Mathematician but not philosopher here. Great to see this fascinating result appear in my inbox! Infinities are wonderful!
There is big leap being made here:
"Consequently to there being dark numbers, many, perhaps in some sense, most, ways the world can be (eg X being some dark number of meters away from Y)"
Leap being assumption that the world is based on real numbers and that distances of real-precision physically exist. Eg there is notion of plank length and seconds. Or universe might be based on rational numbers. Or it could be discrete computations (see wolframs theory of everything project)
In fact, on our current understanding of physics, the universe must only contain a finite number of particles, and as you say, they can only be in finitely many positions
Lovkush and John may well be right about the actual world, which could include a finite cap on possibilities, or might allow continuous possibilities, but only rational ones- but I simply want to say that the space of *possibilities* includes that the distance between X & Y is a dark number.
This also shows that a possible world is not just a maximally consistent set of sentences because there are aleph null possible sentences and beth 2 possible worlds.
Does this mean that the definition of a possible world as a maximally consistent set of sentences is false? Because there are at least Beth 2 possible worlds, as Lewis showed, and only aleph null sentences.
I'm not sure. I find there are many things I can express better in its original language that English simply cannot properly. Maybe it's only me. By=ut when I tried reading Sein und Zeit in English, it was hopeless, the same word "Being" , or maybe to be, or oit could have been a really bad translation, who knows. But I had to read it in German to even begin to understand. But God knows I can't translate from one language to another because the meaning just doesn't mean what it meant. So if I read another language I can't read it and try to translate it into my head. I remember in college I was making straight A's in Greek and the teacher said I was the best student he'd ever had. Then in third year he through me a loop. The tests had been questions in Greek and we wrote answers in Greek, the one day the questions were in English and we had to reply in Greek and I was just absolutely stuck. The professor was really disappointed in me.
The wikipedia page for definable real numbers (https://en.wikipedia.org/wiki/Definable_real_number) mentions that there are different notions of definability, but for any formal language almost all real numbers will be undefinable. But a "dark number" in the sense of a specific number that can't be defined in any language ... well, that doesn't seem quite right without some additional specification, because a language could cheat by giving it a name. The name wouldn't tell us anything interesting about the number though.
To be able to name the number meaningfully, we have to be able to grasp it in some sense in language. Now it could certainly be that we might point to it, for example "The distance between A & B." but we would never know that this distance unknown number.
I parsed the Socratic assertion about knowing nothing thusly: what I know is finite, what there is to know is infinite: the ratio is zero. Thanks for supporting the second axiom, though it may not be true: Planck, you know? Nevertheless, big number, right?
English isn't a formal language, which is why paradoxes like Berry's exist. Consequently, I can talk in English about any dark distance between X and Y by simply referring to the distance between X and Y.
I think what is missing from this analysis is that sentences in languages such as English are not self-contained "expressions of worlds" (all by themselves). Rather, they are actions in the world. Think of a sentence as being similar to pointing with your body. There is an inexhaustible set of meanings to what it could mean for you to make the "identical" bodily movement of pointing. To list just a few random examples, you could be telling someone to appreciate the beauty of the moon, you could be telling someone which pastry you want to order, you could be warning someone about something behind them, you could be teaching someone something from a diagram, etc. Likewise, the meaning of any single sentence depends very much so on its linguistic and extralinguistic context.
Certainly I don't claim that English is capable of all possible artistic manoeuvres, just all propositional manoeuvres. The space of the aesthetic use of language is necessarily larger purely semantic concerns, and includes it!
Mathematician but not philosopher here. Great to see this fascinating result appear in my inbox! Infinities are wonderful!
There is big leap being made here:
"Consequently to there being dark numbers, many, perhaps in some sense, most, ways the world can be (eg X being some dark number of meters away from Y)"
Leap being assumption that the world is based on real numbers and that distances of real-precision physically exist. Eg there is notion of plank length and seconds. Or universe might be based on rational numbers. Or it could be discrete computations (see wolframs theory of everything project)
In fact, on our current understanding of physics, the universe must only contain a finite number of particles, and as you say, they can only be in finitely many positions
Lovkush and John may well be right about the actual world, which could include a finite cap on possibilities, or might allow continuous possibilities, but only rational ones- but I simply want to say that the space of *possibilities* includes that the distance between X & Y is a dark number.
No. For all we know the universe is infinite and could contain an infinity of quantum states. Perhaps you mean within our light cone.
This also shows that a possible world is not just a maximally consistent set of sentences because there are aleph null possible sentences and beth 2 possible worlds.
Does this mean that the definition of a possible world as a maximally consistent set of sentences is false? Because there are at least Beth 2 possible worlds, as Lewis showed, and only aleph null sentences.
I'm not sure. I find there are many things I can express better in its original language that English simply cannot properly. Maybe it's only me. By=ut when I tried reading Sein und Zeit in English, it was hopeless, the same word "Being" , or maybe to be, or oit could have been a really bad translation, who knows. But I had to read it in German to even begin to understand. But God knows I can't translate from one language to another because the meaning just doesn't mean what it meant. So if I read another language I can't read it and try to translate it into my head. I remember in college I was making straight A's in Greek and the teacher said I was the best student he'd ever had. Then in third year he through me a loop. The tests had been questions in Greek and we wrote answers in Greek, the one day the questions were in English and we had to reply in Greek and I was just absolutely stuck. The professor was really disappointed in me.
The wikipedia page for definable real numbers (https://en.wikipedia.org/wiki/Definable_real_number) mentions that there are different notions of definability, but for any formal language almost all real numbers will be undefinable. But a "dark number" in the sense of a specific number that can't be defined in any language ... well, that doesn't seem quite right without some additional specification, because a language could cheat by giving it a name. The name wouldn't tell us anything interesting about the number though.
To be able to name the number meaningfully, we have to be able to grasp it in some sense in language. Now it could certainly be that we might point to it, for example "The distance between A & B." but we would never know that this distance unknown number.
I parsed the Socratic assertion about knowing nothing thusly: what I know is finite, what there is to know is infinite: the ratio is zero. Thanks for supporting the second axiom, though it may not be true: Planck, you know? Nevertheless, big number, right?
The example I use is a relatively small number. There are roughly 7000 languages. I am fluent in, on a good day, one.
What is the difference between unfathomable and infinite?
English isn't a formal language, which is why paradoxes like Berry's exist. Consequently, I can talk in English about any dark distance between X and Y by simply referring to the distance between X and Y.
Thanks, I've added a treatment of these issues at the end.
I think what is missing from this analysis is that sentences in languages such as English are not self-contained "expressions of worlds" (all by themselves). Rather, they are actions in the world. Think of a sentence as being similar to pointing with your body. There is an inexhaustible set of meanings to what it could mean for you to make the "identical" bodily movement of pointing. To list just a few random examples, you could be telling someone to appreciate the beauty of the moon, you could be telling someone which pastry you want to order, you could be warning someone about something behind them, you could be teaching someone something from a diagram, etc. Likewise, the meaning of any single sentence depends very much so on its linguistic and extralinguistic context.
Thanks, I've added a treatment of these issues at the end.
This Borges fellow thinks otherwise:
https://www.reddit.com/r/Spanish/comments/13cg3xv/jorge_luis_borges_speaking_about_key_differences/
Certainly I don't claim that English is capable of all possible artistic manoeuvres, just all propositional manoeuvres. The space of the aesthetic use of language is necessarily larger purely semantic concerns, and includes it!
Do you mean to say, all languages have *identical* capability in this regard?
And then on top of it, there's the many problems in all languages, and how we use them in ways FAR below their capabilities.