is consistent, and so on. The truth of math is measured by what it says on the tin, and the fact that it does that. And here you come dressed in salesmen clothes peddling this new theory, telling us how math has led us astray. How pray tell? It does exactly what it says on the tin. Of course that's the issue. What sort of "truth" are you pitching?Whether or not it "works" is not the issue. I have no doubt that it works. What is at issue is the truth. — Metaphysician Undercover
Deception working isn't a truth criteria for deception.You know, until they're exposed, lies and deception work. Don't you? — Metaphysician Undercover
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. — Metaphysician Undercover
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. — Metaphysician Undercover
Sure, why would I deny this? It's been shown to me in so many different ways. — Metaphysician Undercover
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. — Metaphysician Undercover
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? — Metaphysician Undercover
By definition, division is the inverse of multiplying. — InPitzotl
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
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
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
Math is a language that does what it says on the tin... this follows; that is consistent, and so on. — InPitzotl
I'm not exactly sure what it is you even think it means for aa to be divided by bb. — Michael
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
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
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
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
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
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 — Metaphysician Undercover
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. — Metaphysician Undercover
Color me surprised.I've never seen any such definition of "division". — Metaphysician Undercover
Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse −a for all elements a, and of a multiplicative inverse b−1 for every nonzero element b. This allows one to also consider the so-called inverse operations of subtraction, a − b, and division, a / b, by defining:
a − b = a + (−b)
a / b = a · b−1. — Wikipedia
You're only demonstrating your incompetence, over and over. You're just proving you don't speak the language.You turn a blind eye to the evidence, to insist on a falsity. — Metaphysician Undercover
Wrong. The exact ratio between your circumference and diameter is pi. c/d = pi, pi*d = c. If your circumference is 1, your diameter is approximately 0.318310. If your diameter is 1, your circumference is approximately 3.14159. Division's role here is a red herring; you have to round off for both operations because pi is irrational.But to start with the same diameter, and multiply it by pi, will give you a different number as the circumference, becauseyou'll have to round off pi. — Metaphysician Undercover
Wrong. That's just the decimal system. In base 3, divide one by 9 and you get 0.013. In base 9, you get 0.19. In dozenal, you get 0.1412. 0.013*9=13. 0.19*9=19. 0.1412=112. 1 in each of the bases is the same as 1 in decimal. The reason 1/9 is a repeated decimal has to do with the way placement systems work and the fact that its radix is 10, not some ill-placed conspiracy theory about mathematical deceptions of division.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.... — Metaphysician Undercover
In these cases all you're doing is tripping over your confusions of the decimal system representation of numbers. But clearly you're convinced these are truths.In these cases, when — Metaphysician Undercover
I'm not ignoring your evidence; I'm collecting it. But the evidence doesn't point to your conclusions; it points to your being confused.You ignore the evidence of the fundamental difference between multiplication and division. — Metaphysician Undercover
Integers don't form a field under addition/multiplication; but rationals do.This evidence is that when you carry out an operation of division there is often a remainder. — Metaphysician Undercover
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.There is never a remainder in multiplication, nor do you start with a remainder, — Metaphysician Undercover
Wrong. Conflating requires confusing two unrelated ideas... the units of measurement of lengths are lengths.You seem to be conflating units of measurement, foot, yard, etc., with length, which is the determined measurement of something. — Metaphysician Undercover
Of course, because you're confused.So your argument here really makes no sense. — Metaphysician Undercover
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.You argue that three one foot long rulers — Metaphysician Undercover
I argued that it was by definition, so I provided you the definition.Even if you provide examples — Metaphysician Undercover
Wrong; see above. This is a generic description. It's not about the ends of foot long rulers; it's about the particular length that is a foot. We don't need an 8-inch long ruler to measure 8 inches, nor do we need eight inch-long rulers. We just something 8 inches long, like marked partitions on a bigger ruler.But this is clearly false, because this is just one example of something which measures a yard, three one foot measuring sticks, — Metaphysician Undercover
Fine. Worry about being permanently trapped by the unfalsifiability of false narratives that you've spun out of straw man while being blissfully unaware of this condition.I'm not worried about that, — Metaphysician Undercover
What problems? Zero of the things you've pointed out so far have been problems; all of them without fail have been confusions.the problems in math are glaring. — Metaphysician Undercover
It's about a lack of meta-cognitive awareness on your part of your low degree of expertise on the subject being made apparent to people who actually know about it.What's with the appeal to others? — Metaphysician Undercover
It's not just @Banno, though I have to say based on his posts (in every thread I see him in) I generally love the guy. There is a difference though... I'm giving you the benefit of a doubt; he's ruled you out years ago. I factor that in, but choose to give you the benefit of a doubt anyway. Right now, though, you're stuck in your own web. I don't think much is going to come from this conversation, because you have rigged the false game you're playing. But I don't mind fiddling with the puzzle.Banno was in the same boat as you — Metaphysician Undercover
Nice narrative... why do you suppose you're spinning it? I've been on this forum for less than a year. I learned the math here over 3 decades ago in high school... before my BS math minor. Banno and I agree because we know the material, not because I'm his puppy. In contrast, by your own admission, you have never heard of the definition of division.It's as if when someone comes up to you and pats you on the back saying "your right", this makes you right. — Metaphysician Undercover
Yes.Oh, poor me. Don't you just feel so sorry for a poor soul like myself? — Metaphysician Undercover
No. I want you to realize you're in a trap of your own making, and not as you perceive at the crux of a great uncovering. I don't want to protect you, you're a grown man. But you care about truth. So long as you do, you're harmed by your web.And you want to shelter me, and protect me. — Metaphysician Undercover
Okay, and I should care why? I don't need anything from you, MU. You're the one who needs this.What kind of bullshit is this? You're even worse than Banno. — Metaphysician Undercover
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.... — Metaphysician Undercover
I generally love the guy. — InPitzotl
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
1÷2=0.51÷2=0.5
0.5×2=10.5×2=1
What's the problem? — Michael
0.999... = 1, so you do get 1 by multiplying 0.111... by 9. — 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. — Metaphysician Undercover
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 "...". — Metaphysician Undercover
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. — Metaphysician Undercover
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. . — Metaphysician Undercover
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. — Metaphysician Undercover
You're only demonstrating your incompetence, over and over. You're just proving you don't speak the language. — InPitzotl
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
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
But I don't mind fiddling with the puzzle. — InPitzotl
Again, what's the problem? — Michael
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 getting so lost in what the symbols look like that you're not paying attention to what they mean. — Michael
You seem to be reifying. — Michael
You misuse the word "correct" IMO. — jgill
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
Can you imagine how offended they would be if I addressed something of more importance? — Metaphysician Undercover
Unfortunately for you, that's not how language works. English is the language we speak, but it's also a relationship with England, and a type of spin imparted upon a ball. Words can have multiple meanings (homonyms), and math is no different in this respect. The precise meaning of the word often depends on context.I'm trying to learn the language, and I don't like inconsistency or contradictions within the language I use. ... So I am very careful in learning language — Metaphysician Undercover
Sounds like you're more interested in controlling the language than you are learning it. Unfortunately, that's not how it works.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. — Metaphysician Undercover
You're unqualified to make that judgment. But I'll show you how this works by example. First, let's use integral division with remainders:It really looks like you're the one confused. — Metaphysician Undercover
Wrong. Your original criticism was that I require parts in the way you think about it. This is analogous to demanding that the / operator in C must refer to integral division. The standard does not specify such a restriction; I can indeed do floating point and complex divisions. Likewise, the fact that your pet theories of number having no bearing on how people use numbers suggests not that other people are misusing numbers, but rather, that you don't understand what other people mean by numbers.Whatever you use, sticks or markings on the ground, my criticism holds. — Metaphysician Undercover
That's correct, but the problem is on your side. A foot is simply a specific particular length. The foot ruler is just a tool to measure that length. In fact, by the official definition, a foot is 1/3 of a yard; a yard is 0.9144 meters, and a meter is . Note that a foot is defined as a particular length, but that particular length is not defined in terms of the length of any ruler.You are not distinguishing between a unit of measure, "a foot", and a measured foot on the ground, or foot ruler. — Metaphysician Undercover
In measuring lengths in feet the unit is known as a "foot", and the number 2 represents the quantity of those units that are spanned by the length; that is, starting at one position going to another position, you count the quantity of foot-lengths. We do the same thing when we drive; we can use our odometer to measure the driving distance... we do that by counting 1/10 of a mile each time the odometer ticks up by a tenth; if we want the result in miles we convert the tenth mile units to mile units. This is perfectly well defined. Your complaint is about an irrelevancy that you want to picture numbers as meaning. Counting isn't necessarily (and therefore isn't fundamentally) counting objects... you can count the number of times a bell rings, can you not?Consider that the number "2" is a unit of measurement, rather than a collection of two things. — Metaphysician Undercover
No, as explained. You need to apply the correct definition for the correct context. The context here is clearly understood by speakers of the language; see above.OK then do you agree to what I stated above? — Metaphysician Undercover
It appears you don't understand this, since you're repeating the same error. Equality is an equivalence relation, but it's a specific equivalence relation... not all equivalence relations are equality. Take "modulo 4" for example, which is an equivalence relation defined by having the same remainder when dividing by 4. 7 is equivalent to itself, 3, 11, 15, 19, and so on modulo 4. Clearly all these numbers have different values. But 7 is only equal to itself; that is, it's equal to a particular quantity. That equivalence doesn't indicate the same number is irrelevant, because you're presumably talking about not merely equivalence, but equality. The issue isn't whether equivalence indicates the same number, it is whether equality does. Just as the thing that is the same when two numbers are equivalent modulo 4 is the remainder when divided by 4, the thing that is the same when two numbers are equal is the particular quantity that they refer to. So 0.999...=1 does indeed mean they represent the same number.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. — Metaphysician Undercover
Ah, more narratives, more dysphemisms. The problem here, MU, is that you're derailing a thread and breaking social norms. The paranoid projection that mathematicians are insecure and just can't handle your superior knowledge is a delusion... you have no superior knowledge here. You're not addressing any of the issues with math. You're just confused. But what annoys people here isn't your confusion... it's your attention hogging, derailing, social norm breaking. There's nothing wrong with a good discussion about the limitations of math... about considering say Platonic philosophies, the absurdity of AOC and/or AD, and so on. But this isn't a (mathematically) interesting discussion. It's simply a language barrier.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. — Metaphysician Undercover
As a matter of representing numbers, wouldn't most be fine with 9/9 = 9 × (1/9) = 9 × (0.111...) ? — jorndoe
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
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. — Metaphysician Undercover
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