Dorian Sabaz says:
April 29, 2012 at 6:35 am
Freshman or first years, it matters not. You start teaching false understandings at the beginning of a mathematical education, you end up with just confused and poor mathematicians. Worse yet, you end up with poor engineers and ridiculous physics theories, which is what we have today. Disciplines like Physics are in trouble, this has been so now for the better part of 30 years. There are subtleties that need to be understood, especially now in physics, and these subtleties are being ignored. There is a fundamental problem how we do theoretical physics and the way we interpret mathematics in physics, that is, how we do mathematical physics. I am of a school where we should be doing physical mathematics – that is, learn from observation and let reality guide your mathematics – this is contrary to mathematical physics, where the math is pushing how we want reality. That is why I gave the argument above on the interpretation of what is a theory, lemma or corollary. Perspective is illusory, not mathematical fact.
As a small but very profound illustration, consider how the Hamiltonian is calculated for a particle in a box, or infinite well (it matters not the case). After some effort, the wave functions are determined for the possible states (energy, position, momentum, whatever) that the particle is allowed. But is this really true! Consider what the mathematics is really telling us. We have calculated these possible quexposingzed values for the entire system, in situ and a priori (refer back to my initial commentary for why this is important). So the question begs, how does a particle get to know the entire system, and know how to go into these appropriate quantum states IMMEDIATELY. The mathematics, is insitu! Reality is not. You can bring Relativity into this, it doesn’t change anything, the problem still remains, its just one level higher. This problem becomes self evident once you get into entanglement, but even there, the mathematics we use breaks down.
I’ll end by giving you a first year dilemma, since you are so focused on freshmans and their capacity to understand basic mathematics. Let us see how you can explain this to them. Riddle me this, the inner dot product of two vectors is considered to be a scalar, whereas the cross product is vector. Furthermore, the inner dot product is interpreted by every mathematician and physicist in every university in the world today, as a projection of one vector upon the other, and the cross product illustrates the area (if we take units of distance) of two vectors. But is this true? Why is it that inner dot product has units of area, like the cross product. And if the inner dot product truly does represent an area, what area is it? Furthermore, reinterpreting the dot product to this “forgotten area”, what does this do to our interpretation of, say, Maxwell’s equations? As you can see, if you start off with a poor or misguided understanding in mathematics, it just may take you down a path to paradoxical theories, like Quantum Theory and Relativity. Paradoxes are a sign that it is not your theory that is flawed, but the axioms that the theory is supported upon is in error. Just imagine, two hundred years of the greatest mathematical minds and greatest physicists in history all missed one important and minor thing, like understanding what the dot product really is. Could this really be true? Its not like its ever happened before is it? Hmmm, two and half thousand years of believing that the Earth was the center of the known universe, does come to mind.
Just because all the sheep follow the shepherd, it doesn’t mean that the shepherd knows where he is going.
I gave you an argument earlier that linear thinking is dangerous. For it brainwashes people into thinking one way. I can tell you now, there are very deep problems and mistakes we have made in physics and mathematics, all because, we do mathematical physics and not physical mathematics. There are other ways to look at reality, and if you don’t teach them early in the educational process, you only perpetuate the lies of the past. How many centuries must past before ??? learns these lessons?

https://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/