Against Reducing Newtonian Mass to Kinematical Notions Niels - PowerPoint PPT Presentation

Against Reducing Newtonian Mass to Kinematical Notions Niels Martens Ockham Society Slides available at http://users.ox.ac.uk/~corp2044 25 November 2015

Outline The Project 1 Mach 2 Historical & New Responses 3 Evaluating the argument 4 Bits and bobs 5

Outline The Project 1 Mach 2 Historical & New Responses 3 Evaluating the argument 4 Bits and bobs 5

The Project Main question Mach Reducing to... Historical & New Responses Motivations Evaluating the argument The Project Bits and bobs The main question 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

The Project Main question Mach Reducing to... Historical & New Responses Motivations Evaluating the argument The Project Bits and bobs Reducing mass to... Curvature of space-time Interactions Binding energy Higgs mechanism Kinetic energy Kinematical notions (eg. acceleration) Niels Martens Reducing Newtonian Mass 5/30

The Project Main question Mach Reducing to... Historical & New Responses Motivations Evaluating the argument The Project Bits and bobs 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

The Project Main question Mach Reducing to... Historical & New Responses Motivations Evaluating the argument The Project Bits and bobs The Project Assumptions Newtonian Gravity Equivalence between inertial and gravitational mass The Project 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 Mach’s Project Broader than my project No Mach exegesis Niels Martens Reducing Newtonian Mass 7/30

The Project Main question Mach Reducing to... Historical & New Responses Motivations Evaluating the argument The Project Bits and bobs How far do we go? Fundamental mass determinates, reduction of their quantitative structure only (Dees, Perry) Eliminating any fundamental notion of mass (Mach, NM) Niels Martens Reducing Newtonian Mass 8/30

The Project Main question Mach Reducing to... Historical & New Responses Motivations Evaluating the argument The Project Bits and bobs 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

Outline The Project 1 Mach 2 Historical & New Responses 3 Evaluating the argument 4 Bits and bobs 5

The Project Mach Operational definition Historical & New Responses Mach the Comparativist Evaluating the argument Bits and bobs Mach 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) ( Mach, 1893) Niels Martens Reducing Newtonian Mass 11/30

The Project Mach Operational definition Historical & New Responses Mach the Comparativist Evaluating the argument Bits and bobs Mach’s operational definition 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 = − a 21 a 12 Choose one m i as the standard of mass, to fix all the other masses ( Mach, 1893) Niels Martens Reducing Newtonian Mass 12/30

The Project Mach Operational definition Historical & New Responses Mach the Comparativist Evaluating the argument Bits and bobs Mach: a reductionist and a comparativist This suggests that Mach 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

The Project Mach Operational definition Historical & New Responses Mach the Comparativist Evaluating the argument Bits and bobs Argument against Comparativism � F g = G mM 2 GM v e = r 2 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

The Project Mach Operational definition Historical & New Responses Mach the Comparativist Evaluating the argument Bits and bobs Mach & Absolute Mass Even for the Machian 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 = 2 ar Anyway, reductionism is the core of the Machian project, not comparativism or absolutism Niels Martens Reducing Newtonian Mass 15/30

Outline The Project 1 Mach 2 Historical & New Responses 3 Evaluating the argument 4 Bits and bobs 5

The Project Mach Pendse Historical & New Responses Narlikar Evaluating the argument The argument Bits and bobs Does this generalise to n > 2? 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: n � a k = a k / j ˆ u kj , ( k = 1 , . . . , n ) j = 1 n ( n − 1 ) unknown coefficients in 3n linear equations: n ( n − 1 ) ≤ 3 n 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

The Project Mach Pendse Historical & New Responses Narlikar Evaluating the argument The argument Bits and bobs 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 ) + m 3 ( x 3 − x 1 ) + ... + m n ( x n − x 1 ) r 3 r 3 r 3 12 13 1 n A 12 m 2 + A 13 m 3 + ... + A 1 n 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

The Project Mach Pendse Historical & New Responses Narlikar Evaluating the argument The argument Bits and bobs Diverging from Mach’s aims Mach: the epistemological/reconstructive/descriptive project of humans reconstructing (afer 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

The Project Mach Pendse Historical & New Responses Narlikar Evaluating the argument The argument Bits and bobs The final atempt [whiteboard] There is no unique solution for the masses in terms of the initial accelerations! Niels Martens Reducing Newtonian Mass 20/30

Outline The Project 1 Mach 2 Historical & New Responses 3 Evaluating the argument 4 Bits and bobs 5

The Project Mach No solutions Historical & New Responses Infinite solutions Evaluating the argument Bits and bobs What follows from this? 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

The Project Mach No solutions Historical & New Responses Infinite solutions Evaluating the argument Bits and bobs Case 1: No 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

The Project Mach No solutions Historical & New Responses Infinite solutions Evaluating the argument Bits and bobs Case 2: Infinite solutions Indeterminism! Response 1: Perhaps all sets of initial masses corresponding to one of these sets of initial accelerations produce empirically equivalent models. 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 Niels Martens Reducing Newtonian Mass 24/30