Physics contradicts maths - how is this possible?

There are a couple of simple mathematical arguments that are related to these ideas:x = 0.999...
10x = 9.999...
9x = 9
therefore:
x = 1OR1/3 = 0.333...
therefore:
1 = 0.999...

The answer is quite simple.If x is time and y is distance to the wall then the flies motion is described as:y = 1 - xy = 0 when x = 1. So the fly lands on the wall at one second. All you're doing is taking values that are not 1 and then pointing out that the fly isn't on the wall for any of them. Of course it isn't, because the values aren't 1.Limits would be used if you wanted to show that y(x) is continuous at x = 1. i.e., lim x->1 y(x) = y(1).Zeno's Paradox would more relate to the fact that since we know the fly lands on the wall at 1 second using the standard selection process of just picking 1, your limiting selection process should have 0.9 + 0.09 + 0.009 + ....... = 1.
Basically 0.9 + 0.09 + 0.009 + ....... should equal 1, but does it? It was a question the greeks couldn't answer and a full understanding of the continuum had to wait a long time. However that maths says the fly landing on the wall regardless of this question.

I am sorry I brought that up, it was perhaps off topic, I'll address it below never the less.Now your example shows that you are a little confused about math, let me explain: You are right in the first. We can only make measurements to a certain degree of accuracy, and objects in the real world are subject to countless constraints, so the speed of the fly would never be constant, and it would rarely be exactly 1m/s.The next part of your statement seems to be a little confused though.When you write 0.99999... it is a certain kind of mathematical notation, that means that the infintely repeating fraction defined as9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...Why do you state that the fly cannot have the speed of 1 m/s but it can have the speed of 0.9999... m/s?Now to the fact that 0.999.. = 1remember that when I write 0.9999... what I mean is9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...
That is the infinite sum of the factor written above.A basic fact of real numbers, is that if you add real numbers you get a real number. A more advanced fact is that if you add an infinite real numbers the result is either undefined, infinite or a real number.So the number 0.9999... is a sum of infinitely series of real numbers.Lets on a wild hunch guess that the sum of this series is 1!We do not know this, but when we have a guess we can test if we are right.We do this by looking at the limit of the sum9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...Now if the sum is different than 1 then there must be a difference between the sum, and 1. That is
1-9/10 + 9/100 + 9/1000 + ... + 9/10^n + ... <> 0Now lets look at it piece by piece...1-0.9 = 0.1
1-(0.9 +0.09)=0.01
...
1-(0.9+0.09+...+9/10^n) = 10^(-n)The last statement shows how the sum of the series relates to the number 1. If you have 10,000 elements in the sum, then the difference between the sum and 1 is 10^(-10,000)As you can see the difference between 1 and the sum decreases steadily .This means that the two are the same! Why?Let Z= 1-0.9999...Now if 1> 0.9999... then Z>0, this is a basic fact of real numbers.Assume that Z is a real number greater than zero, that is, assume that 1>0.999....Now since Z is a real positive number we can always find a N such that 10^(-N) < Z.That is we can always find a number smaller than Z. This again is a basic fact of real numbers.But this means that the sum
9/10+9/100+...+9/(10^N) is closer to 1 than Z, or
1-9/10+9/100+...+9/(10^N) < ZBut that is impossible, since we know that
0.999.. > 9/10+9/100+...+9/(10^N)So assuming that Z is larger than zero gives us a contradiction, we must then acknowledge that
1-0.999.. = 0or in other words 1=0.999...

In terms of quantum physics, which you would have to take into account if you are dealing with the minute scales indicated by the problem, the concept of "reaching" a "point" can only be defined through statistical significance. Unlike in the deterministic mathematical model described in the OP, the statement that a particle has reached a "point" only says that there is a probability that a measurement will find the particle at that "point".

There is a simpler proof, in my opinion:10x-x = 9xx=0.999...therefore9.999... - 0.999... = 9xtherefore9 = 9xthereforex=1=0.999...Oh, I see Rick got there first. Nevermind...Edited by Modulous, : d'oh

It's not an acceptable mathematical proof, because you are assuming certain facts about the convergence of the series and convergent series in general.That is why kongstad went to such lengths.

I just assumed that 9/10 - 9/10 = 0 that 9/100 - 9/100 = 0 and that this continues all the way. Thus 0.999... - 0.999... = 0. These are the same assumptions that kongstad used, only presented in algebraic form which I consider simpler to understand.Hopefully you don't feel that x - x = 0 is an assumption that is not warranted do you?

x = 1 + 2 + 4 + 8 + 16 + ....2x = 2 + 4 + 8 + 16 + ....2x - x = -1 (subtracting terms and "going all the way").x = -1That is what can go wrong and why your "proof" is mathematically invalid.

As someone who has had some training in analysis (the branch of mathematics that deals with limits), I can offer an explanation of your paradox.Neither of the series has a limit that is defined, and so neither x nor 2x are defined. Therefore, the subtraction doesn't make sense, since neither x nor 2x represent real numbers.Edited by Chiroptera, : typo

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That is correct. So you first have to prove convergence which is the point I was making.

I should add that if we put a different metric on the rationals, the 2-adic metric, then the completion is what we call the 2-adic numbers. In that casex = 1 + 2 + 4 + 8 + ...does have a limit, and indeed x = -1 in the 2-adic numbers.This is easy to see if I may offer this proof (it is left to the reader to supply the more rigorous steps):x = -1 if 1 + x = 0 (this is what -1 means, after all).1 + x = 1 + 1 + 2 + 4 + 8 + ...
= 2 + 2 + 4 + 8 + ...
= 4 + 4 + 8 + ...
= 8 + 8 + ...and so forth. You notice that at each step the addition "knocks out" the next higher power of two, and "eventually" (defined in a suitably rigorous limit kind of way) nothing is left, so, indeed, 1 + x = 0 and so x = -1.The p-adic numbers have all sorts of weird topological properties that make them fun, which is why I know a little about them.Edited by Chiroptera, : another typo

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It has become fashionable on the left and in Western Europe to compare the Bush administration to the Nazis. The comparison is not without some superficial merit. In both cases the government is run by a small gang of snickering, stupid thugs whose vision of paradise is full of explosions and beautifully designed prisons. -- Matt Taibbi

Another simple way to show this is:1/3 + 2/3 = 11/3 = .3333...2/3 = .6666....3333... + .6666... = .9999...therefore: .9999... = 1

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Right you just proved that 2 x infinity = infinity. Irrelevant to the fact that x-x = 0 for all real numbers. Are you suggesting that π - π does not equal 0?

You can't assume you have a real number until you show your sequence converges. You can't talk of x or x - x until you have shown it is a real number.My example showed what can go wrong if the sequence diverges. Also, "2 x infinity = infinity" is not a valid mathematical equation, so no proof there.You need to open a book on mathematical analysis. You are murdering the subject.You need to first understand the field axioms and what constitutes a logical mathematical argument.

If it's not a real number, then it doesn't matter does it? Yes I can see that. I'm not sure how one could argue that 9/10 + 9/100 +9/1000... diverges. It wasn't meant to be mathematical equation. Replace it with English if it makes things better for you. You proved that the sum of two infinite values is itself an infinite value. I seriously doubt I am murdering that which I am not engaging in. A few lines of algebra that were provided to me via a mathematician hardly constitutes a rigorous but absurdly inaccurate mathematical analysis. Obviously, why would I argue with that? I didn't say my proof was better or more thorough. It was more simple, obviously that means less axioms are explained. The more rigorous proofs could in theory take hundreds of pages of axiom declarations - I am not going to embarrass myself or murder the subject by attempting it.