How real are real numbers? (2004)
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How real are real numbers? (2004)
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Re: How real are real numbers? (2004)
#2This was a light and interesting read.
Here is the direct link: https://arxiv.org/pdf/math/0411418.pdf
"Indeed, the most important thing in understanding a complex system is to understand how it processes information. This viewpoint regards physical systems as information processors, as performing computations. This approach also sheds new light on microscopic quantum systems, as is demonstrated in the highly developed field of quantum information and quantum computation. An extreme version of this doctrine would attempt to build the world entirely out of discrete digital information, out of 0 and 1 bits."
It would indeed be quite a blast to discover we are to a high degree of probability in a simulation.
Re: How real are real numbers? (2004)
#3"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.)
I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never write expressions for - only that in a single symbolic system we can't have expressions for all of them at once.
Besides, if you confuse extant with useful you might end up believing that some random large integers aren't "there!"
Re: How real are real numbers? (2004)
#4"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…
only that in a single symbolic system we can't have expressions for all of them at once.
I don't think that's true. There are only countably many different symbolic systems[1], and as we can only express countably many numbers in each, we don't leave the realm of countable.
[1] - A "symbolic system" must at least come with a procedure to tell whether a sequence is a part of it, and, unless you disbelieve the Church-Turing thesis, there are only countably many procedures.
Re: How real are real numbers? (2004)
#5"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…
> The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never write expressions for
Yes it does: there are countably many numbers we can define, and the reals are uncountable, therefore there are uncountably many numbers that cannot be defined.
Re: How real are real numbers? (2004)
#6This was a light and interesting read. Here is the direct link: https://arxiv.org/pdf/math/0411418.pdf "Indeed, the most important thing in understanding a complex system is to understand how it processes information. This viewpoint regards physical systems as information processors, as performing computations. This approach also sheds new light on microscopic quantum systems, as is demonstrated in the highly develop…
I don't see the connection between figuring out that the universe is discrete and learning that the universe is a simulation. Not even in our universe are all simulations done using digital computers, some are done using analog computers.
Re: How real are real numbers? (2004)
#7"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…
>Besides, if you confuse extant with useful you might end up believing that some random large integers aren't "there!"
In addition to the points already made about the "all at once" claim, every integer (every rational number, even) can be given a finite description (just write it down).
Re: How real are real numbers? (2004)
#8"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…
> The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never write expressions for Yes it does: there are countably many numbers we can define, and the reals are uncountable, therefore there are uncountably many numbers that cannot be defined.
This may be clearer to the parent commenter if expressed as "there are only countably many strings drawn from the Unicode alphabet, so if we fix a coding scheme such that a string expresses at most one number in that scheme, then there are only countably many numbers we could have defined. If you try and get around this by saying that the coding scheme is arbitrary so the symbol L can be made to stand for any real, then you have dodged the question of how you specify that coding scheme; there are only countably many coding schemes which can be expressed by strings of Unicode once a coding scheme is fixed, and in particular there are only countably many coding schemes expressible in English".
Re: How real are real numbers? (2004)
#9Unrelated, but I read a article a while back which said something similar, but it was based on the fact that our entire mathematical system is designed around "base 10" and as such is only relevant in many constructs to our specific human 'ten fingered', interpretation of math.
Re: How real are real numbers? (2004)
#10This was a light and interesting read. Here is the direct link: https://arxiv.org/pdf/math/0411418.pdf "Indeed, the most important thing in understanding a complex system is to understand how it processes information. This viewpoint regards physical systems as information processors, as performing computations. This approach also sheds new light on microscopic quantum systems, as is demonstrated in the highly develop…
I don't see the connection between figuring out that the universe is discrete and learning that the universe is a simulation. Not even in our universe are all simulations done using digital computers, some are done using analog computers.
But in our universe, if it's discrete on a macroscopic level, then it's either inherently a natural number (it can be counted), or it's intelligently created.