wtf, the pedagogical point is why this thread is going on for 18 or so pages (is not about the math).
Mr. Liner, I admire your eponymous brevity 
Agreed, although incomplete. The ellipsis, “…”, is often the contended issue. The ellipsis means that the preceding pattern is continued eternally (in this case). The oft presumption is that it refers to something that “ends at infinity”. Obviously there is no “at infinity”, so it doesn’t refer to such at all.
No. Not only do mathematicians disagree on that issue, but logically (the only thing that counts here) the “sum” is NOT defined as the “limit”.
Sorry for the inconvenience, but let’s keep it clean.
We aren’t concerned with the definition of a “limit” since it is NOT defined to be the sum. A limit is a boundary that either cannot be reached, or cannot be surpassed (depending on your intent). The sum is only indirectly related.
Mathematicians (and other priests) often attempt to simply define words to mean what they wish to be the case (once the RCC actually redefined a small pig as a “fish” so that their rules didn’t starve out a tribe). The problem is that logic supersedes priestly authority when it come to reality. We are discussing reality, not the holy Sanctus Mathematica Covenant.
Then why can’t you find fallacy in the ones that I have provided? Start with this one:
I can find fallacies in your “proofs” (and have repeatedly). Why can’t you find any in mine?
Time or “eternity” is not a factor in this at all. The dots indicate that we have a function from the natural numbers to the set of digits that always returns “9”.
Do you believe in functions? If I tell you that f(x) = x^2 and I ask you what is f(3), you’ll say 9; and if I ask what is f(12) you’ll say 144. I assume that your philosophy of mathematics is perfectly fine with that.
The notation “.999…” stands for a function that inputs a natural number and always outputs 9. So if you ask me what is f(3), it’s 9; and what is f(12), it’s 9 again. It’s a constant function.
What is your philosophical issue with functions?
Nobody is making any presumption of “ending at infinity.” On the contrary, the sequence of 9’s never ends simply because the sequence of natural numbers 1, 2, 3, … never ends.
I asked you a while back if you are perhaps arguing from a finitist perspective, in which infinite sets are denied. But you told me you believe in the hyperreals, which not only require infinite sets, but also require principles of reasoning in excess of those needed to construct the reals. (Specifically, the hyperreals require a weak form of the Axiom of Choice for their construction; while the reals do not).
So I can’t understand why you aren’t willing to acknowledge the sequence of natural numbers, or functions defined on them.
Can you name some mathematicians or provide links of mathematicians who disagree that .999… = 1? Even the estimable Professor Katz acknowledges that he is (1) making a pedagogical point, not a mathematical one; and (2) using alternate definitions in order to obtain nonstandard results.
The standard definition is what it is. You are entitled to make up your own definition but that does not alter the standard definition, which is on Wikipedia and any standard calculus text.
I’m perfectly happy to dialog with you as long as we can keep it clean. Fair enough?
The definition of the sum of an infinite series is the limit of the sequence of partial sums. You (or any reader) can refer to the Wiki page I’ve linked several times; or any account of calculus either written or online. It’s like arguing that Trenton’s not the capital of New Jersey. It’s not a matter of opinion. Trenton is the capital of New Jersey. You can write a fictional novel in which Trenton is not the capital, but that doesn’t alter the facts in the real world. In the real world, the definition of an infinite sum is exaclty as I’ve given it. Any third party can verify that. To save readers the trouble, here’s the Wiki definition again. en.wikipedia.org/wiki/Series_(mathematics#Definition
I get that you have some annoyance or objection to standard math. How far does it go with respect to other sciences? Do you regard physicists as priests? Biologists? Is your objection to standard science limited to math, or does it go further? Trying to understand.
I Googled “RCC fish pig” and found nothing. How does this relate to mathematics? It sounds like some kind of regulatory or political thing.
I get that you have a fundamental objection to math. What’s the source? How far does it go? Like I said, if you don’t believe in the modern mathematics of infinity, that’s a logically and philosophically honorable position. It’s called finitism. But I already asked you, and you said you’re not a finitist. If you believe in infinite sets, you’re basically forced to take all the rest of it. I don’t understand the coherence of your position in accepting infinite sets yet denying the standard definition of an infinite sum.
The fallacy is that you are not using the standard definition of an infinite sum. If you make up your own definition, you can get any result you want; just like Professor Katz. But Katz acknowledges that he is using an alternative definition.
None of which has anything to do with how an infinite sum is defined. “No opportunity to become different” is meaningless. “How long it’s carried out” is equally meaningless, since we are not talking about time. The function f(x) = x^2 happens “all at once.” Put in an input, get out its square. No time is involved. The function is static, defined in one statement.
It’s true that none of the finite partial sums .9, .99, .999, …is 1. But the limit of the partial sums is 1. That’s the definition.
A post or two ago I referred to the completeness of the real numbers. This is the defining property of the real numbers. It means there are no “holes” in the real number line. By completeness (I linked the Wiki page earlier), the set {.9, .99, …} must have a least upper bound. If a number is larger than 1, it’s not the least upper bound of this set. If a number is smaller than 1, it’s not an upper bound at all. It follows that the least upper bound must be 1, and that is taken to be the limit of the sequence.
This is math. This is how it’s done. If you want to do it differently, that’s perfectly ok. I have no objection. But then you’re doing “alternate” math, not standard math. Just like Professor Katz can claim .999… is not 1, by supplying an alternate defininition of .999… And he admits up front he’s doing that. You are providing an alternate definition of a limit, and therefore you can get any nonstandard result you want. But with the standard definition, .999… = 1.
Pedantic arguments don’t work against me. The difference between “always” and “eternally” is moot. You can substitute “infinitely” if you wish.
Not an issue yet.
That isn’t how functions work, but… (irrelevant).
“0.999…” is the result of a function. It is not a function itself.
You and I know that, but not everyone realizes it at first glance (as they have shown in this thread). And the fact that it “never ends” is what tells you that it never sums to a conclusion such as “1.0”.
It is your blind spots that I am not blind to that are preventing me from agreeing with your conclusions.
That wasn’t the issue to which I was referring. And even if it was, mathematicians, like other priests, have to be concerned of their occupation and employment. Honesty does not prevail in science nor society.
I need not make up any definition for the word and concept “limit”. You, on the other hand, do. Mathematicians are not free to invent definitions so as to avoid logical conundrums. We all know what “limit” means. And it doesn’t mean anything different in math merely because someone wants to avoid infinitesimals. As I said, pedantic arguments aren’t going to get you anywhere.
“Clean”, in this case, means that “limit” means a boundary either unapproachable or unsurpassible. It does not mean that whatever the limit is, is whatever the sum is by definition. Since both “limit” and “sum” have their own pre-established definitions (and very relevant ones at that), they cannot be conflated (that would be “unclean” logic).
We don’t care what the definition of a limit is. We only care what the definition of a sum is. Your attempt to conflate and equivocate won’t cut it. In philosophy, a pig is not a fish regardless of the RCC declaration.
Those who only think in defense of their holy doctrine, I refer to as “priests”. You may call them whatever you like.
This is an issue of logic, not doctrine.
No, you really don’t. I only object to fallacious logic, which happens to occasionally include mathematicians.
Mathematics is subservient to logic.
Oh really? Can you explain that further? I don’t see how I have misused the concept of an infinite sum. What I see is you trying to avoid the concept and substitute a rationalizing definition (much like the RCC).
How does that misuse the concept of an infinite sum?
Yes. Therefore again:
Exactly. Get the logic square/clean, and the math will be just fine.
What are the pre-established definitions of limit and sum? I agree that I have some vague intuitions, left over probably from my first encounter with Zeno’s paradoxes or perhaps an illustration of Archimedes’ determination of the area of a circle.
But those intuitions have long been replaced with the formalisms of modern math. So tell me, what class should I have taken instead of Real Analysis, which is where math majors get their brains washed? What should I have read, what should I have studied, to learn the truth about the real number continuum?
For what it’s worth, the modern formalistic approach to math is the result of a historical situation known as the foundational crisis. In the late 19th century, everyone had intuitions about continuity and limits and sums. They began to realize that intuitions were not sufficiently precise to account for all the phenomena that needed to be studied. The need for extremely picky, pedantic rigor became necessary.
You are correct if you note that many people are unhappy at this trend in modern math. The physicists, for example think that math has hopelessly lost its way. But even if the modern trends toward rigor and abstraction are wrong, they are at least understandable in their historical context; a period of time that includes our own.
See web.stanford.edu/~ebwarner/SplashTalk.pdf for a more academic writeup.
Bottom line: I plead guilty to substituting pedantic formalism for reality. Naturally I am interested in reality; and you claim to know reality. You claim to know what a limit or sum is in the absence of formalism. So if the formalisms I’ve been taught are a lie; how may I come to know the truth?
ps – I still have no idea what RCC means. Are we perhaps not in the same neck of the woods? I’m in San Diego, USA. I’m guessing that RCC must be not nearby.
I have already explain the definition of “limit” twice:
Logic 101 would have helped. Ontological Construct would have finished off any confusion.
Thus the need for better foundation in logic and ontological construct. You don’t seriously expect to impress me with such rhetoric, I hope.
Roman Catholic Church.
Oh. Dictionary definitions! I hope you would agree that no specialized domain of human endeavor relies on dictionary definitions. Computer programmers, plumbers, doctors, physicists, economists, oil change jockeys, farmers, you name it. Everyone has specialized jargon. The dictionary is useless for technical terminology.
And it’s also useless for philosophical terminology. WHen a philosopher says, “intention” or “color” or “What it’s like to be X,” or “Mary’s room,” they mean those things in a huge technical context that’s not available to laymen. Each phrase brings to the expert’s mind all the books and seminars and converations and thinking around those things. You can’t get that from a dictionary.
I guess I don’t understand your communication style. Before I go off on you, I want to just ask if this was intended to be condescending, or if it’s intended as a joke, or if I’m just misunderstanding you in some way.
I take this as a very condescending remark.
Am I misreading you? Tell me.
I’m not trying to impress you. I’m trying to have a civil conversation with you. I’m trying to understand your point of view, and explain mine.
Is that something you have an interest in?
Funny how logic and truth don’t care what specialty anyone claims.
What you are suggesting is that only to the elitist mathematicians does 1 = 0.999… In the real world, they are not equal. So when a Doctor says that he has cured your disease even though you are still dying from it, he is right because to the special field of medicine, “cured = treated”. Or perhaps a car mechanic? If the mechanic says it is fixed, it is fixed?
You are relying on excuse making and rationalizing.
The fact of the matter is that it is LOGIC and ONTOLOGY that reveal the truth of the issue. Not having taken such courses, you wouldn’t know that.
As I said, this is an issue of pedagogy not of maths.
When two people have resolved a debate to the point of disagreeing upon a particular thought or principle, what is left to do but to attempt to teach of the correctness of one’s stand?
Math is not some mystical field independent of logical and rational thought. Math is in fact a product of logical thought concerning quantities.
But of course, as history strongly reveals, Man often preaches with grand certainty against logical thought, only to later fall. The issue of infinities and infinitesimals seems to stretch Man’s mind to a dubious limit whereat he chooses passion for preference over restraint to rationality.
Note that the title and OP ask if the concept is “Really” true. The question isn’t about what is currently most popular to believe or what the contemporary experts teach. And to discover what is really true despite what authorities are preaching requires logic. Unfortunately even on a philosophy forum, logic is seldom the guiding light. Such is the Planet of the Apes:
…and btw, Rational Metaphysics is that top square, labeled “Ratiocination”.
Unfortunately, yes. That is very sad.
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Ok, so James says Aristotle solved Zeno …
When the distance to traverse is halved, the amount of time to cross it is also halved. (Twice the speed of the distance) …
This is very similar to my solution for Zeno …
If you can only cover half the distance… Then double the distance and you’ll reach the finish line…
You don’t need to obscure it with the complexity of time…
And actually, James is contradicting himself here…
It doesn’t matter how FAST you do
1/2 of 1/2 of 1/2…
Because it will never converge…
While our solutions are similar… James’s solution contradicts his argument that 0.9… Doesn’t equal 1… As he is using a bound infinity argument to solve Zeno …
Now I need a drink.
Not that my explanation was perfectly written, but you seemed to have left out an important part of it. I explained that there are two views of Zeno’s Paradox; motion and the definition of distance vs convergence of an infinite series. The second is very different from the first.
The upshot is that it doesn’t matter that the infinite series doesn’t converge at the limit, because physical reality, space, is formed of segments of space, not of location points. There are an uncountable amount of potential numbers between any two numbers because numbers don’t actually take up space.
The number of segments is entirely up to the ontology to declare. It can be declared that between 0 and 1 there are an infA amount of segments. And from there, all of the math works out perfectly. But without declaring a specific ontology, one can chase the infinity issue eternally. A standard must be declared, such as infA for the same reason a standard meter must be declared in order to establish consistent measurements.
The important point is that every distance is made of segments, not of numbers.
Conceptualized segments not actual segments as no such segments exist (minor point but critical).
???
Distance “exists” in the sense that it has affect, although isn’t an object but a property of space. Segments are merely portions of distance. They too are not objects, yet still imbue affect.
The overlooked idea, perhaps is, that segments in a convergent sequence reduce to smaller and smaller segments, and are measurable by their size between lines(points)
A
t minimal convergence, the size is no longer measurable because it is news urged between 2 points which are indescribable.
That doesen’t mean they become identical, therefore the differential of the function applies to unbounded progressions, as well.
Therefore the function is no linger differentiable, but that does not mean that the progressive reduction
has transferred the function into an identity.
Here lies the paradox, where spatial derivatives (may act like real mathematical integers.
There is, according to this a tangential relationship between mathematical and spacial determinants, where the act like both.
I am not sure of the idea of affectance, but given the little exposure I had with it, I am sure there is some sort of tie in.
At critical points affects and their objects may not be differentiable, on extremely small sizes.