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March 5th, 2018, 05:05 PM   #11
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Quote:
 Originally Posted by v8archie This doesn't make any sense. Take $\pi$ for example. It is perfectly well defined as the ratio of a circle's circumference to its diameter, but we can't calculate it exactly. The same goes for $\sqrt2$ (the length of the diagonal of the unit square).
This doesn't make any sense. Here you are assuming that $\pi$ and $\sqrt2$ can exist in their entirety on the basis that a perfect circle can exist (with an infinitely thin line of circumference) and that a perfect diagonal of a unit square can exist, with infinitely small points at either end of an infinitely thin line.

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 process-related. 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:
 Originally Posted by v8archie You are focusing on the most complicated definition you've found. Perhaps because it suits your purpose, perhaps because you know no better.
I am focusing on it because it is more often than not claimed to be the definition favoured by most mathematicians.

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:
 Originally Posted by v8archie But a better (and equivalent) definition is that every real number is the limit of at least one sequence of rationals.
But as I said, this means we can only define the limit of a pi sequence as a pi sequence. Again, I am astonished that any mathematician can find this an acceptable state of affairs.

Quote:
 Originally Posted by v8archie The validity of the concept of limits should not be a problem, because they are defined entirely in terms of finite structures.
The limit of an infinite pi sequence is supposedly an infinite pi sequence. There is nothing finite here.

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:
 Originally Posted by v8archie This is trivially not a problem because the equivalence class is defined as those sequences having the same limit.
What do you mean? Are you saying that finding one sequence is equivalent to finding the infinite set of all sequences with the same limit? So if we find one pi sequence then have we effectively found all possible pi sequences? I don't think so.

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:
 Originally Posted by v8archie Moreover, there is (should be) no suggestion that a real number is the sum of infinitely many elements. It's not. A limit is a limit, not a sum.
I have absolutely no idea what this means. For the purpose of clarity, are you saying that the series 0.9 + 0.09 + 0.009 + ... strictly cannot contain an actual infinity of digits? This is a yes or no answer.

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:
 Originally Posted by v8archie You could equally well discard all of science on the basis that old theories have been superseded. It's not an argument that holds any water. It's also important to remember that the decimal representation is not the number. Even if infinite decimals didn't make sense, it wouldn't stop real numbers from existing.
You are missing the point.

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: 1,801 Thanks: 636 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
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Quote:
 Originally Posted by Karma Peny This doesn't make any sense. Here you are assuming that $\pi$ and $\sqrt2$ can exist in their entirety on the basis that a perfect circle can exist (with an infinitely thin line of circumference) and that a perfect diagonal of a unit square can exist, with infinitely small points at either end of an infinitely thin line.
I literally made it this far before I stopped reading. This is absolutely absurd to think that "constructing" a number is required for it to exist. I don't even know what you mean by constructing a number. You claim I can't construct $\pi$, but then I claim you can't even construct 1.

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
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Quote:
 Originally Posted by Karma Peny This doesn't make any sense. Here you are assuming that $\pi$ and $\sqrt2$ can exist in their entirety on the basis that a perfect circle can exist (with an infinitely thin line of circumference) and that a perfect diagonal of a unit square can exist, with infinitely small points at either end of an infinitely thin line. It is impossible to construct a perfect line, let alone a perfect circle.
Do you remember that I said:
Quote:
 Originally Posted by v8archie This suggests to me that you labour under the misapprehension that the mathematical world somehow holds some equivalence to the real world.
The great thing about the human mind is that it can consider things that do not exist and deduce consequences of that. This is exactly what a mathematical system is. Real lines and circles (almost certainly) do not exist, but mathematical ones do.
Quote:
 Originally Posted by Karma Peny 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.
Quote:
 Originally Posted by v8archie There are many equivalent definitions of the real numbers.
I take it that you understand the word "equivalent"? I realise that the existence of different and equivalent definitions might be a surprise to you. But it's not a revolutionary idea. Moreover, being equivalent, it cannot be a case of falling back onto a different definition if one fails because they are all the same. But, as I alluded to before, different definitions can provide different perspectives on a problem. It is sensible to use the one that gives the clearest view.

Quote:
 Originally Posted by Karma Peny this means we can only define the limit of a pi sequence as a pi sequence. Again I am astonished that any mathematician can find this an acceptable state of affairs.
Quote:
 Originally Posted by v8archie Take $\pi$ for example. It is perfectly well defined as the ratio of a circle's circumference to its diameter... $\sqrt2$ [is] the length of the diagonal of the unit square.
As you can see, your position here is simply wrong. No mathematician finds your idea an acceptable state of affairs, because it's not the state of affairs.

Quote:
 Originally Posted by Karma Peny The limit of an infinite pi sequence is supposedly an infinite pi sequence. There is nothing finite here.
You are talking about a different sort of limit, heading into the actual infinities of set theory. These infinities are ones that even finitists presumably have to accommodate. You can't have the natural numbers without having an infinite number of them.

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:
 Originally Posted by Karma Peny 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.
I'm not familiar with Wildberger's output, but a brief Google was enough to make me wary of taking what he says at face value. I don't know of any serious problems with the definition of a sequence, and there is no need whatever for an infinite number of Epsilons. That's a misrepresentation of the concept of limits.

Quote:
 Originally Posted by Karma Peny 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.
As I said before, any suitable sequence is sufficient to define the number. There's no need to look at all of them - and doing so clearly makes life significantly more difficult. As with definitions, it makes sense to pick a sequence that is easy to handle. The sequence is more-or-less immaterial - it's the limit that is important.

Quote:
 Originally Posted by Karma Peny So if we find one pi sequence then have we effectively found all possible pi sequences? I don't think so.
Nor do I. But we have found a way to determine $\pi$ to any desired accuracy, it being the limit of the sequence. The fact that it is the limit of a sequence makes it a real number.

Quote:
 Originally Posted by Karma Peny are you saying that the series 0.9 + 0.09 + 0.009 + ... strictly cannot contain an actual infinity of digits?
You can have an infinite sequence of symbols, but nobody really knows how to add an infinite number of terms. We have defined$$0.9 + 0.09 + 0.009 + \ldots = \sum_{k=1}^\infty \tfrac9{10^k} = \lim_{n \to \infty} \sum_{k=1}^n \tfrac9{10^k}$$but this is just assigning a reasonable value to the sum. It does not in any way involve the adding together of an infinite number of terms.

Quote:
 Originally Posted by Karma Peny 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.
Not at all. Because the limit (and not the series) is the real number, the actual value of the sum is immaterial. Indeed there is no actual sum, because addition is defined only over a finite number of summands. What matters is the value of the limit. The limit is defined without reference to anything infinite (other than the set of natural numbers, but it only uses particular natural numbers which are all finite).

Quote:
 Originally Posted by Karma Peny I proved that the sum of such an infinity of terms cannot equal 1.
From what I recall, your reasoning is flawed - not least because you think that 0.999... represents an actual summation rather than the limit of a sequence of summations that it actually represents.

Quote:
 Originally Posted by Karma Peny You cannot retrospectively change the basis of these arguments.
Why not? You won't find a mathematician who knows the subject that would claim that it is possible to add an infinite number of summands together. Just as you won't find a physicist that claims that everything revolves around the earth.

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
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Quote:
 Originally Posted by v8archie I'm not familiar with Wildberger's output ...
That's why you're wasting your precious life energy on this guy. Here's his website. Extreme Finitism

You ain't gonna out-write him and you're not going to teach him anything.

March 5th, 2018, 07:39 PM   #16
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Quote:
 Originally Posted by Maschke That's why you're wasting your precious life energy on this guy. Here's his website. Extreme Finitism You ain't gonna out-write him and you're not going to teach him anything.
Probably not. But he's making the same mistake everyone does with this. He assumes that the decimal representation is a real-number (ignoring whether or not the representation is infinite). But this is not, in my view, a sensible way to approach the problem. After all, $2 = \frac63 = 2.000...$. They are all the same number.

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
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Quote:
 Originally Posted by v8archie that any limit of a sequence of rationals is a real number.
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
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Quote:
 Originally Posted by Maschke That's why you're wasting your precious life energy on this guy. Here's his website. Extreme Finitism You ain't gonna out-write him and you're not going to teach him anything.
Thanks for the heads up. I had a feeling I was bashing my head against a wall. That link makes it clear that in fact, bashing my head against a wall would actually be a better use of my time. I'm out.

March 5th, 2018, 10:52 PM   #19
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Quote:
 Originally Posted by v8archie Probably not. But he's making the same mistake everyone does with this. He assumes that the decimal representation is a real-number (ignoring whether or not the representation is infinite). But this is not, in my view, a sensible way to approach the problem. After all, $2 = \frac63 = 2.000...$. They are all the same number.
I don't see how this is a problem really. Why can't I define 2.000000... as the real number? It is a perfectly sensible approach to real numbers. Sure, you're gonna have multiple representations like 2.000... = 1.999... = 6/3 = .....
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
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Quote:
 Originally Posted by v8archie The great thing about the human mind is that it can consider things that do not exist and deduce consequences of that. This is exactly what a mathematical system is. Real lines and circles (almost certainly) do not exist, but mathematical ones do.
As I said, 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.

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:
 Originally Posted by v8archie your position here is simply wrong. No mathematician finds your idea an acceptable state of affairs, because it's not the state of affairs.
Professor N J Wildberger & a large number of his fans do for starters. He rejects the idea that $\pi$ and $\sqrt2$ can exist and he rejects that infinite sets can exist. You can get an overview of his views from this video:

Infinities and Skepticism in Mathematics: Steve Patterson interviews N J Wildberger

Quote:
 Originally Posted by v8archie I'm not familiar with Wildberger's output, but a brief Google was enough to make me wary of taking what he says at face value. I don't know of any serious problems with the definition of a sequence, and there is no need whatever for an infinite number of Epsilons. That's a misrepresentation of the concept of limits.
How can you find an infinite limit without doing infinite things? You might claim each step only involves finite things but whatever game you play you cannot avoid the need to go to infinity. To claim that real numbers can be defined without infinities is just a word game.

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:
 Originally Posted by v8archie You can have an infinite sequence of symbols, but nobody really knows how to add an infinite number of terms.
Exactly. The best we can do is line up terms in a way that we hope will all cancel out, all the way to infinity, as is supposedly done in the algebraic proof for 0.999... = 1.

Quote:
 Originally Posted by v8archie From what I recall, your reasoning is flawed - not least because you think that 0.999... represents an actual summation rather than the limit of a sequence of summations that it actually represents.
So let's be clear, consider a random infinite sequence that corresponds to an irrational real number. You agree that its limit can only be described as another infinite sequence, or even the original sequence. In other word, you agree that it is correct to claim it is its own limit. And yet you disagree that limit is the sum of its terms. Then what is it?

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.

 Tags 0.999, 0.999..., 0.999...=1, arguments, fail, finitism, real numbers, usual

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