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.
