MASS COMPARATIVISM AND THE METAPHYSICS OF QUANTITIES
Yifan Li
A thesis submitted to the faculty at the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Master of Arts in the Department of Philosophy in
the College of Arts and Sciences.
Chapel Hill 2020
Approved by:
John.T.Roberts
Marc Lange
c
2020 Yifan Li
ABSTRACT
Yifan Li: Mass Comparativism and the Metaphysics of Quantities (Under the direction of John.T.Roberts)
Comparativism about quantity claims that certain physical quantities, such as mass in Newtonian
Mechanics, are grounded in extrinsic relations obtained among objects instead of intrinsic features of
individual objects themselves. Absolutism about quantity claims the contrary. In this paper, I raise
an argument against comparativism about mass in Newtonian Mechanics. My argument proceeds
by considering whether comparativism can be a part of a plausible general metaphysical theory of
mass. I describe two pairs of incompatible views on two related metaphysical issues about mass:
reductivism versus primitivism, and realism versus nominalism, and I show that there are serious
problems for each of general metaphysical theory we can construct by combining comparativism with
any of these views. My conclusion is that this shall give us a good reason to prefer absolutism over
comparativism for Newtonian mass as well as other physical quantities that are similar to Newtonian
TABLE OF CONTENTS
CHAPTER 1: INTRODUCTION . . . 1
CHAPTER 2: METAPHYSICS OF MASS: MAPPING THE LANDSCAPE . . 3
2.1 absolutism vs. comparativism . . . 3
2.2 realism vs. nominalism . . . 7
2.3 reductivism vs. primitivism . . . 9
CHAPTER 3: AGAINST REDUCTIVISM . . . 12
CHAPTER 4: AGAINST REALIST COMPARATIVISM . . . 18
CHAPTER 5: AGAINST NOMINALISM . . . 23
CHAPTER 6: CONCLUSION . . . 30
CHAPTER 1
Introduction
Quantities are, roughly speaking, attributes that admit of degree: it can be possessedto a greater
or lesser extent by an entity. For example, mass, charge, length, (temporal) duration, all of these are
quantities. Moreover, they are ascribed to all sorts of things in our attempt to describe, predict, and
explain the world: materials bodies have mass, particles have charge, rods have length, events have
duration, etc. It is fair to say that quantities play a central role in our scientific image of the world.
In light of the importance of quantities, it is natural that all kinds of metaphysical questions
would be raised about them. Shamik Dasgupta, in his seminal paper (Dasgupta 2013), does just
this thing. In that paper, he asks the following question: what does a quantity consist in, at the
most fundamental level? Does it consist in intrinsic, determinate properties, or dyadic/polyadic
properties (i.e., relations)? He dubbed the position which takes dyadic/polyadic properties to be
metaphysically fundamental comparativism, and the position which takes intrinsic, determinate
properties to be metaphysically fundamentalabsolutism. After laying out these positions, he set out
to defend comparativism in the case of mass in the context of Newtonian Mechanics.
Dasgupta’s paper opened up an active debate on the issue of absolutism vs. comparativism
about mass.1 In this paper, I will join that debate and argue for absolutism about mass. Following
other philosophers, I will assume that Newtonian Mechanics is true, and use it as the background
physical theory of my discussion; so to be precise, what I aim to defend in this paper is absolutism
about mass in Newtonian Mechanics. While I cannot say that the argument given in this paper can
be generalized straightforwardly to theories like General Relativity or every to other quantities, I
believe that the strategies I employ in this paper are nonetheless potentially applicable to argue for
absolutism about some quantities like mass in more realistic settings of background theories.
In the broadest stroke, my general plan for defending absolutism in this paper is as follows.
1
First, in the next section, I will lay out three pairs of opposing positions in the metaphysics of mass:
absolutism vs. comparativism, realism vs. nominalism, and finally, reductivism vs. primitivism.
As we shall see, each pair of these positions attempt to characterize one different aspect of the
metaphysics of mass. Therefore, if we let these positions recombine with each other, we will end up
with 8 theoretical possibilities in the metaphysics of mass. That sets up the stage for the second part
of my plan: an argument from elimination on these 8 possible positions which shows that absolutism
is the position we should take up. In other words, I will raise objections for these possible positions,
and thereby eliminating them until there is only an absolutist position left. More specifically, this
part of the plan comes in three steps: in the first step, which will take up section 3, I will develop an
objection against reductivism; in the second step, which will take up section 4, I will develop an
objection against realist comparativism; and finally, in the third step, which will take up section
5, I will develop an objection against nominalism. After these three steps, what is left in the 8
possible positions isprimitivist realist absolutism; since this is an absolutist position, the upshot is
CHAPTER 2
Metaphysics of Mass: Mapping the Landscape
As I have mentioned, in advancing my argument in this paper, I will relate absolutism and
comparativism with two other pairs of opposing positions in the metaphysics of mass: the first pair
is what I call realism and nominalism about mass, and the second, reductivism andprimitivism
about mass. In this section, I will clarify and explain each of these three pairs of positions to set up
the stage for the argument to unfold in the following sections.
2.1 absolutism vs. comparativism
Consider the proposition that my laptop is 2kg (at a time t). Suppose that it is true. A
true proposition like this is supposed to describe some portion of reality; we may say that what
fundamentally constitutes that portion of reality is whatultimately grounds the proposition’s truth.
Thus comes the question: what is the thing that ultimately grounds the truth of that proposition?1
Now the ground for that proposition might include particular objects: for example, my laptop (or
the fundamental particles that constitute it). It might also include facts: for example, the fact that
my laptop instantiates certain propertyX (or the fact that the fundamental particles that constitute
it instantiate certain properties). These claims can be verified by seeing that necessarily, whenever
my laptop does not exist, or that it exists but do not instantiateX, the proposition that my laptop
is 2kg would fail to be true. We can restate this modal connection between the proposition and its
ground in the language of possible worlds: for all possible worlds, the proposition that my laptop
is 2kg is true at a possible worldW, only if some of its ultimate grounds exist (for particulars) or
obtain (for facts) inW; moreover, if all of the ultimate grounds of a proposition exist or obtain in a
world, then the proposition must be true in that world.2
1
To avoid awkwardly long sentences, in what follows I will sometimes simply omit the term “ultimate” when talking about what ultimately grounds what.
2
We have said that the ground of the proposition that my laptop is 2kg might include some fact,
and this sounds pretty plausible. We can then ask: what are those facts? The disagreement between
absolutists and comparativists can be shown in their answers to this question. While absolutists
would claim that they are facts about monadic properties instantiated in the laptop, comparativists
would claim that they are facts about how the laptop is related to other objects. In general, then,
we may define absolutism and comparativism in the following way. First, let’s define mass-quantity
propositions as propositions that are either 1) predications of some particular mass magnitude to an
object, or 2) propositions entailed by propositions of type (1) under an appropriate physical theory.
We can then identify absolutism and comparativism as two theses concerning the nature of the facts
that ultimately ground the truth of every mass-quantity propositions: while absolutism claims that
those facts only involve monadic properties (“absolute mass”), comparativism claims that those facts
only involve dyadic/polyadic properties which hold between several objects.
Now perhaps for most people, absolutism is the default position here, because that seems to be
what we get if we take propositions such as that my laptop is 2kg at face value: if the proposition
that my laptop is 2kg is true, then there must be some monadic properties which are denoted by
the predicate “2kg”, which the object denoted by “my laptop” instantiates.3 However, in a recent
paper (Dasgupta 2013), Shamik Dasgupta argues that absolutism about mass implies the following
claim: a possible world which resembles the actual world in every aspect except that everything
is twice as massive would be a physically distinct possible world compared to the actual world.
However, there is no empirical method that allows us to tell the physical difference between those
two worlds. If, on the other hand, we adopt comparativism, we can avoid committing ourselves to
that. This shows that absolutism about mass posits undetectable physical structures in the world,
while comparativism may not posit those structures; thus, applying Ockham’s Razor, we have reason
world must always be the same set of entities; it can be the case that a proposition is grounded by two different sets of entities in two different worlds (as in the case of a disjunctive proposition).
3
to prefer comparativism over absolutism about mass.4
It should be noted that according to the definition of comparativism given in this paper,
Dasgupta’s argument is not really an argument for comparativismper se. This is because according
to the definition in this paper, not every comparativist theory about mass can guarantee that two
possible worlds that are empirically indistinguishable in the relevant ways are physically identical
in their mass facts; However, since Dasgupta’s argument accuse absolutists of committing to the
existence of empirically indistinguishable, but physically (specifically, mass-wise) distinct possible
worlds, it follows that some comparativist theories would be vulnerable to the same accusation. In
fact, the only comparativist theories that can avoid his accusation, and thus be supported by his
argument, are those theories which take the fundamental mass relations in question to be relations
which would remain invariant under “global mass-rescaling”. To be more precise, let’s say that a
worldW∗ is aglobal mass-rescaling of another worldW, if and only if there is a positive real number
k, such that for all propositions of the formm(x) =r, understood as “x has a mass magnitude ofr”,
ifm(x) =r is true onW, then m(x) =kr is true onW∗. Thus, if in this world it is true that my
laptop is 2kg, I am 65kg, etc., then a global mass-rescaling of this world would be a world in which
it is true that my laptop is (2k)kg, I am (65k)kg, etc., for a definite positive real numberk. Now
observe that some relational facts about mass that hold (or fail to hold) between some objects in a
world would still hold (or fail to hold) between the same objects in the global mass-rescaling of that
world. Facts about whether an object is more massive than another are good examples: the fact
that I am more massive than my laptop would not change, if we are just multiplying the numerical
value of our mass magnitudes by the same (positive) factor, so to speak. Let’s call relations that
are involved in these kinds of facts relations that are invariant under global mass-rescaling (for
short,invariant); thus, we see that the relation of “being more massive than” is invariant. Another
invariant relation, the one that figures most prominently in the debate between absolutism and
comparativism, is the mass-ratio relation, and we shall come back to it later. On the other hand,
4
some mass relations are not invariant: for example, the mass-difference-being-63kg relation. It is
a fact in this world that I am 63kg more massive than my laptop, but that fact does not obtain
anymore, if we multiply the numerical value of everything’s mass magnitude by, say, 2.
We have said that only comparativist theories whose fundamental mass relations are invariant
can avoid the problematic implication that there exist empirically indistinguishable but physically
distinct possible world, and we now know what that means. Consequently, Dasgupta’s argument
only supports favoring comparativist theories of that kind over absolutist theories. Recognizing the
importance of this fact, some philosophers in this debate propose that we should redefine the notion
of comparativism, so that it only includes theories whose fundamental mass relations are invariant
(Baker 2014). That is a reasonable idea; however, for my purposes, it does not really matter whether
we define the term “comparativism” more exclusively or not. This is because as long as it is clear
that theories whose fundamental mass relations are not invariant are actually not supported by the
main argument for comparativism, and are thereby not the positions I want to engage with in this
paper, our discussion here on the absolutism and comparativism debate would still be on the right
track, no matter what definition of “comparativism” we adopt. For this reason, I will stick to the
definition of absolutism/comparativism I propose at the beginning of this section.
Finally, to anticipate the argument I am going to make later in this paper, let’s return to
Dasgupta’s argument. It is an argument that relies on Ockham’s Razor; more specifically, the
ockhamist principle Dasgupta employs is an epistemic one, which says that other things being
equal, we should prefer theories which do not postulate empirically undetectable structures to
theories that do so.5 The principle itself is usually taken for granted in this debate; and this
leaves philosophers who wish to defend absolutism from Dasgupta’s argument two options. First,
they can try to show that the additional structure absolutism postulates is not really empirically
undetectable or otherwise empirically surplus6; alternatively, they can try to show that even if
absolutism implies that there are undetectable physical structures in the world, we nonetheless
still have reason to favor absolutism because it is better than comparativism in other aspects (for
5
Dasgupta articulates this epistemic formulation of Ockham’s Razor in (Dasgupta 2016).
example, absolutism about mass has more explanatory power than comparativism about mass)7.
My defense of absolutism falls into the second category. More specifically, I will try to argue that
the most plausible forms of absolutist theory, coupled with the Newtonian laws of nature, is a better
choice than comparativism overall, even if it does commit to the existence of undetectable structures.
Thus, a large portion of this paper will be devoted to the explanation and defense of my conception
of what a plausible form of absolutist/comparativist theory of mass is like; and that is where the
other pairs of positions (reductivism/primitivism and realism/nominalism) come into play. We shall
now turn to the explication of those positions.
2.2 realism vs. nominalism
The distinction between realism and nominalism about mass can be considered as a proper part
of a more general distinction in metaphysics between realism and nominalism about properties.8 As
a first approximation, realism about mass claims that facts about mass involve not only features of
massive objects, but also entities which exist over and above these things, namely mass properties,
andtheir features; by contrast, nominalism is the claim that facts about mass only involve features
of massive objects, and nothing exists over and above these massive objects.
The paradigm example of a mass realist is D.M.Armstrong, who holds that the reference of
predicates like “is 5kg” are universals (Armstrong 1978, Part 5, Chap. 18); the paradigm example of
a mass nominalist is Quine, who insists that only those massive particulars, which fall within the
range of our scientific theory about massive bodies, are to be admitted into our ontology (Quine
1953).
We may make precise our characterization of these two positions by employing the framework
we developed in the last section, which focuses on what constitutes the grounds for the truth of
mass-quantity propositions. This leads to the following definition: Suppose thatpis a mass-quantity
proposition. Then mass realism is true, if and only if the grounds forp’s truth in a world always
7Cf., (Martens 2017, Ch.4).
8Cf., (Loux and Crisp 2017, Ch.1 & 2). However, the terminology I use here is different from Loux’s terminology: my
include some property, which exists in all worlds;9 nominalism is true, if and only if the ground for
p’s truth does not always include some properties.
This definition implies a difference between realism and nominalism about mass which is going
to be important for the paper. To see this, let’s look at an illustrative case in the debate between
realism and nominalism about color properties. Consider the proposition that red is a color. For
realists, this proposition is true, just in case the property of redness is related to the property of
colorfulness in some way; and because the property of redness still exists in possible worlds with no
red things, the proposition that red is a color is still true, and that red is not a color is still false, in
those worlds. On the other hand, for nominalists, the question of when the proposition is true is
less straightforward. A prominent treatment here is to take that proposition as saying the same
thing as the following proposition: if anything is red, then it is colored. This way, nominalists can
argue that they can take the proposition that red is a color as true without committing themselves
to the existence of redness, because the proposition is really only about objects, but not about the
property of redness. However, this strategy has the implication that “red is a color” and “red is not a
color” are both true in worlds with no red things. Another possible treatment for nominalists here
is to take the proposition that red is a color as not equivalent to a material conditional, but the
followingsubjunctive conditional: if anything were red, then itwould be colored. The problem with
this approach is that it is not at all clear what can ground the truth of these subjunctive conditionals.
It might be the case that relations among color properties need to be brought in to serve as their
ground, which makes this approach incompatible with nominalism after all.
In the same vein, realists about mass can account for why some mass-quantity propositions are
true/false in all possible worlds by claiming that they are grounded in the features of properties;
however, for nominalism, they have to say that those propositions might have different truth values
in different possible worlds, depending on how many massive objects there are and what mass-facts
about them are true. For example, consider the proposition that 5kg is not the same mass as 6kg. A
9
realist might want to claim that “5kg” and “6kg” refers to different mass properties, and the truth of
this proposition is grounded in the fact that the property of 5kg and the property of 6kg are distinct
properties. Now because the distinctness between properties is presumably necessary, the proposition
that 5kg is not the same mass as 6 kg would be true in every possible world. However, a nominalist
who wants to avoid commitment to mass properties may need to construe that proposition as saying
that it is not the case that all 5kg objects have the same features as 6kg objects; however, it then
follows that in a possible world without 5kg objects or 6kg objects, this proposition would be false.
Alternatively, they could appeal to subjunctive conditionals about 5kg and 6kg objects to do the
work; but that move comes with the risk of undermining their nominalist commitment.
Now the above is only a toy example that is only supposed to illustrate the idea; in reality, a
realist or a nominalist might reject the interpretation of “5kg is not the same mass as 6 kg” ascribed
to them in above, and thereby avoid disagreeing over whether that proposition is necessarily true.
More generally, precisely which one of the mass quantity propositions are the ones over which a realist
and a nominalist disagree in the above way depend on the details of their theories. Nonetheless,
it seems to me that the existence of such disagreement about some mass-quantity proposition is
unavoidable. At the very least, this is the case for every nominalist theory of mass in the existing
literature, as we shall see in what follows.
2.3 reductivism vs. primitivism
Reductivism about mass is, roughly speaking, a thesis according to which the mass properties
that ground the mass-quantity propositions can be reduced to other physical properties. In the sense
of “reduction” I use the term here, to say that mass properties can be reduced to other physical
properties (for example, spatiotemporal properties) is to say that mass properties are nothing over
and above these (spatiotemporal) properties; in all possible worlds in which these spatiotemporal
properties are instantiated, the corresponding mass properties must thereby also be instantiated in a
determinate manner. It is in this sense of reduction that we see, for example, people claim that the
instantaneous velocity of objects is reducible to the time-derivative of their trajectory. Moreover, this
notion of reduction can be easily generalized to cover the reduction of entities other than properties:
objects, facts, etc. Therefore, our characterization of reductivism is easily generalizable to cover cases
in which we think the grounds of mass-quantity propositions are not mass properties, but entities of
that facts, objects, and whatever else which ground the truth of these propositions are reducible (in
the sense defined above) to some other facts, objects, or whatever else. By contrast, primitivism is
the thesis that these grounds for the truth of mass quantity propositions are irreducible; in other
words, they constitute a portion of our most fundamental reality.10
It might be questioned why we should be concerned about the issue of reductivism and primitivism
at all in this paper, given that the stated aim of this paper is defending comparativism. The quick
answer to this question is that there is a historically influential interpretation of Newtonian Mechanics,
i.e., Mach’s interpretation, that is both reductivist and comparativist about mass.11 Thus, we can
at least show that that particular comparativist theory about mass does not work, if we show that
reductivism does not work. More importantly, if assume that in a reductive theory, the reductive base
for mass is invariant under mass-rescaling, then a successful reduction of mass entails comparativism
about mass. This is because in that case, mass quantity-propositions would be only grounded in
mass-rescaling invariant facts, which fits into our definition of comparativism. This shows that
even though reductivism is logically independent with comparativism, reductivism is nonetheless a
viable strategy for comparativist to pursue their agenda: the thing they need to do is to find the
appropriate reductive base that satisfies the assumption, and then work out the details of how mass
is reduced to that base. Indeed, Mach’s interpretation of Newtonian Mechanics can be viewed in
this light: the reductive base Mach chose for mass, namely acceleration facts, seems to be exactly
those mass-rescaling invariant facts needed by a comparativist. So we see how the plausibility of
reductivism could be potentially an issue in the absolutism/comparativism debate.
So far, we have introduced and clarified three pairs of opposing positions in the metaphysics
of mass. To recall, they are 1) comparativism vs. absolutism, 2) realism vs. nominalism, and 3)
reductivism vs. primitivism. It should be clear by now that each pair of them is concerned with a
different aspect of the metaphysics of mass, and that it is at least apparently possible for them to
10
A question that might arise at this point is whether theories that take mass magnitudes to be some kind of dispositions counts as reductive or not. The answer here is that it depends on the details of the theories in question. If a theory treats the relevant dispositional facts at a world to be metaphysically determined by some other sort of facts (e.g. in theories according to which dispositions supervene on its activation), then the theory would count as reductive. However, if a theory admits the possibility that the relevant dispositional facts may change without the change of any other sorts of facts, then the theory would count as non-reductive. More on this later in the paper.
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freely recombine with each other to form 8 distinct possible metaphysical theories about mass. As I
have mentioned, recognizing this fact sets up the stage for my defense of absolutism. I will start
by defense by an argument against reductivism, because ruling out the possibility of a reductive
CHAPTER 3
Against Reductivism
This section aims at arguing against reductivism about mass. To begin with, let us first ask
ourselves the following question: why do we have the concept of mass in our physics in the first
place? Unlike shape, color, (relative) velocity, etc., mass is not something we can directly observe
with our perceptual apparatus; so why do we take it to be part of the furniture of the physical world?
The answer is that it enters into our view as a theoretical postulate of modern physics. A theoretical
postulate is often introduced in a physical theory to explain some phenomena; and in the case of
mass, it is originally postulated to explain the motion of objects when they are acted upon.1 This
suggests to us a way of evaluating reductivist accounts of mass: if it can reduce mass to something
else without making the motion of objects inexplicable or less explicable, then a reductivist account
should be counted as successful. Otherwise, we should reject a reductive theory of mass and accept
mass as a sui generis, primitive part of the physical reality, just like what we do with all other
theoretical postulates in other fundamental theories.
Following the above line of thought, in the rest of this section, I shall first present a case which
illustrates what sui generis mass facts can help explain, and then argue that reducing mass facts
to facts about other quantities would make some aspects of the case inexplicable. Finally, I will
consider and reject various reductivist responses to that point.
Consider the following example. Suppose that a billiard ball is at rest on a frictionless billiard
ball table. And then, at some later timet0, it felt a certain quantity of force (imposed by the strike
of a stick, say) for a short interval of time, and it moves around for a while until t1. Now let the
billiard ball’s trajectory on the pool table be described by the function f(t), t∈[t0, t1], which maps
from instants in time to spatial points on the pool table. Now we might ask: what fact causally
1
explains the billiard ball’s trajectory?
Presumably, that should include facts about the position of the billiard ball on the table, as well
as itsbeing at rest relative to the table. Furthermore, it shall also include the fact that a certain
quantity of force are felt by the billiard ball aftert0. However, these facts alone are not enough to
give a complete causal explanation of the billiard ball’s trajectory. This is because fixing the facts
about the position and the state of movement (velocity) of the billiard ball, and the force felt by it
after t0, it is still nomologically possible for the billiard ball to move a little bit faster, or slower, and
thus taking on a different trajectory, described by a function different fromf(t). Thus, these facts
taken together cannot causally explain why the billiard ball has the trajectory f(t), but not some
other trajectory. Consequently, a causal explanation of the fact the billiard ball has the trajectory
f(t) but not a different one needs to include some facts about the billiard ball in addition to its
position, velocity, and the force it feels. These additional facts are the mass facts about the billiard
ball. In another word, the postulation of sui generis mass facts about the billiard ball enable us
to give a much more powerful explanation of its trajectory: we can now explain why the ball has
the particular trajectory it has, but not any other trajectories that are nomologically possible for it
to have, given other physical facts (location, velocity, and force) about it. Finally, it is easy to see
how the moral we draw from the billiard ball example can be easily generalized to other physical
objects. The upshot is that a physical theory which postulates sui generis mass facts about objects
is significantly explanatorily superior to a theory which does not postulate such facts, and this should
give us a compelling reason to prefer an ontology withsui generis mass facts to a reductive ontology,
which does not recognize its existence.
There are several ways a reductivist may resist the above argument. First, they might reject the
thesis that fixing the location and velocity of the billiard ball att0 and the force it feels after that, it
is still nomologically possible for the billiard ball to have a different trajectory. However, the proposal
looks rather implausible. The dynamic laws of Newtonian Mechanics seem to straightforwardly imply
that the location, velocity, and felt force of the billiard ball at that time fail to determine a unique
trajectory for it aftert0. Given Newtonian Mechanics and the set up of the case, the trajectory of
the billiard ball after t0 entails that it has a determinate acceleration at the short interval after t0.
However, the value of that acceleration is undetermined by the location, velocity, and force felt of
the total force felt by the billiard ball is represented by F, and mis supposed to be an independent
variable about the billiard ball, the accelerationais therefore undetermined by the location, velocity,
and force felt of the billiard ball at t0.
Here is another way a reductivist might respond to the challenge. The above discussion shows
that the acceleration of the ball at the short interval aftert0, combined with the location of the ball
at t0, is sufficient to determine a unique trajectory for the ball; so a reductivist would have enough
resource to explain why the ball has its actual trajectory but not some other ones, if they take
into consideration the acceleration of the ball. Indeed, one might try to adopt Mach’s operational
definition of mass in terms of acceleration here, and then argues that an explanation of the trajectory
in terms of mass can be re-written as an explanation in terms of acceleration.2
However, the problem with this line of thought is that facts about the acceleration of the
ball during the short interval after t0 are not sui generis facts independent of the instantaneous
velocity of the ball at instants aftert0. In general, acceleration is a measure of the rate of change
in instantaneous velocity of an object; this is shown in the fact that the acceleration of an object
is represented mathematically as the time-derivative of the object’s instantaneous velocity. Now
instantaneous velocity is either simply the measure of the rate of change in an object’s position over
time, or it is asui generis property which explains the change of an object’s position over time.3 In
the first case, the acceleration of the ball at the short interval after t0 is ultimately reducible to the
trajectory of the ball in that interval; this shows that an explanation of the complete trajectory of
the ball after t0 with its acceleration at those times is circular. In the second case, the acceleration
cannot be used to explain the instantaneous velocity of the ball at the instants in the short interval
aftert0 on pain of circularity; however, since it is these velocity facts that explain the trajectory of
the ball throughout that interval, it follows that acceleration cannot explain the trajectory of the
ball throughout that interval either. The upshot is that adding acceleration into the picture does
2
Cf., (Mach 1915). Speaking precisely, the operational definition Mach gives is ofmass-ratio, and it is not straightforward how that can be used here to explain the trajectory of the ball, where there is only one object involved. Moreover, there are also questions concerning whether Mach’s operational definition is even adequate in worlds with more than two particles; see (Martens 2018).
3
not help reductivists to explain the trajectory facts that can be explained by the postulation of sui
generis mass.
Perhaps reductivists can claim that acceleration facts are not reducible to instantaneous velocity
facts, which would allow for the possibility that the acceleration facts about the ball during the
short interval aftert0 can be used to explain the trajectory of the ball during that interval without
circularity. This is indeed a possible position to take; however, I have not seen it defended in
the literature, and it seems that it is an ad hoc position motivated only by the need to save the
reductivist theory. More importantly, this move seems to be in tension with the main consideration
that motivates reductivism in the first place: the minimization of non-observable primitive entities
postulated in our physical theory. In a non-reductivist Newtonian Mechanics with a primitive mass,
accelerations can be defined as the time-derivative of velocity, and is thus ontologically innocent;
however, if reductivists claim that they can reduce away mass, but only with the cost of turning
acceleration into a primitive notion itself, then acceleration would be a genuine ontological price we
need to pay for reducing mass. Moreover, this primitive acceleration, just like primitive mass, is also
an entity that is not directly observable. This is because it is an attribute objects have intrinsically,
and it is connected to the change of position of objects by some sort of law of nature. However,
this means that its connection with an object’s movement is no different than primitive mass: so if
primitive mass is not directly observable, neither is primitive acceleration observable. But then, the
reductivist suggestion of reducing mass by postulating primitive acceleration starts to sound more
like a pushing away of the problem rather than a solution to it. For these reasons, I will not pursue
this possibility further here. Let’s turn to another reductivist strategy to resist the argument from
the billiard ball.4
In the billiard ball example, we require reductivists to explain the trajectory of the ball using
only the location and velocity of the ball at t0, plus the force felt by the ball at a short interval
aftert0. However, a reductivist might want to include more things into the potential explanans: for
example, it should include the location, velocity, and force felt by the ball at instants prior to t0.
On the face of it, this is permissible: the reductivist is still only relying on spatiotemporal and force
4
facts about the ball, which are things they allow in their ontology. Moreover, since they are only
considering facts which obtain beforet0, there is no worry that its causal explanation of the ball’s
trajectory, which exists aftert0, would be circular. Most importantly, it seems that these additional
data could enable a reductivist to give a complete explanation of why the ball has the trajectory it
actually has, by ruling out all the other trajectories as nomologically impossible: this is because
they can give a definition of the mass of the ball at a time prior to t0 with its force and acceleration
at that time, and since in Newtonian Mechanics the mass of the ball does not change over time, it
follows that the force and acceleration of the ball at earlier times would determine how the ball
would move after t0 under any given quantity of force.
Now mathematically speaking, this is correct. It is indeed the case that given the data about the
force and acceleration of an object at a time, we can calculate how much the object would accelerate
under a given quantity of force at a later time, thus working out its trajectory after that. Indeed, as
Niels Martens remarks when dealing with a slightly different problem (Martens 2018), this better be
true, for otherwise there would be no way for us to figure out the mass of an object by measuring
its acceleration under force. However, this does not show that we can simplyexplain, rather than
predict, the trajectory of an object in terms of its previous movement under force. For one thing,
even if an object has not felt any (non-zero) amount of force before a certain time since the beginning
of the universe, there still seems to be an explanation of the object’s trajectory, when some force
applies to it for the first time and causes it to accelerate in a particular rate. For example, consider
the billiard ball again. Now suppose that at every instant before t0, the billiard ball has been at
rest on the pool table. This seems to be a possible situation; however, in that case, the force and
acceleration of the ball at instants prior tot0, being zero at every single instant, would not be able
to determine a unique mass for the billiard ball, and therefore not be able to explain why the ball
has the trajectory it in fact has after t0, but not some other trajectory.
Moreover, even if we reject the possibility of that scenario, and accept that we can define the
ball’s mass in terms of its accelerations at earlier times together with the force it feels at those times,
we may still want to ask what explains the trajectory of the ball after those earlier times. As we
have articulated in our previous discussion, given such an instant earlier in time, the acceleration of
the ball at that instant needs to be understood as the measure of change in instantaneous velocity at
after that instant on pain of circularity. Consequently, it seems that we have to rely on the force and
acceleration fact about that ball atyet another earlier time. Tracing up this chain of explanation
in time, we would either have to say that there are infinitely many instants in the past when the
ball experience non-zero force and acceleration, so that the trajectory of the ball after each instant
can be explained by the force and acceleration of the ball at an earlier instant, or we would have
to say that there is an instant in time after which the trajectory of the ball is unexplained. In the
first case, we seem to be making substantive assumptions about the physical structure of the world
that is ill-motivated; in the second case, we admit the existence of some facts which the reductivist
theory will have failed to explain (and which can be easily explained by postulatingsui generis mass
facts). Either way, the reductivist theory turns out to be explanatorily inferior compared to its
non-reductivist counterpart.
The above discussion has covered a wide range of reductivist proposals that have appeared in the
literature; the upshot is that none of them can successfully eliminatesui generis mass facts from our
ontology without at the same time depriving our physical theory of some of its explanatory power.
There might still be other possible reductive proposals that are not covered by the discussion here;
however, I hope that the discussion here is at least enough to shift the burden of proof to reductivists
to make the case that some of these proposals are immune to the problems raised here.5 So for now,
let us put reductivism about mass aside, and proceed to examine non-reductive theories of mass.
5Zee Perry defends what she calls adynamic-reductive account of mass in (Perry 2016, Ch.4). That account is immune
CHAPTER 4
Against Realist Comparativism
In this section, I will be arguing against (non-reductive) realist comparativism. Recall that a
mass realist, according to the characterization given in the first part of this paper, holds the grounds
for mass-quantity propositions include facts about mass properties, which exist as something over
and above the massive objects that instantiate them. Thus, a realist comparativist would be one
who holds that these mass properties are relations (dyadic or polyadic properties) instead of monadic
properties.
Now although realist positions about mass seem most naturally aligned with absolutism, a realist
comparativist position is nonetheless possible. In fact, the metaphysical theory about mass defended
in (Bigelow and Pargetter 1988) seems to be exactly one of this kind. In that paper, Bigelow
and Pargetter develop a “three-level” theory of mass, according to which massive objects stand in
mass-relations with each other, and those mass-relations themselves stand in various second-order
relations of “proportion” with each another.
To keep the discussion concrete, I will focus on Bigelow and Pargetter’s theory in developing
my objection to realist comparativism. However, the points I raise should be generalizable to other
plausible realist comparativist theories as well.
The objection I am going to develop can be summarized as follows. We have mentioned before that
the main motivation to be a comparativist about mass is that it avoids postulating the undetectable
surplus structure that absolutists are supposedly committed to. However, we shall see that if you
are a realist comparativist, you would also commit yourself to the existence of surplus structure that
does not make any empirical difference; the upshot is that, realist comparativism undermines itself
by taking aways its main motivation.
Now I will unwrap the objection in more detail. Consider a world with three particlesx,y, andz,
and suppose that the mass-ratio betweenxandyis1 : 2, the mass-ratio betweenyandzis1 : 3, and
mass facts that ground the truth of these three mass-quantity propositions are the fact that the
pairs of objects hx, yi,hy, zi,hx, zi each instantiates a different relation (call themR1,R2, andR3,
respectively), and that each of these relations themselves is associated with different second-order
properties of proportions. So far, so good. However, notice that the following is true: given that
(a comparativist version of) Newtonian Mechanics governs the movement of objects in this world,
the fact thatR1 holds betweenhx, yi, and R2 holds betweenhy, zi entails that the universe would
behaveas if R3 holds betweenhx, zi.
Let me explain. First, let us consider a world with only x, and y but no z, and suppose that
R1xy. Moreover, suppose that the world is governed by Newtonian Mechanics. This seems to allow
us to infer howx andy would behave in various ways: for example, we can infer that 1) if we let
them move towards a spring attached to a wall with the same velocity, the distance y compresses
the spring would be twice as much asx. The situation is similar if we consider a world with onlyy
andz but nox: the fact thatR2yz, together with Newtonian Mechanics, entails that 2) if we let
these two move towards a spring attached to the wall with the same velocity, the maximal distance
z compresses the spring would be three times more thany.
Now let’s re-combine these two worlds, such that we have x, y, z together, withR1xy andR2yz.
The inference that we can make based on this information should be the same: namely, we should
be able to infer (1) and (2). However, (1) and (2) together also entail that 3) if you let x and z
move towards a spring with the same velocity,z would compress the spring six times more thanx.
Moreover, since, under the assumption of Newtonian Mechanics, you can reach (3) from (1) and
(2) alone, and (1) and (2) can in turn be inferred from the fact that R1xy and R2yz obtains, it
seems that you can conclude (3) from the facts about the mass-relations that obtain betweenhx, yi
and hy, zi. Now we typically take the phenomena described by (3) to show that some determinate mass facts obtain betweenx andz; however, what we have seen is that we don’t actually need to
postulate such mass facts to predict that the phenomena described by (3) would occur, because the
existence of other mass facts (namely, R1xy andR2yz) are enough for making that prediction.
Now this conclusion has a limited scope, because (3) is a proposition concerning the behavior of
x andz when they move towards a spring; moreover, in the whole time, we are just thinking about
three particles. However, it is not hard to see how this argument can be generalized to any possible
in a populated world. Consequently, in a world with many massive objects, for any observable
phenomena that could show that a mass-relationRi holds between a pair of massive objectsha, bi, the phenomena can be guaranteed (by laws of nature) to occur on the basis of some fundamental
mass facts about other pairs of objects alone. The upshot of this is that as comparativists claim that
all detectable facts about mass are invariant under global mass rescaling, we may also claim that all
detectable facts about mass areinvariant under global mass shrinking: in a realist comparativist
world where there are fundamental mass-relation facts that hold between each pair of objects, any
measurement process that we think can tell us what mass relation obtains between two objects is
guaranteed (by the laws of nature) to produce the same measurement results, even if we “shrink” our
ontology by getting rid of the fundamental mass-relation fact that holds between the pair of objects
we are measuring, as long as we keep a sufficient number of other fundamental mass-relation facts
which obtain between other pairs of objects intact.
This gives rise to a dilemma: either realist comparativists take all the mass-relation facts, which
covers every pair of objects in a world, to be equally fundamental, or they only take a subset of these
facts which are enough to allow us to infer all observable phenomena as the most fundamental. If
they take the first option, then they would commit themselves to empirically undetectable structures,
because all the observable phenomena can be predicted with only a subset of these mass-relation
facts; if they take the second option, then, since in general there could be multiple possible subsets
of these facts that is sufficient to let us infer all the observable phenomena, and that we have no
way to choose between these subsets, arbitrarily taking one of them as the most fundamental is, like
the postulation of absolute reference frame, another way of postulating empirically undetectable
structure.
A realist comparativist who would like to take the first option might try to resist the claim that
some surplus structure is postulated. They might argue that in a world withx,y, andz, ifxis twice
as massive asy andyis three times more massive asz, then ifz’s mass-ratio withxis different from
6, Newtonian Mechanics would give inconsistent predictions regarding how these particles would
behave. Thus, it is impossible forx to be twice as massive asy,y to be three times more massive
as z, and z not to be six times as massive as x. Therefore, there would not no possible world in
which R1xy andR2yz hold while R3zxdoes not hold. This shows that even if the fact thatR3zx
it comes necessarily with the fact of R1xy andR2yz, which are indeed required for explaining and
predicting empirical phenomena.
However, this response cannot work, because it is not necessary that R3zxholds together with
R1xy and R2yz. It is true that on pain of making the laws inconsistent, we cannot posit that
any mass relation obtains betweenhz, xi other than the relation R3; however, the laws would not
be inconsistent, if we claim that no mass-relation hold between hz, xi at all. If no mass relation holds between hz, xi at all, then it cannot entail that any observable phenomena would occur; consequently, the law would be in no danger of inconsistency this way. Now this does not mean that
the mass-quantity proposition of “the mass-ratio betweenz and x is6 : 1” is false; it might simply
be lacking in true value because the notion involved is not well-defined. Another possibility is that
the proposition is true; it is just that it is not grounded in any facts about the fundamental mass
relation that holds between zand x, but rather grounded in the fact of R1xy and R2yz. Thus, the
reductivist cannot reject this hypothesis by saying that it leads to the counter-intuitive situation
where mass-quantity proposition like “the mass-ratio betweenz and xis 6 : 1” should be considered
false.
An analogy with the notion of the mass of a composite object might help here.1 Consider two
elementary particles, each of which has a mass of 1kg. Now let the composite of the two particles to
be A, and we can see that the mass-quantity proposition thatA is 2kg is true. However, in order to
ground the truth of this proposition, we don’t have to postulate that there is some fundamental
mass fact that involve the compositeA; in fact, we can let the ground to be simply the fundamental
mass facts about the two elementary particles that constitute A. Moreover, denying that there
are “emergent” fundamental mass facts about A would not pose any problem for the law at all.
Although it is true that now the law cannot directly be applied to that emergent fact to predict
and explain the evolution of the two-particle systemA, it can still be applied to the two particles
individually and make the same prediction and explanation. Moreover, by refraining from positing
emergent fundamental mass fact for the composite, we manage to avoid exposing the law to the
danger of inconsistency: because if the emergent mass fact does not make true the mass-quantitative
1
proposition thatA is 2kg, the law would entail contradictory predictions about the evolution ofA,
and thus failing to be consistent. I hope that now we can see the similarity between the situation of
the composite object here and the situation of R3zxwe discussed a few paragraphs back: because
it is at least plausible that we can deny that there is any emergent fundamental mass fact about
a composite object beyond the fundamental mass facts about its atomic parts, it should also be
plausible that we can maintain that R1xy and R2yz holds while at the same time deny that R3zx
holds. But this seems to show that it is after all possible forR1xy andR2yz to obtain in one world
withoutR3zxobtaining. The upshot is that the response that it is impossible for R3zxto not obtain
together with the other fundamental mass-relation facts fails.
Let’s recount what has happened in this section. The main topic of this section is realist
comparativism about mass, the position according to which the fundamental mass facts that grounds
mass-quantity propositions involve facts about mass properties which exist over and above massive
individuals, and that these mass properties are dyadic/polyadic instead of monadic. Using Bigelow
and Pargetter’s version of the theory as a model, I argue that realist comparativist theories about
mass would eventually commit to the existence of physical structures that are undetectable, and
thereby undermine its own primary theoretical motivation. I take this to be a strong reason for us
to put realist comparativism off our table for the time being; that accomplished the second step of
my plan of defending absolutism. In the next section, I will execute the final step of my plan and
CHAPTER 5
Against Nominalism
In this section, I will present an argument against (non-reductive) nominalism about mass. There
are many prominent (comparativist) nominalist theories about mass out there in the literature:
the influential measurement-theoretic account of mass given in (Krantz et al. 1971) is one such
theory; the account of mass in Newtonian Gravitational Mechanics defended in (Field 1980) is
another prominent example. Now these nominalist theories about mass in the literature have some
similar features. First, they attempt to ground the truth of mass-quantity propositions onqualitative
relational facts about massive objects; second, thetype of qualitative relational facts about massive
objects they incur is finite. These two features of nominalist theories would play a crucial role in my
argument against them; therefore, here I shall say something to show that these features are not
merely accidental features that do not really matter, but rather crucial for a nominalist theory to be
plausible.
The first feature is that the fundamental mass facts postulated by these theories are qualitative,
but not quantitative: this means that the (relational) predicates that are used to describe these
fundamental facts do not involve any indispensable reference to numbers, or other mathematical
entities. Thus, the fact described by the sentence “y is no less thanx in terms of mass”, or “there
is some z such that z is the mass-sum of x and y” are qualitative facts; on the other hand, facts
which can only be described with the help of numbers or other mathematical entities, such as the
fact described by the proposition “x is twice as massive asy”, are quantitative facts.
The claim I want to make here is that taking quantitative facts to be fundamental is not a viable
way of developing comparativism. The problem with taking quantitative facts as fundamental is that
it implies that what grounds the truth of mass-quantity propositions are unacceptably external to the
objects themselves. To see this, notice that a quantitative predicate which contains an ineliminable
reference to mathematical entities is not really a predicate that only applies to massive objects,
that only involves an individual object and its spatial location. In fact, the predicate of “being
in the philosophy department” is in reality a relational predicate which applies to the philosophy
department and some object; likewise, a quantitative predicate, like “being twice as massive of”, is
really a three-place relational predicate which relates two massive objects with a particular number,
namely 2. Therefore, taking quantitative relational facts about massive objects as fundamental
seems to commit us to the existence of numbers (or whatever mathematical entities that are brought
in). This is at least a significant ontological price to pay. Moreover, numbers are supposed to be
abstract objects that do not exist in space and time; but then, how is being related to that kind
of things somehow grounds the truth of propositions like “my laptop is 2kg”, which seems to be
describing some fact that is solely about my laptop? Moreover, numbers are also supposed to be
things that are not causally efficacious; if that is so, how could it be involved in the explanation of
how the laptop’s interaction with other bodies (e.g., its gravitational attraction to other bodies)? All
these are serious problems with taking quantitative relational facts to be fundamental. Given that,
we should see that the correct way of understanding the relationship between numbers and mass
(and indeed any physical quantities) is the other way round: numbers are not part of what grounds
the truth of mass-quantity propositions; rather, they are representational devices we use to express
these propositions, whose truth are grounded solely in facts about physical objects themselves.
So we have shown that for a nominalist theory about mass, the fundamental facts that ground
the truths about mass should be qualitative, rather than quantitative. How about the second feature,
that there are only finitely many types of qualitative facts? This requirement amounts to the
requirement that the number of primitive predicates is finite. Now it is true that a theory which can
account for a portion of reality with only finitely many primitive predicates is simpler, and thereby
(other things being equal) preferable than one which needs to bring in infinitely many primitive
predicates: thus, we think that a physical theory which postulates only 4 kinds of fundamental
particles and interactions is simpler, and thereby (other things being equal) theoretically more
virtuous than one which postulate infinitely many kinds of particles, each of which interacts with
another in its own ways. This pursuit of simplicity presumably also explains why both Krantz et.
al. and Field’s theories mentioned above include only finitely many primitive predicates. So that is
one consideration to think that we should require a nominalist theory to have only finitely many
about mass with infinitely many primitive predicates would be like. For example, in order for the
theory to be empirically meaningful, it needs to postulate laws or generalizations which allow us to
link these primitive predicates to observational phenomena. However, this seems to imply that the
theory needs to postulate infinitely many laws or generalizations, because there are infinitely many
primitives. This looks extravagantly non-parsimonious. Perhaps there are ways to work out the
idea of a theory with infinitely many primitive predicates that avoids those difficulties mentioned in
above: however, in the absence of such attempt, I shall conclude here that the fundamental facts
about mass be of finitely many kinds is a requirement for any acceptable nominalist theory about
mass.
Now the stage has been set, in what follows I will begin my presentation of the argument against
nominalism.
Recall that nominalism about mass accepts an ontology that is deprived of facts about massive
properties; for nominalists, the only relevant facts that ground mass-quantity propositions are facts
about massive objects. This entails that propositions that describe the quantitative relations between
the mass magnitude of two objects are also grounded in qualitative relations that hold between
massive objects alone. Now that kind of propositions come in many varieties: for example, it includes
propositions that describe the sum, difference, or ratio between the mass magnitude of any two
objects. Moreover, the facts described by these propositions are physically meaningful: the ratio
between the mass magnitude of two billiard balls, for example, is part of the explanation of their
behavior when they collide with each other.
Now the complex pattern of quantitative relations among mass magnitudes implies that whatever
fundamental mass facts ground these mass-quantitative propositions, these facts too must have a
certain kind of structure to “take care of” (so to speak) the pattern. For a nominalist about mass,
this requirement boils down to the requirement that the totality of fundamental mass facts about
massive objects must have a certain structure.
To illustrate this point, consider the representational measurement theory developed in (Krantz
axioms that set some general constraints on the patterns of-facts and◦-facts onA. For example, one of the axioms requires thatis a weak semi-order; another one, called the “Archimedean axiom”, states that for anyx, y in A, if xy, then there must exist some concatenation of a finite number of perfect copy ofx, signified bynx, such thaty≺nx(where≺is the strict counterpart of). That this structure of facts of massive objects is adequate in grounding the mass-quantitative propositions
is shown by the provability of aRepresentation Theorem, which states that there is a homomorphism
that maps hA,,◦ito the semigroup ofR+ which is unique up to arbitrary multiplication.
One way of understanding the representational theorem is to see it as telling us the following:
given a unit for mass (e.g., kg), each truth of the form “x ism kg” is grounded in facts about howx
and the concatenations of xwith other objects are positioned in the mass-ordering of all massive
objects; moreover, for every possible value of m, there is a unique way in whichx is related to other
objects by ordering and concatenation which grounds the proposition “x ism kg”. This means that
the structure of massive objects is discriminatory enough, such that “x is m kg” and “x is n kg”
would always be grounded by different fundamental mass facts about x if m andn are different
numbers.
So far, what we have said about the representational theorem seems to confirm the idea that the
structure of massive objects postulated by Krantz et.al.’s theory can adequately take care of the
complexity of the pattern of true mass-quantitative propositions. There are a lot can be said about
that; however, for the sake of argument, let’s just take it for granted that the structure of massive
objects is indeed adequate. Now let’s go back and consider the structure itself. One interesting thing
about it, pointed out in (Mundy 1987), is that the axioms which set constraints on the patterns of
-facts and◦-facts onA entails thatAmust contain infinitely many elements; sinceA is interpreted
as the collection of all massive objects, we can then see that the axioms require that there are
infinitely many massive objects.1
Upon further reflection, this should come as no surprise. Suppose that A has a finite cardinality
of k. It follows that for an element ain A, there are at most k2 ways in which ahas a different
-order relation with at least some of the elements inA; and there are at mostk3 ways in which
1
a’s concatenation (◦) with some element in A can vary. This entails that there are only finitely many ways in whichacan be related differently to other elements inA by-order or concatenation. However, there are infinitely many distinct possible values that can be associated with the mass
magnitude of an object; so if we claim that the mass-quantity proposition which ascribes a value to
a’s mass magnitude is grounded in the-facts and◦-facts about a, then we would have to concede that there would not be enough candidates of grounds to take care of every possible value a’s mass
magnitude can take, if there are only finitely many massive objects.
Now this “richness problem” of the domain of massive object is presumably a general issue faced
by all otherwise plausible nominalist theory about mass.2 Nominalists want to ground the truth
of mass-quantitative propositions on facts about massive objects. However, for any object, there
are infinitely many sets of mass-quantitative propositions that can be true of it; moreover, as we
have argued above, a plausible nominalist theory about mass should only postulate finitely many
kinds of qualitative fundamental mass facts. These two requirements seem to imply that for the
facts about massive objects to be able to ground the infinitely many possible true mass-quantitative
propositions, the domain of massive objects must be rich: in particular, every nomologically possible
world need to contain at least an infinite number of massive objects, as opposed to merely finitely
many, because otherwise the structure of massive objects would not be rich enough to ground the
complex pattern of mass-quantitative propositions.
However, the problem here is that whether the world contains infinitely many massive objects
does not seem to be a matter of any kind of necessity. It does not violate any laws of nature (or laws
of metaphysics or laws of logic) to conceive of a universe that only contains, say, three point-particles
with each particle having some arbitrary value of mass magnitude.
By contrast, a realist about mass can easily avoid this problem about richness. This is because a
realist can always postulate a rich domain of mass properties; since the existence of mass properties
is independent of the existence of massive objects, even in worlds with only a few objects, there can
still be infinitely many distinct mass properties, and the object is free to instantiate any one of them
and make true different mass-quantitative propositions. Consequently, a realist can claim that even
2
if we look at worlds with only a few objects, objects in this world can still possess infinitely many
distinct mass magnitude values, simply because there are infinitely many mass properties the objects
in these worlds can instantiate.
So we see that nominalists are faced with a difficulty here. This difficulty leaves open two options
for them. First, they can maintain that richness assumptions are only contingently true, and try to
show that it is not problematic. However, this entails that worlds with an ontology with only finitely
many massive objects, frugal worlds, are nomologically possible. In these frugal worlds, because
of their frugality, the structure of fundamental mass facts is correspondingly simpler: as we have
discussed, if we think about a particular frugal world, then we shall see that there are only finitely
many different ways the total mass facts about this frugal world can vary. However, even in these
frugal worlds, there are still infinitely many trajectories the objects in the world can take when they
feel a certain quantity of force. This feature about the possible trajectory of objects is determined
by the geometrical features of Newtonian spacetime, so the fact that these objects inhabit in a frugal
world does not matter for that. Combining these two theses, we can reach the conclusion that there
are some frugal worlds with, say, three objects, such that 1) there is an instant in which the position
of these objects, the force felt by them, and the fundamental mass facts about them all remains
the same, and 2) the nomologically possible paths of the objects after that instant are different.
The upshot is that Newtonian Mechanics fails to be deterministic for these possible worlds, because
two worlds with exactly the same initial fundamental mass, force and position facts can evolve in
different ways, exhibited by the different possible trajectories of objects in them.
This failure of determinism is itself a point that should raise some concern. Moreover, it also
suggests that nominalists’ version of Newtonian Mechanics suffers from explanatory failure. For
example, consider a billiard ball A in a frugal world. Imagine that the ball is hit by a stick, and
it moves according tof(t). When you ask, why does this billiard ball move according to f(t), but
notf0(t), which denotes some different trajectory? The answer to this question must include the
fundamental mass facts about the ball; however, if there are only finitely many different fundamental
mass facts about the ball in worlds that are equally frugal, then there must exist somef0(t)for which
this question cannot be answered, since f0(t) would be consistent with the fundamental mass facts
about the ball according to Newtonian Mechanics. This shows that our physical theory of the billiard
it actually has as opposed to one that it could have; that should be viewed as an explanatory
failure of the theory. In conclusion, if nominalists want to maintain that richness assumptions are
only contingently true, then they are committed to the claim that 1) Newtonian Mechanics is not
deterministic, and 2) Newtonian Mechanics is not as explanatorily powerful as we take it to be.
The second option for the nominalist is to grant richness assumptions the status of laws of
nature. This is a revisionary move: we normally would think that how many objects in a world is
a contingent matter. Indeed, it seems that many of our theoretical discussions about Newtonian
Mechanics are restricted to its finite models, because of their relative simplicity. Nonetheless, the
more problematic aspect of this move is that it does not fully address the problems nominalist
Newtonian Mechanics has: it still leaves the theory with diminished explanatory power compared
to its realist counterpart. This is because the geometrical structure of Newtonian spacetime still
obtains in a frugal world, and it therefore still makes sense to ask why in those worlds, massive
objects move in certain ways but not some other ways. The question is still there, and it seems that
a uniform answer of these questions across frugal worlds and rich worlds would be an explanatorily
desirable feature. However, relying on the richness assumption as a law of nature would not respect
this requirement of explanatory uniformness. Perhaps one way to avoid this charge is to give an
alternative geometrical theory of Newtonian space and time, according to which the structure of
the spacetime depends on the structure of the mass facts in such a way that frugal worlds don’t
even have the same amount of geometrical structure than a rich world. But again, in the absence of
an actual formulation of this kind of theories, it is unknown whether it can satisfactorily solve the
CHAPTER 6
Conclusion
In this paper, I have given an argument in support of absolutism about mass in the context of
Newtonian Mechanics. My argument involves an examination of the various ways absolutism and
comparativism can feature in a comprehensive metaphysical theory of mass: I first introduce two
other pairs of opposing metaphysical positions about mass, namely realism vs. nominalism and
reductivism vs. primitivism, and suggest that these three pairs of positions together give rise to
8 possible metaphysical theories about mass; I then proceed to present objections to reductivist
theories, non-reductive realist comparativist theories, and finally non-reductive nominalist theories,
and show that we have reason to avoid all of them. This leaves us with non-reductive realist
absolutist theories on the table, and I conclude that this gives us reason to prefer absolutism over
