#lambda calculus mentioned!!!!!

[chongoblog]

thoughts on lambda calculus? :3

Gonna be real, it's been a LONG time since I learned about lambda calculus, and I think I've forgotten most of it. I know a fair amount of the more advanced calculus in college was learned for the sake of getting a passing grade rather than because I wanted to learn it, so it didn't really stick.

That's not to say I regret my degree! There's still a lot to math that I adore! Like, I've probably mentioned this, but I know I have a PDF of a Graph Theory textbook that I've been slowly dipping my toes into for a while now.

[m---a---x]

Welcome to the premier of One-Picture-Proof!

This is either going to be the first installment of a long running series or something I will never do again. (We'll see, don't know yet.)

Like the name suggests each iteration will showcase a theorem with its proof, all in one picture. I will provide preliminaries and definitions, as well as some execises so you can test your understanding. (Answers will be provided below the break.)

The goal is to ease people with some basic knowledge in mathematics into set theory, and its categorical approach specifically. While many of the theorems in this series will apply to topos theory in general, our main interest will be the topos Set. I will assume you are aware of the notations of commutative diagrams and some terminology. You will find each post to be very information dense, don't feel discouraged if you need some time on each diagram. When you have internalized everything mentioned in this post you have completed weeks worth of study from a variety of undergrad and grad courses. Try to work through the proof arrow by arrow, try out specific examples and it will become clear in retrospect.

Please feel free to submit your solutions and ask questions, I will try to clear up missunderstandings and it will help me designing further illustrations. (Of course you can just cheat, but where's the fun in that. Noone's here to judge you!)

Preliminaries and Definitions:

B^A is the exponential object, which contains all morphisms A→B. I comes equipped with the morphism eval. : A×(B^A)→B which can be thought of as evaluating an input-morphism pair (a,f)↦f(a).

The natural isomorphism curry sends a morphism X×A→B to the morphism X→B^A that partially evaluates it. (1×A≃A)

φ is just some morphism A→B^A.

Δ is the diagonal, which maps a↦(a,a).

1 is the terminal object, you can think of it as a single-point set.

We will start out with some introductory theorem, which many of you may already be familiar with. Here it is again, so you don't have to scroll all the way up:

Exercises:

What is the statement of the theorem?

Work through the proof, follow the arrows in the diagram, understand how it is composed.

What is the more popular name for this technique?

What are some applications of it? Work through those corollaries in the diagram.

Can the theorem be modified for epimorphisms? Why or why not?

For the advanced: What is the precise requirement on the category, such that we can perform this proof?

For the advanced: Can you alter the proof to lessen this requirement?

Bonus question: Can you see the Sicko face? Can you unsee it now?

Expand to see the solutions:

Solutions:

This is Lawvere's Fixed-Point Theorem. It states that, if there is a point-surjective morphism φ:A→B^A, then every endomorphism on B has a fixed point.

Good job! Nothing else to say here.

This is most commonly known as diagonalization, though many corollaries carry their own name. Usually it is stated in its contraposition: Given a fixed-point-less endomorphism on B there is no surjective morphism A→B^A.

Most famous is certainly Cantor's Diagonalization, which introduced the technique and founded the field of set theory. For this we work in the category of sets where morphisms are functions. Let A=ℕ and B=2={0,1}. Now the function 2→2, 0↦1, 1↦0 witnesses that there can not be a surjection ℕ→2^ℕ, and thus there is more than one infinite cardinal. Similarly it is also the prototypiacal proof of incompletness arguments, such as Gödels Incompleteness Theorem when applied to a Gödel-numbering, the Halting Problem when we enumerate all programs (more generally Rice's Theorem), Russells Paradox, the Liar Paradox and Tarski's Non-Defineability of Truth when we enumerate definable formulas or Curry's Paradox which shows lambda calculus is incompatible with the implication symbol (minimal logic) as well as many many more. As in the proof for Curry's Paradox it can be used to construct a fixed-point combinator. It also is the basis for forcing but this will be discussed in detail at a later date.

If we were to replace point-surjective with epimorphism the theorem would no longer hold for general categories. (Of course in Set the epimorphisms are exactly the surjective functions.) The standard counterexample is somewhat technical and uses an epimorphism ℕ→S^ℕ in the category of compactly generated Hausdorff spaces. This either made it very obvious to you or not at all. Either way, don't linger on this for too long. (Maybe in future installments we will talk about Polish spaces, then you may want to look at this again.) If you really want to you can read more in the nLab page mentioned below.

This proof requires our category to be cartesian closed. This means that it has all finite products and gives us some "meta knowledge", called closed monoidal structure, to work with exponentials.

Yanofsky's theorem is a slight generalization. It combines our proof steps where we use the closed monoidal structure such that we only use finite products by pre-evaluating everything. But this in turn requires us to introduce a corresponding technicallity to the statement of the theorem which makes working with it much more cumbersome. So it is worth keeping in the back of your mind that it exists, but usually you want to be working with Lawvere's version.

Yes you can. No, you will never be able to look at this diagram the same way again.

We see that Lawvere's Theorem forms the foundation of foundational mathematics and logic, appears everywhere and is (imo) its most important theorem. Hence why I thought it a good pick to kick of this series.

If you want to read more, the nLab page expands on some of the only tangentially mentioned topics, but in my opinion this suprisingly beginner friendly paper by Yanofsky is the best way to read about the topic.

[aikoiya]

LoZ: Wild - The 7 Heroines Symbol Analysis

Ya'll, with @alasse-earfalas' help, I realized that dang near every one of the Heroines' symbols all relate to mathematical symbols or phonetic symbols in some way!

To start off, one of them is most definitely a Lambda, which relates back to another theory, but that's for another time.

Anyway, here are the symbols in question!

The one in the upper left is the Lambdah (Λ/ʎ), which is the letter correlating to L in the Greek alphabet.

The Lambda is the 11th character in the Greek alphabet & according to Herodotus, Labda was the daughter of Amphion (demi-god son of Zeus & twin to Zethus, associated with the lyre) & the mother of Cypselus by Eetion. Cypselus eventually established the earliest known tyrant dynasty.

Evidently, it symbolized unity under oppression & is referred to as the Greek letter for “sameness.” It also corresponded with the number 30 in the Greek numerical system. Which, 30 was often associated with maturity, as well as the attainment of a certain level of experience or wisdom, especially considering the lifespan expectations in ancient times.

The origins of the letter is thought to depict a shepherd's staff, meaning that it has themes of guidance & authority. It's shape also resembles an arrow pointing upwards, which conveys an impression of accent or elevation. And this leads to metaphorical themes revolving around progress or advancement.

Not to mention, in physics, it refers to wavelength. But in calculus, it's a "formal system in mathematical logic for expressing computation based on function abstraction & application using variable binding and substitution."

Here are the symbols, as I know them. Λ (lambda, phonetic symbo, Ll; ʎ used in math), ʃ ("esh" & integral; phonetic symbol, formerly used to represent ‘sh’) or ⎎ (technical symbol, hysteresis symbol, correlates to f), ǁ (math, parallel to; phonetic symbol, aveolar lateral consonant), : (ratios or scales), ❜ (phonetic symbol), o (phonetic symbol) or O (math, Landau's symbol, describes the asymptotic behavior of functions or represents the null matrix where all elements are zero), last is the drop-looking symbol that I'm struggling with.

It may be that the colors of them are also important.

Just for reference, asymptote is defined as "a line that continually approaches a given curve but does not meet it at any finite distance."

Now, for the Heroines, here's what we know.

Rotana said that the powers were: "skill, spirit, endurance, knowledge, flight, motion, & gentleness."

However, the Stelae say that they are: "heart, skill, fortitude, wisdom, flight, mobility, & compassion."

Luckily, these are easily condensed: heart/spirit, skill, fortitude/endurance, wisdom/knowledge, flight, motion/mobility, & gentleness/compassion.

Heart & spirit are interchangeable as "heart" is likely referring specifically to one's spiritual heart based on Kokoro (心), which can translate to both.

For now, I believe that the Λ may be the symbol for the Heroine of either wisdom/knowledge or motion/mobility or possibly flight. Though, for the moment, I'm leaning more towards wisdom/knowledge.

And, because of the nature of spirit within Japanese culture, I'm inclined to believe that the symbol of the Heroine of heart/spirit may have been ❜.

Heart = ❜

Skill = "The ability to do something well."

Fortitude = "Courage in pain or adversity."

Wisdom = Λ?

Flight =

Mobility = Λ/O? "The ability to move or be moved freely & easily."

Compassion = "Sympathetic pity or concern for the sufferings or misfortunes of others."

Also, the gifts left for the 8th were a gold rupee, a ruby, a sapphire, a topaz, a spear, a shield, a claymore, & scimitar.

These gifts were all positioned at the foot of a statue with, most likely the 8th being the one in the center.

Starting from front-left (Gerudo Shield), middle-left (Golden Claymore), back-left (Ruby), middle-back (Sapphire?), then going back-right (Topaz), middle-right (Gerudo Scimitar), front-right (Gerudo Spear), & ending in center (Gold Rupee?).

I haven't figured it all out yet, but if anyone has any suggestions or knows what the drop symbol is, I encourage assistance.

---

Edit: I'm sorry.

I just realized that it's very possible that the game already shows you which is which.

Look at the way they are always arranged.

It appears to be the same each time.

And, the Stelae, which may or may not have been carved by the Heroines themselves, write it out as "heart, skill, fortitude, wisdom, flight, mobility, & compassion."

Also, something interesting of note is the characters used for them in Japanese.

Shin or Kokoro (心) = Heart, spirit, mind.

Gi or Waza (技) = Skill, technique.

Tai or Taeru (耐) = Enduring, to resist or withstand (e.g. 耐火 is “fire-proof”).

Chi or Shiru (知) = Knowledge, wisdom.

Hi or Tobasu (飛) = Flight, used in a lot of verbs related to jumping or letting something fly.

Dō or Ugoku (動) = Motion.

Jū or Yawaraka (柔) = Gentleness, softness (it’s the “ju” in judo).

---

Heart = Shin (心; heart, spirit) = ❜ = Shield

Skill = Gi (技; skill, technique) = Λ = Claymore

Fortitude = Tai (耐; endure, resist) = O = Ruby

Wisdom = Chi (知; knowledge, wisdom) = : = Sapphire

Flight = Hi (飛; flight) = ⎎ = Topaz

Mobility = Dō (動; motion) = Drop Sign = Scimitar

Compassion = Jū (柔, gentleness, softness) = ǁ = Spear

...

Did I just figure it out?

... Regardless, even if I am right, I'm still ridiculously curious as to what the drop symbol is, though. Like, is it just the odd one out that they chose just to mess with us?

Are the gifts they left for the 8th in any way related to their Special Powers™ or the way they fight? Are the symbols?

And, if so, how?

Like, I'm pretty positive that, if I am right, then the Heroine of Heart's symbol is, in fact, a reference to the Mitama of Japanese myth & the Heroine of Fortitude's gift may have been a hint to the term taika (耐火) meaning "fireproof."

Though, I noticed that the weapon missing from this line-up is the Golden Bow. But, then again, it's possible that one of the Heroines that gave a gem might've used one. Possibly the Heroine of Flight. I mean, Hi (飛) is evidently used in verbs referring to “letting something fly,” so that might well be it.

I'm not entirely positive of the rest, though...

... That drop sign is gonna erk me until I get confirmation, won't it? 😮‍💨

LoZ Wild Masterlist

[pv1isalsoimportant]

4-12-2024

I've just got home and I'm completely dead so no more math today I guess. I wanted to look at some homework problems but it is what it is. Today's lectures were exciting af tho.

Logic in computer science (lecture, 1.5h) - finished basics of lambda calculus, started arithmetic. I love arithmetic and the way certain fragments of arithmetic formalizes computation (bounded arithmetic, my beloved) so this should be fun. Also Logicomics mentioned!

Selected topics from set theory (lecture, 1.5h) - started square and reflecting stationary sets.

(The picture above is from the logic in computer science class.)

[max1461]

In response to this post by @loving-n0t-heyting.

Well... I guess?

So I don't actually have as much of a dog in this fight as maybe I implied I did, and maybe it's ill-advised to get into this further, but I do kind of feel like this reply misses the point a little?

Right, I might be missing something, but I believe that models of FOL, and models of SOL in the standard semantics, and models of typed lambda calculus, and so on, are all defined in basically the same way? They are all special cases of a more general definition.

Like, I believe that given a first- or second-order language L with constants c_1,...,c_n, functions f_1,...,f_m, and relations R_1,...,R_k, the set of all terms and formulae of L (call this set A) is an example of a free partial algebra. This partial algebra is generated by the variables and the constants, under application of partial functions corresponding to f_1,...,f_m, R_1,...,R_k, the logical connectives, and ∀ and ∃. I might have missed one. Uh the terms of lambda calculus are generated in a similar way. And anyway, a model is just another partial algebra M in the same signature with a structure preserving map φ : A -> M (EDIT: actually, one for each variable assignment function) (EDIT 2: since the constants are part of the signature and A is free, any variable assignment functions automatically induces a structure preserving map A -> M. So a model is just another partial algebra M.). Uh and then we make some demands about the structure of M, like for FOL we force M to be the disjoint union of a set M' and {T, F}, where M' is the universe of the model as traditionally defied. And for lambda calculus we demand that M is the disjoint union of a universe M_t for each basic type t and (maximal) sets of ith order functions between the M_t.

Shit, I should actually work out all the details. I don't think this reply is going anywhere without that. I probably don't have time for that.

Right, the point is that I am pretty sure that all these different definitions of "model" are just special cases of the exact same thing. And the way we define models of simply typed lambda calculus with quantifiers exactly reduces to the way we define them for first- and second-order logic, if you just get rid of all the lambda terms and the stuff of highers types. Uh yeah. I'm like 90% certain this is true. But without working it out on paper I can't be sure. Consider this a beta thought.

Uh but Henkin semantics are not a special case of this. Because you need to fix a Henkin prestructure—the sets of higher-order functions are not maximal, like you're have some subset of M_t^{M_s} for types t and s, or whatever. It's a different thing.

But the point is like: this strikes me as the canonical way to define a model! Far from defining models this way specifically to get the results we want—which seems more like what Henkin semantics is doing, to get a nicer meta-theory—this is just, defining models like we define homomorphisms in algebra: there's one canonical way to do it, and if you do it a different way, that's the choice that students are going to ask for motivation for. Right?

But, yeah, I agree there is not a lot of mathematical content to this.

My point (that FOL is odd in various ways) would maybe have been better illustrated by talking about nonfirstorderable sentences, which I did mention in a reblog. I think it's difficult to deny, re: nonfirstorderability, the claim that "FOL is limited in certain ways by the topology of strings". Which is fine, FOL works fine, but it is weird. I am not sure what argument one could put forth to say that that is not a strange contingency for a logical system to have.

And, actually, this is directly related to the stuff about the Peano Axioms: because the axiom schema of induction in Peano Arithmetic includes one axiom for each first-order sentence, what is firstorderable directly determines what you can do induction to! And I claim that is... strange. That's really funky. That's very funky!

You don't actually need the whole of SOL to remedy this string thing, you can just allow trees of quantifiers or whatever. Various people may have looked into this.

But, no yeah actually, after having written this last bit I am slightly convinced again that there is a genuine (slight) odd thing going on here. Uh it's not just all trivial, motivated reworking of definitions as the linked post sort of implies. Well I don't know. It's late. Maybe I've made an error somewhere here; lord knows when I don't double check my math posts that can happen.

[theohonohan]

Functional Formalism: Const, Identity, etc.

In the development of the Church encoding of natural numbers, a definition of the predecessor function ("subtract one from this natural number, unless it is zero") was hard to come by. The predecessor function can't be represented in simply-typed lambda calculus, and the most popular realisation of pred is a bit difficult to understand.

Before reading this post you might want to explore some of these StackOverflow answers on the topic which provide a range of perspectives.

One version of pred looks like this: \[ \lambda n.\lambda f.\lambda x. n\ (\lambda g.\lambda h. h\ (g\ f))\ (\lambda u. x)\ (\lambda u. u) \]

This contains a few subexpressions, to wit:

\[ \lambda g.\lambda h. h\ (g\ f) \]

\[ \lambda u. x \]

\[ \lambda u. u \]

The second and third expressions I've extracted are the functions usually known as constant and identity.

Strictly speaking, the name const is usually reserved for the form \( \lambda a. \lambda b. a \), in other words, a helper function that creates a constant function that just returns \(a\). Here the expression \( \lambda u. x \) is quite far from the lambda abstraction that binds \(x\), but in some sense, the same thing is going on here as in the usual const function.

The identity function, usually referred to as id, is very simple. It just returns its argument.

The theme of this post is that the presence of these functions should clue you in to the nature of this invention. The constant function and the identity function are so simple as to be almost meaningless, but they come in handy as components to be used in syntactical plumbing. This is formalism, and it resists meaningful interpretation. One of these expressions (for example, the lambda term for pred) just is what it is. As somebody says on StackOverflow, talking about the Const functor (of which more later): "It is good for similar things that the const function is good for! Passing to higher order functions, etc."

Talking through the workings of pred is a bit tricky. There is an attempt on Wikipedia, and there are the previously mentioned StackOverflow answers: "The predecessor definition here really is just hard to comprehend, and the best way to see how it works might be to step through the evaluation manually to get a feel for what's going on."

I wanted to try explaining it myself to improve my understanding.

This term, \( \lambda g.\lambda h. h\ (g\ f) \), is designed to apply its arguments \(g\) and \(h\) (in that order) to a copy of \(f\). The term gets repeated \(n\) times. Each copy (they're nested) contains an \(f\), which is important: there are \(n\) of them, and we want \(n-1\). I said that they're nested: In other words, the term gets applied to the return value of another instance of itself. The initial argument is the constant function containing \(x\). The term is a sort of 'applicator'—Wikipedia describes it as a 'value container'.

So \( \lambda g.\lambda h. h\ (g\ f) \) takes a function (\(g\)) and "returns" a function. The returned function is typically something like \(\lambda h. h\ x\) or \(\lambda h. h\ (f\ x)\).

The constant function \(\lambda u. x\) is the argument to the innermost instance of \( \lambda g.\lambda h. h\ (g\ f) \). Beta-reduction results in the term \(\lambda h. h\ ((\lambda u. x)\ f)\) which reduces further to \(\lambda h. h\ x\). This is actually the crucial step in achieving the effect of pred: the specially prepared constant function has thrown away one of the instances of \(f\), or rather, replaced it with \(x\). Since it's the innermost copy of \(f\), there is no problem with the rest of the process of function composition, and we end up with \(f\) applied to \(x\) \(n-1\) times.

The final result is obtained by applying the 'applicator' to the identity function at the very end of the pred term. This just extracts the contents of the container and returns the expression we were after.

It's worth noting that this idea of an 'applicator' that takes a function as an argument, and applies it right away, comes up again in the definition of Church pairs. Perhaps this is inevitable, since there's nothing much else that can be done with functions in the lambda calculus. In particular, the representation of a pair is an applicator, and so are the functions that project the first and second elements of the pair. See Church pairs on Wikipedia. [Note: by "a term which takes a function as an argument and applies it right away", I essentially mean a term which is in continuation-passing style: the argument comes first in the body of the lambda abstraction.]

It's probably best to limit myself to one topic per post, but I just want to gesture here towards the use of Const and Identity functors in Twan van Laarhoven's implementation of Haskell lenses. The Const and Identity functors are very similar to the Haskell functions const and id, but, being functors, they are a way of encoding data rather than of transforming it. The Const functor has two type parameters and ignores the second, representing simply a value of the first type. The Identity functor, as you might expect, takes a single type parameter and encodes a value of that type. Van Laarhoven's trick is to use these two functors as adapters to unify the types of getter and setter lenses. To get there, he transforms the straightforward getter type \(s \rightarrow a\) into continuation-passing style: \((a \rightarrow r) \rightarrow s \rightarrow r\). This brings its type close enough to the type of the setter that the functors can be used to unify the two types. (The type of setters is \((a \rightarrow b) \rightarrow s \rightarrow t\)).

Apart from the explanation van Laarhoven's own blog post, there is a very clear StackOverflow answer on the topic and there even exists a video of a presentation on lenses by Simon Peyton-Jones, which you can probably track down, if you like that sort of thing. (For what it's worth, I've never written any code that uses lenses, and I gather their unergonomic aspects can be a problem.)

The simple observation that connects pred and van Laarhoven's types for lenses is that simple, dumb functions like const and id can become useful components, in the right circumstances. And maybe you want to avoid those circumstances, because it really is a case of formalism, where id comes to have some (possibly obscure) syntactic role rather than its straightforward semantic one.

[hydralisk98]

Creativity for the immediate future and past (WIP)

Some poll / vote / survey and my lexicon of useful technical terms (adding their relevant definitions soon enough) about my workflow to better grasp what I am gonna talk about.

Poll section:

Glossary of terms

Human being

Humanity

Humanoid

Sapient

Sapient-kind

Sentient

Agent

Golem / agent construct

Abstract agent

Imagination

Innovation

Infrastructure

"Natural"

Agency

Court

Species

Phenotype

Clade

Cladogram

Domain

Specialty

Alignment

Stance

Interest

Ethos

Worldview

Perspective

Opinion

Magick

Eldritch

Entity

Free-will

Choice

Morphological Freedoms

Charter of Rights

Progress

Liberty

Harmony

Adaptation

Personality

Tulpa

Archetype

Demographic

Psyche

Organization

Authority

Commune

Communion

Faction

Tribe / Hive

Nucleus

Unity

Union

Curation

Supernatural

Divine

Hellish

Feywild

Generation

Iteration

Value

Register

Reference

Address

Instance

Constant

Variable

Pattern

Construct

Modifier

Parameter

Modularized

Piece

Song

Album

Collection

Series

Season

Pipeline

Workflow

Feed

Data

Connectors

"Closeby" future

"Mid-sized scale future"

"Farther future"

"Far far away future"

Million

Billion / Milliard

Trillion / Billiard

Degenerate era

Dying Universe

Black Holes era(s)

Iron Stars era

Heat death

Judgment day

Cosmogony

Ecological niche

Plane

Realis

Realm

Planet

Galaxy

Sector

Spatial Hash

Verlet Integration

Lambda Calculus

Pāṇini grammar

Universal Turing Machines

Esolang

Esoteric

Mythos

Storytelling

Science

Science-ism

Ideology

Philosophy

Valence

Continuum

Ancestor

Organized religion

Folk religion

Animism

Tradition

Dogma

Doctrine

Scientific method

Secularism

Free Thought

Freedom(s)

Query

Question

Learning

Wisdom

Erudition

Intelligence

Von Neumann Cellular Automaton

Cellular Automata

Voronoi Diagram

OpenSimplexNoise

RandomNoise

Model Synthesis

Wave Function Collapse

Markov Junior

Quantum Entanglement

Quantum Superposition

Entropy

Stand-still

Optima

Pessima

Average

Median

Series

Collection

Set

Node

Connection

Relationship

History

Stories

Historical record

Generative

General-purpose Artificial Intelligence

Android individual

Instrument-tier

Robotic-tier

Droid-tier

Synthetic-tier

Biology

Toymaker

True Polymorph

Skinwalker

Shapeshifter

Changeling

Witch

Wizard

Sorcerer

Warlock

Patron

Sponsor

Supporter

Peer

Subordinate

Delegate

Slave

Serf

Servant

Wage worker

Autonomous worker

Entrepreneur

Rule

Hint

Guideline

Suggestion

Order

Obligation

Duty

Mandate

Responsability

Unit Cell

Grid Cell

Vector

Raster

Symbolic

Digital / Numeric

Alphanumeric

Analog

Analogy

Container

Child

Sibling

Parent

Cousin

Root

Tag

Category

Deque

Grid

Stack

Queue

Loop

Wrap

Joint

Teleport

Jump

Reality shifting

Bitwise shift

Bitwise rotate

Mathematical remainder

Mathematical modulus

Rotor

Linked list

Unordered list

Ordered list

Logical connectives

Reason

Common sense

Negation

Tautology

Contradiction

Inclusive Conjuction

Exclusive Conjuction

Disjunction

Lightyear

Statement

Affirmation

Fact

Verb

Subject

Object

Topic

Comment

Structure

Data

Spacetime

Energy

Exclusive OR

Jonction gate

Assign

Compare

Load

Store

Branch

Add

Substract

Truth

Falsehood

Gender

Role

Identity

Sex

Romance

Sexual attraction

Mutual intelligibility

Unidirectional intelligibility

Dialect

Language

Linguistic family

Linguistic group

Protocol

Standard

Formation

Experience

Understanding

Grasp

Know-how

Knowledge

Social norms

Respect

Social awareness

Social skills

Context

Contextual clues

Implicit

Explicit

Opaque

Transparent

Free

Libre

Open

Honest

Useful

Interesting

Competency

Curiosity

Confidence

Charisma

Sanity

Prestige

Still going to send much previews as I work on it but yk, the larger pieces I got in mind need more time in the oven before I start alpha-reading and eventually beta-reading. Will link so much more workflow insight and processing tutorials as well as original content suggestions queue thread for the very near future. So stay tuned for those.

[wuggen]

@lady-inkyrius:

Curious on why this is, having never tried to use a fixed point combinator outside of Haskell? I definitely can imagine running into problems writing something like a Y combinator in most languages but I don't know exactly what those problems would be.

Rust's operational semantics absolutely allow for it; you can use closures to introduce the necessary lazy evaluation and it'd work just fine. The issue is in type inference. Simply put, fixpoint combinators tend to produce some really fucking weird and unresolvable types. Like, for instance, Haskell will let you do this:

fix :: (a -> a) -> a fix f = f (fix f)

and that's fine, but if you try to do the more usual lambda calculus Y combinator,

fix2 :: (a -> a) -> a fix2 = \f -> (\x -> f (x x)) (\x -> f (x x))

you get this error:

Couldn't match type ‘t0’ with ‘t0 -> a’ Expected: t0 -> a Actual: (t0 -> a) -> a, In the first argument of ‘x’, namely ‘x’, In the first argument of ‘f’, namely ‘(x x)’

'f' must take an 'a', so '(x x)' must be an 'a'; but then the type of 'x' must be '([type of x] -> a)', which is infinitely recursive. The usual Y combinator simply cannot be typed in most type systems.

That first one ('fix f = f (fix f)') only works operationally because of Haskell's lazy-by-default semantics; an eagerly-evaluated language would infinitely recurse here before any of the actual intended evaluation could happen. You need something like the usual Y combinator, with explicit closures to introduce laziness, in order to define a fixpoint combinator in an eagerly-evaluated setting (although the usual Y combinator doesn't work for call-by-value either, you have to expand it to '\f. (\x. f (\v. x x v)) (\x. f (\v. x x v))'; this is still not typable, but works operationally).

Now, in theory, Rust's type system could actually type some of these kinds of things:

fn fix<F, A>(f: &F) -> A where F: Fn(&F) -> A, { f(f) }

That type could work just fine! Granted, if it were expanded to its full extent, F would indeed be an infinitely recursive type, but as long as we can cut that off by just referring to it as F we can still deal with it tractably. Give that thing a closure-returning-closure for laziness purposes and you've got yourself recursion via fixed-point! And Rust's operational semantics would evaluate it just fine!

And in fact, if you just compile this function without using it, the compiler doesn't complain at all! But it gives the following error when you actually try to use it:

error[E0644]: closure/generator type that references itself note: closures cannot capture themselves or take themselves as argument; this error may be the result of a recent compiler bug-fix, see issue #46062 https://github.com/rust-lang/rust/issues/46062 for more information

According to that issue, the problem is with type inference. Now granted I'm not super familiar with type inference systems, but my assumption is that, in the absence of complete type information, the compiler has to fully expand that infinite type to infer the remaining information. And closure types in Rust are almost always fully inferred; it's not actually possible to name or explicitly define a concrete closure type, only the traits that they implement. So inference must be kept tractable there.

Now that issue does also mention that you can get around this by indirectly referencing the closure's type via virtual calls/trait objects. This is very enticing to me and I might have to experiment, a Rust fixpoint combinator might be possible after all 👀

Very upsetting that Rust doesn't let you make a fixpoint combinator. Like I understand why, but it still sucks 😔

[transsexualhamlet]

asmr i psychoanalyze my favorite war criminal, aka calling out norman the essay

basically all of my thoughts on norman on one callout post because i care him (both manga and anime are discussed)

LINK TO RAY PSYCHOANALYSIS:  https://chaoticgaymess.tumblr.com/post/646749875570196480/ray-81194-the-long-explanation 

this is going to be ungodly long so here’s a keep reading, essay below the cut

((tw for suicidal ideation and self harm, brief discussion of eating disorders))

Disclaimer: no shipping is included here this is just about norman also they’re kids who call each other siblings

Thoughts: So you may be thinking, Rowan, why do you yell about the colorless war criminal so often? Well the answer lies in your honor the court hates to see a girlboss winning. Norman is a girlboss :) Yes norman is a tiny twink who can't lift a milk jug. And he is a girlboss :) Obviously I don't condone, um, eugenics and all, but that's not the point the point is that he satisfies my need for more characters like Levi motherfucking Calder from Unwind because I’m apparently an edgy 13 year old. Also all of his problems are violently things I can fix and I keep him around as a pet project because someone needs to give him a hug and slap him on the face

I diagnose him with things: 

-pisces man :pensive:

-is he albino? Not literally. Is his skin so pale he would catch fire if he went outside at noon? Yes.

-autism: Yes I’m aware that calling him autistic makes him, problematic rep by perpetuating the autism unfeeling savant stereotype whatever but have you considered i’m autistic and I’m projecting also he’s L with standards? Anyway traits of AuTism he has: hyper   fixation, canonically breaks and fixes things over and over because like ofc he does, doesn’t understand Emotion, hyperaware of body language at the same time as it all somehow flying over his head, low empathy, sensory experiences™, min maxed in certain areas, and I don’t think he’s got social interaction quite right? There’s something off about it

-gifted kid (derogatory) This is self explanatory but basically him being the smartest and the best in a competitive environment caused most of his issues, such as the perfectionism, the need to succeed, the lack of self esteem and ridiculously high expectations on himself, giving himself no breaks or time to relax, the “i must be productive with every second of my day or i will die” deal, the “peaked at 11” thing, the way in which he goes through life like there’s going to be a fucking test on it

-Eldest Daughter™ lmao. Norman’s always had to be mature, he’s always had to be the best, he’s always had to do the things Ray got out of bc he’s a snitch and Emma got out of because Isabella likes her. Norman gets respect from Isabella only if he excels, and her bar for him is astronomical. He doesn’t have the Mommy Issues that Ray has, but it’s because for him Isabella basically just reflected his expectations on himself, whereas with Ray it was more personal.

-low empathy (part of the autism thing): this one needs more explanation, but it’s not a bad thing in and of itself. Cognitive empathy is a thing and he can use it, but he does not instinctively understand other people’s emotions, or even recognize them properly, especially when the person is not like himself. This is obvious in Emma. Man has no fucking clue what’s going on in her head or why she does what she does, but he can predict what she will do in any given situation very well. He could understand the suicide attempt from ray he predicted more because Ray’s an easier equation to solve, and someone who’s more similar to him. I know he gets it because, well, motherfucker’s just as self desctructive as him, just in a more dignified manner.

-he’s got some sort of chronic illness. This is also me projecting and a headcanon but he’s got something going on, even before lambda pumped him full of growth hormones or whatever which they maybe should have Not Done but oh well. (I assume this just didn’t happen in the anime, since he’s still so fucking short) But he's So weak. He passed out when it was too hot. He passed out when it was too cold. He can’t open a pickle jar. His skin is too pale and he’s skinny af. He’s much more prone to sickness and probably has asthma too? But in the case that he did actually have something going on, I don’t think grace field would see the need to treat it, if it didn’t impact the quality of his meat? Isabella’s probably just “you have chronic pain and you get migraines? Great, take some tylenol and do some calculus.” Can’t say that probably helped anything.

personality type: ISTJ

Basically, he’s the most boring personality type to exist, and personally as an enfp i do not respect him. But basically this means he’s a fucking nerd that gets his projects done for school the day they’re assigned, is probably the president of the Anime Student Council™, and could probably get away with premeditated murder (ok actual istjs this is a joke don’t skin me)

The only trait that norman doesn’t have on the istj thing is telling the truth. Yeah, he values the truth, but like, that doesn’t apply to him, clearly. Bitch is a notorious liar.

The only other personality type he has any similarity with is intj, which is the same except it’s more rare and a purple theme instead of a blue theme. Sadly, that’s not him though, because although he can care more about some kinds of philosophy overall this isn’t the case and ray already occupies this personality type tbh. 

strengths and weaknesses: This one’s kind of obvious, but he is aside from the crazy insane intelligence good at planning. Extremely good at planning. He can predict any outcome and figure out how to prevent it, using all his resources. For example he’s physically weak and someone could literally just walk up and stab him, but it doesn’t impede his progress on his goals because he’s surrounded himself with strong, mentally inferior people who would die for him in a heartbeat. He never gets stuck in some “everything is shit and i can’t do anything” deal like Emma and Ray do, he always works through it and has confidence in his abilities (in as much as he will solve the problem or die™. Weaknesses other than his twink body include his Low Wisdom score. It’s funny how he’s often associated with an owl, the mans is 14. He thinks he knows what he’s doing. He doesn’t. Plus obviously his fundamental misunderstanding of so much of everything going on around him, the fact that he lies not just to the world but himself, his refusal to take care of himself and his incredible cowardice. His achilles heel is being forced to, actually confront his actions.

what he likes about himself: He does pride himself on his mental abilities, and his judgement, which in his opinion is the only correct opinion and the only correct way. In the past, he likes being seen as a leader, he likes being responsible for other people. He likes his ability to manipulate and lie, because he sees it as an asset, and I honestly think he enjoys being william minerva more than he enjoys being Norman. He prides himself on his unhealthy expectations and the fact that he is able to meet them. Honestly, he does think he’s better than everyone else, mentally, though it’s humbled by his self hatred. Cursed thought: If Norman had self esteem he would be light yagami. 

what he doesn’t like about himself/insecurities: Oh god, nearly everything. His appearance, his status, his superiority, his physical inability, his own mess of a mind, also have I mentioned his appearance. He’s obsessed with self control. He wants everything he sees wrong with himself gone. And I understand why having control of everything is necessary and appealing, everything for him has always been rigid and planned out from moment one, he was even more regulated in lambda, and though he desperately wants to Not Be Food, he has no idea what to do with the chains now that he’s broken out of them. So he just wraps them around himself. Regulates to an unhealthy degree when he sleeps, what he eats, when he actually takes even minimal care of his own problems, what he looks like, how much of himself he lets show, the expressions on his face, the literal thoughts inside his own head he will shut down if they are not Correct. It’s literal self harm. Norman, please stop it.

motivations/goals in life/general philosophy: To be honest, I’m not sure he knows what he wants. He sure thinks he does, he could sure give you a memorized answer, but it means nothing. He wants to excel. He wants Emma to be happy. He wants to be perfect and for that to make everything perfect. But he doesn’t realize everything he’s working towards will do pretty much the opposite of that. He’s a crippling perfectionist, and pretty much everything he does is motivated by his fear of failing. He picks the certain path, he doesn’t wait for anyone else, he doesn’t care if it’s not nice. Emma foils that a most of the time because he cares about her, but it can only go so far, especially after he’s had so much time without her to develop a Complex. His philosophy is very contradictory, basically the tokyo ghoul “everything bad that happens to you stems from a lack of ability”. All of his problems are his fault. All the world’s problems are his to fix. If he can’t fix them, it’s his fault, it’s because he wasn’t strong enough, and not being perfect condemns someone forever, including himself.

how he’s perceived by others vs how he actually is: In most people there wouldn’t truly be much of a difference, but with Norman things are different, because, well, most of his personality in grace field is a put on, as well as the tough guy dictator thing he radiates after lambda. How he appears to someone is determined by the context of their meeting- the kids at grace field see him as a nerdy, weakish, pretty boring kid who is really caring and kind. The researchers at lambda see an obedient, beaten down and perfectionistic boy. The lambda kids see him as an infallible leader, ruthless and genius, a good man who knows what’s right. But in truth none of that is him. It’s a fucking chess game to him, putting on different faces, lying and pretending and treating everyone differently. In truth? He’s a fucking coward. He’s scared out of his mind and he’s tired and he can’t take pain, he’s obsessed with reaching some goal he deems is necessary that in the end is going to be his death because he doesn’t want to face the consequences of his actions. He’s taken on the role of someone evil, though deep down he’s not, he feels it’s easier to live that way because it strips him of his conscience. 

interpersonal relationships: In general, Norman sees all relationships in a pretty dim light. He sees everyone as black and white, for the most part, and other people make no sense to him intuitively, he has to figure them out like a puzzle. He’s manipulative and not particularly kind, but he follows all societal expectations to a T, overly focused on his appearance and placing the person he’s interacting with into a Category™. So he can be truly kind, to people he feels deserve it, to people who he values and doesn’t see flaws in. He gets incredibly attached to people he loves, protective, though he often doesn’t take their own feelings on the matter into consideration, and he’s ruthless with anyone who he deems a bad person. With people he understands and relates to, though, things can be different. If he sees someone as like himself, he will drop all the social interaction police bullshit and cut to the chase of whatever he wants or needs from them, and he’s not very forgiving in any manner, if he thinks what someone did is actually bad.

Emma: Norman obviously cares a lot about Emma, and honestly views her as better than anyone else. He realizes her moral integrity and all of the things she has and he doesn’t, and admires it. Because of his black and white view, Emma is like an angel to him. She couldn’t do anything wrong if she tried. But he comes to treat her as something to be protected instead of respected, and although he realizes she wouldn’t like what he’s doing, he fundamentally cannot empathize with her and doesn’t try to understand her. Their personalities are very literally opposite. Norman really needs to fucking listen to her. And Emma needs to understand that Norman doesn’t have a single ounce of empathy and you really do need to spell it out for him. Emma can only convince him when she has logical reasons for her actions, which she, doesn’t often have. And Emma gave Norman too much slack, because she didn’t see past the surface, and Ray never wanted to warn her, even though he knew the dude was showing a bunch of red flags, because you know. It was kind of an unspoken deal between them. (on ray’s part)

Ray: His relationship with Ray is a lot more complicated than with Emma. He understands Ray, where he doesn’t understand Emma, and he can see right through anything Ray does. And this makes things really tense between them, because Ray doesn’t, take kindly to being psychoanalyzed. If someone perceives him he will deck them and Norman is just there silently perceiving him at all times when Emma doesn’t see it. They are both constantly in competition with each other, but they care about each other a lot, though it’s kind of in a derogatory way. They both recognize each other as fundamentally fucked up, and silently agree never to bring it up with Emma. They’re nice to each other when she’s around, but all pretenses disappear when she’s gone. Ray is always frustrated with Norman, because Norman’s never been intimidated by him, and though he tries his best not to be vulnerable around him, Norman can always see through it, whereas Ray can’t crack Norman’s fake fucking smile no matter what he does. Norman will always take Emma’s side, and doesn’t see Ray as a good person at all, but he still understands and can excuse him, he takes measures to be… worse than Ray, which is better in his mind, because it’s rational, and ‘not selfish’.

Isabella: She has always had ridiculously high expectations for Norman, and treats him kind of harshly compared to the others. Bitch has heat stroke and Isabella’s first question is a calculus problem instead of like, “are you ok”. She knows he doesn’t complain about anything ever and she doesn’t stop him from being Terrible to himself, because it makes her job easier. They want smart kids, not mentally adjusted kids. She does really care for all of them, but she basically overrides it, she gives them what they want, not what they need, lets them be exactly what they’re making themselves. Isabella is distant with Ray but gives him anything he wants, she’s close and super nice with Emma, but Norman is… it’s weird. Isabella is proud of him because he meets her astronomically high bar. But at the same time, Norman never really cared for her that much and has never pretended to. Once they discover The Thing, though, he has a revelation, and it doesn’t take him long to switch his entire perspective about her. He’s pretty much like. Oh. She’s like me. That explains it, time to treat her like I treat myself: fucking brutally. Passive aggressive as hell. The kind of energy the :) emoticon at the end of an email gives. He does like just go “yeah we should kill her” at one point, which. You know, ok. When he got shipped out it was hhhh really interesting because Isabella knew full well he knew he was walking to his death and Norman was like “are you Truly Happy?” and just went :) and she was like h u h and tried to get him to talk while they were walking there because she feels Bad about it and he just. Did not. He didn’t say a single word just kind of smiled menacingly at her and I think it was half a sort of rebellion and half because he viewed her as similar to himself and therefore felt no need to put up any front with her, no words were necessary for him to impart exactly how he felt about it

Lambda kids: His relationship with the lambda kids is weird and bittersweet. I think he really truly does care about them, they were in a similar situation to his and he wants them to get what they want. However it is not a healthy or beneficial relationship, they see him as a god and don’t realize that he’s killing himself to give them what they want, he’s basically adopted them when out of anyone norman’s the one that should least be in charge of kids. I think he’s honestly younger than them but I’m not sure if they even know. He acts like their fucking mom, and that’s from what he thinks mothers are like… like isabella?? Giving them what they want, not what they need, lying to them, showing a front, caring deeply for them but at the same time using them for his own ends. And it’s not helpful for him. He thinks he knows what they need, but what he’s doing is what they want. What they need is therapy,(and so does norman), and he doesn’t think there’s anything wrong with using them as weapons because they love him. It makes him feel good, to be seen as perfect, to have people who don’t know how weak he really is. But it’s only making him worse, and he’s enabling everything the lambda kids are doing wrong as well. They need like, Yuugo and Lucas. Some actual adults who are actually wise and have the ability and the knowledge to take care of them and understand their mental problems and maybe actually address them. And actually be nice to them. But um sadly. 

what he’s doing wrong: It’s pretty obvious, but… Norman, you maybe *shouldn’t* commit genocide? You’re not helping emma, you’re not making anything better. You’re not helping the lambda kids, you’re enabling them. You’re not helping your friends from grace field, you’re ignoring what they want. You’re not helping the world, you’re eradicating an entire race from the face of the earth and murdering the poor for the crimes of the fucking 1%. You’re not being a martyr, you’re a selfish piece of shit liar you little coward, you just want an easy way out and you want to die on your bloody fucking hill instead of admitting you’re wrong. Grow up, cringe little man.

why he went wrong: I think most of the reason this happened was the way he was raised combined with the kind of person he is. Norman would have turned out fine, if there has been good adults in his life who actually cared about his well being. Instead he got people who just wanted to control him and make him what they needed, and family who largely didn’t realize there was anything wrong. Ray being an ass to him most of forever probably didn’t help but well, that’s just Ray. Even then, he would have managed alright if he escaped with the rest of the kids because he would never have been separated from the experiences that caused the rest of them to realize demons weren’t all evil. In lambda he didn’t have anyone supporting him or telling him when things went too far, so he fell into relying on himself alone, pushing himself further with absolutely no limits. All he saw was enemies and allies, and things got stratified. He never had a lucas or a yuugo or mujika when he would have needed it, instead he found children who wanted him to be in charge and a world that made it so he had to be. Everything was an echo chamber for his worst thoughts, so they just became more and more dominant.

what he needs: To put it simply, he needs Emma and Ray to cut to the chase and slap him across the face and make him take care of himself. He needs to be forced to see everything for what it really is- this edgy 14 year old committing atrocities to feel better about himself? He needs to be told that what he’s doing is irrational, because in reality, it is. There are better solutions that he’s ignoring, both to his own suffering and the demons, and the way he’s going now no one will truly be happy because of it, that there is no requirement that things be perfect and this bullshit doesn’t make him stronger. He needs someone responsible to take the fucking dagger out of his hands. He also needs someone to babysit him and make him go to bed at a reasonable time.

i describe his personality through songs on my spotify playlist for him:

-outrunning karma by alec benjamin: this one super applies because it calls him out for making shitty decisions, being manipulative and a liar, and having blood on his hands in a very calm and subdued manner, that he knows this is wrong and yet he chooses to keep running faster and faster towards destruction, that he means to escape it through death

-empty by boyinaband and jaiden: yes this is a song about anorexia yes it also applies to norman i’m not saying norman literally has an eating disorder (but honestly it wouldn’t be far out of character if he did) but metaphorically this applies to his method of ignoring his needs, both emotional and physical, in favor of seeming in control 

-toxic thoughts by faith marie: this one speaks to his gifted kid trauma. Man’s got perfectionism running his entire soul. He’s terrified of failing, because he’s always been at the very top, he’ll beat himself up over any miniscule mistake and forces himself to keep at bad habits that keep him Productive, but he won’t ask for help no matter how much he’s suffering because that would be failing, he fights with his mind, this song basically tells him “yeah i feel you but you need to stop that”

-no time to die by billie eilish: ignore the romantic overtones but this is emma and norman, emma who trusted norman and was lied to, betrayed, for norman’s greater good, and norman who refuses to feel or hurt because of it, who refuses to apologize or see himself as wrong, pushes forward because he’s going to Pass Away

-achilles come down by gang of youths: hhhhh it's like. His vibe. Obviously you can disregard the lifestyle specific shit but it's. It's achilles come down you have to understand it’s like the same deal as friend, please just like french and longer

-friend, please by 21 pilots: i feel like i don't have to explain this one but it’s more to the manga (not the anime where he kind of figures out he done did wrong by himself instead of committing unforgivable sins and still going yeah this is valid before emma is like holy fuck). He is like sorry emma I cannot fix anything I’m going to die :) *coughs blood* and emma going like stop it stop it stop it fuck you see you fucked up and i forgive you just stop don’t walk away while he’s like “no<3”

why im a repressed little norman kinnie even tho he’s my exact opposite: I don’t generally kin ppl like norman, honestly he’s an infj I have no clue how it happened but I’m pretty sure it’s because of my intense desire to project onto a little man who cannot lift a milk jug and has chronic pain and decides you know what I AM tired of being nice i DO wanna go apeshit. Also he’s a twink. A little bastard. He’s a terrible person and I go mood every time he does anything. I said mood when he fell out of a tree. Don’t know what this says about me, I swear I wouldn’t commit no genocide. He’s like the inverse of Yoichi Saotome, and somehow i kin him too. Damn.

Miscellaneous headcanons:

-man’s SO attached to his william minerva cloak. He’s a wispy little bitch, you know he’s wearing that thing inside the house, he’s fucking cold. It also makes him Look Important he can retreat into it like an emo middle schooler with an oversized sweatshirt

-although you could probably get Mad street cred from having two whole brands you know he’s not gonna whip it out and show off his lambda thing he’s incredibly self conscious and his chest hasn’t seen the sun in years

-norman’s got MAD laundry skills to be able to wear like, all white all the time while constantly murdering people. I think he’s the only one who knows to do the laundry. And Ray is the only one who knows how to cook.

-but even then there’s gotta still be a few questionable stains on that thing, but if anyone asks he’s like “ketchup” “I’ve literally never seen you eat anything with that much color” “ketchup :)” *coughs blood*

-he’s probably thought “well i have not literally coughed blood yet today so I am not legally obligated to take care of myself”

-He probably adopted much of his current personality from taking on the persona of william minerva. I’m calling him out for being like me, he’s a blank motherfucker, he absorbs personality traits from characters he plays! He’s just not in theatre so it’s a bit more intense!

-the first time he sees barbara Eating Demon Meat he kinda stares and goes oh cool! not for me and violently exits the room. Like it's hilarious bc he thinks that's really gross on a moral level though he understands why she would do it 

-Which is even funnier bc I’m not sure about the canon on this but there was That Chapter Cover that one time that kinda seemed to imply norman eating demon meat which i absolutely latched onto because I’m terrible. He was just politely eating it. With a knife and fork like why dude. As to a possible reason for him doing that I can come up with, of course barbara does it out of spite, but man we don’t know the properties, if it had some sort of painkilling aspect to it or it was like, caffeine, you know he would, but he would Definitely not talk about it

-I kinda disagree with what the anime did in episode eight? It was good I liked it and the imagery was fantastic but also have you considered Norman could not kill someone with his own hands if he tried, or even physically injure them? That’s what his minions are for shawty. That doesn’t make it any less bad, of course, but the manga captured it perfectly by the fact of he carries around a dagger and a scepter in the capitol battle, but he never even raises it out of more than intimidation. He walks through calmly like he’s not scared at all but he makes sure all the lambda kids do all the actual murder, he just stands there impartially, clearly The Mastermind, as the kids fucking murder the queen of the demons. And I think that’s more profound because he’s, a coward. And he doesn’t realize being the one who orders the strike makes you just as responsible as the one who sticks the knife in someone. The knife is just there to Compensate™  for the fact that he weighs like eighty pounds.

-he’s more of like lady macbeth (because he’s a girlboss) than macbeth himself. He has blood on his hands, but it’s the kind of blood that you can’t wash off. He never killed anyone himself, and he cannot admit he never would have been able to.

-the last thing is that there are definitely epic things about the anime, episode 8 was my favorite so far, goddamn that imagery and the bitch walking through the city while it burns down with the screaming asmr going on behind him my god. We stan. But like the downside of, letting Emma and Ray get to him before he commits first degree murder makes the whole thing lose a lot of his value. In the manga (oh my god look at me being a pretentious manga fan please) it fit more of his ideas- he never backed down, and he planned for Emma coming and trying to stop him. Of course he wanted Emma to stop him, he wanted it with all his fucking heart he was pleading for it to happen but the man wouldn’t give himself what he wanted if he was held at gunpoint. He knew she’d come and he made absolutely sure she wouldn’t be able to stop him. So when she came and he said “you’re too late”??? It kind of said it all, in the fact that he was disappointed that he got his way. He still thought he did the right thing, but deep down there where he shoved all his thoughts and feelings he desperately wanted to be saved from himself.

So yeah, those are my thoughts. Feel free to eviscerate me if these are not Correct he is just my favorite girlboss who I feel the need to yell at

[esoteric-codes]

Interview with David Madore

David Madore is responsible for one of the best-known and most-confounding esolangs of all time: Unlambda. The language is based on the SKI combinator calculus, a super-minimalist computational system used in the mathematical analysis of algorithms, but considered impractical for coding. In Unlambda everything is a function that takes a single variable, so there are no indicators like ()s to take parameters. Like the SKI calculus, it entirely eschews variables and the lambda indicator, and so is described as lambda-without-the-lambda. If this is entirely new to you, the second half of this post is a nice introduction. Madore, an accomplished mathematician, is also responsible for languages that play at the edges of infinity. His main web presence is his webpage from his undergrad days thirty years ago, which remains a fascinating portrait of the sincerity, optimism, and banalities of the early Web. He is also on Twitter with a handle derived from his engagement with The Hackers' Zen.

In response to a flurry of questions about creating Unlambda, Madore gives us some notes:

I think I came up with Unlambda in 1999. I knew very little about other esolangs at the time (and I still don't know much about them), that is to say, I knew a bunch of names but didn't try programming in any of them.

I was reading a book on logic that mentioned the Hilbert-style axioms (S: (A⇒B⇒C)⇒(A⇒B)⇒A⇒C; K: A⇒B⇒A; and I: A⇒A) and I was simultaneously thinking about the Curry-Howard correspondence between proof systems and typing systems, so I thought about what these axioms meant in terms of functional programming and how we could use them to re-explain the lambda-calculus without lambdas: this is how I came up with Unlambda. The 'c' function is because I was also fascinated by call/cc at the time (and still am, I guess), especially as it also has a nice interpretation in the Curry-Howard correspondence as Peirce's law (((A⇒B)⇒A)⇒A). The 'd' is clearly an error of mine, I should never have included it, it's unnecessary (I thought it might be, but it was a misunderstanding on my part of how normalization proceeds) and it makes the language far less elegant (on the other hand, it makes the interpreter's job a bit more complex, which is maybe fair for an obfuscated programming language).

At the time, I was thinking about proposing another obfuscated programming language in a somewhat similar flavor, where everything would be written in continuation-passing-style (no function ever returns), but never got around to it.

I think Unlambda got some attention from various teams writing compilers with multiple front-ends or stuff like that, or in compilation courses, because it can be an interesting test case, toy example or illustration.

I once had a page in Wikipedia because of Unlambda, but it was later decided that I wasn't notable enough. :-(

[Ed note: while Madore's page is gone, Unlambda's lives on]

READ MORE...

[grammar-lab]

28/03

Because of a small incident that happened yesterday (that I will not mention here) I decided to stay and work from home today.

And, I felt way better, mostly because there were ideas and things I was too afraid to do in the lab (Like work on my own lambda calculus interpreter, when they have ACGtk) But in the end, I was very very productive and laid a solid foundation for my program. I also re-read the computational semantics course in its entirety and noticed that there were a few small sections here & there that I did not completely understand.

I also felt more relaxed as I took breaks whenever I wanted and was able to simply watch videos and laugh during those breaks (this couldn't happen in the lab, we never laugh :( )

Anyway, it was a great day, and I've decided to stay and work from home tomorrow as well.

[rafaeltheraven]

idk just gonna put some thoughts here since I just watched the last episode (very excited for season 2)

Pantheon was a very welcome surprise. Except for /co/ I didn't really see anybody talk about it, but those that did absolutely love it. The same thing happened with Severance, which was also a brilliant sci-fi show about what it means to be human stuck on a shit streaming service (insert overused meme here).

I remember when Mr Robot came out people were talking about how realistic and accurate it was, but Pantheon does computer science way better imo. Yeah, a lot of it is sci-fi magic, you just have to accept that. But instead of just going "Oh you use Gnome? I use KDE!" It shows at least a surface level knowledge of actual computer science subjects (a discussion on dining philosophers, lambda calculus being mentioned at all, showing htop and irc chats etc.). From what I read, there's also a lot of specific references to current affairs that I'm not particularly aware of, but it shows that the writers really did their research, which is more than I can say of most shows.

It's just nice to have an (american) adult animated series which both uses the animation to show things you could not show in another medium and also portrays a thought-provoking drama. I'd probably put this up there with Venture Bros when it comes to western animated shows, but even if we include anime Pantheon ranks pretty high. I'll probably try to read the original short stories the show was based on now.

Just wish more people knew about it.

Pantheon is an upcoming adult animated series based on a collection of science fiction short stories by Ken Liu. Maddie, a bullied teen, receives mysterious help from someone online. The voice cast includes Daniel Dae Kim, Katie Chang, Paul Dano, and many more.

Pantheon begins streaming September 1 on AMC+.

[kanchanawijerathne-blog]

Tech For Today Series - Day 1

This is first article of my Tech series. Its collection of basic stuffs of programming paradigms, Software runtime architecture ,Development tools ,Frameworks, Libraries, plugins  and JAVA.

Programming paradigms

Do you know about program ancestors? Its better to have brief idea about it. In 1st generation computers used hard wired programming. In 2nd generation they used Machine language. In 3rd generation they started to use high level languages and in 4th generation they used advancement of high level language. In time been they introduced different way of writing codes(High level language).Programming paradigms are a way to classify programming language based on their features (WIKIPEDIA). There are lots of paradigms and most well-known examples are functional programming and object oriented programming.

Main target of a computer program is to solve problem with right concept. To solve problem it required different concept for different part of the problems. Because of that its important that programming languages support many paradigms. Some computer languages support multiple programming paradigms. As example c++ support both functional programming and oop.

On this article we discuss mainly about Structured programming, Non Structured Programming and event driven programming.

Non Structured Programming

Non-structured programming is the earliest programming paradigm. Line by line theirs no additional structure. Entire program is just list of code. There’s no any control structure. After sometimes they use GOTO Statement. Non Structured programming languages use only basic data types such as numbers, strings and arrays . Early versions of BASIC, Fortran, COBOL, and MUMPS are example for languages that using non structures programming language When number of lines in the code increases its hard to debug and modify, difficult to understand and error prone.

Structured programming

When the programs grows to large scale applications number of code lines are increase. Then if non structured program concept are use it will lead to above mentioned problems. To solve it Structured program paradigm is introduced. In first place they introduced Control Structures.

Control Structures.

Sequential -Code execute one by one

Selection - Use for branching the code (use if /if else /switch statements)

Iteration - Use for repetitively executing a block of code multiple times

But they realize that control structure is good to manage logic. But when it comes to programs which have more logic it will difficult to manage the program. So they introduce block structure (functional) programming and object oriented programming. There are two types of structured programming we talk in this article. Functional (Block Structured ) programming , Object oriented programming.

Functional programming

This paradigm concern about execution of mathematical functions. It take argument and return single solution. Functional programming paradigm origins from lambda calculus. " Lambda calculus is framework developed by Alonzo Church to study computations with functions. It can be called as the smallest programming language of the world. It gives the definition of what is computable. Anything that can be computed by lambda calculus is computable. It is equivalent to Turing machine in its ability to compute. It provides a theoretical framework for describing functions and their evaluation. It forms the basis of almost all current functional programming languages. Programming Languages that support functional programming: Haskell, JavaScript, Scala, Erlang, Lisp, ML, Clojure, OCaml, Common Lisp, Racket. " (geeksforgeeks). To check how lambda expression in function programming is work can refer with this link "Lambda expression in functional programming ".

Functional code is idempotent, the output value of a function depends only on the arguments that are passed to the function, so calling a function f twice with the same value for an argument x produces the same result f(x) each time .The global state of the system does not affect the result of a function. Execution of a function does not affect the global state of the system. It is referential transparent.(No side effects)

Referential transparent - In functional programs throughout the program once define the variables do not change their value. It don't have assignment statements. If we need to store variable we create new one. Because of any variable can be replaced with its actual value at any point of execution there is no any side effects. State of any variable is constant at any instant. Ex:

x = x + 1 // this changes the value assigned to the variable x.

          // so the expression is not referentially transparent.

Functional programming use a declarative approach.

Procedural programming paradigm

Procedural programming is based on procedural call. Also known as procedures, routines, sub-routines, functions, methods. As procedural programming language follows a method of solving problems from the top of the code to the bottom of the code, if a change is required to the program, the developer has to change every line of code that links to the main or the original code. Procedural paradigm provide modularity and code reuse. Use imperative approach and have side effects.

Event driven programming paradigm

It responds to specific kinds of input from users. (User events (Click, drag/drop, key-press,), Schedulers/timers, Sensors, messages, hardware interrupts.) When an event occur asynchronously they placed it to event queue as they arise. Then it remove from programming queue and handle it by main processing loop. Because of that program may produce output or modify the value of a state variable. Not like other paradigms it provide interface to create the program. User must create defined class. JavaScript, Action Script, Visual Basic and Elm are the example for event-driven programming.

Object oriented Programming

Object Oriented Programming is a method of implementation in which programs are organized as a collection of objects which cooperate to solve a problem. In here program is divide in to small sub systems and they are independent unit which contain their own data and functions. Those units can reuse and solve many different programs.

Key features of object oriented concept

Object             - Objects are instances of classes, which we can use to store data and perform actions.

Class                - A class is a blue print of an object.

Abstraction     - Abstraction is the process of removing characteristics from ‘something’ in order to reduce it to a set of essential characteristics that is needed for the particular system.

Encapsulation - Process of grouping related attributes and methods together, giving a name to the unit and providing an interface for outsiders to communicate with the unit

Information Hiding - Hide certain information or implementation decision that are internal to the encapsulation structure

Inheritance     - Describes the parent child relationship between two classes.

Polymorphism - Ability of doing something in different ways. In other words it means, one method with multiple implementation, for a certain class of action

Software Runtime Architecture

Languages can be categorized according to the way they are processed and executed.

Compiled Languages

Scripting Languages

Markup Languages

The communication between the application and the OS needs additional components. Depends on the type of the language used to develop the application component. Depends on the type of the language used to develop the application component.

This is how JS code is executed.JavaScript statements that appear between <script> and </script> tags are executed in order of appearance. When more than one script appears in a file, the scripts are executed in the order in which they appear. If a script calls document. Write ( ), any text passed to that method is inserted into the document immediately after the closing </script> tag and is parsed by the HTML parser when the script finishes running. The same rules apply to scripts included from separate files with the src attribute.

To run the system in different levels there are some other tools use in the industry.

Virtual Machine

Containers/Dockers

Virtual Machine

Virtual machine is a hardware or software which enables one computer to behave like another computer system.

Development Tools

"Software development tool is a computer program that software developers use to create, debug, maintain, or otherwise support other programs and applications." (Wikipedia) CASE tools are used throughout the software life cycle.

Feasibility Study        - First phase of SDLC. In this phase gain basic understand of the problem and discuss according to solution strategies (Technical feasibility, Economical feasibility, Operational feasibility, Schedule feasibility). And prepare document and submit for management approval

Requirement Analysis - Goal is to find out exactly what the customer needs. First gather requirement through meetings, interviews and discussions. Then documented in Software Requirement Specification (SRC).Use surveying tools, analyzing tools

Design                         - Make decisions of software, hardware and system architecture. Record this information on Design specification document (DSD). Use modelling tools

Development             - A set of developers code the software as per the established design specification, using a chosen programming language .Use Code editors, frameworks, libraries, plugins, compilers

Testing                        - Ensures that the software requirements are in place and that the software works as expected. If there is any defect find out developers resolve it and create a new version of the software which then repeats the testing phase. Use test automation tools, quality assurance tools.

Development and Maintenance - Once software is error free give it to customer to use. If there is any error resolve them immediately. To development use VMs, containers/ Dockers, servers and for maintenance use bug trackers, analytical tools.

CASE software types

Individual tools

Workbenches

Environments

Frameworks Vs Libraries Vs Plugins

Do you know how the output of an HTML document is rendered?

This is how it happen. When browser receive raw bytes of data it convert into characters. These characters now further parsed in to tokens. The parser understands each string in angle brackets e.g "<html>", "<p>", and understands the set of rules that apply to each of them. After the tokenization is done, the tokens are then converted into nodes. Upon creating these nodes, the nodes are then linked in a tree data structure known as the DOM. The relationship between every node is established in this DOM object. When document contain with css files(css raw data) it also convert to characters, then tokenized , nodes are also formed, and finally, a tree structure is also formed . This tree structure is called CSSOM Now browser contain with DOM and CSSOM independent tree structures. Combination of those 2 we called render tree. Now browser has to calculate the exact size and position of each object on the page with the browser viewpoint (layout ). Then browser print individual node to the  screen by using DOM, CSSOM , and exact layout.

JAVA

Java is general purpose programming language. It is class based object oriented and concurrent language. It let application developers to “Write Once Run anywhere “. That means java code can run on all platforms without need of recompilation. Java applications are compiled to bytecode which can run on any JVM.

Do you know should have to edit PATH after installing JDK?

JDK has java compiler which can compile program to give outputs. SYSTEM32 is the place where executables are kept. So it can call any wear. But here you cannot copy your JDK binary to SYSTEM32 , so every time you need to compile a program , you need to put the whole path of JDK or go to the JDK binary to compile , so to cut this clutter , PATH s are made ready , if you set some path in environment variables , then Windows will make sure that any name of executable from that PATH’s folder can be executed from anywhere any time, so the path is used to make ready the JDK all the time , whether you cmd is in C: drive , D: drive or anywhere . Windows will treat it like it is in SYSTEM32 itself.

[felord]

CSE505 Problem2-Lambda Calculus  Solved

CSE505 Problem2-Lambda Calculus Solved

  Two important reduction strategies in the lambda-calculus are leftmost-outermost (LO) and leftmost-innermost (LI). For each of the two operations mentioned below, compare LI and LO with respect to the number of reduction steps needed to compute the normal form as a function of the size of the input Church numerals.  Assume the Church numeral for number n is of size n. The ‘add’ operation – for…

[methinking]

The rise of Functional Programming languages

When I was starting college, the language used to teach the first computer science class was SML or Standard ML. It was a simple but powerful language which was strongly functional. But unfortunately I did not know what a functional language was back then. Soon the classes moved to the more main stream imperative languages like C, C++ and Java and I ignored the functional programming paradigm. Fast forward a years of working and functional programming is hard to ignore. 

It is not the latest rage in programming but definitely an on-going development. Functional Programming languages are being adopted at an ever increasing pace and are considered part of  "modern languages" at times. Functional programming has its roots in lambda calculus and Lisp. It was mostly academic for a long time and only in last decade has it began getting used in commercial software development.

Functional programming is a programming paradigm. Hence a lot of languages which are not designed to be functional first can be used to implement functional concepts. Languages like Scala live in the area between functional and imperative languages. F#, for instance, lives on the other extreme and strongly pushes for functional paradigm. All the most profound languages, C++, Java have been getting functional features in their latest upgrades. 

Functional programming has a few primary features. First and foremost being functions as first class citizens. That usually translates into being able to define functions anywhere in code and being able to pass them around as you would any other value.  For example:

def addTwo(number): return number + 2 def addFunction(number, fn): value = fn(number) return value print(addFunction(5, addTwo)) # returns 7

Second biggest feature for me would be pure functions. Pure functions are functions without any side effects. A simple example of no side effect vs side effect would look something like this:

class Number: def __init__(value):   self.value = value def addTwo(number): number.value = number.value + 2 def addTwoNoSideEffect(number): value = number.value + 2 return Number(value)

Functional programming languages are usually also accompanied with higher order functions out of the box. This usually makes most code bases smaller and that is also one of the reason the industry is seeing a movement towards functional languages. For instance, if I want to multiply all the numbers in a list by 2:

nums = [1, 2, 3, 4] #imperative for i in range(len(nums)): nums[i] *= 2 #functional nums = map(lambda x: x*2, nums)

Such higher order functions are also a reason why some of the functional languages are used for scientific purposes in terms of data manipulation and churning.  All these come together to make a powerful combination. But there are some drawbacks too. It is important to know the pros and cons of a choice before making it. To simply note, the no side effect way of doing things can lead to temporary high memory usage. If that is not allowed for your application, maybe if it has extensive memory constraints, then functional programming might not be the correct choice. There are other systematic choices, some handled by compiler optimizations and hence not that impact full and some are more obvious as mentioned above. 

I have been working with functional programming lately with a combination of Scala, Java8, Python and F#. Some are more functional than others but all have the feature set to allow for functional first code. For any developer, I highly recommend checking out functional programming even if your language of choice is not functional first. As everything, knowledge only helps guide our future decisions. 

[sineofpsi-old]

Classical E&M III: The Electric Ambiance of Continuous Distributions

Multidimensional objects are more complex than points, but there is a method of relating the two. We can break multidimensional objects into tiny pieces that are qualitatively point-like. As the pieces become infinitesimally small, this relationship becomes exact:

In this way, multidimensional objects are translated into collections of points. We know the electric properties of point charges, and through that knowledge we'll derive the electric properties of generic charge configurations.

I. The Electric Field of a Line Charge, Infinitesimals

Consider a curve $C^\prime$ in $3$-space with total length $\ell^\prime$:

To begin, let’s assume this curve is uniformly charged.

“Charged” means it emits photons, whereas

“Uniformly” means it possesses only one type of charge (positive or negative) and each point of $C^\prime$ is equally likely to emit photons

For the time being, we restrict ourselves to positive-type charges. This means we don’t have to worry about positive and negative charges cancelling each other out. $C^\prime$ as a whole will be emitting photons at a certain rate $r_{C^\prime}$ proportional to its total charge $Q^\prime$.

Our intent--as mentioned in the intro--is to chop $C^\prime$ into tiny pieces and approximate each piece as a point charge. Thus, we partition $C^\prime$ into some large number $N^\prime$ of equal-length pieces:

Each partition piece has length $\Delta \ell_i^\prime = \ell^\prime /N^\prime$. In order to reproduce the total rate of photon emission (and because the charge is distributed uniformly over the curve), each piece needs to provide an emission rate equal to $r_{C^\prime}/N^\prime$. This implies a curious result: as $N^\prime$ gets larger, the number of pieces increase, and every piece contributes less to the total emission rate. Because emission rate is proportional to total charge (because there’s no positive/negative cancellations), any individual piece possesses less charge. In the limit that $N^\prime\rightarrow \infty$, the curve becomes equivalent to infinitely-many point charges, where each point has no charge!

Charge is evidently an inappropriate local quantity for a uniformly charged curve. This presents a roadblock on our path to continuous charge configurations. We can circumvent it by finding an alternative local quantity to work with. As usual, the trick lies in the problem: charge vanishes as the length of our partition pieces go to zero, so instead of considering charge outright let’s consider the linear charge density $\lambda_i^\prime$ per piece:

This has units of charge per length. For a uniformly charged curve cut into $N^\prime$ equal-length pieces, each piece has length $\Delta \ell^\prime_i = \ell^\prime /N^\prime$ and charge $\Delta q^\prime_i = Q^\prime /N^\prime$. Therefore,

where $\lambda_{C^\prime}$ is the total linear charge density of $C^\prime$, a finite result! Linear charge density let’s us rewrite the charge of each partition piece as,

We found this expression by considering a uniformly-charged, positively-charged curve, but--like we did when developing the integral calculus--we can reweight the charge contribution of each piece, so that $\lambda^\prime_i$ is a function along $C^\prime$. In doing so, we can access nonuniform line charges which possess differing amounts of (positive and negative) charge along their lengths.

Specifically, as the partition becomes infinitely fine, each point $\vec{r}^{\text{ }\prime}$ along the curve can be uniquely identified with an infinitely-short piece of curve possessing a certain linear charge density. The linear charge density thereby becomes a real-valued function $\lambda^\prime(\vec{r}^{\text{ }\prime})$ along $C^\prime$.

Now is as good as any time to introduce infinitesimal quantities. As $N^\prime \rightarrow \infty$, the charge of each piece technically vanishes ($\Delta q^\prime_i\rightarrow 0$). However, we can imagine “stopping short” of infinity and--in doing so--obtain a point-by-point identification of curve pieces yet somehow without fully removing the charge of each piece. If this seems too good to be true, it’s because it is. We must be careful to finish off that $N^\prime\rightarrow \infty$ limit before calling our calculation complete.

Warning: Infinitesimals are used widely throughout physics, oftentimes when juggling multiple limits. For example, sometimes we’ll break time into infinitesimal intervals while simultaneously breaking shapes into infinitesimal pieces. These are technically two different limits, so mixing them can cause trouble (recall how cautious we had to be when manipulating the double-limit in the Dirac delta post). Exercise caution when encountering infinitesimals in the wild.

It’s in this spirit that we talk about the infinitesimal charge $dq^\prime$ and the infinitesimal length $d\ell^\prime$ of each curve piece, which are related through the line charge density function $\lambda^\prime(\vec{r}^{\text{ }\prime})$ along $C^\prime$:

Hence, the infinitesimal limit associates every point $\vec{r}^{\text{ }\prime}$ with a point charge of strength $dq^\prime(\vec{r}^{\text{ }\prime})$. Of course, point charges produce electric fields, and $dq^\prime(\vec{r}^{\text{ }\prime})$ is no exception:

Because the charge is infinitesimally small, the electric field is also infinitesimally small. This encourages us to write the electric field with the lowercase $d$ we associated with other infinitesimals, yielding the label $d\vec{E}$. We also change the subscript on $d\vec{E}$ to reflect our intent to add up all of the electric field contributions along the curve:

We can rewrite this as a line integral contribution via the line charge density. In this case, an individual contribution becomes,

Note that this is a vector-valued object. In our integral calculus series, we discussed integrals of real-valued functions, but not vector-valued functions. How do we proceed?

II. Integrating Vector-Valued Functions

It turns out integration of vector-valued functions follows without much trouble if we assume that linearity of integrals plays nice with $3$-space vector operations.

Particularly, any element of $3$-space can be uniquely associated with three real numbers, called rectangular coordinates:

These coordinates are useful because they allow us to write vector addition and scalar multiplication with ease. Given two $3$-vectors $\vec{r}_1\equiv (x_1,y_1,z_1)$ and $\vec{r}_2\equiv (x_2,y_2,z_2)$, we can use vector addition to build a new $3$-vector labeled $\vec{r}_1+\vec{r}_2$, defined as,

Furthermore, given a real number $\alpha$, we can use scalar multiplication to transform $\vec{r}=(x,y,z)$ into a $3$-vector $\alpha\vec{r}$ defined as,

We will assume that integrals behave nicely with these properties, and treat the unit vectors $\hat{x}$, $\hat{y}$, $\hat{z}$ as constants. (This is true of our rectangular coordinates; there exist other coordinate systems of $3$-space which aren’t so lucky.)

Given a function $\vec{f}$ that maps $3$-vectors to $3$-vectors, we may decompose $\vec{f}$ into components,

Then we define the integral over a vector-valued function $f$ as a component-by-component operation, like so:

Although I’ve written this as a volume integral, the sentiment holds equally true for surface integrals and line integrals.

With this definition in mind, we can now add up the electric field contributions due to all the point charges along $C^\prime$, so that the electric field at $\vec{r}$ due to a line charge $\lambda^\prime$ equals,

We then find the force experienced by a target charge $q$ at $\vec{r}$ due to $\lambda^\prime$ by multiplying $\vec{E}_{C^\prime}(\vec{r})$ by $q$, so that we find,

III. The Electric Field of a Surface Charge

Let’s move on to the equivalent analysis of a surface charge. Again, by evenly distributing a finite amount of positive charge $Q^\prime$ over a surface $S^\prime$ we ensure that any point of $S^\prime$ contains only infinitesimal amounts of charge, thereby eliminating charge as a useful local metric of electric behavior.

Instead, a more appropriate local measure is the surface charge density $\sigma^\prime$. Its definition proceeds just like the line charge density: we divide $S^\prime$ into infinitely-many infinitely-small two-dimensional regions, so that each point $\vec{r}^{\text{ }\prime}$ on $S^\prime$ is uniquely identified with its own infinitesimal region. If the infinitesimal piece at $\vec{r}^{\text{ }\prime}$ has area $dA^\prime$ and charge $dq^\prime$, then the surface charge density $\sigma^\prime$ at $\vec{r}^{\text{ }\prime}$ is defined as,

such that it has units of charge per area. This allows us to transform the charged surface $S^\prime$ into point charges, each contributing an electric field amount equal to,

which combine to yield the electric field at $\vec{r}$ due to a surface charge $\sigma^\prime$:

as well as the force experienced by a target charge $q$ at $\vec{r}$ due to $\sigma^\prime$:

IV. The Electric Field of a Volume Charge

Finally, we distribute charge over a volume $V^\prime$. The appropriate local measure of charge is volume charge density, labelled $\rho^\prime(\vec{r}^{\text{ }\prime})$. To define volume charge density, we break $V^\prime$ into infinitesimals, each possessing some volume $d\tau^\prime$ and some charge $dq^\prime$. Subsequently, $\rho^\prime(\vec{r}^{\text{ }\prime})$ is defined as the multiplicative factor between the two.

Volume charge density has units of charge per volume. With this definition in place, each point charge contributes some infinitesimal amount of electric field,

which collectively generate the electric field at $\vec{r}$ due to a volume charge $\rho^\prime$:

and enable us to write the force experienced by a target charge $q$ at $\vec{r}$ due to $\rho^\prime$:

V. Combining Charge Densities into Generic Configurations

A fully generic charge density may be a combination of all of these objects. Suppose we have a charge configuration $CC^\prime$ made of point charges $q_{i_0}^\prime$ at $\vec{r}^{\text{ }\prime}_{i_0}$, line charges $\lambda^\prime_{i_1}$ on curves $C^\prime_{i_1}$, surface charges $\sigma^\prime_{i_2}$ on surfaces $S^\prime_{i_2}$, and volume charges $\rho^\prime_{i_3}$ on volume charges $V^\prime_{i_3}$. The electric field generated by $CC^\prime$ must be, by the principle of superposition,

If we place a point charge $q$ at a position $\vec{r}$, it will experience a force due to $CC^\prime$ equal to,

There we go: we now know how to calculate the electric properties of any distribution of charges throughout $3$-space. It’s worth noting that if we wanted, we could utilize Dirac deltas to express all charge densities as volume charge densities. Given a volume $V^\prime$ that encompasses a point charge $q^\prime$ at $\vec{r}^{\text{ }\prime}$, a line charge density $\lambda^\prime$ on $C^\prime$, and a surface charge density $\sigma^\prime$ on $S^\prime$, physicists will sometimes write,

Note that the units work out appropriately for these to be volume charge densities. However, these expressions are technically incorrect because Dirac deltas may only exist within integrals. Only in that context may these expressions be utilized without jeopardizing mathematical consistency.

And so, we’ve generalized the “source” aspect of electrostatics. Next week, we generalize our target charges and calculate the force between two generic charge configurations. See you then!

Thanks for reading today’s post! Follow sineofpsi.tumblr.com for new physics content every Friday. Have questions about anything we’ve talking about? Send me an ask. I’m wishing you the best!