The Math Protocol

A mathematical view

For every type of action, there's a well defined range of outcomes, this aludes to the way Set Theory works in math. Set theory studies operations like union, intersection, and difference, and relations such as membership (whether an element is in a set) and inclusion (whether a set is a subset of another). Moreover, in category theory, the primary objects of study are not just the elements (like in set theory) but the morphisms (arrows) between objects. These morphisms represent relationships or processes rather than mere membership, where in ordinary life the same occurs through actions and general state changes.​

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From a cognitive science or psychophysics perspective, humans naturally categorize their sensory experiences into discrete concepts to make sense of the world. We tend to impose structure on continuous experiences by dividing them into categories. This is a philosophical parallel to how Category Theory imposes structure on diverse objects via morphisms. Both in Aristotle’s view and in mathematical Category Theory, categorization is essential for organizing knowledge and perception.

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The aim of defining a protocol here, as in a computer network, will work as seen in functors, where given an input, there's a set of expected outputs. Speaking of real-life daily activities, aside from subjective experiences, there's a rational ground which operates following certain formal rules. There are certain rules that are generally seen as bureocracy and others, less strict, as etiquete. It's possible to categorize all actions under a set, such as daily household activities, or business activities in general as a super set and its variations as subsets, etc. Jumping into Category Theory, for every action or morphism, it's possible to categorize its structure as well, in other words, we would have sets of functors/morphisms.

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This concept also applies to cognitive and phenomenological frameworks, where each individual operates as a closed information system, with subjective information access constrained by their cognitive state. Drawing parallels to phenomenology and solipsism, each person can be seen as possessing an internal “information set” that no one else can fully access. Communication, therefore, becomes the process of transmitting limited subsets of information across boundaries—what's referred as a protocol.

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In this view, communication involves passing subsets of total information between individuals. Just as physics limits information transfer, so too does psychology and phenomenology. In game-theoretic terms, these transmissions involve probabilistic choices and outcomes, where participants engage in a dynamic, strategic exchange of information across these barriers. Communication, thus, mirrors a game of constrained rationality, with the possible paths dictated by the "moves" of information exchange.

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We also want to have a more clear definition of perception, meaning that perception is the first act of any inference, where elements are distinguished in a given context. This very basic concept of distinguishing and categorizing elements is how indeed perception works intuitively. And going forward with the relation between these elements, constitutes the protocol or operation rules for that given context. The same approach can be seen in Leibiniz's Monads or Eclidean points, meaning we tend to categorize elements that exists on its own and, going even further, the very act of categorize has its structure as seen in the semiotic triangle.

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Speaking of natural language and inferences, the aim of this proposal, is that every concept with its inferences, containing words, sentences and whole paragraphs, are categorized under sets and categories that are linked to primary concepts, which means having an ontology, and that the relationship between them such as inferences in argument maps have its set of possible outputs, or the permutation of words given its context. With that in place, we could predict every inference in any context, or at least an approximation of the real-world interactions if enough data is provided.