Public posts from @abuseofnotation@aethy.com
But I guess the masturbation metaphor isn’t quite accurate, as the only real purpose of these explanations manifested itself only when they were shared with other people, and only insofar as they were recognized by them (what makes mathematics so useful was that it had this property that anyone who is sucked deep enough in it would be completely unable to reject any of its fundaments). The issue…
“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies…
The category of Limitation is often misunderstood because this triad is based on truth. Classical logic is bivalent: propositions are either true or false, yet this triad of quality has three categories. So, what is the third? -- https:// abuseofnotation.github.io/univ ersal-knowledge/index.html
We haven’t covered product categories, however some pages ago, when we covered monoids and groups, we talked about the concept of a product group. The good news is that product categories are a generalization of product groups… -- https:// abuseofnotation.github.io/cate gory-theory-illustrated/11_natural_transformations/index.html
Many, if not most, of the of the things that we said about space, can also be said about time. Like space, it is universal and unique - we can only perceive one timeline, that is, for every two events we can always say that one comes before the other. Space and time are the two components that shape our reality - space allows objects to co-exist, and time enables their states to form a succession…
Aside from facilitating code reuse by bringing in all standard functions of simple types in a more complex context, map allows us to work in a way that is predictable, courtesy of the functor laws, which in programming context look like this. -- https:// abuseofnotation.github.io/cate gory-theory-illustrated/10_functors/index.html
In deterministic systems, non-deterministic events can only be explained by “hidden variables” theories. These theories posit that any failure to predict an outcome is due to a lack of knowledge or understanding. While it is good stance to adopt for evolutionary reasons, as it drives us to analyze and systematize our surroundings, it becomes problematic when we take it too seriously. -- https://…
Of course, he came to talk about work. Mirko would never come to me for any other reason. He started briefing me about the situation he was in with his current project. The whole story was very convoluted and I was absent-mindedly waiting to hear when he would start telling me how all this was related to the stuff I was working on. But that moment never came. After a couple of minutes he stopped…
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 B is made up, there is no reason to think that it should apply to every object in the universe. This is…
3. At stage 2, we can make random observations or discard observations as they lose relevance. At stage 3, however, every observation must connect with all the other ones (it is this connection which is necessary). New observations must align with our worldview, and we cannot disregard established observations without impacting our entire conceptual structure. -- https://…
1. At stage 1 and 2, we operate on a phenomenological level, dealing only with things we directly perceive. In stage 3, however, we extend to imagining entities and concepts based on what we perceive but which cannot be directly observed. -- https:// abuseofnotation.github.io/univ ersal-knowledge/index.html
To keep the suspense, before we discuss the relationship between monoids and categories, we are going through see some simple examples of monoids. -- https:// abuseofnotation.github.io/cate gory-theory-illustrated/03_monoid/index.html
Realize how ridiculous this looks in the grand schema of things: someone comes up with marvelous new technology that solves all your issues and your first reaction as a society is “but what would the people who used to work on that issue do for a living?” -- https:// abuseofnotation.github.io/capi talism-innovation/index.html
Like we said at the beginning, with isomorphisms, going back and forth brings us to the same object, while with equivalence the object is just isomorphic to the original one. This is truly all there is to it. -- https:// abuseofnotation.github.io/cate gory-theory-illustrated/11_natural_transformations/index.html
This is probably the most important example you will see if you want to understand what is FP all about. Not all functions can really be represented as data structures (many of them have infinite inputs), but all of them can be reasoned about as such. One way to grasp the power of this concept is to imagine how the function-data isomorphism scales. Imagine a simple function like the one you see…
This particular equivalence (a set as a category with no morphisms) is, however, rarely useful. Not because it’s incorrect in any way, but rather because category theory is all about the morphisms — if the arrows in set theory are nothing but a connection between the sets that serve as their source and a destination, in category theory it’s the objects that are nothing but a source and destination…
Task 7: Try to extend the natural transformation in the other direction (by taking $baralpha$ to be identity). -- https:// abuseofnotation.github.io/cate gory-theory-illustrated/11_natural_transformations/index.html
Unlike all other categories, the categories of modality are not concerned with the observed phenomena (objects, appearances etc.) but with the observer. This is why Kant says that they are not objectively synthetic (as all other categories), but it’s subjectively synthetic. -- https:// abuseofnotation.github.io/pure -reason/index.html
Like with categories, the object mapping of an order is represented by a function between the orders’ underlying sets. -- https:// abuseofnotation.github.io/cate gory-theory-illustrated/10_functors/index.html
By the way, the way Elon acts provides for me an irrefutable, proof that capitalism is dead — if the world’s richest person cannot just use their money to enjoy themselves and instead prefers to continually make a fool of themself and be ridiculed by the whole world on a daily basis, then what’s the point of it all? -- https:// abuseofnotation.github.io/elon -capitalism/index.html