So these two men get along well and enjoy a cup of coffee together. This article took a long while to get going but in the end was quite interesting.
I agree that there are problems with mathematics but to throw logic out of the door to fix it sounds a bit extreme. If it just boils down to making the mathematics more flexible that is a different matter but picking and choosing when to be flexible and when not is a bit arbitrary.It doesn't sound right to just pick the mathematics that will fit.
I think the problem is that the mathematics we have now, has been developed from the way in which we perceive the world, which is our subjective reality.The mathematics fits the way we see things.This anthropocentric view of reality has directed the development of science and the mathematics to describe it.
An example being past and future directed time-like vectors within the Lorentz transformation. The past, present,future idea being the Historical concept of time which is imaginary, rather than objective reality. It is a construct of the mind that enables sequential spacial experience to be comprehended by the brain, memory and anticipation and preparation. These are important for survival of the organism.
The way in which the mind processes information has been turned into mathematical models.Those models do not then always fit with other data subsequently obtained from objective reality.
All of the reality we perceive is a construct of the mind and mathematical models models which process data received from objective reality.
Is it possible to distinguish which parts of that reality are exact,or very close analogues of objective reality and which are subjective reality analogues that give a false perspective or opinion of objective reality? I believe the answer is sometimes, when there are paradoxes or the (subjective reality derived) mathematics will not work.
These are no longer just questions of mathematics but of neuroscience, psychology and biology.Physics is at a limit where mathematics alone will not provide the answers. Where the question "what is reality?" has to be asked. Is it the reality experienced by the mind? or the underlying reality that allows that experience? For analogy consider the computer game on the screen and the software producing it. What is real? Both, although they are very different. The software can not be explained by playing the game.Objective reality can not be explained by using only subjective experience and ideas derived from that experience.
Not directly related to the article,
Chaotic and dynamic processes within 4 dimensional space can lead to many kinds of development of matter.This will be fascinating to investigate. The "Vacuum" must have substance which provides no data to inform our subjective reality.That does not mean that it is not there, only that it can not yet be detected. Perhaps some of the dark matter particles only now being detected are a part of this substance. (But not "the dark" matter holding galaxies together which is probably galactic matter in afore space.)