Considering our extensive discussions with Dick Stafford on math and reality, some of you may find the following paper of interest:
Towards a necessary change in the mathematical principles of natural philosophy
Authors: Mauricio Ayala
Comments: LaTex File, 28 pages, 10 figures, first version in Spanish
Subj-class: General Physics
Being mathematics a natural language to Mankind and to physics, it must be constantly adapted to our necessities and our natural perception. Then, mathematical concepts are not absolute to reality. Although mathematical theories are constructions of our mind, and the existence of objects in such theories is a matter of consistency or coherence of the theory (this allows us to study them independently of other theories), in physics its not enough that mathematical structures are coherent or consistent, these must have a much more tangible reference to reality. Therefore any mathematical model not necessarily is appropriate in physics. So, what sense has the concept of a point in physics, that elementary notion used to represent the objects of the reality.
If we consider that in a good model of reality, the existence of a mathematical object with a physical reference must be equivalent to the possibility of its construction, perception and verification of physical existence, where the refutaction of the non-existence of an object does not necessarily mean that it is possible to find a test of its existence. Then, one obtains that the mathematical language becomes richer in structures, where the point concept is replaced by more intuitive and perceivable concepts. On the one hand, this possibly can lead us to a physics without infinites, and without paradoxes, on the other hand, one restricts the use of mathematics and the use of mental experiments in physics.