# Is there a Kantian a priori deduction (post-hoc) of Newtons Second Law?

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).

• What do you mean by Kantian ?
– sure
Aug 22, 2015 at 16:32
• 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 sure.zhln.eu/wp/?p=126
– sure
Aug 22, 2015 at 16:36
• I've added a link which adds some explanation. Aug 22, 2015 at 16:39
• @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. Aug 22, 2015 at 19:27
• 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? Aug 22, 2015 at 20:04

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).

• 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. Aug 23, 2015 at 10:28
• They're natural of course because they're associated with dimensions one, two and three. Aug 23, 2015 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.
– user9166
Aug 23, 2015 at 16:25

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, 2015 at 7:59
• 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.
– user9166
Aug 23, 2015 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, 2015 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.
– user9166
Aug 23, 2015 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.
– user9166
Aug 23, 2015 at 16:09