[HN Gopher] Digital Infinity ___________________________________________________________________ Digital Infinity Author : drdee Score : 30 points Date : 2023-03-10 03:29 UTC (19 hours ago) (HTM) web link (en.wikipedia.org) (TXT) w3m dump (en.wikipedia.org) | hgsgm wrote: | The atomic/elemental model of matter is also a digital infinity, | and it existed (unknown to humans, but existed) before language. | edgyquant wrote: | This doesn't make any sense. How could a model used by humans | for predictions exist before human language? | ben_w wrote: | What is human language? Where is the boundary before which | it's primate vocalisations and displays? Did those distant | ancestors not model their environment and predict things? | dofnskd wrote: | I think what is being alluded to here is the "particulate | principle", in other words the fact that atoms combine to | produce a compound with emergent properties, rather than | simply a blend of the properties of the constituents. https:/ | /www.sciencedirect.com/science/article/abs/pii/014017... | | In chemistry, the distinction is often described as being | between _additive_ and _emergent_ properties. | | The resulting range of possible chemical compounds is not | quite an infinity, perhaps, but the particulate principle is | clearly in the same category as the "digital infinity" | concept. | originalcopying wrote: | I had this idea but I did not know about this portrayal of it | | that there are two ways to 'have' an infinity: | | - by a lack of something. the classic original infinity. there is | not a biggest number, they just keep going. it's a 'negative' | definition; infinity because NOT finite. | | - by construction. intuitionist or constructivist infinity (?). | like a cycle or a going-back and forth never stopping. or with a | self-referential _next-state_ arrow. | | but I'm a bong smoking graduate student. as to the connection | between all this and intutionism and/or constructivism? I really | wish I knew or where in a position where I can discuss this with | people; however I also think that internet randos like me need to | await for whiter, wealthier, and more european academics from | truly prestigious universities to decide what's what. Which does | get in the way of getting myself into a position where I can | understand this. | qsort wrote: | You're probably trying to say something but I don't understand | what. | | The way you 'have' infinity in math is by postulating there | exists an inductive set (axiom of infinity in both ZF and NBG), | and constructing other infinite sets using that as a building | block. | | Your first point is a definition of an infinite set (there are | a bunch of equivalent ones), your second point is a statement | of the axiom of infinity I assume? | opnitro wrote: | For natural numbers you can form that first one in terms of the | second one. Natural number can be through of as an inductive | construction. | | Either: | | - Zero | | - 1+ (another natural number) | 0x69420 wrote: | even if going by chomsky's choice of qualifier -- "biological | system" -- DNA still fits the bill for an abstraction as broad as | "digital infinity" | | pointing to it as a profound and unique aspect of language feels | like something a college student could accomplish after | misreading some wittgenstein, ripping the bong, looking up at the | stars, and going ~whoa dude~ | | don't get me wrong, there's no shame in seeing the concept of | discrete signals having some ~corporeal form~ as profound, but | you should be open to seeing more instances of your abstraction, | even if that ceases to give you a pretext to imbue your chosen | field (say, lingustics) with some vaguely spiritual aesthetic | meroes wrote: | Ya, if you want to see this cashed out here's Stuart Kauffman | https://www.npr.org/sections/13.7/2011/04/04/135113346/there... | | "IS THERE A FINITE PRESTATABLE SET OF BIOLOGICAL "FUNCTIONS?" | That is, is there a finite prestatable list of features of | organisms that MIGHT serve a selective function in some | selective environment? | | I think the deep answer is NO. | | The unexpected uses of features of organisms, or technologies, | are precisely what happens in the evolution of the biosphere | and econosphere, and the analog happens in cultural evolution | with the uses of mores, cultural forms, regulations, | traditions, in novel ways. In general, these possibles are | novel functionalities, in an unbounded space of | functionalities, and so are not mathematizable and derivable | from a finite set of axioms. " | | I think Kauffman is not against finite origins, but that they | can capture infinity afforded by them is the impossible task. | | What's misunderstood about Wittgenstein though? Didn't he say | the uses of language are limitless? I think he also would add | in some form for finite starting point. | dimatura wrote: | Interesting article touching on the meaning of the "digital" in | digital computers, and beyond that, how digital computation | relates to thought (and even, what is "computation"?). My | personal guess is that the Church-Turing hypothesis is true, and | that digital computers are enough for AGI (not anytime soon), | although I'm aware of various thinkers positing that digital | computation is insufficient for AGI and/or consciousness and some | kind of analog "computing" is needed. | feoren wrote: | > I'm aware of various thinkers positing that digital | computation is insufficient for AGI and/or consciousness | | Those thinkers are being silly. They're basically saying that | consciousness is literally magic. Totally agree that digital | computers are absolutely capable of consciousness, and also | that we're nowhere near that yet, despite the ChatGPT hype. | causi wrote: | _the use of finite means to express an unlimited array of | thoughts_ | | This is statement so ridiculous it verges on not even being | worthy of being called unfalsifiable. It's plain old personal | incredulity that because Chomsky cannot comprehend there are | limits to human thought, human thought must be infinitely | various, and because he uses language to describe thought, | language must also be infinitely various. Thought is not | infinitely various and language is even less so. This is | blatantly obvious from even every day experience. How could | language express an unlimited array of thoughts when language | cannot even fully express the experience of eating a chicken | nugget? Can language express all your thoughts when looking at a | sunrise? When embracing your lover after a hard day? Language is | the best tool we have but that doesn't stop it from being an | _extremely_ limited form of communication. | popctrl wrote: | The limit in your examples is more time than language. Ample | language could exist to describe a sunset. Language is | extremely flexible and good poets constantly find ways to make | words describe deeper and deeper concepts. The problem is that | you could spend a hundred years expressing your thoughts of a | sunset and only scratch the surface. | | Your argument seems similar to saying that infinite numbers are | "so ridiculous it verges on not even being worthy of being | called unfalsifiable" because you can't count that high. | humanistbot wrote: | "Unlimited" does not mean "all" in this context, it means | something more like "inexhaustible" or "infinite." And some | infinities can be larger than others: the set of all real | numbers is larger than the set of all integers, even though | both are infinite in size. | | Of course language is limited in terms of its ability to | represent reality, Chomsky doesn't deny that. All ways of | representing reality are limited, that's pretty much axiomatic | in whatever definition of "representation" you mean. | opnitro wrote: | In fact Chomsky is repeatedly on the record of believing | there are computational limits to humans. | qsort wrote: | This isn't even consistent. You can't disprove that language | can express infinitely many thoughts by showing a thought | language can't express. Likewise, the fact you can show that | there exist functions a universal Turing machine can't compute | doesn't disprove that a universal Turing machine can compute | infinitely many functions. | | You're also just plainly wrong, e.g. because recursion exists. | cecilpl2 wrote: | There are an infinite number of real numbers, and yet "blue" is | not a real number. | | A set can be infinite and yet not contain all things. | ben_w wrote: | > How could language express an unlimited array of thoughts | when language cannot even fully express the experience of | eating a chicken nugget? Can language express all your thoughts | when looking at a sunrise? When embracing your lover after a | hard day? | | I think your attempts at counterexamples are bad. | | None of those experiences (modulo being vegetarian so no | chicken) seem to me to be hard to express in language. Slow, | perhaps, but not hard. | | _However_. | | I believe that words are mere references to experiences, and | without shared experience the meaning of any given word will | generally differ somewhat between any two minds, and therefore | while my words can model my experiences I can be sure that | those same words will not create in your mind more than a | merely similar experience, not even if you can visualise all | the same senses, which you may not: if you have aphantasia, me | saying "red" will never convey red in quite the same way, and I | assume all other senses have equivalents though I do not know | their names. | OliverJones wrote: | Dredging up my old college math. A mathematician named Kantor | proved that the number of rational numbers (fractions) of postive | integers is the same as the number of positive integers. His | proof involves COUNTING the fractions. And that kind of infinity | is called, well, countable or aleph sub(0). It's like O(n) in | algorithms. And in that world O(polynomial) and O(n) are both | countable. | | But our favorite transcendental numbers, you know them, pi, e, | psi, that lot, are not part of that. Neither are multiples, or | fractions, of those numbers. There's an uncountable infinity as | well, holding them, and it's strictly larger than countable | infinity. Maybe that what Walt Whitman was thinking when he wrote | "I contain multitudes"? | | At any rate, possible physical distances are uncountable. Yup. | There's more of them than there are of integers. And living | things with brains have a (probably) countable number of neuronal | interconnections, each of which depends on uncountable physical | distances. | | (We know this in the computer industry: we have all sorts of | hardware and software that quantizes the physical stuff going on | in chips and conductors to extract bits -- to make the | uncountable countable.) | | My question: is this digital infinity countably infinite? Or does | it go beyond that? | | Do people who model -- information-theorically -- living brains | and the minds they hold consider this issue? Does this | countability matter to our understanding? | A_D_E_P_T wrote: | Physical distances are "uncountable" only if physical space is | infinitely divisible. If there's no continuum, and if reality | is granular -- even at a resolution well below the Planck | Length -- then all physical distances in space are countable. | | Digital infinity is by definition countable. There's no reason | to assume that anything in our universe is actually uncountable | -- as far as we know, it can all be simulated mathematically | without invoking Cantor's hierarchies. | | This isn't necessarily a finitist position. It's just to say | that the uncountable infinities don't necessarily interact with | any known universe -- digital or otherwise. | emmelaich wrote: | I suspect it's not infinitely divisible. | | My stupid argument is to ask whether you can be say pi metres | away from something else. You'd think so because as you move | somewhere between 3.142 and 3.143 metres away from something, | you'd pass pi and therefore land right on it. | | But how do you find where to stop at this transcendental | position? Having granular space would solve this because | there would be no such position. | atleastoptimal wrote: | I'd wager it's uncountably infinite. | | Here's my very vague justification. Something countably | infinite proceeds towards infinity in one direction. Let's say | we were at a store containing an infinite number of grocery | items, there would be an infinite number of words signifying, | so in a language which could only be the expression of listing | items in that store, it would be countably infinite. | | The thing about real languages though is that there is an | infinite number of possible interrelations between any two | words based on context. This is similar to the uncountable | infinite of the real numbers, in which any two rational numbers | have an infinite number of real numbers between them. | feoren wrote: | > A mathematician named Kantor | | Cantor. Georg Cantor. | | > And that kind of infinity is called, well, countable or aleph | sub(0). It's like O(n) in algorithms. | | It's not really at all connected to O(n), and only tenuously | connected to Big-O notation at all. Big-O notation works over | integers or reals, or even some other (possibly finite) sets. | It doesn't make sense to say O(n) is countable any more than it | makes sense to say that the line "y = 2x + 7" is countable. | What would that mean? Especially if x and y are real numbers? | | > our favorite transcendental numbers, you know them, pi, e, | psi, that lot, are not part of that. Neither are multiples, or | fractions, of those numbers. | | True for pi and e, but what is psi? Do you mean phi, the golden | ratio? Or do you really mean psi, the sum of the reciprocals of | the Fibonacci numbers (I had to look this one up)? The golden | ratio (phi) is not transcendental: phi * (1 - sqrt(5)) is -2. | It doesn't look like it's known whether psi is transcendental | or not. | | > At any rate, possible physical distances are uncountable. | | There's no particular reason to believe this is true, and some | reason to believe it's not. Look up the "Planck length"; below | this length it's not clear whether the concept of "distance" is | even meaningful. | | > And living things with brains have a (probably) countable | number of neuronal interconnections ... | | Not just countable neuronal interconnections: literally finite. | Neurons have finite size and your brain isn't infinitely large | (sorry). | | > ... each of which depends on uncountable physical distances. | | Pseudoscientific mumbo jumbo. Not even wrong. Literal nonsense. | | > My question: is this digital infinity countably infinite? Or | does it go beyond that? | | It is countably infinite by definition. It's isomorphic to the | free monoid over the (finite) digits. | | > Does this countability matter to our understanding? | | No. Uncountability is a curious feature of our model of real | numbers. All models are wrong, but some models are useful. | There's no real evidence that the uncountability of reals is an | actual useful feature of that model, and not just a curious | edge-case artifact. Most likely there is no physical analogue | to uncountably infinite sets (my opinion, obviously). | | Am I nitpicking you? Details matter. You seem pretty careless | with your facts here, which is a great way to accidentally | spread disinformation. Maybe try to be more careful in the | future. ___________________________________________________________________ (page generated 2023-03-10 23:01 UTC)