#isomorphism

[courtling]

I need to be punished by an Octoberfest in May.

[nablavii]

Sinceramente, una de las fotografías a las que más afecto le tengo.

ft. @ras-al

[lesbianslovebts]

I got excited about linguistics again after that last ask, so I want to elaborate on how I view translation as a mathematical mapping between two sets. Observe the following sentences written in Korean, Japanese, and English, respectively.

First, going from Korean to Japanese, you will notice that the function (translation) is bijective (word-for-word) because there exists a map between the two sets (sentences) where each element (word or word-part) in the first set (Korean) is paired with exactly one unique element of the second set (Japanese), each element of the second set is paired with exactly one unique element of the first set, and there are no repeats. Moreover, the mapping preserves the relations between elements in the sets and is, therefore, a homomorphism. In other words, the translation preserves the grammatical part of speech of each word or word-part (i.e., the direct object remains the direct object). Thus, the translation of this sentence from Korean to Japanese is a grammatical isomorphism. Obviously, the same can be said of the Japanese-to-Korean translation.

On the other hand, the translation from Japanese to English is not even a proper function because there exist no elements (word or word-parts) in the codomain (English) for two elements in the domain (Japanese): the topic particle は and the direct object particle を. Likewise, in the translation from English to Japanese, there exist no elements in the codomain for two elements in the domain: the definite article "the" and the possessive adjective "my." Additionally, there are two elements in the domain that map onto two elements each in the codomain: "went" to 行き and ました, and "with" to と and 一緒に. There is no function, let alone isomorphism, between English and Japanese. By extension, there is none between English and Korean, either.

In conclusion, neither English-Japanese-English nor English-Korean-English translation is as direct or complete as Korean-Japanese-Korean translation, which is a grammatical isomorphism, at least in the case of the sentence pictured above.

[nord-thoughts]

Coincidentia oppositorum: the union of opposites

[lesbianslovebts]

My therapist asked me to explain what my autistic brain does when studying languages, and I answered, "Math." She didn't quite get it, but it really is math! When I see the same sentence in different languages, it's like solving an algebraic equation. And I love how translation can be surjective, injective, or bijective! Korean and Japanese are as close to a grammatical isomorphism as you can get, in my opinion.

나는 친구하고 같이 공원에 점심을 먹으러 갔었어요.

私は友達と一緒に公園に昼ごはんを食べに行きました。

As opposed to English:

I went to the park with my friend to eat lunch.

[isomorphismes]

Kai Behrends, on the way to explaining stacks with easy examples, explains groups, groupoids, indexing sets, families, and equivalence classes

[isomorphismes]

the pigeonhole principle, as applied to the enter() and data() verbs of d3.js

by Sebastian

[isomorphismes]

Galileo observed that the set of positive integers can be put into 1‒1 correspondence with the set of square integers, even though the set of nonsquares seems more numerous than the squares. He deduced from this that "the attributes =, >, < are not applicable to infinite quantities".

Stan Wagon