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On axioms and assumptions. How changing them can change the world.
- Axioms and assumptions play a crucial role in various fields such as mathematics, physics, and economics.
- Logical axioms are generally valid in different settings and are considered true across domains.
- Non-logical axioms, while specific to a particular domain, are essential for advancing understanding within that field.
- Aristotle and Euclid were instrumental in establishing foundations for logical and non-logical axioms in their respective works.
- Challenges to long-standing axioms, like Euclid's Parallel Postulate, have led to the development of non-Euclidean geometries and advancements in other fields.
- Systems define the truthfulness of axioms, allowing for the creation of models tailored to specific domains.
- Despite the limitations of axioms, choosing appropriate ones can drive progress and innovation in various scientific disciplines.
- Changing axioms can lead to significant discoveries and advancements, particularly in emerging fields of study.
- Gödel's theorems illustrate the inherent limitations in our systems constructed with axioms, highlighting fundamental obstacles within defined systems.
- While some axioms may not translate to the real world, others, like those in quantum physics and game theory, offer valuable insights and practical applications.
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