Geometry And Physics

One stone is one stone, and no one needs to investigate the external world with their senses to discover this fact.  However, can knowing that mathematics must apply to physical objects reveal the material composition, weight, or subjective physical feel of a given stone?  Of course not!  Logic reveals only necessary truths, some of which pertain to numbers and shapes, not contingent facts about particular objects that can be perceived with the senses.  Geometry and physics are intertwined, but they are far from synonymous.

Physics inherently relies on mathematics, which is itself founded on the laws of logic, but the mathematical aspects of geometry are not revealed by the empirical investigation of matter, nor are any other mathematical truths.  The fact that one equals one is not grasped because of scientific inquiry.  No matter how many times a scientist examines an object, it will always be governed by the law of identity, the law of non-contradiction, and numerical truths because logic inescapably transcends the whole of science.  One can know this before even opening one's eyes or physically analyzing an object.

The logical factors that define shapes likewise apply to all physical objects without exception.  Even if a person never once intentionally uses the scientific method to dissect a repeatable sensory observation, they can easily prove to themselves that nature is confined by logic, including the numeric logic of mathematics.  It is attributes like density, weight, size, and chemical makeup that cannot be discovered solely by rational reflection (not that science can unveil any facts about these things outside of the present moment, and even then the results are only known to reflect one's perceptions).  To obtain information about these characteristics of a given object, some form of sensory investigation is required.

It is only the extreme overemphasis on physics in modern culture that drives many to go beyond this and overlook that geometry, while it does overlap with physics, is separate from science in the ultimate sense.  Geometry is the application of logic to the concept of shapes, which can in turn be applied to science.  Science is therefore not the foundation of geometrical facts.  Sensory experiences are not the basis for one's comprehension of numbers and shapes; numbers and the laws of logic as a whole are the basis for one's comprehension of sensory experiences.