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Why didn't Newton just propose the 2nd Law ($F=\dot{p}$) and leave it at that? The 2nd Law implicitly contains the first, doesn't it? If so, it seems he wasn't following his own Rule #1 of Book 3 of his Principia: "We are to admit no more causes natural things than such as are both true and sufficient to explain their appearances." There are intro textbooks (e.g., this one) that study momentum first and then force $F=\dot{p}$ based on momentum. Cf. also this SE post and comments.

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It doesn't contain the third. –  Ron Maimon Jun 24 '12 at 6:52
    
Thanks I corrected it –  Geremia Jun 25 '12 at 16:10
    
@Geremia: If you want Ron to read your comment you have to write "at username" in your post. Also, I don't think it's good to edit a false assumption out of the question after answers have been given - the people who will come here will be confused about the answers. –  NiftyKitty95 Jun 29 '12 at 12:51
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2 Answers

Some of the problems regarding the first law are already pointed out and discussed in the question you linked to, so I'm not entirely sure where that part of your question comes from.

I understand theoretical physics as the activity of coming up with mathematical models to give a structure to the behaviour of quantities, which the consensus agrees to be part of reality. One does this to thereby "understand" reality and to be able to calculate predictions (and also to make money and/or have some fun). From that perspective it's very possible to argue that the first law is redundant. The argument is that you have to declare an explicit form for the force for every model anyway, e.g. $\boldsymbol{F}_{ab}={q_1q_2\over4\pi\varepsilon_0}{\boldsymbol{\hat{e}_{a}}\over r^2}$, and at that point the first law truly becomes just a special case. In its history, there are other ways to formulate it or read new meaning into a first law but from my perspective they don't help much in doing physics in the sense above.

The third axiom is not already captured by the second, because at the very least, if you really only take space, time, at least two point particles and Newton's equations of motion into account, then nothing would hold you from investigating a mathematical structure with $\boldsymbol{F}_{ab}\ne -\boldsymbol{F}_{ba}$. This is then not Newtonian mechanics. Moreover, having that relation guaranteed also makes it possible to conclude general results about the theory for all the systems you might consider, e.g. conservation laws.

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However the third and second law could be condensed into an Only Law: $m\ddot{q} = -\nabla V(q).$ –  user2617 Jun 25 '12 at 16:25
    
@user2617: No it can't. You need that V is equal between different objects, and only proportional to |q_1-q_2|. This is Newton's model of the world, but it contains more information than the laws. –  Ron Maimon Jun 25 '12 at 18:41
    
Fair enough, although you wouldn't need to postulate the third law as a separate "law"; it would be a consequence of the symmetries of the potentials you plug in. –  user2617 Jun 25 '12 at 20:50
    
@user2617: If you want to answer Ron, add at symbol + username, otherwise he won't get a message. Also, the point is that choosing a force that can be represented as a gradient field of a scalar function $V(q)$ is a loss of generality. E.g. does the Lorentz force or friction always look like that? –  NiftyKitty95 Jun 25 '12 at 22:00
    
"The third axiom is not already captured by the second..." Newton himself argues in the Principia that the third law is not completely logically independent of the others. See the scholium after the statement of the laws, at "In attractions, I briefly demonstrate..." –  Ben Crowell May 29 '13 at 23:37
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Newton wanted to sound like Aristotle. Aristotle gives physical laws in two parts--- there are the "natural tendencies" of objects, so that Earth wants to be at the center of the universe, fire wants to be on the sun, air wants to be in the middle, and water wants to be down, but above the Earth and below the air.

In order to make this happen, things go in straight lines at constant velocity. So Earth falls down, water falls down, air goes nowhere, and fire goes up. This is to make the natural tendency work.

But then there are external forces that make things deviate from their natural state. All of this is ridiculous nonsense, but this is what people called "physics" in Newton's time, and Newton had to deal.

When he is writing, he formulates the laws of motion to persuade the idiots, since he knows they won't read past page 3. So he says right up front:

  1. Hey, bozos, there is a new notion of "natural state". The "natural state" of all things is to move with a constant motion, no matter what it is. Sorry, I mean "all objects remain in motion forever unless disturbed".
  2. Hey, douchebags, the deviations from the natural state is by forces, which give you the time rate of change of the momentum. Sorry, I mean, "the impressed force is the magnitude of the deviation from the natural state".
  3. Oh yeah, dipshits, look at this amazing new thing: the forces are equal and opposite! So that in any system, as much momentum leaves one body as enters another, and this is a universal law! Ahem, I mean, "every action has an equal and opposite reaction."

So now he has stated things Aristotle style (in Latin no less), and redefines the notion of "natural state" and "deviation from the natural state" so that first and foremost, it is correct, unlike Aristotle's bullcrap, and secondly, so that it is philosophically as compelling as Aristotle idiocies, if not more so.

Remember that Aristotle's physics had things that people considered laws of nature, like

  • Nature abhors a vacuum: if you try to make empty space, things speed up to infinity to fill it.
  • Nature doesn't make jumps: everything is continuous. No atoms.
  • Nature proceeds in regularity: Everything moves in lines and circles

All this stuff is complete laughable nonsense, but philosophers thought it was deep. The "nature abhors a vacuum" meant that if you have a vacant position in the leadership, people clamor to fill it. Nature doesn't make jumps means you don't screw with tradition. And nature proceeds in regularity means you execute your radicals. Aristotle didn't care much about physics. He just wanted to justify the stupid politics of his time.

Anyway, Newton gives you new laws that have just as much political resonance:

  • Everything keeps moving of its own accord: so nature keeps changing spontaneously.
  • A force will change the velocity, not produce the velocity. This is resonant with the idea of "shifting the discourse slowly by political pressure".
  • Nothing acts without being acted upon: there are no things that can do something without changing in the process of doing it.

These philosophical points, as trite and stupid as they are, are what gave Newton's proclamations weight with the philosophers. In this way, Newton beats Aristotle, and Newtonian mechanics (aside from being correct) is a suitable replacement for the Aristotelian garbage.

The philosophical nonsense is so dated, it is hard to take it seriously. Aside from the maxim "nothing acts without being acted upon" (which is a general law of universal validity, useful for other arguments in physics), the rest is just an embarassing demonstration that Newton was a politician as well as a physicist.

Today, we can dispense with the politics, and say "Newton's laws are the statement that momentum is conserved, and angular momentum too, if you assume that action at a distance between points occurs along the line of separation of the particles". Note that Newton didn't state "the world is made of particles that attract and repel along their lines of separation" (even though he believed this), he confined himself to macroscopic laws he was sure could be justified empirically without making hypotheses regarding microscopic constituents. In this regard, he was a true scientist. As regards the three laws, remember they are the content of the first 3 pages. Read the rest of the book, that's where all the interesting stuff is.

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-1. [citation needed] [POV check] Is it quite evident that Newton's personal foundational conception of physics put the convervation of momentum first, that he did not concieve of matter having a "natural state" which exhibited deviations, and that his formulation was perhaps "philosophically conservative" precisely in order to be digestible by the establishment? Even if these things are so, perhaps you might indicate what tell-tale signs there are that this is so, and do less creative anachronistic interpolation of Newton's attitudes and vernacular. This answer simply isn't very constructive. –  Niel de Beaudrap Jun 24 '12 at 21:32
    
It's all POV, of course. What isn't POV is that Newton had a model of particles pushing and pulling along their lines of separation, and asymptotically attracting--- this is because otherwise angular momentum conservation in the Newtonian model is mysterious, and he clearly believes it from his argument for angular momentum. This is not the content of his laws, however. He states these to make it clear that things move forever if they are left undisturbed, that deviations from the natural state of motion are due to forces, and the forces are such that momentum is conserved. –  Ron Maimon Jun 25 '12 at 4:31
    
I don't argue that "momentum is conserved" is the bottom line of the Three Laws. But as Newton had no way of explaining why momentum changes e.g. due to gravitation (which he explicitly refused to explain, i.e. 'frame hypotheses' about), the heuristic notion of force continued to be useful, and still does today. And because so much of our prejudice about the world is friction-infused, it remains pedagogically useful even to people not immersed in Aristotle to point out Newton's First Law explicitly. But it is the revisionist angle you take which I cricitize, not the physical meaning. –  Niel de Beaudrap Jun 25 '12 at 15:46
    
@NieldeBeaudrap: What's the "revision"? That Newton was writing for people with wrong philosophy? He is obviously trying to make sure people understand that this is not Aristotle. And friction is not really the problem, it's the idea of something continuing forever uncaused that is the problem. People see an effect, motion, and they want a cause. Newton is saying there is no cause. –  Ron Maimon Jun 25 '12 at 18:42
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