Absolutism vs. Comparativism about Mass in Newtonian Gravity Niels - - PowerPoint PPT Presentation

Absolutism vs. Comparativism about Mass in Newtonian Gravity Niels Martens

Lugano Qantities Conference

Slides available at martensniels.wordpress.com

16 November 2019

www.history-and-philosophy-of-physics.com Lichtenberg Group for History and Philosophy of Physics University of Bonn @hppBonn

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NCMM’17 Regularity Comparativism about Mass in Newtonian Gravity Philosophy of Science 84(5):1226-1238 NCMM’18 Against Laplacian Reduction of Newtonian Mass to Spatiotemporal Qantities Foundations of Physics 48(5):591-609 NCMM’20a Machian Comparativism about Mass The British Journal for the Philosophy of Science NCMM’20b The (Un)detectability of Absolute Newtonian Masses Synthese

Laplace’s problem

Initial Variable & Parameter Problem What is the minimal choice of initial variables and parameters that corresponds to a well-posed initial value problem (in Newtonian Gravity)—that is, the associated determinate values, together with the laws of Newtonian Gravity, determine a unique evolution?

(Poincaré, 1902; Skow, 2007)

Does a Mass Scaling lead to an empirical difference?

(Active) Leibniz Mass Scaling A uniform scalar multiplication of each of the absolute mass magnitudes, ceteris paribus. Realism about Absolute Masses Absolute masses are empirically meaningful/relevant Anti-Realism about Absolute Masses Absolute masses are empirically meaningless/irrelevant

Absolutism vs. Comparativism

(Weak, Metaphysical) Absolutism The determinate mass ratios obtain in virtue of determinate absolute masses.

(Armstrong, 1978, 1988; Mundy, 1987; Lewis, 1986; Sider, ms)

(Weak, Metaphysical) Comparativism The determinate mass relations do not obtain in virtue of determinate absolute masses.

(Russell 1903; Mach, 1960; Ellis, 1966; Field, 1980; Bigelow et al., 1988; Arntzenius, 2012; Dasgupta, 2013; Eddon, 2013; Baker, ms; Perry, 2016; Roberts, ms; Sider, ms; Wolff, ms)

Absolute fundamentality

Strong (Metaphysical) Absolutism

1

Weak (Metaphysical) Absolutism

2

Mass is fundamental. That is, the determinate absolute masses do not themselves obtain in virtue of anything else. Strong (Metaphysical) Comparativism

1

Weak (Metaphysical) Comparativism

2

Mass is fundamental. That is, the determinate mass relationships do not themselves obtain in virtue of anything else.

NCMM’20b

Assumptions

Newtonian Gravity Equivalence between gravitational and inertial mass Scale-invariant mass relations: ‘Mass ratios’ (Baker, ms)

Definitions

Absolute mass magnitudes Set of monadic properties Cardinality: 2ℵ0 Totally ordered & Concatenation structure (‘addition’) Transworld identity (quiddities) → totally ordered semi-group Mass relations Set of binary relations Cardinality: 2ℵ0 Totally ordered & Concatenation structure (‘multiplication’) Transworld identity (quiddities) → totally ordered group

Kinematic Comparativism

Kinematic Comparativism ( ⇐ ⇒ dimensionfulness) For any dimensionful determinable, such as mass, the magnitude predicated of any particle can only be meaningfully reported or expressed in terms of how this magnitude relates to the magnitude

• f another particle having the same determinable property.

Therefore, absolute mass magnitudes need to be represented by a numerical quantity times a unit. This representation is non-unique (conventional choice of unit).

(Hugget, 1999)

Naive argument for comparativism

Kinematic comparativism → (metaphysical) comparativism Metaphysical comparativism requires us to prove: Dynamic Comparativism Physics depends only on the mass ratios, not on further absolute masses in virtue of which those ratios obtain. In other words, metaphysical comparativism is empirically adequate.

Comparativist Argument

Pdyn Dynamic Comp: (Metaphysical) comp is empirically equiv- alent to (metaphysical) abs. Pocc Occamist norm: All other things being equal (i.e. Pdyn), we should favour theories that are metaphysically more parsimonious. Ppar (Metaphysical) comp about mass is metaphysically more par- simonious than (metaphysical) abs. C (Metaphysical) comp about mass should be favoured over (metaphysical) abs.

Comparativist Argument

Pdyn Dynamic Comp: (Metaphysical) comp is empirically equiv- alent to (metaphysical) abs. Pexp Explanatory Adequacy: (Metaphysical) comp is at least as explanatorily adequate as (metaphysical) abs. Pocc Occamist norm: All other things being equal (i.e. Pdyn ∧ Pexp), we should favour theories that are metaphysically more parsimonious. Ppar (Metaphysical) comp about mass is metaphysically more par- simonious than (metaphysical) abs. C (Metaphysical) comp about mass should be favoured over (metaphysical) abs.

Outline

1

Absolutism vs. Comparativism

2

Empirical Adequacy Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

3

Metaphysical parsimony & Explanatory adequacy

4

Eliminating mass altogether?

Outline

1

Absolutism vs. Comparativism

2

Empirical Adequacy Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

3

Metaphysical parsimony & Explanatory adequacy

4

Eliminating mass altogether?

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

Dasgupta’s Comparativism

fq = mq · aq fq = Gq m1,qm2,q r2

q

(L1) For any material thing x, (a) For any reals r1 and r2, if x has mass r1M and acceleration r2A, then x has force r1r2F acting on it. (b) For any real r3, if x has force r3F acting on it, then there are reals r4 and r5 whose product is r3, such that x has mass r4M and acceleration r5A.

(Dasgupta, 2013, p.130)

Niels Martens Absolutism vs Comparativism about Mass 16/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

Dasgupta’s Comparativism

fq = mq · aq fq = Gq m1,qm2,q r2

q

(L2) For any material things x and y [in the same world], (a) For any reals r1 and r2, if x is r1 times as massive as y and is accelerating at r2 times the rate of y, then x has r1r2 times as much force acting on it as y. (b) For any real r3, if x has r3 times as much force acting on it than y, then there are reals r4 and r5 whose product is r3, and such that x is r4 times as massive as y and is accelerating r5 times the rate of y.

(Dasgupta, 2013, p.130-1)

Niels Martens Absolutism vs Comparativism about Mass 16/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

Comparativism’s bucket

Fg = G mM

r2

ve =

• 2GM

r

v0 v0 F F

Double Mass

v0 v0 F F

(Baker, ms; NCMM’20b)

Niels Martens Absolutism vs Comparativism about Mass 17/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

Where to go from here?

Absolutism The comp bucket shows that absolute masses are real, i.e. empirically meaningful → meta- physical absolutism Regularity comp Accept that the comp bucket proves realism about absolute masses, but insist that those can be grounded in mass ratios (and other non-mass facts). (NCMM’17) Machian comp Modify the syntax (i.e. equations) such that the comp bucket is avoided (whilst retaining empirical equivalence to abs) → anti-realism about absolute masses (NCMM’20a)

Niels Martens Absolutism vs Comparativism about Mass 18/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

Regularity Relationalism

Response to i.a. Newton’s bucket (i.e. inertial effects) Core Idea: It is merely the truth of Newton’s laws in certain frames that privileges those frames, not the structure of absolute space. (Van Fraassen, 1970) Regularity Protocol: Assume a relational Humean mosaic (with intrinsic masses). Consider all possible reference frames that are naturally adapted to that mosaic: only in some frames will the best axiomatisations be Newton’s laws. Claim: those are the best laws overall. → Inertial frames & laws supervene as a package deal.

(Hugget, 2006)

Niels Martens Absolutism vs Comparativism about Mass 19/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

Regularity Comparativism

If not in spirit, then at least in leter

Core idea: Absolute mass scale is privileged because of the truth of Newton’s laws (incl. Gravitational Law) for that choice

• f scale, not because that scale is grounded in absolute masses.

Liberalisation: Replace the absolutist Humean mosaic by a mosaic consisting of fundamental mass ratios. ‘Coordinates’: Consider all possible choices of an absolute mass scale. Regularity Approach: Claim: Only for one choice of the absolute mass scale will the laws be the best axiomatisation, and those laws are Newton’s laws & the Gravitational Law. → Mass scale & laws supervene as a package deal.

(NCMM’17)

Niels Martens Absolutism vs Comparativism about Mass 20/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

Problem

Slippery slope: throwing away the massive baby with the bathwater

(Narlikar, 1939)

Regularity Eliminativism (Hall, ms) Super-Humeanism (Esfeld & Deckert, ms)

Niels Martens Absolutism vs Comparativism about Mass 21/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

Problem

Niels Martens Absolutism vs Comparativism about Mass 21/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

Machian Comparativism

Modify the absolutist law by substituting Newton’s Constant for a variable—across possible worlds only, not across space and time. G = G(W) = γ/

k

mk Fgrav,ij = γ

mimj r2

k

mk

v <

• γ

r

k mk mj

NCMM‘20a

Niels Martens Absolutism vs Comparativism about Mass 22/31

Outline

1

Absolutism vs. Comparativism

2

Empirical Adequacy Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

3

Metaphysical parsimony & Explanatory adequacy

4

Eliminating mass altogether?

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Qantitative Parsimony? Qalitative Parsimony? Conspiracy

Measuring Metaphysical Parsimony

Naive intuition: absolutistism acknowledges both absolute masses and mass relations while comparativism only recognises the later → comparativism has a ‘lower metaphysical bill’ Both views fundamentally commit to different types of building

• blocks. How to compare metaphysical parsimony?

Qantitative Parsimony?

Absolutism: n absolute masses Comparativism: n2 or n(n − 1) mass relations

NCMM‘20a

Niels Martens Absolutism vs Comparativism about Mass 24/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Qantitative Parsimony? Qalitative Parsimony? Conspiracy

Qalitative Parsimony?

Absolute mass magnitudes Set of monadic properties Cardinality: 2ℵ0 Totally ordered & Concatenation structure (‘addition’) Transworld identity (quiddities) → totally ordered semi-group Mass relations Set of binary relations Cardinality: 2ℵ0 Totally ordered & Concatenation structure (‘multiplication’) → totally ordered group Machian comp: additional concatenation structure (‘addition’)

Niels Martens Absolutism vs Comparativism about Mass 25/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether? Qantitative Parsimony? Qalitative Parsimony? Conspiracy

The conspiracy of mass relations

Transitivity constraint required on mass relations if they are to be interpretable as mass ratios at all Either: meta-relations required to ensure that the constraint holds → loss of quantitative and qualitative parsimony Or: if the mass relations conspire to behave as if they obtained in virtue

• f absolute masses, one should

infer to the best (i.e. only) explanation: absolutism

x2 x2 x7

(NCMM’20a; Roberts, ms)

Niels Martens Absolutism vs Comparativism about Mass 26/31

Outline

1

Absolutism vs. Comparativism

2

Empirical Adequacy Dasgupta’s Comparativism Regularity Comparativism Machian comparativism

3

Metaphysical parsimony & Explanatory adequacy

4

Eliminating mass altogether?

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether?

Can mass be eliminated altogether?

Can mass—as it features in Newtonian Gravity (NG)—be reduced to spatiotemporal quantities (i.e. distance, velocity, acceleration and higher-order derivatives) at t0 without loss of the predictive and explanatory power of NG? That is, can we solve Laplace’s problem without any notion of mass? No.

NCMM‘18

Niels Martens Absolutism vs Comparativism about Mass 28/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether?

Reducing mass

a1,x(t = t0) = m2(x2 − x1) r3

12

+ m3(x3 − x1) r3

13

+ ... + mn(xn − x1) r3

1n

a2,x(t = t0) = m1(x1 − x2) r3

21

+ m3(x3 − x2) r3

23

+ ... + mn(xn − x2) r3

2n

. . .

NCMM‘20a

Niels Martens Absolutism vs Comparativism about Mass 29/31

Absolutism vs. Comparativism Empirical Adequacy Metaphysical parsimony & Explanatory adequacy Eliminating mass altogether?

Failure

Gm = a G =       α12 · · · a1n α21 . . . . . . ... . . . αn1 · · · · · ·       αij = xj−xi

r3

ij

|G| = |GT| = | − G| = (−1)n|G| For odd n: |G| = −|G| = 0 There is no unique solution for the masses in terms of the initial accelerations!

NCMM‘18

Niels Martens Absolutism vs Comparativism about Mass 30/31

Conclusion

1

‘Machian’ comparativism is the most viable form of comparativism—it successfully responds to the bucket argument without admiting the empirical meaningfulness of absolute masses.

2

However, like all forms of comparativism it fails to explain the transitivity of mass ratios. Moreover, its metaphysical parsimony is even more questionable than other forms of comparativism.

3

Mass cannot be eliminated altogether and is hence fundamental → strong abs/comp

References I

• D. Armstrong (1978), A Theory of Universals: Volume 2,

Cambridge: Cambridge University Press

• D. Armstrong (1988), ‘Are quantities relations? A reply to

Bigelow and Pargeter’, Philosophical Studies 54:305-316

• F. Arntzenius (2012), Space, time, & stuff, Oxford: Oxford

University Press D.J. Baker (manuscript), ‘Some Consequences of Physics for the Comparative Metaphysics of Qantity’ D.J. Baker (manuscript), ‘Comparativism with mixed relations’

References II

• J. Bigelow, R. Pargeter & D.M. Armstrong (1988), ‘Qantities’,

Philosophical Studies 54:287-316

• S. Dasgupta (2013), ‘Absolutism vs Comparativism about

Qantity’, Oxford Studies in Metaphysics: Volume 8, Oxford University Press

• S. Dasgupta (2015), ‘Inexpressible Ignorance’, Philosophical

Review 124(4):441-480

• B. Ellis (1966), Basis concepts of measurement, Cambridge:

Cambridge University Press

• M. Eddon (2013), ‘Qantitative properties’, Philosophy Compass

8(7):633-645

References III

• M. Esfeld & D-A. Deckert (manuscript), ‘What there is. A

minimalist ontology of the natural world.’ H.H. Field (1980), Science without Numbers: A defence of nominalism, Oxford: Basil Blackwell

• N. Hall (manuscript), ‘Humean Reductionism About Laws of

Nature’

• N. Hugget (1999), Space from Zeno to Einstein: Classic Readings

with a Contemporary Commentary, Cambridge, MA: MIT Press

• N. Hugget (2006), ‘The Regularity Account of Relational

Spacetime’, Mind 115:457

• E. Mach (1960/1893), The Science of Mechanics, T.J. McCormack

(transl.) The Open Court Publishing Co.

References IV

• D. Lewis (1986), Philosophical Papers, volume ii, Oxford: Oxford

University Press

• D. Lewis (2009), ‘Ramseyan Humility’, in D. Braddon-Mitchell &
• R. Nola (eds.) Conceptual analysis and philosophical naturalism,

p.203-222, Cambridge, MA: MIT Press N.C.M. Martens (DPhil thesis, 2017), ‘Against Comparativism about Mass in Newtonian Gravity –a Case-Study in the Metaphysics of Scale’, Magdalen College, University of Oxford

• T. Maudlin (1993), ‘Buckets of water and waves of space: Why

spacetime is probably a substance’, Philosophy of Science 60, 183-203

• B. Mundy (1987), ‘The Metaphysics of Qantity’, Philosophical

Studies 51(1):29-54

References V

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

• Z. Perry (2016), Physical Qantities: Mereology and dynamics,

PhD thesis, New York University

• H. Poincaré (1902), ‘Science and Hypothesis’, Dover (1952), New

York, translated by W. Scot J.T. Roberts (manuscript), ‘A Case for Comparativism about Physical Qantities – SMS 2016, Geneva’, htps://www.academia.edu/28548115/A_Case_for_Comparativism _about_Physical_Qantities_–_SMS_2016_Geneva

References VI

• B. Russell (1903), The principles of mathematics, New York:

W.W. Norton & Company

• T. Sider (manuscript), The Tools of Metaphysics and the

Metaphysics of Science

• B. Skow (2007), ‘Sklar’s maneuver’, British Journal for the

Philosophy of Science 58: 777–786. B.C. Van Fraassen (1970), An Introduction to the Philosophy of Time and Space, New York: Columbia University Press J.E. Wolff (manuscript), The metaphysics of quantities

Extra Slides

Three Approaches to Empirical Adequacy

1

Symmetry Approach: Are Leibniz Scalings symmetries of Newtonian Gravity, or not?

2

Undetectability approach: Are absolute masses undetectable? (Dasgupta, 2013)

3

Possibility counting: Does comparativism correctly generate the set of empirically distinct possible worlds allowed by Newtonian Gravity?

Problems with the Undetectability Approach

1

Equivocating fundamentality and reality Realism about Absolute Masses Absolute masses are empirically meaningful

2

Inexpressible what would be detected

3

Inexpressible what would remain undetected

4

Empirical access to G · m only

(Maudlin, 1993; Lewis, 2009; Dasgupta, 2015; NCMM’20b)

Empirical Equivalence as Correct Possibility ‘Counting’

Empirical Equivalence: The comparativist laws of nature uniquely (i.e. deterministically) evolve each set of initial conditions allowed by the theory into a dynamically possible model, such that the whole set of empirically distinct dynamically possible models generated by the absolutist theory is reproduced (i.e. completeness) and no models that are empirically distinct from each of the absolutist solutions are generated (i.e. soundness). T1 (absolutism) T2 (comparativism)

P1 (possible models) P2 (possible models) S1 (dynamically possible models) S2 (dynamically possible models)

NCMM’20a

Empirical Equivalence as Correct Possibility ‘Counting’

Empirical Equivalence: The comparativist laws of nature uniquely (i.e. deterministically) evolve each set of initial conditions allowed by the theory into a dynamically possible model, such that the whole set of empirically distinct dynamically possible models generated by the absolutist theory is reproduced (i.e. completeness) and no models that are empirically distinct from each of the absolutist solutions are generated (i.e. soundness). T1 (absolutism) T2 (comparativism)

P1 (possible models) P2 (possible models) E1 E2 E2 E1

NCMM’20a

Empirical Equivalence as Correct Possibility ‘Counting’

Empirical Equivalence: The comparativist laws of nature uniquely (i.e. deterministically) evolve each set of initial conditions allowed by the theory into a dynamically possible model, such that the whole set of empirically distinct dynamically possible models generated by the absolutist theory is reproduced (i.e. completeness) and no models that are empirically distinct from each of the absolutist solutions are generated (i.e. soundness). T1 (absolutism) T2 (comparativism)

P1 (possible models) P2 (possible models)

φ

S1 S2

NCMM’20a

Empirical Equivalence as Correct Possibility ‘Counting’

Empirical Equivalence: The comparativist laws of nature uniquely (i.e. deterministically) evolve each set of initial conditions allowed by the theory into a dynamically possible model, such that the whole set of empirically distinct dynamically possible models generated by the absolutist theory is reproduced (i.e. completeness) and no models that are empirically distinct from each of the absolutist solutions are generated (i.e. soundness). T1 (absolutism) T2 (comparativism)

P1 (possible models) P2 (possible models)

φ

S1 S2

NCMM’20a

Varying Newton’s ‘constant’?

“Leibniz Mass Scaling is ill-defined until we are told what happens to the strength of the law (as represented by Newton’s constant)” Motivations:

1

When changing mass units, we also change the units of G.

(Roberts, ms)

2

Empirical access only to G · m.

There is no problem: ceteris paribus clearly means keeping the laws the same.

Chain comparativism & Inter-world parsimony

(a) Web of relations (b) Chain of relations

? ?

(c) Removing a particle

Different chains lead to a plurality of distinct but indistinguishable worlds corresponding to each single Web-world → loss in inter-world metaphysical parsimony