Dear Alfredo Oliveira,
There many fascinating threads in this contest. Another thread dealt with "the unreasonable effectiveness" of math for physics, and you made some excellent points which I quote below.
I have addressed Wigner's quote in my ref.5. The key physical fact underlying our metaphysical reasoning is that the universe behaves logically. This can be exemplified by the creation of 'logic gates', AND and NOT, and subsequent sequential operation of these gates to construct all (finite) logical structures. In his 2009 FQXi essay Marcel-Marie LeBel noted that
"Maths are the metric extension of logic. Logic is therefore more primitive, more fundamental than mathematics."
It is not difficult to show that from logic gates one can easily construct counters to produce [finite] numbers, and comparators to test for relations (less than, equal, greater than]. From Kronecker we have reason to believe that, given the numbers, all else follows. Grossberg's mathematical model of neural nets allows us to construct similar logic and to sequence it, and to do so with 3-D structures. Given consciousness [!!] we become aware of these math relations, but without awareness of the material source of the logic, we may do as Robert Godwin says:
"One begins by abstracting from concrete existence, and ends by attributing concreteness to the abstraction."
Instead, Alfredo Oliveira notes:
"Mathematics is a logic language, strictly logic; however, to where it leads depends on the hypothesis and assumptions on which it is applied. Because it is logical, it leads to 'understandable' models provided that the hypotheses and assumptions are "understandable"...
"Mathematics has also the possibility of fitting whatever set of data - it is just a matter of considering enough parameters." [... such] mathematical models are usually "not-understandable", they present logical inconsistency and parameters that obviously cannot represent a physical entity."
"However, many consider that these models of data are correct models of reality, and so they consider that the universe is "non-understandable". That seems to be the case of Wigner,..."
I believe that Oliveira has perfectly stated the situation.
I try to further clarify "the unreasonable effectiveness" as follows: My vehicle was to teach a robot how to derive a theory of physics from measurements. The general approach, group the numbers via inter-set and intra-set distances to derive feature vectors, is summarized in my endnotes. Thirty years later Schmidt and Lipson applied this theory via pattern recognition algorithms to
"automatically search motion tracking data captured from various physical systems..."
Whereas I had treated little more complicated than trajectories of rocks, etc, Schmidt and Lipson treated complex systems such as weights on springs and the double pendulum, systems with predictable regularity. Based on their pattern recognizing robot they found:
"Without any prior knowledge about physics, kinematics, or geometry, the algorithms [the robot] discovered Hamiltonian's, Lagrangians, and other laws of geometric and momentum conservation."
This agreed with my theory. However what I found most fascinating was that the 'type' of law that the system found was determined by what variables were presented (to the robot observer). They discovered:
"... if we only provide position coordinates, the algorithm is forced to converge on a manifold equation of the system's state space. If we provide velocities, the algorithm is biased to find energy laws. If we additionally supply accelerations the algorithm is biased to find force identities and equations of motion."
I find this absolutely fascinating. This comment does not address life or consciousness. The consciousness is in the mind of the programmer of the robot and the designer of the pattern recognition algorithms. Yet the result is one that I had not expected, namely that the type of data determines the type of physics in the derived model. It makes sense when one thinks about it, but it's still an impressive fact. It addresses the question of the "unreasonable effectiveness" of mathematics, and falls on the side of "complete reasonableness" of mathematics, as it depends from logic, which can be demonstrated physically. To dispute this I believe requires that one demonstrate physically something that is not logical. And such a demonstration should not depend upon mathematical structures that have been projected onto physical reality, as described in my essay.
While this in no way detracts from the beauty of mathematics, or the mystery of life and consciousness, it does, I believe, remove some of the mystery from mathematics as applied to physics.
I thank you, Alfredo, for so succinctly stating your arguments above.
Edwin Eugene Klingman
