Apparently Einstein died wishing he knew more math. Actually he should have investigated the very foundations of mathematics.
Mathematics is a subject that is extremely consistent. One is not going to find any inconsistency in math. But Einstein should have been curious about the ideas on which mathematics is based.
The Foundations of Mathematics
Einstein struggled to understand Quantum Mechanics all his life. He felt a fundamental disagreement with its basis but he could not put his finger on it.
Experimentations at quantum level have shown that there is wave-particle duality. This means that there is no such thing as a “point-particle.” Yet the quantum theory still looks for a point-particle in terms of probability through a wave function.
Maybe it is not a point-particle that is being assumed. Maybe it is a “point-location” and a “point-time” that is being taken for granted. This is possible because mathematics itself is founded on the ideas of “point” and “unit.” The truth is that our very thinking is constructed upon the ideas of “point” and “unit.” Mathematics is simply a refinement of logic.
Quantum Mechanics shows us that the classical thinking is no longer valid. The mass of a particle cannot be reduced to a precise center of mass . Nor is there an ideal disturbance with no center of mass. Infinite sine and cosine wave representations of disturbance are mathematical constructs only. The truth is somewhere in between. the center of mass gets increasingly diffused as we investigate the quantum levels.
The inconsistency seems to be that space and time are still being perceived as independent of the disturbance (wave) and the mass (particle). It is being believed that there is a “point-location” and a “point-time” even when there is no “point-particle.” Looks like reality is being confused with mathematical thinking.
Can the very space and time be “diffused” like the quantum “particle”?
Maybe space and time are not independent of disturbance and mass. Maybe space and time are as diffused at quantum levels as are the observations of waves and particles.
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Comments
Because of this uncertainty in position and time, I doubt if the ideas underlying differentiation and integration can be applied to phenomena at quantum level.
I have come to doubt the validity of the Shroedinger’s equation itself.
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QM seems to be 5% experimental knowledge and 95% interpretation. We even call it Copenhagen interpretation.
It is not difficult to start out from the actual experimental knowledge and then explore outwards from there keeping in mind that mathematics is just a tool. If we put too much reliance in mathematics without checking against reality now and then, then it can be akin to being brainwashed by mathematics.
We cannot idealize a quantum object as a particle that has a precise location. It is more like a wave packet (snake) of finite bounds. So, it doesen’t make sense to compute the probability of some particle as to its location. That would be like computing the probability of “part of a snake” at a certain location. Why not compute the location of the whole wave packet (snake)?
The situation is that we cannot fit the whole “snake” in a precise point location as was done for a particle. We have to think in terms of location of the whole wave packet, and the probability of that is always going to be 1. However that location is not going to be a point.
So problem gets to be the fixed idea of thinking in terms of “point” locations only. That is what mathematical integration is doing. I believe that integration no longer works at quantum levels. We need new mathematical tools for better understanding.
Maybe it is time to look at Georg Cantor’s ideas related to infinity and infinite sets, because it seems that the composition of wave packet belongs to a different infinite set then the infinite set in which it is moving.
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