Are Newton's laws just definitions? I have read a bunch of articles online regarding my question but none have helped.
Newton's Laws:


*

*In an inertial reference frame, an object's momentum doesn't change unless the object is acted upon by a force.


*In an inertial reference frame, the force on an object equals the time derivative of its momentum.


*In an inertial reference frame, the total momentum of every isolated system is conserved.

I have explicitly mentioned "inertial reference frame" in all three statements since the force on an object is defined only in an inertial reference frame. Also, the law of conservation of momentum is completely equivalent to the usual statement of Newton's third law. It's just much easier to work with the law of conservation of momentum.
My observations:

*

*[1] follows directly from [2]. It contains no more information than [2] does, so we can scrap it.

*[2] is a definition, but it is not complete. We have no way of knowing whether a frame is an inertial reference frame or not.

*[3] makes a real statement, but it is incomplete. We still have no way of knowing if a frame is an inertial frame.

If we assume that the total momentum of every isolated system is conserved only in an inertial frame, then we can use [3] to determine if a frame is an inertial frame. We just check if the total momentum of every isolated system remains constant to determine whether our frame is an inertial reference frame.
But then, [3] gives us no information. It only defines what an inertial reference frame is. [2] doesn't give us any information, it just defines what force is and it's incomplete without [3].
To my understanding, Newton's laws are just definitions and don't make any real claims about this world.
So, how am I wrong? Please note that I have no problems with defining mass through experiment. I have seen many posts where people claim that Newton's laws are "circular" because they don't define mass without using force, but this has nothing to do with that.


Unrelated sidenote:
I had posted this exact same question on physicsforums a while back, but I was accused of being a troll (and was banned) for signing up with my completely legitimate and verified cock.li email address. Also, a so-called mentor edited some of my replies after I had posted them. It was a very upsetting experience.

 A: From a mathematical point of view, they are definitions: they relate mathematical abstractions. But from a physical point of view, they are not definitions: they capture real behavior of real physical objects through the metaphors that relate the mathematical abstractions to the phenomena.
Other definitions might yield perfectly good math, but physics is more constrained.
A: Within Newton's Principia, Newton follows the Euclidean tradition of presenting mathematical proofs by first providing a set of definitions which are then followed by a set of axioms (i.e., laws) that are assumed to be true.  Definitions in themselves do not admit the existence of anything, but rather provide terminology that is later recognized as a particular condition as a result of a set of axioms.
Newton presents his laws of motion based on a collection of known observations regarding motion (in particular, he cites Galileo's inclined plane experiment and hypothesis that objects only slow down due to air resistance or other friction forces, which is counter to Aristotelian physics), and carries forward with the logical consequences of these laws in a series of propositions, which so happens to match experimental evidence.
Once the laws are provided, then particular effects of the laws can be matched with the definitions he provided.  For example, his definitions of quantity of matter, quantity of motion, vis insita, etc., only make sense within the framework provided by his laws.
In short, no, the laws are not "definitions" since definitions are only understood within the assumed truth of the laws.
Also, you can't just scrap Law 1 since it provides a framework for justifying that the motion is operating in an inertial frame.  If we only accept law 2 without law 1 and the requirement for inertial frames, then pseudo-forces can appear; however, by restricting law 2 to only apply to inertial frames, then you're de facto admitting a further underlying axiom that's more fundamental than law 2, in other words, law 1.  In other words, if an axiom requires additional further underlying admissions or assumptions, then there exists an additional axiom that's more fundamental than the original axiom.
For example, consider Euclid's 5 postulates (i.e., specific axioms) for geometry.  In it, postulate 2 states "any straight line segment can be extended indefinitely"; however, this would admit that the line would have points as being more fundamental to the line.  Therefore, postulate 1 "a straight line segment can be drawn joining any two points" is given as the most fundamental axiom since nothing can be more fundamental than connecting a collection of points.
A: Your statements starting with "in an inertial reference frame" are indeed trivially true, because of the definition of the inertial frame: inertial frame is a frame where the first and the second law are true. You're using modern formulations here that are not the original Newton laws.
It is a sin of physics pedagogy when laws get reformulated into the thought scheme of the day, but resulting new statements are still referred to using the original inventor/name supplier. This happens with Newton's laws, but also, for example, with Kirchhoff's circuital laws in electromagnetism (circuit theory). Often approximate laws derived from observations become definitions or mathematical truths in the new scheme, which can be confusing. But the physics law ingredient is still somewhere in there, just in different statements.
You need to remove "in an inertial frame" and then the laws become independent non-trivial generalizations of experience. Also, they stop to be universally true (as is the rule for physics laws). The original formulations are indeed physics laws, not definitions. The concepts of mass, force in there are assumed to have meaning independently of Newton's laws.
The first law, or the law of inertia, states that compact bodies (that do not lose parts) do not change their velocity spontaneously, there has to be an external actor that initiates any change of velocity.
The second law states that when change of momentum of such compact body indeed is happening, rate of change of momentum of this body is proportional to net external force acting on the body (sum of all impressed and constraint forces acting on the body).
This is not definition of the concept of impressed or constraint force or net force; the concept has some established meaning (actor-subject, vector, superposition of forces, measure of interaction of bodies, deformation, etc.) even without the knowledge of the 2nd law. For example, one can introduce and define the concept of impressed force using examples where nothing accelerates. Instead, body position, orientation and deformation is involved. Or force can be defined via formula, such as the gravity force of a mass point center. Acceleration is not needed to define force.
Also, 1st law is not a special case of 2nd law. 2nd law on its own does not require that body at rest will stay at rest. In certain indeterminate systems such as the Norton dome, 2nd law alone allows for the body to spontaneously start to move from rest at any point of time. 1st law does not allow for this. In other words, second law involves only the first derivative of momentum, but 1st law involves all of them. Admittedly, this example is somewhat artificial.
The third law states that every force $\mathbf F_{BA}$ due to body B acting on the body A is accompanied by another force $\mathbf F_{AB}$ due to the body A acting on the body B, and
$$
\mathbf F_{BA} = - \mathbf F_{BA}.
$$
This law is sometimes violated by magnetic forces, which Newton probably did not know.
A: I think Feynman tackles pretty much this exact issue in his lectures, and I'm not sure why people in that thread you linked were so hostile to a question that Feynman thought was very profound indeed :
https://www.feynmanlectures.caltech.edu/I_12.html

Suppose we have a law which says that the conservation of momentum is
valid if the sum of all the external forces is zero; then the question
arises, “What does it mean, that the sum of all the external forces is
zero?” A pleasant way to define that statement would be: “When the
total momentum is a constant, then the sum of the external forces is
zero.” There must be something wrong with that, because it is just not
saying anything new. If we have discovered a fundamental law, which
asserts that the force is equal to the mass times the acceleration,
and then define the force to be the mass times the acceleration, we
have found out nothing.


We could also define force to mean that a moving object with no force
acting on it continues to move with constant velocity in a straight
line. If we then observe an object not moving in a straight line with
a constant velocity, we might say that there is a force on it. Now
such things certainly cannot be the content of physics, because they
are definitions going in a circle. The Newtonian statement above,
however, seems to be a most precise definition of force, and one that
appeals to the mathematician; nevertheless, it is completely useless,
because no prediction whatsoever can be made from a definition. One
might sit in an armchair all day long and define words at will, but to
find out what happens when two balls push against each other, or when
a weight is hung on a spring, is another matter altogether, because
the way the bodies behave is something completely outside any choice
of definitions.

From what I understand, he says that just by considering the definition of "force" and its relation to momentum and acceleration, we don't have any new knowledge, just a potentially good way to analyze a system of moving objects. Sure, you know now that F = ma, but you don't know anything about what the force actually is

"but the specific independent properties that the force has were not completely described by Newton or by anybody else, and therefore
the physical law F=ma is an incomplete law."

It's only when we get some law for the forces itself(friction, drag, molecular, gravity, coulomb's, nuclear, etc etc) that we can actually "break out" of the cycle and do some predictions about how the bodies actually move.
The point is that you may very well have another definition if you like, say "Morce = Volume^16 x sqrt(Velocity) x Heat^pi". Great, this is a definition of "Morce" that, just by looking at it, is as valid as a definition for "Force", and as useless for making actual predictions(by itself) as F = ma is. The big difference of course is that it doesn't give us a very good program for analyzing nature, as Feynman puts it, while Newton's laws do.
Now, I think that one common intuition for "Force"(outside of Newton's laws) was(and is) that a body's motion is altered only when it's acted upon another agent in its vicinity.

For example, in dealing with force the tacit assumption is always made
that the force is equal to zero unless some physical body is present,
that if we find a force that is not equal to zero we also find
something in the neighborhood that is a source of the force. This
assumption is entirely different from the case of the “gorce” that we
introduced above. One of the most important characteristics of force
is that it has a material origin, and this is not just a definition.

This is the key here, that connects a bunch of definitions to the actual world and gives us a physics program at last. Even though it's not included in Newton's laws, most considered it "common knowledge" that when I push something with my hands or pull it with a rope, I am applying force to it, and when something is in outer space far away from anything else, no force is applied to it. Those are not mathematical theorems of any kind, those are just things we know by observing the real world. They are not part of Newton's laws, but I assume they were a starting point for figuring out the specific characteristics of different "kinds of force", after Newton gave us a good program of relating force to motion.
Based on those common intuitions, one could see a real difference between Aristotelian and Newton's dynamics : Aristotelians believed that, when not acted upon an agent, all bodies slow down and eventually stop(they had a pretty convoluted explanation for how arrows keep moving after being fired). Essentially they believed that a body needs constantly to be acted upon in order to even keep moving, not merely accelerating.
Newtonian mechanics OTOH, based on the same intuitions, would predict that, in the absence of air drag, gravity, or another body colliding with it, if I fired an arrow it would go on in a straight line forever. Conversely, if I noticed the arrow diverged from its straight line, then I would suspect some source of force should be in its vicinity, and I could proceed to either postulate the existence of that source based on the existing laws(that's how Neptune was discovered), or, much more rarely obviously, postulate the existence of a new type of force or interaction and work on figuring out its particular characteristics.That is, as long as I'm observing the arrow from an inertial frame of reference, otherwise I could see it diverging from a straight line because a force is applied to me.
So, all in all, yes, a pretty good program for analyzing nature indeed.
A: Newton's first law is usually taken as the definition of an inertial frame. Hence it is not a special case of the second law, and resolves your concern of the second bullet point regarding the second law.
Your statements "Also, the law of conservation of momentum is completely equivalent to the usual statement of Newton's third law. It's just much easier to work with the law of conservation of momentum" and "If we assume that the total momentum of every isolated system is conserved only in an inertial frame, then we can use [3] to determine if a frame is an inertial frame" seems to be a subjective matter of perspective, no? In my experience the problem will determine whether it is easier to apply Newton's third law or the conservation of momentum. One is not always superior to the other.
Also see the answer here.
A: 

*

*In an inertial reference frame, an object's momentum doesn't change unless the object is acted upon by a force.





*In an inertial reference frame, the force on an object equals the time derivative of its momentum.


The laws are interconnected and they depend heavily on what is meant by force. It is heavily implied in this formulation, that the force should be a measure of interaction between bodies. The clue is In an inertial reference frame, the force.... Time derivative of momentum is defined all the time in any frame. Why would we state specifically that it works only in an inertial frame, if we didn't mean by force something else than just the time derivative of momentum?
No, force is not the time derivative of momentum. It is a measure of interaction that is supposed to equal to time derivative of momentum. For gravitational interaction of two bodies it is $\vec{F}=-G\frac{m_1m_2}{r^3}\vec{r}$ and there is no time derivative of momentum in sight. When we have the interaction law, we can use the second law to predict movement. If we do not have the interaction law, we can use it to find it. The whole point is that the force should arise from the interaction, not from the movement of bodies.
With second law understood in this way, first one is basically a subset of it. But it is not without a pedagogical reason. In the first law of motion, you do not need to know the formula for forces or really anything specific about them. This law provides the minimal necessary information to try to find an inertial frame and usually should be a starting point when building a theory.
Physics is very holistic. The whole theory must work together and cannot be really understood by separating its parts. It cannot really be axiomatized. We are trying to find force law using equations of motions. On the other hand, equations of motions in a certain sense defines what we mean by forces. We have a circular reasoning that refers to itself. How is this then useful?
Well, because of generalizations. Kepler used motion of one planet - Mars to determine his laws. He also used only data for finite time interval. He then said, the same laws applies to all other planets and it will be the same in the future. The fact that nature is so kind and we can extrapolate to situation we did not experienced yet is why Physics work. Without it we would be trapped in an endless circles of A being defined B and B being defined by A.
A: The first law is a definition of inertial reference frames. Sometimes it is stated as
There exist such reference frames, where an object's momentum doesn't change 
unless the object is acted upon by a force.* 

It can be also considered an equivalent of the first postulate of relativity, that in all inertial reference frames all physical laws have the same form.
The other two laws hold in only in the inertial reference frames defined by the first law. In fact, these are the actual laws that hold in the inertial reference frames, defined by the relativity postulate (aka "first law"). The second describes the effect of the force acting on an object in rigorous mathematical terms: $m\ddot{x}=F_x$, etc. The third relates the forces with which the objects act on each other (note that the 2nd law deals with one object relating the force and the acceleration for the same object, the 3rd deals with a pair of objects and forces applied to different objects).
A: The Newtonian laws as you have 'quoted' them differ from what Newton actually wrote. I suggest that to obtain effective answers to questions such as yours, it is indispensable to read the original.
So, first, some sources:  Newton's 'Principia' has two respected English translations, one from 1729 ('Mathematical Principles of Natural Philosophy', translation attributed to A Motte) and the other from 1999 (translation and introduction by I Bernard Cohen et al). The 1729 translation is out of copyright and readily available online, e.g. in 2 volumes at (https://books.google.com/books?id=Tm0FAAAAQAAJ) and (https://books.google.com/books?id=6EqxPav3vIsC). The 1999 version can be found on publishers' and bookshop websites. There have been other English versions, e.g. a 1934 attempted 'cleanup/modernisation' of the 1729 edition, by F Cajori & R T Crawford, even regarded for some time as a standard, but critics have found material to object to in this and other versions, and it is probably better to look at those from 1729 and/or 1999.
General plan of the work: If you move a step back and take a broad look at the work, you can see how Newton constructed what may be called an almost purely mathematical model (in Book 1), and then compared it (in Book 3) with generally accepted astronomical phenomena, showing how the model accounts for the phenomena, and developing the model some more to show how its development fits several specialized types of phenomena.
The 'laws of motion' preceding Book 1 are actually entitled 'Axioms  or laws of motion'. Newton was not concerned to prove these in the book, either physically or mathematically. But he did take evident care, seen for example through his papers, to ensure that they match physical reality as well as he could manage. Indeed they do so to an extent that was remarkable for his time. Also notable is that Newton did not take credit for the laws of motion as his own, he ascribed them or their confirmation largely to others (see the 'scholium' that follows the laws and corollaries). Here he was (arguably) excessively modest, and succeeding generations have seen that while yes, they do owe multiple elements to predecessors, they are also more than that, and as a whole they amount to an original interactive compilation which is highly truthful to physical reality: hence the (very posthumous) label 'Newton's' laws of motion that has customarily been applied to them.
So the answer that I would offer to your question is that 'Newton's laws of motion' are not just definitions, nor intended to be. They were intended to be physically realistic, to axiomatize physical properties of the real world in a way that could then be mathematically treated. Book 1 is then mostly a large extension of that mathematization, and Book 3 seeks (with a large measure of success) to show how astronomical consensus-phenomena confirm the match between the developed model and physical truth.
Book 2 is of a rather different nature (see for example this answer on hsm) so I'll stop here except for one further type of point.
If you have been accused of 'trolling' it is likely because your question has adopted some heavily anachronistic assumptions about Newton and his time and work, without apparently attempting to see whether these have any justification at all.  Your question says (for example) that it is easier to consider the law of conservation of momentum than Newton's third law. Or that one law (as you have misquoted it) contains no more information than another.  But have you considered -- for example -- whether the concepts of conservation even existed in Newton's time? They are (in historical fact) post-Newtonian constructs. Their construction over the centuries owes much to the groundwork that Newton laid. Equally, your other criticisms appear to pay no attention to Newton's laws as he wrote them, they concern extraneous matters. So in the event that indeed you are interested in enquiry into this topic, then I believe you would be concerned to look into the genuine sources for the material you seek to criticize, and I hope you will find utility in my references to sources and to the aspects of the plan of Newton's work that I have invited you to consider.
