Mathematics views reality as fundamentally discrete. It even approximates continuity through infinity of vanishing discrete units. It then finds itself becoming increasingly complex when it tries to deal with the reality at the quantum level.
Understanding seems to decrease with Increasing complexity.
At the quantum level we find the wave nature coexisting with particle nature. From the viewpoint of abstraction, this is “continuity” coexisting with “discrete-ness”.
Physics has been using the model prescribed by Euclidean geometry in its application of mathematics. The “probability function” at quantum level assumes locations in space to be like dimensionless points of Euclidean geometry. This approach is adequate to model mass particles in space, but when it comes to space with no mass particles in it, the Euclidean approach becomes questionable.
How can a point that, by definition, has a boundary, be devoid of volume or mass? It is at best an idealization for a vanishing particle, but when it comes to defining continuity, that supposes no boundary at the level of “point” this idealization does not seem to work. The boundary comes from the mass of a particle. When there is no mass there is no boundary. Pure energy has no point like boundaries. If pure energy has boundaries at all, then they seem to enclose a dimension equal, at least, to the wavelength of energy.
When it comes to modeling continuity of space, the dimensionless point presents a logical inconsistency.
We expect the atomic electron with “diffused mass” to have a “large” location. When we try to approximate that location with dimensionless points it seems to introduce arbitrary assumptions. We base the “probability function” of quantum mechanics on Euclidean geometry, and this basis is questionable. It is no wonder that there are so many suppositions, conclusions and conjectures related to quantum mechanics that lead to incomprehensible oddities.
We desperately need to model the quantum phenomenon such that it moves our understanding toward greater simplicity. It may need mathematics that can conceive of “points” that have variable dimensions. Such mathematics may define photons by “points” that have dimensions equal to the wavelength of the corresponding electromagnetic wave.
We may need a mathematical model for quantum phenomenon that shows continuity condensing toward increasing “discrete-ness”.
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