Classical Physics Axioms

One of the first courses that I took as an undergraduate student in the Department of Mathematics was the "Introduction to Mathematics". One of the very important things that I have learned in this course were the Peano axioms that give a formal definition of the natural numbers. Having defined natural numbers, everything becomes easy - integers, rational/irrational and real numbers. We can define different (complicated) operations, but everything simply follows once we have the mathematical atoms.

Similarly in an abstract algebra we define a group - a set with an operation satisfying four requirements known as the group axioms. And from there we get rings, field etc.

In geometry - we have a point, a line and five Euclid's postulates (axioms) for plane geometry.

In a similar way, while studying quantum mechanics, one learns the postulates of quantum mechanics - a set of rules that we one has to follow in quantum word.

An axiom -- or postulate --  is a starting point of reasoning.

And how it is in classical world? What is fundamental in classical mechanics? Newton's Laws? Let's look then at the second law. We have:

And what is m? Well, one can say that there is the International Prototype Kilogram that can be used as a reference for mass.

How about acceleration? It is the rate at which the velocity of a body changes with time. What is velocity? It is the rate at which the position of a body changes with time. How do we measure position? There is a meter that is an international unit of length and is equal to the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second. 

So what is fundamental in classical physics? I started to think about it while watching the video from last Thursday where teachers are discussing the definition of heat vs. thermal energy (I will put the clip of that conversation in a post that I was working on when I got distracted by axioms). 

There is not really good way to define what energy is. Does it mean that energy should be thought of as an axiom? What are the axioms or postulates of classical mechanics? I was trying to look it up in some textbooks but couldn't find anything like postulated of quantum mechanics...