Sabbatical said:
How exactly does a physical scientist or mathematician go about modeling the way that a phenomena works or appears to work. For example, how do I know when it's appropriate to introduce something like --> ∂ or even the integral instead of something else?
Alternatively, maybe I'm asking, at what point does 'theoretical' become 'practical', but fundamentally, what happens before I can model out a mere thought as theoretical?
Most valuably, once I do understand this process, how do I go about having a mindset to be able to map out the things that I think about, mathematically?
Hey Sabbatical and welcome to the forums.
There is no general answer to this IMO, but usually what happens is that ideas form, they are shared, revised and refined and once it gets to a point where it has a lot of clarity it is worked on and extended by mathematicians working in that particular field.
In terms of when something becomes 'practical' from 'theoretical' this is a tough question.
Usually what happens is that pure mathematicians or theoretical mathematicians like theoretical statisticians will try and work on things that encompass lighter constraints: that is instead of working on specific cases like an engineer would, they work on more general results.
The result of the above is that if a mathematician ends up proving results for a large general class of phenomena, then when the engineer, applied mathematician or scientist runs into a problem that requires such methods, they might find that X has already solved this problem and then use it for their purposes.
In terms of understanding processes in terms of mathematics, you need to understand what the symbols really mean from a viewpoint that is able to explained in english rather than mathematics.
For example many applied problems deal with analyzing the change locally rather than globally and for this reason, many applied problems are formulated in terms of differential equations or difference equations because the local changes are easier to process and understand whereas the global changes and classifications are often too hard to understand and this is the goal of the person doing the modelling to go from 'local' characteristics to 'global' characteristics in terms of understanding and in the context of what they are actually trying to do.
The key thing is to put away the math and ask the fundamental thing: what are you trying to do?
For example in asymmetric cryptography, the key thing that people are trying to do is to create a function that is easy to do but hard to undo given certain conditions. The function has to have a way of being undone otherwise you wouldn't get the data back, but the way has to have the above property.
It turns out that number theory provides this kind of functionality and so a lot of these algorithms are based on number theoretic results but again: if another way existed that had better properties for what is in mind then there is no reason that it should or would not be used.
Usually when we are trying to understand phenomena that we are not purposely engineering (cryptography as opposed to something like physics or biology) then the ways to analyze these kinds of things involve not only calculus (which is able to model how changes between variables are linked) but also statistics.
The reason we use statistics is that due to the amount of disorder (think complexity) of the results, it's not even easy in many circumstances to decipher what the local changes should be when you are studying something that is so complex that appears to be random.
It's not an exact science and many people are developing new methods as we speak to take some phenomena and try and make sense out of it.
