Comments
These are comments on the effort to derive physics from logic as described here.
Why is it desirable to derive physical laws from logic alone? If actual physical reality cannot be derived on principle alone, then you cannot stand on principles if they are irrelevant. You cannot talk about what is right and wrong in the world if right and wrong, true and false have nothing to do with reality.
Yet, there is hesitation among some people to accept any attempt to derive physics from logic. There are objections based on such attempts not being "science" as they understand it. And there are objections based on the incompleteness of mathematics. The claim is that science is a process of producing theories that predict events that can be confirmed by experiment, and the data gained from experiment can be used to refine theories. But theories based on logic alone bypass data and experimental confirmation and so cannot be science. This perspective might make it difficult to publish derivations from logic in the physics department; they might say such efforts belong more in the philosophy or math department. And those departments might say it has more to do with physics. But if such a derivation has a good chance at producing a valid theory, all departments should be concerned. And since it concerns the ultimate justification of physics, I would think it should belong in the physics department.
A derivation from logic alone does not exclude confirmation from experiment. Confirmation would certainly be welcome, but it would not be a surprise, since it would be an undeniable conclusion. If logic predicted certain events that experiment proved did not occur, then logic itself would be held suspect, and we would have no reliable means to believe anything. And the discomfort over this possibility probably causes some to hesitate. But to think that logical reason does not have the final authority about reality can be equally distressing.
A derivation from logic differs from the usual in that it does not start with data from any particular observation, but it starts with the most general of principles. It is possible, however, to view logic as the first physical theory. The ancient philosophers used physical situations as the premises to conclude the existence of other physical situations. Later this was generalized to cause and effect, and later to premise and conclusion of abstract argumentation. The ability of logic to make conclusions was then used to deduce the truth and falsity of purely abstract constructions such as mathematics. It is this ability of logic to apply equally well to the abstract as well as the concrete, to fiction as well as to fact, that makes some think that physics cannot be derived from logic. You'd never know whether you're deriving fact or fiction. But just like every other theory of physics, the laws would only tell you what type of results are possible if the right experiment is set up; they don't tell us that the experiment will be set up.
For most scientists physical circumstances always remain a contingency in the equations of logic - a proposition, describing reality, whose truth or falsity is not ascribed, but whose consequence follow if it is true. It is claimed the ancient Greeks failed in their attempt to derive everything from logic, and it should not be attempted now. But the ancient Greeks failed because everything remained a contingency in their logic. But whatever is a contingency remains by definition unexplained because a contingency is not something whose existence is necessarily asserted as true. Stating a theory that given these physical circumstances as a premise these conclusions must follow as a result does not say that the premises are required; it does not give reason why the premise must exist. The only way to explain everything is that everything derives from reason itself. A theory of everything (TOE) cannot exist except as a derivation from logic. Otherwise, if physical entities are contingent on other entities which themselves are contingent on something else, etc, then your theory rests on a contingent fact which beg for further explanation.
Theorists must face the dilemma that theory will ultimately not be able to be confirmed by experiment. For it would take all the energy in the universe to create an experiment to confirm a theory of how the universe was created. There is no choice but to build a TOE on principle alone. It only remains to be seen on what principles to rely on. Since logic is based on the algebra of true and false, and since theory must ultimately be true or false, logic has to be the ultimate basis of the ultimate theory.
And then there are arguments against this effort that rely on the incompleteness of mathematics. Godel's incompleteness theorem argues that any axiomatic mathematical system cannot be both consistent and complete. A consistent system is one in which no statement in that system can be proven by that system to be both true and false; that system always proves its statements to either be true or false, not both. Complete means each and every statement in that system can be proven by that system. Godel seems to have proven that any system advanced enough to use math cannot be complete. This means that formulas can be written using the rules of math but cannot be proven or derived by math. The argument is that any theory of physics is a mathematical system and so it cannot be complete; you can't prove or derive everything that's physical using math. And some go so far as to use this to explain life, or consciousness, or ghosts, or even quantum mechanics.
However, the attempt to find a physical theory of everything is not the same as the effort to find every mathematical formula possible. In fact we are trying to find only one formula to describe physics, not every formula. Godel himself has proven that propositional logic is complete and consistent; math is only used here to label the propositions for accounting purposes just like in financial accounting. And no one argues that financial accounting is incomplete or inconsistent because math only enters the picture in a limited fashion as is done here.
If each separate physical entity required a unique mathematical formula to account for it, then you could argue that we will never be able to describe all of physics with math since math is incomplete. But that's not what is being attempted here. We are trying to find only one unique formula to describe the kinds of events or interactions that might occur. This is not an attempt to derive the position and momentum of every particle in the universe. The effort is to describe typical kinds of interactions and very broad aggregate properties which everything in the universe must have.
I've used only the very basic concepts of propositional logic and math. What could be more basic than describing things with propositions and counting them? But perhaps this is not new. Didn't Einstein describe each "event" with coordinates, and didn't he require a causal relationship between events? Then I introduced reality as a conjunction of facts. Perhaps this is actually a physical requirement imposed on the logic. It seems like an obvious assumption, but I've not seen a conjunction of statements used as a requirement for any other abstract axiomatic system. I then manipulate the conjunction into a disjunction of "paths", where each path is a conjunction of implications, where the previous fact implies the next. The logic is correct, but does a conjunction of implications really describe moving along a parameterized curve in space? It's interesting to note how replacing each implication with an exponential function so that when all of the exponents are added up in the conjunction they form the integration of a path. If that integral were differentiated, it describes a parameterized curve through space. It seems strange, though, to represent an implication with a Dirac delta function. But then perhaps as a premise gets closer in description to its conclusion, then the truth that this premise prove this conclusion become more and more obvious.
Have I achieved a theory of everything (ToE)? No, not yet. What I have so far is not much more than a curiosity. But I am seriously encouraged by what I have so far. It seems I have made contact with accepted laws of physics (the path integral) from principle alone. And it seems difficult for me to believe that this is only a coincidence and that no further progress can be made. After all, if each part of physical law is consistent with every other part, then the part I have must be consistent with the rest. It only remains to show how the rest of it is connected. What remains to be shown is how to get Quantum Field Theory from this method. And it also remains to be shown is how General Relativity can be obtained just on principle. The trick will be to find principles and methods that rely on math and logic without resorting to physical concepts. I expect that the more general principles will be found first, such as the principle of least action and the requirement of a Hilbert space and a form of General Relativity independent of dimensions, etc.
This effort is not finished and may take some time to complete. In the mean time, I have a TO-DO list, outlining areas that need more attention. Perhaps some readers might have comments and suggestions along these lines of thinking. Contact information is on that page.