Chapter 1: The Dilemma of Unanswerable Questions

In life, certain questions remain perpetually unanswered. Is there a divine being? Is the concept of time merely a human construct? What lies beyond the edge of the Universe? And let's not forget the classic debate: which came first, the chicken or the egg? (Spoiler alert: it's the egg.)

These profound inquiries about existence, science, and the potential presence of a deity are emblematic of the unresolvable dilemmas we face. In the early 20th century, the logical positivists in Europe dominated philosophical discourse. They argued that questions lacking empirical verifiability were essentially devoid of meaning.

This school of thought dismissed ethical discussions, claiming that moral assertions could not be empirically validated. How could one design an experiment to determine the "right" course of action? Consequently, concepts like ethics were rendered meaningless, while metaphysical discussions were likened to sorcery.

However, the decline of logical positivism reveals a critical flaw: limiting rational discourse stifles meaningful dialogue. Just because we cannot empirically validate theories surrounding ethics or the existence of God does not mean we should abandon these discussions altogether.

Additionally, the feasibility of verification itself came under scrutiny. Philosopher Karl Popper proposed an alternative framework centered on falsification. Instead of asserting that a theory can be proven true, he contended that theories could only be disproven. For instance, claiming "All swans are white" cannot be verified without observing every swan, which is impossible. Therefore, Popper's focus shifted to finding evidence that contradicts a hypothesis.

The relevance of this discourse to the concept of axiomatising our dilemmas is significant, as the reality is far more nuanced.

Chapter 2: The Fragility of Scientific Foundations

Science, contrary to popular belief, lacks a solid foundation. We grapple with the uncertainty of the assumptions we hold about the world, and the complexity of scientific theories defies binary classifications of true or false. Each theory builds upon previous ones in a seemingly endless chain, often referred to as "turtles all the way down."

Consider climate models as a case in point. We have a consensus that the Earth is warming due to human activities, yet the margins of error in these models are substantial. Questions abound regarding the extent of warming, timelines, and the behaviors responsible. These models are probabilistic and rely on a network of localized models, which are themselves grounded in complex theories of temperature, fluid dynamics, and chemical interactions.

The scientific endeavor can be likened to constructing a skyscraper: if a single brick is misplaced in the foundation, the entire structure might collapse — but we will only discover this once the building is complete. Currently, science appears to yield reliable and useful insights, but what if fundamental theories about matter, heat, or acidity are fundamentally flawed?

Perhaps science is not a robust skyscraper but rather an intricate castle built on clouds, or a house erected on unstable piles in a swamp. Popper himself articulated this notion, suggesting that the piles are driven into the swamp from above without reaching any solid ground. Construction halts not upon finding a firm base, but when the builders are satisfied that the piles can support the structure — at least for now.

In the first video, Terminology of Axiomatic Systems, the discussion delves into the foundational language and principles that underpin axiomatic systems, essential for understanding the structure of mathematical theories.

Chapter 3: The Limitations of Mathematics

You might argue that all scientific inquiries can ultimately be distilled into mathematics. If climate science leads to quantum theory, which is based on mathematical principles, one might assume science is more akin to a "castle on solid ground."

While this perspective is understandable, it overlooks the complexities of mathematics itself, particularly Gödel's incompleteness theorems. Kurt Gödel, a pioneering logician and mathematician, made groundbreaking contributions to the field at a remarkably young age.

To comprehend Gödel's insights, it is crucial to recognize that mathematics operates as a theoretical system grounded in axioms. The Peano axioms serve as a fundamental example:

1. 0 is a natural number.
2. For every natural number x, x = x (reflexivity).
3. For all natural numbers x and y, if x = y, then y = x (symmetry).
4. For all natural numbers x, y, and z, if x = y and y = z, then x = z (transitivity).
5. For all a and b, if b is a natural number and a = b, then a is also a natural number (closure under equality).
6. For every natural number n, S(n) is a natural number (closure under successor).
7. For natural numbers m and n, m = n if and only if S(m) = S(n) (injection).
8. For every natural number n, S(n) = 0 is false (no natural number has 0 as its successor).
9. If K is a set such that (a) 0 is in K, and (b) for every natural number n, n being in K implies that S(n) is in K, then K contains every natural number.

These axioms represent fundamental truths in mathematics, yet they need not be the only valid foundation. Gödel's first incompleteness theorem reveals that no matter which axioms are adopted, some mathematical truths will remain unprovable, indicating that mathematics can never achieve completeness.

The second incompleteness theorem states that one cannot ascertain the self-consistency of a mathematical system solely from within that system. In essence, it becomes clear that mathematics, like science, is inherently incomplete.

The second video, (AGT4E12) [Game Theory] Axiomatization of Shapley Value, explores the axiomatic foundations of game theory, shedding light on the methods of establishing meaningful value assessments in strategic interactions.

Chapter 4: Lakatos and the Evolution of Scientific Thought

Imre Lakatos, a Hungarian philosopher of science, initially studied under Popper but eventually developed his own philosophy. He viewed scientific theories as comprising a dual-layered structure: a "hard core" of established principles and a "protective belt" of adaptable hypotheses.

Consider Newtonian physics; its hard core consists of Newton's laws of motion and the law of universal gravitation, while the protective belt encompasses all other variables that sustain a Newtonian framework.

A historical example illustrates this approach: when discrepancies in Uranus's orbit emerged, physicists did not discard Newton's principles. Instead, they hypothesized the existence of an unknown planet affecting Uranus's orbit, leading to the discovery of Neptune, which validated Newton's theories.

However, as scientific understanding evolved, quantum mechanics challenged the foundational principles of Newtonian physics, underscoring the dynamic nature of scientific paradigms.

Chapter 5: Axiomatising Problematicity

Science, philosophy, and mathematics are complex domains riddled with assumptions that are challenging to prove or disprove. Among philosophers of science, Lakatos stands out for his ability to encapsulate the essence of human intellectual behavior.

For instance, in the realm of religion, some individuals may take the existence of a deity as an axiom, surrounding it with flexible hypotheses like "God is mysterious" or "God is just." Similarly, in economics, core Marxist principles might form the foundation of a revolutionary's beliefs, protected by ad hoc explanations for any critiques.

This pattern of establishing a hard core of beliefs while defending it with adaptable hypotheses is ubiquitous in human thought. We accept axioms in mathematics, despite Gödel's assertion of their insufficiency, and we rely on the uniformity of nature in science.

When we construct elaborate theories, or when we cease to dig deeper into the unknown, we are not committing a philosophical error. Instead, we are simply navigating the complexities of being human in a world rife with uncertainty.