The Anthropic ETH algorithm apparently needs more than 10^(10^8) variables in order to be better than brute force search.
That means, being an Ultrafinitist pays off again, I simply do not believe this number is real, so it cannot hurt me.
@sophieschmieg I love it.
How do you ultrafinitists feel about the decimal expansion of irrational numbers?
Oct 28, 2026, 9:18 PM@demofox @sophieschmieg I assume there’s no such thing as an irrational number to an ultrafinitist, they’re all rational (if you can’t write the digits it’s not a number?)
@bnut @sophieschmieg I wonder if they consider the problem to be the decimal expansion? Because if you have a right triangle with side lengths of 1, you get sqrt(2) for the hypotneuse which is definitely a real existing thing.
@bnut @sophieschmieg the pythagorean cultists killed a guy over it even!
@demofox @bnut
One important thing (even in other philosophies of mathematics) is that the decimal expansion of a number isn't really the same as the number itself. It is merely an infinite series that converges to the number that we have built a convenient shorthand around, but just like 1/2 = 2/4 is the same number, with a different notation, we have things like 0.999… = 1, since 0.999… is just a shorthand for sum_{k=1}^inf 9*10^(-k), which converges to 1.
As such, you can reason about e.g. pi, without ever computing all its digits. For example, in university, we just proved that 3 < π < 4 and called it a day, it's good enough.
@sophieschmieg @bnut @demofox … she says, immediately before using infinity in a sum in another branch of the thread. 😉
Oct 28, 2026, 11:11 PM@sophieschmieg @demofox the big numbers part is very interesting to me. iirc a lot of definitions around infinities and cardinality use proofs by induction, but maybe you can’t do a proof by induction without infinity? What’s the last number that’s not too big?
@sophieschmieg @bnut @demofox the ultrafinitist opinion about computability is that the busy beaver would just get tired
Considering that the irrationals are dense in R, I'd assume they DO believe in them.
They kind of have to!
Now tetrations of high order, yeah those might as well not even be real.
Damn! Not even the reals. I guess the Borels are out too then. Damn. I need to look more into this philosophy. Naively, I just assumed everyone used whatever our texts required, but then again: axiom of choice, etc.
I can dig it.
@rythur @bnut @demofox
The whole thing emerged after Gödel incompleteness and due to the inconsistencies in set theory.
One way to avoid the problems Gödel poses is via Constructivism, where only things that can explicitly constructed are given reality, basically avoiding proof by contradiction.
The more rigorous way of doing Constructivism, at least in a finite world, is Finitism, where only finite objects are considered real.
Ultrafinitism is the even more extreme position where "finite, but arbitrarily large" is no longer considered a real object.
The real numbers as we usually understand them are out already at the first step, only computable numbers are constructible, and computable numbers are countable, so they are a proper subset of the reals.
The more popular philosophical reaction to Gödel was Formalism, also known as "shut up and calculate". There are ways of combining Platonism with Gödel as well, depending on how flexible one wants to be. The main philosophy that is completely incompatible with Gödel is Hilbert's Positivism "we can know, and we will know".
In a certain way (applied) cryptography as a branch of mathematics is somewhat inherently connected to Ultrafinitism, more so than many others. In the end, we do not believe that brute forcing a 256 bit number is possible, which is similar to saying that numbers larger than 2^256 do not exist as genuine, real, objects. (You can see them as symbols and manipulate those symbols, but you cannot count to them, so are they real?)
I have always loved Gödel's result. It still blows my.mind!
Thanks for this thorough explanation too. Very interesting. Essentially ole Gödel made so many formalizations in a single result. I'm apt to agree with some of these as measures of (mathematical) reality, but then there's the fact that I can essentially choose from an infinite number of positions between my nose and foot, and that's real.
To avoid paradoxes/contradictions makes sense though. ...
Is this not all, on the whole, contingent upon our technology?
I picture an altered (and very unnatural) brain capable of somehow "counting" beyond that. Picture handling 10k values in each "count"
Ugh, oh how great concepts breed great concepts.
Cheers and thanks again.
@rythur @sophieschmieg @bnut @demofox There are two problems with counting to 2^256 (~1.158*10^77): time and energy.
• Even if a computer could go through one number every Planck time (~10^-43 seconds), it would still take ~10^34 seconds to go through all of them. There are 3.154*10^7 seconds per year, so that impossibly fast computer would take about 3*10^26 years. The universe is believed to be ~1.38*10^10 years old.
• The energy required to go through all of the state changes in an ideal 256-bit register would be greater than the total expected lifetime output of the sun. I forget the math on just how much greater.
@bob_zim @sophieschmieg @bnut @demofox
Quite badass an approach to it's impossibility, but I mean something like "batches over time" or "X multicounting per batch" -- still, is this even counting. When we think of it, that's one, two, three; but it could be all three at once to some other being somewhere, or an artificial machine. Energy is still an issue here, unless we employ, as I've said elsewhere "cooling tech" whereby the tech works by essentially cooling itself. Cheers o the impossible!
Oct 29, 2026, 6:12 AM@bob_zim @rythur @sophieschmieg @demofox in case you’re interested, it’s believed that we can continue to produce entropy (do stuff) for another 10^100 years. So I guess we’d need 501 bits to count stuff sequentially (See https://en.wikipedia.org/wiki/Heat_death_of_the_universe#Time_frame_for_heat_death ). Although there’s only 3.28x10^80 particles in the currently observable universe, so who knows what we’re counting.
@bnut @bob_zim @sophieschmieg @demofox
Now you have me picturing a film in which the world/universe is saved against entropy, like Interstellar did with time. In general then, and in the absence of being saved in this way, we can only count so high, which of course goes back to the beauty of finitism.
So, our path to living forever has to be Climate > Time > Entropy.
I bet there's still much we don't understand at all about the exhaustive counts of things, like particles. Ugh!
Oct 29, 2026, 6:41 AM@rythur if you like this sort of thing you may like to read “the 3 body problem”
I've definitely read about the three-body, was working on coming up with a more complicated variant, involving negative energy balances, when I started noticing my house being broken into. Now, I'm targeted and homeless. Lol. Maybe someone didn't like the diagrams???
Either way, thanks!
The N-Body is just plain super-monkey stuff BTW.
Cheers!
Oct 29, 2026, 7:04 AM@rythur I’m sorry that happened. Just to clarify though, the three body problem I’m referring to is a fictional book that heavily features the math problem
Ohhhhh. Didn't know there was a book. I'll scope it out. But, yeah, coordinated harassment is terrible, those doing it have no soul. One will take pics of the car, the next will get in, and then Stasi damages. Real unfortunate times. But I'll scope the book!
Thanks :-)
@bnut With a count that large, obviously you have to recount all the particles a few times to be sure!
Oct 28, 2026, 9:53 PM@rythur @demofox @sophieschmieg depends on the basis of the philosophy, I guess? When you get down to plank lengths etc. everything is countable, and I suppose Reals may not exist? I have no idea how black hole singularities work there, but I assume with time dilation it only asymptotically approaches a point (and the plank length is still locally fixed for a specific reference plane).
So maybe we should just take black holes to be the philosopher walking to and from a door?
On the real, though, think, much like ship wrecks, just how many important real numbers we haven't discovered yet, like pi before pi was found.
This is an awesome thread!
Oct 28, 2026, 9:18 PM@sophieschmieg brute force search starting with 8-bit numbers always wins!
@sophieschmieg A one and a hundred million zeros IS a very big number .. 🤔
Does the life of the universe itself allow for that search algorithm to be useful? 🤔🤔
I feel this answer created more questions than it solved. 🤔🤔🤔
@gimulnautti you don't have enough atoms in the observable universe to write down that many variables, much less a non trivial circuit that mentions all of them.
@sophieschmieg that's not an algorithm. That's an idea for the shape of an algorithm.
In the Land of Random Nonsense, all AI Hype can appear.
Surely a smaller model generated this.
@sophieschmieg Are the LLMs cheating at O(notation) again, using the ever popular trick of hiding "n" in the O(1) operations? (This is a great flaw in the way O(notation) is generally defined and makes it somewhat useless for real computing)
@jripley it's just a large constant that's hidden in the Landau O.
@sophieschmieg i kinda wish the P=NP problem gets solved in a similar way (ideally not by “AI”): either someone finds an algorithm running in O(n^{a lot}) with a prohibitive additive constant for solving NP problems (officially fast, realistically slow), or conversely someone manages to solve NP problems in time O((1 + {something small})^n) (realistically fast, officially slow). both would render the question somewhat moot 🙃
@sophieschmieg i mean, SAT/ILP solvers already do some nontrivial magic around that, so really in some applications the question is already irrelevant i think…
@sophieschmieg big variables georg energy
@sophieschmieg remember that attacks always get better. It'll be down to 10^(10^7) in no time
@sophieschmieg this is the problem with only doing asymptotic big-O analysis.
Constants are real and they can hurt you :p
@sophieschmieg like there are plenty of real world cases where a simple linear search on a very small dataset is faster than something with sublinear performance that has a high setup overhead
@sophieschmieg ditto for matrix multiplication, the simple O(n^3) algorithm is still better for smaller matrices than the newer ones with lower asymptotic complexity
Oct 29, 2026, 6:28 AM@sophieschmieg Wait okay there ARE other ultrafinitists!
Do you have, like, names or links to people explaining (a) their ultrafinitism and what it means to them and (b) how it impacts how they do/interact with mathematics?
Like, I listened through a bunch of Wildberger's videos, which... do give HIS answer to those, but he is, like, a full-on conspiracy theorist and so I don't necessarily expect him to represent the non-conspiracy-theorist ultrafinistist positions.
@gaditb I would consider myself somewhere between a formalist and ultrafinitist. To me the main thing about ultrafinitism is that at a certain size, numbers stop making sense, as a "quantity is a quality of its own" thing. It also jells well with the type of Wittgenstein inspired philosophy of reality being best understood through socially influenced models that can only ever approximate, but never reach a true, independent reality, to the point of denying the existence of such a reality as being more real in the first place.
Oct 29, 2026, 6:04 PM@sophieschmieg Does it influence how you, like, /do/ mathematics practically, or just (I assume this at least) how you approach mathematics emotionally/philosophically?
Like, constructivism has a practical impact on what moves are legitimate in mathematical proof and discovery.
And Wildberger's take on Ultrafinitism drove him away from using real or even algebraic numbers at all, building up his own parallel, differently-behaving, constructions of geometry and functions (e.g. his concept of "circles" and "lines", he will say that a line with a point inside a circle and a point outside a circle will intersect the circle at 2, 1, or 0 pointa) -- but, his is a conspiratorial take on it, so maybe a /non-/conspiratorial take wouldn't require any changes to practice.
Oct 29, 2026, 6:05 PM@sophieschmieg (e.g. "if we reach some object and describe some properties with a proof, then THAT object has a formal existence in THAT respect, sure, but any concrete consequences like "has a digit there" or "allows this computation" require their own arguments about reachability as concrete outcomes, and don't come inherently through implication from the more-abstract level."
Which I don't... /think/ would make doing-the-math-itself (which is.. always a concrete act in its own right? right?) still work the same, and it's just the reaction/outlook on that math that changes? But also I don't actually know anything this is just guesses.)
@gaditb no it doesn't really change anything about how I do mathematics, which is why I call myself at least partially formalist. I do prefer explicit constructions, and certainly when it comes to algorithms, don't quite see the point of algorithms that only exist by contradiction, or require impossibly large numbers.
The most complicated situation is almost feasible algorithms, which are a risk factor for cryptography, but don't quite work without further optimization.
@sophieschmieg aren't all numbers just imaginary "objects"? Isn't infinity just another imaginary object?
Oct 29, 2026, 7:13 PM@sophieschmieg Oh, you like ultrafinistism? Name the 10 most finite numbers.