## Abstract

The principle of mass additivity states that the mass of a composite object is the sum of the masses of its elementary components. Mass additivity is true in Newtonian mechanics but false in special relativity. Physicists have explained why mass additivity is true in Newtonian mechanics by reducing it to Newton’s microphysical laws. This reductive explanation does not fit well with deducibility theories of reductive explanation such as the modern Nagelian theory of reduction, and the a priori entailment theory of reduction that is prominent in the philosophy of mind. Nonetheless, I argue that a reconstruction of the explanation that incorporates distinctively philosophical concepts in fact fits both theories. I discuss the implications of this result for both theories and for the reductive explanation of consciousness.

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## Notes

- 1.
In special relativity the mass \(\hbox {m}_{\mathrm{c}}\) of a composite composed of \(N\) particles each with energy \(E_i\) and momentum \(p_i\) is \(m_c =\left[ {\left( {\mathop \sum \nolimits _{i=1}^N \frac{E_i }{c^{2}}} \right) ^{2}-\left( {\mathop \sum \nolimits _{i=1}^N \frac{p_i }{c}} \right) ^{2}} \right] ^{1/2}\). See Okun (1989, p. 632; 2000, p. 1271).

- 2.
The name comes from Dizadji-Bahmani et. al. (2010, Sect. 3.1) who contrast this view with Schaffner’s (1993, ch. 9) own so-called

*generalized reduction-replacement model*. The latter is stronger as it involves specific claims about bridge principles that the former stays neutral on. See also Schaffner (2012). - 3.
- 4.
“As regards active specialists they answer in perfect unison insofar as their scientific work is concerned: the mass is independent of velocity, it is not additive [...] there is no disagreement among researchers on the definition of mass. [...] According to modern terminology, both terms, ‘relativistic mass’ and ‘rest mass’, are obsolete” (Okun 2000, p. 1270). Compare Field (1973, p. 469) and Feyerabend (1962, pp. 80–1).

- 5.
- 6.
E.g. let two one-dimensional particles have unit mass, and be located two meters apart at \(-1\) and \(+1\) respectively so that the origin is at 0. In that case (15) puts the composite at 0. But move the origin so the coordinates of the particles are now 99 and 101 and (15) puts the composite at 200\(\sqrt{2}\).

- 7.
See Lewis (1983, pp. 370–373; 1984). Naturalness is not an external constraint on reference but is imposed by our conventions: Schwarz (2014). A formal account of the semantics of theoretical terms governed by such conventions can be given in terms of the so-called Unique Best Deserver semantic theory, see: Elliott et al. (2013, Sect. 4.1).

- 8.
The choice of subscripts comes from Dizadji-Bahmani et al. (2010).

- 9.
One might object that ‘composite’ is not contained in the reducing microphysical theory, but is clearly in the statement of mass additivity. But here one could treat unrestricted composition (or one’s preferred composition principle) as a logical or a priori truth. Alternatively one could follow Dizadji-Bahmani et. al. (2010, p. 404) and treat the correct composition principle as an entity association law that is part of the reducing theory, as opposed to a property association law, which can’t be.

- 10.
Even this is not clear given the Force Additivity Problem. Thus Nagelians require my solution to the Force Additivity Problem, or something similar, to work.

- 11.
- 12.
And indexical truths (p. 408)—a complication I ignore for present purposes.

- 13.
- 14.
To make the analogy even more vivid, let M\(_{\sim A}\) instead say ‘something is massive’ and let P stand for a simple microphysical description of some electromagnetic radiation composed entirely of

*zero-mass*photons, given in the language of special relativity. In that case (d’) is shown to be false in Gabovich and Gabovich (2007). - 15.
- 16.
Prima facie to deduce \(F_1 =-F_c \) just infer that C exerts \(F_1 \) on particle 1 in virtue of C’s components exerting \(F_1 \)on particle 1. Then apply the third law to C and particle 1 to deduce the force on C: \(-F_1 \). But the third law cannot obviously be applied to more than two particles: K&B (2004, p. 7) and the applicability of the third law to composites is something we should deduce, not assume. So let’s try another option.

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## Acknowledgments

I would like to thank David Chalmers, Dan Marshall, Tim Maudlin, Daniel Nolan, Jesse Robertson, Raul Saucedo, Craig Savage, Wolfgang Schwarz, Jonathan Simon, Michael Simpson, Jonathan Tapsell, and two anonymous referees, for helpful feedback. This publication was made possible in part through the support of a Grant from Templeton World Charity Foundation. The opinions expressed in this publication are those of the author.

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## Appendix: A solution to the force additivity problem

### Appendix: A solution to the force additivity problem

K&B successfully deduce (6) and therefore also: \(F_1 =-(m_2 +m_3 )a_c\). But to deduce (7) we must first deduce: \(F_1 =-F_c\). Here I provide a defensible suggestion.^{Footnote 16}

Recall (5): \(m_1 a_1 =-m_2 a_2 -m_3 a_3\). Substitution using the second law gives: \(F_1 =-\left( {F_2 +F_3} \right) \). So we must deduce: \(F_2 +F_3 =F_c\) or “the additivity of force” as that will give: \(F_1 =-F_c\). I do this in four steps. Step one deduces the claim that C exerts some force on particle 1. Step two deduces that this force must be \(F_1 =-\left( {F_2 +F_3} \right) \). Step three deduces the claim that particle 1 exerts some force on C. Step four deduces that this force must be equal but opposite to the force that C exerts on 1 i.e. \(F_2 +F_3\).

*Step one:* necessarily, removing C from the microphysical explanans removes particle 2 or particle 3 or both. This removes force \(F_2\) or \(F_3\) or both. But this reduces the force on particle 1, and so given 1’s constant mass and \(F_i =m_i a_i\), 1’s acceleration reduces. So C induces an acceleration on 1 so C exerts some force on 1.

*Step two:* deduce the value of \(F_{1c}\) i.e. the force that C exerts on 1. We prove by reductio that \(F_{1c} =F_1 =-\left( {F_2 +F_3} \right) \). We know that particles 2 and 3 exert \(F_1 =-\left( {F_2 +F_3}\right) \) which (by stipulation) is the *total* force on particle 1 and we know that \(F_{1c} \ne 0\) (from step one). Now assume that \(F_{1c} \ne -\left( {F_2 +F_3} \right) \). But if \(F_{1c} \ne -\left( {F_2 +F_3}\right) \) then the total force on 1 is not equal to \(-\left( {F_2 +F_3}\right) \), hence a contradiction. So C exerts the force that its components exert, that is: \(F_{1c}=F_1 =-\left( {F_2 +F_3}\right) \).

*Step three:* to deduce the claim that particle 1 exerts a force back on C we appeal to considerations similar to those used in step one. If we remove 1 we affect C’s acceleration. Since removing (or even changing the state of) particle 1 affects C in this way there is a force on C due to 1. The remaining question is what the value of this force is.

*Step four:* to deduce the value of \(F_{c1}\) we appeal to considerations similar to those used in step two. We prove by reductio that \(F_{c1} =-F_1 =F_2 +F_3\). We know (by stipulation) that \(F_2\) is the total force on particle 2 and that \(F_3\) is the total force on particle 3, and we know (from step three) that \(F_{c1} \ne 0\). Now assume that \(F_{c1} \ne F_2 +F_3\). Since there can be no changes in C without changes in particles 2 and 3 then \(F_{c1}\) must to some extent apply to particles 2 and 3. But then \(F_2\) is not the total force on 2 or \(F_3\) is not the total force on 3 (or both), hence a contradiction. So C experiences the sum of the forces that its components experience, that is: \(F_{c1} =F_2 +F_3\).

Since C is only interacting with particle 1, \(F_{c1}\) is the total force on C. So \(F_{c1} =F_c =F_2 +F_3\) such that \(F_1 =-F_c\). We now return to \(F_1 =-m_2 a_2 -m_3 a_3\) and substitute: \(F_c=m_1 a_1+m_2 a_2\) Assuming \(a_1 =a_2\) we then deduce (7) and solve the Force Additivity Problem. (Note that in deducing force additivity we have also deduced that the third law scales up to composites.) Thus, mass additivity is deducible from Newtonian microphysics when components have identical accelerations.

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McQueen, K.J. Mass additivity and a priori entailment.
*Synthese* **192, **1373–1392 (2015). https://doi.org/10.1007/s11229-014-0627-7

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### Keywords

- Reduction
- Explanation
- Mass additivity
- Nagel
- Newtonian mechanics
- A priori entailment
- Scrutability
- Consciousness