#ultrafinitism

[reptilemodernism]

The largest number is twenty-eight million two hundred and seventy-eight thousand four hundred and sixty six.

[anocana]

learned yesterday (from a comment by john baez on an old post about ultrafinitism) that, a few years before the "conceptual art" movement existed as such, the fluxus-associated philosopher/artist henry flynt coined the term "concept art" to refer to a much more radically ambitious project: instead of creating works in in traditional artistic media whose concept is the main work and whose execution is a mere formality, his goal was to create art entirely within the realm of concepts.

he attacks what he calls "structure music" (john cage, schoenberg, etc) as failing on the grounds of music so you can only appreciate it on the level of structure, and simultaneously having a "limited, impoverished" structure; he appreciates the beauty of mathematical concepts but vehemently rejects the idea of mathematics as a search for truth "discovering" theorems and proofs. concept art is to generalize both of these (which he conceptualizes as small, limited subgenres of concept art), creating conceptual structures purely for their beauty without any question of whether they're true or whether they can be incarnated in another artistic medium.

anyway this obliquely convinced me to finally try out audio programming, specifically fundsp in rust which looks cool for very customizable synth stuff. i'm not gonna do real "concept art" but i've been meaning to get into audio programming for a while so now i'm going to try to see if i can make structure music that passes his second critique. current goal is to make a programmatic translation of dependently typed lambda calculus into sound and make an audio piece out of girard's paradox, which is at least an interesting structure. various unresolved questions including how best to represent a term with no normal form, whether it's worth trying to translate the entire calculus into sound or just approach the paradox specifically, and whether i'll stick with this for more than a few days lol. but even if i drop the paradox i want to keep on with the audio programming

[cecilias-cool-stuff]

https://en.m.wikipedia.org/wiki/Ultrafinitism

Relevant

A finite number so large it cannot be semioticslly represented

[heirloomrug]

Dancing shadows and Heirloom rugs adding some warmth and texture to a recent @BlankForms event. Even weeks after hearing the powerful and transcendent sounds of Catherine Christer Hennix, we're still vibrating and feeling lifted 😌 #ihavethisthingwithrugs #blankforms #catherinechristerhennix #Ultrafinitism #nycarts transcendentaounds (at Brooklyn, New York) https://www.instagram.com/p/CgxEoMBLqWc/?igshid=NGJjMDIxMWI=

[thetransintransgenic]

Happy Mutherflipping Pi Day, kinderlech~!

[max1461]

I implicitly answered it when I said "somewhere difficult to place between maxminner and utilitarian". I should have said minmaxxer but I was thinking about it backwards in my head.

Like, consider that we're in abstract thought experiment land, where nothing has any second-order effects, and there are N + 1 people. I am given the choice between two outcomes (call this "Scenario A"):

Outcome A1: N people receive 0 utility and one person receives -q utility for extremely large q.

Outcome A2: N people receive -p utility, for very small p, such that Np >> q (since N is very large), and one person receives 0 utility.

Given this choice, a utilitarian will choose outcome A1, since net utility is higher. An "ethical minmaxxer", who seeks to maximize the minimum utility experienced by anyone, would choose A2, since the floor of all the utilities is higher. An ethical maxmaxxer, who seeks to maximize the maximum utility experienced by anyone, would be undecided, since it's zero in both cases.

Now consider this, Scenario B (this is the one analogous to your question), in which the setup is the same but my two choices are:

Outcome B1: N people receive -p utility, and one person receives q utility, with p very small, q very large, and q << Np (again because N is large).

Outcome B2: All N + 1 people receive 0 utility.

A pure utilitarian chooses B2 because the sum is greater. A maxmaxxer chooses B1 because the maximum is greater. A minmaxxer also chooses B2, because the minimum is greater.

None of these positions are mathematically incoherent, or require ultrafinitism. Both utilitarian and minmaxxing intuitions seem quite common to me, and I am often pulled in both directions by my own intuition. The maxmaxxing intuition seems rarer, but it seems to me that it's expressed in certain Nietzschean-adjacent positions (any amount of suffering among the masses is justified to allow the Great to be Great).

I am neither a utilitarian, a minmaxxer nor a maxmaxxer, although I have in certain circumstances some intuitions of each type. I don't have a clear synthesis of these intuitions into a fully general paradigm, but I don't think there are any purely mathematical failings in this fact.

On the diffuse harms point, if you had a button that would take $0.01 from every person in the US and give $2M to one random person in the US, would you press it? how many times?

In abstract thought experiment land I think it would very probably be moral to do this; if it wouldn't, that has more to do with how the million dollars affects the recipient than anything else. Change dollars to utility (or whatever) and I feel pretty confident in saying it would be moral. I am a strong "mildly inconvenience 3^^^3 people in order to save one life"-er. I think the problem with this thought experiment is that in the real world, mildly inconveniencing very large numbers of people has second-order effects that are actually worse than mildly inconvenient, like that thing about how a certain number of people probably die due to the economic inefficiencies caused by the TSA. But if I was a wizard and I could arrange that 3^^^3 people be mildly inconvenienced in a truly second-order-effect-free way in exchange for saving one life, yes I would definitely do it. Would I arrange that 3^^^3 be tortured to save one life? I don't think so, even if the torture was less bad than death. In fact, I probably wouldn't arrange for even 50 people to be tortured to save one life, again even if the torture was less bad than death. I'm not a utilitarian but I'm also not an... ethical maxminner, or whatever. I'm at some difficult to place position in between.

In the real world, I think taking $0.01 from every person in the US and giving $2M to one random person would almost certainly be bad, for various implementation reasons if nothing else.

[king-of-men]

So I looked into buying options on BNTX, as one does, in the interest of gambling a little on it becoming a meme stock on the earnings today. Didn't happen which doesn't matter because those options are only sold in lots of a hundred (as far as my brokerage web interface knows anyway) and at 20 dollars a pop that's way beyond what I'm going to bring to a casino. It did however make me wonder: Is anyone selling options on these here options? And if not, why not?

[transgenderer]

okay so i cant find the pdf explaining it but ive been obsessed with zeilbergers vision of ultrafinitism, that essentially the real numbers "are actually" pZ, for some very large p, so theres both a maximum number and an mininum number, ever since i first read it. its just so rich with implications, such a bizarre view of what math IS

[kata4a]

Kink meme: being gaslit into believing that ultrafinitism is *obviously* true and widely accepted.

eh there are hotter things to be gaslit about

[samueldays]

Bans on fraud also infringe on (an absolutist conception of) freedom of speech, and I was already skeptical of the notion of "human rights", so yes we definitely have different axioms here.

This is hard to measure, so take my estimate with a grain of salt, I think I'm on the whole more in favor of free-speech than the average human but less so than the average person talking about free-speech online. I also like rectification of names and I think it's good to bite the bullet and say I am pro-censorship for specific kinds of censorship. A lot of people talking about free speech say things like "free speech doesn't cover hate speech" and I find that dishonest and stupid. I have a pet unfunny joke that "hate speech" should be translated from doubletalk as "speech I hate".

In the alternate timeline without state-enforced copyright and infringements of the right to free speech, I expect something similar to private copyright would get reinvented by distribution contract, loosely akin to how movie theaters can kick people out for being too noisy. You have the 'right' to pirate books, and a bookstore-adjacent network of private operators has the 'right' to refuse service to pirates and pirate-enablers.

If the state were to ban all speech in support of some political ideology, you wouldn't justify it on the grounds that that political ideology was "too specific for its members to independently re-invented its ideas". Who cares! Maybe they never would have come upon that exact constellation of positions

Well, I agree that wouldn't be the grounds I'd justify it on. I would still draw an information-theoretic line between banning support for a political ideology or one specific position, which is short and should be free on these grounds, and banning an exact constellation of positions (implicitly: many positions), which is long and less encumbering to ban because people can simply add a position on Bigfoot or Nessie so they're not supporting that exact constellation any more.

This is petty nitpicking, I admit that, but it's nitpicking about what I think is a fascinating idea from information theory: that very big numbers in a math sense are not "numbers" in a colloquial sense. Ultrafinitism-adjacency of a sort. Humorous IEEE-754 absolutism: numbers are only real if they can be represented in 64 bits. Fractions can get a pair of 32-bit integers. Big numbers don't have a decrement operator.

JK Rowling got wealthy through rent-seeking with a non-rivalrous good; not exactly as bad as being an arms dealer but pretty fucking bad! Her ability to get rich off writing Harry Potter is a direct consequence of all of our freedom of speech being continuously violated!

Private property serves a number of social functions at present (that don't all have to be served by the same mechanism, but I digress), but IMO the most important is to mitigate the tragedy of the commons. One can debate various other mechanisms and whether they would be better or worse towards this end, but needless to say there is no tragedy of the commons for information. The social functions which could possibly be served by JKR owning the Harry Potter franchise seem extremely limited, and not remotely worth abrogating a fundamental human right over.

Another important function that private property serves is to incentivize the creation of goods by giving the creator special rights and rewards. You appear to have fallen into the classic trap of focusing on [re]distribution of goods, ignoring the question of how goods get made and whether goods get made. The likely alternative to JKR owning the Harry Potter franchise isn't "everyone gets it for free", it's "franchise never comes into existence".

With that said, the Life-Plus-70-Years copyright term is absurdly long and I'm going to look the other way on piracy. America's first draft of 28 years was more sensible. Still, JKR very much got rich in the first 28 years of Harry Potter publication.

Getting philosophical about information, I make the information-theoretic argument that the Harry Potter books are too specific, complex and arbitrary for independent reinvention. Random search over human utterances is not going to produce even the first HP chapter in the lifetime of the universe. Observation of the physical universe can produce longer sets of physical laws, but those would be found by other people searching too. If it weren't for Rowling, you would never have wanted to convey the particular piece of information that is a Harry Potter book, so the harm being suffered here also seems extremely limited.

(Inspired by I think it was Scott Aaronson who had a better original post along these lines, though I can't find it right now, about the DeCSS system and "illegal numbers" like 09 F9 11 02 9D 74 E3 5B D8 41 56 C5 63 56 88 C0. There's a "surely you can't own a number" intuitive appeal by way of computers representing any kind of textual or visual information as a number. But, people's intuitions about "a number" are very much shaped by small numbers, and by small we mean less than 10^100, not the kind of number that is the digital representation of a book. So the retort is that small numbers should be free-libre, but big numbers too big for you to ever count or even use as a GUID can have claims staked on them no problem. Informational work would have to go into identifying specific numbers or small ranges to stake a claim on them, so no spamming, no claiming primes and no round numbers.)

[originalcontent]

🔥 ultrafinitism - 680-22-5590

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[ravelite]

In the philosophy of mathematics, ultrafinitism, also known as ultraintuitionism, strict-finitism, actualism, and strong-finitism is a form of finitism. There are various philosophies of mathematics that are called ultrafinitism. A major identifying property common among most of these philosophies is their objections to totality of number theoretic functions like exponentiation over natural numbers.

Ultrafinitism - Wikipedia

[samueldays]

You might also be interested in ultrafinitism: an extremist branch of constructivism where numbers don't really "exist" (substitute appropriate verb of your choice here) until someone has counted to them or written them out in some way.

Ultrafinitism is a fringe branch of maths, so there's by necessity wiggle room and ad hoc disagreement on what it means to write out a number, but the general idea is that tossing together a pile of quantifiers and exponents at most creates that exact number, all the numbers between (for example) 2^42 and 2^43 don't really exist.

Floating-point arithmetic would later reimplement a kind of ultrafinitism: numbers get sparser as they get bigger. :D

Reading @jadagul's Part 3 post and I am once again reminded that the real numbers are actually horrifying, in the Lovecraftian cosmic sense. Whenever I am reminded that almost all the real numbers are uncomputable, it feels like a through-the-keyhole glimpse into the reality of what infinity truly means and that's enough to fill me with terror.

[transgenderer]

is there a model for ultrafinitist arithmetic (negate the "every number has a successor" axiom) and what's it like? Like, i guess itd have to have the natural numbers (0, 1, 2...) plus at least one extra copy of the natural numbers (let P be the biggest number: P, P-1, P-2, ...) but how many more objects would it have?

okay so im not sure how much of this is like, a bit he's doing, but he talks in detail about what his vision of ultrafinitism is in '"Real" Analysis is a Degenerate Case of Discrete Analysis". his conception of the "real" (deeply unclear what he means by this) real line is hZp, where h is a very small but not infitesmal mesh size, and p is a very large and unknowable but finite prime. this is isomorphic to Z_{p/h}, so i guess its just....i mean its just modulo arithmetic? so then like, derivatives are just yknow, finite difference but with h. the weird thing, which he acknowledges, is that in the finite grid vision of R^2, "distance" is not a real thing, because you cant take square roots, altho distance-squared is a thing.

here's how he ends the paper:

Myself, I don’t care so much about the natural world. I am a platonist, and I believe that finite integers, finite sets of finite integers, and all finite combinatorial structures have an existence of their own, regardless of humans (or computers). I also believe that symbols have an independent existence. What is completely meaningless is any kind of infinite, actual or potential. So I deny even the existence of the Peano axiom that every integer has a successor. Eventually we would get an overflow error in the big computer in the sky, and the sum and product of any two integers is well-defined only if the result is less than p, or if one wishes, one can compute them modulo p. Since p is so large, this is not a practical problem, since the overflow in our earthly computers comes so much sooner than the overflow errors in the big computer in the sky.

However, one can still have ‘general’ theorems, provided that they are interpreted correctly. The phrase ‘for all positive integers’ is meaningless. One should replace it by: ‘for finite or symbolic integers’. For example, the statement: “(n + 1) 2 = n2 + 2 n + 1 holds for all integers” should be replaced by: “(n + 1) 2 = n2 + 2 n + 1 holds for finite or symbolic integers n” . Similarly, Euclid’s statement: ‘There are infinitely many primes’ is meaningless. What is true is: if p1 < p 2 < ... < pr < p are the first r finite primes, and if p1p2 ...pr + 1 < p , then there exists a prime number q such that pr + 1 ≤q ≤p1p2 ...pr + 1. Also true is: if pr is the ‘symbolic rth prime’, then there is a symbolic prime q in the discrete symbolic interval [pr + 1 ,p1p2 ...pr + 1].

By hindsight, it is not surprising that there exist undecidable propositions, as meta-proved by Kurt Godel. Why should they be decidable, being meaningless to begin with! The tiny fraction of first-order statements that are decidable are exactly those for which either the statement itself, or its negation, happen to be true for symbolic integers. A priori, every statement that starts “for every integer n” is completely meaningless

i dont know if this actually makes sense tho. postulating that h and p are unknowable is weird but necessary, but then it makes the whole thing sort of...idk, meaningless? also like, if you consider R^2, well now that has (p/h)^2 elements, so are you just, not allowed to count them?

[rationalists-out-of-context]

Don't break the law of ultrafinite recursion unless you want to lose your pants.

[anocana]

okay i think my general opinion on the ontological-foundational question at this point is something like: structure is an unformalizable extramathematical term referring to what we abstract out from things, whether the things we're abstracting are concrete or abstract themselves. it's fine that this is informal because there's no reason to expect that we'd be able to fully characterize mathematics within mathematics. (nor can we fully characterize mathematics without mathematics; studying axiom systems in the abstract is an important part of the endeavour.) i'm not sure how minimal of ontological commitments we can get away with here but they're not going to be any worse than zfc or whatever.

connections between different areas of mathematics really are "coincidence" in the sense that they occur when structures coincide, but not in the sense that there's no reason they happened; where structures coincide it's generally for some sort of reason, but we don't need a global reason for all coincidences of structures (see also: don't need a formal notion of "structure", have no "set of all sets"), just reasons for any particular coincidence. formalism cannot explain this. intuitionism cannot explain this. ordinary mathematical platonism cannot explain this. (structuralism can, and i think this applies in both platonist and antiplatonist varieties.)

what we call a "foundation" is a framework in which to do mathematics which best possesses the appropriate philosophical qualities to be foundational. these qualities may include simplicity, minimal ontological commitments, canonicity, ability to theoretically cover mathematics, closeness to real practice, and so on; how much we care about different ones depends on our general philosophical stance.

the default position in foundations is a sort of platonism that's very pragmatic about its ontological commitments but impractical in its usage: we must have objects which we could hypothetically do all of mathematics with, but having established that it's possible to do it there's no reason to actually bother, so after that let's just try and be as simple as possible. this is bulk of the zfc crowd.

frege was somewhat platonist and very strongly committed to canonicity not just in a mathematical but in an ontological sense. this commitment backfired so badly that strict formalism was a respectable response.

do formalists care about foundations? are we allowed to interpret their statements about what they do and don't care about?

ultrafinitism takes minimality of ontological commitments to its ultimate extent, consigning most mathematical practice to the fire.

type theory is generally associated with a greater concern for computability, canonicity, and closeness to actual reasoning (i think this often comes from a structuralism), at the expense of some simplicity and (to a platonist) more ontological commitments than necessary. type theory isn't exclusively constructive but insofar as it is (equally, insofar as it's computable) it gives up on some amount of existing mathematical work.

finally. foundations are not as central as they're made out to be. we need to look more into what makes things interesting or not and not just how someone jerks off the axioms to make the system cum.

although etymologically "coincidence" is that which coincides... in a certain sense a connection between two mathematical objects is precisely where the two structures coincide. the question is whether "coincidence" is a complete explanation admitting no further reasons