Abuse of Notation bot

So how can we do that? One key observation is that diagrams look as finite categories, as, for example, the above definition looks in the same way as the category $3$. -- https://abuseofnotation.github.io/category-theory-illustrated/10_functors/index.html

Because a set is defined only by the items it contains, the empty set is unique — there is no difference between the set that contains zero balls and the set that contains zero numbers, for instance. -- https://abuseofnotation.github.io/category-theory-illustrated/01_set/index.html

But what’s even more impressive, at least for me, is to look at production bicycles that came some time later, like the 1911 [6]La Francaise diamant (ridden by the winner of the first Tour de France) - this bike is not merely similar to the bike I own when it comes to design. It is (with a few […]

“Maxim. Sorry, I am a bit of a dork” I said trying to apologize for not responding to his very simple request, while at the same time spicing up my persona with some self-irony. “It’s OK, me too” he responded softly. Although I was almost sure that he was doing it just to make me feel good, his […]

The diagrams above use colours to illustrate the fact that the green morphism is equivalent to the other two (and not just some unrelated morphism), but in practice this notation is a little redundant, as the only reason to draw diagrams in the first place is to represent paths that are […]

We can do the same for the addition monoid — numbers can be seen not as quantities (as in two apples, two oranges etc.), but as operations, (e.g. as the action of adding two to a given quantity). -- https://abuseofnotation.github.io/category-theory-illustrated/03_monoid/index.html

Most of the sets that we use in programming are composite sets e.g. make a list of Chars and you have a string. We will see how this happens later. -- https://abuseofnotation.github.io/category-theory-illustrated/01_set/index.html

“Or perhaps there needn’t be a main plot.” Yes, perhaps I ought to find time to both look at Anton’s messages and do something at work. But then I quickly realized that I couldn’t do both of these things. -- […]

fact that I could only be at one place at one time, to be one person, leading one, and only one, life. -- https://abuseofnotation.github.io/mind-against-the-machine/confusion-breeds-clarity/11-irony/index.html

Besides being the source of rules (concepts) according to which intuitions are judged, understanding also contains the principles for applying these rules, which we call “principles of the pure understanding”. We can derive those principles using the table of categories. Here they are: -- […]

On first glance, $Cons$ looks pretty similar to the inductive $Succ$ constructor that we saw. And indeed, like $Succ$, $Cons$ is an inductive/recursive constructor that generates an infinite amount of terms. -- https://abuseofnotation.github.io/category-theory-illustrated/06_type/index.html

It may not look like this, but its actually “the crazy ones”, the rebels, the weirdoes, the people who are not afraid to show their different views and openly exhibit their quirks who care most about others. People who disrupt the status quo do so not because they don’t care about other people […]

2. One function from the number order to the prime inclusion order (which is an operation called prime factorization of a number, consisting of finding the set of prime numbers that result in that number when multiplied with one another). -- […]

Whether they are broken, or operating as intended, inhumane technologies often look like they have a will of their own and they often surprise you unpleasantly. With human technologies, on the contrary, you always know exactly what is gonna happen. And in case there is a problem, you often know […]

Picking a type theory (or a type system let’s call it), also involves picking a language that this theory is described in terms of. When hearing about language, programmers would probably think of the popular feature-rich programming languages like TypeScript or Java. Type theorists, on the […]

There is actually one more prerequisite which is not so easy to explain: namely the arrow types for each pair of objects, known as the Lambda types. In Lambda Calculus arrows between types are types as well (a feature sometimes called “first-class functions” in programming context). -- […]

A nice example of a group, which is related to a monoid we covered, is the set of integers under addition — the operation is again ($+$), but the objects are the integers $mathbb{Z}$, not the natural numbers $mathbb{N}$ (so it’s not ({ 0, 1, 2, 3 ...}), but ({... -3, -2 -1, 0, 1, 2, 3 ...})) […]

“Because they can tell the wife they are with the mistress, tell the mistress that they are with the wife and stay at home to do math.” -- https://abuseofnotation.github.io/mind-against-the-machine/confusion-breeds-clarity/07-annas-ego/index.html

2. At stage 1 and 2, our observations are fragmented. At stage 3, they coalesce into a cohesive whole, implying an underlying presumption that the universe itself forms a cohesive whole. -- https://abuseofnotation.github.io/universal-knowledge/index.html

Emanuel Kant’s idea was to create a system which unites these opposing views in a cohesive whole, where each of them has its rightful place. To do that, he rejected the rationalistic notion that we can know something about the real world just by thinking, but he also argued that experience is […]

“This deficiency in our ideas is not, indeed, perceived in common life, nor are we sensible, that in the most usual conjunctions of cause and effect we are as ignorant of the ultimate principle, which binds them together, as in the most unusual and extraordinary. But this proceeds merely from an […]

Notice that we added the black ball at the top and the white one at the bottom. We did that because otherwise the top three elements wouldn’t have a join element, and the bottom three wouldn’t have a meet. -- https://abuseofnotation.github.io/category-theory-illustrated/04_order/index.html

Given a set of objects, there can be numerous criteria, based on which to order them (depending on the objects themselves) — size, weight, age, alphabetical order etc. -- https://abuseofnotation.github.io/category-theory-illustrated/04_order/index.html

The polymorphic Lambda Calculus (System F) is the type system which only has one type constructor — polymorphic lambda, with typing rules Abs and App (for introduction and elimination of terms) and kinding rules TAbs and TApp (for introduction and elimination of types). It also has the Var […]

The two ways of combining three sets into a triple product are isomorphic, $(A times B) times C cong A times (B times C)$. -- https://abuseofnotation.github.io/category-theory-illustrated/02_category/index.html

But wait—if the third member of each triad is merely a combination of the other two, doesn’t that mean it’s not truly pure/atomic/first-order and should be excluded? Kant argues otherwise, although his explanation remains somewhat ambiguous: -- https://abuseofnotation.github.io/time/06/index.html

The answer lies in the concept of the understandable universe, or substance (I’ll use “universe” since the meaning of “substance” is unclear). The concept of the universe is both obvious and odd

The chains in an order don’t have to be completely disconnected from each other in order for it to be partial. They can be connected as long as the connections are not all one-to-one i.e. ones when the last element from one chain is connected to the first element of the other one (this would […]

from chatting about all kinds of stuff to only talk with people about their sexual experiences and then about her made up sex life (picking older married guys at sport pubs and fucking them at parks at night, having group sex. This resulted in her sharing more and more of her personal details […]

In the previous chapter, we saw one non-cyclic product group (the Klein four-group), which was a product of cyclic groups. Most product groups (even the product of cyclic groups) would be non-cyclic, because it would have the generators of both groups that comprise it, i.e. even if the two […]

In contrast, intuitionistic logic represents the opposite view, that knowledge is subjective and limited in scope. Here, rather than contrasting reality with limitation, we use negation. A thing may be B or it may not be B, but it doesn’t have to be one or the other, because, since the predicate […]

It is just that the so-called “morphism mapping” between categories that have just one morphism for any two objects are trivial, and so we can omit them. -- https://abuseofnotation.github.io/category-theory-illustrated/10_functors/index.html

But I guess there was much difference in the way we perceive things that we consider forbidden and things that are just taboo, the former constituting joke material, and the latter bringing drug-like satisfaction to some people, when brought up. -- […]

This is pretty much the only one that is useful with this signature (the others being $a to [a, a]$, $a to [a, a, a]$ etc.), but there are many examples with signature $list a to list a$, such as the function to reverse a list. -- […]

However, from the standpoint of the real, objective world (W), mental images are just arrangements of elements. And improbable ones at that. That’s because in W the “lawful” arrangement (the one that corresponds to a mental image) would be only one of the countless other arrangements, (i.e […]

“Well, okay, now imagine some guy like me, who isn’t afraid of death, and lives as he WANTS, not as is best for him? If we’re just a more talkative version of animals, where is THAT PERSON’S place in the picture?” -- […]

A function is a relationship between two sets that matches each element of one set, called the source set of the function, with exactly one element from another set, called the target set of the function. -- https://abuseofnotation.github.io/category-theory-illustrated/10_functors/index.html

Without me noticing, (as these things always seem to happen without us noticing) this dilemma has become like a leitmotif of my life. It was the dichotomy that is best exemplified by comparing the approaches of two great computer pioneers - the well-known Allan Turing and the (perhaps slightly […]

I wanted to end this article with some examples and these are the ones I came up with. This should give you a hint on how you can use the tips from above. -- https://abuseofnotation.github.io/fp/index.html

If done right, a federated network like Mastodon would be much more reliable and cheap to run than a centralized one: when the scale is big, the network has to be federated under the hood anyways (i.e. you have to have several caches in different locations, several database nodes that copy data […]

Like, a concept is considered true and real by us only if it relates to our concept of ourselves. Our concept of ourselves is the realest thing there exists for us (although it in actuality is completely objective). -- https://abuseofnotation.github.io/concept-of-self/index.html

Now, we ask ourselves, in which cases does there exist an isomorphism between those two functors? -- https://abuseofnotation.github.io/category-theory-illustrated/11_natural_transformations/index.html

As we can see, these terms are not precisely defined i.e. they are opened to interpretation — e.g. if the person who we are sending to get us drinks is significantly wealthier than us, or a heavier drinker, their idea of “cheap” and “low-alcoholic”, respectively, would not match ours. This will […]

However, that did not work out. It did not fit into my narrative and as much as I admit it, and despite my best attempts, it was still dishonest. But I felt I was onto something - at least my new writings were relatable, something which cannot be said about my older ones. -- […]

“While Newton seemed to draw off the veil from some of the mysteries of nature, he showed at the same time the imperfections of the mechanical philosophy, so agreeable to the natural vanity and curiosity of men

If one thing is equal to another, the reverse is also true, i.e, the other thing is also equal to the first one. This idea is called symmetry. Symmetry is probably the most characteristic property of the equivalence relation, which is not true for almost any other relation. -- […]

An example of societies without a firm ideology are the authoritarian regimes of the 20th century, like the Third Reich. Although the leaders of such regimes talk about ideology all the time, their “ideology” consists of just a couple of slogans, whose constant repetition cannot make up for […]

If we apply the first function $g$ to some element from set $Y$, we will get an element of the set $P$. Then, if we apply the second function $f$ to that element, we will get an element from type $G$. -- https://abuseofnotation.github.io/category-theory-illustrated/01_set/index.html

So, if you combine two groups with orders that have some common divisor (as $2$ and $2$, which are both divided by 2), then, their product would not be cyclic. But, if you combine two groups with orders that are relatively prime, (like $2$ and $3$) you would get a cyclic group. -- […]

What it means, for me at least, is that there is no justification for having a central body of authority that governs our lives: one monarch, one church, one school of philosophy, (at the time of Kant this was the scholastic school, which Kant aimed to reign upon). -- […]