lilith-hazel-mathematics

[lilith-hazel-mathematics]

via @swatercolor [insta]

[lilith-hazel-mathematics]

its really convenient that the normal reaction to being violated and scrutinized and disbelieved for a sustained period of time is to just lose the ability to maintain consideration and cautiousness with one's words. i bet making a woman go fucking insane and pretending it happened for no reason feels soooo good when you're a man

[lilith-hazel-mathematics]

Pure functions

We've talked a whole lot about various formal mathematical theories, but I don't think we've ever talked about our actual all-time favorite. It's a (roughly equivalent) alternative to ZFC, where instead of sets, we say everything is a function. Specifically, single-input functions whose domain is the entire universe. To avoid Russel-like paradoxes, our functions return 0 for almost all inputs (their support is set-sized).

A theory of functions easily reasons about sets. Select any nonzero object, and call that 1. Now define a "set" to be a binary-valued function; an S such that S(x)=0 or S(x)=1 for every x. We define the membership relation "x∈S" to mean "S(x)≠0". The ontology of ZFC then embeds directly into ours, namely the domain of ZFC is exactly the class of all pure sets. A set is said to be pure if it admits a transitive superset which only contains other sets. Stated recursively, a pure set is a set whose members are also pure sets. The axioms of our theory are designed specifically to make this interpretation of sets work. We have eight axioms, each a direct parallel to one of the ZFC axioms.

Extensionality: If two functions agree for all inputs, they are equal. (∀x, f(x)=g(x)) ⇒ f=g

Foundation: Our universe of functions is wellfounded, and contains at most two atoms (functions which output themselves). S≠{} ⇒ ∃(f∈S), ∀x, [x∈f⇒x∉S]∧[f(x)∈S ⇒ f(x)=f∈{0,1}]

Specification: Given sets X,Y and a formula φ defining a class function X→Y, there's an object function f implementing φ. (∀(x∈X),∃!(y∈Y),φ(x,y)) ⇒ ∃f, ∀x, (x∈X ⇒ φ(x,f(x))) ∧ (x∉X ⇒ f(x)=0)

Union: If S is a set of sets, then it admits a union U. ∃U, ∀(s∈S), ∀(x∈s), x∈U

Replacement: Given a set X and formula φ defining a class function on the domain X, the image of F over X forms a set Y. (∀(x∈X),∃!y,φ(x,y)) ⇒ ∃Y, ∀(x∈X), ∃(y∈Y), φ(x,y)

Infinity: There's an infinite set! ∃W, {}∈W ∧ ∀x, x∈W ⇒ {x}∈W

Powerset: For any S, there's a set P containing all subsets of S. ∃P, ∀Z, (∀x, x∈Z ⇒ (x∈S ∧ Z(x)=1)) ⇒ Z∈P

Choice: If S is a set of nonempty sets, it admits a Choice function. ∃c, ∀(s∈S), s≠{} ⇒ c(s)∈s

The first three of these are specific to functions, but the rest are literally direct ripoffs from ZFC. Each of these function-theoretic axioms directly implies its corresponding set-theoretic variant, as restricted to pure sets, so we interpret ZFC. Conversely, ZFC can also interpret our theory of functions (this is nontrivial), so the two theories are equiconsistent.

There's several cute features which makes this theory much more user friendly than ZFC. For example, an ordered pair (a,b) is just a function p obeying p(0)=a and p(1)=b, hence it's just a function with domain {0,1}. This is dramatically simpler than the most common set theoretic definition. Our theory also greatly simplifies the meaning and usage of set exponentiation. Given two sets A,B and a function f, we say f:A→B precisely when all x∈A has f(x)∈B, and each x∉A has x∉f. Then, the exponent B^A is defined to be the set of all functions A→B. Defining 2:={0,1}, the powerset is given exactly by Pow(S) = 2^S, and similarly the Cartesian square given by S×S = S^2. Those last two equalities are necessarily false in ZFC, for a variety of annoyingly subtle technical reasons.

Foundation is certainly our most opaque axiom, with the most non-obvious consequences. It's practically useless, in that you don't need it to create any of your favorite objects. However, it pulls incredible weight in clarifying the overall structure of the universe. Roughly speaking, the axiom lets us partially order the universe by complexity, where sets are more complex than their members, and functions are more complex than their outputs. Everything is built from strictly simpler objects, down to the atoms, and we insist that there's only at most two atoms. This implies 0={}, since there must be a minimally complex object f, which can thus only output itself, but since f∉f then f(f)=0, hence f=0 is the unique minimally complex object. For similar reasons we're also forced to have at least one nonzero atom, which we might as well define to be 1, and it turns out that 1={0}. Similar to ZFC's cumulative hierarchy, we can construct a hierarchy of functions starting with F[2]={0,1} and recursively F[α]=∪{F[β]^F[β] : β<α} for all ordinals α>2. Our Foundation axiom implies that every function appears within this hierarchy. This pins down the structure of our universe quite thoroughly, to the extent that (NBG proves) our theory admits a unique maximal standard model, up to isomorphism.

[lilith-hazel-mathematics]

Our counting to 100 challenge was actually a test of the dead internet theory. Modern LLMs can't consistently count past 4 using full explicit Von Neumann ordinal notation, so the fact that we got to 16 is proof that there's at least 12 real people on Tumblr... Well, except our blog was 3 of those, so I guess 9 people besides ourselves.

[lilith-hazel-mathematics]

under communism we'll restrict cars to just be used to make cool tricks and drifting and stuff like that. until we develop bullet trains that can drift

[lilith-hazel-mathematics]

Me-first ideologies drag societies backward.

See: Republicans, Texas, Florida

[lilith-hazel-mathematics]

The conversation surrounding cultural appropriation has been so severely mutilated by white “allies” that the original intention behind that conversation has become almost unrecognizable in most social contexts.

[lilith-hazel-mathematics]

If any of you wanna know how sweden is racist, basically here's what my Pakistani friend told me about the way employment and residence permits function in this country that's deemed bastion of progressivism in the west.

To live in sweden as someone outside of the EU you need to have a job permit, that job permit is tied to your current employment so if you lose your job you will lose right to live in sweden. It doesn't matter if you fall sick or if you get fired because they deemed you unprofitable, you will lose your right to live in sweden, and here's the most coercive bullshit I've ever heard in my life: if you want to switch your job, you need to go back to your country of origin and THEN apply for work from there and go through whole immigration process AGAIN and work through all of this bs again, applying for residency permit etc etc.

This ensures you have no way of getting back into the country if you lose your job. It doesn't matter if you have been here for 10 years and worked your ass off and everything and evertone you have is here. This process also keeps you tied to corporations whims. The pay isn't good enough? Tough shit you can't switch work or you're risking getting deported. "Legal immigrant" is a nice way to describe a person on coercive bullshit job contracts that corporations exploit knowingly. They keep the money low because you can't quit. You can't leave, you're stuck. It's bullshit

[lilith-hazel-mathematics]

so long as we're back to social justice 101 on this stupid website, u need to be aware of the feedback loop that emerges from disproportionate scrutiny: any social group that is placed under extra scrutiny, regardless of the actual prevalence of any particular behaviour, will appear to engage in that behaviour more often.

you see this most blatantly with racialised groups (more cops in black neighbourhoods = more arrests in black neighbourhoods = "omg look at all the crime in these neighbourhoods!" = more cops in black neighbourhoods etc). even if the rate of crime is the same (putting to one side the criminalisation of poverty which is also an important related factor), one group gets away with it way more often and a new generation of racists is indoctrinated with the crime statistics which "prove" that some groups are simply more criminal in nature. we see a similar phenomenon online with particular groups (trans women being a huge example) being subjected to mass stalking, their every move documented by weirdos and broadcast as representative of the group as a whole.

tl;dr - overscrutinising groups based on existing bigotries creates a recurring feedback loop, reproducing those bigotries across generations and nominally justifying them. this is bad, and you need to remember that you are not immune to it.

[lilith-hazel-mathematics]

speaking of unhinged stupid shit I've read recently

[lilith-hazel-mathematics]

For no reason at all, here's a list of American Universities with inexplicable ties to slavery. Among the 100 something list, plenty are from here up north.

The article from the Guardian that prompted this link was from yesterday, about Jordan Lloyd. A descendant of two slaves of a Harvard founding member. I'll share the article and headline. Read the whole thing.

Get even angrier with me.

[lilith-hazel-mathematics]

Hilbert's Epsilon

One of my biggest pet peeves about formal First-Order Logic is just how limited term formation is. To clarify the terminology, a "term" is an expression in some formal language which corresponds to an object, like a number or a set or something. Contrast with "formula", an expression which corresponds to an assertion such as a proposition or a relation. FOL has tons of versatility in constructing complicated formulae, because the connectives and quantifiers are just so powerful. However, it has almost no architecture for constructing terms.

To give an explicit example, in Peano Arithmetic, every term must be constructed from addition and multiplication, along with the constants 0 and 1, hence the terms of PA are all polynomials. By contrast, there's an arithmetical formula ψ(x,y) encoding the assertion "y=10^x". It's a very long and complicated formula, but its existence shows that PA can reason about exponentiation, which is non-polynomial. This puts us in a situation where "y=10^x" refers to a well-defined arithmetical formula, but "10^x" is technically meaningless and refers to nothing. Annoying!!

Of course, there is such a thing as definitorial expansion, where we include a new symbol like a relation or function, which gives access to more terms. For relations it's very slick: given a formula φ with free variables among x1,...,xn, we may include a new relation symbol R and assert that R(x1,...,xn) is equivalent to φ. This is nice for saving space, since φ might be really long and complicated. If you want to unravel the definition, you just string-replace R out for φ, very simple. By contrast, definitorial expansion for functions is really ugly. You can't just do a string-replace for some longer well-defined term because, again, term formation in FOL is absurdly restrictive.

They should include Hilbert's epsilon as a default feature of formal First Order Logic. It makes term formation at least as powerful as formula formation, and it makes definitorial expansion for functions identical to how we do it for relations. Denoting Hilbert's epsilon by "𝓔", the axioms for it are really simple too.

Given any formula ψ, then "𝓔x,ψ" is a term with x bound.

(∃x,ψ) ⇒ ψ(𝓔x,ψ) for any formula ψ

Optional: (¬∃x,ψ) ⇒ 0=𝓔x,ψ

Optional: ψ(y) ⇒ (𝓔x,ψ(x)) = (𝓔z,(z=y ∨ z=𝓔x,ψ(x)))

In English, you can translate "𝓔x,ψ" to mean "an x satisfying ψ", or perhaps "the simplest x satisfying ψ". If no such object exists, then the meaning of "𝓔x,ψ" is undefined, but optionally the third axiom above gives a definition for those cases. Mechanically, the 𝓔 operator works like a choice function, choosing a witness for each satisfiable predicate. The fourth axiom above (also optional) converts it into the minimization operator of a global wellorder. Something neat is that the existential quantifier ∃x,ψ(x) is logically equivalent to ψ(𝓔x,ψ), so you can actually eliminate all quantifiers from your language and just use 𝓔 instead.

Despite how it seems, the axioms about 𝓔 don't actually imply the Axiom of Choice. In fact, Hilbert's epsilon is a conservative extension to classical logic! That's a rather subtle observation though, and someone who objects to Choice would probably still reject Hilbert's epsilon. We can weaken our assumptions by only allowing 𝓔 to select an object when the selection is forced via uniqueness. I'll write "𝓔!" to denote this uniqueness operator, which has very similar axioms.

Given any formula ψ, then "𝓔!x,ψ" is a term with x bound.

(∃!x,ψ) ⇒ ψ(𝓔!x,ψ), for any formula ψ

(¬∃!x,ψ) ⇒ 0=𝓔!x,ψ

Optional axiom: (𝓔!x,ψ) = 𝓔x, (ψ ∧ ∃!x,ψ)

In English, "𝓔!x,ψ" means "the unique x satisfying ψ". In the case ψ is unsatisfiable, or if it's satisfiable but not uniquely, then "𝓔!x,ψ" returns 0 as a default. Of course, "𝓔!" is redundant if you already have "𝓔", as in the last bullet above, but it's easier to commit yourself to the meaningfulness of "𝓔!" since it's well-defined. This operator can be used to define function symbols in a very straightforward way. Namely, given any function symbol y=F(x) defined by a first-order formula ψ(x,y), we can consider "F(x)" to be a mere shorthand for the expression "𝓔!y, ψ(x,y)". Very simple!

For all the same reasons as 𝓔, the "𝓔!" operator is a conservative extension to classical logic. Unlike the 𝓔 operator however, the "𝓔!" operator doesn't even increase the expressiveness of our language! That is to say, every formula is logically equivalent to some other formula which omits "𝓔!" entirely. The full proof is a little technical, so I'll omit the fine details, but for example we might rewrite the formula φ(𝓔!x, ψ) into ∃x,ψ∧φ(x). The fact that it doesn't increase the expressiveness of the language just goes to show how natural "𝓔!" is, as a component of logic.

[lilith-hazel-mathematics]

crazy that in the 1970s they were like, "fine, women can play sports. but because they're innately less athletic than men, only in a special ghettoized League For The Frail And Delicate where they get paid less 😊". And not only is that still the system in 2023, but viciously lashing out at the smallest challenges to that system gets framed as Feminist Praxis

[lilith-hazel-mathematics]

Imagine the incredibly warped view of Jewish people you must have if you believe that the statement "Stop Killing Children" is anti-semitism.

[lilith-hazel-mathematics]

What a beautiful day to violate copyright and generally disregard any and all notions of intellectual property

[lilith-hazel-mathematics]

[lilith-hazel-mathematics]

i think we should all try to spend less time reading the words of stupid people we hate, and if we find some stupid people we hate on the internet, we should generally try not to share their words with each other (in the form of screenshots, etc). possible exceptions for stupid people we hate who hold immense worldly power. but even then it should be particularly funny or important