University math is different than high school math. Profs have a lot of possibility to be crazy, both ways.
Very true. There is also a huge difference in "math" math and "applied" math; i.e. between math as a research subject and math as a tool.
Math is science in its pure abstracted form.
Well, I'd say that math is the language in which we express science; science itself being the systematic application of the scientific method to find answers and solutions to all sorts of questions and problems. (This is why I also prefer to call it a tool instead of a language.)
We are in agreement. I would "refine" it one step further (but by doing so, it also expose where I am fuzzy myself). Math is a language for science when it comes to quantification. Where quantification is not required, math does not seem necessary.
There is math and logic for qualification as well; it's just in the abstract math-about-math domain. Other tools, like dimensional analysis (which considers types or units instead of quantities), can be combined with math to handle non-quantification type problems. (Thus, I do agree.)
Electron flow in LED creates photon. No one needs math to clarify that. But one cannot describe the relationship between electron flow vs number of photons created without mathematics.
Right; to describe the interactions of the subatomic particles (electrons and photons in this case), we can use for example
Feynman diagrams.
I believe it is important to realize this in all subject matters where math is an important tool: it can be used in different ways (and one can approach it in so many different ways, it is important to find the way that works efficiently for oneself), but properly "applying" it with other, non-math methods, is crucial.
For electronics,
network analysis and
equivalent impedance transforms are excellent examples of such.
To prove that "there is something required to making the forces balance" requires math. So if one can accept it exists without proof, one can forgo math. But is that really "understand"?
One of the reasons I love
computational materials physics is that every time you run a simulation, the first question both before and after, is always
"Does this make any sense?"Because math is only a tool, it cannot tell you how and when to apply it. Science, on the other hand, gives us a method (the
scientific method, obviously) to direct and guide a questioning mind to examine the phenomena. Models (usually in the form of a
theory or
law, often called "
something analysis" by mathematicians) bridge the two.
Newtonian physics is one such model. We know the domain where it applies to an amazing accuracy, and where it fails.
We can even start with
ab initio simulations of electrons (using
density functional
theory), and derive the electrical properties of semiconductor materials; in fact, this is what a lot of computational physicists do in practice.
Although this stack, science + methods + math, has completely arbitrary definitions (I'm sure many of those reading this post will disagree with my definitions), it is useful in the same way a well-organized workshop is: it
works in practice, because it gives a logical place to store everything one uses.
And just like workshop tools, there are different approaches to math that one can choose. Some woodworkers only use hand tools; this is roughly equivalent to those that say that you need to be able to derive the
Bessel functions yourself, to truly understand and correctly use them (in say solutions to
Laplace's equation in cylindrical coordinates). Some only use a hammer, because all problems can be described as nails of various sorts, and so on.
Just because an approach is popular, does not mean it is optimal/best; just that a lot of humans
like it. It is important to get past that, and examine
oneself, to find out which approaches work best for themselves. This is particularly important in "higher learning", be that in an university, or in just-for-my-own-amusement type of thing.
A particularly good example of that, in my opinion, is in geometry or linear algebra: how to rotate 3D coordinates. A lot of people learn how to do that by using
Euler angles, but it is actually a horrible tool for that: it is vague (there being a couple of dozen different ways to define the angles, and practically nobody bothers to specify which one they mean!) and subject to
gimbal lock. Very few are familiar with
versors or
unit quaternions, which can describe any rotation; allows adding, subtracting, or interpolating between rotations; is numerically stable when "stacking" any number of rotations; has unambiguous forms for converting to and from rotation matrix form; and while the mathematical operations (like multiplying two versors is done using
Hamilton product) look complicated, is in practice very easy and simple to use.
I would go as far as claim that anyone doing 3D computer graphics and using Euler angles, is a fool: doing a lot of hard work for unimpressive results, when a simple, efficient, and problem-free (no gimbal lock) solution exists. (If someone is using Euler angles in a microcontroller dealing with 3D orientation or rotations, I'd say they are an idiot. Harsh, but in my opinion, one must acknowledge crap is crap, even if a billion flies love it.)