Reference: Essays on Substance
Mathematics and Physics
The postulates of Mathematics do arise from real experience, but they are then extended into abstract concepts that become distant from reality. For example, numbers, and their relationships arise from our experience with counting and accounting of distances and directions, but they are then extended to ideas, such as, zero and infinity that are abstract concepts only, and can be interpreted in many ways.
Mathematics has developed along the lines of establishing consistency among its postulates. The abstract postulates have become part of its woof and warp. When mathematics is applied to physics, its abstractions have to tested against real observations. Establishing consistency between the mathematical abstractions and real observations benefits both mathematics and physics.
A scientific hypothesis starts from real observations that are not quite consistent with established theories. The scientist generates a hypothesis to resolve that inconsistency. He may create a mathematical model to flush out as many inconsistencies as possible from his hypothesis.
The hypothesis may make new postulates. These postulates need to be properly justified. When we talk about subjectivity, we are talking about unjustified, arbitrary postulates.
So the scientist develops experiments to test the postulates of his hypothesis. These experiments can be conducted using physical equipment in a lab. The resulting measurements are then compared against the predictions from the hypothesis. A consistency between the two helps the hypothesis to be accepted as a theory.
When mathematical interpretations of the hypotheses and theories start to become complex, unreal, or simply pointless then it is time to develop thought experiments to further test the hypotheses and theories for inconsistencies. The theory of relativity and quantum mechanics are very mathematically oriented. They have espoused ideas about space and time that conflict with reality. Such ideas need to be carefully examined with well designed thought experiments that use live logic.
The thought experiments shall examine the postulates of the hypothesis for consistency with established principles. It will also examine the logical continuity among the ideas and observations leading to those postulates. Finally it will examine the harmony, which these new postulates bring to the broad scene of scientific principles.
The resolution of inconsistencies arising between a hypothesis and the reality of scientific principles may also bring into view basic concepts that are missing both in mathematics as well as in physics. This is what Faraday was insisting in his essay quoted below:
Faraday 1857: On the Conservation of Force
.
