Against Reducing Newtonian Mass to Kinematical Notions

Transcription

1 Against Reducing Newtonian Mass to Kinematical Notions Niels Martens Ockham Society Slides available at 25 November 2015

2 Outline

3 Outline

4 The main question Main question Reducing to... Motivations Can mass as it features in Newtonian Gravity be reduced to kinematical notions (i.e. distance, velocity, acceleration and higher-order derivatives)? Probably not. Niels Martens Reducing Newtonian Mass 4/30

5 Reducing mass to... Main question Reducing to... Motivations Curvature of space-time Interactions Binding energy Higgs mechanism Kinetic energy Kinematical notions (eg. acceleration) Niels Martens Reducing Newtonian Mass 5/30

6 Motivations Main question Reducing to... Motivations Search for a final theory (inter-theoretical reduction) Empiricism and/or ontological parsimony (intra-theoretical reduction) Absolutism vs. Comparativism debate Niels Martens Reducing Newtonian Mass 6/30

7 Main question Reducing to... Motivations Assumptions Newtonian Gravity Equivalence between inertial and gravitational mass Choose a kinematic ideology (r,v,a,...) and laws referring only to that ideology Obtain unique solutions to the corresponding initial value problems Generate the complete set of empirically possible models of NG s Project Broader than my project No exegesis Niels Martens Reducing Newtonian Mass 7/30

8 How far do we go? Main question Reducing to... Motivations Fundamental mass determinates, reduction of their quantitative structure only (Dees, Perry) Eliminating any fundamental notion of mass (, NM) Niels Martens Reducing Newtonian Mass 8/30

9 Main question Reducing to... Motivations A bad argument against reducing mass Famous 1953 axiomatisation of NG with mass as a primitive notion (McKinsey, Sugar & Suppes, 1953) For mass to be reducible to the other primitives of the theory, it should be impossible to find two models of NG that differ solely with respect to the primtiive masses, but not with respect to the other primitives. Proposed counter-example: two models consisting of one particle each, at rest at all times, with different mass values Response: These models are empirically equivalent Turning things around: it counts against the mass theory that it recognises empirically indistinguishable distinctions Niels Martens Reducing Newtonian Mass 9/30

10 Outline

11 Operational definition the Comparativist Precursor of logical empiricism: the task of physics is merely the abstract quantitative expression of facts concerning the relations between observable phenomena Operational definition of mass in terms of acceleration (ratios) (, 1893) Niels Martens Reducing Newtonian Mass 11/30

12 s operational definition Operational definition the Comparativist Two particles (alone in the universe, or dynamically isolated) Third law: F 12 = F 21 Second law: F 12 = m 1 a 12, F 21 = m 2 a 21 m 1 m 2 = a21 a 12 Choose one m i as the standard of mass, to fix all the other masses (, 1893) Niels Martens Reducing Newtonian Mass 12/30

13 Operational definition the Comparativist : a reductionist and a comparativist This suggests that is not only a reductionist about mass, but also a comparativist, since the absolute masses are only conventions. Any justification? Niels Martens Reducing Newtonian Mass 13/30

14 Argument against Comparativism Operational definition the Comparativist F g = G mm r 2 v 2GM e = r F F v 0 v 0 Double Mass F F v 0 v 0 (Baker, manuscripts; NM, manuscripts) Niels Martens Reducing Newtonian Mass 14/30

15 & Absolute Mass Operational definition the Comparativist Even for the ian project expressing quantitative facts and their relations this is something that needs to be accounted for. Could we use the two-particle escape velocity scenario as an operational definition for the mass scale, once the mass ratios have been fixed (as well as the length and velocity)? Escape velocity inequality: v 2 > v 2 e = 2ar Anyway, reductionism is the core of the ian project, not comparativism or absolutism Niels Martens Reducing Newtonian Mass 15/30

16 Outline

17 Does this generalise to n > 2? Pendse Narlikar The argument No. Argument from counting d.o.f. s (Pendse, 1937) Data: acceleration ratios at t 0 Claim: For systems of more than 4 particles, the data does not determine the mass ratios. Proof: a k = n a k/j û kj, (k = 1,..., n) j=1 n(n 1) unknown coefficients in 3n linear equations: n(n 1) 3n Stronger claim: acceleration ratios at any number of instances will not fix the mass ratios if n > 7. (Pendse, 1937) Niels Martens Reducing Newtonian Mass 17/30

18 Pendse Narlikar The argument Including other kinematical notions Include distances, and the gravitational law (G 1) (Narlikar, 1939) Data: distances and acceleration ratios at (n-1) different instants (+ gravitational law) Claim: The data fixes the mass ratios Proof: a 1,x (t = t 0 ) = m 2(x 2 x 1 ) r m 3(x 3 x 1 ) r m n(x n x 1 ) r 3 1n A 12 m 2 + A 13 m A 1n m n = X 1 Repeat for a total of (n-1) different instants: (n-1) linearly independent equations (supposedly): solve for m 2, m 3,...m n. Fix m 1 via single additional acceleration component at a single instant. Niels Martens Reducing Newtonian Mass 18/30

19 Diverging from s aims Pendse Narlikar The argument : the epistemological/reconstructive/descriptive project of humans reconstructing (after the fact!) the masses from the 4D world generated by nature/god. This project: the metaphysical project of explaining our actual world by deterministically evolving forward the initial conditions (i.e. playing God) Moving forward: Using Narlikar s insight (besides acceleration we may also use r,v, and the laws), but sticking to data at t 0 only Niels Martens Reducing Newtonian Mass 19/30

20 The final attempt Pendse Narlikar The argument [whiteboard] There is no unique solution for the masses in terms of the initial accelerations! Niels Martens Reducing Newtonian Mass 20/30

21 Outline

22 What follows from this? No solutions Infinite solutions A non-zero determinant would have proved reductionism right. It is less straightforward whether the vanishing of the deterinant rules out reductionism. Either no solutions, or infinitely many solutions. Niels Martens Reducing Newtonian Mass 22/30

23 Case 1: No solutions No solutions Infinite solutions [whiteboard] Could we somehow eliminate these deviant sets of initial accelerations that do not correspond to a set of masses? Niels Martens Reducing Newtonian Mass 23/30

24 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0-1,5-1 -0,5 0 0,5 1 1,5 Case 2: Infinite solutions No solutions Infinite solutions Indeterminism! Response 1: Perhaps all sets of initial masses corresponding to one of these sets of initial accelerations produce empirically equivalent models. Niels Martens Reducing Newtonian Mass 24/30

25 Case 2: Infinite solutions No solutions Infinite solutions Indeterminism! Response 1: Perhaps all sets of initial masses corresponding to one of these sets of initial accelerations produce empirically equivalent models. Response 2: Use the y and z components of the accelerations as well, to get unique solutions. Response 3: Indeterministic laws Niels Martens Reducing Newtonian Mass 24/30

26 Outline

27 Additional argument Other issues Additional argument against reductionism Even if the initial accelerations fix the masses, the accelerations have more d.o.f. s than the masses (for D > 1), so it is conspiratorial that all of these match up in exactly the right way as to correspond to a set of masses. Niels Martens Reducing Newtonian Mass 26/30

28 Other issues Additional argument Other issues The proof is only for an odd number of particles. Even if the reductionist attempts above had worked, they would have only fixed the mass ratios, not the absolute masses. Have we ruled out all possible types of kinematic reduction? We haven t used v anywhere? Cf. the suggested operational definition of the absolute mass scale Humeanism about laws of nature Niels Martens Reducing Newtonian Mass 27/30

29 Conclusion I have argued that it is not possible to reduce Newtonian mass to kinematical notions (i.e. position, velocity, acceleration). That is, those kinematical notions at an initial time do not serve to fix and thereby explain the observable evolution of the system. Moreover, the reductionists never had any strategy to account for the absolute mass scale over and above the mass ratios.

30 References D.J. Baker, Some Consequences of Physics for the Comparative Metaphysics of Quantity, Manuscript M. Jammer (2000), Concepts of Mass in Contemporary Physics and Philosophy, Princeton University Press E. (1960 [1893]), The Science of Mechanics, translated by T.J. McCormack, The Open Court Publishing Co. N.C.M. Martens, Transfer & Confirmation of Status Dissertations, Oxford University, Manuscripts

31 References - continued J.C.C. McKinsey, A.C. Sugar & P. Suppes, Axiomatic Foundations of Classical Particle Mechanics, Journal of Rational Mechanics and Analysis 2(1) V.V. Narlikar (1939), The Concept and Determination of Mass in Newtonian Mechanics, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Series 7, 27(180):33-6 C.G. Pendse (1937), A Note on the Definition and Determination of Mass in Newtonian Mechanics, Philosophical Magazine, 24: