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Traditionally, mathematical work is presented in a linear fashion. Books, papers and articles are single streams of text meant to be read sequentially, from beginning to end.

However, mathematical content often has a not-so-linear underlying structure. Sometimes it can be imagined to be tree-like, with nodes being results and directed edges being dependencies.

Question

Is there a format for presenting maths that is faithful to some underlying logical structure of the work? The 'logical structure' could be defined by the author. Using digital devices, we are obviously not restricted to linear text anymore.

Have you seen such an 'untraditional' format being used?

Prototype

Imagine a PDF-viewer that can collapse and expand certain blocks of texts, as defined by the author and with the possibility of nesting.

In proofs there are often steps which are very unclear to some readers and trivial to others. These steps could get elaborated on in an expandable block -- providing the necessary details for the people who want it while maintaining reading flow and brevity for the others.

Using layers in LaTeX something similar can be achieved as described in this question

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  • $\begingroup$ Many IDEs (integraged development environments) do that for computer programs. Overleaf does it for $\LaTeX$ source. $\endgroup$
    – Ethan Bolker
    Sep 28 at 15:15
  • $\begingroup$ I like this idea. It feels like Prezi vs PowerPoint. For implementation purposes, wouldn't you only need to arrange a table of contents in this way? $\endgroup$
    – Chris Lewis
    Sep 28 at 15:18
  • $\begingroup$ True, that prototype is basically a finer table of contents in wich the reader can easily skip around. But it can not capture parallel structures; for instance when in one text ideas get developed in parallel, independent of each other. So one would need some other format to express this structure. Maybe just put expandable boxes next to each other? $\endgroup$
    – Alex
    Sep 28 at 15:39
  • $\begingroup$ I had the same ideas, including the nesting collapsible text. I also thought about putting the content in the hyperbolic plane, to make room for branching. $\endgroup$
    – mr_e_man
    Sep 28 at 18:02
  • $\begingroup$ Isn't the purpose of parentheses to simulate branching in linear text? $\endgroup$
    – mr_e_man
    Sep 28 at 18:07

2 Answers 2

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I toyed with the idea during my PhD, and spend quite a lot of time on the searching for the best type of solution.

To me this essentially changing from writing a personal (or shared) wiki rather than a pdf document. There are many tools for it.

During my PhD, I used tiddlywiki for this https://tiddlywiki.com/. It is very friendly, can compile latex with either latex or mathjax. I used this to organise my notes during qualification exam. It allows for collapsing and so on. But it has its own little problems.

Nowadays, I use vimwiki, which is a vim plugin, just because of convenience and speed.

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I could see something like your prototype being useful when we have large, branching proofs. E.g., lets say I have statements $A,B,C,D$

My argument/proof could be: $A \cdot B \cdot C \implies D$

In this case, we could show this argument as having $D$ with three child "nodes" $A,B,C$, so it's clear from the structure of the PDF how the overall argument flows. I can see such a visualization being even more useful for diagramming monster proofs like Wile's of Fermat's Last Theorem.

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