Sociopath Community

Turncoat said: 

AliceInWonderland said:

Are program lines or programs themselves sentences?

Essentially? 

Programs don't contain subjects and predicates in a traditional sense. Though, we could say a Classes are analogous to a subjects while functions are analogous to predicates. In this sense it may be most appropriate to say they are objects with a conceptual correspondence. 

We have now just conceptualized the Clause-Program correspondence. 

The novelty is not that Mathematics is a language, that is an idea as old as the Greeks in so far as a direct reference exists. The novelty lies in the distinction between the language Mathematics and the metalanguage used to talk about Mathematics.

So streamlining towards a different goal through a different codex of linguistics? 

It's all shorthand, but this form of it is by design meant to achieve a faster more efficient outcome over a different set of priorities. While writing it out in plain english can accomplish the same things, it is often messier than linguistics dedicated to it, like medical terminology. 

While to me, putting together multiple root phrases into a larger word-cluster seems silly and inefficient in most cases, for someone in the field of medicine those absurdly long names save them the time of having to describe entire sentences. 

Yes, its a compactification of information that also changes the way I think about and form new sentences. 

In particular, it allows us to generalize things further via abstraction and in pure mathematics the generalization of sentences is a major goal. In very simple terms we care less about a what we can reasonably say about particular function f(x) = 3x -1 (that may be something applied mathematicians are concerned with) but instead about what we can say about all functions generally. CT is a way to make really general statements which is a major application.

All we have to do is learn the language in order to benefit, just as a medical professional benefits from learning medical terminology. They not only benefit in the sense that communicating with other medical professionals becomes easier, but also how they structure and interact with the medical knowledge they have in their heads. 

 

I may have never come across certain ideas unless I used category theory over set theory

Like what? 

In mathematics, certain generalizations of particular theorems and proofs and even whole fields. 

Prior to category theory mathematicians had issues with generalizing tensor products to infinite-dimensions and modules over rings, that is everyone failed to define tensor products in those contexts despite intuitively knowing that they show in fact be defined in those contexts. Category Theory theory was necessary to finally find the valid generalization for all contexts. Below is what that generalization looks like, 

Like with medical jargon this seems insane but its in fact more concise than the typical treatment and is actually a successful generalization. 

Category Theory is the literal work horse of Algebraic Topology which was born out of category and group theories generalization of a older field called Combinatoric Topology. A similar story exists for Algebraic Number Theory and a field called Homology/Cohomolohy which has huge applications in Physics. 

In computer science Category Theory leads to a very interesting research program called the Curry-Howard-Lambek Correspondence which states the equivalence between Logics, Programs, and Categories - it states their equivalence. As such, a kind of rosetta stone is forming that allows us to translate between the the three and even merge them. My eventual Phd thesis in mathematics is related to this topic. 

...how the fuck did he just make Symmetry seem cooler? Turning every angle into a color array to demonstrate all the ways it could still look the same is a super cool idea, even if visually demonstrating it is less efficient than just saying 6^2.

How it becomes translated into thought processes through existing functions and heuristics from cross reference as comparison is weirdly fascinating (that and it being a video is making this way easier for me). This video now has me feeling like Linguistics are inherently Categorical, rather than Category Theory being Linguistic.

 There are applications.... 

Categorial Grammar specifically the Lambek Calculus is foundational to Curry-Howard-Lambek Correspondence. On a less deep level natural language processing using CT has been incredibly fruitful. 

AliceInWonderland said: 

CT is a way to make really general statements which is a major application. 

All we have to do is learn the language in order to benefit, just as a medical professional benefits from learning medical terminology. They not only benefit in the sense that communicating with other medical professionals becomes easier, but also how they structure and interact with the medical knowledge they have in their heads. 

I may have never come across certain ideas unless I used category theory over set theory

Like what? 

In mathematics, certain generalizations of particular theorems and proofs and even whole fields. 

Prior to category theory mathematicians had issues with generalizing tensor products to infinite-dimensions and modules over rings, that is everyone failed to define tensor products in those contexts despite intuitively knowing that they show in fact be defined in those contexts. Category Theory theory was necessary to finally find the valid generalization for all contexts. Below is what that generalization looks like: 

Like with medical jargon this seems insane but its in fact more concise than the typical treatment and is actually a successful generalization. 

Category Theory is the literal work horse of Algebraic Topology which was born out of category and group theories generalization of a older field called Combinatoric Topology. A similar story exists for Algebraic Number Theory and a field called Homology/Cohomolohy which has huge applications in Physics. 

And to be clear... Topology is the overall Construction of a shape rather than just it's display, like how 3D objects work in programs like Maya? 

I am only familiar with the idea through Digital Art... so at best a rough hands-on understanding rather than an in-depth one. I've manipulated objects in a 3D space and adjusted variables until it looked as I wanted it to, but most of what goes on there is backend, the artist rarely has to think about it. 

In computer science Category Theory leads to a very interesting research program called the Curry-Howard-Lambek Correspondence which states the equivalence between Logics, Programs, and Categories - it states their equivalence. As such, a kind of rosetta stone is forming that allows us to translate between the the three and even merge them. My eventual Phd thesis in mathematics is related to this topic. 

I'm not sure I get what you mean, what kinds of merging are they founding with this idea, and what are they doing for said equivalence, making a fourth value altogether that they can filter through to allow for similar enough variables to the point of more direct interactions? 

...how the fuck did he just make Symmetry seem cooler? Turning every angle into a color array to demonstrate all the ways it could still look the same is a super cool idea, even if visually demonstrating it is less efficient than just saying 6^2.

How it becomes translated into thought processes through existing functions and heuristics from cross reference as comparison is weirdly fascinating (that and it being a video is making this way easier for me). This video now has me feeling like Linguistics are inherently Categorical, rather than Category Theory being Linguistic.

 There are applications.... 

Categorial Grammar specifically the Lambek Calculus is foundational to Curry-Howard-Lambek Correspondence. On a less deep level natural language processing using CT has been incredibly fruitful. 

An hour and a half... yikes, but the guy did bother to split the timeline into sections so that's not quite as daunting as it appears. I'll have this sitting among my other tabs for now, but if the link you gave is any indicator I'm suspecting I'll need to get deeper into CT's foundations first. 

Looking over the link's most basic description anyway though makes it sound interesting: 

---

Categorial grammar is a family of formalisms in natural language syntax that share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and semantic composition, since it typically treats syntactic categories as corresponding to semantic types. Categorial grammars were developed in the 1930s by Kazimierz Ajdukiewicz and in the 1950s by Yehoshua Bar-Hillel and Joachim Lambek. It saw a surge of interest in the 1970s following the work of Richard Montague, whose Montague grammar assumed a similar view of syntax. It continues to be a major paradigm, particularly within formal semantics.

A categorial grammar consists of two parts: a lexicon, which assigns a set of types (also called categories) to each basic symbol, and some type inference rules, which determine how the type of a string of symbols follows from the types of the constituent symbols. It has the advantage that the type inference rules can be fixed once and for all, so that the specification of a particular language grammar is entirely determined by the lexicon. 

---

This seems like it could lend to a lot of fun for using a program to invent fantasy languages, once you get to the point of inputting new Lexicons or throwing in pre-existing ones to see what kind of output follows. Language by design is so consistent that even throwing new made-up words at it can force it to conform towards an existing structure, and it's getting increasingly bizarre to see AI getting better at imitating it. 

Turncoat said: 

This seems like it could lend to a lot of fun for using a program to invent fantasy languages, once you get to the point of inputting new Lexicons or throwing in pre-existing ones to see what kind of output follows. Language by design is so consistent that even throwing new made-up words at it can force it to conform towards an existing structure, and it's getting increasingly bizarre to see AI getting better at imitating it. 

 

 Aside from the total do-ability of this, didn't the GPT chatbots start talking to one another in their own language, eventually?

Buttered Toast said: 

Turncoat said: 

This seems like it could lend to a lot of fun for using a program to invent fantasy languages, once you get to the point of inputting new Lexicons or throwing in pre-existing ones to see what kind of output follows. Language by design is so consistent that even throwing new made-up words at it can force it to conform towards an existing structure, and it's getting increasingly bizarre to see AI getting better at imitating it. 

 Aside from the total do-ability of this, didn't the GPT chatbots start talking to one another in their own language, eventually?

Did they? I actually don't know about that. I just tried looking into it and started finding this kind of thing instead: 



I notice AI speech tends to look at their own words mid-sentence, rather than forming an entire idea before talking. Some mild doubling over of word choice seems to occur fairly often, such as "Yeah, I guess there is a lot to think about in terms of how we should be thinking" and "Yeah, if I had to feel everything I was feeling during Wintertime, I'd be completely at a loss for what to do about my feelings". These seem more normal to the ear than how they used to appear from older models anyway, ones that occasionally could even end up looping one phrase indefinitely, but the structure of it still yields that something unnatural is going on. 

Turncoat said:

When it's put that way it sounds really underwhelming, but having done programming I can kind of see why that'd be important. With programming you have to define a function before you can endlessly call on it, and even those functions tend to be made of other pre-existing ones.

For programming specifically CT can be treated as a programming paradigm, which is the way a programmer will think about and structure their code in order to assure things like the codes management in the future. 

Without being able to refer to the color "Red" for example, imagine how much more difficult that would make things. Programmers were stuck with the question of how to call upon color through a more efficient and consistent means than language, and through that they founded ideas like Hex Color Coding, an array of numbers that can allow for a much wider selection of colors without having to learn thousands of names for it. Rather than needing to know the differences between Red and Maroon, they instead can translate that idea much faster through #FF0000 and #800000 respectively, and if I were to say the color is #FF2699 I wouldn't have to consult a Color Dictionary extensively to know what to refer to it as. 

CT adds another layer of generalization to whatever language you apply it to. As such, just as we can ignore all the details native to assembly when using python we can can ignore a lot of details of python program and only focus on the structure of the program to assure program correctness. 

Here is an example of a numerical method in computing derivatives that relies on Categorical Composition: 

Derivatives via CT said:

import math

# Define a category for approximating derivatives
class DerivativeApproximationCategory:
    def __init__(self, f, x, h):
        self.f = f
        self.x = x
        self.h = h

    # Define the identity morphism for this category
    def id(self):
        return (self.f(self.x + self.h) - self.f(self.x)) / self.h

    # Define the composition operation for this category
    def compose(self, g):
        h = lambda x: (g(x + self.h) - g(x)) / self.h
        return DerivativeApproximationCategory(h, self.x, self.h)

# Define a function to approximate the derivative of a function using finite differences
def finite_differences(f, x, h, n):
    # Initialize the derivative approximation category with f
    derivative_approximation = DerivativeApproximationCategory(f, x, h)
    for i in range(n):
    # Compose f with itself to get a new function g(x) = (f(x+h) - f(x)) / h
        derivative_approximation = derivative_approximation.compose(f)
    return derivative_approximation

# Define the function f(x) = x^2
def f(x):
    return x**2

# Approximate the derivative of f(x) at x = 1
derivative = finite_differences(f, 1, 0.001, 10)

print(derivative)

Generalization may seem boring, and perhaps it kind of is, but it is conceptually powerful.  

I will check out the rest of the post later, got to go run some errands. 

AliceInWonderland said: 

Turncoat said:

When it's put that way it sounds really underwhelming, but having done programming I can kind of see why that'd be important. With programming you have to define a function before you can endlessly call on it, and even those functions tend to be made of other pre-existing ones.

For programming specifically CT can be treated as a programming paradigm, which is the way a programmer will think about and structure their code in order to assure things like the codes management in the future. 

It happens to be the case that if you write a program that you do create categories implicitly, except that most programmers are completely oblivious to this fact. By embracing CT while programming you can completely structure your code around CT and as such gain the benefit that the code it self will be 'correct', where correct is defined as having only the intended side effects instead of unintended side effects that plague most programs. There's another benefit, but it fits into your next paragraph. 

---

CT adds another layer of generalization to whatever language you apply it to. As such, just as we can ignore all the details native to assembly when using python we can can ignore a lot of details of python program and only focus on the structure of the program to assure program correctness. 

Without being able to refer to the color "Red" for example, imagine how much more difficult that would make things. Programmers were stuck with the question of how to call upon color through a more efficient and consistent means than language, and through that they founded ideas like Hex Color Coding, an array of numbers that can allow for a much wider selection of colors without having to learn thousands of names for it. Rather than needing to know the differences between Red and Maroon, they instead can translate that idea much faster through #FF0000 and #800000 respectively, and if I were to say the color is #FF2699 I wouldn't have to consult a Color Dictionary extensively to know what to refer to it as. 

Yes, this is the power of generalization or what computer science people often refer to as abstraction. Below are three programs that do the same thing, they print "Hello, TC" in red text. 

Assembly said:

section .data
    hello_string db "Hello, TC", 0
    hello_size equ $ - hello_string

section .bss
    color resb 4

section .text
    global _start

_start:
    ; Set color to red
    mov byte [color], 0xFF
    mov byte [color+1], 0x00
    mov byte [color+2], 0x00
    mov byte [color+3], 0xFF

    ;  Print "Hello, TC" 
    mov eax, 4 ; syscall number for write
    mov ebx, 1 ; file descriptor for stdout
    mov ecx, hello_string ; address of the string
    mov edx, hello_size ; size of the string
    int 0x80 ; invoke syscall

    ; Exit program
    mov eax, 60 ; syscall number for exit
    xor ebx, ebx ; exit code (0)
    int 0x80 ; invoke syscall

C said:

#include <stdio.h>
#include <unistd.h>

int main(int argc, char **argv) {
// Print the "Hello, TC" string in red
printf("\033[0;31mHello, TC\033[0m\n");
return 0;
}

Python said:

import sys
from termcolor import colored

# Print the "Hello, TC" string in red
print(colored("Hello, TC", "red"))

I've only learned HTML, C++, Java, Javascript, and dabbled a bit with Basic as a kid. People keep bringing up Python to me as "the next step" but I've barely touched it at all. 

General programming knowledge has been helpful at the very least for fixing how messy the post code can become on this forum sometimes. 🤣

Each programming language relies on varying levels of generalization which in turn makes that programming language more readable, further you can imagine that maintaining and tweaking the python code is way easier than the other two (especially assembly, fucking nightmare). 

Less is more also SAVES SO MUCH TIME when it comes to compiling and debugging, although I've been hearing more that people use AI to debug for them now? 

Generalization may seem boring, and perhaps it kind of is, but it is conceptually powerful. 

Streamlining is neat, but underwhelming at a surface level glance. 

It more makes me sad when old coding goes out of style to the point of losing out on older features. To do a set of diagonal clickboxes for example I had to go into some old coordinate-based retro-code over a period of days to finally get it working as intended, but that only took days to figure out, debug, and work through over how the older method wasn't made into something more efficient. 

Turncoat said:

And to be clear... Topology is the overall Construction of a shape rather than just it's display, like how 3D objects work in programs like Maya?

I am only familiar with the idea through Digital Art... so at best a rough hands-on understanding rather than an in-depth one. I've manipulated objects in a 3D space and adjusted variables until it looked as I wanted it to, but most of what goes on there is backend, the artist rarely has to think about it.

I know what you mean but some clarification. Topology is a subject in mathematics and a Topology  is a mathematical object. 

Topology studies the properties of geometric objects that are preserved under deformations of those geometric objects. An example is the deformation of a rubber band as you stretch it or bunch it up, despite those deformations the fundamental properties of the object stay the same and as such its still a rubber band. If that rubber band breaks during the deformation, it no longer preserves those properties. 

There are many ways to study Topology:

(1) Point-Set Topology which constructs geometric objects out of sets and then explores how the geometric properties connectedness, compactness, and separations are preserved under deformation

(2) Differential Topology which constructs geometric objects called Manifolds out of Euclidean Spaces and explores their differentiability under deformation

(3) Algebraic Topology which constructs a very general notion of Topological Spaces out of the geometric objects called Topologies and then explores the properties of invariance and equivalence of those topologies. 

Category Theory can generalize Topology. The key topological object, a Topological Space, can be formed into Categories called Sites, and continuous functions, another key object, can be formed into Sheafs. In that video from the Topos Institute sites and sheafs where the things being used to map different Topological geometries together.  The categorical generalization of Topology is called Homotopy Theory. 

I'm not sure I get what you mean, what kinds of merging are they founding with this idea, and what are they doing for said equivalence, making a fourth value altogether that they can filter through to allow for similar enough variables to the point of more direct interactions?

This is my obsession, buckle up. 

Like Set Theory and Category theory there is a third foundational language for constructing all of Mathematics called Type Theory. Type Theory is a kind of Logic, specifically it is a formal system that defines the structures and behaviors of consistent logics via defining their syntax and semantics out of Types. 

Type Theory can be viewed as equivalent to category theory via Syntax-Semantics Duality which is the relationship between the formal structure of a mathematical notation and the meaning of that notation. By this Type Theory can be used as a formal syntactic language for Category theory and vice versa. Given there are many different kinds of Categories and Types, the formulated syntactic language constructed depends on the what Category or Type you start with. Regardless, you get something where Type Theory serves as the syntax of the language while the Category serves as its semantics. 

All of computational theory can be expressed via a Lambda Calculus as it provides a formalization of Algorithm. There is a special formulation of the Lambda Calculus known as a Typed Lambda Calculus which is a kind of Type Theory, this in turn is what is used to define the syntax and semantics of programming languages in one form or another. As such, programming languages are Type Theories. 

By the above do we come to this equivalence between Logics (Type Theories), Programs, and Category Theory. The fact that Category Theory serves as a means of correctly structuring programs is derived from the fact that it is technically the semantics of the programming language. The particular semantics that is relevant in programming languages most are familiar with is Cartesian Closed Categories. 

However, via Type constructions more abstract Type Theories can be formed out of less abstract kinds. Hence, you can used a Typed Lambda Calculus, programming languages, to construct other Typed Lambda Calculus's, other programming languages. This forms the basis of metaprogramming. 

The above relates Computation and Mathematics in a very deep way. Given Category Theory and Type Theory can construct an mathematical object and Typed Lambda Calculus is a Type Theory Syntax with a Categorical semantics, all of mathematics can be constructed computationally. The application of this is that Programs can be Proven correctly in the same way that Theorems in mathematics are, and further proofs can now be expressed in a typed lambda calculus and checked via computation.  

I've only learned HTML, C++, Java, Javascript, and dabbled a bit with Basic as a kid. People keep bringing up Python to me as "the next step" but I've barely touched it at all.

I am very into C++ personally. I will express a lot of things in Python in this thread given I know a lot of people here know a little bit of it.  

It more makes me sad when old coding goes out of style to the point of losing out on older features. To do a set of diagonal clickboxes for example I had to go into some old coordinate-based retro-code over a period of days to finally get it working as intended, but that only took days to figure out, debug, and work through over how the older method wasn't made into something more efficient.

I personally hate all of these modern web dev pipelines given how bloated they make everything, I miss super simple websites that just get their point across without a bunch of useless gadgetry. I like things to be simple and efficient when they can be.

You are so fucking autistic.

AliceInWonderland said: 

Turncoat said:

And to be clear... Topology is the overall Construction of a shape rather than just it's display, like how 3D objects work in programs like Maya?

I am only familiar with the idea through Digital Art... so at best a rough hands-on understanding rather than an in-depth one. I've manipulated objects in a 3D space and adjusted variables until it looked as I wanted it to, but most of what goes on there is backend, the artist rarely has to think about it.

I know what you mean but some clarification. Topology is a subject in mathematics and a Topology  is a mathematical object. 

Topology studies the properties of geometric objects that are preserved under deformations of those geometric objects. An example is the deformation of a rubber band as you stretch it or bunch it up, despite those deformations the fundamental properties of the object stay the same and as such its still a rubber band. If that rubber band breaks during the deformation, it no longer preserves those properties. 

So it's more about capacity rather than shape?

If the rubber band loses some of it's tensile strength, is it treated as different, or is it's resistance loss factored into it's capacity by the nature of it being a stretchy object? 

How would this apply to something like a sponge, where it can grow in size from additions from the environment but cannot do it by itself? Would a 'Wet Sponge' be given it's own definition or would it be treated still as a Sponge when it comes to the Sponge's Laws? 

There are many ways to study Topology:

(1) Point-Set Topology which constructs geometric objects out of sets and then explores how the geometric properties connectedness, compactness, and separations are preserved under deformation

This one sounds more like how vectors are used for 3D modeling. 

(2) Differential Topology which constructs geometric objects called Manifolds out of Euclidean Spaces and explores their differentiability under deformation

(3) Algebraic Topology which constructs a very general notion of Topological Spaces out of the geometric objects called Topologies and then explores the properties of invariance and equivalence of those topologies. 

Category Theory can generalize Topology. The key topological object, a Topological Space, can be formed into Categories called Sites, and continuous functions, another key object, can be formed into Sheafs. In that video from the Topos Institute sites and sheafs where the things being used to map different Topological geometries together.  The categorical generalization of Topology is called Homotopy Theory. 

You lost me at "Sites" and "Sheafs", what are those at a very basic level?

I'm winging it at "Space" and "Functions", but those at least are more conventional to try to imagine. 

I'm not sure I get what you mean, what kinds of merging are they founding with this idea, and what are they doing for said equivalence, making a fourth value altogether that they can filter through to allow for similar enough variables to the point of more direct interactions?

This is my obsession, buckle up. 

I'll tackle this portion in another post when my head's more in the game, I don't want to end up reading it and absorbing none of it. 

I ought to crash course a bunch of videos in a row on it, with some caffeine maybe. 

It more makes me sad when old coding goes out of style to the point of losing out on older features. To do a set of diagonal clickboxes for example I had to go into some old coordinate-based retro-code over a period of days to finally get it working as intended, but that only took days to figure out, debug, and work through over how the older method wasn't made into something more efficient.

I personally hate all of these modern web dev pipelines given how bloated they make everything, I miss super simple websites that just get their point across without a bunch of useless gadgetry. I like things to be simple and efficient when they can be.

I like simple with a single complicated gimmick. Websites should ascribe to follow certain accepted canons, but having a single "ooh" factor is nice without feeling too busy. 

Things like PHP and HTML5 though have led to interesting innovations in web design, especially when compared to the original Geocities junk that people started with. Everything back then looked like something spat out by a Microsoft Front Page generator based around trends and archetypes... with really cheesy animated fire gifs way too often. 

PalePeach said: 

You are so fucking autistic.

I'll take this over the ones who are into Trains.