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.
Bhartrihari was a king, philosopher, and poet. Scholars argue whether they were the same person or not, but I don't care, as scholars also argue about if Socrates really existed or not.
He wrote 300 verses in Sanskrit. And they are on three different topics: sensuality and pleasure, policy and ethics, and finally renunciation.
In Shringar Shatakam (100 verses on sensual pleasures), he writes:
“Casting aside envy, considering the matter carefully, let the noble ones tell us, with due propriety: Which ought one to frequent — the slopes of the mountains, or the buttocks of women whose smiles are stirred by Love?”
and
“Why all this elaborate, pointless talk? There are only two things worth attending to in this world: the fresh, wine-intoxicated youth of beautiful women, heavy with their breasts - or the forest.”
But in the final book, he realizes the folly of the senses, and writes:
“Sensual objects will inevitably leave us, even after remaining with us for a long time. What difference is there between losing them and voluntarily abandoning them? When they depart against our will, they cause unbearable anguish; but when we ourselves abandon them, they produce the infinite happiness of inner tranquility.”
A REASON FOR RENUNCIATION
Possessions leave us at the end,
However long they stay;
Then why not cast aside, my friend,
What leaves us anyway?
And if they leave against our will,
The heart takes time in mending;
If given willingly, they fill
That heart with joy unending.
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)
I have the A.N.D.Haksar, Purohit Gopinath translations of all the Satakas and Swami Madhavananda's translation of the "Vairagya Shatakam". Need to get the others ;-)
I couldn't add the Haksar translation as it's still under copyright, but the site has the other two. Plan to add others like M. R. Kale's (https://github.com/shreevatsa/bhartrhari/issues/11) — just need a chunk of time one of these weekends.
One thing you might want to add after each author's name is which categories (viz. Niti, Shringara and Vairagya) they have translated. Due to excessive prudishness many have omitted the Shringara satakam which is quite silly (this is what makes him "Human"). I got the Haksar and Gopinath editions specifically because they include all three categories. A.N.D.Haksar in particular has translated many of the works in Sanskrit literature into easy English (including the Kama Sutra) and does not censor anything.
Also i suggest that you add author names of all known translations of the work whether you have access to their actual text or not (due to copyright etc. reasons). That way your site can be a one-stop portal to Bhartrhari's Sataka-Trayam.
PS: In case you don't already know of it; there is a much larger work in the Tamil language named Tirukkural which is also divided into three similar categories (viz. Aram, Porul and Kamam) the whole having a total of 1330 couplets (133 chapters of 10 couplets each). There are many English translations available of which the original Penguin edition titled "Kural" translated by P.S.Sundaram is pretty good and done in the original couplet style. For a more detailed study see the 2-vol translation with commentary by S.M.Diaz.
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.
„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.
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.
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
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.
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.
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).
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.
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.
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.)
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.
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.
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.
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).
This is basically why art exists — painting, music, and poetry can point at things without having to name them. Language traps itself; other forms don't.
What you say is valid only when you restrict yourself to Western definitions/conceptions of language/linguistics i.e. study of Phonetics/Morphemes/Syntax/Semantics/etc.
In Bhartrhari's Philosophy (and other Hindu philosophies) "Language" has a much broader definition which can encompass Art/Dance/Music/Painting/etc. Any medium of communication which can bring forth a "burst of meaning" (called Sphota) in one's consciousness is a language.
Natural Spoken language based on Sound (aka Sabda in Sanskrit) is considered the most fundamental since you can have languages without a written script/symbols/diagrams.
In Hindu philosophy, a "Language" is said to have four stages, only the last of which is the gross manifestation in the physical world;
1) Para - This is the latent undifferentiated potential which exists in everybody.
2) Pashyanti - This stage is where intuitive holistic meaning (of what you want to convey) exists.
3) Madhyama - This stage is where you have differentiated the thought/intention into an object and the means of representation for its communication.
4) Vaikhari - In spoken language, this is the manifest stage where you utter sentences according to established syntax/semantics to convey meaning.
Note that the first three stages are internal and only the last is the medium of expression in the physical world. It should now be obvious that the last can be any medium (eg. Dance/Painting/Music/Written-Language/Sign-Language/etc.) as long as the receiver "gets" the intended meaning.
You didn't name the object when you said "let this thing be X"; you actually had already identified that "thing", and that process of identification was the process of naming it. You then defined some syntax ("X") and said that it was a name.
But there are things for which you can't even say "let 'this thing' be…". For example, ZF proves that there are uncountably many reals. There are only countably many names, so there must be unnameable reals. You can talk about "generic" reals (you can say "let x be a real" and do all sorts of interesting things with a generic x), but there are specific reals you will never be able to name specifically enough to distinguish them from their uncountably-many brethren. That doesn't make them "vague, undefined or ephemeral"! They're just so numerous that you can't describe the distinctions between them.
(Even hardcore constructivists usually accept enough Choice to prove the reals uncountable, although https://arxiv.org/abs/2404.01256 made headlines when it was shown not to be necessarily true.)
> that process of identification was the process of naming it.
No, it wasn't. Entities can be identified without being named, by relationships to other entities and class and such. That identification requires words. Not all denotational words and phrases constitute names.
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.
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.
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).
Bhartrhari (https://en.wikipedia.org/wiki/Bhart%E1%B9%9Bhari) is a pretty difficult philosopher who seems to be enjoying a revival now due to the ascendancy of AI LLMs and the question of whether they can be considered as having "consciousness".
His central idea (highly simplified) is that since Language is the only way we can name objects and discuss relations between them it is synonymous with "Reality" and "Consciousness". Sort of like how the properties of an object define that object (ADTs anyone?). One can imagine that the use of language by LLMs gives birth to appearance of both consciousness and reality as "emergent phenomena" in it. In his theory of "Sphota" he posits that "meaning bursts forth" (in consciousness) as an indivisible whole when a complete sentence/sentences is/are uttered (is this what happens when LLMs do reasoning and generate text within a "context window"?) Perhaps Epistemology and Ontology are just two sides of the same coin.
4) Sabda: A Study of Bhartrhari's Philosophy of Language by Tandra Patnaik. This is particularly scholarly with the author comparing western authors (like Frege and Wittgenstein) model of language with Bhartrhari - https://test.dkprintworld.com/product/sabda/
Excellent resources, thank you. Discussions of Sphotavāda are limited to the non-English Indosphere, largely, but the word itself is also used in high register Hindi to appreciate an unexpected but brilliant off-hand remark. Sort of like the English expression of calling something inspired; it both appreciates the speaker while taking away from their absolute agency.
It's very easy to give examples. Often when someone coins a word, they are giving a name to something that previously had no name. Therefore the thing is an example of something that did not have a name prior to the coining.
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.
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.
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.
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.
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.
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.
Zen koans don't have a meaning at the manifest (Vaikhari) and differentiated (Madhyama) stages. So you are forced to go back to the Intuitive/Holistic Meaning (Pashyanti) stage and thus realize "a burst of meaning" aka "a flash of insight" aka "Satori".
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?
> 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?
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.
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".
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."
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.
https://3quarksdaily.com/3quarksdaily/2014/03/boundaries-and... This is a beautiful article on the subject.
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.
Countable and Uncountable Infinities - https://philosophicaltreasures.com/countable-and-uncountable...
i think there must be a scheme of naming that is uncountably infinite too.
Bhartrihari was a king, philosopher, and poet. Scholars argue whether they were the same person or not, but I don't care, as scholars also argue about if Socrates really existed or not.
He wrote 300 verses in Sanskrit. And they are on three different topics: sensuality and pleasure, policy and ethics, and finally renunciation.
In Shringar Shatakam (100 verses on sensual pleasures), he writes:
“Casting aside envy, considering the matter carefully, let the noble ones tell us, with due propriety: Which ought one to frequent — the slopes of the mountains, or the buttocks of women whose smiles are stirred by Love?”
and
“Why all this elaborate, pointless talk? There are only two things worth attending to in this world: the fresh, wine-intoxicated youth of beautiful women, heavy with their breasts - or the forest.”
But in the final book, he realizes the folly of the senses, and writes:
“Sensual objects will inevitably leave us, even after remaining with us for a long time. What difference is there between losing them and voluntarily abandoning them? When they depart against our will, they cause unbearable anguish; but when we ourselves abandon them, they produce the infinite happiness of inner tranquility.”
Amazing character.
The third verse you quoted (https://shreevatsa.net/bhartrhari/web/K157.html) has a nice translation into English verse by Ryder:
The other two you quoted (https://shreevatsa.net/bhartrhari/web/K084.html https://shreevatsa.net/bhartrhari/web/K085.html) have also been translated, though IMO not as successfully.No one argues about whether Socrates really existed. He is attested by multiple writers of the time.
The debate is solely about how much the Socrates of Plato's dialogues represents the views of the historical figure.
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)
Nice.
I have the A.N.D.Haksar, Purohit Gopinath translations of all the Satakas and Swami Madhavananda's translation of the "Vairagya Shatakam". Need to get the others ;-)
Also see https://news.ycombinator.com/item?id=49486845
I couldn't add the Haksar translation as it's still under copyright, but the site has the other two. Plan to add others like M. R. Kale's (https://github.com/shreevatsa/bhartrhari/issues/11) — just need a chunk of time one of these weekends.
One thing you might want to add after each author's name is which categories (viz. Niti, Shringara and Vairagya) they have translated. Due to excessive prudishness many have omitted the Shringara satakam which is quite silly (this is what makes him "Human"). I got the Haksar and Gopinath editions specifically because they include all three categories. A.N.D.Haksar in particular has translated many of the works in Sanskrit literature into easy English (including the Kama Sutra) and does not censor anything.
Also i suggest that you add author names of all known translations of the work whether you have access to their actual text or not (due to copyright etc. reasons). That way your site can be a one-stop portal to Bhartrhari's Sataka-Trayam.
PS: In case you don't already know of it; there is a much larger work in the Tamil language named Tirukkural which is also divided into three similar categories (viz. Aram, Porul and Kamam) the whole having a total of 1330 couplets (133 chapters of 10 couplets each). There are many English translations available of which the original Penguin edition titled "Kural" translated by P.S.Sundaram is pretty good and done in the original couplet style. For a more detailed study see the 2-vol translation with commentary by S.M.Diaz.
Tirukkural - https://en.wikipedia.org/wiki/Kural
Tirukkural translations - https://en.wikipedia.org/wiki/Tirukkural_translations
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.
„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.
"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.
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.
What does "naming" mean?
Assigning a symbol? But who said that the set of symbols must be countable?
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
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.
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.
The Beauty of Bézier Curves
https://youtu.be/aVwxzDHniEw?si=tPPRId1y5L7U2_5L
Name_I = a*lambda + b (1-lambda)
lamda is a real number.
How is it a paradox? Isn't this just a proof that there are many unnamed things, but no unnameable ones?
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.
Agreed.
This is the stuff of magic and folklore, and neatly resolved by Ursula K. LeGuin in _A Wizard of Earthsea_.
See the original paper linked to here - https://news.ycombinator.com/item?id=49478566
> 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).
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.
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.
> 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.
> 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.)
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.
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.
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.
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).
This is basically why art exists — painting, music, and poetry can point at things without having to name them. Language traps itself; other forms don't.
What you say is valid only when you restrict yourself to Western definitions/conceptions of language/linguistics i.e. study of Phonetics/Morphemes/Syntax/Semantics/etc.
In Bhartrhari's Philosophy (and other Hindu philosophies) "Language" has a much broader definition which can encompass Art/Dance/Music/Painting/etc. Any medium of communication which can bring forth a "burst of meaning" (called Sphota) in one's consciousness is a language.
Natural Spoken language based on Sound (aka Sabda in Sanskrit) is considered the most fundamental since you can have languages without a written script/symbols/diagrams.
In Hindu philosophy, a "Language" is said to have four stages, only the last of which is the gross manifestation in the physical world;
1) Para - This is the latent undifferentiated potential which exists in everybody.
2) Pashyanti - This stage is where intuitive holistic meaning (of what you want to convey) exists.
3) Madhyama - This stage is where you have differentiated the thought/intention into an object and the means of representation for its communication.
4) Vaikhari - In spoken language, this is the manifest stage where you utter sentences according to established syntax/semantics to convey meaning.
Note that the first three stages are internal and only the last is the medium of expression in the physical world. It should now be obvious that the last can be any medium (eg. Dance/Painting/Music/Written-Language/Sign-Language/etc.) as long as the receiver "gets" the intended meaning.
PS: See also this comment of mine for further resources - https://news.ycombinator.com/item?id=49486845
The claim is false. There is not a thing that cannot be given a name. If it is a thing, it can be given a name.
You take that thing into the discourse and say, "let this thing be X", and now it has a name.
What cannot be given a name cannot be discussed in any way; it is completely vague, undefined or ephemeral.
The inability to name it is not what is at the core of not being able to pin it down; it is a byproduct or corollary.
You didn't name the object when you said "let this thing be X"; you actually had already identified that "thing", and that process of identification was the process of naming it. You then defined some syntax ("X") and said that it was a name.
But there are things for which you can't even say "let 'this thing' be…". For example, ZF proves that there are uncountably many reals. There are only countably many names, so there must be unnameable reals. You can talk about "generic" reals (you can say "let x be a real" and do all sorts of interesting things with a generic x), but there are specific reals you will never be able to name specifically enough to distinguish them from their uncountably-many brethren. That doesn't make them "vague, undefined or ephemeral"! They're just so numerous that you can't describe the distinctions between them.
(Even hardcore constructivists usually accept enough Choice to prove the reals uncountable, although https://arxiv.org/abs/2404.01256 made headlines when it was shown not to be necessarily true.)
> that process of identification was the process of naming it.
No, it wasn't. Entities can be identified without being named, by relationships to other entities and class and such. That identification requires words. Not all denotational words and phrases constitute names.
naming schemes can produce uncountably inf names
If you can't name it you can still describe it.
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.
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.
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).
The Herzbergers' paper: https://sci-hub.ru/10.1007/BF00202726
Thank You.
Bhartrhari (https://en.wikipedia.org/wiki/Bhart%E1%B9%9Bhari) is a pretty difficult philosopher who seems to be enjoying a revival now due to the ascendancy of AI LLMs and the question of whether they can be considered as having "consciousness".
His central idea (highly simplified) is that since Language is the only way we can name objects and discuss relations between them it is synonymous with "Reality" and "Consciousness". Sort of like how the properties of an object define that object (ADTs anyone?). One can imagine that the use of language by LLMs gives birth to appearance of both consciousness and reality as "emergent phenomena" in it. In his theory of "Sphota" he posits that "meaning bursts forth" (in consciousness) as an indivisible whole when a complete sentence/sentences is/are uttered (is this what happens when LLMs do reasoning and generate text within a "context window"?) Perhaps Epistemology and Ontology are just two sides of the same coin.
Some resources for further study;
1) Bhartṛhari’s Linguistic Idealism - https://loc.closertotruth.com/theory/bhart-hari-s-linguistic...
2) Bhartrhari on Language, Perception, and Consciousness - https://academic.oup.com/edited-volume/27982/chapter-abstrac...
3) The Word And The World: India's Contribution To The Study Of Language by Bimal Krishna Matilal - https://archive.org/details/wordandtheworldindiascontributio...
4) Sabda: A Study of Bhartrhari's Philosophy of Language by Tandra Patnaik. This is particularly scholarly with the author comparing western authors (like Frege and Wittgenstein) model of language with Bhartrhari - https://test.dkprintworld.com/product/sabda/
5) The Sphota Theory of Language by Harold Coward - https://www.mlbd.in/products/sphota-theory-of-language-harol...
6) Sphota - https://en.wikipedia.org/wiki/Spho%E1%B9%ADa
Excellent resources, thank you. Discussions of Sphotavāda are limited to the non-English Indosphere, largely, but the word itself is also used in high register Hindi to appreciate an unexpected but brilliant off-hand remark. Sort of like the English expression of calling something inspired; it both appreciates the speaker while taking away from their absolute agency.
I'm open to the idea that some things are unnameable but would need an example :)
I'll write you as soon as I can
sounds like the paradox that _could_ illustrate Gödel's incompleteness theorems
https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_th...
Russell's Paradox -> Hilbert's Program -> Godel's Theorems - https://news.ycombinator.com/item?id=49287595
Bhartrhari's Paradox can be said to be analogous to Russell's Paradox (though of course the latter is specific to mathematics/logic).
It's very easy to give examples. Often when someone coins a word, they are giving a name to something that previously had no name. Therefore the thing is an example of something that did not have a name prior to the coining.
c.f. the Berry paradox https://en.wikipedia.org/wiki/Berry_paradox
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.
(This is an older subthread which I moved hither because a different submission happened to make it onto the frontpage)
The proof is simple: there are countably many names, but uncountably many real numbers. Hence, some real numbers must be unnameable.
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.
names have to be finite in length. i think that's pretty obvious
I don't see how that's any more obvious than the suspicious claim that numbers can have only so many digits.
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.
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
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.
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.
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.
> This reminds me of how (I think) Zen koans are designed to make no sense at all.
True in a way.
To understand how, note first the four stages of language given in my comment here - https://news.ycombinator.com/item?id=49491764
Zen koans don't have a meaning at the manifest (Vaikhari) and differentiated (Madhyama) stages. So you are forced to go back to the Intuitive/Holistic Meaning (Pashyanti) stage and thus realize "a burst of meaning" aka "a flash of insight" aka "Satori".
counterargument:
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 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?
> 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?
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.
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!
It’s not a foolish question, you got to learn about Cantor’s diagonal argument!
guess I won't respond now :( "what is a real number, anyways" is one of my favorite questions, not foolish at all
Jeff jeff jeff, jeff jeff jeff jeff! Jeff? Jeff.
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
I see what you did there.
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."
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.
"The Tao that can be told is not the eternal Tao"
> There are some things that are unnameable
Like what?
Oh wait…