Why Do We Assume That Mathematical Proof Is Universally Applicable Throughout All Domains and Universes?
The relationship between mathematical proof and the physical universe is often a topic of debate in both mathematics and philosophy. One common belief is that mathematical proof, once established, is universally applicable, irrespective of physical laws and domains. However, this notion can be challenged when considering that mathematical axioms and the theorems derived from them are fundamentally independent of the physics of any actual universe.
Independence of Mathematical Proofs From Physical Laws
Mathematical axioms and the theorems derived from them are not tied to the physical laws of our universe. For instance, the laws of logic are universal, meaning they apply in all contexts, whereas the laws of physics might vary from one universe to another. The foundational aspects of mathematics, such as numbers and geometry, are discovered rather than invented. Therefore, mathematical proof can be seen as a universal truth that transcends physical boundaries.
The Universality of Mathematical Proof
Mathematical proof is considered universal because once a theorem is proven, it remains true under the same axiomatic framework. Theorems like the Pythagorean theorem are proven within a specific mathematical context and remain valid without the need for empirical evidence. This is because mathematical proofs do not rely on physical observations, but instead, they are based on logical axioms and definitions. The validity of a theorem is relative to its context, not to any particular physical universe.
Universal Validity and Context Dependency
It is crucial to understand that no theorem is assumed to be valid in all domains across all universes. The domain of applicability for any theorem is a key aspect that must be considered. For example, the zeros of a holomorphic function of a single complex variable are isolated, while this is not true for a function of two or more complex variables. This illustrates the importance of context in determining the validity of mathematical theorems. The context, defined by a set of axioms, dictates the scope of application for a given theorem.
Special Cases and Boundaries of Mathematical Theorems
Mathematical concepts often have specific conditions under which they are valid. For instance, infinity is not well-defined in the fields of real or complex numbers, leading to contradictory results when treated as a number. However, in certain contexts, such as the extended complex numbers, infinity can be treated as a number. This highlights the importance of understanding the context and boundaries within which a mathematical theorem applies.
The Role of Infinity in Mathematics
Infinite properties of mathematical objects cannot be directly determined, as mathematics cannot provide information about such properties in our physical universe. Similarly, observing infinity through empirical methods is not possible, as it would require an infinite amount of time and resources. Infinity is a concept used in a theoretical sense within mathematics, rather than a directly observable phenomenon in the physical world.
Mathematics vs. Physics: Measurement and Existence
The focus of physics is on measurement and observations, not on the abstract concept of infinity or other theoretical constructs. All physical measurements are finite, which is why many physicists argue against the existence of infinity in the physical universe. By defining the universe of physics to be the universe of measurable phenomena, physicists can avoid the complexities and contradictions associated with infinite properties.
Conclusion
The assumption of mathematical proof as universally applicable spans across different domains and universes is not a universally accepted belief in the field of mathematics. While mathematical proof is indeed universal, its applicability is context-dependent and governed by the axiomatic framework in which it is established. The principles of logic and mathematics are universal, but their application within specific areas and domains can have nuanced limitations and conditions. Understanding these distinctions is crucial for both mathematicians and physicists in advancing their respective fields.