Can Newton's laws of motion be proved (mathematically or analytically) or are they just axioms? Today I was watching Professor Walter Lewin's lecture on Newton's laws of motion. While defining Newton's first, second and third law he asked  "Can Newton's laws of motion be proved?" and according to him the answer was NO!
He said that these laws are in agreement with nature and experiments follow these laws  whenever done. You will find that these laws are always obeyed (to an extent). You can certainly say that a ball moving with constant velocity on a frictionless surface will never stop unless you apply some force on it, yet you cannot prove it.
My question is that if Newton's laws of motion can't be proved then what about those proofs which we do in high school (see this, this)?
I tried to get the answer from previously asked question on this site but unfortunately none of the answers are what I am hoping to get. Finally, the question I'm asking is: Can Newton's laws of motion be proved?
 A: They are an approximation to General Relativity, so yes, they can be proven using general relativity.
A: If you want to prove something, you have to start with axioms that are presumed to be true.  What would you choose to be the axioms in this case?   
Newton's Laws are in effect the axioms, chosen (as others have pointed out) because their predictions agree with experience.  It's undoubtedly possible to prove Newton's Laws starting from a different set of axioms, but that just kicks the can down the road.  
A: One can derive the laws of classical mechanics from quantum mechanics. Classical mechanics can be reformulated in terms of the principle of least action. The time evolution of a system is such that a quantity called the action is minimized. According to quantum mechanics, the system will evolve in a probabilistic way, the probability of finding a certain outcome is given by a certain integral involving the action that is over all possible paths. For a system in the regime where classical mechanics should be a good approximation, what happens is that the contribution to the path integral path will be dominated by those paths that are at or near the minimum of the action.
A: In some sense, Newton's Second Law can be "derived" from the assumption that a system's evolution is determined only by its initial position and velocity.   This is the argument put forward at the beginning of V.I. Arnold's Mathematical Methods of Classical Mechanics.   He starts Chapter 1 with the following "experimental facts":

  
*
  
*Our space is three-dimensional and euclidean.  Time is one-dimensional.
  
*There exist a set of coordinate systems (called "inertial") possessing the following two properties:  (a)  All the laws of nature at all moments of time are the same in all inertial coordinate systems.  (b)  All coordinate systems in uniform rectilinear motion with respect to an inertial one are themselves inertial.
  
*The initial state of a mechanical system (the totality of positions and velocities of its points at some moment of time) uniquely determines all of its motion. 
  

Suppose that our system is determined by $N$ real numbers, which we can assemble into a vector $\mathbf{x}$.  Since "experimental fact" #3 says that all properties of the motion are determined by positions and velocities, the acceleration of $\mathbf{x}$ (in particular) is determined by these quantities.  We can then conclude that there exists a function $\mathbf{f}: \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R} \to \mathbb{R}^N$ such that
$$
\ddot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \dot{\mathbf{x}},t).
$$
This can be viewed as defining $\mathbf{F}$ for a given system;  if we multiply each component of $\mathbf{f}$ by the "mass of each point" (in some appropriate sense), we would get "the force on each point."  So the fact that initial positions and velocities determine the motion implies the existence of Newton's equation for some function $\mathbf{F}$.
Note that this implication goes the other way as well.  If we assume that this function $\mathbf{f}$ exists, there are theorems from the field of ordinary differential equations that guarantee the existence and uniqueness of the solutions $\mathbf{x}(t)$ to this equation.  (In other words, we don't need to define an independent function for $\dddot{\mathbf{x}}$ or some higher derivative to determine the motion;  one function that determines the second derivative is sufficient.)  Thus, the "experimental fact" that initial positions and velocities completely determine the motion is entirely equivalent to the statement of Newton's Second Law.
A: ZFC is one of the axiomatic systems we require for mathematics. 
In physics however, we need to assume some external conditions which are set in the universe (which may change in other universe). I would not call them axioms, but rather conditions/constraints which can only be found experimentally. 
It's like asking what is the color of a ball kept kept in a box, you can only find out by observing it, since it is just an information.
The interesting part is what is the fundamental theorem/law which governs what will be the connection between various physical quantities? Where is all this coded? In other words, why should $V = I R$ and not $V = kIR^3$ for some constant $k$. Until we gain a better understanding, these must be found by experiments.
A: tl;dr: No, but basically they're equivalent to the conservation laws of total energy and momentum of a system.
The first law (object has constant momentum unless a force acts on it) is, in my eyes, merely a special case of the second law (since acceleration is per definition the change of velocity and the second law gives its connection to force) - so nothing to prove there except for some thinking about the connection between momentum and velocity.
The second law (change of momentum of an object is proportional to the force acting on it) is basically a definition of force assuming you already know what "mass" is. A definition is nothing you can nor need to prove.
The third law (actio = reactio) is the most interesting one. It states that this "force" defined above, not only happens to stem from an interaction between two (or more) objects (i.e. $\vec F_i = \sum_{j\neq i}\vec F_{ij}$) —which is again just a definition— but that this works in both directions with opposing signs ($\vec F_{ji} = -\vec F_{ij}$). Can we "prove" that?
Not exactly. But let's show it is equivalent to something that (to me) seems quite intuitive:


*

*Assume no "external" forces (they can be explained away by assuming the objects causing them possess infinite mass).

*If the force is free of vortices ($\vec\nabla\times\vec F_i(\vec x_i) = 0$) and only depends on the object's position, it can be expressed as the gradient of a scalar potential $V_i(\vec x_i)$ such that $\vec F_i(\vec x_i) = -\nabla V_i(\vec x_i)$. The third axiom is then equivalent to requiring that the individual potentials only depend on the distances between the objects, i.e. $V_{ij} = V_k(\vec x_i - \vec x_j)$ (where the index $k$ merely serves the purpose of allowing for different kinds of potential between different objects, e.g. electrostatic or gravitational).

*More complicated dependency of the force involves more complicated potentials, but still boils down to the same thing: All interaction depends on the distance (and optionally relative velocity+).


Via Noether's theorem, the independence of those forces from absolute positions in space (and time) implies the conservation of total momentum (and energy) of all objects together. And that seems very intuitive to me.

+ Note that the force acting on an object $i$ can depend on its position $\vec x_i(t)$ and its velocity $\vec v_i(t) = d/dt\ \vec x(t)$, but not on its acceleration, since that can be fixed by redefining force. Arguing why the force also should not depend on higher derivatives of $\vec x_i(t)$ is more complicated* but also irrelevant.
* I'd start with relativity and metrics only depending on $x$ and $dx$, but that goes too far here...
A: Logic is essentially a tautology -- saying exact same thing twice in different ways. To prove something is the process that ensuring a statement (what you trying to prove) is equivalent to the previous statements (the premise or the axioms). 
In this light, asking about a mathematical proof of a fundamental physics law is a bit pointless. We can NEVER be sure a physics law is absolutely right by sole mathematics - all we are doing is just rephrasing something else (sometimes adding a bit extra assumptions) -  we can only increase our confidence in it by repeating experiments on it. 
Moreover, there are a few tricky points,

Newton's I law: It defines what inertial frames are -- You can't
  prove a definition, you can only state a definition.
Newton's II law: It is partly a definition, partly an empirical law. 
Newton's III law: It is an empirical law.

Empirical laws can be seen as a general declare about our nature - something that can only be disproved by experiments. And by mathematics, we calculate their consequence, or derive them (rephrase them) from other given laws (e.g. F=ma from Lagrange).
A: To ask for a proof of a law is silly. A law is something which is given to explain a phenomenon. It is valid as long as something does not contradict it and it is able explain things correctly. As far as Newton's laws are concerned, they are already contradicted by Einstein. So it is not valid as the basic axioms used by it like the consistency of time intervals and length in different frames of reference are disproven by the theory of relativity. The fact that it uses Euclidean geometry which is already disproven along with its axioms clearly disproves Newton's laws themselves. Even then it is well and easy to apply in speeds negligible with respect to the speed of light and hence we use it. At last I would say that any law in science does not need a proof. Also if it will be proved, will not it become a theorem?       
A: Your bigger question is probably: "What is the relationship between physics and mathematics?".  Richard Feynman has an excellent talk on that relationship between physics and math.  You can find it here: Richard Feynman - The Character of Physical Law - 2 -The Relation of Mathematics to Physics.  He begins the discussion on this point at 22:55.
Paraphrasing Feynman feels a bit like heresy, but his point is that even if you were to approach physics axiomatically, there would be a lot of choices you could make on which idea was the axiom and which was the theorem.  Geometry has a similar problem.  The most pragmatic thing to do then is to have a set of principles that are both useful for working things out, and ones that you have a lot of confidence in.  That confidence comes from being simple enough to have clear implications and having checked as many of those different implications as you can.
A: I always thought that only the third law is an axiom; and it is fairly intuitive that you can exert a force only "pushing away from something else".
All the rest of classical physics — including the two other laws, and the conservation of energy — follows when one assumes that time and space are the same for all concerned (yes, that may be considered kicking the can down the road ...).
A more abstract wording of this fundamental principle could be: "All interaction is — as the word suggests — mutual, and happens in a common frame of reference."
To be honest, I'm not so sure about the common frame of reference in relativistic physics; although intuitively I would say that this fundamental sentence still holds: the common frame of reference just gets more complicated.
A: I was encouraged to develop my comment into a full answer.
There have been various (very approchable to the novice) papers that develop special relqtivity from first principles, including reports that Galileo could have figured it out without calculus.
This is elaborated upon in this answer.
These papers show that given symmetry, further developed from the idea of reciprocity (if A sees B move at velocity X than B sees A move at velocity −X), the general form that velocity addition must have can be determined.  This includes Galileo’s fixed time and space as a special case.  It’s commonly said to derive special relativity from first principles: so does it also derive Newton’s laws as a special case, or are those assumed as input to the process?
The starting point of assuming symmetries will give you Newton’s First Law: a base case stating that inertial reference frames exist.  However, just because you know how velocities add doesn’t mean that you are handed the concept of forces and momentum.
So, you can postulate Newton’s first law, or you can postulate space and time symmetry which is more precise in meaning.
To prove Newton’s other laws you need different axioms to start with; they don’t just appear out of nothing (beyind showing that they are a possible consistent set of rules within the established symmetries). 
Other answers here do this.  Start with something we now consider to be more fundamental. You need to postulate the idea that objects have different resistance to motion, can exchange momentum etc. and the main law that nature minimizes some quantity, and you can derive the specific formulas for the laws.
A: Newton's First Law effectively states that momentum is conserved.  Newton's Second and Third Laws can be derived from the First Law, for either of two possible definitions of momentum.  The derivation uses the Reynolds' Transport Theorem.
The two possible definitions of momentum can be derived analytically, based on one assumption about what relative motion is.  The first definition ignores the speed of light.  The second definition is the modern relativistic definition.  The second definition has been verified experimentally.
