Mathematical Girl §
- When something is found that cannot be explained by the traditional definition, it becomes a new concept or discovery.
- Examples:
- In the definition of natural numbers (Peano Axioms), it cannot explain something that becomes 0 when one is added to it.
- Therefore, negative numbers were introduced.
- In the definition of rational numbers, it cannot explain something that becomes 2 when squared.
- Therefore, √2 was introduced.
- In the definition of real numbers, it cannot explain something that becomes -1 when squared.
- Therefore, i was introduced.
- In the definition of finite sets, it cannot explain the bijection with Infinity (the expression feels wrong, will be corrected later).
- Therefore, infinite sets were introduced.
- The names “negative,” “irrational,” and “imaginary numbers” reflect the struggle faced when encountering something that cannot be explained by the traditional definition (it seems).
- (/icons/understood) (takker)
- This struggle was resolved by the concept of “complex numbers,” which emerged as a result of understanding the “two-dimensional numbers.”
- This is part of the The Function of Generalization in Mathematics essay, specifically the section on Semantic Interpretation.
To be covered in the TOK essay. §