QED.I've already given my proof — Metaphysician Undercover
Sure, yet we do know some things at least, and can reason to some extent if careful.The sorry fact is, that we cannot either describe or simply cannot understand infinity as clearly as we would want. — ssu
(y) (I'd hit "Like", but this will have to do)Much ado about very little. : — jgill
I didn't claim this is a problem, that was Pitzotl''s misinterpretation. I said that if the same thing has two distinct names, there is a reason for that. — Metaphysician Undercover
A ninth is the specific particular quantity corresponding to dividing one into nine equal units. — InPitzotl
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. — InPitzotl
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. — InPitzotl
Meta did not directly address this, or any other such proof. Instead he went to an irrelevance, his claim that 1/9 is not a number. — Banno
Even if 1/9 were not considered a number, the proof would stand. — Banno
Which premise is false? — Michael
How you choose to define "number" has no bearing on whether or not 0.999... = 1. — Michael
The result will be to show in even greater relief that this is a thread about Metaphysician Undercover, not about maths. — Banno
You don't seem to understand. — Metaphysician Undercover
My point was that ".999..." has a different meaning from "1". InPitzotl insisted that it is two names referring to the same thing. Clearly it is not, because .999... is derived from 1/9 in the op, and 1 has a simple meaning without any such baggage. — Metaphysician Undercover
So the issue I've pointed to is whether 1/9 is a representation of a number, or not. I've argued that it is a representation of a ratio and therefore not necessarily a number. Some ratios are impossible to represent as a number. That is where we get the term "incommensurable". — Metaphysician Undercover
This rules out that you understand the language and refuse to speak it. You genuinely don't speak the language of math.As a result, I have no idea what you're talking about. No one mentioned multiplication, the issue was division. — Metaphysician Undercover
(a) 1/9 of nine is(a) 1/9 can be one if thewholegroup is nine, (b) it can be two if thewholegroup is eighteen, (c) it can be three if thewholegroup is 27, (d) it can be four if thewholegroup is thirty six, and so on and so forth. — Metaphysician Undercover
Two major problems with this MU: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
It appears you just haven't taken the time to understand what I was saying. — 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.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
In the long run, this intellectual ferment led mathematicians to let go of the vague notion of “number” or “quantity” and to hold on, instead, to the more formal notion of an algebraic structure. Each of the number systems, in the end, is simply a set of entities on which we can do operations.
This standard "proof" is of course bullpucky. It's true, but not actually a proof at this level. Why? Well, as you yourself have pointed out, the field axioms for the real numbers say that if x and y are real numbers, then so is x+y. By induction we may show that any finite sum is defined. Infinite sums are not defined at all.
To define infinite sums, we do the following:
* We accept the axiom of infinity in ZF set theory, which says that there is an infinite set that models the Peano axioms. — fishfry
Also it seems to me that what you call "numbers" mathematicians call "natural numbers" (or maybe "integers"; do you consider negative numbers as numbers?). There's more than just natural numbers in mathematics; there's rational numbers that include the commensurable fractions like 1919, real numbers that include irrational numbers like 2–√2, and more. — Michael
don't see what purpose there is in saying that non-natural numbers aren't numbers, and latching onto the OP saying "as a matter of representing numbers" completely misses the point of this discussion. — Michael
This rules out that you understand the language and refuse to speak it. You genuinely don't speak the language of math. — InPitzotl
(a) 1/9 of nine is 19×9=119×9=1
(b) 1/9 of eighteen is 19×18=219×18=2
(c) 1/9 of 27 is 19×27=319×27=3
(d) 1/9 of thirty six is 19×36=419×36=4
Do you see the multiplication now? — InPitzotl
When I slice one pizza into eight slices, it's still one pizza. — InPitzotl
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. — InPitzotl
Also I think you're putting the cart before the horse. We don't start with some definition of "number" and then see which things satisfy that definition. Instead we have the mathematical terms 11, 1919, 2–√2, etc. which mathematicians place in sets that they decide to name "natural number", "rational number", "real number", etc. and then lexicographers try their best to come up with an adequate description of what the word "number" means when they write their dictionaries. — Michael
That's putting the cart before the horse. Before deciding which items go into which set, we need to define the conditions of the set. No one puts a whole bunch of random terms into one set, then names the set "numbers". if that were the case, why wouldn't we put "house" and "car" into that set called "numbers" as well? — Metaphysician Undercover
Of course. If you're just now noticing, I refuse to use that deceptive language, — Metaphysician Undercover
^-- One of those two things is a lie. Most charitably, you're incapable of using the language.As a result, ->I have no idea what you're talking about<-. No one mentioned multiplication, the issue was division. — 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.Of course. If you're just now noticing, I refuse to use that deceptive language, — Metaphysician Undercover
Sure, but that's just division:(e) To divide nine into nine parts, or (f) to divide eighteen into nine parts is very clearly division. — 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!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
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.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
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:and not just randomly decide to call this a part, and that a whole,
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.because that would get nowhere. — Metaphysician Undercover
You don't seem to understand. "One" does not represent a quantity which can be divided. — Metaphysician Undercover
...and here is the gift the crackpot gives to the world. Occasionally. — Banno
Exorcisms cheap. Or, go home and read my book, How to Keep One Step Ahead of Your Mind and you'll feel better in the morning. Or, best, three ounces or so of a decent single-malt Scotch. — tim wood
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