You're just distracting from the topic. — Metaphysician Undercover

Oh that's rich.

my question concerns the principle by which you divide a number — Metaphysician Undercover

^^ 1/9 is that thing.

is not the same as infinite. — Metaphysician Undercover

^^ distraction.

FYI, under the rationals, there's no such thing as 1/infinity.

It doesn't answer my question, and it's false. First, my question concerns the principle by which you divide a number, not the act by which you represent this, which is what your example describes. Second, your example is false and invalid because "ten's of G's of symbols" is not the same as infinite. — Metaphysician Undercover

So we can't divide a pizza into 9 slices because the slices don't weigh the same, and we can't divide 1 into 9 because 9 isn't infinity.

Sorry MU, but I'm not interested in playing Calvinball with you.

It's not the case that I don't understand — Metaphysician Undercover

It is the case that you do not understand.

how do you propose that it might be divided. — Metaphysician Undercover

By just doing so. I gave you an example, which is quite relevant, to help you understand. You ignored it. But it's still there. If you're going to ignore what I say, I'm not going to pretend we're having a conversation.

If you're genuinely interested in this:

So I really want to know what principles you are applying to divide 1 — Metaphysician Undercover

...then you need to understand that example.

I didn't ever blame the mathematicians for "not finding the remainder", I blamed then for hiding the fact that there is a remainder. — Metaphysician Undercover

So, the answer to the question of what the remainder of 0.(142857) is, is that there is in fact a remainder, it's clearly evident, mathematicians ought to respect that it is impossible to divide one by seven, and mathematicians are using smoke and mirrors to hide the fact that there is a remainder?

You're ignoring a key part of your Wikipedia definition, "These concepts may originate in real world concerns...". — Metaphysician Undercover

You're ignoring a key part of your Wikipedia definition: "may", not to mention the bolded part.

... then to simply assert "no it's doesn't" is completely insufficient. — Metaphysician Undercover

Multiple examples provided in link of utility following math.

An object, represented as "1", can be divided in an infinity of different ways. — Metaphysician Undercover

Of course. But the object represented as 1 is a mathematical object, not an onion. Nobody is claiming you can chop an onion into infinite pieces. But we can subdivide 1 indefinitely; there's no "math-atom" we run into. We can apply arithmetic coding for example to encode 10G of text, which means we can generate an arbitrary precision number with on the order of ten's of G's of symbols. So it turns out, you're the one confused, not me; you think 1 is a pizza or an onion. It's not. It's a number.

Regardless, I think I have found the core issue here. Your theories of where math originates suggests you think math is about just physical objects and, since it's not, you find counterexamples. But rather than taking this as being proven wrong, you double down, positing that mustachioed mathematicians conspire to lie about the nature of physical objects to themselves and others... something like that?

The LEM is used to come up with the list in the first place. — Samuel Lacrampe

No, your imagine did. There are people who argue against LFW on the basis that there can only be determined and random things... if it's determined it's "forced", if it's random it can't be will (or so it goes). But they get this list because they enumerate: not determined = random, not random = determined.

(a)They get two things, in their enumerated list. They applied LEM.
(b) You have three things, in yours. You applied LEM.
(c) Supposed there's a fourth thing; maybe it's just "p-zombies". Does that violate LEM? Nope.

So, you're arguing that LEM makes this list. But some people apply LEM and get two mechanics. You apply it, and you get three. If there were a fourth, LEM still holds. So apparently LEM is consistent with discarding your souls, with accepting your souls, and with accepting yet another possibility (p-zombies in this case). So how can you argue that LEM tells you there are three things when LEM works perfectly fine with 2 or 4 things? You're not using LEM to make this list... you're using your imagination. You're just enumerating, and stopping when you can't think of anything else.

If theories are supported by hypotheses, and hypotheses are supported by testing, then theories are supported by testing. — Samuel Lacrampe

It's a bit more subtle than that. The "more general" things are still theories before they are tested (and accepted). String Theory's a prime example.

To relate it to the "split-brain video", it is possibly the same "I" — Samuel Lacrampe

...if it were distinct "I"'s after the split, would that be critically damaging, or just interesting?

Well, if you think that I haven't already seen how math works, then you're wrong. — Metaphysician Undercover

What I mean is something like this:

I haven't seen that definition. — Metaphysician Undercover

I have no idea what you're talking about — Metaphysician Undercover

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

...

I've made from what I've seen is that a healthy dose of skepticism — Metaphysician Undercover

I can't see how this makes any relevant point. You've just demonstrated another smoke and mirrors method to hide the fact that there is a remainder. — Metaphysician Undercover

...no; blaming the mathematicians for your not finding the remainder is not healthy skepticism.

Where's the problem? — Metaphysician Undercover

You didn't answer the question, and I think the reason you didn't is because the question doesn't make sense. That carries over to your previous claim that 0.111... has a remainder.

I don't deny any of this, that's how math works, conventions are followed, and that's what convention has us call "dividing". — Metaphysician Undercover

...so... 0.(1)=1/9?

The question is on what principles do we say that the conventions are right or wrong. — Metaphysician Undercover

Logical deduction based on the axioms.

Do you agree that for any particular way that an action is carried out (an action being the means to an end), in this case a mathematical operation, it is possible that there might be a better way? So even if — Metaphysician Undercover

Sure, why not? Something like: 97*104=97+(100-3)(100+3)=97+10000-9=10088?

You seem to think that mathematics works because people dream up random axioms, then the axioms are applied, and voila, mathematics works. — Metaphysician Undercover

That's quite a fair description of Finite Geometry.

I start with the fundamental principle of "pure mathematics", which states that — Metaphysician Undercover

No it doesn't:

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. — Pure mathematics (wikipedia)

Part of the reason I post this definition (and rearrange this) is context for the response below:

I think that mathematics works because people design the axioms so as to be applicable to the real world. — Metaphysician Undercover

Examples of pure mathematics becoming useful (exact opposite of what you just said) here.

You start with the opposite (and what I claim backwards) position, that the fundamental "unit" is divisible any way one can imagine, an infinity of different ways. — Metaphysician Undercover

I start (frontwards) with the axioms.

First, I will argue that this annihilates pure mathematics — Metaphysician Undercover

I wouldn't go that far... it's just useful.

Second, I will argue that it leads you to believe, as you've demonstrated in this thread, that any object is divisible in any way imaginable, i.e. an infinity of different ways. — Metaphysician Undercover

Interesting... you claim that I've demonstrated that I believe any object can be divided an infinity of ways, and yet, in the same post, you quote me as saying there's about 10⁸⁰ atoms in the universe. Ladies and gentlemen... MU's healthy skepticism!

This demonstrates my point. — Metaphysician Undercover

No clue what you mean by that, so I'll just generically offer that arithmetic encoding works based on the same concepts we're discussing here... converting fractions (representing ranges of relative symbol frequencies) into placement systems (representing the coding).

As it turn out, no. Meta has revealed that one cannot subtract from a whole. Subtraction only works if you have more than one individual. And division leads to the heresy of fractions. — Banno

Meta's still playing with rocks while the rest of us have pointy sticks.

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

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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) — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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? — Metaphysician Undercover

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.

What are you doing then? — Metaphysician Undercover

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? — Metaphysician Undercover

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. — Metaphysician Undercover

Your criteria for failure amuses me. I have 9 happy people. You have 6 pizzas in the trash.

You seem to misunderstand. — Metaphysician Undercover

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 — Metaphysician Undercover

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. — Metaphysician Undercover

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? — Metaphysician Undercover

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? — Metaphysician Undercover

Yes; I'm learning about how you think.

Would you consider the proposition that certain numbers just cannot be divided by each other? — Metaphysician Undercover

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,... — Metaphysician Undercover

If pure math can have no fractions, what is this?:

reasserting it will not persuade me to agree. — Metaphysician Undercover

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 — Metaphysician Undercover

Therefore it is impossible that 2/3 represents an exact portion of a unit. — Metaphysician Undercover

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 — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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 — Metaphysician Undercover

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 — Metaphysician Undercover

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, — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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? — Metaphysician Undercover

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, — Metaphysician Undercover

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. — Metaphysician Undercover

(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, — Metaphysician Undercover

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).

energy in a simple closed system with nothing else cannot increase — Samuel Lacrampe

That's quite hedged... "closed" seems to imply not getting energy from somewhere else, and "simple" can mean anything. Regardless, there's not necessarily a "place" where dark energy is "coming from", and a principle with a generic out ("simple") isn't a fundamental principle. What you're really doing is pattern matching to save your definition, not applying a principle. To demonstrate, let's just table this...

...instead, we'll look at entropy. Entropy actually increases in time. But that means that effects can indeed be greater than the cause, even in a closed system. In fact, that's just how entropy works. You have a closed system with some amount of entropy, and through causal evolution of states, it yields an effect with greater entropy. I'm sure you're not surprised or even shocked by this, but by the definition of PoSR currently on the table this does indeed violate it. So you have a problem.

To resolve this you need a better definition. If entropy increasing doesn't violate PoSR, then some effects can indeed be greater than causes in some sense of the word "greater". But if all we're doing is finding some things that cannot increase and saying "See? PoSR is true!", and other things that can increase and saying "Well yeah, but that doesn't really count", then PoSR is just a slot machine theory (SMT). "There are two senses of greater... where effects cannot be greater than causes, the PoSR predicts that effects cannot be greater than causes. In senses where effects can be greater than causes, we say that's allowed because x." Do you see the problem?

PoSR has to be able to conceptually fail else it's vacuous.

A Little Bang has insufficient causal power on its own to explain a Big Bang. This would be creating something out of nothing. — Samuel Lacrampe

You have to first explain why Little Bangs being Little implies they're being Lesser in the right way before you can apply PoSR here. Because entropy increases, either there is a Lesser in the wrong way, or PoSR is evidentially untrue. If the only way we can tell which is which is to see which ways are consistent with the PoSR, then PoSR is vacuously true like Slot Machine Theory. This is why you need a functional definition of greater.

How does your example of colors rely on the LEM? — Samuel Lacrampe

It doesn't... that's the point.

You can however categorize all things into red things and non-red things, and this is exhaustive. — Samuel Lacrampe

Red, yellow, green, blue... done. That's all the colors I can imagine. If a color isn't yellow, green, or blue, then by LEM and a bit of deduction, it must be red. I can apply that same argument to yellow; to green; and to blue. Therefore, I'm using LEM to show that my color list is complete. Right? I say, wrong. There could be a fifth color, call it orange. If there is, that's not a matter of LEM being violated; it's a matter of my being wrong about the list being complete. The problem is that my imagination failed me.

So here's what you're saying. Determinism, randomness, original cause... done. That's all the mechanics you can imagine. If something isn't deterministic or random, it must have free will; if it's not free will and not random, it must be deterministic, and so on. So by LEM, the list is complete. Right? I say, wrong. If there can be these three mechanics, there can be a fourth. The problem of showing this list is complete is not a problem of applying the LEM. It's a problem of proving there's no fourth mechanic. Make sense now?

Randomness is hard to define in itself. One might think it's just randomness and determinism that's exhaustive; in fact, people have actually made that argument against LFW. But I'm granting original causation is a third category. But the whole problem is, can there be more, some that you maybe just haven't imagined? LEM is the wrong tool here; that there are more colors than I imagine doesn't violate LEM, and that there are exactly that many doesn't follow from LEM. You cannot argue this is a complete list by applying LEM.

Theories are built from hypothesis testing. From this link: "The scientific method involves the proposal and testing of hypotheses, [...] if it fulfills the necessary criteria (see above), then the explanation becomes a theory." — Samuel Lacrampe

First off, there's subtlety here... see the plato.stanford.edu page. Second, that does not address my comment. See e.g. here:

Both are statements about the physical universe. Hypotheses are more specific and theories are more general. Theories tend to have many hypotheses incorporated into them. ... The difference between hypothesis and theory is not one of “certainty”. Hypotheses do not “grow up” to be theories. — Paul Lucas PhD, quora

It's worth jumping over and reading through Lucas's quora answer... he gives specific examples.

Split personality does not entail split soul. — Samuel Lacrampe

Alien hand syndrome (AHS) is not split personality. Incidentally, I don't think this really shows your religion is flawed (nor intend to do so)... if a soul can be split into two separate souls, that would just be... a fact. We could simply say that there's this interesting tidbit of doctrine you're just mistaken about. You can choose to hang your hat on it if you like, but that's up to you. What I find interesting though, is that this (corpus callosotomy induced AHS) evidentially appears to be a genuine thing.

If one side of the brain holds a memory that the other side does not, then this could be sufficient to explain a change in behaviour.

Yeah, but that sounds like duct tape and bubble gum. The less your souls have to do with subjective experiences and phenomena like making apparent free will choices as it relates to situations like AHS, the more it sounds like it's more about preserving a belief than being correct.

But I don't see how expressing the remainder as a fraction resolves the issue of the remainder. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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.

With everything you're saying about numbers and division and the like, I honestly want to know what is going on in your head. — Michael

@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.

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.

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.

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.

I would like to see this principle applied in practice to see how effective it is in disarming people who make blind guesses about things just for the sake of making blind guesses about them. — Frank Apisa

It just sounds interesting. :smile: I think we wind up at the same spot. You're correct IMO... I don't think we disagree... I think we're seeing the same thing, just expressing two perspectives of it.

Anyone can justify an opinion...and the strength of the justification is almost always strong to the person doing the justification. — Frank Apisa

But that's a false front... people with "opinions" like this don't generally hold the opinions because of the justifications they give; rather, they give justifications for the opinions because they hold the opinions. The primary error people commit here is the high burden they place on discarding their opinions and the low burden they place on keeping them. So they tip the scales of the strength of justification, generally, to the highest degree possible; they require no justification to keep the opinion; but they require the highest burden to discard it. That's what's conveyed with "change my mind"... it's the notion of "I don't need to justify my belief to keep it... rather, you need to make it impossible to believe it before I discard it". That is the opposite of belief warranted based on the strength of justification; it is, rather, belief warranted for its own sake.

It is my OPINION that one way to improve philosophical discussions would be to insist that the words "believe" and "belief" not be used except to comment on the absurdity of using them. — Frank Apisa

Maybe there's something to this. But I'm not necessarily convinced this would thwart the problem I see above. I would like to see this principle applied in practice to see how effective it is in disarming people of belief for its own sake.

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.

Of course. If you're just now noticing, I refuse to use that deceptive language, — Metaphysician Undercover

As a result, ->I have no idea what you're talking about<-. No one mentioned multiplication, the issue was division. — Metaphysician Undercover

^-- One of those two things is a lie. Most charitably, you're incapable of using the language.

Of course. If you're just now noticing, I refuse to use that deceptive language, — Metaphysician Undercover

^-- This is straight up paranoia. Deception has two parts... the advertised meaning, and the true meaning... the advertised meaning must be what you want to trick the other person to believe... the true meaning must be something different. We don't have that here... we only have one part... the usage.

The problem is that you don't understand the language, therefore, you spin this meaningless narrative that people are trying to deceive you (sprinkled with paranoia). It's meaningless, because there's nothing you can say that you're being deceived to believe... it's a working language, you just don't understand it. This is trivial to show in your debates; you don't even bother to debate what doesn't work. This is perceived on your part as non-compliance: "I refuse to use that deceptive language", but in terms of truth, that's hollow... your only possible genuine complaint is that the language doesn't work... to show it doesn't work you have to know how to speak the language, in order to construct the contradiction.

Otherwise, the only possible complaint you have left, the one you keep whining about, is that it's not the same language as your uninteresting one.

(e) To divide nine into nine parts, or (f) to divide eighteen into nine parts is very clearly division. — Metaphysician Undercover

Sure, but that's just division:
(e) $\displaystyle 9 \div 9 = 1$ <- "divide nine into nine parts"
(f) $\displaystyle 18 \div 9 = 2$ <- "divide eighteen into 9 parts"
...and this is using a fraction:
(g) $\displaystyle \frac{1}{9} \times 9 = 1$ <- " 1/9 of 9"
(h) $\displaystyle \frac{1}{9} \times 18 = 2$ <- "1/9 of 18".
Going back to our loaves:
(i) $\displaystyle 3 \div 24 = \frac{3}{24}$
There's no deception here, there's only confusion... on your part. Nobody who uses this language is confused. This is just how the language works. 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.

If this is true, then we need to define how to distinguish a whole from a part, so that we are not referring to the part as "one", when it is really 1/8 of the whole, and we are not referring to the whole as "eight" parts when it is really one whole. — Metaphysician Undercover

We do it by applying a unit. A slice is a part of a pizza. One pizza. Eight slices. It's so easy, everyone but you does it all the time!

I would enter a discussion of parts and wholes with you, so long as we have principles whereby we can distinguish one from the other, — Metaphysician Undercover

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. Parts are transitive; an inch being part of a foot being part of a yard means an inch is part of a yard. We call the "whole" we're talking about a unit, and we just specify it... that's all there is to it. I say the yardstick is one yard long. That is three feet long, 36 inches long, and 288 eights of an inch long.

and not just randomly decide to call this a part, and that a whole,

Why not? That's how you use language. You have to specify the thing you're talking about, even if it's a part. I drive my car. I drive it into traffic. I turn the steering wheel. There's no problem doing this, outside of you having a problem with it, but that's not our problem. Let me rephrase this so that it sinks in:

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

Nobody has to talk to you before they use an IEEE-754 64-bit float in a program and, even if they do talk to you, you're giving them absolutely no reason to care. Let me rephrase that... nobody cares about your impoverished language. The reason I'm talking to you is that I care about you.

because that would get nowhere. — Metaphysician Undercover

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.

Aha! I think I've got it:

As a result, I have no idea what you're talking about. No one mentioned multiplication, the issue was division. — Metaphysician Undercover

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

Okay, let me show you where the multiplication is. Let's revisit this:

(a) 1/9 can be one if the whole group is nine, (b) it can be two if the whole group is eighteen, (c) it can be three if the whole group is 27, (d) it can be four if the whole group is thirty six, and so on and so forth. — Metaphysician Undercover

(a) 1/9 of nine is $\displaystyle\frac{1}{9}\times 9 = 1$
(b) 1/9 of eighteen is $\displaystyle\frac{1}{9}\times 18 = 2$
(c) 1/9 of 27 is $\displaystyle\frac{1}{9}\times 27 = 3$
(d) 1/9 of thirty six is $\displaystyle\frac{1}{9}\times 36 = 4$
Do you see the multiplication now?

Since I think we've finally nailed down the problem, I'll keep to just the key parts.

If you think that the quantity represented by "one" can be divided in any way that you please, then you deny the meaning of "one" as a single thing, — Metaphysician Undercover

Two major problems with this MU:

• When I slice one pizza into eight slices, it's still one pizza. So no, I don't deny singularity by saying that something can be divided.
• Ignoring the fact that I can still say it's one pizza, I can slice a pizza into multiple parts anyway. If there's some sense in which that denies your concept of singularity, then your concept of singularity is broken, because those pizzas can be sliced

(Of course, you could always deny pizzas are real).

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

On an interpersonal note:

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

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

^^-- this makes you look irresponsible and lazy. You're blaming me for not understanding you, and blaming me again for you not understand me. This conveys the message that you think your time is extremely valuable and my time is worthless. That's... not great optics.

Saw something fly by about adding zeros — jorndoe

I think you're referring to the discussion I was having which was in ℝ.

A lot of the stuff you're saying has no meaning to me. The way my mind works, words are just words, so telling me what something is called just gets filtered out into a "there is a category x". Outside of that I have some basic questions about what you're saying.

The soul is the seat of your perfect consciousness — BrendanCount

Is this a definition of the soul? I'm good for consciousness, so I'm fine with a soul being a seat of consciousness, but I'm not quite sure in what sense my consciousness is perfect.

in the present moment forever and ever..in eternity — BrendanCount

...it seems there's the possibility of gaps in my consciousness; specifically under anesthesia (I'm okay with saying I'm conscious while sleeping just in a different "way", I just cannot define consciousness in this specific circumstance). I'm very literal, so I would count those as cessations... is that... fair to you?

The next topic of concern is a witness to the chief rites..of the world soul..which literally IS the Creator..being smaller than a circle the size of a foot in diameter — BrendanCount

Not sure what you mean by chief rites, but it sounds like you're describing the soul as having spatial location (in contrast with Sam's description)... is that correct?

The Adam Kadmon is the soul itself..and is located in Chochmah — BrendanCount

Before this has meaning to me, you would have to explain what the Adam Kadmon is and what Chochmah is.

as the eternal will power being Holy of Gods and Goddesses on Earth..Including Gaia.. — BrendanCount

FYI, just as a commentary... I'm not a big believer in the gods, but I'm good just interpreting this as meaning that the will is sacred in a less literal sense.

In reality it is contradictory trivial to divide one into any parts, because then you are saying that it is not one, but however many parts you are dividing it into. — Metaphysician Undercover

To slice a pizza into equal slices, try a pizza cutter. Ten paragraphs of nonsense gibberish can be refuted with one kitchen appliance.

To most people, cutting one thing into multiple pieces is a triviality. Where's the contradiction you're describing? Cut one pizza into four equal slices, and you have four slices, each of which is one fourth of the pizza. What you're saying translates that there's a contradiction here because I'm saying that there are four slices and one pizza. How's that a contradiction? You would have to change the concept of "contradiction" to something that can actually be done, and the concept of "impossible" to something you can actually do, to make this argument stick, because we damned well can slice a pizza into four slices.

We can also cut each of those four slices in half, making eight slices. One of the four slices we had before is the same quantity of pizza as two of the eight slices we produce; i.e., it's a "particular quantity". It all works, MU. There's no contradictions except in your fictitious world where you can't use pizza cutters. In reality, we have pizza cutters and we can and do slice pizza.

That ain't no OED. — Banno

The former has no attribution. The latter says "powered by Oxford" (OUP). I personally was granting that maybe he had one of those ancient analog thingies made of trees; can't quite trace it further than that (though TBH I didn't try too hard past those two).

It's a slow day, so i dug out both the Concise and the Shorter OED. — Banno

I found it on some random russian vocabulary site: (https://slovar-vocab.com/english/fundamental-vocab/number-6810737.html) (ETA: Unhiding this link). An expanded version can be found here.

Your point, as I understand it, is that the PoSR is (1) too generic to be falsifiable, and (2) not substantial enough to make an impact on our reasoning. Is that it? — Samuel Lacrampe

Sort of. The current definition you have for it has both these properties.

If so, I respond that (1) It is indeed hard to falsify, due to my claim that it is a first principle, which means it cannot be judged by appealing to any prior principles; very much like the LNC. But also like the LNC, it can be posited from induction and the criteria for self-evidence. — Samuel Lacrampe

You're missing the point. You're trying to apply PoSR in a particular way. But if your application of PoSR can be wrong without violating PoSR, then there's giant questions as to whether PoSR is meaningful enough to apply. Go back to that CoE thing. You said according to PoSR energy cannot increase. Energy, it turns out, does indeed increase. So that's wrong. But PoSR was true anyway. How can you claim such a thing is inductive or useful?

(2) it does impact our reasoning in hypothesis testing. E.g. "What caused the Big Bang? Maybe a Little Bang?" This hypothesis would be automatically rejected on the grounds that it does not sufficiently explain the phenomena. — Samuel Lacrampe

Why?

The first premise should be changed to "All physical things are either deterministic or random". We can defend this claim either by observations, or by appealing to the Law of Excluded Middle, as previously described here in the last paragraph. — Samuel Lacrampe

Let's say LFW exists, and QI is a thing, and somehow we wind up showing both. Then I observe deterministic like things (computers), random things (wavefunction collapse), and original cause things (people). So here's the big question... why can't all three be physical?

by appealing to the Law of Excluded Middle — Samuel Lacrampe

I think you're confused. The LEM means that you either have something or you don't. But you're treating it as a guide to whether you've enumerated everything or not. It can't be used for the latter. I can't say that because I can only think of four colors, therefore there are only four colors due to the LEM. You enumerated determined things and random things, and you also have this other category of original causation (reminder... my label... this is something I picked up while talking with libertarians). LEM doesn't tell you that these are the only categories.

Are you a proponent of Scientism? The PoSR is a principle of metaphysics which transcends science. — Samuel Lacrampe

You're reaching. You applied "greater" in your definition of sufficient to say that energy never increases, remember? But it turns out it actually does increase, due to dark energy. Being charitable to PoSR, this alone proves that your definition of "greater" is insufficient to be practically used. This has nothing to do with whether I'm a proponent of scientism.

From this page: "For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it." The criteria is not how it can be verified, but if it can be verified in principle. — Samuel Lacrampe

FYI, a hypothesis and a theory are different kinds of things.

I'm fairly sure the two names are interchangeable... But this is not the case — Samuel Lacrampe

...so there's your answer... science isn't distinct from philosophy... it's intermingled with it.

Physical things can be destroyed, in the sense of spatially split in pieces. — Samuel Lacrampe

Okay... that's a bit bad news then, because it would appear people are splittable into pieces (link: youtube, Ramachandran) (at least two).

No one has demonstrated how 1/9 represents any particular quantity, — Metaphysician Undercover

Wrong. I have. Also, Banno has:

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

...using your own dictionary. And your dictionary has, as demonstrated by Banno. Your problem is that you don't understand the language; that's confounded by the fact that you think you do.

because as I've explainedasserted, it does not represent a quantity.

FTFY. A ninth is the specific particular quantity corresponding to dividing one into nine equal units. That's why your same dictionary that you quoted the definition of a number in says that a fraction is a number.

So how do you claim they are equalor equivalent? — Metaphysician Undercover

By proof, such as the one given in the OP.

I see no such definition. — Metaphysician Undercover

We're using your OED definition.

According to the definition of "number" which I provided they do not represent numbers. Where's your definition of "number"? — Metaphysician Undercover

Wrong. According to that definition, they are numbers. You just don't understand that definition... see above.

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

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

Ha ha, that's ridiculous. — Metaphysician Undercover

It's only ridiculous to you, because you don't speak the language.

1/9 can be any quantity you want, depending on the size of the whole which is being divided nine ways. — Metaphysician Undercover

We've already addressed this... you're saying nothing about fractions that doesn't also apply to counting numbers. To have a point you must special plead it.

1/9 can be one if the whole is nine, it can be two if the whole is eighteen, it can be three if the whole is 27, it can be four if the whole is thirty six, and so on and so forth. — Metaphysician Undercover

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

And if mathematics speakers really use the term in this way, then I'd have to say that they really do not know what they're talking about. Don't you agree? — Metaphysician Undercover

Nope. I would say you had some severe misunderstanding of math.

How can something (1/9), which can be absolutely any quantity whatsoever, be said to be a particular quantity? — Metaphysician Undercover

1/9 is only a ninth of 1. But you can take 1 of anything, including groups.

And how can you not see the ridiculousness of the claim that it is a particular quantity? — Metaphysician Undercover

Because I understand how it makes sense, because I understand it.

The disagreements are flat Earther stuff. — Michael

Hang on... isn't the flat earth two dimensional?:

Space cannot be represented as distinct dimensions, as the irrationality of these two dimensional figures demonstrates. — Metaphysician Undercover

The reason the unit circle — EnPassant

Not talking about the unit circle... just unity on the number line, and the idiom "going in circles" which means to retrace your paths over and over.

But if you draw the x - axis and mark off one unit, there you have it. The sum of dimensionless points add up to a unit. — EnPassant

You're going in circles. 1 is one of the possible things that sum can be. Pause for a second and think about this; otherwise this could continue forever.

But you still have 0 + 0 + ... = something — EnPassant

If you do. It can also be 0. It can also not be anything. It can also be anything.

Say 0 + 0 + 0 + ... = 50 units. — EnPassant

Okay, so that sum is 50 units.

(0 + 0 + 0 + ...)/50 = 1 — EnPassant

You can only say that if you're literally talking about that 50 unit thing, because:

But 0/50 = 0. — EnPassant

Yes, and 0+0+0... can be equal to 1. And 50. And a billion. And negative 7. To recap, that sum is undefined.

Yes, because it can't be defined in terms of calculus but the question remains, what is it? — EnPassant

It's undefined! :wink:

Yes, but that is arbitrary as the unit can be taken as any width, as in geometry - the unit can be 1 inch or 1 light year. — EnPassant

No, it's not arbitrary. It's just infinitely non-specific. That sum genuinely is sometimes 1 inch, sometimes a light year, sometimes 0, sometimes negative. So it's undefined. You can't reduce the sum to 1 inch if it could be negative or a light year. So it's useless to ponder whether it "truly" is 1 inch or "truly" is a light here, because your infinite sum doesn't give you the information to distinguish any length from any other.

Sometimes you can get that information elsewhere. But from just this, you just can't say.

It's the same idea 0.999.. - 1/10 + 1/100 + ... — EnPassant

So you're talking about 0.111...? Then @Pfhorrest's post applies:

That doesn’t sum to 1, that sums to 1/9. — Pfhorrest

What seems to be happening here is that 1/x = 0 at infinity. — EnPassant

There is no "at infinity" here though. Every term here is a finite number; there's just an infinite number of finite numbers. Think of it intuitively this way... imagine the set you're trying to picture... it has "at infinity" in it, and maybe some other things. Remove every infinite-th step from this; we only care about finite steps. But we do want all of the finite steps. Now you still have an infinite set, but it only has finite terms in it. That is the thing we're describing.

So 0 + 0 + ... = 1 after an infinity of terms.
But most mathematicians probable would not accept this. — EnPassant

If we see 0 repeated an infinite number of times in a sum, we tend to say that the result is undefined.

But they add up to 1 unit. How do da? — EnPassant

But you could do the same thing with a segment of length 2, 50, 0, and -7. So that infinite sum could also add up to 2, 50, 0, or -7, or any other value. This is what undefined refers to.

Every time a mathematician draws a graph on the x,y axis they are implicitly accepting that 0 + 0 + ... = 1 because they are working under the assumption that an infinity of dimensionless points add up to extension; the unit. — EnPassant

No, they aren't. The unit position is not defined in terms of infinite additions of 0. That would be useless, since infinite additions of 0 is undefined.

Ok, but isn't this what happens with 1/2^c in the sum 1/2 + 1/4...? — EnPassant

No.

Series 1
Step 1: 1/2
Step 2: 1/2+1/4
Step 3: 1/2+1/4+1/8
...
Step 10: 1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024
Step 11: 1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024+1/2048
...

Step with all finite numbers: 1/2+1/4+1/8+1/16+...

Series 2
Step 1: 1/2
Step 2: 1/4+1/4
Step 3: 1/8+1/8+1/8
...
Step 10: 1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024+1/1024
...

Step with all finite numbers: ? + ? + ? + ? + ? + ...

In series 1, at the "step with all finite numbers", the first term is 1/2. 1/2 is finite, and 0 < 1/2 < 1. In series 2, what is the first term at the "step with all finite numbers"? It's just that "step with all finite numbers" that has an infinite number of terms. Can you name the finite number x such that ? is 1/x? Can you say that 0 < ? < 1? Can you even say what ? is?

Or in terms of 0.999..you can't, by this criterion, say 0 < 1/10^c < 1 — EnPassant

Sorry, what is c here and how does that relate to 0.999...?

Ramanujan summed the natural numbers and got -1/12. — EnPassant

Yes, by using a different definition for a divergent infinite sum. I toyed with that here:

So does this mean ...999.999... = 0? — InPitzotl

...since ...999=-1 in 10-adics.

Ok, but can't this be also said for 0.999...? — EnPassant

.999... is an infinite string of 9's. There's no problem with that per se.

Adding terms and then saying 'at infinity'. — EnPassant

We don't have to say "at infinity"... it's an infinite string of 9's. We add one 9 for all finite numbers, there are an infinite number of finite numbers, therefore there are an infinite number of 9's. But I think you misunderstand what the problem is...

You can't have (b)

In the infinite string .999...., I can say that the first digit after the decimal is 9. That's how I construct that infinite string. It's done homogeneously... it's always there at all steps.

In contrast, you cannot say what finite value x is when you have your infinite case, or that 0 < 1/x < 1 is true in that case. Your x changes every time you increment even by 1. What x is is inhomogeneous; it's diferent every time.

9 is the n-th digit in my string at all steps in the construction of it starting at n, going on indefinitely. x is a different value at every step in your construction. When I extend my string to infinity, 9 is still the first and nth digit. When you extend your inequality to infinity, x isn't finite, and you can't say 0 < 1/x < 1 for an infinite x. You never have an infinite number of a finite 1/x where 0 < 1/x < 1.

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

MU, you're pretending here to be making an argument about .999... = 1:

As I've explained to fishfry already, that two things are equivalent does not mean that they are the same thing. Therefore what is on the left side of the "=" (which indicates equivalent) does not provide a definition of what is on the right side. — Metaphysician Undercover

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

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

...you're still not talking about the relevant point. You're still not advancing any reason why you think equality represents different values, or why you think the same value representing a particular quality can actually wind up representing different qualities. Nor are you even trying to make this point; you're just, instead, playing hide-the-ball.

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

$\displaystyle\sum_{i=1}^{\infty}{a_i} = \lim_{n\to\infty}\sum_{i=1}^{n}{a_i}$
When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. — Wikipedia

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

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

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

I made my point — Metaphysician Undercover

Yes, but it's all gibberish nonsense.

I made my point, symbols such as 1/2, 1/3, 1/9, are representative of ratios between quantities, they do not represent any particular quantitative value. — Metaphysician Undercover

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

The argument is very clear in my discussion with Banno. — Metaphysician Undercover

What argument?

You just cannot grasp the first premise, that 1/9 does not represent any particular quantity, and therefore it is not a number.

Your "premise" isn't a premise... it's a pointless language game. It shows you cannot speak the language (or at the very least, refuse to). You invented some niche and uninteresting alternate meaning for "particular quantity" that mathematics speakers do not use. The way mathematics speakers use the term "particular quantity", 1/9 is indeed one of those things. So you're not really advancing a "view" of quantities, you're promoting a language that's uninteresting. Therefore, your real burden is to show what's wrong with the language of math; you can't just say, "I don't 'believe' 1/9 is a particular quantity"... you have to say, "saying '1/9' is a particular quantity leads to the following problem" and say what that problem is.

As the number of terms taken increases 1/x decreases but never becomes zero. — EnPassant

That's true for all finite x. But you need it to be true for an infinite x. To see the problem, here's a "troll proof" that infinity is finite. 1. 1 is finite. 2. For all x where x is finite, x+1 is finite. 3. By 1 and 2, and infinite recursion, infinity is finite, QED.

Think of this "troll proof" analogous to your conjecture. Your inequality holds for all finite x's, no matter how big the x is. But also, no matter how big the x is, you only have a finite number of terms. But to apply Theorem 1, you need two things: (a) a value x such that 0 < 1/x < 1, (b) an infinite number of those values. You can't have (b) with any finite number. You can't say (a) "at infinity". Since you need both, and never have both, you cannot apply Theorem 1.

I said these do not represent any particular quantity, and ought not be considered as numbers. — Metaphysician Undercover

But I note that your OED definition talks about values referring to the same particular quantity. And I note that you've chosen of your own will in this post to not actually argue the relevant point... which was that .999...=1 is equivalence under equality, and under equality equivalence implies having the same value.

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

It is the belief that they are numbers which is what I consider to be a problem.

Your problem is your problem though, not mine.

The difference between these two is a difference which does not make a difference, for them, so they say that it is the same value. — Metaphysician Undercover

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

But that doesn't prevent me from arguing that the claim that there is a difference which doesn't make a difference is a contradictory claim.

You can do whatever you wish, but I'm under no obligation to take you seriously, especially at your word.

My argument, if you've read what I posted, is that .999... does not represent a particular quantity. — Metaphysician Undercover

That's not an argument, it's a claim.

And I told you why that's inconsistent with the views you presented. It's still there in the post. To help you out, I repeated it at the top. But the barrier between us (and also you and many others) goes far deeper than this. You're trying to have a conversation without speaking the language. That's made even worse by your refusal to even consider speaking it, which is made even worse by your having unfalsifiable "opinions" on how the language should even work. All of this is a grand recipe for having pointless arguments, but nobody is interested in having pointless arguments with you. We have to clear this barrier before it's even possible to have a conversation with you.

I suggest that you come back when you've got an argument to make. — Metaphysician Undercover

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