I need to spend some time on primitive axiom terms. In mathematics these are terms that we agree upon to use, but which have no further explanation, such as a point is without length width or depth. Or a line has length only. Equality and equivalence are also primitive terms.
In physics, when we have the Maxwell Equations with the facts of Chemistry as the axioms, there are primitive axiom terms involved, such as duality, rest-mass, electricity, magnetism, energy, time, space, momentum, charge, spin.
I need to spend a little bit of time discussing the primitive term equality and equivalence of mathematics and its misuse and understanding in mathematics and then its horrible flagrant misuse in physics.
In mathematics, equality is sameness. So to say that 2 = 2 means one is the same as the other. But what about 2 = 1 + 1? Well, in mathematics, since math is abstraction and not physical objects, that 2 = 1 + 1, is just as acceptable as 2 = 2. So what about equivalence? In math, equivalence is something like 1/2 equivalent to 3/6. Now in physics, if we have a pie cut in 2 parts and a pie cut in 6 parts, that 1 part of the one pie is the 3 parts of the other pie. Which is not exactly sameness of math but similarity of math, but for physics, equality and equivalence are used when there is no equality nor equivalence.
Now I am not here to straighten out math and physics as concerns equality and equivalence. I am here to draw attention to this thorny troubling problem that is in both mathematics and physics. There is no clear way of dismissing as to what is equal and what is equivalent, because 1/2 = 3/6 and 1/2 equivalent to 3/6.
But major problems do occur in physics when we think that equality is everywhere in physics. The equality or equivalence in E = mc^2. It is not really equality at all, nor is it equivalence.
In physics, often the concept of "proportional" is used. Now let us try that on mathematics. Is it better to say 1/2 is proportional to 3/6, rather than say it is equal or equivalent. It is safe to say that 2 = 2, because equality is identity. But in physics is it safe to say the Coulomb force is a equality? Or is there equality in the Maxwell Equations?
When physicists use "proportional" they still have some linear factor to include before they can put in a equal sign. So that in Faraday law emf is proportional to -dB/dt. Once we include the N windings do we have equality, emf = -N dB/dt.
But do we really have equality in physics as in mathematics? I would say no, because mathematics is not physical objects and physical structure but rather abstraction divorced of physical objects and structure, whereas physics is all about objects and structure. An electron at point A in space is different from electron at point B in space, simply because they are at different locations. The share the same charge of -1 and other attributes but they are not the same and identical.
So that in physics we just borrow the concept of equality from mathematics to allow us to make deductions.
So I guess I can help a little bit, by saying there are terms better suited than equality or equivalence for physics. Terms such as "transforms" or "proportional" or "becomes" or "changes" or "dual".
So that in the formula E = mc^2 we can say energy changes to mc^2 or the neutron in this formula:
Neutron = proton + electron + neutrino we can say Neutron tranforms into proton + electron + neutrino
In the formula E = mc^2, perhaps the best term for that is that Energy is the dual of mc^2. So if we symbolize duality with a O symbol and a arrow in that O such as the arrow in -->, that the equation looks like this E Q mc^2, (sorry the keyboard cannot do the symbol, but the letter Q imagine that slash bar as a arrow in the circle), meaning that energy is the dual of mass * c^2.
And so the four Maxwell Equations have no equal sign but a sign of duality.
I do not want to spend much time on this because someone can devote their entire life on just this topic with no guarantee of success or improvement.
I do want to point out the gray and prickly area of logic of "primitive axiom terms".
And it should be obvious to all who master physics that although we should not use equality symbol in physics, we do so anyway, since it is the most practical means of going forward. So as long as we keep it back in our minds that equality is a symbol for mathematics, but not really for physics.
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: