On the mathematical duality of reading and writing

1A brief note about duality

Perhaps I don't know what I'm talking about here.

Here I'm loosely using "duality"¹ in the mathematical sense: Roughly, I think, two things are dual if each one can be derived from the other.

Duality enables us to buy two for the price of one.

An important benefit of duality is that if we know how to read well, then we can derive how to write well from that without much thought.

I want to show that, if you can read, then you can write.

2Can we derive writing methods from reading methods?

Yes!

If we have a reading method, we can derive a writing method by reversing the steps of the reading method.

If readers best read from the first section to the last section, then writers best write from the last section to the first section. For example, the conclusion appears last but should be written first; whereas the introduction appears first but should be written last.

Here is a more concrete example.

Keshav 2007 [1] has a three-pass reading method:

The key idea is that you should read the paper in up to three passes, instead of starting at the beginning and plowing your way to the end. Each pass accomplishes specific goals and builds upon the previous pass: The first pass gives you a general idea about the paper. The second pass lets you grasp the paper's content, but not its details. The third pass helps you understand the paper in depth.

Reversing those steps gives us a three-pass writing method for free:

1. Write the parts that require the deepest understanding.
2. Write the parts for grasping the gist without drowning in details.
3. Write the general ideas.

Another example.

This is a possible reading method.

1. Repeat as necessary:
   1. Read some of the table of contents.
   2. Read the first sentence of some paragraphs.
   3. Read some paragraphs.

The writing method that is the dual of that reading method is:

1. Repeat as necessary:
   1. Write some paragraphs (ignoring the quality of their first sentences).
   2. Write (or rewrite) the first sentence of some paragraph.
   3. Write some of the table of contents. (Group the paragraphs into sections.)

If a reading method is desirable, then its dual (the writing method derived by reversing the steps) should also be desirable.

3Formalism of methods and their duals

A method is a statement.

A statement is any of these, where E means an expression and S means a statement:

• read E
• write E
• repeat S
• S then S

Note that the above definition is recursive.

The "then" operator associates to the right: "X then Y then Z" means "X then (Y then Z)".

The dual of a method is defined as follows:

dual(read E) = write E

dual(write E) = read E

dual(repeat S) = repeat dual(S)

dual(S then T) = dual(T) then dual(S)

It can be proven by structural induction that the "dual" function is an involution, that is, dual(dual(S)) = S.

Indeed duality enables us to buy two for the price of one: Everytime we come up with an enhancement to our reading method, we get the dual enhancement to the dual writing method for free.

4Derivations?

4.1What are desirable reading and desirable writing?

A desirable reading is inspiring and enjoyable. However, the process of writing to produce a desirable reading may be boring and toilsome.

4.2Can reading techniques and writing techniques inform each other?

4.3Can readers teach writers how to write, and writers teach readers how to read?

5Systematic reading and systematic writing?

Readers read text in several passes, from shallow to deep. Writers write text in several passes, from deep to shallow.

Writing is the inverse of reading. Readers read top-down: They begin with table of contents, and then they decide which fragments interest them. Writers write bottom-up: They begin with scattered fragments that interest them, and then they reorganize those fragments into a table of contents.

Readers build up thoughts: If topic Q requires P, then readers expect to read P before reading Q. Writers break down thoughts: If topic Q requires P, then it would be easier for writers to write Q before writing P. (Really?)

It is easier for readers to read sequentially. It is easier for writers to write fragments randomly and sometimes serendipitously according to what comes to their minds.

The easiest way to write is the hardest way to read. The easiest way for writers to write fragments is in reverse to the order readers would find easiest to read. If it would be easy for readers to read A-B-C in that order, then it would be easy for writers to write C-B-A in that order.

Thus, in order to make a writing that is easy to read, write from the end.

Reading and writing are deserialization and serialization of knowledge, of internal mental models, of internal beliefs.

Writers also read to enrich their writings.

Readers scan and skim. Writers also do a similar thing called writing an outline or a draft or a skeleton. scan-write and skim-write.

How do we read systematically? https://libraryguides.griffith.edu.au/systematic-literature-reviews-for-education/systematic-reading And how do we derive a way to write systematically from the way to read systematically?

Why Do We Read? - Purposes & Advantages https://study.com/academy/lesson/why-do-we-read-purposes-advantages.html

6Bibliography

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1. https://en.wikipedia.org/wiki/Duality_(mathematics)↩