## 0.999... = 1

• 242
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.
Prove it.
• 932
The "unending 1s" indicates that there is a remainder
P2. No matter how many 1s you write

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

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

Hm regarding abstract thinking, in analogy: suppose we want to prove p; then by some other means we find that we can prove that p can be proven; well, then we're done with our initial task (unless we're curious).
• 8.9k
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.

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

I think the most important thing here is, what is MU's criteria for truth? MU made an actual truth claim here to counter a proof. Can MU offer a proof in return, or does MU think he has a better truth criteria? Either way, I want to see the proof or this better criteria.
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Thanks for the lesson in terminology InPitzotl. But I don't see how expressing the remainder as a fraction resolves the issue of the remainder. The fraction is just an expression of an unresolved division problem. So in expressing "24÷9=2 rem 6", as 24÷9=2 6/9" or "2 2/3", all you are doing is replacing the remainder with an unsolved division problem. It's not really any different than having a remainder because what you are doing is saying the division can be carried out to this point, but the rest remains not divided.

Prove it.

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

So this is the state of the art on what's known about the consistency of ZF.

If you have found an inconsistency, you will be famous. I'd be glad to help you express it mathematically and we can both be famous.

The problem is that so far you have not demonstrated an inconsistency in ZF. You've only made a sequence of increasingly bizarre and nihilistic assertions about mathematics, none of which are remotely true as concerning that discipline.

To show ZF inconsistent, here is what you must do: Produce a proposition $P$, a well-formed formula of the first order predicate calculus plus the axioms of ZF; such that there is a proof within ZF of both $P$ and of $\neg P$.

You made a bold claim. That's what you need to do to back it up. I'll be glad to help with the translation of your idea to math; if you actually have an idea.

I wonder what claim you think it being asserted by .999... = 1.

I don't think any claim is being asserted beyond the fact that the equation is derivable line by line from the axioms of set theory and predicate calculus. You're the one who thinks it "means" something. I have no idea what you are even thinking. The equation refers to nothing in the real world and I never claimed that it does. You're punching at a strawman.

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.

Of course. You could use a different model of the real numbers such as the hyperreals. But .999... = 1 is a theorem in the hyperreals as well. You could try intuitionist math. .999... = 1 is most probably a theorem of intuitionist math but I confess ignorance on this point. I can never make sense of the intuitionists and it's not for lack of trying.

So if you want to work in some alternative framework I'm perfectly open to it. There are in fact a number of interesting variants of chess, too. Like the 3D chess they play on the Enterprise.

This is due to inconsistency in the rules of mathematics. Look at how many different systems of "numbers" there are.

There's no general definition of "number" in mathematics.

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

You know @Meta, you seen to deny any understanding of math as a social activity of humans. But that's exactly what it is. Perhaps there's a Platonic math out there and perhaps not; but either way, mathematics included the history of people who do mathematics, going back to the first cavedweller who put a mark in the ground when he killed a mastodon.

in any event there are dictionary definitions of number, but there is no general mathematical definition of number. Particular kinds of numbers, yes. Number in general, no.

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.

You're confusing pure math with applied math. And it's true that chess doesn't apply to the world; but I could pick a better analogy. Take sailing. Recreational sailors are playing a game that has no actual consequences outside of the game. But their game arises out of the accumulated knowledge of thousands of years of sailing, most of which was done for trade and exploration. So that's a formal game, if you like, with connections to the real world.

But really, you are saying that there is no math other than applied math. You miss a lot from that perspective. And a lot of abstract pure math becomes very practical hundreds or even thousands of years later. So you can't really make the distinction you are making. Euclid studied the factorization of integers into primes; but it wasn't till the 1980s that someone had the idea of applying prime factorization to the security of digital communications. Today number theory underlies the security of the Internet. If you'd been in charge back then you'd have told Euclid to stop fooling around and build a wheel or something, and humanity wouldn't have learned any number theory and would not today be able to secure the Internet.

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

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.

You're arguing that BECAUSE math is sometimes useful, it may ONLY exist if it is useful. What's your evidence for that proposition?

That's like saying that abstract art is ok as long as it's useless; but the moment anyone uses a painting to cover a hole in their wall, only practical art is permitted. You know you are speaking nonsense.

Physicists and others find math useful. That doesn't place any limits on what math can be or what mathematicians can do.

Euclid wasn't trying to solve the problem of Internet security 2200 years ago; but that's where his mathematical thinking led. You simply never know when a piece of math will eventually be indispensable, as they say, in the world.

Nobody claims that math = physics. That hasn't been true since Riemann and others developed non-Euclidean geometry in the 1840s. Surely you must know a little about this.

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.

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

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

Man you are a nihilist true.
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But I don't see how expressing the remainder as a fraction resolves the issue of the remainder.
Good... you're caught up then.
The fraction is just an expression of an unresolved division problem.
Wrong. The fraction part of a mixed number specifies an exact portion of a unit. We can only say that 3/9=1/3 insofar as both of these fractions represent that specific quantity of a unit.
So in expressing "24÷9=2 rem 6", as 24÷9=2 6/9" or "2 2/3", all you are doing is replacing the remainder with an unsolved division problem.
Wrong. Assume I start with 24 pizzas. 2 rem 6 simply means that if I give each of 9 people just 2 pizzas, that I would have 6 left over.

By contrast, the fraction specifies an exact quantity. It means a specific thing to give one person 2 2/3 pizzas. If I give each of 9 people 2 2/3 pizzas, then I have none remaining.
It's been proven. It's called inductive reasoning.
You didn't provide reasoning until just now; you just asserted it. Now let's go over your argument:
"'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.' — Metaphysician Undercover;
Prove it. — InPitzotl"
...Every time someone adds another 1, there is still a remainder.
Correct. But also, every time someone adds another 1, is a time. Algorithmically, that time is a step; we can count the steps. Specifically, each step is a finite step. Might I remind you, though, that your truth claim is explicitly about "even after we put an infinity of 1s (whatever that means)".
And never ever is there not a remainder.
"never" applies to all steps in the process. And all steps are finite. So this does not apply "after" we put an infinity of 1s. (I would argue there's no such thing as that after).
And since the nature of the numbers stays the same, we can conclude that this will always occur.
We can conclude that for all steps. But we cannot conclude that "after" we put an infinity of 1s, which is the very thing you're making a truth claim about.
I don't see what makes you think that at some point we'll have enough 1s that there'll suddenly be no remainder.
I don't think that at some point we'll have enough 1s. I think you're once again speaking about something you have no clue about. There is no last finite counting number; there's no "point" "after" you have an infinite number of 1's. But there are an infinite number of counting numbers. I don't think you proved anything, except what you explicitly admitted to here... that you don't know what this means.

How can you say you proved something when you don't know what it means?
The criteria for truth is honesty. I provided my argument, and you disagreed with the second premise, that no matter how many 1s you place after the decimal point there will still be a remainder.
You're by context including infinite strings. The literal string .111... refers to an infinite string starting with .111 and followed by a 1 for every finite ordinal position; that is, if you count the first 1 as 1, the second as 2, and so on, there is no finite n such that the nth position does not have a 1 in it. There is no point in this string that is "the last 1" for the same reason there is no last counting number. Your argument hinges on the hidden assumption that there is such a step... but that's a confusion on your part.
I think that this premise is true and I am honest in this claim.
Your honesty and sincerity is not in question; your claims are. Your proof falters because it does not apply to the one thing you're making a claim about. .111... is an infinite string; that is the thing under discussion. Your proof applies only to finite steps, which ipso facto is not infinite. By the way, because I use correct reasoning, I will not claim that your proof being wrong means there's no remainder; it does not mean that at all. What your proof being wrong means, instead, is that your reasoning that there is a remainder is invalid. I explicitly mention that because you make that mistake here:
If you claim that you do not think that this premise is true, I think you are being dishonest.
My honesty and sincerity is not in question; your claims are. Your proof still falters for the reason specified above.
If that is the case, then your claim is false.
Wrong. Your claims stand or fall on their own merits; it has nothing to do with me. This isn't a relevant argument, it's a psychological response. You cannot conclude anything about the veracity of your claim based on presumed character flaws you guess I have.
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I agree that just because you argue from certain premises doesn't mean you agree with them. But you are being disingenuous here. I could easily go back to our older discussions and show you where you accepted the rationals in order to deny 2–√2. I don't take this as a serious remark. Your prior posts don't support your claim that "I was only kidding about the rationals." You are retconning your posts and I'm not buying it.

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

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

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

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

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

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

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

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

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

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

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

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

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

But, proof or not, this should be intuitively clear. You understand? If yes, then you're free to suggest a means to communicate this unambiguously, or you can follow typical conventions like $0.\bar{1} = 0.(1) = 0.111\ldots$.
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• 8.9k
I've seen some mathematical proofs, and as I've shown that the axioms are full of inconsistencies and contradictions. A lot of these so-called "proofs" are smoke and mirrors built on false premises and therefore unsound.

You didn't really answer the question though. Do you believe that generations of mathematicians are aware of this, and yet for some reason continue to use them, or are they unaware, and you're just smarter than everyone else?
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The fraction part of a mixed number specifies an exact portion of a unit.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

So there's no issue of anyone being smarter than anyone else, I don't know how you would even judge such a thing. It's a matter of where one's attention is directed. I happen to have an interest in music, and musicians work with a fundamental unit called an octave, along with divisions and multiplications, using frequencies to produce harmonies and dissonance. So the matter of what can and cannot be divided into equal parts is interesting to me. The issue of the acceptable divisions of a unit has never been resolved. And to claim as InPizotl seems to, that a unit can be divided in any way one pleases is totally unrealistic. However, notice that my interest in the problems of division is piqued by my interest in music, such that the pure side of my math interest is still guided by the applied side. And this is why I do not accept fishfry's proposed division.
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I don't think you can cast a net of generality on all mathematicians in that way. Some follow the discipline in such a way that they would apply the principles without being aware of the underlying issues. Fishfry might call this applied math. Some question the underlying principles, as indicated by jgill. Fishfry might call this pure math. If I understand fishfry's proposed divisions.

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

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

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

If they are then why do they use them and not "fix" them? If they're not then how are you, a mathematical layman, able to notice what the experts can't?

There is of course a simpler explanation. You don't know what you're talking about.
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I'm sure my intuition is quite different from yours
Hmm... No pattern recognition...? Odd.

Doesn't the second statement directly imply the falsity of the first?
Nope. Arithmetic works fine regardless of notational conventions.

Intuitions and conventions aside ...
We can prove that all the procedure does here is give us 0, decimal point, followed by endless 1s.
And we can prove that without writing down 0, decimal point, followed by endless 1s — it's an artefact of the procedure, and the proof involves mathematical induction and such.
Doesn't really matter much whatever anyone makes of it, that's how the arithmetic works.
We can prove things about switching 1/9 to decimal form without doing it (↑ stands on its own).
You understand...?

I wonder, do you ever balance checkbooks, file taxes, etc? :)

(Side note: like @fishfry, I don't know if intuitionist mathematics blocks anywhere, but offhand I kind of doubt it.)
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reasserting it will not persuade me to agree.
We've been through this MU. We're not debating... you're under the delusion that we're having a debate... that my goal is to persuade you, that I'm trying to do so, that something is riding on your agreement, and that it actually matters that I persuade you. We're not, I'm not, I'm not, it isn't, and it doesn't. But I have to say... it's all kinds of adorable that you think we're debating!
The fact that there is a repeating decimal
Therefore it is impossible that 2/3 represents an exact portion of a unit.
The decimal is a red herring; 2/3 is a fraction, not a decimal point number. You're conflating notation with representation.
What you have argued is that you can define "one" or "unit" however you please
But I can. I can use a pizza cutter, I can slice a pizza into 3 equivalent parts, and then I have thirds of a slice.
and that's just contradiction plain and simple.
It's no contradiction; we just specify the units. I slice 6 of the pizzas into three slices each. Now I have 18 pizzas and 18 thirds-of-pizzas. I give each of 9 people 2 pizzas and 2 thirds-of-a-pizza. This is something I can actually do in real life. Keeping track of two kinds of things (pizzas, slices-of-pizza) is child's play. If you have difficulties doing that, that's your problem not mine.
You're intentionally avoiding the point, and I must say, lying, when ... Sorry, no offence meant, but I feel it's necessary to point this out.
Well likewise no offense intended MU on my part but, I can't possibly take any of your arguments seriously, including your ad hominem conspiracy theories about me. So there's no way you offend me by this... all you're managing to accomplish is to expose your own irrationality.
You have no qualifications here
That is literally untrue. I have a BS minor in math; I'm pretty sure that qualification covers mixed numbers, since that's a grammar school topic.
to stipulate the size of the pizza
...is exactly as relevant when talking about thirds as it is when talking about wholes. Now that I described what that process is, let's compare notes. There is a real thing I can do to distribute 24 pizzas among 9 people (above). Your objections fail to describe or affect that procedure; worse, they fail miserably to account for the fact that I wind up with no pizzas instead of six.
Actually my truth claim was that
There's no disagreement about finite decimals, but it's irrelevant to 0.111....
Now, can you give me an honest answer to how you think the remainder is dealt with then,
There is no "then" here... belief is not a mandate. You cannot defend a false belief on the basis that nobody offered you an alternative...
such that we can end up with an "exact quantity".
There's nothing special about the decimal system with respect to number values. 1/9 is 0.013 exactly, no remainder. 1/9 as a fraction is a value in and of itself. The question under concern is whether 1/9 can be represented exactly by the decimal system. It can, if your decimal system includes (or is extended to include) repeated decimals, and if you use the definition I supplied earlier for repeated decimals. The OP provided a proof of this.

But for now you're choking on the fact that we can meaningfully talk and reason about infinite strings/repeated decimals (ironically, while talking about and reasoning about such things). So let's do this.

Imagine we write a computer program to calculate using standard long division on decimals. As a primer, let's do 1/8 first. Our program then winds up doing the following (if the following confuses you, you know what long division is (?)... do long division yourself and read along and you should see what's going on):
• Takes 1 and 8 from input (or hard coded, doesn't matter) and stores it for next step.
• It notes that 8 goes into 1 0 times; emits "0." (. due to special "at-unity" rule). Then it multiplies: 8*0=0; subtracts: 1-0=1; tests for halt: 1!=0 so continues. Then, it shifts-carries: 1 becomes 10+0=10, and loops back.
• It notes that 8 goes into 10 1 time; emits "1". Then it multiplies: 8*1=8; subtracts: 10-8=2; tests for halt: 2!=0 so continues. Then, it shifts-carries: 2 becomes 20+0=20, and loops back.
• It notes that 8 goes into 20 2 times; emits "2". Then it multiplies: 8*2=16; subtracts: 20-16=4; tests for halt: 4!=0 so continues. Then, it shifts-carries: 4 becomes 40+0=40, and loops back.
• It notes that 8 goes into 40 5 times; emits "5". Then it multiplies: 8*5=40; subtracts: 40-40=0; tests for halt: 4!=0 so halts.
This program is now done; it has emitted 0.125. So we say that 1/8=0.125.

Take the same program and use it to calculate 1/9. Here's what happens
• Takes 1 and 9 from input and stores it for next step.
• It notes that 9 goes into 1 0 times; emits "0.". Then it multiplies: 9*0=0; subtracts: 1-0=1; tests for halt: 1!=0 so continues. Then, it shifts-carries: 1 becomes 10+0=10, and loops back.
• It notes that 9 goes into 10 1 time; emits "1". Then it multiplies: 9*1=9; subtracts: 10-9=1; tests for halt: 1!=0 so continues. Then, it shifts-carries: 1 becomes 10+0=10, and loops back.
• It notes that 9 goes into 10 1 time; emits "1". Then it multiplies: 9*1=9; subtracts: 10-9=1; tests for halt: 1!=0 so continues. Then, it shifts-carries: 1 becomes 10+0=10, and loops back.
At this point we can pause the program, because we note that the machine is in the same state twice. Because programs are deterministic, if a machine gets to the same state twice, we immediately know it's in an "infinite loop" (that's literally the jargon). But in getting from that state to the next instance of the state, the program will emit another "1". For that reason we can say that the program will emit 0.11 followed by an infinite number of 1's. We know this immediately because we can reason and we can recognize symmetric recursion.

We can represent what the first run does as emitting the string "0.125". We know that's the complete output of the program because we can wait for it to halt. We can write down "0.125" because it's just 5 symbols. The "last digit" here is 5, because that is the thing that the program emitted just prior to halting.

The second run is qualitatively different, but we can still represent what it does. We know that there's no complete output of this program because we know it will never halt; but we know the program keeps generating 1's in perpetuity. We know we cannot write down the full output here, because we know it is infinite, but we know that its output will keep spitting 1's because it did so for a couple of steps and because the nature of the infinite loop is that of symmetric recursion. So we can represent the output as 0.111... meaning it never stops, and will always spit out 1's. "An infinity of 1's" is just a shortcut for saying the same thing. "One repeating" says the same thing as well; 0.(1) refers to the same thing.
No, .111... cannot refer to an infinite string, because we've agree that we cannot put an infinite string there.
Oh, thank you MU. It saves a lot of time when you make a claim but accidentally prove by contradiction that it's false (underlined). If you can describe this string as "an infinite string" and reason about what that implies, then I can refer to the same string as "0.111..." and reason about what that implies. As a bonus points, you've demonstrated that you yourself are just confused about this, which is something I keep saying.
You are now claiming to do what we've agreed is impossible.
No, you're confused. I'm claiming to do what you (accidentally) proved by demonstration is possible.
Which do you accept as the truth, can we put an infinite string there or not?
That's easy. "We cannot put an infinite string there" is a true statement.
If you say that .111... refers to an infinite string that is somewhere else other than there,
Silly MU. I only claim that "..." refers to an infinite string. Call it 0.(1) if that helps. It all refers to the same idea... that the program generating this string never halts and always repeats as demonstrated by symmetric recursion.
then how is it relevant?
It's trivially relevant, because it's the infinite string referred to by "0.111..." that you are trying to object to.
You've refuted my proof by proposing that (a) there cannot be an "after" one (b) puts an infinity of 1's there, and then (c) going and putting an infinity of 1's there.
(a) yes, the program never halts. (b) Not really; we use "0.111..." to refer to the fact that it never halts (and that all of the symbols in it are 1's). I got to this from the program by running just a few steps, and recognizing symmetric recursion was going on. (c) No, we used reason to conclude (a) and (b) and call this an infinite string of 1's. And I'll add (d), that your proof only applies to terminating decimals, because we can only say that a result has a remainder if we're "left" with one "after" we're done, and there's only such a thing as "done" for terminating decimals.
Now we are at the point of after you put the infinity of 1's there,
If there's no after putting an infinity of 1's, there's no such thing as the remainder you're left with when you do. To talk about such an entity you have to either reify it, or prove it actually exists. Good luck talking about the remainder at the final execution step just prior to halting, in the context of a program that never halts.
so all you have done is disproven the premise of your refutation.
You have this backwards. I don't have to prove your proof doesn't prove something, your proof has to prove it. We have an object 0.111... that describes the output of a program that never halts. You applied fallacious reasoning akin to my previously mentioned troll proof that infinity is finite; "it has a remainder at each step, therefore the infinite string has a remainder" is exactly analogous to "it's finite at each step, therefore infinity is finite". We trivially know this doesn't apply, because there is no such thing as a last step to have a remainder at. There is no counting number that represents the count of the counting numbers. There's no "after" to writing an infinite number of 1's. There's no "end" to a program that never halts.

ETA: Or try this one. 0.111... represents a string with infinite 1's. Let's "prove" that there's a remainder:
"Proof" A: 0.111... is the result of dividing 1/9. When dividing 1/9, we get a remainder of 1 at step 1 (the tenths digit). At any step n, if we start with a remainder of 1, then there is a remainder at step n+1. Apply infinite induction, and we generate 0.111..., and are left with a remainder of 1.
...now let's prove the exact opposite:
"Proof" B: 0.111... is the infinite result of adding 1 digits after "0.". When we follow this procedure, at step 1 we wind up with 0.1=1/10 exactly, no remainder. At any step n, given the value is p/q exactly, adding a digit gives us (10p+1)/10q exactly at step n+1, no remainder. Apply infinite induction, and we generate 0.111..., and are left with no remainder.
...compare to the troll proof:
"Proof" C: 1 is a finite number. If any number n is finite, n+1 is also a finite number. Apply infinite induction, and we get that infinity is finite.

The same problem occurs with proofs A, B, and C. Infinite recursion is fine for proving a property exists at all finite steps (in Proof A, all steps have a remainder of 1; in Proof B, all steps have a remainder of 0, in Proof C, all such numbers are indeed finite as is the following number), but cannot prove anything (at least in this fashion) about the property of the infinite extension (in Proof A, you cannot say 0.111... has a remainder of 1, just as in Proof B, you cannot say 0.111... has a remainder of 0, and in Proof C you cannot say infinity is finite).
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@Metaphysician Undercover has deified the individual. The prime commandment is that One must be kept whole. At the alter he sacrifices all of mathematics beyond addition.
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This doesn't answer my question. Do you think that mathematicians are aware that "the axioms are full of inconsistencies and contradictions. A lot of these so-called 'proofs' are smoke and mirrors built on false premises and therefore unsound"?

Some are, some aren't.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Realising this sort of thing happens as one moves to the concrete operational stage according to Piaget. There's a phenomenon called "irreversibility", in which "two rows containing equal numbers of blocks are placed in front of a child, one row spread farther apart than the other, the child will think that the row spread farther contains more blocks". Pushing the block together seems to the child to decrease the number of blocks. Similarly,
...a four-year-old girl may be shown a picture of eight dogs and three cats. The girl knows what cats and dogs are, and she is aware that they are both animals. However, when asked, "Are there more dogs or animals?" she is likely to answer "more dogs". This is due to her difficulty focusing on the two subclasses and the larger class all at the same time.

Talking to you has similarities to talking to a pre-operational child.
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What are you doing then?
Multiple things. Playing a game. I'm trying to see how much perspective I can give you about your lack of competence in this area... that you're uncooperative makes it a bit challenging. But I'm being quite honest here; I don't take you seriously.

I'm not trolling though... I'm just not debating. If you like you can treat this as a debate; the form is the same (to some approximation). But I don't want to give you the impression that I actually believe you can be convinced if I give you good reasons, nor that I really need you to "believe in math".
And what do you say when we weigh the slices and find out that they are not exactly equal?
Well seeing as the pizzas themselves wouldn't be exactly equal either, why would we care? It doesn't affect the definition of fractions, and 9 people are getting more pizza than they would if you threw 6 in the trash. I'm pretty sure 6 pizzas in the bin because you have some sort of deep rooted aversion to fractions is more significant mathematically speaking than guy 3 getting a few tenths of an ounce more pizza than guy 4 because we're approximating fractions (not to mention hand waving that whole pizzas weigh the same, for some mysterious reason).
Sorry, but your example is what fails miserably.
Your criteria for failure amuses me. I have 9 happy people. You have 6 pizzas in the trash.

You seem to misunderstand.
Don't I?
What I'm arguing is that an infinite string of 1's. following a decimal point, following a zero, does not represent a solution to one divided by nine.
Sure, but as I said, there's still a remainder
But you failed to prove there's still a remainder in an infinite string of 1's following a decimal point following a zero.
Exactly! Without an end the problem is not resolved.
Without an end, when do you have a remainder? (Did you not see where I pointed out the flaw in using your argument to show there was a remainder? That's "Proof" A, it's still in the post, countered by "Proof" B, and satirized by "Proof" C).
And, as you say we are not debating this, nor are you trying to persuade me of your point of view, so why continue?
I'm doing multiple things at once; debating just isn't one of them. I'm trying to see, as a challenge, how much perspective of math community you will take in while being paranoid about it. I'm attempting to reverse engineering your jaded views of the math community.
Are you learning anything yet?
Yes; I'm learning about how you think.
Would you consider the proposition that certain numbers just cannot be divided by each other?
If we're talking about integers, sure. If we're talking about fields, no. It's intriguing to me that you take this sort of integral and/or whole and/or counting number realism to such extreme deepisms that you both transport the properties of such things into other number systems and trick yourself into thinking you've done something profound, but I have no actual interest in the broken theories that lead to this. I am however interested in the psychological aspects of why you're so committed to these deepisms... but not being a psychologist I'm content with just what I can piece together with reverse engineering.
You know that the value of "one" is that of a whole, a single unit,...
If pure math can have no fractions, what is this?:
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I'm learning about how you think.

"two rows containing equal numbers of blocks are placed in front of a child, one row spread farther apart than the other, the child will think that the row spread farther contains more blocks". Pushing the block together seems to the child to decrease the number of blocks.

A child would be taught reversibility by being given different examples of this sort, until they learned to talk about the number of blocks in a suitable way. Meta is like a child who, when shown the blocks pushed together, insists that "Yes, I see that there are the same number of blocks when they are spread out and when they are pushed together. But there are still more blocks when they are separated".

"You seem to misunderstand", the child says. "Look, when they are spread out you can see that there are more of them. Are you learning anything yet?"
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So, @Metaphysician Undercover, am I to understand that you agree (or understand) that

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

2. say, $(4 + \frac{1}{2}) \times \frac{1}{9} = 0.5$, and $9 \times \frac{1}{9} = 1.0$

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

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

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

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

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

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

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

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

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

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

Without an end, when do you have a remainder?

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

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

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

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

This is what I disagree with. Instead, I think that one divided by nine is an impossible procedure, provable by induction.
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Some are, some aren't.

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

Can you show me a mathematician who has questioned rational numbers like $\frac{1}{9}$?
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In other words, you're trying to persuade me.
Wrong. To persuade is to convince someone that something is true. You are, in my estimation, unpersuadeable; you've invested huge chunks of your time developing your weird theories and creating narratives to rehearse... your own personal thought terminating cliches (I've seen them), and you're not going to give that up. To see a perspective is entirely different; that is simply to understand what another's view is. As I've said multiple times, I could care less whether you believe the math or not. The only thing I'm giving a shot at is for you to see how the math works.
This is why the example, as proposed, is not useful.
Quite the contrary... it's the epitome of utility. Each of 9 people are getting dramatically closer to an equal portion of the 24 pizzas with this method than they are with 6 pizzas in the bin.
We are dealing with numbers, not with pizzas, and discussing the basis (principles) upon which we divide quantitative values.
Wrong. You are dealing with integers or some subset thereof, arbitrarily calling that numbers, ignoring the concepts laid out before you while making deepist excuses and deluding yourself into thinking that by doing so you've actually made some sort of interesting fundamental point.
I've tried to make this clear to you, but you keep going back to these examples. We are dealing with numbers, not with pizzas, and discussing the basis (principles) upon which we divide quantitative values.
There's nothing to make clear to me; this is illusory insight. The examples demonstrate that there is another concept here. Along with those six pizzas with not-quite-equal slices going into the bin you're chunking out perfectly valid mathematical ideal slices of ideal equal weight into the bin, with excuses. The excuses give you the illusion that you're being rational, but they are irrelevant with respect to throwing away the principles of rationals. They are, however, relevant to what mathematicians talk about.
As soon as we come to a mutual agreement about the divisibility of quantitative values (abstract numbers)
But you have an illusory insight with no valid truth criteria. You're in essence making an idol of integers, arbitrarily calling that number, and pretending you've done something fundamental.
I gave you an inductive proof and you refused it.
The problem isn't that I refused it. The problem is that it didn't prove what you claimed it proved.
Each time the machine is forced to "loop back" it is because there is a remainder which must still be divided.
Try this... instead of 1/9, let's do 1/7. Now our description has to change, because we get 0.(142867). So yes, each "time" the machine is forced to "loop back" it's because there's a remainder. But what is the remainder to 0.(142867)? Is it 3, 2, 6, 4, 5, or 1? Note that "each time the machine is forced to 'loop back'" it is because there is exactly one of these left as a remainder. Is there exactly one of those left as a remainder to 0.(142867)? Can you even answer these questions... do they have an answer? I'll await your reply before commenting further.

But if we can't say which remainder this is, we can still talk about the same thing using an alternate view. Suppose we run our long division program and we're told that the result is 0.125. Then what can we say about the ratios it was dividing? I claim we can say it was dividing k/8k for some k. Now likewise suppose we run our long division program and we're told the output is 0.(142857) using the description given by a symmetric recursion and infinite loops. Now what can we say about the ratios it was dividing? I claim we can say it was dividing k/7k.
Good, we're making some progress toward principles of agreement.
No, we're not. No mathematician denies that division is not closed in the integers; if you look back, you'll see where I actually posted the same thing in a prior post. You're denying that we can divide at all, and field division by definition can do so. The real discussion then is whether we're doing integral division using decimals or rational division, and since decimals are driven by powers of tens (including powers of tenths), it's immediately apparent it's rational division. But because you worship the idol of the integers, you're incapable of using the appropriate language for the appropriate context.
If you recognize that there are some restriction which may apply to the division of a unit, due to the nature of the unit, then you ought to understand that the conditions are derived from the real particulars of the application.
You've got it backwards. They're derived from the axioms of the system you're using. The axioms define various relationships between undefined terms. The application demands use of an appropriate axiomatic system whereby the mappings of the undefined terms have the relationships described by the axioms.
Why would you think that there is any type of thing, like a field or whatever, which would admit to any possible division imagined, whatsoever?
Because we define it. Incidentally in terms of application we can use this in arbitrarily complex ways. There are some 1080 atoms in the universe, but we can practically get far smaller than 10-80 by applying arithmetic coding to text. Note also that machines can far exceed what we can do, so the limits of what we can do are not bound by some smallest unit of some extant thing... they're bound by the furthest reaches of utility we can possibly get from machines. We can get much further not limiting our theories in silly inconsistent ways. But even without all of this, just for the math is all of the required justification.
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This is what I disagree with. Instead, I think that one divided by nine is an impossible procedure, provable by induction.

And yet 1 is deductively provable. even went through the troubles of outlining the start of a proof by induction. Thus, your disagreement ain't right.

What we can't do here, is write down 0 decimalpoint and endless 1s on paper. And we don't have to, because we can reason about 1/9 nonetheless as shown, like we can for other numbers.

Is writing down 0 decimalpoint and endless 1s on paper the whole of your troubles/denial here?

(Don't confuse/equivocate base 10 algorism, decimal representation, and numbers; it so happens that 1/2 = 0.5base 10 (decimal) = 0.222...base 5, artefacts of procedures.)
• 148
Interesting. Can 1/9 be represented non-repeatedly in base 9 arithmetic - or some other base?
• 8.9k
Interesting. Can 1/9 be represented non-repeatedly in base 9 arithmetic - or some other base?

$\frac{1}{9} = 0.111..._{10} = 0.1_{9}$
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At the alter he sacrifices all of mathematics beyond addition.

I thought all mathematics is addition, with just some techniques for doing it efficiently.
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I thought all mathematics is addition, with just some techniques for doing it efficiently.

That may be true of multiplication, exponentiation, tetration, etc, but the inverse operations break that closure. The numbers you can get by starting with 1 and then doing those operations are all the same, the natural numbers.

But if you subtract (undo addition to) a natural from another, you might get something that isn’t a natural: a negative number. So okay, we call the naturals and their negatives integers.

But if you divide (undo multiplication to) an integer by another, you might get something that’s not an integer: a fraction. Okay, so the integers and all their fractions are the rationals.

But if you take the root or log of (undo exponentiation to) a rational, you might get something that isn’t a rational... etc.
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