Section 1.1: The Principles of Existence

Let’s define some terms to frame their physical nature, inviting a formal examination of their interrelations. Nature’s wildness is encapsulated by two key principles:

• Principle of Plenitude: □𝒯(◊Φ ⟶ Φ) (Over time, if something is possible, it becomes actual.)
• Principle of Wastefulness: □𝒯(∃Φ 𝒰 ¬∃Φ) (Over time, something exists until it ceases to exist).

In essence, nature's wildness arises from a force, ⊗, that ensures both principles hold true universally. Their convergence results in a peculiar plenitude, 𝕻. Time is filled with all conceivable entities, yet these entities are contingent, indicating that nature ultimately consumes each of its creations.

The outcome of this relentless cycle of creation and destruction is a wild environment, where 𝕻 embodies two sets of existential traits derived from the two principles. From plenitude, we observe majestic variety and sublime grandeur, while wastefulness yields godlessness, absurdity, and monstrousness. Nature executes both principles without bias, generating a plethora of beings while simultaneously eliminating those that might disrupt the balance.

To further illustrate, consider how 𝕻 contrasts with pure chaos. Nature is inherently constructive, demonstrating a natural order that allows for the emergence of higher orders and properties. In a chaotic state, however, one would find no stable patterns or creative force to impose order. This is distinct from chaos theory, which suggests that chaotic systems can self-organize, indicating that even seemingly disordered systems contain some underlying order.

If we envision pure chaos, 𝓒, while excluding ⊗, we encounter a paradox. One cannot conceive of something emerging without a coherent structure to define its properties. For instance, if a frog randomly materializes in space, it may seem chaotically placed within that environment, but its internal structure must remain orderly to define it as a frog.

This brings us back to the contradiction of pure chaos. It simply does not exist as a viable metaphysical option; rather, we confront the complexities of 𝕻.

Now, drawing an analogy with black holes, we can visualize a concentration of ⊗, resulting in the evolution of life and, ultimately, consciousness. This leads to the emergence of personhood.

In summary, we can differentiate between the impersonal generation of forms and the conscious struggle for survival among living beings. The former reflects a cosmic force, while the latter represents deliberate action to reproduce and shape future generations.

Here, we define the strictly physical aspect of 𝕻 as "𝓐R," characterized by automation and devoid of intent. Conversely, the living aspect is termed "𝓜R," indicative of mental properties such as autonomy and ambition. Thus, we can see that 𝓐R < 𝓜, highlighting how consciousness enables a deeper engagement with the natural world.

This inequality underscores the parasitic relationship that 𝓜 fosters, as living entities absorb stimuli from 𝓐 (e.g., scientific observations) and create extensions of their consciousness, culminating in the artificial constructs of civilization, 𝓔.

Thus, we can express this transformation as 𝓜(𝓐 ⇒ 𝓔), where the thick arrow symbolizes the potential to convert the wild into a domesticated space for minds. Yet, this domestication is merely another instance of 𝕻's self-undermining nature, as it perpetuates the cycle of creation and destruction inherent to wildness.

In essence, 𝓜 intensifies ⊗, akin to how the core of a black hole intensifies gravity, distorting the very fabric of spacetime. Without mental faculties, physical entities can only dismantle one another incrementally, while the conscious mind can deliberately engage with nature through symbols and concepts.

From this, we can infer that 𝓜 = 𝓐². Moreover, 𝓜(𝓐 ⇒ 𝓔) serves as a targeted manifestation of the cosmic cycle of self-overturning inherent in nature.

This video, "The Mathemagician," showcases Art Benjamin, who combines mathematics and magic to reveal the beauty and wonder of numbers. His performances illustrate the surprising connections between mathematics and magical feats, challenging our perceptions of both disciplines.

Section 1.2: The Illusion of Mathematical Formality

But what was the intention behind this formal attempt? It wasn't to validate any claim, but to contrast formal discourse with philosophical inquiry, raising questions about the contexts in which mathematical reasoning thrives. Can any subject be treated mathematically? If so, does this stem from the intrinsic nature of subjects or the character of the mathematical method itself?

Mathematics focuses on forms or structures, striving to expose underlying features with precision, much like reverse engineering a natural process or designing a system from a blueprint. The artificiality of mathematical language aims to eliminate ambiguity, creating rigorous logical frameworks where each proof builds upon its predecessor without omitting any steps. Consequently, one could argue that patience in mastering these artificial concepts leads to an inevitable acceptance of the conclusions drawn. Rationality seemingly demands submission to this process.

Yet, could it be that mathematical proofs function similarly to magic tricks? While mathematical presentations may be more intricate than my previous example, the thoroughness could be seen as a form of overcompensation.

Consider how a stage magician insists that everything is above board and transparent, allowing spectators to witness each step of the trick unfold while still delivering a result that defies expectations. Similarly, could mathematicians inadvertently obscure elements of their proofs?

Take, for instance, the sequence of natural numbers. Once "1" is established, it becomes apparent how to derive subsequent numbers through simple addition. This process appears straightforward and fair, with no hidden tricks at play.

However, what happens when we include zero in the natural number series? How do we logically transition from 0 to 1? If we start with zero bananas and add one, we transition from having none to one. But the state of zero is misleading, as it suggests an absence that wasn't truly there.

Zero behaves differently than other numbers; adding 1 to itself yields a higher number, while adding 0 to itself leaves you with 0. Thus, the logical application of addition to 0 becomes murky, revealing a potential trick where the mathematician's exhaustive transparency falters in the transition from no numbers to some.

We need not solve this metaphysical puzzle to utilize numbers effectively. However, it raises the question of whether the meticulous presentation of formal systems is deceptively simplistic.

If the aim is not to mirror reality but to explore formal possibilities, one can invent concepts and rules at will, transforming mathematics into a game. Yet, what kind of game is as serious and tedious as that depicted in math textbooks? Perhaps a misleading one—akin to a trick.

This suspicion has lingered since my early encounters with mathematics in school. I had numerous philosophical inquiries, but time constraints in math class left little room for exploration. Students had to follow the laborious steps laid out in textbooks or risk falling behind, as authors claimed to reveal everything.

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Section 2.1: The Illusion of Complexity

The core issue lies within the style of mathematical discourse, particularly in educational contexts. The emphasis on meticulous detail, the reliance on unnatural concepts to eliminate human subjectivity, and the insistence that no step is overlooked in a proof mask the fact that mathematics is fundamentally a constructed discipline.

Indeed, certain mathematical ideas are grounded in natural patterns, proving useful for tracking regularities, aiding scientific exploration, and benefiting industries. However, when strictly addressing forms or structures, every mathematical concept represents a significant simplification. Plato posited that formal reality is obscured by the complexities of the natural world.

Mathematicians face two constraints: the necessity for practical tools to navigate the environment and the preference for simplicity—often referred to as the "beauty" of elegant proofs. Mathematicians can adjust definitions or introduce assumptions to achieve desired outcomes, akin to how a government can print currency. Yet, if the results lack utility, elegance, or are contrived, the proof may be dismissed.

The pragmatic role of applied mathematics is subjective, as our interest in quantifying and tracking phenomena is inherently human. The appeal to beauty is even more questionable, as the mathematical endeavor often appears excessively nerdy and male-centric. This notion of beauty may serve as a form of overcompensation, similar to a magician who employs an attractive assistant to enhance their allure.

However, if these "beautiful" proofs are intended to apply to nature, their beauty must be subjective, reflecting the mathematician's choice to disregard complicating factors. The true complexity of natural patterns is often better captured in terms of horror rather than romance. While arithmetic may possess elegant proofs, counting itself is unsettling, bounded by inexplicable events that frame the cosmos.

Nonetheless, math textbooks rarely emphasize these subjective choices—the hidden elements or traps—focusing instead on the tedious rule-following that suggests mathematicians merely follow orders without any personal stakes. Such textbooks are scarce. Mathematics is defined by what the magician allows you to see, not by what lies concealed.

Moreover, if a person presents you with ten dollar bills, one of which is a forgery, that constitutes fraud. Conversely, if someone hands you a similar stack without any forgeries, you can confidently claim to have received ten legitimate bills. This illustrates how arithmetic functions correctly.

Yet, as Friedrich Nietzsche posits in "On Truth and Lies in a Nonmoral Sense," counting is superficial because it assumes equivalence among instances, imposing our subjective categorizations. The dollar bills would only be equal for specific purposes, as closer examination reveals discrepancies among them. When do differences among instances matter, and when can they be overlooked? Only we, as counters, can answer this, as the broader world remains indifferent to our classifications.

My contention is not that counting or any mathematical operation lacks utility. Rather, we risk self-deception if we mistake the apparent objectivity of formal presentations for true objectivity.

In this respect, mathematics resembles a formal occasion—such as a wedding or gala—where attendees must adhere to social conventions to avoid being deemed informal. The rules governing these events are constructs of human imagination, much like mathematical regulations.

Are mathematical rules merely artificial? This question does not negate their practical applications in navigating reality. Instead, it raises an aesthetic and ethical inquiry: can the exercise of logic lead formalists to arrogance, causing them to overlook the likelihood that all mathematical concepts and rules are subjective idealizations?

If you engage in a game, enjoy it and abandon the pretense of merely following orders, as though you're not subtly pulling tricks behind the scenes.

My attempt to formalize my philosophical explanation was enjoyable, akin to the joy of writing a novel—the creative thrill of constructing narratives and exploring their implications.

To address the earlier question, while it’s conceivable that the world is fundamentally structured or formal, as proposed by Platonism or the idea of a metaphysical simulation, it is more plausible that nature itself does not conform to those notions.

As a mode of inquiry, mathematics embodies formality, which signifies a societal stance. This formality reflects an intention to dominate nature. When we speak of a thing’s "physical formality," it implies that the entity is exploitable due to its lack of consciousness, rights, or human characteristics. In essence, "form" or "structure" translates to "dumbness" or some form of automatism.

The moment scientists fully articulate the human mind’s structure, formalizing every aspect of thought, emotion, and behavior, we will have effectively enslaved ourselves—demonstrating, through our elaborate games, that we are merely natural beings bound to obey external rules.

In this video, Art Benjamin discusses how he turns mathematics into magic during his performance at the USA Science & Engineering Festival, illustrating the wonders of math and its captivating nature.