Finite
Zeno's Paradoxes illustrate various reductio ad absurdum arguments for various manners of discussing motion. The dichotomy paradox states that to walk a distance X one must first walk a distance X/2. To walk the distance X/2 one must first walk the distance (X/2)/2. In english, to travel 4 meters one must first travel 2 meters. To travel those 2 meters one must first travel 1 meter. On and on in an infinite regress, rendering a person incapable of movement. This description of movement is obviously flawed, as I am fully capable of walking to walls, through doorways, etc. Keep Zeno in mind.
Yesterday on the PA forums a thread appeared discussing whether .999... equals 1. I am certain that .999... does not equal 1. Why? Read on. Keep Zeno in mind.
1 ≠ .9
1 ≠ .99
1 ≠ .999
1 ≠ .9999
If we continually write this sequence no number of 9s to the right of the decimal will put the value to the right of the decimal equal to 1. It won't happen. Yet mathematicians maintain that when the value "goes infinite", when we cease writing 9s and instead denote an unending sequence of 9s something happens which puts the two values equal. But how? What happens? Remember this, as I will now talk about Anselm's ontological argument for the existence of God.
Anselm defined God as aliquid quod maius non cogitari potest (That than which a greater cannot be thought). By defining God to be aliquid quod maius non cogitari potest Anselm argues that God must necessarily exist, since God is aliquid quod maius non cogitari potest, and aliquid quod maius non cogitari potest must necessarily exist for if it did not exist it would not be that than which a greater cannot be thought, but would be that than which a greater can be thought, which is a contradiction. This type of argument is called an Ontological argument, an argument based upon the nature of the thing. It is not an argument for existence based upon proof, but rather is based upon definition and the nature of the being. As many philosophers have argued, the ontological argument is nonsense. "Existence" is not the stort of thing one may gain via definition.
As I have described Zeno's Paradox, the inequality of .9 and 1, and Anselm's ontological argument I will now discuss Unicorns.
Unicorns are often used in philosophy to illustrate non-being. Stated simply, Unicorns do not exist but we can discuss unicorns. We can draw pictures of unicorns. We can discuss the temperment of unicorns, the feeding habits of unicorns, the properties of unicorns. We have a wealth of information for these things which do not exist.
Now, combine all of that.
Infinity does not exist. Everything is finite. The universe, existence, being, apples, all are finite. So the problem with the notion of .999... is that it does not exist. And yes, we can describe it (like unicorns) and we can argue for its existence (like aliquid quod maius non cogitari potest) but in actuality infinity is nonsense.
Think of an orange. If I cut an orange into thirds do I have three unending, infinite fractions of orange? No. Oranges are finite. 1/3 of an orange is not .333... of orange. That is nonsense. Much like Zeno's paradox, infinity is a bad and inaccurate description of reality. There is no infinity. Like unicorns, we fabricated the notion.
Sequences of numbers stop when we stop writing them, stop computing them. The notation .999... doesn't actually mean anything, is not representative of anything. Surely we can write .999... and gesticulate wildly as we articulate the properties and qualities of it. But we can do the same for aliquid quod maius non cogitari potest, for unicorns. We can argue, as Zeno did, that there is an infinite amount of space between myself and the wall, myself and the door. But then we walk to the door and open it, we cut the orange into thirds and consume it. All is finite. The infinite, unending sequence is an innaccurate, flawed, and bad description.
And that's why 1 ≠ .999... There is no .999... There is just the decimal and as many 9s as we can write or calculate. But when we stop writing, stop calculating, the sequence stops.
Because it is finite.
32 comments:
"And that's why 1 ≠ .999... There is no .999... There is just the decimal and as many 9s as we can write or calculate. But when we stop writing, stop calculating, the sequence stops."
Calling bullshit on this part. It's the equivalent of calling the Infinite finite, because it only has eight letters.
The argument that the form of expression somehow limits the concept is absurd beyond belief.
Did you even read what I wrote?
nice work J.
and you didnt even have to offend anyone.
.9...
That works in the same way as an infinite combination in a MTG duel. One does not state the method of their combination each time they use it. One simply states the number of times it is to be performed. Thus it is done here, except I shall say that I will never stop writing 9's.
1.0
-0.9
----
0.1
Correct?
1.0
-0.9999
--------
0.0001
Correct?
1.0
-0.9...
-------
0
That's the way subraction works. If I never stop writing 9's the difference can never be anything but 0. If there is no diference then they are the same.
1 = .9...
Also note: Numbers do not exist of themselves. They are a form of abstraction which we use to generalize in the world.
so it doenst exist.
you can not have .9 repeating out side of your mind. its like a unicorn. it doesnt exist. i hate math.
but meh. its all greek to me.
or is it arabic?
Thus it is done here, except I shall say that I will never stop writing 9's.
But that can't happen. You will eventually stop.
Infinity doesn't exist. Because everything stops.
But infinity DOES exist, in the realm of the theoretical.
and this entire discussion is about the self-same!
Further, you have to prove that everything stops, because frankly, I'm not willing to accept that.
But that can't happen. You will eventually stop.
Yes.
But in so saying, I am introducing infinity as a concept.
Does it not have existence, like a unicorn, as a concept in the way that our andrew pointed out?
It exists in the same way that geometric circles exist.
It exists as 1 exists. One does not exist in the world. You cannot go and find oneness; you may find one of something and presume on that basis that oneness exists, but then you are creating an absraction and not observing oneness in the world.
Singularity is to the same degree an abstraction as infinity.
Neither exist except as abstractions, yet you grant the existence of 1 and deny the existence of infinity... very troubling indeed.
just because you can't wrap your mind around the concept of infinity, doesn't mean that limits don't exist as numbers approach it...
The problem you have, it appears after reading your post is not with 0.9... but instead with a basic understanding of decimal arithmetic.
As Caleb pointed out numbers do not exist, they are just an abstraction to describe quantity.
At some point in time, mathematicians decided to write excess fractions of numbers after the whole number, because this saved space. The numbers after the decimal point represent a fraction of
n/10^(m | m is the number of digits after the decimal)
Unfortunately, base 10 (all bases have this issue in some way or another) cannot evenly divide n all the time, so the decimal then becomes an approximation of n/10^m.
Unfortunately, the approximations do not always translate directly to what they are, thus we create proofs to prove that they are what they are. There are hundreds of thousands (I can actually produce infinite given infinite resources) of proofs out there to prove that 0.9... = 1. If you refuse to believe it, then it is a result of your refusal to accept the foremost mathematical system used by western culture.
Bah, I don't even know what my final point was anymore, trying to explain this to you... so I'm just going to stop mid reply...
Here's what I don't get.
1 ≠ .9
1 ≠ .99
1 ≠ .999
1 ≠ .9999
1 ≠ .99999
That will continue to be the case for as long as we continue to write 9s to the right of the decimal. If we have ten-billion, ten-trillion 9s the two quantities will still not be equal.
So what the hell happens when we say it "goes infinite" and there are an infinite number of 9s to the right of the decimal? No number of 9s to the right of the decimal will make the two quantities equal:
1 ≠ .9
1 ≠ .99
1 ≠ .999
1 ≠ .9999
1 ≠ .99999
So what happens when we stop writing nines and invoke the idea of infinity, which itself doesn't make any sense?
Put another way, no amount of nines to the right of the nine in the tens place is going to change the zero to the left of the decimal into a one.
Why does the concept of infinity not make any sense?
What .9... represents is an infinite sum...
the sum of one to infinity on n of 9/(10^(-n)). Sums such as this, that have a limit (read do not approach inifinity), can prove useful to mathematicians. In this case this sum approaches 1 as n approaches infinite.
If you refuse the concept of infinite, you must at least accept the premise of approaching this thing doesn't exist. If it is the case that you also deny the concept of infinite then you must also not make statements regarding the equality of infinite sums.
As far as the world is concerned 0.9... = 1. This is an undeniable fact given the existence of the concept of infinite. What you are arguing is that no finite amount of 9's to the right of the decimal point equals 1, which is true, but this does not correlate to an infinite amount of 9's, thus the mathematical proofs.
We have a flawed representational system of our number system, and it doesn't always reflect what we need it to, thus we have a defined way to prove things to be true in mathematics. You argument is one of failed inductive reasoning. You did the first two steps properly, but inductive reasoning relies on the infinite case to finish, i.e. the limit as n approaches infinite.
You can not apply philosophy to mathematics and assume it will be correct. They are two mutually exclusive abstracts and therefore are not applicable to one another.
Sorry repost... submitted early on accident..
Had you read the entire wikipedia article you linked me you would know that your arguments are flawed.
From wikipedia
"... reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example that each real number has a unique decimal expansion, that nonzero infinitesimal quantities should exist, or that the expansion of 0.999… eventually terminates."
You need to tread carefully here J, you're arguments are bordering on conservapedia like content. Denying what experts in the field know to be true, just because you don't like the answer, and it doesn't sit comfortably with you.
Ow... Kyle, put away the heavy artillery. We all know this conversation would best be served taking place in a basement somewhere wherein we can talk over one another, and point fingers with vehemence.
Preferably while something I have no desire whatsoever to see is on the tv. I'm looking at you here, Mikey, and your filthy, filthy Noggin.
Why does the concept of infinity not make any sense?
Ultrafinitism Finitism
I think "infinity" is nonsense because...
Think of .9... What does that mean? Does it mean that, somewhere, there is a non-terminating sequence of 9s? Well, that can't be infinity, because that is not infinite but rather a continually increasing finite number.
To "be" infinite it must have a quality of "being", and "being" is finite. "Being" only happens at a finite point in time, a finite location (with regard to both physical beings and concepts). And if we are to divorce mathematics from both time and space then what are we talking about and how are we talking about it?
And even if we ignore all of that we still have the problem of whether infinity is a non-terminating sequence, which isn't infinity but rather a process of continual growth, or we have an infinite number which is not non-terminating, but is always, at all times, in all ways, always already unending, but not in a sense that it is continually increasing, but rather it always already is as unending. It exists as perfectly unended, yet not perpetually increasing.
Because if it is perpetually increasing then it's just a big finite number that continues to increase ((((N+1)+1)+1)...). And if it is not unending but rather, itself, infinite, a number which itself is unending by virtue of always already having not ended, then, what is that?
Think of a timeline. A timeline is not infinite. A timeline is continually growing, but at each point is finite. An infinite timeline would itself be unending always, at all times. An infinite timeline would not be a sequence of points next to one another which is ever-expanding, because ever-expanding means that it is always at a finite point which then increases to the next point, but rather an infinite timeline would itself already exist as unended yet not increasing.
Regardless of whether or not you agree with me, does that make sense?
And if one is to argue that math is a priori and therefore divorced from time, space, and causality one will will be saddened to know that time, space, and causality are themselves a priori.
So they are subject to one another, as their existence is co-dependend.
According to Kant, that is.
I would think you of all people would know the difference between existence and abstracts...
Also since you don't believe in infinity, then you should probably stop driving your car, playing video games, using your computer, or any other form of modern technology since they have some sort of reliance on calculus which has a reliance on the concept of infinity.
A time line is infinite, though, unless you can give me a concrete starting point?
Ignoring our representation of time, but instead looking at the abstract itself, this is not possible because there was always a point in time before every given point in time, as there is also a point in time after every given point in time, thus time is infinite.
Except for when time stops, and when time began. (One cannot say "after" time stops or "before" time began. Cause "after" and "before" can only happen within time.)
I don't think the abstract of infinity makes any sense because it's so completely estranged from everything.
A time line starts when you start drawing it and ends when you stop drawing it. You can replace "drawing" with "calculating" or "computing" or etc.
And I still want someone to explain what happens that puts that really long sequence of 9s equal to 1.
Nothing has to happen to the end of that string. Becuase it's fundamentally equivalent to one.
.9rep is the same written concept as 1 as is the same written concept as One.
Further, you reason you can't see when time stops and ends? Becuase it's a infinite contiuum.
You can concieve of a bounding upon time, an artificial beginning point and ending, but that's not time, it's a period OF time. This is a fairly important distinction to make. A bound period of time is no more the entirety of time, than a bound set of 9s is .9rep.
.9rep is the same written concept as 1 as is the same written concept as One.
That's insane. You're insane.
And yet, Wiki, The source of truth and light in internet conversations and work avoidance, agrees with me.
Who is insane now, Mr. Hamman? (strokes evil cat) Hmmn?
did you not read my wiki quote at all?
here let me do it again and put just the parts you need
"... reasoning for denying ... the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example that each real number has a unique decimal expansion..."
What this basically says is that is a fallacy to assume that each numerical concept can be expressed in only one way.
I'm fine with multiple expressions of a single number. 4/2 is the same as 2. That's keen.
But
1 is not equal to .9
So how the hell is it that by throwing a hell of a lot of nines to the right of that first nine we somehow get a value in the ones place to the left of the decimal?
You mentioned an orange earlier, and that you cut it into thirds.
You took 1 orange, and separated it into 3 portions, each .333... of the whole. If you were to reconstitute your cut orange (including the imperceptible portions lost by the act of cutting), it would then be .999... of an orange?
Did separating the orange and then reforming it reduce the amount of orange by .00...1?
Math, Time, and Space are broken.
but its what we have at the moment. I dont think my current concept of time and space works, but i don't know another way to think about it. i think it is the same with infinity.
but im just a crazy person.
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