In modern philosophical and scientific discourse, fundamental sentiments and mathematical definitions occupy a central role, serving as the starting point for the formation of a unified theory.

Foundations of Thought: The Role of Self-Evident Axioms in Science and

In modern philosophical and scientific discourse, fundamental sentiments and mathematical definitions occupy a central role, serving as the starting point for the formation of a unified theory. It is the self-evident propositions, accepted without the need for additional proofs, that provide the solid foundation upon which all knowledge is built. Separating the emotional tone from intuitive representations allows these ideas to be transformed into strict, formal axioms, which greatly simplifies the process of analysis and the further development of new ideas.

In mathematics, this approach is particularly evident: adhering to immutable definitions such as point, line, and plane ensures the unambiguous and logically coherent development of theories. The acceptance of basic premises without proofs helps form an underlying framework in which every subsequent thought and conclusion finds its justification. This method not only minimizes the need for endless proofs but also endows constructions with the precision and integrity that are so highly valued in scientific analysis.

Thus, the use of fundamental sentiments and mathematical definitions as foundational axioms becomes a reliable tool for building complex theoretical constructs. This approach maintains internal logical coherence in argumentation and allows for the dynamic development of new theories built on unchanging, yet remarkably flexible, principles.

Why are basic sentiments and mathematical definitions often accepted as axioms in philosophical and scientific discussions?

In these discussions, fundamental sentiments and mathematical definitions are often taken as axioms because they serve as the starting point for reasoning and the subsequent development of theories. They are seen as self-evident propositions that require no proofs, thus becoming the fundamental elements on which the entire system of knowledge is constructed.

As noted in one source, "It is necessary to pay attention to the common definition of an axiom as a self-evident proposition accepted without proof. If one removes the affective tone from this definition, it can be considered quite precise" (source: link txt). In other words, when fundamental sentiments or intuitive representations are stripped of their emotional evaluations, they become strict axioms that can serve as the basis for formal reasoning.

At the same time, mathematical definitions—an integral part of the logical structure of science—are accepted as axioms to provide a clear and unequivocal foundation for further constructions. For example, axioms are used to define basic concepts such as point, line, and plane in Euclidean geometry, which, when accepted without proofs, create a limited yet complete context for theory formation (source: link txt).

Thus, the choice of basic sentiments and mathematical definitions as axioms is driven by their ability to represent fundamental, self-evident starting points. This approach avoids the endless demand for proofs and imposes a rigorous logical framework on the system. It simplifies the process of reasoning, making subsequent conclusions internally coherent and well-founded within the established context.

Supporting citation(s):
"It is necessary to pay attention to the common definition of an axiom as a self-evident proposition accepted without proof. If one removes the affective tone from this definition, it can be considered quite precise" (source: link txt)
"Axioms are statements accepted without proof. The set of axioms of a given theory is both a condensed formulation of that theory and the context that implicitly defines all the concepts contained therein" (source: link txt)