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

1

The Project

2

Mach

3

Historical & New Responses

4

Evaluating the argument

5

Bits and bobs

Outline

1

The Project

2

Mach

3

Historical & New Responses

4

Evaluating the argument

5

Bits and bobs

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

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

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

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

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

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

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

1

The Project

2

Mach

3

Historical & New Responses

4

Evaluating the argument

5

Bits and bobs

The Project Mach Historical & New Responses Evaluating the argument Bits and bobs Operational definition Mach the Comparativist

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 Historical & New Responses Evaluating the argument Bits and bobs Operational definition Mach the Comparativist

Mach’s operational definition

Two particles (alone in the universe, or dynamically isolated) Third law: F12 = −F21 Second law: F12 = m1a12, F21 = m2a21

m1 m2 = − a21 a12

Choose one mi as the standard of mass, to fix all the other masses (Mach, 1893)

Niels Martens Reducing Newtonian Mass 12/30

The Project Mach Historical & New Responses Evaluating the argument Bits and bobs Operational definition Mach the Comparativist

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 Historical & New Responses Evaluating the argument Bits and bobs Operational definition Mach the Comparativist

Argument against Comparativism

Fg = G mM

r2

ve =

• 2GM

r

v0 v0 F F

Double Mass

v0 v0 F F

(Baker, manuscripts; NM, manuscripts)

Niels Martens Reducing Newtonian Mass 14/30

The Project Mach Historical & New Responses Evaluating the argument Bits and bobs Operational definition Mach the Comparativist

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

• perational definition for the mass scale, once the mass ratios

have been fixed (as well as the length and velocity)?

Escape velocity inequality: v2 > v2

e = 2ar

Anyway, reductionism is the core of the Machian project, not comparativism or absolutism

Niels Martens Reducing Newtonian Mass 15/30

Outline

1

The Project

2

Mach

3

Historical & New Responses

4

Evaluating the argument

5

Bits and bobs

The Project Mach Historical & New Responses Evaluating the argument Bits and bobs Pendse Narlikar The argument

Does this generalise to n > 2?

• No. Argument from counting d.o.f.’s (Pendse, 1937)

Data: acceleration ratios at t0 Claim: For systems of more than 4 particles, the data does not determine the mass ratios. Proof: ak =

n

• j=1

ak/jˆ ukj, (k = 1, . . . , n) 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

The Project Mach Historical & New Responses Evaluating the argument Bits and bobs 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: a1,x(t = t0) = m2(x2 − x1) r3

12

+ m3(x3 − x1) r3

13

+ ... + mn(xn − x1) r3

1n

A12m2 + A13m3 + ... + A1nmn = X1 Repeat for a total of (n-1) different instants: (n-1) linearly independent equations (supposedly): solve for m2, m3, ...mn. Fix m1 via single additional acceleration component at a single instant.

Niels Martens Reducing Newtonian Mass 18/30

The Project Mach Historical & New Responses Evaluating the argument Bits and bobs Pendse Narlikar The argument

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 t0 only

Niels Martens Reducing Newtonian Mass 19/30

The Project Mach Historical & New Responses Evaluating the argument Bits and bobs Pendse Narlikar The argument

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

1

The Project

2

Mach

3

Historical & New Responses

4

Evaluating the argument

5

Bits and bobs

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

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 Historical & New Responses Evaluating the argument Bits and bobs No solutions Infinite solutions

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 Historical & New Responses Evaluating the argument Bits and bobs No solutions Infinite solutions

Case 2: Infinite solutions

Indeterminism! Response 1: Perhaps all sets of initial masses corresponding to

• ne of these sets of initial accelerations produce empirically

equivalent models.

0,1 0,2 0,3 0,4 0,5 0,6 0,7

• 1,5
• 1
• 0,5

0,5 1 1,5

Niels Martens Reducing Newtonian Mass 24/30

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

Case 2: Infinite solutions

Indeterminism! Response 1: Perhaps all sets of initial masses corresponding to

• ne 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

Outline

1

The Project

2

Mach

3

Historical & New Responses

4

Evaluating the argument

5

Bits and bobs

The Project Mach Historical & New Responses Evaluating the argument Bits and bobs 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

The Project Mach Historical & New Responses Evaluating the argument Bits and bobs Additional argument Other issues

Other issues

The proof is only for an odd number of particles. Even if the reductionist atempts 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

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.

References

D.J. Baker, ‘Some Consequences of Physics for the Comparative Metaphysics of Qantity’, Manuscript

• M. Jammer (2000), Concepts of Mass in Contemporary Physics

and Philosophy, Princeton University Press

• E. Mach (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

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, 24:1012-1022