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raincom 2 hours ago [-]
If 'it' is unnameable, there is no way to circumscribe or even describe what 'it' is. Even to show that what it refers to is an empty set, we need its description. If we use concepts like intention and extension, we can sketch out four scenarios:
extension, no intension (yes, we can point out things, which we can't describe)
extension, intension (we point out, and we describe)
no extension, intension (Yes, we can imagine and describe things vividly, but no referent in the world. Here, one can say these things exist in a Platonic world, but not the world we live in; this is where numbers, sets, ideas can exist. Neo-Platonism in Philosophy of Mathematics)
no extension, no intension (this paradox falls in this area).
ngvrnd 10 hours ago [-]
„Wovon man nicht sprechen kann, darüber muss man schweigen“
but if the thing can be interacted with, it can usually be mapped and defined and then named. But there will always be things which do not have names, at least in mathematics -- think of the real numbers.
abnry 9 hours ago [-]
I suppose if the number of nameable things is countable (because humans can only enumerate, and it is humans who name), then it trivially follows that some real numbers are unnameable. Which proves the existence of such entities.
The argument is something like the set of possible thoughts about physical objects is larger than the physical objects themselves. It feels similar in flavor to the idea of unameable objects.
There's something special about a name. The name of the God of the Bible is special. Christians are to call upon _the name_ of the Lord. We pray, hallowed by _thy name_. It is somehow denotes the summary essence of the thing being named, even if it doesn't give specific details of its characteristics.
bryanlarsen 9 hours ago [-]
"Unnameable" and "unnamed" are two different things, in my opinion. Are there real numbers that are unnameable or are those just unnamed?
There are some that unnameable with my mathematical understanding, but that's not saying much.
amavect 8 hours ago [-]
No surjective function exists from names to real numbers (diagonalization). With any naming scheme, some unnamed real numbers always remain.
On the other hand, given any real number, I can name it. I'll run out of unique names, since no injective function exists from real numbers to names.
So, some unnamed real numbers will always remain (non-constructively), I can make real numbers that escape a naming scheme (constructively), and no unnameable real numbers exist.
ithkuil 59 minutes ago [-]
What does "naming" mean?
Assigning a symbol? But who said that the set of symbols must be countable?
amavect 8 minutes ago [-]
My hidden assumption: I said the set of names must be countable! I assumed you would know that naming means assigning a finite string (in the Ithkuil writing system of course). and don't nitpick further or else I'll have to write a proof in Agda or Rocq lol
10 hours ago [-]
lordnacho 8 hours ago [-]
This reminds me of the 6 degrees of separation thing.
People tell you that you can connect more or less everyone by 6 degrees. But what struck me was, for anyone you try this with, you know their names, so you've already restricted yourself in how far out someone can be.
GPerson 10 hours ago [-]
If we assume the real numbers exist, then perhaps the paradox resolves because there are uncountably many reals and only countably many nameable things, but then perhaps the paradox does not resolve because we assume ZFC is true and we can well order the reals, hence name the first unnameable real.
mark_something 10 hours ago [-]
The reals can be ordered, just use x < y. I think you mean that if ZFC is true, we could enumerate unnameable reals (choose one with the axiom of choice, remove it, choose another one, etc.), but you could not enumerate them all. But it is true that you could get a "first" unnameable real.
pdonis 10 hours ago [-]
> The reals can be ordered, just use x < y.
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 < y: for example, the set of all reals > 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 must be a well ordering of the reals, even though no one has found one. Issues like this are why not all mathematicians accept the axiom of choice.
Smaug123 5 hours ago [-]
> no one has found one
More than that: there is no way to build one (assuming the word "build" means some concrete construction), because it's consistent with ZF that the reals admit no well-ordering. Indeed, you can use forcing to construct a model of R in which there is an infinite but Dedekind-finite subset of R; and you can't well-order such a set, because a well-ordering would turn it into an ordinal, and any Dedekind-finite ordinal is finite. You must use some sort of choice principle to construct a well-ordering. (Of course, it's consistent that they can be well-ordered, too, as you say; or e.g. under the hypothesis V=L, where there's even a canonical well-ordering given by the lexicographic well-ordering L comes with.)
simonh 9 hours ago [-]
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.
pdonis 6 hours ago [-]
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.
luc4 8 hours ago [-]
We just used the standard ordering < to define the set, it has nothing to do with the candidate well-ordering. If that's confusing, consider the set { 10^-x | x \in N } instead. It also has no minimum element in the standard ordering.
GPerson 9 hours ago [-]
A well ordering on a set is a total order such that all non empty subsets have a minimum element with respect to this order. The standard ordering of the reals is not a well ordering, but the axiom of choice is equivalent to the statement that all sets possess a well-ordering. A well-order of the reals would probably look pretty chaotic though.
To prove that there are unnamable concepts, he uses cantor's diagonal argument. There are countably infinite names. Any subset of these names is a concept, which is same as the powerset of the set of names, and through cantor's diagonal argument, there are uncountably infinite concepts, most which are not namable.
Yes you can take a specific concept, and name it, but there are uncountably infinitely many, so even with infinite time, you cannot name them all.
BiraIgnacio 7 hours ago [-]
sounds like the paradox that _could_ illustrate Gödel's incompleteness theorems
But then by describing it you are committing it to a set of conditions this unnameable thing satisfies.
But then if you go beyond a narrow interpretation of that paradox and accept that naming and describing are both accomplishing the same fundamental thing, that being committing a thing to a condition (like a name) or set of conditions (like a description), you do run into the same problem.
Hmm, interesting. Now back to this E2E testing stuff I've been avoiding.
LanceH 9 hours ago [-]
In mathematics, there are infinitely many "computable" numbers. That is, numbers which can be describe using any mathematics available. Then there are far more "non computable" numbers, which can't be described by anything finite.
I think there is some analogy to be made here.
bwfan123 10 hours ago [-]
Names are like variables in a function. you can name variables anything you want from a human understanding point of view (final cause), but the compiler doesnt care about that. The compiler only cares about the efficient cause of that variable in the sense of what it represents (stack/heap etc).
10 hours ago [-]
svat 8 hours ago [-]
Incidentally, it is a matter of some debate whether Bhartṛhari the philosopher and Bhartṛhari the poet are the same person or two (or more, in the case of the anthology of verses). Oral tradition holds them to be the same person, scholars have debated back and forth. I have a collection of the poems here: https://shreevatsa.net/bhartrhari/web/ (will clean it up someday)
snapcaster 11 hours ago [-]
I'm open to the idea that some things are unnameable but would need an example :)
loa_in_ 10 hours ago [-]
I'll write you as soon as I can
Xcelerate 8 hours ago [-]
Pretty sure ZFC proves such things exist (and that it also can’t pinpoint any individual instances of course). Now, whether syntactical “∃” in the formal language of set theory corresponds to the platonic existence of some “thing”, who knows.
another way to put this is that it's natural to take the paradox as a reductio.
amavect 9 hours ago [-]
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 math. Does an unnameable (and undefinable) thing actually exist? If a tree falls in a forest, but no one can hear it, does it make a sound?
onraglanroad 7 hours ago [-]
> 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?
xelxebar 51 minutes ago [-]
It's because most real numbers are uncomputable. That means, most of the time, the only way to check that two numbers (i.e. names) are the same is to spend infinite time looking at all their infinite digits.
An unknowable name isn't a very good name, IMHO.
onraglanroad 6 hours ago [-]
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!
amavect 5 hours ago [-]
guess I won't respond now :( "what is a real number, anyways" is one of my favorite questions, not foolish at all
quickthrowman 2 hours ago [-]
It’s not a foolish question, you got to learn about Cantor’s diagonal argument!
crimsonspy 9 hours ago [-]
Jeff jeff jeff, jeff jeff jeff jeff! Jeff? Jeff.
miksteyp 8 hours ago [-]
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.
delecti 8 hours ago [-]
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".
miksteyp 6 hours ago [-]
And this is why analytical philosophy should be kept out of mathematics - Some 18th century mathematician before Cantor
curtisblaine 10 hours ago [-]
How is it a paradox? Isn't this just a proof that there are many unnamed things, but no unnameable ones?
arjie 5 hours ago [-]
Many ancient paradoxes are not really paradoxes. Zeno's ones are resolved today with infinite series.
But this is a real one. Is it possible to describe an arbitrary real number? Almost all reals are not describable. But you cannot find a single such number.
The 'paradox' is that the search itself is self-failing - a broken strategy. Of course, now we have the language of sets and functions between them and cardinalities and we resolve this for us in a way that is meaningful. But still now you know the 'existence' of this thing? Can't be described.
It's interesting because of the property of creating with finite words universes of infiniteness.
WillAdams 10 hours ago [-]
Agreed.
This is the stuff of magic and folklore, and neatly resolved by Ursula K. LeGuin in _A Wizard of Earthsea_.
bwfan123 9 hours ago [-]
> 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.
curtisblaine 7 hours ago [-]
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).
Zhyl 9 hours ago [-]
The Way that can be walked is not the eternal Way.
The name that can be named is not the eternal name.
-- Lao Tzu, Tao Te Ching
goodmythical 6 days ago [-]
Are 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 indeed a кот, gato, Felis catus, and cat, then Phenomenon 8x306Q can certainly be Penomenon 8x306Q as much as it is "familial bonds strained by misdeeds" or whatever the things we're naming is.
dang 5 hours ago [-]
(This is an older subthread which I moved hither because a different submission happened to make it onto the frontpage)
nofriend 6 days ago [-]
The proof is simple: there are countably many names, but uncountably many real numbers. Hence, some real numbers must be unnameable.
goodmythical 4 days ago [-]
Are there countably many names? Countably sayable, perhaps, but I don't recall seeing any limitation to word length in the spec. See: "Below is the full 189,819-lettered word for 'titin':" at https://cw39.com/wp-content/uploads/sites/10/2020/09/longest... as an example.
Surely if we can have a 189,819 lettered word we can have a million billion trillion lettered word or an uncountably lettered word. They'd be used in the same way as the really long numbers in that we'd give them some other handier name that collides when not given context. e.g. In spoken language pi/pie are often confused if the conversation does not already have a mathematical context and no one says 3.1415926535... conversationally just as no one uses the lenghtier version of chitin and no one would use the uncountably long name for some uncountably long number.
nofriend 4 days ago [-]
names have to be finite in length. i think that's pretty obvious
goodmythical 3 days ago [-]
I don't see how that's any more obvious than the suspicious claim that numbers can have only so many digits.
nofriend 3 days ago [-]
A number is not in the first place its digit sequence. A number like pi is in the first place the ratio of a circle's diameter and its circumference, and only incidentally a certain (infinite) decimal expansion. A name is in the first place something you say, hence the thing you say has to be (at least theoretically) sayable.
whack 4 hours ago [-]
Can't this also be used to justify things that are obviously nonsensical. Like for example: "I possess an immense undetectable sphere. How can I prove this? Well, any proof I offer you would by definition violate the undetectability of the sphere. So there's no way for me to prove it, I guess you'll just have to trust me bro."
amavect 4 hours ago [-]
I love philosophy Calvinball, so I would counter by asserting that undetectable implies no possession, an immediate contradiction. Or go further and assert that undetectable implies nonexistence. We all possess an immense undetectable nonexistent sphere. No bounds on assumptions means I can make up anything to annoy the interlocutor.
So, you're right. This shows why we should use formal math, so we can agree on the result yet bicker about the interpretation. Some folks point to Cantor's diagonalization theorem to show that some unnameable things exist, when the theorem doesn't say that at all.
dredmorbius 6 hours ago [-]
I see what you did there.
ogogmad 7 hours ago [-]
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.
cyanydeez 10 hours ago [-]
Sounds akin to the complexity inherent in cellular automata. We know via the rules how to mutate successive generations, but backwards propagation, algorithmic simplification, etc may exist but not traceable from any given ruleset.
extension, no intension (yes, we can point out things, which we can't describe)
extension, intension (we point out, and we describe)
no extension, intension (Yes, we can imagine and describe things vividly, but no referent in the world. Here, one can say these things exist in a Platonic world, but not the world we live in; this is where numbers, sets, ideas can exist. Neo-Platonism in Philosophy of Mathematics)
no extension, no intension (this paradox falls in this area).
but if the thing can be interacted with, it can usually be mapped and defined and then named. But there will always be things which do not have names, at least in mathematics -- think of the real numbers.
This reminds me of a YouTube video I watched this week titled "A counting argument for why mind comes before matter": https://youtu.be/AtduNjJV-6E?is=nBZ9ztZsyeoVhCj1
The argument is something like the set of possible thoughts about physical objects is larger than the physical objects themselves. It feels similar in flavor to the idea of unameable objects.
There's something special about a name. The name of the God of the Bible is special. Christians are to call upon _the name_ of the Lord. We pray, hallowed by _thy name_. It is somehow denotes the summary essence of the thing being named, even if it doesn't give specific details of its characteristics.
There are some that unnameable with my mathematical understanding, but that's not saying much.
On the other hand, given any real number, I can name it. I'll run out of unique names, since no injective function exists from real numbers to names.
So, some unnamed real numbers will always remain (non-constructively), I can make real numbers that escape a naming scheme (constructively), and no unnameable real numbers exist.
Assigning a symbol? But who said that the set of symbols must be countable?
People tell you that you can connect more or less everyone by 6 degrees. But what struck me was, for anyone you try this with, you know their names, so you've already restricted yourself in how far out someone can be.
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 < y: for example, the set of all reals > 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 must be a well ordering of the reals, even though no one has found one. Issues like this are why not all mathematicians accept the axiom of choice.
More than that: there is no way to build one (assuming the word "build" means some concrete construction), because it's consistent with ZF that the reals admit no well-ordering. Indeed, you can use forcing to construct a model of R in which there is an infinite but Dedekind-finite subset of R; and you can't well-order such a set, because a well-ordering would turn it into an ordinal, and any Dedekind-finite ordinal is finite. You must use some sort of choice principle to construct a well-ordering. (Of course, it's consistent that they can be well-ordered, too, as you say; or e.g. under the hypothesis V=L, where there's even a canonical well-ordering given by the lexicographic well-ordering L comes with.)
To prove that there are unnamable concepts, he uses cantor's diagonal argument. There are countably infinite names. Any subset of these names is a concept, which is same as the powerset of the set of names, and through cantor's diagonal argument, there are uncountably infinite concepts, most which are not namable.
Yes you can take a specific concept, and name it, but there are uncountably infinitely many, so even with infinite time, you cannot name them all.
https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_th...
But then by describing it you are committing it to a set of conditions this unnameable thing satisfies.
But then if you go beyond a narrow interpretation of that paradox and accept that naming and describing are both accomplishing the same fundamental thing, that being committing a thing to a condition (like a name) or set of conditions (like a description), you do run into the same problem.
Hmm, interesting. Now back to this E2E testing stuff I've been avoiding.
I think there is some analogy to be made here.
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.
Really, the blurb about "seems impossible to verify this by giving positive instances" contains the tension between constructive math and non-constructive math. Does an unnameable (and undefinable) thing actually exist? If a tree falls in a forest, but no one can hear it, does it make a sound?
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?
An unknowable name isn't a very good name, IMHO.
But perhaps that's not such a bad thing that I can get answers to my foolish questions!
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".
But this is a real one. Is it possible to describe an arbitrary real number? Almost all reals are not describable. But you cannot find a single such number.
The 'paradox' is that the search itself is self-failing - a broken strategy. Of course, now we have the language of sets and functions between them and cardinalities and we resolve this for us in a way that is meaningful. But still now you know the 'existence' of this thing? Can't be described.
It's interesting because of the property of creating with finite words universes of infiniteness.
This is the stuff of magic and folklore, and neatly resolved by Ursula K. LeGuin in _A Wizard of Earthsea_.
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.
The name that can be named is not the eternal name.
-- Lao Tzu, Tao Te Ching
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 indeed a кот, gato, Felis catus, and cat, then Phenomenon 8x306Q can certainly be Penomenon 8x306Q as much as it is "familial bonds strained by misdeeds" or whatever the things we're naming is.
Surely if we can have a 189,819 lettered word we can have a million billion trillion lettered word or an uncountably lettered word. They'd be used in the same way as the really long numbers in that we'd give them some other handier name that collides when not given context. e.g. In spoken language pi/pie are often confused if the conversation does not already have a mathematical context and no one says 3.1415926535... conversationally just as no one uses the lenghtier version of chitin and no one would use the uncountably long name for some uncountably long number.
So, you're right. This shows why we should use formal math, so we can agree on the result yet bicker about the interpretation. Some folks point to Cantor's diagonalization theorem to show that some unnameable things exist, when the theorem doesn't say that at all.
Like what?
Oh wait…