My Math Forum Selecting a Natural and a Real Uniformly at Random

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April 17th, 2017, 01:48 PM   #11
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 Originally Posted by studiot Interesting that you should bring in Physics here because random is one of those terms that the Physicist and the Mathematician interpret differently. The Mathematician is mostly interesting in the result (the kolmogorov definition) whereas the Physicist is mostly interested in the journey.
There are a number of attempted applications in probability theory that allow for selecting a real randomly from the interval [0,1]: throw a dart, use a binary sequence like I do above, etc. In reality, there are no practical ways to do this. We are clearly in the abstract here.

I am hoping someone can read the above and say what is wrong specifically. I was expecting Maschke to say something like "the mappings in part 3 of the process aren't uniform because 5 has a better chance of being selected than 7." I think they are uniform, but just trying to come up with something. Can anyone do this as opposed to spouting of the typical "there is no uniform probability on the naturals" argument. I am well aware of that argument, the proof for it, and why people say that. What in my proof specifically goes wrong, if anything? Two days almost and nothing here or at the physics forums where, in my experience, people will ream you quite quickly for posting pure rubbish.

 April 17th, 2017, 02:05 PM #12 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,306 Thanks: 2443 Math Focus: Mainly analysis and algebra As I said, selecting an infinite binary sequence with equal probability is circular reasoning. You can only do it in theory. Any attempt in practice is non-terminating. Just like throwing a dart and measuring where it lands. In each case you assume that you have selected uniformly one of an uncountable number of items, so obviously it is possible to do so. Thanks from AplanisTophet
April 17th, 2017, 02:15 PM   #13
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 Originally Posted by v8archie As I said, selecting an infinite binary sequence with equal probability is circular reasoning. You can only do it in theory. Any attempt in practice is non-terminating. Just like throwing a dart and measuring where it lands. In each case you assume that you have selected uniformly one of an uncountable number of items, so obviously it is possible to do so.
Everything involving real numbers is theoretical since there are no real numbers instantiated in the physical world (as far as we know). That's no argument against the existence of a uniform probability distribution on the unit interval, which looks just like the line $y = 1$ between $0$ and $1$.

In my opinion, complaining that you can't physically instantiate some mathematical idea is like saying that we can't be sure $\frac{1}{2} + \frac{1}{2} = 1$ because we can't exactly divide a physical substance in half and perfectly measure the result. After all, all physical measurement is approximate. This is a very disingenuous argument that utterly fails to understand the nature of mathematics. I'm not sure how it creeped in here.

Last edited by Maschke; April 17th, 2017 at 02:24 PM.

April 17th, 2017, 02:29 PM   #14
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 Originally Posted by AplanisTophet Two days almost and nothing here or at the physics forums where, in my experience, people will ream you quite quickly for posting pure rubbish.
Evidently their manners have improved over there. A few years ago your post would be deleted, now it's just being politely ignored.

It's a matter of genuine disappointment to me that you've decided that I'm being personally mean to you rather than making a mathematical judgment that your argument is convoluted and incoherent. A few weeks ago I spent many many hours patiently working through your flawed mathematical argument to help you clarify your understanding. I'm sad that in this case you don't realize that I'm doing exactly the same.

At this point I've been through your argument about six or eight times and can't make heads or tails out of it. When I take the time to tell you that your argument is convoluted, incomprehensible, and doesn't even begin to show what you claim it does, I am doing you the service of allocating my time to helping you. I am very personally disappointed that you would rather conclude that I'm treating you with "disdain," your word. You have clearly let your emotions get the better of you here and you can't even see who your friends are.

For that reason I need to step back and let you sort this out on your own. I would suggest that you make an attempt to write a much clearer version of your argument. That's a supportive suggestion, not an insult. I'm sad that you can't see it that way.

Last edited by Maschke; April 17th, 2017 at 02:38 PM.

April 17th, 2017, 02:33 PM   #15
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Quote:
 Originally Posted by v8archie As I said, selecting an infinite binary sequence with equal probability is circular reasoning. You can only do it in theory. Any attempt in practice is non-terminating. Just like throwing a dart and measuring where it lands. In each case you assume that you have selected uniformly one of an uncountable number of items, so obviously it is possible to do so.
CRGreathouse was the one that told me you could select an infinite binary sequence with equal probability. See the bottom of post #93:

Probabilities and Infinite Sets

In my OP I say "given" this binary sequence, then...

So, you are not willing to give me that binary sequence because even though it's possible in theory, it's not in practice. Are you not willing to entertain the possibility of doing it in theory even if not practical as CRGreathouse was? Why not?

April 17th, 2017, 02:52 PM   #16
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 Originally Posted by AplanisTophet There are a number of attempted applications in probability theory that allow for selecting a real randomly from the interval [0,1]: throw a dart, use a binary sequence like I do above, etc. In reality, there are no practical ways to do this. We are clearly in the abstract here. I am hoping someone can read the above and say what is wrong specifically. I was expecting Maschke to say something like "the mappings in part 3 of the process aren't uniform because 5 has a better chance of being selected than 7." I think they are uniform, but just trying to come up with something. Can anyone do this as opposed to spouting of the typical "there is no uniform probability on the naturals" argument. I am well aware of that argument, the proof for it, and why people say that. What in my proof specifically goes wrong, if anything? Two days almost and nothing here or at the physics forums where, in my experience, people will ream you quite quickly for posting pure rubbish.
I am not Maschke and I feel quite affronted by this casual dismissal of my comment designed to help.

How can you say my view is not relevant when you don't know what it is or take any steps to find out?

And then you complain that no-one is speaking to you anywhere.
Do you wonder at that?

April 17th, 2017, 02:53 PM   #17
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 Originally Posted by studiot I am not Maschke
Can you prove that?

On a serious note, OP is quite correct to be dismayed by your claim that "... selecting an infinite binary sequence with equal probability is circular reasoning." That's entirely incorrect. It's certainly possible to select a random real in the unit interval. I understand that you meant to say that you can't do such a thing in the real world, but neither can you divide $1$ into two copies of $\frac{1}{2}$. Physical considerations are not relevant here. As OP is struggling with a construction based on the Vitali set, I truly fail to understand how physical considerations would enter into this conversation and I understand OP's dismay.

Last edited by Maschke; April 17th, 2017 at 02:58 PM.

April 17th, 2017, 02:55 PM   #18
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 Originally Posted by studiot I am not Masche and I feel quite affronted by this casual dismissal of my comment designed to help. How can you say my view is not relevant when you don't know what it is or take any steps to find out?
I thought your view was relevant. I don't understand. It was helpful. I'm sorry if that's what you took from that. I thought you were pointing out that physicists, being concerned with the journey, wouldn't read past the "select a real in [0,1] at random part." Mathematicians, being concerned with the theoretical, would accept it.

April 17th, 2017, 03:17 PM   #19
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 Originally Posted by AplanisTophet I thought your view was relevant. I don't understand. It was helpful. I'm sorry if that's what you took from that. I thought you were pointing out that physicists, being concerned with the journey, wouldn't read past the "select a real in [0,1] at random part." Mathematicians, being concerned with the theoretical, would accept it.
I actually said almost the exact opposite.

Tell me what you think a random number is?

For instance are 1 or 0 random numbers?

April 17th, 2017, 03:23 PM   #20
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 Originally Posted by Maschke Evidently their manners have improved over there. A few years ago your post would be deleted, now it's just being politely ignored.
Maybe.

Quote:
 Originally Posted by Maschke It's a matter of genuine disappointment to me that you've decided that I'm being personally mean to you rather than making a mathematical judgment that your argument is convoluted and incoherent. A few weeks ago I spent many many hours patiently working through your flawed mathematical argument to help you clarify your understanding. I'm sad that in this case you don't realize that I'm doing exactly the same.
I don't think you're being mean. It appeared to me as though you were relying on the classical proof that the sum of probabilities on a uniform distribution over $\mathbb{N}$ cannot sum to 1 as a means of not taking the OP seriously. You said you didn't read it, but now:

Quote:
 Originally Posted by Maschke At this point I've been through your argument about six or eight times and can't make heads or tails out of it. When I take the time to tell you that your argument is convoluted, incomprehensible, and doesn't even begin to show what you claim it does...
So, if the problem is that you don't understand my OP because you feel it is "convoluted" and "incomprehensible," then please quote the first part of the OP that you feel fits that description and I'll address it. Fair enough?

On a side note, I was genuinely confused. You have been very kind in the past with your time. It's not something I expect of you and it is appreciated. Do you understand that I thought this time you were simply electing to blow me off? I stayed up late so as to hide easter eggs for my kid and made the OP, but I was exhausted working tax season. I wouldn't have posted this in the first place if it wasn't for the conversation in the other thread.

 Tags natural, random, real, selecting, uniformly

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