Dear Peter,
I greatly enjoyed your essay. Its wonderful.This is one of the most relevant essays I have found here on this site.
The reasons are obvious.
As you have mentioned
("We consider mathematics as fundamentally digitised geometry, so well able to approximate natures 'non-linearity'. As Galileopointed out;
"He who undertakes to deal with questions of natural sciences without the help of geometry is attempting the infeasible."
We cite various tricks which mislead us, not the fault of mathematics itself but of it's poor application due to our limited conceptual understanding.Reliance on mathematics as the 'language of physics' became pragmatic necessity when we were unable toclassically rationalise findings. Many now believe no classical rationale is
possible at quantum scales. John Bell describing that view as 'sleepwalking.')
I too believe that Mathematics,numbers ultimately come down to Geometry. The main reason why maths mislead us in any scenario is not the fault of mathematics but we don't understand the compatibility of mathematics with physics/natural sciences. There are certain laws of invariance behind mathematics itself,being a geometrical phenomenon and what is generally done is that we don't try to check whether its intrinsic law of invariance matches with that of the physical scenarios we are dealing with, and this leads to the mutual conflict and friction that can mislead us.
The true reason why mathematics is effective in physical scenarios : first of all we have limited physics to certain criterias to be able to explain it mathematically. Further,once we have done it, its infact the hidden laws of invariance behind the mathematical setup (geometry) which explains the laws of invariance of physical setup, even if the scientist/mathematician may not be aware of this intrinsic invariance behind the mathematics itself. This is the key point.
In context of Skolem paradox , as I have mentioned in my essay how model theory succeeds to explain some aspects of aphysical scenario while fails in other aspects. This is because of the instrinsic compatibility & incompatibility of their hidden laws of invariance.
The main focus should be to peep into that hidden laws of invariance behind the mathematical setup itself . This is greatly revealed i probably the most important problem of mathematics"Riemann Hypothesis. David Hilbert said - if he wakes up after 1000 years , the 1st question he will ask - has RH been resolved ? I personally have devised a method to attack RH that its true and the clue is to reveal the intrinsic invariance behind the mathematical geometrical setup itself.Numbers are nothing but Geometry.
As you have mentioned
(Mathematics can show that multiple inversely proportional 'complementary' sine and cosine curves naturally and physically exist for various qualities, as they do for the Dirac paired inverse spinors in so called 'unphysical' quantum mechanics.
Yet some higher order curvature is always superposed, at reducing scales but ensuring no entirely linear and precise mathematical description can be possible beyond the limits identified by Gödel. Many will find the visualisation methods revealing the relationships difficult, but they are powerful conceptual tools which
can be taught.)
You have also dealt with The Filter paradox.
The concern is paradoxes. the paradoxes reveal the fundamental discrepancy lying at the root of mathematics. They arise because of underlying structural conflicts.
I am also focusing on Godel incompleteness and Inconsistency theorems.
Now, if they arise, there are two ways.Either, considering it as the the end and accepting the limitation and secondly,trying to resolve the limitation by making some structural and fundamental change in the way way formalistic mathematics has been developed. I am concerned with this shift in the formalistic mathematics by allowing time and reference dimension to mathematics to sort out the paradoxes like physics. What we do is we take physics to be in time and reference dimension and take mathematics to be in timeless absolute dimension. This is the cause of paradoxes.If physics is in timeless , absolute, then so is mathematics and if physics is within the periphery of time and reference frame and the so is mathematics. This is because Physics and Mathematcis both are the cause of vibrations.
As you have mentioned :
(
We suggest then that 'new ways of looking' at large sectors of physics may then be
possible by using mathematics in fundamentally different ways, improving
understanding. However it seems that those 'different ways' are not visible using
just the 'lens' of current mathematics. We need formalisms to avoid the various
fallacies and hidden tricks and invalid proofs of a system which can prove 2 =1
via 'division by zero' easily disguised for instance by any term with a value of 0)
We certainly need formalistic mathematics which can sort out the fallacies,paradoxes,inconsistencies to deal with physical scenarios in a improved ways beyond conventional approaches.
Hence,as I have mentioned that we need to restructure formalistic mathematics itself based on their intrinsic characteristics to look at physics in fundamentally different and improved way. we need to operate the laws of invariance behind mathematical(geometrical )set up itself before explaining to physical /other natural sciences.
Anyway you have written great essay.
Thanks,
Pankaj.