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I would like to share my philosophical methods and results and explore further how they might be relevant for math and its applications. I will start with ideas on network theory which I have found fruitful.
In general, I am interested to document the mental structures that define the limits of my imagination. Such cognitive or metaphysical foundations, if they can be ascertained, should be meaningful in characterizing what math can do. So I would like to develop a "science of math" to describe the "implicit math" that we use in our minds in our mathematical activity and that would explain how we figure things out, but also why we find math beautiful, understandable, insightful, meaningful, practical and so on. It's possible to consider many bases, for example, neurological, but I will focus on "cognitive math".
In 1998, I visited Kestas Augutis, a hermit in the Lithuanian countryside. He had a dream that the goal of a student's high school education should be for them to write three books to show that they could think in sequences, in hierarchies and in networks. The sequence would be an archive of their works, much like today's blogs. The network would be an encyclopedia and the hierarchy would be a thesaurus.
I am curious to collect and ground such distinctions. Also, I thought that creating a software tool for writing in all three ways would be a good endeavor for my business, Minciu Sodas, (Orchard of Thoughts), to serve and organize independent thinkers around the world.
As one of my hundreds of investigations that I've done, I collected examples of how we organize our thoughts so as to verify that we use sequences, hierarchies and networks. To my surprise I discovered that we never use them in isolation! We always use them in pairs: when one structure grows too robust, then we restructure it with another structure. This makes for six visualizations, as I describe in my paper: Organizing Thoughts into Sequences, Hierarchies and Networks.
These are visualizations, external structurings. There is also an internal structuring, like a row of fields in a table, a scheme or template, which is not visualized in the same sense.
So my claim is that any visualization, any diagram, will fit into one of the six categories, cognitively, in our minds. For example, I think a Petri net would be a Sequence (from input to output) that grew too complicated and so it was restructured with a Network to allow for branchings, loops, etc.. There can be more going on there. But I am claiming that our minds approach a Petri net as a Handbook, as I call it. Here I'm not concerned what's explicitly written on the paper but what's taking place implicitly in our mind that's interpreting the paper. Of course, that can be very murky so it takes investigation to develop, check and sharpen the result.
One step I could take now is to consider this anew and collect a fresh set of examples to try to categorize as above or otherwise. One set of diagrams to consider is the Yed gallery. Another would be all of the diagrams in John Baez's Network Theory pages and here.