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?

What makes you think that I believe in any sort of mathematics? — Metaphysician Undercover

:up:

What I believe is that it's about time for a good dose of healthy skepticism to be directed at mathematical axioms. — Metaphysician Undercover

It must be brutal that few in the mathematical community seem concerned. But I do agree that the axiom of choice is an unhealthy pathology. :cool:

There is a problem, dividing is clearly not the inverse of multiplying. — Metaphysician Undercover

By definition, division is the inverse of multiplying.

The evidence of this is the existence of irrational numbers, which are derived from dividing, but not derived from multiplying. — Metaphysician Undercover

Silly MU. Given any integer a; and any nonzero integers b, c, d:
$\displaystyle \frac{a}{b} \div \frac{c}{d} = \frac{a\times d}{b\times c}$
...and since a, d are integers, and b, c are nonzero integers, the result is also rational.

This means three things:

• Irrationals are not derived by division
• The proof is trivial
• Ergo, you're unqualified to have this discussion

For a mathematician to say that dividing is simply the inverse of multiplying is like a physicist — 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.

These are measurements, and what you are describing is equivalencies. — Metaphysician Undercover

Yes, they are measurements.

A "yard" is equivalent to three feet, and a foot is equivalent to twelve inches. — Metaphysician Undercover

Yes! Let's math this using the equivalence symbol.
$1\ yard \equiv_{?} 3\ feet$.
There. Now what is the question mark here? We need the sense of equivalence... I wonder where that comes from? :chin:

Each term refers to a particular length, — Metaphysician Undercover

Thank you! So let's math that:
$1\ yard \equiv_{length} 3\ feet$

and the length is one unit, without parts. — Metaphysician Undercover

Without parts you say? Interesting:
$1\ yard \equiv_{length_{w/o\ parts}} 3\ feet$
Great! With you so far. But one more thing... we already know these units indicate length. So we can drop the qualifications here and just say:
$1\ yard = 3\ feet$
See anything interesting? We're using the number 3 in a sense that, by your own admission, does not require parts by your definition (you said it yourself).

If a yard, or a foot consisted of parts, — Metaphysician Undercover

But by your own words, we have an equivalence relation without your parts. So we have something that works already.

there would have to be something within that unit to separate the individual parts, one from another. — 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).

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

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.

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 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.

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

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.

Or are you so naive to actually believe that there is no more to division than an inversion of multiplication? — Metaphysician Undercover

Dysphemisms and appeals to my alleged gullibility isn't an argument.

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

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?

You know, until they're exposed, lies and deception work. Don't you? — Metaphysician Undercover

Deception working isn't a truth criteria for deception.

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

So clearly you don't understand mathematics. Let me set the record straight; mathematicians can do calculations with any kind of $\frac{a}{b}$ where $a$ and $b$ are integers and $b\neq0$. That $a$ can be divided by $b$ just is that they can do these calculations. I'm not exactly sure what it is you even think it means for $a$ to be divided by $b$.

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

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.

If I've caught you in a little inconsistency, or your thinking has changed, or if I'm misunderstanding this point, I'd like to understand. Specifically with respect to 1/9 and its rationality.

Point 2: Now your deflection is interesting. You changed the subject to claim that I have no reason to believe you believe in mathematics. But it's perfectly obvious that you do. I don't believe in tennis. It doesn't interest me. I don't hang out on tennis forums and tell players that their game is nonsense and their rules are unsound. I simply don't watch tennis matches and don't click on tennis-related news. The last tennis match I paid attention to was Bobby Riggs versus Billie Jean King. So when I see you passionately arguing your point -- whatever it may be, since your mathematical nihilism is hard to fathom -- I assume you must care a lot about mathematics.

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?

Sure, why would I deny this? It's been shown to me in so many different ways. — Metaphysician Undercover

But that's great. Then you and I are in absolute agreement. The proposition here is that ".999... = 1 is a logical consequence of the ZF axioms." You are agreeing with this. From a formalist perspective, it is no different than saying that a particular chess position can be legally reached.

But if you agree with this, then you and I have no disagreement. Because I make no other claims!

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.

So tell me: What extra secret sauce are you imbuing the symbolism with? Why do you think there's some "true" meaning out there in Platonic land? Do you think there's a real way that the knight moves and the rules of chess have got it wrong?

Can't you see that if you agree that the formalism is valid, then we're in agreement. I myself make no claim of the soundness of mathematics; only its validity. You're arguing against a strawman of your own invention.

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

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?

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

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?

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

Sweet.

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.

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

However, modern, abstract mathematics may be more like the game of chess and less likely to describe our world. For a number of years it's been fashionable to move away from the kinds of mathematics that we normally associate with physical reality and into "higher" levels that are increasingly abstract and generalize concepts and processes to the extent that ideas and technicalities peculiar to classical math don't even appear on the radar.

But then I wonder, perhaps what seem like total abstractions really do point to some underlying aspects of reality that we have not reached the point of comprehending. One might think this possibly true with regard to QM, although most of the math used there is fairly traditional. Even oddities like virtual particles are really just mathematical terms in the series solution of difficult integrals. Or so I am told.

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

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

$1\div2=0.5$

$0.5\times2=1$

What's the problem?

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

Color me surprised.

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 turn a blind eye to the evidence, to insist on a falsity. — Metaphysician Undercover

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

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. 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.

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

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.

In these cases, when — 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.

You ignore the evidence of the fundamental difference between multiplication and division. — 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.

This evidence is that when you carry out an operation of division there is often a remainder. — Metaphysician Undercover

Integers don't form a field under addition/multiplication; but rationals do.

There is never a remainder in multiplication, nor do you start with 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.

You seem to be conflating units of measurement, foot, yard, etc., with length, which is the determined measurement of something. — Metaphysician Undercover

Wrong. Conflating requires confusing two unrelated ideas... the units of measurement of lengths are lengths.

So your argument here really makes no sense. — Metaphysician Undercover

Of course, because you're confused.

You argue that three one foot long rulers — 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.

Even if you provide examples — Metaphysician Undercover

I argued that it was by definition, so I provided you the definition.

But this is clearly false, because this is just one example of something which measures a yard, three one foot measuring sticks, — 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.

I'm not worried about that, — 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.

the problems in math are glaring. — 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.

What's with the appeal to others? — 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.

Banno was in the same boat as you — 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.

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

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.

Narratives are not arguments.

Oh, poor me. Don't you just feel so sorry for a poor soul like myself? — Metaphysician Undercover

Yes.

And you want to shelter me, and protect me. — 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.

What kind of bullshit is this? You're even worse than Banno. — Metaphysician Undercover

Okay, and I should care why? I don't need anything from you, MU. You're the one who needs this.

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

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

$0.111...=\frac{1}{9}\\\frac{1}{9}\times9=\frac{9}{9}=1$

I generally love the guy. — InPitzotl

Ooooh... that's sweet.

Yeah, I gave up on @Metaphysician Undercover about a twelve-month ago. I'd previously thought that there was something interesting going on, but it's not forthcoming. Even here, there is something about definitions, meanings, that is just wrong. Here it is something about what it is to be an individual that excludes its being divisible. that sits just behind his thinking but is never articulated.

In the end that is not a philosophically interesting position, but an odd piece of personal psychology that prevents Meta from participating in the discussion. Instead of progressing our thinking about mathematics we find ourselves stuck on the Meta Treadmill, pointing out the same errors repeatedly.

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

A beginner question...

If this proof is fine, then why the proof in the PDF?

If this proof is fine, then why the proof in the PDF? — Banno

I think because it doesn't explain what is meant by 0.111...

Ah... Cute.

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. .

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

There's a remainder in my example:

$1\div2=0.5\\0.5\times2=1$

I don't understand why this is a problem.

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

$1\div9=0.\overline{1}\\0.\overline{1}\times9=1$

Again, what's the problem?

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

$13$, $\frac{65}{5}$, $XIII$, and $11_{12}$ are the same number. You don't seem to understand that different symbols can be used to refer to the same thing.

And $0.999...$, $0.\overline{9}$, $\frac{9}{9}$, and $1$ are the same number. You're getting so lost in what the symbols look like that you're not paying attention to what they mean.

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

I explained here how mathematical objects are separated. I also explained here that mathematicians have moved away from the vague notion of "number" and use different terms instead. Your continued insistence that we must have some formal definition of the English word "number" that allows it to refer to every kind of thing used by mathematicians when they perform calculations is ridiculous. You seem to be reifying. Read some Wittgenstein. Words are just useful tools. Don't make them into something more significant than that.

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

That's just the point. Perhaps we don't really understand "our world" that well. Odd looking axioms should not be cavalierly discarded simply because they are "not true" to our limited view of reality. You misuse the word "correct" IMO.

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.

It's time, gentleman - and any ladies present - to consider and weigh what continued participation in this thread means.

One of us is clearly of diminished capacity with respect to the subject matter. In this respect the image of a man in a cage or a trap is apt, except he refuses to be freed. All that's left is poking at him, and while a bit of that is always part of the game at TPF, at some point it becomes ungracious and unseemly. And a waste of the time, knowledge, and skills of some of us. Consider not posting here and letting this thread quickly fade away. Substantive matter and worthwhile discussion await in new threads.

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?

Can you imagine how offended they would be if I addressed something of more importance? — Metaphysician Undercover

I know, I know. You'd be driven from your castle in the dead of night by an angry mob of mathematicians waving their torches and holding their frothing mastiffs on chains. They are an uncivilized and ignorant bunch, so it serves them right you are withholding precious knowledge. :scream:

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

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 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

Sounds like you're more interested in controlling the language than you are learning it. Unfortunately, that's not how it works.

In the C programming language there is an operator /, which is used to instruct the underlying machine to perform a division. But loosely speaking there are three distinct types of divisions: integral division, floating point division, and complex division. In a well formed program, the type of division being performed in any application of the / operator is defined by the type system as specified by the standard, but it can nevertheless be one of these three types. Now all of this is describing what we call "the C language", and the C language by official definition is the language specified by the C standard. Given this, it would be quite silly of me to argue that the standard is lying to me because, as I rationalize, I like my languages to be consistent and have no contradictions for fear that I might misunderstand what / is or even be deceived. The rationale here is actually irrelevant to what the standard is specifying, i.e. what the language is. It is incumbent upon me as a user of the language to learn what type of division is being performed based on the context. Any misunderstandings is not a fault of the C language; it's a fault of my not understanding what the language is.

We have a similar situation here in math; the meaning of division depends on the context. In the context of interpreting the meaning of 0.999... in the statement 0.999... = 1, we apply field operations under a normative application of addition, subtraction, multiplication, and division, as applied to the reals (or at a minimum the rationals); a definition of a repeated decimal; and the mathematical interpretations needed to apply the definitions. Fractional notation can easily be added and mixed in at will.

It really looks like you're the one confused. — 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:
(a1) $\displaystyle 24 \div 9 = 2\ rem\ 6$
Some terms... 24 here is the dividend, 9 is the divisor; 2 in this form is the quotient, and of course 6 is the remainder. Now let's do the same operation using mixed numbers.
(a2) $\displaystyle 24 \div 9 = 2\ \frac{6}{9} = 2\ \frac{2}{3}$
More terms... the top portion of the fraction in bar form is the numerator... the bottom portion is the denominator. Note that the numerator in the fractional part of the first mixed number is the remainder from a1, not accidentally. And the denominator of this fractional part is the divisor, also not by accident. Now I chose this example precisely because it reduces to make a second point... the 6/9 fraction reduces to the 2/3 fraction by means of an equivalence relation. 6/9 is equivalent to 2/3 in a specific sense... it represents the same portion of a unit. The meaning of that equivalence is that if you split the unit into 9 equal pieces and take 6 of those ninths, you wind up with the same portion of a unit as you would if you split it into 3 equal pieces and take 2 of those thirds. In other words, the fraction represents a particular quantity; viz, a specific portion of a unit.

Whatever you use, sticks or markings on the ground, my criticism holds. — 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.

You are not distinguishing between a unit of measure, "a foot", and a measured foot on the ground, or foot ruler. — 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 $\displaystyle c\times \frac{1}{299792458}\ sec$. 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.

Consider that the number "2" is a unit of measurement, rather than a collection of two things. — 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?

OK then do you agree to what I stated above? — 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.

I was considering just amending this claim to something agreeable, but as it is presented, I cannot see an easy edit. The important thing to preserve here is the intended meaning, but the meaning isn't so much in the rules for calculation or the representational system, as it is in the mapping of how the division operator transforms the particular quantities of its operands into the particular quantity of its result (or by extension, how the field under consideration works). For example, if $(1\div 9) \times 9 = 1$, and $1\div 9=\frac{1}{9}=0.111...$, then $\frac{1}{9}\times 9 = 0.111... \times 9 = 1$.

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

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.

Just because I brought this up earlier doesn't mean you resolved it. You didn't. Committing the same error twice doesn't make you correct, it just makes you still wrong.

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

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.

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

So, @Metaphysician Undercover, I picture you sitting down with a piece of paper and a pen, and start writing out 1/9 using the simple mathematics you were taught in early elementary school, subtraction remainders repeat all that... 0.1 ... 0.11 ... 0.111 ... and, presumably, you catch on after a short while. "My god, it's full of 1s." (Is Strauss appropriate here?)

It's a fairly simple procedure (incidentally, one that I've had to implement on a computer to calculate digits of π, like many before me).

The interesting part is now what we can prove about that procedure without even keep running it: following the procedure just results in unending 1s.

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...".

Conversely, once we take the opposite approach, rewriting this result concisely as a sum and a limit, we can also prove that we end up with 1/9 — consistency within the mathematics. (y)

By the way, if you really want something more concise about the numbers themselves, a constructive approach, then maybe check the doc Michael posted, looks neat to me anyway.

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.

I'm becoming increasingly astonished that this thread continues.

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 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.

It’s a shame @Devans99 got banned because this thread is clear evidence that infinity exists.

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

Or, another way of writing all that is "..."