Some Philosophy of Science
This effort to derive physics from logic alone is a relatively new idea. And some have argued that this is a hopeless cause doomed to failure because it is by no means clear on how to proceed. They argue that logic is too general to predict something as specific as the laws of physics. But now that I have some math to back up this claim, and this effort seems to be predicting physical laws such as quantum mechanics, quantum field theory, and symmetries of the Standard Model, perhaps it is not such a far fetched hope to expect that all of physics can be derived from logic.
The current paradigm of science is to construct an hypothesis, and make predictions based on this hypothesis, and then design experiments and compare results with predictions. And if the results do not match predictions, then modify your hypothesis and start over. This has worked well over the years, and I don't see any reason to change it. The only thing my efforts may offer is on how to construct hypothesis to begin with.
Presently, hypotheses are constructed by noticing patterns in nature. And mathematical equations are sought to predict the patterns that are observed. Experience with the math suggests means to extend the theory with new mathematical entities, and this new math makes predictions that are further tested. Essentially this is simply a curve-fitting exercise that attempts to find mathematical equations that best fit the data from experiments. These mathematical entities can be very sophisticated, such as symmetries hidden in the patterns. And I do have respect for these efforts; they can be very ingenious. But the underlying reason for why this math and not some other kind of math is not even addressed with this method. It seems as though there is a lack of overall guiding principles; in reality it's just trial-and-error guess-work. It may eventually produce results, but you can never know if you are done, and you'll never know why it's that way, or where it all comes from to begin with, or why.
Physicists will say that you can never prove that a theory is correct; you can only possibly prove it wrong if you come up with an experiment that contradicts it. Theories are provisionally correct until proven otherwise. Confidence in a theory is gained only when many experiments in various circumstances don't falsify the theory. With no means to definitively prove a theory correct, we cannot even begin to address questions of the meaning and purpose of the universe because we don't even have a starting premise for the structure that we observe. We are left wondering why it exists, why it is this way and not some other way, or what exactly is driving the universe towards some ultimate goal.
All this is especially true for quantum mechanics. In Isaac Newton's work, a mathematical structure of physics is built on the observation that force is responsible for making things move. And this makes an intuitive sense since we are all accustomed to applying force to things to make them move. We can actually feel forces on our bodies. And in Einstein's work physics is built on the premise that the speed of light is constant for all observers. And this makes an intuitive sense in our minds because if we were stuck in a closed box, our observations, which depend on the speed of light, could never tell us how fast we are moving, or whether we are accelerating or in a gravitational field. But what gives rise to the quantum nature that we observed in the lab?
Until now, there doesn't seem to be any underlying principle on which to construct quantum theory. It seems to entirely be a curve-fitting exercise. The developers of quantum theory simply noticed that the energy levels of electrons in an atom come in discrete states, and that predictions of black body radiation work better if one assumes a discrete nature to the electromagnetic waves bouncing off the walls. With the prior knowledge that discrete spectrums can be described by eigenvalues of operators, they eventually developed the math of Schrodinger's equation with its Hermitian operators, eigenvalues, and Hilbert spaces. But those who study the foundations of quantum theory have not converged on an underlying reason for it, and professors that teach the subject are loath to answer questions on foundations. If pressed to explain why nature is this way or what gives rise to quantum theory, the usual mantra has been "shut up and calculate", and don't ask such questions. They use it because it works.
This has allowed all sorts of wild speculations about what gives rise to quantum nature and what it proves about our ultimate fate. Some have even asserted that the spirituality of the mind gives rise to quantum theory, that this accounts for ghosts, or angels, or God, or the heaven to come. And the lack of foundations has lead to the frustration of students over the use of imaginary numbers to predict physical quantities. This has caused some students to give up on physics altogether. So I think my efforts can go a long way to alleviating this confusion and frustration and give students a sense that there are good reasons for why the world is as it is.
As I show in detail here, my efforts derive at least quantum theory from logic alone. I make no assumptions about the definition of force, or the speed of light, or whether there are particles, or fields, or mass, or energy or even any structure of spacetime. Everything is derived from the premise that all the facts, whatever they may be, exist in logical conjunction with each other. This conjunction can be formed into alternate paths of implication, where the conclusion of one implication is the premise of the next. I explain how the Dirac delta function can be used to give a mathematical representation to implication. This allows logical OR's to be mapped to addition and logical AND's to be mapped to multiplication, just as in probability theory. When the gaussian form of the Dirac delta function is used, the formula produces the Feynman Path Integral, which is the wave function of quantum mechanics. So the wave function of quantum mechanics is really just a mathematical representation of logical implication.
The math is straightforward; I make no assumptions nor assert any postulates whatsoever. I employ only the typical math in the subject. The immediate result is quantum mechanics, the process is iterated to give quantum field theory, imaginary numbers enter to describe the algebra of multiplying distributions, the Born rule for probabilities is easily understood, and the symmetry of the Standard Model is derived to explain the higher algebras involved. There are even hints at the underlying symmetry of the spacetime metric of special relativity. It's still a work in progress, but I think this is definitely encouraging.
But this presents a potential problem. What happens if the logic is undeniable but experiments contradicts it? That would seem to be a dilemma that would suggest that reality is illogical. Which one would you believe, your eyes or your faculties of reason? Either one could be wrong. Is logic falsifiable? Do experiments lie? Perhaps this explains some of the aversion to believing that physics can be derived from logic.
But I don't think this should stop us from trying to derive all of physics from logic alone. In fact it seems we are headed in that direction anyway. Our theories are becoming ever more abstract, until eventually our theories will be built on abstract principle alone. The question is will we be able to recognize the logic in these abstract mathematical principles? So when will the search for answers stop? Will we stop if we discover this particle or that symmetry? Or will we continue to search for answers? I don't think there is any alternative but to continue to ask why until answers are derived from reason alone. For at that point the only thing left would be to question our faculties of reason. The only alternative seems to be to simply stop asking why.
It's beginning to look as though the energy levels necessary to confirm our theories are so great that in all practicality we will never be able to directly test predictions. We will never be able to create another Big Bang in order to directly confirm our theories of creation. Instead we are being forced to rely more heavily on the internal logical consistency of a theory as the deciding criterion of its validity. So why not start with that? I think this is completely consistent with the goal of science which is to explain everything with reason. So far my efforts predict quantum mechanics, quantum field theory, and the symmetries involved with the Standard Model. Can a theory based solely on consistency be partly right and not completely right?
But what are we really dealing with? What is maintaining the consistency between facts? Things in reality are described with propositions which are true. True is assigned to propositions that do describe things in reality; false is assigned to propositions that do not describe things in reality. There are no false facts in reality. Yet, if the universe is logical, in the propositional logic sense, then there is some logic taking into account both true and false in order to determine what is true only. This is beyond the realm of the reality of true facts only; it exists everywhere, determines everything, and is always justified in whatever it does. I'll leave it to you to think about what that could be.
If you have any comments, please let me know. Thank you.