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Kant provided an a priori analysis or deduction of Newton's third law - the law of action and reaction.

This leaves the first and second law; it's an easy observation that the first law follows from the second (but not vice-versa); for one sees that no force means no acceleration, and this means being at rest.

Hence, one can ask is there an a priori deduction of the second law, following the pattern shown by Kant; in part, or in whole.

Is there? (I mean within classical Newtonian Mechanics).

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Mozibur Ullah
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  • What do you mean by Kantian ? – sure Aug 22 '15 at 16:32
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    In any case, I'm "currently" writing a construction of general relativity starting from "nothing. I've almost finished the parts concerning classical mechanics based on first philosophical principles, even though its still a draft. Feel free to read the first five parts of this (I guess eventually 15 parts) series there http://www.sure.zhln.eu/wp/?p=126 – sure Aug 22 '15 at 16:36
  • I've added a link which adds some explanation. – Mozibur Ullah Aug 22 '15 at 16:39
  • Kant in "Metaphysical foundations of natural science" states and "proves" the first of Newton's axioms (A119-120) and the third of Newton's axioms (A121f), but he does not consider the second of Newton's axioms. Hence I suppose, the answer to your question is "no". – Jo Wehler Aug 22 '15 at 17:28
  • @jo wehler instead of just assuming that because kant didnt lay out the 2nd law it must not have an a priori deduction, can you show a counter example showing why an a priori deduction is not possible with the 2nd law? And what makes the 2nd different from the 1st and the 3rd? That would be I think what the asker would want, closure in one direction or another. – hellyale Aug 22 '15 at 19:20
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    @wehler: sure, if I was asking whether Kant himself provided such a deduction; but I'm not - I'm asking whether anyone else inspired by Kants example provided one for the second law. – Mozibur Ullah Aug 22 '15 at 19:27
  • @MoziburUllah Sorry, I misunderstood your question. I do not know the answer to your question. - The second axiom of Newton is more difficult to detect. It states that two physical quantities are proportional, i.e. the axiom considers all possible values of acceleration, not just the value zero. In addition, the proportionality constant, i.e. the measure of inertia, is equated to the mass. That's a physical quantity which is defined in quite a different manner. – Jo Wehler Aug 22 '15 at 20:01
  • @wehler: this might be why Kant left it alone... – Mozibur Ullah Aug 22 '15 at 20:04
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    I am confused as to what one is allowed to assume. Friedman reviews all of Kant's a priori arguments for natural laws in Kant and Exact Sciences, in late works he even attempted to prove that ether is a priori. All of them illicitly insert empirical input in one place or another, and all are unsound because classical laws are strictly speaking false, and modern physics is a counterexample. E.g. is one allowed to assume Euclidean geometry and strict causality as the necessary condition of experience in time, etc.? Or are you asking if someone actually did it historically? – Conifold Aug 22 '15 at 20:04
  • @hellyale Why an a priori deduction is not possible? An a priori deduction has the same truth claim as a mathematical proof. Now we know by General Relavitiy that the second axiom of Newton is an approximation to the more general result that particles in arbitrary gravitational fields move on geodesics. Because the second axiom of Newton does not consider curved spacetime manifolds - which exists near big masses like the sun - the axiom does not hold in general. But that's not a criticism of Newton, rather of metaphysics :-) – Jo Wehler Aug 22 '15 at 20:22
  • @conifold:possibly; I'd suggest that the aether, as the mechanical medium of transmission of light was *identified* eventually with the EM field; and the outcome of Einsteins work was to identify spacetime as the mechanical medium of transmission of gravity; I say mechanical because both fields warp, stretch and have curvature; I haven't read Friedmann but I'd also suggest that the a priori content is that they both act as mediums; for how can a signal go from here to there, when there is a void in-between? – Mozibur Ullah Aug 23 '15 at 11:39
  • @wehler: I'm not sure that 'a priori' has the same sense in different disciplines; this is why I said in the 'pattern shown' by Kant. – Mozibur Ullah Aug 23 '15 at 11:42
  • I've modified my question to say 'in part or whole'; I wasn't expecting that the whole of it to have an a priori deduction; but to discover what is its a priori content. – Mozibur Ullah Aug 23 '15 at 11:45
  • @MoziburUllah I always consider it part of Einstein's radicality that he completely abolished the ether hypothesis: The electromagnetic field does not need any carrier, it propagates through vacuum. Before certain exotic mechanical properties were ascribed to ether and had to be explained. Similarly, embedding gravitation into geometry relinquishes any mechanical carrier. Both steps are examples how to apply Occam's razor. ad "How can a signal go ...?": A bit sloppy: We have to adapt our intuition to the properties of nature, not vice versa. – Jo Wehler Aug 23 '15 at 13:41
  • @wehler: 'the EM field does not need any carrier': it's easy enough to imagine spacetime without the EM field; but not vice-versa - this is one sense on which spacetime carries the EM Field; also, is it the field *itself* that *propagates*; if so, then where? It must be to where there is no field already - so are you suggesting that there are parts of spacetime that has no associated EM Field? – Mozibur Ullah Aug 23 '15 at 13:51
  • Isn't it rather that there are ripples in the EM field; rather like waves on the sea, but the water of the sea itself remains still? – Mozibur Ullah Aug 23 '15 at 13:54
  • How 'does embedding gravity in geometry' relinquishes mechanical properties? When we talk about curvature and the stress-energy tensor? Perhaps the problem is what is meant by *mechanical* here. – Mozibur Ullah Aug 23 '15 at 13:58
  • @Mozibur Ullah If at all, then the ripples are the EM. The ripples are not in(!) the EM. But I consider it a perverted metaphor, because there is no carrier of the ripples alike to wave propagation through media like water or air. - Yes, there are parts of spacetime without EM - why not? – Jo Wehler Aug 23 '15 at 14:07
  • 'A bit sloppy': it's just a manner of philosophical style - not physical. – Mozibur Ullah Aug 23 '15 at 14:09
  • Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/27307/discussion-between-mozibur-ullah-and-jo-wehler). – Mozibur Ullah Aug 23 '15 at 14:12

2 Answers2

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Acceleration is a proportion. Proportion is not one of the Categories. There is a reason.

We obviously only find things proportional when we measure them. And measurement is subject to the shape of space, whether you take that to mean the space that is an aspect of humanity in Kant, or you take that to mean space as currently understood by General Relativity.

So as given, the statement is not expressed in a form we can reasonably make any a priori deduction about, for pretty much the reason we have ultimately given it up (since gravity in general relativity is an aspect of space, and not a force per se).

Equal, Opposite and Like in Kind are basic or derived Categories. So in some sense the Third Law lies at a far more basic level of logic.

  • Which statement is objectively false? I'm not sure to get it. – sure Aug 23 '15 at 07:59
  • Relativity makes the second law false. A greater force might have a lesser effect if applied at a different place in space. You cannot claim this is due to other forces, because in general relativity, gravity is not a force. –  Aug 23 '15 at 13:14
  • I still can't understand your answer or the point you're trying to make (regardless of the fact that the "true/false" terminology is really shitty when it comes to describe theories) – sure Aug 23 '15 at 13:54
  • How about just the Kantian side, stripped of analogies. Statements that cannot be captured in Categories will have different interpretations for different species. So they can never have a priori proofs, because the thing to be proven would not mean the same thing for the different contexts. And an a priori proof should equally be proof for all cases. –  Aug 23 '15 at 16:05
  • The Second Law of Motion cannot be expressed this way because it involves the notion of proportion, since acceleration is a ratio of measures. The notion of proportion involves space, a concept specific to animal species (according to Kant). So the Second Law of Motion cannot possibly have an a priori proof. –  Aug 23 '15 at 16:09
  • The relativity side is, in theory, irrelevant, but 'cute'. When physics decided measure depends upon space very directly, because gravity is not a force, but an effect of spatial curvature, they then see force as more effective certain places (open space) and less effective other places (in strong gravitationsl fields). The Law suggests that a greater force given the same mass should always produce more acceleration -- but it might not, if the former is in open space and the latter is in a strong gravitational field. –  Aug 23 '15 at 16:14
  • It is somehow 'deep' to me that both arguments depend upon the subjectivity of the model of space. –  Aug 23 '15 at 16:15
  • I think that I just don't understand the Kantian notion of category. Any kind of reasoning strictly speaking depends upon the subjectivity of your way of thinking and making a proof (that is, a logic). The point is, even though the second law doesn't hold globally in GR, there should be a way to formulate such second law for other kind of forces happening "on" space-time and allowing your observer to follow a non geodesic worldline. Newton's second law basically means "to have a non trivial change, you need something that is acting upon you". This is actually close to first and third law. – sure Aug 23 '15 at 16:28
  • It is also stronger, because it means that the first non trivial change happens with 2nd order of motion : acceleration. First law is a consequence of second law, and strictly speaking, the third law is also encoded in the second law if you do things correctly without cheating at any places. – sure Aug 23 '15 at 16:29
  • Then in this case, Kant is wrong not to understand that the very existence of an "inertial structure" is as absolute as the first and third law. Now, certainly, there are different manner to try to explain it or encode it. In any case, this "inertial structure" is the key point of the second law, and is still here in GR (geodesics). In particular, the very existence of an inertial structure can be seen quite easily if you try to study the behaviour of a uniformly accelerated observer and suppose that the latter is equivalent to a guy "not moving on a fictively existing space". – sure Aug 23 '15 at 16:36
  • Given your statement, you understand, you just disagree in principle. Kant thought we needed to accord intelligence to non-temporal beings, for whom thinking was not subjective, and those beings would share all a-priori concepts with us, when they can be translated into neutral terms (the categories). It would be really hard for a non-temporal being to latch unto acceleration as a concept. –  Aug 23 '15 at 16:38
  • Kant is wrong about any number of things, but the question was about Kant. –  Aug 23 '15 at 16:40
  • Sure, but it's nice developing why this view is too naive to be taken as "good metaphysics of knowledge". (In particular, the very existence of interaction - what describes third law - is related to existence of time. No time, no interaction, no third law) – sure Aug 23 '15 at 16:41
  • I am not sure it is so much naive as compromised by his overreaching attempts to remain consistent with Christianity. Kant wanted to preserve idealism at all costs. So there are times when he gives positions that are too weak on purpose. When writing as a physicist per se, he generally got the physics (of his day) right. –  Aug 23 '15 at 16:43
  • Well, no time but still a dynamic of good and evil, then you have a different understanding of equal and opposite force. Interaction is not necessarily the frame for non-humans... (Again, not me, Kant.) –  Aug 23 '15 at 16:46
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In Newton's early (around 1664) manuscript called Waste Book we can find some "conjectures" [see folio 10v : Axiomes & Propositions, 4 and 6] regarding the proportionality between force and change in quantity of motion produced.

The simplest assumption :

force directly proportional to change in quantity of motion

would be the "most natural" assumption to be tested experimentally in order to achieve a quantitative determination of force.

But things may have gone differently ... We can comapre with the Law of universal gravitation : the "simplest" proportionality would be to decrease with the distance; the "natural" one would be to the decrease with the cube of the radius (the force spreads out in space).

See : John Herivel, The Background to Newton's Principia : A Study of Newton's Dynamical Researches in the Years 1664-84 (1965).

Mauro ALLEGRANZA
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  • Those are two natural possibilities; the other natural possibility is to spread out as a surface; like the surface of a balloon being blown up. – Mozibur Ullah Aug 23 '15 at 10:28
  • They're natural of course because they're associated with dimensions one, two and three. – Mozibur Ullah Aug 23 '15 at 11:17
  • You can look at the reason for two dimensions being that time divides one of them out. The expanding shell 'flows' away from the source. But that is not 'natural' if you don't think field effects 'flow' or 'spread out' from the source. Newton would not have seen it that way, because he presumed instantaneous effect. So the cube would have been more natural. However, Newton already knew orbits were conic sections, so squares had to be the real case. –  Aug 23 '15 at 16:25