The Fundamental Axiom

An axiom represents a statement accepted as true without requiring further validation. In mathematics, for example, we understand that if a = b and b = c, then a must equal c—no additional proof needed. In the realm of science, empirical observations serve as axioms, such as the observable order in the universe that allows us to formulate conclusions. These axioms often serve as foundational components for scientists and mathematicians to develop systems, which in turn yield theorems.

For a system to demonstrate completeness, it must possess all the necessary theorems to address any inquiry. This is, undoubtedly, a lofty expectation. Furthermore, a genuinely complete system should only yield proof of one truth, excluding its opposite. For instance, if a lawyer can establish their client’s guilt in a specific case, they must also be unable to prove their innocence. If both could be proven true, it would imply that one proof is false, thereby destabilizing the entire system, including the theorems and the axioms that form their basis. Systems that maintain internal consistency are referred to as consistent.

Gödel’s Incompleteness Theorems

In the early 20th century, Kurt Gödel, a notable logician and mathematician, proposed two theorems that elucidate the limitations inherent in any axiomatic system.

However, there is a prevailing belief that Gödel’s theorems apply solely to mathematics. Why is this the case, considering that science also operates on axioms? The answer lies in a deeper exploration.

The first theorem asserts that a system can only be complete or consistent, but never both. This assertion holds true. There are no systems capable of providing answers to every conceivable question without some contradictions arising. For instance, if I could prove that humans are both warm-blooded and cold-blooded, my system would be considered complete, even though one or both assertions could be incorrect. Such inconsistency is deeply troubling for inquisitive minds.

But what if a system is consistent yet incomplete? Surprisingly, this scenario is not catastrophic. In fact, we would prefer a system that cannot address every question, provided it remains free from contradictions. This concept mirrors reality itself—no system can encapsulate every truth, leading to the notion of "incompleteness."

Yet, a significant challenge arises: how does one establish consistency? Suppose you operate within a singular system. How does it prove its own consistency? The answer is "no." We do not claim that gravity exists solely based on gravitational theorems; instead, we recognize its existence due to its observable effects, which invokes a second system: physics, specifically the Second Law of Motion. In essence, no consistent system can validate its own consistency, as articulated by Gödel’s second incompleteness theorem.

The Distinction Between Science and Mathematics

If mathematics serves as a framework that science utilizes to interpret observed phenomena, why isn't science subject to Gödel’s Incompleteness Theorems? The distinction lies in the nature of their axioms.

Mathematical axioms are universally true, regardless of circumstances. This is because mathematics was designed as a tool to elucidate observed realities. For a mathematical model to accurately convey an observed phenomenon, it must be firmly grounded in axioms. A flawed tool—like a shovel compromised by termites—will fail at its task. Similarly, mathematical constructs that lack a solid foundation cannot effectively explain anything.

Scientific axioms, however, differ significantly. While mathematical axioms are foundational, scientific axioms may be limited or even absent. Why? Science is inherently flexible. Whereas mathematics adheres to strict rules, science constantly adapts and evolves. A scientific axiom may assert that the Earth is the universe's center, but this understanding can shift dramatically with new discoveries—something a true axiom cannot do.

To clarify, science employs postulates, which share similarities with axioms but allow for greater flexibility. These postulates act as presumptions that may or may not hold true, providing a stable framework for exploration through the scientific method. In mathematics, postulates equate to axioms, while science operates on logical assumptions.

What happens to Gödel’s Incompleteness Theorem in the absence of true axioms, as seen in science? Indeed, it becomes inapplicable.

The Quest for Knowledge

The pursuits that endure are those that resonate deeply with humanity. In essence, the endeavors most aligned with our human experience will persist as long as we do.

Science exemplifies such a pursuit. From agriculture to scientific inquiry, our quest for knowledge has been integral to our existence. Indeed, the pursuit of knowledge is among the fundamental aspects of being human.

If you were to ask whether science might someday be rendered irrelevant, my answer would be a resounding "no." This isn't merely because scientific facts are valuable—facts evolve as our exploration deepens. Science will remain relevant as long as we continue to seek knowledge. Until there is no more to discover, or until humanity deems knowledge itself worthless, science will endure.

The first video, "95% Of Jobs Will Become Obsolete By AI | Ben Goertzel," explores the implications of AI on the job market and its potential to render many professions unnecessary. This topic intersects with the ongoing dialogue surrounding the future of various fields, including science.

The second video, "Is Data Science Becoming OBSOLETE? What Do You Think?" invites viewers to contemplate the evolving role of data science in a rapidly changing technological landscape.

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