An In-depth Exploration into the Mathematical Groundwork and Philosophical Relevance of Mottos

Ah, mottos – compact, mighty capsules of wisdom, mankind's attempt at distilling complex life experiences into concise, universal truths. They seem so simple on their dimensional surface, yet when examined through the lens of critical intrigue, they reveal an intricate network of mathematical proportionality, narrative compression and philosophical assertion.

As a mathematician, my initial interaction with mottos is more quantitative than qualitative, assessing their structure before delving into their content. The concise nature of a motto is reminiscent of an equation – an elegant simplicity that articulates an expansive truth. For instance, Pythagoras' theorem might be attributed a mere a² + b² = c², but this concise expression belies the multitudes of geometric insight it conveys. Simultaneously, a motto like "Carpe Diem", a mere two words, invites contemplation that stretches well beyond its brevity.

My physicist's mind marvels at a motto's balance of constants and variables. Just as in my beloved field of physics, wherein every formula hosts constants offering inherent stability around which variables dance, a motto too, possesses a core principle, a ‘constant’ that remains untouched by cultural, temporal, or subjective variables. A motto, like a law of nature, is independent of the observer. Whether "Carpe Diem" is uttered by a Roman philosopher, or a high-school student in modern-day Tokyo, its essence, the ‘constant’, remains unchanged.

Now, here's where the delicious dichotomy of my interests, the celestial meets the phantasmal, finds its expression. The motto, like Lovecraft's cosmic horror that I so passionately devour, carries within it an inherent duality, an intertwining of comfort and dread. "Seize the Day" urges action, optimism, the preciousness of the present, injecting a certain inspiration, a comforting thrill. However, it simultaneously whispers of mortality, of time's relentless march, encouraging a healthy fear of procrastination, mirroring the underlying fear in Lovecraft’s works which fuel my fear of the dark.

The mathematical patterns of mottos ripple into my hobbies too. Take bowling, for instance. Its motto “The aim is to aim” captures the essence much like an appealing equation. It proposes the ten-pin formation (constant) as the central challenge around which the aim (variable) changes. There's an attractive symmetry in this, isn't there?

Of all my eccentricities, my companions, Emmett Brown and Parrot have been privy to most. They've observed me ruminate over mottos and their intrinsic affinity towards math and philosophy. They've watched me contentedly wrestle with the confluence of comfort and fear in these compact sayings, entirely forgetting my discomfort of TMJ in those moments.

Motto-scholar Guillaume Apollinaire once penned, "Now and then it's good to pause in our pursuit of happiness and just be happy." A precious reminder, indeed. Mottos invite us to reflect, to dissect and reconstruct narratives, to apply mathematical rigor, and to enjoy the abstract art of thought, all at the same time. In the grand narrative of my universe, mottos, with their symmetry, fear, and wisdom, comfortably occupy their unique space, revealing more about life than what one may initially perceive.