March 5th, 2018, 05:05 PM  #11  
Member Joined: Oct 2014 From: UK Posts: 61 Thanks: 2  Quote:
It is impossible to construct a perfect line, let alone a perfect circle. The idea that $\pi$ and $\sqrt2$ can exist as valid lengths is just a figment of the imagination. Even Plato conceded that such perfect forms could not exist in the real world, hence his imagined third realm of perfect forms. To say that these things exist because they can be imagined is equivalent to saying unreal things must exist because someone says (s)he can imagine them. When we evaluate $\pi$ and $\sqrt2$, we follow a process of ongoing refinement. So these symbols are inherently processrelated. There is no constant value ('limit') that these processes are 'approaching', they simply go on for as long as you want in order to find an approximation. This is an approximation to a finite value because any circle or unit square we construct will be granular in nature, so it will consist of a number of smallest parts. We calculate an approximation because we don't know how small the smallest parts are. Quote:
While we are on the subject, why can't the mathematics establishment decide on a single definition for a real number rather than confuse matters with several different ones? It feels like an insurance policy to me. If problems emerge with one definition then we'll just switch to talking about a different one. I fail to understand how any mathematician can find this complete lack of clarity acceptable. Quote:
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As Norman Wildberger points out in 'MathFoundations111', the lack of a clear and precise definition of a 'sequence' is problematic also the need for an infinite number of Ns for an infinite number of Epsilons is also a problem. This limit definition is then extended to include all sequences that appear to be converging to the same value as our sequence. And so the definition that I was using is just an extension of the limit definition. Quote:
For an equivalence class to contain different sequences, then there must exist different sequences. They are in the same class because one particular property of them, such as their limit, is being used as the basis for equivalence. We could have a different equivalence class where the basis for equivalence is the 1st term of the sequence. The point is that we need to be able to refer to different sequences and their properties. We cannot claim all sequences in a particular class are exactly the same, because they are not; they just have one particular property that is the same. Quote:
I know that you are trying to avoid this key question because it suits your position to claim we don't need to worry about this in order to find the limit. But it is important because if it contains a finite number of terms then the limit will be less than 1. So it is necessary that 0.999... contains an actual infinity of digits, and in my 'argument 7' I proved that the sum of such an infinity of terms cannot equal 1. We now have an inconsistency because the limit of a finite sequence equals the sum, but this does not hold for an infinite sequence. Quote:
Arguments 1 to 7 were made when the meaning of 0.999... was supposedly the value returned from adding up all of its terms. So arguments 1 to 7 effectively start with "if 0.999... is the sum of 9/10 + 9/100 + 9/1000 + ... then we can argue that it equals 1 because blah, blah ...". You cannot retrospectively change the basis of these arguments. Worse still, by assuming 0.999... and 1 are real numbers, you nullify the arguments completely. Now all the arguments effectively start with "if 0.999... and 1 are merely symbols for the same number, then try to prove they are not the same number ...". Note: Just like you, I have avoided quoting all of your comments, but I think I have covered them all. Last edited by skipjack; March 5th, 2018 at 06:14 PM.  
March 5th, 2018, 05:14 PM  #12 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,533 Thanks: 513 Math Focus: Yet to find out. 
@Karma Penny, have you read many expositions on mathematics itself? Books such as: "The Art of Mathematics"  Jerry King and "Mathematics the man made universe"  Stein come to mind, also classics like G.H. Hardy's essay. Or perhaps something more 'applied' like David Ruelles  "A Mathematicians brain". The obsession with these 'real world' requirements seems to indicate that you haven't understood or have a feeling for what mathematics 'is' exactly.
Last edited by Joppy; March 5th, 2018 at 05:17 PM. 
March 5th, 2018, 05:23 PM  #13  
Senior Member Joined: Sep 2016 From: USA Posts: 299 Thanks: 155 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
Also, you seem overly concerned with geometric intuition so let's define $\sqrt{2}$ as follows: Define $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = x^2  2$. Note that $f$ is continuous, $f(0) < 0$, $f(3) > 0$, and $\mathbb{R}$ is a complete metric space. Then the intermediate value theorem implies there exists some $x_0 \in (0,3)$ satisfying $f(x_0) = 0$. We let $x_0 = \sqrt{2}$ denote this solution. No geometry needed. If you like, you can argue in the same way for $\pi$ as the root of $\sin x$ on the interval $(1,4)$. Last edited by skipjack; March 5th, 2018 at 05:43 PM.  
March 5th, 2018, 06:45 PM  #14  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,214 Thanks: 2410 Math Focus: Mainly analysis and algebra 
Your post illustrates a huge misunderstanding about mathematics as a whole. Quote:
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The limits used to define the real numbers are those defined in the $\delta\epsilon$ theory of limits. The definitions do not involve infinities at all. You clearly need to read up on this to understand it. Quote:
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The reason SDK doesn't like those arguments in facour of 0.999... = 1 is that they are akin to claiming that leeches are some sort of panacea in medicine. It's seriously out of date stuff that we know to be false. I would say that the decimal representations 0.999... and 1.000... represent two different (infinite) sequences of natural numbers. (It is trivial to see that any decimal representation must be a real number, because every decimal representation represents a convergent sequence which therefore has a limit). We can calculate the limits of these two sequences and they turn out to be 1. So 0.999... = 1.000... = 1.  
March 5th, 2018, 07:14 PM  #15 
Senior Member Joined: Aug 2012 Posts: 1,763 Thanks: 480  That's why you're wasting your precious life energy on this guy. Here's his website. Extreme Finitism You ain't gonna outwrite him and you're not going to teach him anything. 
March 5th, 2018, 07:39 PM  #16  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,214 Thanks: 2410 Math Focus: Mainly analysis and algebra  Quote:
For me, you have to start from the point that any limit of a sequence of rationals is a real number. Then, his own arguments give you the result without any need to mention "infinity".  
March 5th, 2018, 07:47 PM  #17 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,533 Thanks: 513 Math Focus: Yet to find out.  If I had a dollar for every time this gets said.. Last edited by skipjack; March 5th, 2018 at 08:32 PM. 
March 5th, 2018, 09:46 PM  #18  
Senior Member Joined: Sep 2016 From: USA Posts: 299 Thanks: 155 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
 
March 5th, 2018, 10:52 PM  #19  
Senior Member Joined: Oct 2009 Posts: 263 Thanks: 90  Quote:
But if you define numbers as Cauchy sequences, you have this problem too. Only cuts (of the usual approaches) kind of avoids using multiple representatives. But I don't see multiple representatives as necessarily something bad. That said, I am very sympathetic towards ultrafinitism and towards any "exotic" philosophy of mathematics really. Finding out a version of math that uses only finite objects is very intriguing, but I haven't yet found any that really works well. That said, the usual standard mathematical theory doesn't take the ultrafinitist approach, so where I think the OP is somewhat misguided is in thinking that it does and in thinking that the ultrafinitist philosophy somehow has any implication on the standard way of thinking. Of course it does not since they're completely separate.  
March 6th, 2018, 02:00 AM  #20  
Member Joined: Oct 2014 From: UK Posts: 61 Thanks: 2  Quote:
The human brain is a finite structure and so it can only process finite quantities. As Prof N J Wildberger says, we cannot imagine an 'infinite' set; we only imagine we can imagine one. In his video "The law of logical honesty and the end of infinity" he asks if we accept the existence of finite sets, does this mean other sets must exist that are ‘not finite’? Surely not he claims, because if we can see a visible hotel it doesn’t mean an invisible one must exist, and if we have a moveable kitten it does not mean an immovable kitten must exist. Rather than trust our imagination or our intuition, we could examine what we can represent with physical materials. This way our results can be objectively validated as opposed to being introspective beliefs. Any claim that we can imagine an infinite decimal or an idealised length is just an introspective view that cannot be validated by others. It is just a belief system with no basis in reality, just like any belief in supernatural phenomena. But my arguments hold regardless of your platonic beliefs. We only started discussing this because you mentioned the geometric definitions for pi and the square root of two. There are many other series that do not have similar geometric derivations and so this in no way refutes my main argument. Quote:
Infinities and Skepticism in Mathematics: Steve Patterson interviews N J Wildberger Quote:
I can only presume that you find it a completely satisfactory state of affairs that the limit of a pi sequence can only be defined in terms of another pi sequence. As such, a pi sequence is effectively defined as being its own limit. The infinite sequence and the limit are one and the same thing. If this holds for an irrational then it should also hold for 0.999... But in my 'argument 7' I show that if they are one and the same thing then they cannot equal 1. Quote:
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You are grasping at straws to avoid admitting that a limit must be the addition of its terms by necessity. As we are now repeating ourselves there is little point in continuing this discussion. Thank you for your comments; you have influenced me  I changed my article in response to one of your comments. I hope you will give Professor Wildberger's position more consideration in future, and thanks once again for an enjoyable chat. Last edited by Karma Peny; March 6th, 2018 at 02:02 AM.  

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