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?)
- Oct 28, 2026, 9:18 PM
Replies
demofoxdemofox@mastodon.gamedev.placeOct 28, 2026, 9:30 PM@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.
- demofoxdemofox@mastodon.gamedev.placeOct 28, 2026, 9:31 PM
@bnut @sophieschmieg the pythagorean cultists killed a guy over it even!
Sophie Schmiegsophieschmieg@infosec.exchangeOct 28, 2026, 9:41 PM@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.
- Sophie Schmiegsophieschmieg@infosec.exchangeOct 28, 2026, 9:35 PM
Zimmiebob_zim@infosec.exchangeOct 28, 2026, 10:48 PM@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?
Richard Barrell0x2ba22e11@unstable.systemsOct 30, 2026, 1:09 PM@sophieschmieg @bnut @demofox the ultrafinitist opinion about computability is that the busy beaver would just get tired
Rythurrythur@mastodon.socialOct 28, 2026, 9:45 PMConsidering 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.
- Sophie Schmiegsophieschmieg@infosec.exchangeOct 28, 2026, 9:49 PM
- Rythurrythur@mastodon.socialOct 28, 2026, 9:56 PM
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.
- Sophie Schmiegsophieschmieg@infosec.exchangeOct 28, 2026, 10:08 PM
@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?)
- Rythurrythur@mastodon.socialOct 28, 2026, 10:28 PM
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. ...
- Rythurrythur@mastodon.socialOct 28, 2026, 10:30 PM
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.
- Zimmiebob_zim@infosec.exchangeOct 29, 2026, 3:05 AM
@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.
- Rythurrythur@mastodon.socialOct 29, 2026, 6:00 AM
@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.
- Rythurrythur@mastodon.socialOct 29, 2026, 6:20 AM
@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”
- Rythurrythur@mastodon.socialOct 29, 2026, 6:49 AM
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
- Rythurrythur@mastodon.socialOct 29, 2026, 11:12 AM
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 :-)
- Zimmiebob_zim@infosec.exchangeOct 30, 2026, 1:37 PM
@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).
- Rythurrythur@mastodon.socialOct 28, 2026, 9:59 PM
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!