A Hard Question (Partially) Answered, Badly

21 Jul 2023

I have managed to not post anything here for a whole month. :<

Every night, I am challenged with the question of whether to keep on watching videos on YouTube that I know I’ll forget by the next morning or whether not to sleep. As a big fan of psuedomathematics - mathematics applied in the least non-rigorous setting for the most fun possible - I’m going to try to apply that here. I’ll go over in the end why pseudomathematics is (maybe?) not a complete waste of time.

Getting Started

Assuming everything in your life is predetermined (big assumption, I know) - we can assume the functions of \(life(t)\) or \(l(t)\) is set. This function will return your set of experiences \(e\) each of which can be evaluated for a happiness score \(H(e(t))\) and a memorability score \(M(e(t))\). For now, let’s assume that \(e(t)\) is a vector-valued function with values that are interpretable in the same way of a latent space. The first element of this vector is the time at which this experience starts. Given this, the total happiness in your life \(Q\) from \(t=0\) to time \(T\) over a total of \(E\) experiences

\[Q(T) = \sum_{i=1}^{E} M(e_{i}) * H(e_{i})\]

or with cheesy calculus,

\[Q(T) = \int_{t=0}^{T} M(l(t)) * H(l(t)) dt\]

But we are not done yet! Keep in mind that the memorability \(M(e(t))\) and its happiness \(H(e(t))\) will actually vary throughout your life. An event may be forgotten about but then remembered, or may seem happy at first but then become sad. Not to mention both \(M\) and \(H\) are probably correlated, which we can ignore to make things simpler. For now, let’s revise the calculation of happiness to be for a single experience.

\[Q(e_{i}, T) = \int_{t=e_{i}[0]}^{T} m_{i}(t) * h_{i}(t) dt\]

where the \(i\)th event will have it’s own memorability and happiness functions for each second - \(m_{i}(t)\) and \(h_{i}(t)\) respectively. Because “life is nothing more than experiences” we can just sum up the experiences given by \(l(t)\) at each second.

\[Q(t) = \int_{t=0}^{T} Q(l(t), T)dt = \int_{t=0}^{T} [\int_{s=t}^{s=T} m_{i}(t) * h_{i}(t) * ds] * dt\]

This also means the change of your happiness every second is given by

\[Q'(t)= \int_{t}^{T} m_{i}(t) h_{i}(t) dt\]

Of course this can all be repeated for pain/sadness, which we’ll just assume is the negative of happiness here. But the basic tradeoff of memorability vs. happiness is still there.

Back to the Question

So basically this view of happiness leaves a question of whether its better to have higher happiness and less memorability or vice versa. We’ll look at each case before translating our rationale to math.

Case 1

A situation with higher happiness and less memorability looks like an overworked hedge fund manager choosing to have a year long vacation (at its extreme).

Case 2

The most extreme example of living with no memorability but constant happiness would be partying every single day, watching YouTube every single day, not working a mundane job, etc.

The goal of contemplating this is to understand the behavior of the functions \(m(t)\) and \(h(t)\). Which scenario seems better? Of course, I think most people would choose Case 1 - the common argument would be that Case 2 is aimless and vainful. Both memorability & happiness of a given experience \(i\) decay past that event’s starting time.

It’s also imperative to remember that given the current mathematical formulation, we are trying to maximize happiness greedily as we are only looking at \(Q(t)\) of the current time and not considering long-term effects. This sways much more heavily to case 2 - that happiness leaves slower than memorability.

This actually, does make sense. When going to a restuarant and eating a dish we’ve never seen before we will likely only remember a few things - 2 minutes of conversation, the first time we saw our food, etc. Happiness of eating the dish when we are hungry would last a little longer.

Therefore, I think it’s fair to say that because memorability of an experience after its start date is naturally focused on non-mundane points (beginning, end) whereas happiness of an experience seems to last longer we can rationale that happiness of an event dies down at a lesser power than memorability. Even further, because memorability focused on specific points (1-2% of the actual lifetime), that means that \(m(t)\) is very likely 0. Therefore, counterintuitively we should be optimizing memorability over happiness.

Back to Problem Again

Because \(m(t)\) is going to be 0 a lot more than \(h(t)\), we need to increase \(m(t)\) as much as possible to make \(h(t)\) actually mean something. Therefore uniqueness of experience and variety matters more than feel-good indulgence.

Stop watching YouTube at 4am lol.

Why use Pseudomath

What I just did is a horrible way to solve a problem.

Pseudomath is a continual reminder that math can’t solve all our toughest problems. Problems so hard a book didn’t come with them. Furthermore, if I had not used pseudomath how would I have thought about the tradeoffs of memorability and happiness? Maybe I would have instead said optimizing happiness is better because we only have one life, or that we don’t forget the happy moments in our life.