Thoughts: I think this is largely because the gray area between the two has shrunk: in order to discover new things, we must know how do to the work of the field we want to discover new things in.

Thoughts: How incredibly odd to presume that, as an individual tasked with understanding things underlying all that there is to understand, that you have the ability to say what is and isn't natural, or probable.

Thoughts: One of the great things about Mathematics is how things get built upon each other, into wildly new fields, yet retaining fundamental concepts, and being relatable to other field: $c$ and $d$ are functions of $ax + by$ and $ax + by + cz$ respectively, and this also relates to the function focused aspect of the fundamentals of calculus, in that we can say $y = f(x) = ax+by+cz$. Also, we know that we are dealing with real numbers, due to the nature of the coordinate system not being complex, so we can assume that $a,b,c,x,y,z \in \mathcal{R}$, which is the set of real numbers for which they have membership, so there are even reminisces of set theory hidden within this. This makes sense though, because the foundations of mathematics have been set theoretic since the early 1900's.

Thoughts: The irony is that everything physics studies is effectively an improbable coincidence: in fact, that's the why of Physics, because arguably nothing should be, yet everything is. I would almost expect the true fundamental theories of Physics to include improbable coincidences.

Thoughts: With all due respect for Aristotle, I have to disagree with this, due to the fact that while fire may rise "up", the up that it rises into is already inhabited by air- though I may be misunderstanding his conception of air in the fundamental, elemental sense.

Thoughts: Some may argue that the most fundamental concept in mathematics is the set, the discrete collection of elements, especially given that functions are mappings between sets, sets are required for the concept of the function to be applied- the domain of a function is simply the set of acceptable inputs to it.

Thoughts:

Thoughts: The Breeding Horses argument, seems super valid to me, but I don't think this is what is taking place in Physics.

Thoughts: This is the reason why Radian Labs has to develop an educational curriculum that allows for people to learn not just via studying and doing projects, but being able to do so under the wing of established professionals.

The assumption of superposition between states leads to a mathematical theory within which the equations that define states are linear in terms of the unknowns. The consequence of this is that people have attempted to establish analogies between this and classical systems, such as vibrating strings or membranes (Thoughts: it is super interesting that the fundamental objects of String Theory were around in Dirac))'s day).