This reminds me of how (I think) Zen koans are designed to make no sense at all. They are designed to teach you the limits of words and language and pure thinking.
Bhartrhari's Paradox
41–50 of 84 posts
Re: Bhartrhari's Paradox
#42How is it a paradox? Isn't this just a proof that there are many unnamed things, but no unnameable ones?
> Isn't this just a proof that there are many unnamed things, but no unnameable ones? Where is it a proof that there are many unnamed things ? I could only see it as an argument that there are no "unnameable" things.
It depends how you define "named", but for example, not all the grain of sands you see on a beach are named (yes, they are named collectively, but not individually. If "collectively" is valid, then that's further proof that "unnameable" things can't exist, because they already have a collective name).
Re: Bhartrhari's Paradox
#43counterargument: 1) let x be a thing 2) I name x "Jeff" 3) all things are nameable (from 1 and 2) another way to put this is that it's natural to take the paradox as a reductio.
You proved that definable implies nameable, and also unnameable implies undefinable. Obviously true. However, the idea of undefinable real numbers closely resembles a modern version of the paradox. No surjective function exists from definitions to real numbers. Really, the blurb about "seems impossible to verify this by giving positive instances" contains the tension between constructive math and non-constructive mat…
> No surjective function exists from definitions to real numbers.
I'm not really up on maths so this is possibly a stupid question, but can't any real number be written as an ASCII string, which is basically an integer number, so there is a direct mapping there?
Or is it because the ASCII number wouldn't be in order that makes the difference?
Or is it that you can't write that mapping as a mathematical function perhaps?
Re: Bhartrhari's Paradox
#44sounds like the paradox that _could_ illustrate Gödel's incompleteness theorems
https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_th...
Re: Bhartrhari's Paradox
#45Earlier quoted context omitted.
The problem with such sleight of hand counterargument is that you haven't even defined what "a thing" is nor "all things" are in this world. And such discussions will just come back to set theory, ZFC, axiom of choice and real numbers.
That isn't a problem with the counterargument, because the "paradox" as-stated also uses the word "thing". For that matter, the paradox is self-resolving. By labeling the entities it is concerned with as "unnameable things", it has named them. As a collection, entities not otherwise named can be simply referred to as "Bhartrhari's things".
And this is why analytical philosophy should be kept out of mathematics - Some 18th century mathematician before Cantor
Re: Bhartrhari's Paradox
#46Earlier quoted context omitted.
You proved that definable implies nameable, and also unnameable implies undefinable. Obviously true. However, the idea of undefinable real numbers closely resembles a modern version of the paradox. No surjective function exists from definitions to real numbers. Really, the blurb about "seems impossible to verify this by giving positive instances" contains the tension between constructive math and non-constructive mat…
> No surjective function exists from definitions to real numbers. I'm not really up on maths so this is possibly a stupid question, but can't any real number be written as an ASCII string, which is basically an integer number, so there is a direct mapping there? Or is it because the ASCII number wouldn't be in order that makes the difference? Or is it that you can't write that mapping as a mathematical function perha…
Actually, and perhaps sadly, I asked an LLM and I understand now.
But perhaps that's not such a bad thing that I can get answers to my foolish questions!
Re: Bhartrhari's Paradox
#47I see what you did there.
Re: Bhartrhari's Paradox
#48Earlier quoted context omitted.
> The reals can be ordered, just use x That ordering is not a well-ordering , which is what the GP specified. A well ordering requires that every non-empty subset has a smallest element. That's not true for the reals ordered by x 0 has no smallest element. No one has explicitly shown that the reals can be well ordered, but it's a consequence of the axiom of choice that every set can be well-ordered. So in ZFC there m…
Not a mathematician, so this question may be a bit thick. I see the problem with the set of reals > 0, but is it perhaps that in this case > 0 is the problem and for sets specified as >= 0 it's fine because 0 is a nameable real and the smallest element. Obviously you can't just exclude certain expressions arbitrarily though, so I don't know how you could justify that mathematically.
If there is any non-empty subset that has no smallest element, then the ordering in question is not a well-ordering. You can of course define some subsets of the reals that do have a smallest element in the standard ordering, for example all of the reals that are greater than or equal to 0. But there are also subsets that do not have a smallest element, and that is enough to show that the standard ordering on the reals cannot be a well-ordering.
Re: Bhartrhari's Paradox
#49Earlier quoted context omitted.
> No surjective function exists from definitions to real numbers. I'm not really up on maths so this is possibly a stupid question, but can't any real number be written as an ASCII string, which is basically an integer number, so there is a direct mapping there? Or is it because the ASCII number wouldn't be in order that makes the difference? Or is it that you can't write that mapping as a mathematical function perha…
Actually, and perhaps sadly, I asked an LLM and I understand now. But perhaps that's not such a bad thing that I can get answers to my foolish questions!
guess I won't respond now :( "what is a real number, anyways" is one of my favorite questions, not foolish at all
Re: Bhartrhari's Paradox
#50Are there actually things that cannot be named? Any such thing could easily be assigned some such "Phenomenon 8x306Q". If any two people agree to call it that and use that to succesfully discuss the thing, then that is a name for the thing. Otherwise nothing can be named. Is the cat in your house really a cat, or is it a Felis catus? How can we be certain that it's not a gato or a кот? If the cat in your house is ind…
(This is an older subthread which I moved hither because a different submission happened to make it onto the frontpage)