Digital Infinity
en.wikipedia.org
en.wikipedia.org
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
"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.
Unless you consider category or set theory "real".
I forget whether it was on Complexity podcast from SFI, or on Prof. Sean Carroll's Mindscape podcast, but this insight can be attributed to Dr. Sara Imari Walker.
In principle it is also true for atoms, or even just atomic nuclei, being composed from a finite set of elementary particles, but they decay too quickly beyond a certain size.
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.
Either:
- Zero
- 1+ (another natural number)
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?
Those thinkers are being silly. They're basically saying that consciousness is literally magic. Humans are seemingly desperate to feel like they have a special unique place in the universe, but we just don't. Sorry.
Totally agree that digital computers are absolutely capable of consciousness, and also that we're nowhere near that yet, despite the ChatGPT hype.
"...[the strong CT thesis] is widely believed to be wrong, primarily because quantum computation stands as a highly likely counterexample to the thesis." [0]
If consciousness is even a little bit quantum, then those thinkers aren't so silly after all.
If we humans never existed, what we refer to as the universe, our very shared experience and interpretation of it may not have even happened?
So I think when people say things like "oh we're not special", you might actually receive push back for other reasons besides the fact people have a large ego. Maybe it's because it seems like a kind of unwise suggestion given how little we know about actual experience and what it means to conscious.
Let's say a computer, or quantum computer, or whatever became conscious, would it also not be special? I mean it's a totally subjective statement to make?
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?
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.
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.
The Planck length is ~10^-35m.
The diameter of the electron is not more than 10^-22m. It may be as small as 10^-88m. (The Planck length sets limits on our ability to measure at such resolutions, but doesn't imply that nothing smaller can exist.)
If "bedrock reality" is granular, it may have an ultimate resolution of roughly 10^-90 to 10^-100m. This is small, but it's very, very far from infinitely small.
Now imagine a circle. In a universe that is not infinitely divisible, there's no such thing as a perfect circle. A circle of any sort will always have "pixels" at, say, a resolution of 10^-95m. Thus we dispense with pi.
And, in any case, you can't divide physical reality past that point -- so if you take a needle with a tip that measures 10^-95m, and you move that tip around a coordinate space, it will always be in a definite location that can be represented as some form of countable number.
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.
This is a mental trap when thinking about these things. In fact the rationals are also countably infinite (a/b, with a and b integers), as are the "complex integers", that is, the set (a + bi) with a and b integers, and i = sqrt(-1). The "2-dimensional" set of tuples (x, y) with x and y integers is countably infinite; in fact any n-dimensional set of tuples of integers is also, for finite n. Many more things are countable than most people realize.
> The thing about real languages though is that there is an infinite number of possible interrelations between any two words based on context.
The set of all possible books is countably infinite. Think about that for a bit. If your "context" can be encoded in any number of millions of books, encyclopedias, and dictionaries, then I can simply append that context onto whatever else I'm saying, and now I have my text plus all context.
Even if you say "but books can't capture the subtle tonalities and facial expressions of human expression", you still have to realize that we have finite photorecptors in our eyes, and finite hairs in our ears, and finite neurons to process all that information. So the set of signals a human could ever possibly process as distinct must be (at most) countably infinite as well.
You cannot get uncountably infinite language without infinitely large brains.
I know we want to all feel like we're magic and special, and this mysterious uncountable infinity feels like it leaves lots of room for us to have magical consciousness and a soul and an afterlife and X-Men superpowers, but it just isn't there. It just doesn't work. Besides, don't underestimate the size of countable infinity. That's not exactly something to get claustrophobic about.
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.
Uncountability starts at the uncomputable reals: these are numbers that only exist as infinite collections of digits and don't have a more concise definition. This includes things like Chaitin's Constant, which is the probability that a randomly-generated Turing Machine will halt. The Cantor diagonalization argument also worked with uncomputable reals - again, infinite sequences of digits. Computable reals are countable.
Physical distances are not uncountable. At the very least we know space cannot be infinitely subdivided: resolving any distance shorter than a Planck length creates black holes. So that would limit us to countable infinities.
The digital infinity is likely not even infinite; just "way too big a state space for humanity to ever feasibly exhaust absent concerted effort to do so". Though, if you want a story that plays around with this, check out Melancholy Elephants[0].
Shannon entropy and the concept of the 'bit' mean it's a measurable, finite amount.
We can put an actual number on the amount of information that has been expressed in every book ever written. Or on the ideas being exchanged every minute on the internet. Or on the rate at which humans produce new utterances.
It's big, but it's absolutely finite.
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.
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.
You're also just plainly wrong, e.g. because recursion exists.
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.
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.
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.
A set can be infinite and yet not contain all things.