Magical Realist
Why is it ok to posit an axiom based on no evidence whatsoever if it only makes your equations work?
It isn't. An axiom based on no evidence is pretty useless and likely wrong. You mentioned Unicorns, it would be foolish to assume they exist as an axiom since we've never seen one or any evidence of one.(careful, Unicorns are not the only assumptions in that category, the basis for many(if not most)people's world view is also on shaky ground)
But lightspeed? It has been MEASURED as always travelling at an invariant speed through a vacuum. EVERY TIME IT IS MEASURED, IN ALL FRAMES. So the axiom that light speed is invariant is not based on "no evidence whatsoever" and is a perfectly good assumption to make, unless and until evidence indicates otherwise. The fact that it fits on our map(IE in our equations or our models)indicates our map may be accurate(provisionally).
There are only two axioms in science that must be accepted without prior evidence...
1. That the Universe exists, that it obeys it's own internal rules and is as we see it.
2. That we are capable of understanding the Universe.
All other axioms are provisional and subject to correction.
If the axiom "Unicorns exist." made everything in the universe explainable in a consistently logical way it still wouldn't make that axiom one bit more true.
Wrong. The evidence would indicate that Unicorns do exist(in fact MUST exist)in your hypothetical. If you replaced Unicorns with invariant light speed that is exactly why we consider that invariance as fact. Dark Matter and Dark Energy were discovered by observations indicating that they MUST exist, they are both now assumed(as axioms)in every facet of Cosmology.
Or are we less concerned with the truth of the axioms and more interested in how they allow theories to explain things?
There are two paths of logic in common use. The Top Down paradigm is that truths(axioms)are handed down from on high and are infallible(the Truth(tm)). The Bottom Up paradigm is that axioms are provisional, a work in progress, to be changed or discarded according to the evidence. The Religions of the world are Top Down, their axioms must be accepted on faith. Science is Bottom Up, it's axioms are only as good as the evidence indicates and should never be accepted on faith. Top Down is concerned with the truth of it's axioms. Science tries it's best to prove them false(and expects them to be false if they do not explain the evidence), we call it Falsification. Theories are maps(we call them models), to be corrected as the evidence indicates. Truths(tm)are maps too, but they are expected to be accepted despite what the evidence indicates(IE on faith)and are considered unchangeable, true in all cases. There are no Truths(tm) in science, scientists are not concerned with the "truth of the axioms", but by the accuracy of our models, as indicated by reality. Axioms are only a beginning for logic in science, subject to revision. Axioms are the end of logic in religion(maybe I should just say "non-scientific thought"), questioning them is heresy, the "truth of the axioms" assumed and defended from change. It really is two separate and incompatible ways of thinking, much of your problem is you are trying to understand(or argue about)science by using a non-scientific way of thinking.
As for light being a particle and a wave at the same, no that is not logical.
It may not be intuitive, but it is what we see, so our intuition is wrong. That has happened a lot in science, especially around Relativity and the nature of light. If your "logic" doesn't conform to reality, it is your "logic" that is at fault(even if the logic is valid, if it is applied incorrectly or in ignorance the conclusion is not valid).
By all standards of aristotlean logic, something cannot be in two opposite states at the same time.
So our modern scientific world view should be constrained by what Aristotle thought was true? He was a brilliant logician, but ignorant of reality. And the wave/particle nature of light is perfectly logical, given the thousands of years of scientific inquiry since Aristotle.
Waves and particles are not mutually exclusive or opposites in any way, they are simply two aspects of the photon(an ocean wave is made up of particles, you know).
It's like saying the figure is both a square and a triangle. It's just not rational. Thus not only is the duality of light illogical, but quantum superpositions are too. The math may support it. The evidence may support it. But reason and logic do not support it.
Don't mistake your misunderstanding and/or ignorance for a failure in logic. What is illogical to an ignorant man is perfectly logical on closer examination by a scientific one. The FACT is that light has a dual nature, both wave and particle behaviors. If your "logic" says that cannot be true, it is your logic that is wrong because it IS true. You are the one clinging to a map that has been shown to be inaccurate, you are the one giving too much weight to your assumptions(axioms)and refusing to correct them given the available evidence showing them to be false. You are a Top Down thinker, pretty much useless in any scientific endeavor. Only Bottom Up thinkers need apply. It's like the "Authority" argument we had in another thread. Their are no Authorities in science, including Aristotle. Authority is where the Top Down thinker thinks validity or Truth(tm) comes from, it's how they see the world. A square contains two right triangles, you know, so any subset of it is both in a triangle and in a square. It's all in how you look at it.
Grumpy