[HN Gopher] Digital Infinity
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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.
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