True Axioms of a science like physics would allow for some replacement; biology in terms of physics.
One of the features lacking in biology, throughout biology is the importance of electricity and magnetism in biology. Very little if any is ever discussed or taught in biology that the living organism is so dependent on electricity and magnetism. I think the reason for this omission and the reason that no-one in physics was smart enough in the 20th century to place the Maxwell Equations as the axiom system of physics, is that the Maxwell Equations are so hard to understand, even for the best of physicists. Their understanding of the Maxwell Equations is mostly memorization, not innate understanding. For example, it was said of Tesla, unlike the physicists of the 20th century, that someone could draw the most elaborate and complex electrical wiring and circuits and Tesla would instinctively know if it would work or not work and point out the errors.
So, given the case of Tesla, that we can perhaps expect only one person per century to be comfortable with electricity and magnetism and not some Phd rote memorization of the Maxwell Equations.
Now when you have the true axioms of a science, then there is some replacement, substitution that can occur. For example the geometry axioms as given by Euclid and then revised by Hilbert. Now we can do some replacement such as a plane instead of a line in the axiom system. For that we can still retain many of the truths such as two nonparallel lines intersect in a point, whereas two nonparallel planes intersect in a line. Or another example is that line A parallel to B, and B parallel to C is that A is parallel to C can be replaced with Plane A, B, C where plane A is parallel to C.
The point I am making is that a true axiom system of a science subject allows for replacement or substitution of the terms of the system, terms more complex than the original system. Now some truths of lines cannot be generated from planes simply because of the increased dimension.
But let me now transfer that understanding over to the Maxwell Equations as axioms over all of physics to that of Maxwell Equations with substituted biology.
So in the case of physics of Faraday law we have a nucleus of an atom as a capacitor, a proton as a capacitor of stored energy of magnetic monopoles and a electron traveling around the proton capacitor as a closed loop of wire with moving electric charge in the closed loop of wire.
In biology we have a living cell and a nuclear region of DNA. DNA is a capacitor of stored information, stored energy and going around the nucleus is messenger RNA.
Alright, I do not know how far I want to dive in and elaborate on this analogy. What I do want to do is to refer to the fact that the true physics axioms must be so that its terms and equations can be stepped up in complexity and thus give us other sciences. In fact, my pursuit of the geometry of the nucleus and proton and electron should be guided by biology more than I have so far done.
And, the true axioms also will reveal how very much complex the Maxwell Equations are, even though we do not step up in complexity. Here I am speaking of the concept of duality, in that our human minds are not able to go below duality, what some have called the principle of Uncertainty, but which is really the principle of duality. The Maxwell Equations derives the principle of Duality and the Uncertainty principle.
But let me tell it so that science illiterates can even understand it. The entire observable Universe that we see is just a single electron of a single plutonium atom. So the stars, galaxies, planets and life are fragments, bits and pieces of one electron. Yet in the Maxwell Equations we view an electron as a tiny object of rest-mass 0.51 MeV with a negative charge, yet that single electron is the entire observable Universe of stars and galaxies and planets.
But, an axiom system can do that, because a plane is far more complex than a line and we can replace a line in the axioms of geometry with planes, to some extant. In physics, we can talk about the electron and proton as what the Faraday law is all about, but we can also talk about the Faraday law as if it was the axiom of biology where the DNA instructs the messenger RNA to commit some biological metabolism.
In physics, we have a whole bunch of terms such as capacitor, closed loop wire, magnetic field, electric field, bar magnet, battery, emf, voltage, solenoid, etc etc. In biology we have a lot of terms such as DNA, RNA, metabolism, meiosis, mitosis, cell wall, nucleus of cell, etc etc.
But if you examine physics with biology, you begin to realize that the very same axiom system that delivers all of physics, when you step up the complexity of terms, that same system that gives all of physics, gives all of biology.
What the principle of duality or Uncertainty enter the picture, is that the terms of physics cannot go lower in meaning than what they already are. If the Maxwell Equations tell us that the proton is a parallel plate capacitor storing magnetic monopoles, then I cannot go any lower in meaning, but that the proton is so very complex, far more complex than our human mind could ever understand. Even though a capacitor appears to be a simple mechanism. The electron as the entire observable universe, yet the electron as a closed loop of wire is the complex to the simple, but that is how axioms work. They allow the most complex to be contained into a simple term-- electron.
Google's archives are top-heavy in hate-spew from search-engine- bombing. Only Drexel's Math Forum has done a excellent, simple and fair archiving of AP posts for the past 15 years as seen here: