Deterministic and probabilistic mathematical theories have in common that they construct mathematical representations of real-world phenomena. On a basic level this can be regarded as a type of explicit problem-solving. This involves presenting the problem in ‘abstract form’ in symbols (often numbers, letters, or geometrical elements). These symbols are then manipulated in accordance with precise rules: Strings of symbols in sets of equations come to represent ideas. The construction of mathematical theories involves testing whether experimental observations fit the postulated ‘mathematical rules.’ If they do not fit then the ‘mathematical rules’ may be refined, extended, or new ones may be formulated. Newly mathematically formalized ideas are validated by testing whether they align with observations but also by examining whether they are consistent with other, previously established, mathematical rules.