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# Cognitive math: sequences, hierarchies, networks

edited January 2017

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.

• Evolution = Hierarchy reorganized with a Sequence (branching out over time)
• Atlas = Network reorganized with a Hierarchy (global and local views)
• Handbook = Sequence reorganized with a Network (rerouting with loops and branches)
• Chronicle = Sequence reorganized with a Hierarchy (grouping into time periods)
• Catalogue = Hierarchy reorganized with a Network (crosslinks)
• Tour = Network reorganized with a Sequence (traveling about the network)

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.

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edited December 2016

I've been skipping around, reading the network theory posts by John Baez and Brendan Fong, such as this one: Part 11: Noether's Theorem for stochastic mechancs. Somewhere you pointed out that a Markov chain is a matrix multiplication taking us from inputs to outputs, which is to say, it is a Sequence (multiplying $U$, $U^2$, $U^3$, etc.) of a Network (whose edges $u_{i,j}$ are given by the entries of the matrix $U$). So that, in a way, conforms and clarifies my comment above about Petri nets, which are a special case of Markov chains.

Comment Source:I've been skipping around, reading the network theory posts by John Baez and Brendan Fong, such as this one: <a href="http://math.ucr.edu/home/baez/networks/networks_11.html">Part 11: Noether's Theorem for stochastic mechancs</a>. Somewhere you pointed out that a Markov chain is a matrix multiplication taking us from inputs to outputs, which is to say, it is a Sequence (multiplying $U$, $U^2$, $U^3$, etc.) of a Network (whose edges $u_{i,j}$ are given by the entries of the matrix $U$). So that, in a way, conforms and clarifies my comment above about Petri nets, which are a special case of Markov chains.
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I submitted the following abstract to the workshop Consequence and Paradox: Between Truth and Proof, 2–3 March 2017, in Tübingen, Germany. Today is the deadline.

Inferring What We Don't Know: A Taxonomy of Paradox

I identify six themes among paradoxes with six different ways that we reorganize thoughts. Paradox is inherent in the gap that arises when we reorganize our thoughts. It is the gap between signifier and signified, as characterized by six equivalences which yield six qualities of signs. Our minds leverage the tensions in these equivalences to shift our attention from what we know to what we don't know. We thus variously clarify what we don't know.

In 1998, I was intrigued by educator Kestas Augutis's vision that every high school student write three books (a chronicle, a thesaurus, and an encyclopedia) so as to master three kinds of thinking (sequential, hierarchical, and network). I thus collected dozens of examples of how we organize our thoughts. Surprisingly, we never use sequences, hierarchies or networks in isolation. Instead, we use them in pairs:

• Evolution: A hierarchy (of variations) is organized with a sequence (of times).
• Atlas: A network (of adjacency relations) is organized with a hierarchy (of global and local views).
• Handbook: A sequence (of instructions) is organized with a network (of loops and branchings).
• Chronicle: A sequence (of events in time) is organized with a hierarchy (of time periods).
• Catalog: A hierarchy (of concepts) is organized with a network (of cross-links).
• Odyssey: A network (of states) is organized with a sequence (of steps).

In general, a first, large, unified, comprehensive structure grows so robust that we restructure it with a second, smaller, different structure of multiple vantage points.

In 2012, I analyzed and grouped all of the paradoxes listed in Wikipedia. This yielded the following six themes:

• Concepts may be inexact. (The paradox of an evolution.) We can't specify exactly at what point in the womb a child becomes conscious, or at what point in evolution two species diverge.
• The whole is not the sum of the parts. (The paradox of an atlas.) If we replace all of the parts of an automobile, and then build a copy with all of the old parts, which is the original?
• Our attention affects what we observe. (The paradox of a handbook.) Achilles can never catch a tortoise if we keep measuring the distance between them.
• There may be a limited contradiction. (The paradox of a chronicle.) How can we reliably learn from one who has ever made a mistake?
• We cannot make explicit all relevant assumptions. (The paradox of a catalog.) 10+4 may equal 2 if we happen to be thinking about a clock.
• We can choose differently in the same circumstances. (The paradox of an odyssey.) "I am lying when I say 'I am lying.'"

Each type of paradox brings to light the essential gap between the (seemingly infinite) primary comprehensive structure and the (manifestly finite) secondary structure which organizes our vantage points. Our mind visualizes a qualitative but illusory relationship between the primary and secondary structures. Upon closer inspection it becomes apparent that there is no definitive way to match up the two structures.

However, these six mental illusions do allow our minds to make tangible that gap, which is to say, that which we do not know. This may be considered as the gap between signified and signifier. Consider four levels of knowledge (whether, what, how, why) in terms of Peirce's types of signs (the thing itself, icon, index, symbol). Then pairs of these four levels yield six qualities of signs. A sign can be:

• Malleable: Equivalent icons refer to the same thing. (Equivalency defined by the paradox of an evolution.)
• Modifiable: Equivalent indices refer to the same thing. (Equivalency defined by the paradox of an atlas.)
• Mobile: Equivalent indices refer to the same icon. (Equivalency defined by the paradox of a handbook.)
• Memorable: The same symbol refers to equivalent indices. (Equivalency defined by paradox of a chronicle.)
• Meaningful: The same symbol refers to equivalent icons. (Equivalency defined by the paradox of a catalog.)
• Motivated: The same symbol refers to equivalent things. (Equivalency defined by the paradox of an odyssey.)

Paradoxes thus heighten and reveal six ways that our minds foster our mental freedom, ever shifting from the structures we know to that gap which models what we don't know.

Comment Source:I submitted the following abstract to the workshop <a href="http://ls.informatik.uni-tuebingen.de/CPTP/">Consequence and Paradox: Between Truth and Proof</a>, 2–3 March 2017, in Tübingen, Germany. Today is the deadline. <b>Inferring What We Don't Know: A Taxonomy of Paradox</b> I identify six themes among paradoxes with six different ways that we reorganize thoughts. Paradox is inherent in the gap that arises when we reorganize our thoughts. It is the gap between signifier and signified, as characterized by six equivalences which yield six qualities of signs. Our minds leverage the tensions in these equivalences to shift our attention from what we know to what we don't know. We thus variously clarify what we don't know. In 1998, I was intrigued by educator Kestas Augutis's vision that every high school student write three books (a chronicle, a thesaurus, and an encyclopedia) so as to master three kinds of thinking (sequential, hierarchical, and network). I thus collected dozens of examples of how we organize our thoughts. Surprisingly, we never use sequences, hierarchies or networks in isolation. Instead, we use them in pairs: <ul><li>Evolution: A hierarchy (of variations) is organized with a sequence (of times). <li>Atlas: A network (of adjacency relations) is organized with a hierarchy (of global and local views). <li>Handbook: A sequence (of instructions) is organized with a network (of loops and branchings). <li>Chronicle: A sequence (of events in time) is organized with a hierarchy (of time periods). <li>Catalog: A hierarchy (of concepts) is organized with a network (of cross-links). <li>Odyssey: A network (of states) is organized with a sequence (of steps).</ul> In general, a first, large, unified, comprehensive structure grows so robust that we restructure it with a second, smaller, different structure of multiple vantage points. In 2012, I analyzed and grouped all of the paradoxes listed in Wikipedia. This yielded the following six themes: <ul><li>Concepts may be inexact. (The paradox of an evolution.) We can't specify exactly at what point in the womb a child becomes conscious, or at what point in evolution two species diverge. <li>The whole is not the sum of the parts. (The paradox of an atlas.) If we replace all of the parts of an automobile, and then build a copy with all of the old parts, which is the original? <li>Our attention affects what we observe. (The paradox of a handbook.) Achilles can never catch a tortoise if we keep measuring the distance between them. <li>There may be a limited contradiction. (The paradox of a chronicle.) How can we reliably learn from one who has ever made a mistake? <li>We cannot make explicit all relevant assumptions. (The paradox of a catalog.) 10+4 may equal 2 if we happen to be thinking about a clock. <li>We can choose differently in the same circumstances. (The paradox of an odyssey.) "I am lying when I say 'I am lying.'"</ul> Each type of paradox brings to light the essential gap between the (seemingly infinite) primary comprehensive structure and the (manifestly finite) secondary structure which organizes our vantage points. Our mind visualizes a qualitative but illusory relationship between the primary and secondary structures. Upon closer inspection it becomes apparent that there is no definitive way to match up the two structures. However, these six mental illusions do allow our minds to make tangible that gap, which is to say, that which we do not know. This may be considered as the gap between signified and signifier. Consider four levels of knowledge (whether, what, how, why) in terms of Peirce's types of signs (the thing itself, icon, index, symbol). Then pairs of these four levels yield six qualities of signs. A sign can be: <ul><li>Malleable: Equivalent icons refer to the same thing. (Equivalency defined by the paradox of an evolution.) <li>Modifiable: Equivalent indices refer to the same thing. (Equivalency defined by the paradox of an atlas.) <li>Mobile: Equivalent indices refer to the same icon. (Equivalency defined by the paradox of a handbook.) <li>Memorable: The same symbol refers to equivalent indices. (Equivalency defined by paradox of a chronicle.) <li>Meaningful: The same symbol refers to equivalent icons. (Equivalency defined by the paradox of a catalog.) <li>Motivated: The same symbol refers to equivalent things. (Equivalency defined by the paradox of an odyssey.)</ul> Paradoxes thus heighten and reveal six ways that our minds foster our mental freedom, ever shifting from the structures we know to that gap which models what we don't know. 
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My proposal was rejected. :( But I like my results so far and will explore them further someday.

Comment Source:My proposal was rejected. :( But I like my results so far and will explore them further someday.
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By the way, titles of posts on the Azimuth Forum should have a single capital letter, like "Cognitive math: sequences, hierarchies, networks". That's just the convention - not a big deal, but we like orderly behavior here. :-)

Comment Source:By the way, titles of posts on the Azimuth Forum should have a single capital letter, like "Cognitive math: sequences, hierarchies, networks". That's just the convention - not a big deal, but we like orderly behavior here. :-) I've fixed your title, which had every word capitalized.