In this text, we describe different mathematical concepts for the joy of mathematical structures, in the sense that, though they can be described purely symbolically, we will focus on describing their spatial structures through intuitively understandable examples.

This is done because, even though we recognize the practical utility of relying only on symbols when doing computations, we acknowledge that there is a greater joy, as well as a deeper and often quicker understanding obtained, by experiencing them through their spatial representations, as we can relate them to intuitions of spatial structures from our own life.

In addition, we wish to also unveil the practical utility in leveraging formal mathematical concepts in less formal practical situations, to improve our understanding and intuition of our everyday life, such that we can make better informed decisions and improve our lives.

Markov Chains

In this section, we describe Markov chains, which are sequences of probabilities for events, where the probability of each event only depends on the previous event.

For example, if you’re walking around a city and at each corner throw a dice to decide which corner to walk to next, then the sequence of corners visited, constitutes a Markov chain; this is because the probability of your getting to the next corner, depends only on the corner you are on, but not on the corners you’ve visited to get there.

As a counter example, if you’re walking around a city in the same manner, but avoiding corners that you’ve been to before, then the sequence of corners visited, does not constitute a Markov chain; this is because the probability of your getting to the next corner, will depend on whether you have been to any of the adjacent corners before, and therefore not just on the corner you are on.

Practical Probability Chains

Probability chains like Markov chains are just one specific instance of, can also be used to clarify or deepen our understanding and intuition of everyday practical situations; for instance, most of us might experience that the probability of one going directly from work or school to the gym, is greater than the probability of one first going home and then to the gym, while the probability of staying home once there is even larger.

This might seem like a trivial example, and most of us might indeed have an intuitive hunch that this is how we function; yet we find ourself leaving work or school to go home, thinking we’ll rather visit the gym later, even though we know we have just made a decision for the now, that we know will make it less likely that we do what we wanted, later.

Therefore, we believe there is a reason of existence for a text like this, where we somewhat pedantically elaborate on practical uses of mathematical concepts for concrete situations, so as to counteract our human tendencies to avoid responsibility for meeting even our own needs and goals, especially when in deficiency states, like being tired after work.

Section further or separately describing virtues of applications, indirectly at least from own life, to brainwash myself further (overeating sugar traveling, underworking while spending, myopically enjoying infatuations; teleopically enjoying enjoyments (plural), general comment on existentially tilting at fatal delusions) – jeg kan ikke huske sist jeg var absorbert i for-the-sake-of-it-arbeid (tapt i tid, mange mange timer)

Ergodic Probability Chains (Source States), or Smoke Will Fill All of a Smoke-filled Room

A ergodic probability chain is one in which no event has probability zero; this captures the intuition of smoke inevitably filling a smoke-filled room, since there are no places where the smoke can escape the room, while there are no places in the room where the smoke can not go.

In the context of personal practical situations, this amounts to living or having habits in such a way that none of your actions lead to a zero probability action, meaning there’s no ultimate shrinkage or death in your action space; this could also be referred to as having healthy habits.

Non-ergodic Probability Chains (Sink States), or Water Will Fill a Boat With Holes

The opposite of an ergodic probability chain, is a non-ergodic probability chain, which contains sink states: these are states with zero probability transitions, which means a process through the chain might terminate. In practical personal situations, this amounts to living a life or having habits that can shrink to no actions, or death, in your action space; which can also be thought of as having unhealthy habits.

For instance, being overly reliant or repeating singular habits for the same needs, means more paths will lead to sink states if those habits can fail, like if one relies on just one thing, like a partner, for joy, and not contain habits of multiple sources for satisfaction of the need for joy.

Ergodic Imperative of Fostering Source States surplanting Sink States (or Death)

This points to an imperative for spontaneity together with a wider set of habits to satisfy each need, like for instance fostering habits of doing things that brings you joy (like writing about mathematical structures for myself, or traveling to new places more or less every other week, or drinking), in addition to singularly joyful habits like spending time with a partner, which is risky on a mathematically structured basis as seen here.

X: Alkohol. Sex. Novelty; myopic – Non-ergodic; sinking.

X*: JMS. Forhold. Bøker. Kultur. Mote; Teleopic – Ergodic; sourcefull.

Elaborte further, deeper. Surplant sinks. Jms.

– Walking far and eating is source state. On.

– Walking 10k daily is source state. On.

– Phone is sink state. Off.

JMS is source state. On.

Travel is source state. On.

Books are source states. On.

Unbounded relationships are sink states. Off.

Bounded relationships are source states. On.

Running is source state. On.

Sugar binging is sink state. Off.

Alcohol binging is sink state. Off.

Culture is source state. On.

Fashion is source state. On.

Intuitively eating nutritiously is source state. On.

Financial security is source state. On.

Financial insecurity is sink state. Off.

Sedentariness is sink state. Off. Travel.

Travel is source state. On.

Elaborate on what this means: that proceeding through source results in matrix of increased probabilities, while proceeding through sink results in reduced probabilities; color code temperature probabilities; and