In an earlier Blog on this post, the distinction between the Empirical Sciences and the Theoretical Sciences was postulated (confer: “Epistemological Difficulties”). The Principles of the Theoretical Sciences ought to be contrasted with Statements written in vernacular Prose.
For example, “The area of a circle equals its radius squared multiplied by pi” is much easier said by simply stating: “A = π × r2.”
The Principles of the Theoretical Sciences seem to exist independently of any conscious observer. It doesn’t matter who performs the calculation “7 + 5.” If they get any other solution besides the number “12” it is an incorrect solution. Likewise with “7 × 5 = 35.”
A certain ambiguity arises when the conventionally held philosophical alternative between what are called “Synthetic a priori Propositions” and “Analytic a priori Propositions” is made. The word “analytic” means taking things apart and the word “synthetic” means putting things together.
For instance, the Proposition “All bodies are extended” was held to be Analytic; whereas, the Proposition “All bodies are heavy” was held to be Synthetic. These kinds of arbitrary dichotomies obviously are obsolete.
To allay this confusion, what were formerly called Synthetic a priori Cognitions ought to be replaced by a contrast between the Principles of the Theoretical Sciences and other types of scientific statements expressed in Prose. These Theoretical Sciences are comprised mainly by Logic, Mathematics, and Geometry.
The most compelling feature of the Theoretical Sciences is their structure. Simply put, this structure is composed of Axioms, Rules of Inference, and Consequences. The Consequences of a Theoretical System can be derived in a decidable manner from the Axioms and the Rules of Inference.
The Science of Physics bridges the gap between the Theoretical Sciences and the Empirical Sciences. This explains the alleged “Pre-established harmony between Mathematics and Physics.” Yet recently contemporary physics seems to be running up against a glass ceiling of philosophical discourse. The metaphysics of materialism is clearly no longer a viable hypothesis. Certain notions such as the multiverse or relativistic space can be analyzed using the methodology of Ontology. Clearly, physics needs philosophy now more than ever.