Suddenly got hypnotized by this book, out of probably the next century.

https://ncatlab.org/nlab/files/ShulmanHomotopicalTrinitarianism.pdf

John De Goes is Programming Without Type Classes

Type classes have become a cornerstone of statically-typed functional programming, powering abstractions like monoid and monad. Yet, type classes often have generalized names, which don’t reflect their purpose in specific domains; and they incur higher learning costs, especially when emulated in languages without them.

In this presentation, John De Goes discusses the pros and cons of using type classes, and presents a systematic method for building powerful functional components that do not require type classes. Attendees will learn how to preserve the benefits of principled, algebraic functional programming, without forcing type classes on coworkers or users.

Here

https://gumroad.com/l/lcbk02

Chapter 1. Functions

General Ideas
Main Definitions
Special Classes of Functions
Binary Relations
Binary Operations

Chapter 2. Abstractions of Algebra

Monoids
Semigroups
Magmas

Chapter 3. Partial Orders, Graphs, and DAGs

Chapter 4. Boolean Logic

The Language of First-Order Logic
Valid and Sound Arguments
Proofs
Normal Forms

Chapter 5. Non-Boolean Logic

The Meaning of Non-Booleanness
Proof in Intuitionistic Logic

Chapter 6. Quantifiers

Universal Quantifier
Existential Quantifier
Connectives and Quantifiers in Boolean Logic
Connectives and Quantifiers in Intuitionistic Logic

Chapter 7. Models and Theories

Algebraic and Geometric Theories
Models

Chapter 8. Category: Multi-Tiered Monoid

Chapter 9. Working with Categories

Arrows in a Category
Initial and Terminal Objects

Chapter 10. Manipulating Objects in a Category

Product, Sum
Equalizer, Coequalizer
Pullback, Pushout

Chapter 11. Relations Between Categories

Functors
Building New Categories
Product of Two Categories
Sum of Two Categories
Contravariant Functor
Variance in Programming Languages

Chapter 12. Relations Between Functors

Natural Transformations
Adjoint Functors
Limits

Chapter 13. Cartesian Closed Categories

Basic Ideas
Examples
Definition: Bicartesian Closed Category

Chapter 14. Monads

Main Ideas
Every Adjunction Gives a Monad

Chapter 15. Monads: Algebras and Kleisli

Category of Algebras
Free Algebras as Functors
Forgetting and Freedom
Kleisli Category

The search of a "universal foundation" of math is actually the search of a universal model for all possible math theories.

Now the idea looks totally weird. It's been 90 years now since Gödel showed something to the world. And still...

There's no universal foundation! A foundation is just another theory.

Geometric Model Theory - hard reading, manifolds, algebraic geometry
Model Theory and Category Theory - high level, but easy reading

Solving the mystery behind Abstract Algorithm’s magical optimizations - here the author implements, in lambda, algorithms with unusually fast performance. Explanations follow (links inside)

A link to the author's github project

Pocket Set Theory

PST also verifies the:

Continuum hypothesis. This follows from (5) and (6) above;
Axiom of replacement. This is a consequence of (A4);
Axiom of choice. Proof. The class Ord of all ordinals is well-ordered by definition. Ord and the class V of all sets are both proper classes, because of the Burali-Forti paradox and Cantor's paradox, respectively. Therefore there exists a bijection between V and Ord, which well-orders V. ∎

The well-foundedness of all sets is neither provable nor disprovable in PST.

https://mathoverflow.net/questions/37329/is-there-a-known-natural-model-of-peano-arithmetic-where-goodsteins-theorem-fai

Like, here, on mathoverflow .

Leinster wrote in 2012: "rethinking".

People went crazy.

And it seems like nobody fucking knows there's a difference between a theory and a model.
That's how Chaitin was stating that there are "mathematical truths" that are true but don't follow from axioms.

For god's sake, it's been 90 years since Gödel explained it all. And? Anybody asked, how come ZFC is not based on ZFC? (sigh)

John Baez writes in his tweets (go ahead and look up, or, better, subscribe to his amazing tweets).

Do you know what "continuum hypothesis" is? It's about whether there is an intermediate size set between a countable (ℵ₀), for example, natural numbers, and 2^countable (ℵ₁). It's been proven over 50 years ago that neither the existence nor the non-existence follows from the axioms of Zermelo-Fraenkel. So, when mathematicians say that they base their absolutely strict and correct theorems on set theory (I don't believe them), we can always ask - which one?

Now the things got more serious.

Suppose you are a serious "machine learning data scientist", and you want to base your tea-leaves guesses on a solid math. That is, figure out the theory behind taking billions of pictures of cats and dogs and detecting cats on them (my former colleagues was focusing on figuring out whether he has a cat or a mouse, and figured that if the fur is uniform gray, the "algorithm" says it's a mouse. Do you have a Russian Blue?)

So, what we do, while "detecting", is a kind of data compression. It's closer to something like mapping, 2^N -> N.

Now, surprise. The feasibility of this operation, in general settings, is equivalent to having a finite number of intermediate sizes between ℵ₀ and ℵ₁.

Details are here: https://www.nature.com/articles/s42256-018-0002-3

Learnability can be undecidable

"The mathematical foundations of machine learning play a key role in the development of the field. They improve our understanding and provide tools for designing new learning paradigms. The advantages of mathematics, however, sometimes come with a cost. Gödel and Cohen showed, in a nutshell, that not everything is provable. Here we show that machine learning shares this fate. We describe simple scenarios where learnability cannot be proved nor refuted using the standard axioms of mathematics. Our proof is based on the fact the continuum hypothesis cannot be proved nor refuted. We show that, in some cases, a solution to the ‘estimating the maximum’ problem is equivalent to the continuum hypothesis. The main idea is to prove an equivalence between learnability and compression."

Just take a look at their bullshit: https://mathoverflow.net/questions/171809/building-a-product-of-two-categories

OTOH, Andreas Blass was rather good at bashing CT, like.

Update. https://mathoverflow.net/questions/319622/sets-are-a-popular-construction-in-programming-languages-is-there-a-solid-theor?noredirect=1#comment796756_319622 - меня попрекают отсутствием веры в математику! Вера в математику, Карл! Тут что-то такое кроется, о чем я не догадывался. Хоть я и работаю иезуитом.

https://en.wikipedia.org/wiki/Chaitin%27s_constant

A probability, no digit of which can be calculated.

package math.cat


import scala.reflect.ClassTag

trait NonEnumerableSet[T] extends Set[T] {
  private def notEnumerable = 
    throw new UnsupportedOperationException("This set is not enumerable")

  override def isEmpty: Boolean = notEnumerable

  def iterator: Iterator[T] = notEnumerable

  override def toArray[S >: T : ClassTag]: Array[S] = notEnumerable

  override def +(elem: T): Set[T] = notEnumerable

  override def -(elem: T): Set[T] = notEnumerable

}

So, undirected graph is a groupoid... I mean, a groupoid of paths.

Cool!

A heap

$(H,t)$ is a nonempty set $H$ equipped with a ternary operation $t : H \times H \times H\to H$ satisfying the relations

t(b,b,c) = c = t(c,b,b)

t(a,b,t(c,d,e)) = t(t(a,b,c),d,e)

More generally, a ternary operation in some variety of algebras satisfying the first pair of equations is called a Mal'cev operation. A Mal’cev operation is called associative if it also satisfies the latter equation (i.e. it makes its domain into a heap).

heap to monoid is like affine space to linear space

http://www.cs.man.ac.uk/~pt/Practical_Foundations/html/index.html

Practical Foundations of Mathematics

Paul Taylor

on SO

TLDR: there's no way you can define in set theory a function that does not return a result. (Actually you can, but then the whole sets axiomatics should be overhauled.)

"Curry’s use of combinators was connected very closely to his philosophy of mathematics: for him, a formal system was not a description of some pre-existing objects, but simply a structure built by induction from some primitives using certain rules, and he did not demand that a semantics for such a system be a total function."

src

So, this guy: Pierre Wantzel, had solved the problem of trisection, in 1837, before Galois. His solution, though, did not attract any attention at all, until 50 years later, when it was casually republished. Only probably 50 or more years later people understood the algebraic importance of it.

So there.

Dunning-Kruger with Math.

Just occurred to me. Reducing facts of real life to math entities is a kind of Dunning-Kruger disease. You can't get the complexity and all the aspects, but math you know, so kaboom - a monad! a linear space! a probability! a derivative! an integral!

Sad.

Same with logic, by the way.

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Juan-Carlos Gandhi

patryshev.com

August 2025

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Entries tagged with math

Wolfram, A New Kind of Science

Майкл Шульман о троице

tuesday event

my book with Table of Contents

just occurred to me

a bunch of links

Solving the mystery behind Abstract Algorithm’s magical optimizations - here the author implements, in lambda, algorithms with unusually fast performance. Explanations follow (links inside)

amazingly simple set theory

is there an arithmetic where Goodstein fails?

there seems to be a big scandal about sets

news of the century

Learnability can be undecidable

I'm rather pessimistic about mathematicians mathoverflow

Chaitin's constant

thinking whether I need a test for this trait

groupoid for an undirected graph

heap

книжечку присоветовали

Practical Foundations of Mathematics

Paul Taylor

talking about functions in Haskell

foundations of math

trisection of an angle

DKM