Is 1 = 0.999... ? Really?

Maybe because a hidden implied confusion that needs a more definitive clarification, ? In order to escape a feedback i & o ? , in order to relate them . I hazard.

This broke my parser.

Sorry. But to avoid a tautology. Such implication implies a referential collapse into it.('self)

Sigh.

Yes. Obviously.

I just told you I would be glad to (in your words) “put to rest this joke of a conversation between the two of us” - not for the first time by any means(!), and each time you continue it or try to misrepresent me because you know I’ll respond to correct you. But no, as usual, you conclude incorrectly the first thing that pops into your head that it’s “ego issues” - I killed that a long long time ago. I just can’t stand stupidity being stupidly attributed by stupid people to people who are not stupid.
It’s not just a personal distaste, if others stupidly take stupidity seriously it can have a detrimental effect on many people, only fuelling the fire that is stupid people, which I am trying my best to douse whenever I can in a desperate attempt to feel like the wider impact of collective stupidity might be lessened even to the slightest extent. I’d be quite happy for others to perceive this attempt as pathetic - as I said, my “ego” doesn’t give a shit what you think of me, - but the problem is that stupid people tend to confuse “pathetic” with being as worthless as stupid. Only non-stupid people recognise that it doesn’t matter who is delivering an argument, so it shouldn’t matter that it’s an officially tested and confirmed fact that statistically I’m almost always the smartest guy in the room, but this doesn’t guarantee that stupid people are able to recognise this. In fact, the more people there are who are stupider than you maximises the number of people who are relatively too stupid to understand this fact - not least due to cognitive biases and poor education. This takes its toll over time - there’s a reason why I’ve singled you out to be impatient with. If you had eyes, you’d notice I’m being perfectly patient with EVERYONE else. This is even if they disagree with me. This is why you’re stupid for thinking I’m an impatient and uncooperative person in general, when this is highly exceptional behaviour on my part. Obviously none of this explanation will mean anything to you, because even though it’s the correct explanation, you are locked into ignoring correct explanations in favour of the first prejudicial assumption that came to your mind that best feeds your cognitive biases.

To quickly sweep up the above mess of a quote, as I have already repeatedly pointed out, quantities can refer to either the general abstract notion of “quantity” or a specific concrete instance of “a quantity”. “Number”, as a representation of quantity, can refer to the shape of “a number” i.e. the symbol that communicates this representation (as well as a shapeless sound of a word, or the shapeless abstract notion of number).

Just to unravel your blanket equivocation that “all numbers are shapeless”…

And yes, this is all factual and true, unlike “all horses are numbers”, which is contingent upon a drastic shift away from convention, and only to the extent of representation rather than direct identity.
There’s just so much oversight and truncated thought processes in everything you say…

I can handle that just fine in somebody who doesn’t relentlessly present themselves as someone who is indisputably correct about a subject on which they know they lack expertise, and as though someone who does know the subject is plainly wrong in face of the first thing that pops into your head. I care far less when people are simply wrong, it’s HOW you (in particular) are wrong that makes all the difference here. Again - if you had eyes you’d have noticed it’s only you who is a problem out of all the other people who aren’t agreeing with mathematical fact.

My problem, which again you incorrectly identify so predictably in line with common cognitive biases, is that I understand your points too easily. They’re all the same things I already considered when initially coming to understand this topic. To you it’s all advanced, insightful stuff, but if you weren’t stupid you would at least be able to comprehend in the abstract that this isn’t the same for everyone. And again this isn’t my ego, clearly I don’t care about what you think of me or I wouldn’t be such an asshole to you, and in front of everyone else. I care that you’re stupidly spreading stupidity and stupidity is an epidemic problem with the world. I care that people aren’t too stupid to understand correct facts and that they don’t present their incorrectness as correct with such certainty and persistence. These facts that I care about aren’t “me” - I care what people think about facts (whoever is sharing them) and their approach to talking about them - nothing more.

Yes, as I said, that’s what a “given” is. It’s not what a truth is. You can posit falsity as a given for the sake of valid argument. It just won’t be sound argument.
It’s not a fact that a horse is a number even if you posit it as a given. Stop equivocating.

It should be quite obvious that for (f(x)=0), (f(x)) represents the number (0). It also represents a function - more specifically the “way” (a doing) to get to the number (a being) that it already represents. Depending on the number in the set with which there is this binary relation - as in your copy/paste from Wikipedia - the function might represent a different number, but it represents a number regardless.

I really don’t see any point in continuing this, you’re not going to get it, nor ask questions if you don’t agree, which you won’t.

Wow.

Even in the realm of mathematics, this sort of truculence goes on.

Some might suppose that one of them is right and the other is wrong. Or that one of them comes closer to whatever the necessarily correct answer finally is.

All this and nary a trace of dasein, conflicting goods and political economy. Or none that I can discern.

That’s one of the points of our disagreement. I have no idea how to resolve it.

What I’m going to do now is merely restate my position:

(0.9 + 0.09 + 0.009 + \cdots) is an expression that is meaningful and an expression that was meaningful long before the concept of limit was introduced.

Here’s something that supports that claim:

We all know that (0.9 + 0.09 + 0.009 + \cdots) is greater than (0.9). Moreover, we can know this without knowing anything about limits. I hope you agree with this. How can that be the case if the expression is truly meaningless? How can you say that (x) is greater than (0.9) if (x) is truly meaningless? You can’t.

To wtf:

Here’s a hypothesis as to what you’re trying to do.

What you’re trying to do here is you’re trying to force the expression to be something other than it is.

You are asking “What terminating decimal is equivalent to this non-terminating decimal?” The answer to this question is “None”. But you can’t accept this because you find it unsatisfying. You want it to be something else because you have a need for something else.

Consider a different example.

You want to know the result of (5 \div 2) but you want it to be an integer. The result is not an integer but you nonetheless want it to be an integer. You know that (5 \div 2) is (2 + 5 \div 10) but since you want the result to be an integer you need to do something about it.

The RIGHT way to approach this problem is to substitute (2 + 5 \div 10) with an integer that is closest to it. That’s either (2) or (3) and the one to choose is to be decided by your needs.

The WRONG way to approach this problem is to fail to recognize that the result of (5 \div 2) is NOT an integer and to instead insist that the result is undefined and thus in need of being defined e.g. by changing the definition of division to that of integer division.

In the same way that (5 \div 2) is not undefined merely because it is not equal to an integer, (0.\dot9) isn’t undefined merely because it isn’t equal to a terminating decimal.

Ooo nice word, I actually didn’t know that one and had to look it up :smiley:

Eh, it’s been building up for a long time now. I began civil as always, but this one guy has been “I’m not a mathematician, but I know for certain that the mathematical consensus is wrong, and everyone who doesn’t agree with me is definitely flatly wrong and their arguments either never existed or are irrelevant” for way too long by now, without learning anything…

Would you say that dasein affects how correct mathematical equations are?

No, you don’t. And we shall see that shortly.

What makes you think that when I said “All numbers are shapeless” I actually said “All written symbols that we use to represent quantities are shapeless”?

That’s all I need to show you that you’re wrong: tell you that what I said is not what you think I said.

But since you love conventions, I’ll add that if you go to Wikipedia you can see that they make a very clear distinction between numbers and numerals.

en.wikipedia.org/wiki/Number

Is it so strange that the word “number” most commonly refers to actual numerical quantities and not to symbols (written or spoken) that refer to numerical quantities?

When I said “All horses are numbers” I did not mean “All horses (as conventionally defined) are numbers”. What I meant is “All horses (as I defined the term here in this post) are numbers”.

I did not count how many times you misinterpreted the simplest of my sentences but I believe it’s a lot of times. This goes against your claim that you understand my points too easily. You don’t.

The point of that post of mine (where I spoke of how horses can be used to represent numbers) was to show how changes in definitions can lead to logically invalid arguments. I wasn’t discussing conventions.

Being an asshole is a common sign of having an over-sensitive ego. You are that kind of guy. The overly sensitive guy who has all kinds of silly rationalizations for his irrational behavior. But once again, that’s not the subject of this thread, so let me keep this nice and short.

Nope. Not even close. Dasein is an existential contraption relevant only in regard to moral and political value judgments in the is/ought world. Again, if I do say so myself.

Mathematics [I presume] is anchored wholly in the either/or world.

And, that being case, I presume further that there is in fact one correct answer. But the problem here for mathematically challenged dummies like me, is that I can’t connect the dots between the arguments and the world of actual material/phenomenal interactions.

Human or otherwise.

Some interesting stats. (16%) of Silhouette’s response to the question “Can you give me an example of a number that has a shape?” deals with the actual question and (84%) of it is about who he is. There are (4,601) characters in total.

Guess which subject is more interesting to him. The subject of who he is.

That’s because there is no connection!

Mathematics, especially modern math, is highly abstract and not bound by any physical considerations. Nobody is saying – well at least I’m not saying – that .999… refers to anything in the real world. On the contrary, I maintain that it does not. Nor does 1! The fact that .999… = 1 is a formal exercise in pure math; and has no point of contact with the real world as described by contemporary physics.

Physics (\neq) Math.

Now of course you will say, well isn’t math used in physics and biology and economics and everyday life at the grocery store and so forth.

Yes. And this is a philosophical mystery. Why is it that math, which is ethereal, way out there, and free of any “ontological burden” to be about the real world; nevertheless turns out to be supremely useful, in fact essential.

This question is discussed in the famous paper of Eugene Wigner titled, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

The title says it all. Math is so abstract that it is unreasonable to think it applies the world; but it does.

So the best thing to do is remember that math has nothing to do with anything except itself. If others find it useful. the mathematicians are happy but that’s not their reason for doing math. Of course I’m exaggerating a position but I hope I’m making my point.

It’s the job of philosophy to explain why math, a purely conceptual enterprise, is useful at all. Much has been written.

Historically, people used to believe that math described the real world, and that math = physics.

The split between math and physics was caused by the advent of non-Euclidean geometry in the 1840’s. That’s when everyone realized that math could not provide certainty about the world; and that if math couldn’t, then NOTHING could.

That was the birth of postmodernism and the cultural relativism of contemporary society. If rationality can’t tell us what’s true, then maybe rationality’s just a tool of oppression and not a path to truth at all. This all started with non-Euclidean geometry.

Look up the field axioms for the real numbers and it’s resolved. Addition is a binary operation defined on a set. It inputs two elements of the set and outputs a third. We only define x + y for real numbers x and y. By induction we can extend the definition to arbitrary finite sums. But there is NO definition or meaning to an infinite sum.

As I recall Wiki doesn’t have a very good article about this but I found another writeup. Note that addition is a function that inputs two variables and outputs a third. Under this system infinite sums don’t exist or have meaning. We have to define their meaning.

web.math.ucsb.edu/~moore/2axiomsforreals.pdf

Its truth value would not change with repetition.

Think about what it means for addition to be defined as a binary relation or a function that inputs two variables and outputs a third.

You are historically correct. Infinite sums were used by Newton in the 17th century 200 years before the modern theory of convergence was developed.

But even then, when “everyone knew” what infinite sums were, .999… was still 1. There was never a historical time when it was anything else. All that changed was that we figured out how to prove it logically from first principles.

I understand very well that addition is a function that inputs two numbers and outputs a third number.

However, I don’t see how such a definition (together with other relevant definitions) isn’t enough for us to logically derive the meaning of the expression that is (0.9 + 0.09 + 0.009 + \cdots).

Your conclusion (that this isn’t impossible) looks like a non-sequitur.

I believe you will have to make your argument a bit more explicit because as it is it is very hard, if not impossible, to evaluate its validity.

How does the conclusion (that addition is not defined for infinite sums) follow from the premise (that addition is a function that takes two numbers and returns a single number)? Where’s the link?

Hopefully, you are NOT saying that just because addition is a function that takes TWO inputs and not AN INFINITE NUMBER of inputs it follows that addition is not defined for infinite sums. Using that kind of logic, one can also conclude that addition is not defined for any sum that involves any number of terms other than (2) which you agree is not the case.

The point was to make my position clear and visible so that other people can address it and not address a strawman.

Did I explicitly state that you or anyone else is stupid? I don’t think so. Indeed, I don’t know what would be the point of doing that. To disagree with someone is not to say that they are stupid.

Yes, I do have eyes, and yes, there is a reason you singled me out to be impatient with. However, what you think is the reason you singled me out is something that can be disputed.

By the way, note that I did not say that you are impatient ALL OF THE TIME and with EVERY person who disagrees with you.

The way I see it, you have no problem with people who disagree with you so as long they do not threaten your ego or otherwise frustrate you. Is that right?

I don’t think I present myself as “indisputably correct”. What I did here in this thread is 1) I presented my opinions and 2) I did not change them in response to what you and other people had to say. This does not mean I am correct, of course, it can simply mean that I fail to see the point that other people are making despite their efforts. But also, it can mean that none of you have any arguments whatsoever.

I don’t think the things we’re discussing here are advanced and insightful. And whether you or anyone else thinks the subject of this thread is advanced and insightful is, to be perfectly honest with you, completely irrelevant.

Note that these kinds of things are typically said by people who are insane.

Maybe you don’t see the point in continuing this discussion but I do.

Consider a word such as “bow”. The word “bow” has multiple meanings. Here are two of them:

  1. a long wooden stick with horse hair that is used to play certain string instruments such as the violin

  2. a weapon to shoot projectiles with (e.g. a bow and arrow)

Here’s a question:

Does this mean that a long wooden stick that is used to play certain string instruments such as the violin is the same thing as a weapon used to shoot projectiles with?

Of course not. They are two different things.

Similarly, (f(x) = 0) might have more than one meaning. For example:

  1. a function that maps every natural number to (0)

  2. (0)

Does that mean that a function that maps every natural number to (0) is the same thing as (0)?

Of course not. They are two different things.

In one sense, functions are not numbers. In another sense, functions are numbers. In this other sense, the word “function” means no more than “number”. This is not what the word “function” normally means and what it originally meant. It’s a subsequent development. And this kind of thing (taking existing words that have an established meaning and then giving them a different meaning) can lead to equivocation.

There is of course this age-old debate:

sciencefocus.com/science/wa … iscovered/

And, if you google mathematics and nature, there are many scholarly inputs — attempts to at least connect the dots:

scholar.google.com/scholar?q=ma … i=scholart

So, aside from it being a “philosophical mystery”, I’m not entirely sure what you mean here by “no connection”. But, as you note, out in the either/or world there seem to be countless connections.

Though I suspect that here we may be getting closer to the points that folks like Wittgenstein raised. The relationship between words and worlds. Language itself.

Besides, just on a day to day practical level, it’s important to accept that if John owns two horses and buys two more, he now has four horses and not five.

But 1 horse vs. .999… horse? That’s the part I can’t wrap my head around. Though I am perfectly willing to acknowledge that this revolves far more around my ignorance in regard to mathematics on this level.

No, where dasein comes in for me is when someone will insist that no one should be permitted to buy and sell horses. Period. That owning animals is immoral.

Indeed, I know someone who insists that this includes cats and dogs.

For purposes of discussion, suppose I grant you that point. Then show me any such logical derivation.

Previously you’ve said that no definition is needed because the meaning is known to everyone or obvious or intuitive or whatever, I apologize if I’m not quoting you perfectly but this is the feeling I get.

But now you are saying is that yes, you AGREE that infinite sums aren’t defined UNTIL we LOGICALLY DERIVE their meaning.

So, I do ask you to simply outline such a logical derivation.

What do you think modern real analysis is, other than a logical derivation from first principles of what limits and infinite sums are?

Addition is defined ONLY for finite sums. One of the axioms for the real numbers says that:

  • The sum of any two real numbers is a real number. In symbols, if (x) and (y) are real numbers, then so is (x + y).

So when we see an expression, or string of symbols such as (x_1 + x_2 + x_3 + \dots), we have no rules with which to untangle or parse that string. If it’s the sum of 47 elements, it’s the sum of the first 46 plus the 47th, and that’s an instance of adding only two things. So by induction, any finite number of additions are possible.

But the theory says nothing about infinite sum. No meaning is ascribed to them. We need additional ideas and formalizations, namely the modern theory of real analysis.

Of course if you have a different way of doing it, I’m all ears. I have said that I am openminded. I’m not defending mathematical orthodoxy, only describing it. If you have another way to define infinite sums, let’s hear it.

If you have a vending machine that lets you put in two coins, you’ll break it if you try to stuff in infinitely many coins.

LOL. But that’s exactly what I’m saying. And if you disagree, then tell me what is the definition of an infinite sum, based on the axiom that the sum of any TWO real numbers is a real number?

A simple inductive argument serves to generalize addition to any finite number of summands, as I outlined a couple of paragraphs ago.

It’s very much like parsing a computer language. If we see x + y + x + w we know we can parenthesize (x + y + z) + w to reduce the problem to a sum of two things, which we have a procedure for.

But if we have an infinite sum, the procedure never terminates. We do not have a computable function that allows us to determine the sum of infinitely many terms.

Do you see that? A machine could compute 1/2 + 1/4 + 1/8 by continually breaking it down into smaller sums, a process that is guaranteed to terminate. But if the input is an infinite sum, the evaluation never terminates and no such sum can be reached. We do not have a procedure or algorithm to calculate an infinite sum that is guaranteed to terminate after finitely many steps.

That’s why we need a new idea. The idea of limits. The theory of limits goes beyond the theory of computation, you can look at it that way. Indeed, the difference between arithmetic and analysis is exactly that. Arithmetic is everything you can do with finite operations. Analysis deals with infinitary operations.

In fact, finite addition can be defined in Peano arithmetic. The theory of limits requires the axiom of infinity.

You are failing to appreciate this conceptual leap. Finite sums do not require the axiom of infinity. Infinite sums do.

Then by all means restate your thesis; rather than provide argument or evidence.

That’s not an example of anything. The integers are a ring. In a ring you can add, subtract, and multiply, but not necessarily divide.

If you enlarge the ring to a field, then you can always divide (except by 0). This is perfectly well understood. Your post was a little off the mark IMO.

Now if one is interested in integer division, we know that 5/2 is 2R1; that is, quotient 2 and remainder 1. That’s the domain of number theory.

You seem to be going off into unproductive ideas about this.