Good water

I love my mother tongue not because our Telugu teacher made us scream our lungs out: Telugu is the greatest language (which, I learned, upon returning home, from my Peddanannagaru, is a patently dumb statement to make not to mention the implicit sickening chauvinism), but may be because of the feeling that the word-by-word translation of the compound word for drinking water—manchi neellu—evokes:

Water is good.

In reading the above sentence, we say: water is the subject, good is the object of the sentence.

A study of sentences involves, in addition to analysis into words, synthesis into paragraphs.  There are usually quite a few sentences in any given paragraph, each of which saying something about one or several things (thinking, feeling, or nothing of that sort).  One approach to paragraphs begins with a pair of sentences; later on we can see how much of a paragraph (in general) we can understand in terms of the pair-wise “interactions” between sentences constituting paragraphs.

sentence1: Water is clean.

sentence2: Clean is good.

subject of sentence1 = water

object of sentence1 = clean

subject of sentence2 = clean

object of sentence2 = good

If, at this point, you are thinking that the above pair of sentences is like the pair of arrows satisfying:

source of arrow2 = target of arrow1

then you must be JLo who sneaked into my mind.

In celebrating the incidence relations of pairs of arrows, we cataloged 11 ways in which two arrows can relate to each other.  Once we identify ‘sentence’ with ‘arrow’, ‘subject’ and ‘object’ (of sentences) with ‘source’ and ‘target’ (dots, respectively, of arrows) we can readily restate the 11 graphs of a pair of arrows in terms of pairs of sentences.

Graph1

S1: Sky is calm.

S2: Streets are quite.

subject (S1) = sky

object (S1) = calm

subject (S2) = streets

object (S2) = quite

Graph2 (think sum

S1: Water is still.

S2: I am still.

subject (S1) = water

object (S1) = still

subject (S2) = I

object (S2) = still

object (S2) = object (S1)

Don’t panic—I’m not going to write down the remaining graphs (I am not that sadist though I have been called that to get my attention, I guess .

Oftentimes I find myself wanting to read blogs in languages I am still beginning to learn.  So, as you would expect, I request Google to translate the text for me.  What Google does when it translates a paragraph from Romanian to English is top-secret, I’m told.  Thankfully translation is an open-book.  Reading this book wide open in our minds, we find something along the following lines:

In translating a paragraph (1) into a paragraph (2)

translation: Paragraph1 -> Paragraph2

1. sentences are mapped to sentences
2. words are mapped to words

Given a paragraph:

Ravana kidnapped Sita.  Rama killed Ravana.

a translation, based on the above 2 guidelines, might read as follows:

Rama entführt Sita.  Ravana getötet Ravana.

Not only is the result confusing but of more present concern is Hindu priests issuing fatwas against me if they chance upon my translation.  So I rush to rewrite our method of translation:

1. Map sentences to sentences.
2. Map words to words in a way respectful of the structure of sentence.

According to (2), if I map a sentence A to a sentence B, then I must map the subject of A to the subject of B and the object of A to the object of B.

Let us, to be clear, consider a collection S of sentences, say,

S = {Ravana kidnapped Sita.

Rama killed Ravana.}

Of course the words constituting these sentences can also be thought of as a set i.e.

W = {Ravana, kidnapped, Sita, Rama, killed}

Here comes the best part:

The subject of a sentence is a function

s: S -> W

assigning to every sentence in S, the word in W which is its subject.

For example,

s (Rama killed Ravana.) = Rama

So is the case with object.

The object of a sentence is a function

o: S -> W

assigning to every sentence in S, the word in W which is its object.

For example,

o (Ravana kidnapped Sita.) = Sita

Not to be forgotten in translating is the incidence relation: Rama killed the guy who kidnapped Sita i.e.

o (Rama killed Ravana.) = s (Ravana kidnapped Sita.)

Sita

 ^

|

Rama -> Ravana

Now let’s see where we are:

A translation or interpretation of one paragraph into another (perhaps in another language) should at least be a morphism of graphs.  But it should preserve more than just the “subject of a sentence” and “object of a sentence” incidence relations…

Toposes of Generalized Graphs (see page 276 – 7)

How about translating differential equations into matrix multiplications?

What about interpreting equations as lines?

Where’s the mapping from geometry to algebra?

Then there’s the one between as is and as if!

I know… it’s not like we have conferences to rush to, audiences to perform for; all too content stay-at-home science