## 1. Varieties of Fundamentality



There are many senses in which a thing may be said to be
fundamental--some technical, some relatively intuitive. A very
common way to think about fundamentality is in terms of independence,
whereby for any notion of dependence *D*, an entity is
*D*-fundamental if and only if it does not
depend*D* on anything else (or on anything else that
does not depend on it). This independence-based characterization of
fundamentality will be discussed in sections
 1.1
 and
 1.2.
 There are also other ways to understand fundamentality; these include
fundamentality as a complete description of reality
 (section 1.3)
 and as primitive
 (section 1.4).
 So, there are many ways to understand fundamentality and whether
there is any single idea of fundamentality that these different ways
are trying to capture is a substantial issue. But even if there is no
unified sense of fundamentality, one interesting question is whether
there are any fundamental entities, where "fundamental" is
understood in one of the different ways that will be specified below.
We might also ask whether we need such entities, whether given
theories are committed to their existence, and what the role of
fundamentalia is in explanation. Before we get started, a few
preliminary issues need to be mentioned.



For present purposes, we are interested in the notion of a fundamental
*thing*, or type of thing. The candidates for fundamentality
may include objects, such as electrons, but they may also include
properties, or facts. The choice regarding the relevant type of entity
may depend on one's preferred account of fundamentality.
However, there is often an acceptable translation between different
 views.[1]
 For instance, let's say that we take the fact <electrons
have unit negative charge> to be fundamental. Someone who does not
wish to ascribe fundamentality to facts (e.g., because they do not
have facts in their ontology), could understand this to be saying that
the property *unit negative charge* is fundamental and
instantiated in electrons. The translation here is from *electrons
have unit negative charge* to the fundamental property *unit
negative charge* that is instantiated in electrons (and only
them). So, the disagreement between those who consider fundamentality
to concern facts and those who regard objects and properties as
fundamentalia may not be as serious as it looks.



One important distinction in the usage of the notion of fundamentality
is worth highlighting: fundamentality as a general metaphysical notion
and fundamentality in physics and physical theories (see Morganti
2020a, 2020b; Allori 2022). These two ways to use the notion are of
course related, but the focus of this entry is on the first, broader
notion of fundamentality. The remit of the latter notion includes
questions such as whether physics is a more fundamental science than
the other natural sciences, what makes a physical theory fundamental,
and what particular physical theories to regard as fundamental
(Morganti 2020b). The notions of fundamentality in metaphysics and
fundamentality in physics can overlap, especially for naturalistically
inclined philosophers who wish to turn to our best science as a guide
to the structure of reality. However, there are at least two important
differences in the usage of the notion of fundamentality in these two
contexts. First, when we consider fundamentality in physical
*theories*, there is a further question about how we link what
the theory describes or represents to reality. Second, some
discussions of fundamentality in physics are framed in a more
epistemic fashion. For instance, Crowther (2019) suggests that a
theory should be considered fundamental if and only if it possesses a
particular set of epistemic features, such as being predictive at all
scales (see Morganti 2020b for discussion).



We should also distinguish between the fundamentality of entities
belonging to a certain ontological category and the fundamentality of
the ontological category itself (see Hakkarainen 2022). It is one
thing to say that certain properties are fundamental; another to say
that the category *property* is a fundamental category. We are
mostly concerned with the former issue, but the debate about which
ontological categories are fundamental and how many such categories
there are has been lively throughout the history of Western
philosophy. For instance, Aristotle may have thought that substances
are the only category that is *separate from* and hence more
fundamental than other categories, such as the category universals
(Aristotle *Physics* 185a31-32; *Metaphysics*
1029a27-28; on the status of the category of substance in
contemporary metaphysics, see O'Conaill
 2022).[2]
 Another example is the category of *natural kinds*,
which is one of the four fundamental categories in Lowe's (2006)
*four-category ontology*, and definitive of the view that we
may call *natural kind fundamentalism* (Tahko 2021a: Ch. 4.3).
Some philosophers (e.g., L. A. Paul 2017) have also defend a view
according to which there is just one fundamental category, by way of
collapsing the distinction between substance and property. On this
view, substances would be mereological complexes of properties and
hence not an ontological category in their own right (see
Keinanen & Tahko 2019 for discussion).



Since one key task for the notion of fundamentality is to help us
articulate the view that there is a *hierarchical structure* to
reality, it is usually assumed that the dependence relation we are
dealing with must be *asymmetric*. Typically, the relation
would also be considered *transitive* and *irreflexive*,
hence producing a *strict partial ordering*. However, each of
these formal characteristics can in fact be questioned, albeit whether
this undermines the layered conception is up for debate (see Rabin
 2018).[3]



We have noted that relative fundamentality is a very important notion,
since the second key task for fundamentality identified above concerns
the hierarchy of fundamentality. Moreover, relative fundamentality
could arguably be used to define absolute fundamentality (some entity
*x* is absolutely fundamental if and only if it is not
relatively fundamental to any entity *y*). In contrast,
relative fundamentality cannot be defined in terms of absolute
fundamentality, so there are reasons to think that we should focus on
relative fundamentality, insofar as the notion makes sense. Given
this, it is perhaps surprising that there are relatively few explicit
accounts of relative fundamentality in the literature so far, but we
will refer to it where relevant, and there are signs that the
literature is maturing in this regard (see Wilson 2012, 2016; Zylstra
2014; Koslicki 2015; Bennett 2017: Ch. 6; deRosset 2017; Correia
2021a, 2021b, 2021c; Werner 2021). One complication, as observed by
Correia (2021c), is that the notion of relative fundamentality comes
in two varieties: *being more fundamental than* and *being
as fundamental as*. The notion of *metaphysical grounding*
has been used to characterize both notions of relative fundamentality
(Correia 2021b, 2021c; Werner 2021), but interestingly, Correia
(2021a) has also explored the possibility of characterizing the notion
of metaphysical grounding in terms of relative fundamentality. In what
follows, a variety of grounding-based characterizations of
fundamentality will be discussed, but this is not the only way to
characterize fundamentality. In any case, the systematization of
relative fundamentality is now proceeding at a fast pace, especially
due to the recent efforts by Correia and Werner, who are partly
reacting to Bennett's (2017) earlier attempts (see also Shumener
2019 and Wilson 2019 on Bennett's views).



Finally, the notion of *naturalness* and the related notion of
*sparseness*, familiar especially from Lewis's work (see
Lewis 1986, 2009; Schaffer 2004; Dorr & Hawthorne 2013; McDaniel
2013, 2017; Thompson 2016a; Sanchez forthcoming) is sometimes
connected with fundamentality. Sider's (2011) influential notion
of *structure* is closely related to the notion of naturalness
(cf. Fine 2013, Mathers 2019, Tahko 2020). Perfectly natural
properties might seem to be good candidates for absolutely fundamental
entities. Naturalness is no doubt a close cousin of fundamentality,
but there are some reasons to think that the two notions cannot do the
same jobs (see Bennett 2017: Ch. 5.7). Most strikingly, there may be
perfectly natural entities that are dependent in ways that are clearly
ruled out by many of the definitions of fundamentality that we will
shortly
 consider.[4]
 For further discussion, see the supplement on
 the natural/non-natural distinction
 in the entry on
 David Lewis's metaphysics.



We should also briefly address another, related notion, familiar from
Sider's work, which links the notions of structure and
metaphysical grounding: *purity* (Sider 2011: Ch. 7.2).
According to the principle of purity, fundamental facts involve only
fundamental notions. Sider gives the following example, where
*C\** is a predicate giving a complete microstructural account
of the city of Hamburg (cf. Sider 2011: 108):


* (C) The proposition
that there is a *C\** grounds the proposition that there is a
city.



Since the notion of *city* is clearly not fundamental, purity
dictates that (C) is not fundamental. Since *C\** is here taken
to be a fundamental fact, the status of (C) becomes problematic if we
accept purity - (C) must either be fundamental itself or
something needs to ground it. If it is fundamental, it violates
purity. But if it is not fundamental, then it needs to be grounded.
What could possibly ground (C)? This is the problem of
*meta-ground*, as discussed, e.g., by Dasgupta (2014a), Litland
(2017, 2018b), and Correia (forthcoming). Unless a satisfactory answer
can be given, one must either abandon purity or sever the link between
grounding and Sider's conception of fundamentality via structure
(see Tahko 2020 for discussion).


### 1.1 Absolute Independence



The first definition of fundamentality to be considered may be labeled
Absolute Independence:


* (AI) *x* is
*absolutely independent* if and only if, for all metaphysical
dependence relations *D*, there is no such *y* that
*Dxy*.



What is included in *D*? We can give an open-ended list of
candidate relations: grounding, dependence between wholes and their
parts (known as mereological or compositional dependence),
realization, existential dependence, essential dependence, and,
controversially, even causal dependence (on the link to causation, see
Bernstein 2016; Koslicki 2016; Schaffer 2016b; Shaheen 2017). The list
is open-ended partly because there is disagreement about what counts
as a metaphysical dependence relation in the relevant sense and partly
because there may be metaphysical dependence relations that we wish to
rule out. A possible criterion (i.e., the "relevant
sense") to be included in the list is that the dependence
relations in question must have at least some features in common, such
as, perhaps, transitivity. Furthermore, two philosophers may disagree,
say, about whether compositional dependence is a genuine metaphysical
dependence relation. So, for these two philosophers,
 (AI)
 will produce a different definition of fundamentality, because the
scope of *D* is different. We will not attempt to give a
complete list or to define all these different kinds of metaphysical
dependence, but for further details on some of them, see the separate
entries on
 ontological dependence
 and
 metaphysical grounding.[5]



What is important is that
 (AI)
 is an attempt to define fundamentality in the most general terms
possible, using a very broad notion of metaphysical dependence. There
is, however, at least one kind of dependence that we will likely wish
to exclude from *D*, namely, *modal dependence*
=df Necessarily, *x* exists only if *y*
exists. The reason for this is simple: nothing will be independent in
the sense of (AI) if modal independence is required (cf. Wang 2016).
This is evident if we consider some necessary existents, such as
numbers (assuming that numbers exist necessarily), for it is
necessarily the case that the number 2 exists if Socrates does. Hence,
the existence of Socrates necessitates the existence of the number 2.
Moreover, the number 2 necessitates the existence of the number 3 and
the other way around. This obviously generalizes, resulting in no
entity whatsoever being "absolutely existentially modally
free", as Wang (2016) puts it.



Even if we rule out modal dependence,
 (AI)
 is a very strong sense of fundamentality, probably much too strong.
Accordingly, it may not be a very popular option. Indeed, it is
difficult to find a direct endorsement of (AI) in the literature. That
being said, there are some potential if controversial candidates for
absolutely independent entities, such as God. Perhaps it is also
possible to think of the universe as a whole as absolutely independent
in this sense. This might reflect something like Georges
Lemaitre's theory of the "primeval atom" or
the "Cosmic Egg" hypothesis--an idea now better known
as the Big Bang theory. Lemaitre's hypothesis was that the
observed expansion of the universe may have started from a single
point, the primeval atom, which would have contained the entire mass
of the universe. Now, one may of course postulate that the primeval
atom itself could depend on something else, such as God, but this
nevertheless gives us an entirely naturalistically motivated idea of
absolute independence. The metaphysical position that comes closest to
this idea would be a type of *monism*, perhaps motivated by
considerations emerging from quantum holism (Calosi 2013; Ney 2015;
Ismael & Schaffer 2020; Calosi 2020; Kovacs 2021; and the entry on
 monism).



Why does fundamentality defined in terms of
 (AI)
 seem too strong? One major reason for this is that many things that
we might normally regard as fundamental turn out to be dependent on
other things in one sense or another. For instance, let us suppose
that there is a mereologically fundamental level, that is, there are
mereological atoms that do not have any proper parts. If mereological
dependence runs from the wholes to their parts contra *priority
monism* of the type defended by Schaffer (2010a), these
mereological atoms are clearly independent in a *mereological*
sense and most philosophers would probably want to regard them as
fundamental. But even if these mereological atoms are mereologically
independent, they could still metaphysically depend on other entities
in another sense.



Consider symmetry principles in physics. Symmetries can be understood
as a type of invariance under transformation. Some of the simplest
cases of symmetries can be found in familiar examples from the natural
world, such as sunflowers, where we observe both radial and bilateral
symmetry. Symmetry principles play an especially important role in
fundamental physics, such as in conservation laws: for each conserved
quantity, such as energy or linear momentum, there is a corresponding
symmetry property or invariance. This gives rise to a complicated
network of dependencies among the variables that feature in
conservation laws. How strong are these dependencies? In particular,
are they merely causal or perhaps necessary in some stronger sense?
This may depend on the type of symmetry in question, as some
symmetries do allow violations. However, in the case of symmetries
associated with conservation laws, Lange (2016) has argued that there
is a stronger type of necessity in effect, stronger than the natural
necessity usually associated with laws of nature. Whether or not Lange
is correct about this, it clearly important to consider the
possibility of non-causal dependencies such as those imposed by
symmetry principles because they may hold between entities that are
independent in some other sense (such as the mereological sense).



Relatedly, Giannotti (2021a) has discussed the possibility of
*fundamental yet mutually grounded entities* in more detail,
mentioning the case of quarks as well as Leibnizian monads as
potential examples (cf. also Wilson 2016: 192-193; Wilson 2020:
293). As Giannotti correctly observes, supposed examples of
fundamental yet mutually grounded entities are controversial, and he
even labels the possibility of such entities as a
"heresy", given the departure from the standard view,
whereby an entity is fundamental if and only if it is ungrounded.
Giannotti argues that even if there are fundamental yet mutually
grounded entities, this does not prevent a grounding-based account of
fundamentality. However, Giannotti's solution requires accepting
*non-asymmetric grounding*, which itself goes against the
grounding "orthodoxy" (see Raven 2013). Giannotti (2022b)
has recently expanded on this and proposed the labels of *radical
brutalism* for the view according to which the fundamental facts
are those which are *wholly ungrounded* and *moderate
brutalism* for the view according to which the fundamental facts
are those which are *partially ungrounded* (see also Bader 2021
on bruteness and fundamentality). The upshot of Giannotti's
discussion is that the choice between radical and moderate brutalism
is currently underdetermined by empirical evidence, leading Giannotti
to recommend *pluralistic brutalism*, according to which some
fundamental facts are moderately brute whereas others are radically
brute.



Whichever way we go here, it appears that many intuitive candidates
for fundamentalia would not be fundamental on the
 (AI)
 definition of fundamentality. A key issue, as with Giannotti's
analysis, concerns the possibility of fundamentalia being mutually
grounded or symmetrically dependent on each other (see also Priest
 2018).[6]



There are two further issues to note. Firstly, as it is formulated,
 (AI)
 rules out fundamental entities that depend on themselves. It could be
easily revised to accommodate such entities, but this is an issue that
the proponent of fundamentality should consider further, and there may
be reasons to leave room for self-dependent entities (see Bliss &
Priest 2018b).



Secondly, there is a speculative class of entities that would be
fundamental according to
 (AI),
 but are in tension with other plausible theories. These entities are
sometimes known as "idlers" (e.g., Lewis 2009: 205;
Bennett 2017: 123). For Lewis, idlers are fundamental properties,
which are instantiated in the actual world, but "play no active
role in the workings of nature". So, idlers are at least
causally isolated. However, we can further postulate that
"absolute idlers" are absolutely isolated: they depend on
nothing, and nothing, at least nothing concrete, depends on them.
Absolute idlers might thus be absolutely independent in the sense
required by (AI). Whether there are any such absolute idlers is of
course another question. If there are any absolute idlers, they are
rather uninteresting entities--indeed, they are presumably
completely beyond our ken given their isolation. Moreover, for the
idea of absolute idlers to have any plausibility, we would surely have
to restrict their isolation in such a way that they may participate in
abstract constructions such as sets. This is because classical set
theory, and indeed many other theories, entail that all entities have
at least some entities that depend on them. This may be a good reason
to think that no idler could be absolute in the strongest possible
sense.


### 1.2 Restricted Independence



The second definition of fundamentality to be considered is much more
versatile and weaker than
 (AI).
 We call it *restricted independence*. This produces a
*relativized* sense of fundamentality, where fundamentality is
relative to a certain variety or varieties of metaphysical dependence.
One should not confuse this with *relative* fundamentality,
which concerns the priority ordering between two (non-fundamental)
entities. So, restricted independence starts from the idea that for
each metaphysical dependence relation there is a corresponding notion
of fundamentality and we must relativize the notion of fundamentality
accordingly:


* (RI) *x* is
*restrictedly independent* if and only if there is one or more
metaphysical dependence relation(s) \(D\_{1}\), \(D\_{2}\) ...
\(D\_{N}\) such that there is no *y* such that \(D\_{1xy}\) or
\(D\_{2xy}\) or ... or \(D\_{Nxy}\).



(RI) is restricted to concern one or more specific kinds of
metaphysical dependence, because it is plausible that some
metaphysical dependence relations, such as modal dependence (as we saw
in
 section 1.1),
 would immediately rule out fundamental entities. Notice that (RI)
allows the possibility of including several dependence relations, but
we can easily define a more restricted sense of dependence for each of
\(D\_{1}\), \(D\_{2}\) ... \(D\_{N}\).



We have already discussed some kinds of metaphysical dependence that
 (RI)
 could apply to, such as grounding and mereological dependence. (RI)
concerns the subset of those metaphysical dependence relations that
are considered relevant to fundamentality, so here two philosophers
could disagree about which dependence relations to include in that
subset. Variations of this conception of fundamentality can be found
throughout the literature and it is probably as close to a standard
conception as we are likely to find (a few examples: Schaffer 2009:
373; Dixon 2016: 442; Bennett 2017: 105; see also Tahko 2015: Ch. 6;
Bliss & Priest 2018b). There are, of course, many differences
between the various accounts. Variations of (RI) are standard enough
to have been picked up by most of those critical towards this
conception of fundamentality as well (Bliss 2013: 413; Morganti 2015:
559; Raven 2016:
 608).[7]



Both
 (AI)
 and
 (RI)
 could be understood as putting forward the idea that fundamental
reality needs a *relational underpinning* (on the notion of a
relational underpinning, compare Fine 2001: 25). In other words,
whatever fundamentality amounts to, it must be the case that one (or
several) of the various relations of metaphysical dependence may be
used to define fundamentality. But since (RI) leaves completely open
which of the metaphysical dependence relations are in fact relevant to
fundamentality, we should mention some examples. At one point in the
early 2000s, it may have been the case that most philosophers using
the notion of fundamentality had in mind what we might call
*mereological fundamentality* whereby the relevant kind of
dependence is mereological dependence (see especially Schaffer 2003).
A concise definition of mereological dependence is offered by Kim:



> 
> 
> the properties of a whole, or the fact that a whole instantiates a
> certain property, may depend on the properties and relations had by
> its parts. (2010: 183; see also Markosian 2005; Thalos 2010, 2013)
> 
> 
> 



However, even though mereological dependence is still sometimes
considered to be relevant to fundamentality, it is becoming less
common to view it as the *only* relevant kind of dependence
(Wilsch 2016; Bennett 2017: 8-9). Be that as it may, there is
certainly an esteemed history for this type of idea, given that
(mereological) atomism of the type defended already by ancient
philosophers such as Leucippus and Democritus would appear to be an
instance of the view (see the separate entry on
 ancient atomism
 for details, and compare with Schnieder 2020a, where Bolzano's
version of atomism is discussed).



It is important to see that the thesis that some things are
mereologically fundamental does not entail a commitment to atomism.
Here it may be helpful to introduce the distinction between (priority)
*pluralism* and *monism*. We have already seen that a
monist might be attracted to a view about fundamentality where there
is just one fundamental entity, such as the cosmos or universe as a
whole (Schaffer 2010a is a good exposition and defense of this
Spinoza-inspired view; see also Newlands 2010). This view does not
directly entail anything about the relative fundamentality relations
among the non-fundamental, but it enables us to better understand that
the converse of mereological dependence, as Kim defines it, might
hold: instead of wholes depending on their parts, the parts could
depend on the whole. This contrasts with the more familiar idea, often
associated with but not entailed by atomism, that mereological
dependence must run from the larger to the smaller, mereological atoms
being the fundamental entities. So, the choice concerning the
direction of the relevant dependence relation is often reflected by
the choice between priority pluralism and monism, although strictly
speaking these issues are independent (Miller 2009; Trogdon 2009;
Cotnoir 2013; Steinberg 2015; Tallant 2015). Two proponents of
 (RI),
 even if they agree on which proper subset of metaphysical dependence
relations is relevant to fundamentality, may disagree on the direction
of the relevant dependence
 relation(s).[8]



Moving on to a different subspecies of
 (RI),
 we see a clear link between grounding and fundamentality, where
grounding is understood as expressing a non-causal connection between
two things. For example, a certain act might be considered evil
because it causes harm. The "because" in this statement
does not express a causal link; instead, it tells us what
*grounds* the evilness of the act. Similarly, one might think
that mental states hold in virtue of neurophysiological states or that
a substance is prior to its tropes. The notions of "in virtue
of" and "prior to" in these cases may be understood
in terms of grounding. What is likely to be the most common
contemporary understanding of (RI) is that the most important (if not
the only) relation of metaphysical dependence that is relevant to
fundamentality is grounding. Note, however, that grounding as well
could be understood as a family of dependence relations (Trogdon
2013), and there is an on-going discussion about the relationship
between grounding and ontological dependence more generally (e.g.,
Schnieder 2020b; Rydehn 2021). Furthermore, since our
discussion of fundamentality has focused on *entities*, it is
worth noting that bridging fundamentality and grounding suggests a
commitment to *entity grounding*, whereby all sorts of entities
can stand in grounding relations. However, some prefer to restrict
grounding to *facts* only, and this would accordingly require a
somewhat different formulation of the relevant fundamentality claims
(see the separate entry on
 metaphysical grounding
 and especially the section on
 regimentation
 for further details).



On the grounding-based characterization of fundamentality, the
fundamentalia are *ungrounded* entities: everything is either
ungrounded or ultimately grounded in the fundamental, ungrounded
entities (Schaffer 2009: 353; Audi 2012: 710; Dasgupta 2014a: 536;
Raven 2016: 613). For instance, Audi (2012: 710) explicitly
distinguishes between the *explanatorily fundamental* and the
*compositionally fundamental* where the first is associated
with grounding and the second with mereological dependence. As Audi
correctly notes, one would think that often these two notions of
fundamentality would overlap. But recall that we've just
observed that the converse of mereological dependence, running from
the smaller to the larger, could hold. For the priority monist, this
dependence relation running from the smaller to the larger is
grounding, so it would also produce a different understanding of
fundamentality.



If explanatory fundamentality and compositional fundamentality are
genuinely two different notions corresponding to two independent
notions of fundamentality, then
 (RI)
 does allow the overlap of explanatory and compositional
fundamentality. But there could certainly be other considerations that
count against this. For instance, if (RI) is understood to mean that
to be fundamental is just to be at the termination point of some
dependence relation, then the debate between monism and pluralism
could just be a debate about the direction of compositional
fundamentality, where compositional fundamentality terminates in the
mereological atoms and its converse terminates in the whole cosmos. In
other words, this could be understood as a debate about whether
explanatory fundamentality aligns with compositional fundamentality or
its converse. Whichever way we go here, we would seem to end up
aligning some notions of fundamentality and misaligning others.



An important question arises: should we indeed postulate several
independent notions of fundamentality, relativized to each of the
metaphysical dependence relations, or should we aim to define only one
sense of fundamentality, either to be defined in terms of just one
relation of metaphysical dependence or in terms of some privileged
proper subset of these relations?



Given that there are good reasons to exclude some notions of
dependence, such as modal dependence and perhaps causal dependence, it
would appear that there is an additional question about why only some
notions of dependence are such that we want to define a corresponding
notion of fundamentality for them. There is an obvious challenge for
the view that we should postulate several relativized notions of
fundamentality. The challenge is simply that the notion of
fundamentality would seem to do little in addition to the various
relativized notions of (in)dependence. Indeed, this is likely to cause
confusion, because the notion of fundamentality is sometimes used in
the literature without any mention of which relativized sense of
independence is in question. Moreover, since different types of
metaphysical dependence have different formal properties (e.g., some
are strict partial orders, but some might be symmetric, reflexive or
non-transitive) and they can perhaps even run in different directions,
it is difficult to see what could *unify* the different notions
of fundamentality, i.e., the proper subset of those dependence
relations that we consider to be relevant for
 fundamentality.[9]



Those who think that grounding is strongly unified often appeal to its
formal properties (but for a different approach, see Trogdon 2018a).
These properties may be debated, but if grounding is strongly unified,
then one could hold that for a relation to count as grounding, it
should at least fit these properties. Relevant formal properties
include the following: grounding is a strict partial order,
*non-monotonic* in the sense that we cannot add arbitrary
grounds and expect that grounding still holds (i.e., if *A* is
grounded in *B*, then it does not follow that *A* is
grounded in *B* and *C*), and it is thought that grounds
metaphysically necessitate what they ground (although see Leuenberger
2014; Skiles 2015 against necessitation). In contrast, Bennett's
*building relations* 
(see note 10)
 do not share all their formal properties. She holds that
they're all antisymmetric and irreflexive, but not necessarily
transitive (Bennett 2017: 46). Another example of this type of
"multi-dimensional" approach to fundamentality can be
found in Koslicki's (2012, 2015, 2016)
 work.[10]



It's important to see that on a multi-dimensional view whereby
the various dependence relations relevant to fundamentality might even
run in different directions, one must be very careful indeed to
specify which relativized notion of fundamentality one has in mind. If
it turns out that some entity is independent in all the relativized
senses of independence relevant to fundamentality, then it would seem
that we are back to absolute independence
 (AI)
 (or independence "full stop", as Bennett 2017: 106 calls
it).



Here we would do well to systematically distinguish between those
proponents of
 (RI)
 who hold that only one metaphysical dependence relation, e.g.,
mereological dependence or a strongly unified notion of grounding, is
relevant to fundamentality and those who think that a proper subset of
these relations is relevant to fundamentality. Sometimes the labels
*monism* and *pluralism* are used to distinguish between
the singular fundamentality view and the multi-dimensional view, but
in the interest of clarity we should introduce different terms, as we
have already used these labels for another
 purpose.[11]
 So, let us use the labels (RI-one) and (RI-many) to distinguish
between those who think that there is only one notion of
fundamentality and those who think that there are several. Typically,
there is an easy translation between these views. Let us say that a
proponent of (RI-one) thinks that only mereological dependence is
relevant to fundamentality. Well, a proponent of (RI-many), provided
that mereological dependence is one of the relations they consider
relevant to fundamentality, can simply translate the (RI-one) notion
of fundamentality into their (RI-many) notion of mereological
fundamentality. Hence the disagreement is that from the (RI-one) point
of view, there is a single notion of fundamentality, and they would
think that (RI-many) mistakenly holds other notions of fundamentality
to be genuine, whereas from the (RI-many) point of view, there are
many relativized notions of fundamentality, and (RI-one) mistakenly
picks just one, or indeed none depending on whether the relevant
(RI-one) relation is
 included.[12]



We conclude this section by posing a further question to all
proponents of
 (RI):
 is there a relativized notion of fundamentality for *every*
metaphysical dependence relation, or only some of them? Further, if
only a proper subset of metaphysical dependence relations are relevant
to fundamentality, then what makes these dependence relations relevant
in this way? A proponent of (RI) may refuse to answer this question,
since it would be an additional commitment to privilege some
metaphysical dependence relations in this way. However, given that one
major task for the notion of fundamentality is often thought to be
related to the idea of the hierarchical structure of reality, it seems
that we may have to accept only a proper subset of metaphysical
dependence relations as being relevant for fundamentality if this task
is to be preserved (of course, it could also be abandoned!). For
instance, perhaps only those metaphysical dependence relations that
are asymmetric and transitive would qualify. Still, there may also be
dependence relations that are relevant to fundamentality that fail to
be transitive. At this point, it could be helpful to move on to the
third potential definition of fundamentality, as something along the
lines of this definition is often used to characterize fundamentality
defined in terms of (RI) as well.


### 1.3 Complete Minimal Basis



The conception of fundamentality to be considered in this section is
often used to explicate the second key task for the notion of
fundamentality, that of the fundamental entities acting as the basic
building blocks of reality. According to this approach, the
fundamental entities determine everything else. By giving a complete
list of the fundamental entities, we can provide a minimal complete
description of reality. So, we are now shifting our focus from that
which everything else depends on to that which, as it were, supports
everything else. This idea is often invoked to characterize
fundamentality, but not necessarily to define it (Schaffer 2010a:
39n14; Sider 2011: 16-18; Jenkins 2013: 212; A. Paul 2012: 221;
Tahko 2014: 263; Wilson 2014: 561; Raven 2016: 609; Bennett 2017:
107ff.). The idea is that fundamentality can be understood in terms of
a *complete minimal basis*:


* (CMB) *x* is
fundamental if and only if *x* belongs to a plurality of
entities *X* and *X* forms a minimal complete basis that
determines everything else. The complete basis is minimal if no proper
sub-plurality of the entities belonging to *X* is
complete.



The obvious notion to be clarified in (CMB) is
"determines". This should be understood as a placeholder,
which is meant to encompass the various ways in which fundamentalia
may give rise to higher-level phenomena. So, "determines"
could be replaced, e.g., by "grounds",
"realizes", "composes", or
 "builds".[13]
 Importantly, this type of determination is supposed to be more than
mere necessitation or supervenience, although the idea is closely
related to traditional discussions of a minimal supervenience base
(see Schaffer 2003; compare also with Schaffer's notion of
"generation" in Schaffer 2016b:
 54).[14]



There are some preliminary issues to be specified before we can move
on to a more general discussion about
 (CMB).


1. (CMB) is compatible both with monism and the idea that reality may
have an *irreducibly plural underpinning* whereby if *X*
forms a complete minimal basis, no proper sub-plurality of *X*
will be
 complete.[15]
2. We have included a *minimality* condition in the definition
of (CMB). This is an important addition, because otherwise we could
take the plurality of all the entities in the world and call it
complete, given that this plurality as well would include all the
fundamental entities. So, according to (CMB), the complete minimal
basis must include all and only the fundamental entities.
3. There is an open question concerning *uniqueness*. Could
there be several distinct pluralities that are minimal and complete?
In other words, could there be distinct minimal pluralities that are
each complete and hence capable of determining everything else?
Bennett (2017: 112ff.) leaves the possibility of distinct minimal
complete pluralities open but makes some use of the notion of a
*unique minimally complete plurality*, whereas Tahko (2018)
speculates about the possibility of several "ontologically
minimal descriptions", dropping the requirement of
 uniqueness.[16]
4. It is, in principle, possible to define relativized versions of
(CMB), just as we did in
 section 1.2
 with
 (RI).
 So, one could distinguish between *absolute completeness* and
*restricted completeness*. It is easy to see how the restricted
notion of completeness is supposed to work. For instance, the
*mereologically minimal complete basis* is the minimal complete
plurality of mereological elements that determine everything
mereological. This could leave out something determined, say, by a
potential *non-mereological* notion of composition or parthood,
such as Armstrong's notion of states of affairs (see Armstrong
1997: 118-127). However, taking this route does not only invite
the challenges observed with regard to (RI), but may also diminish the
potential initial attractiveness of (CMB) (Bennett 2017: 110). So, we
will set aside the relativized sense of (CMB).



We can now move on to some further
 issues.[17]
 One of these issues concerns how, exactly, we should interpret
 (CMB).
 We can find a common line of interpretation gestured at by Schaffer
(2003: 509), Jenkins (2013: 212), Raven (2016: 609), and Tahko (2018)
whereby the complete minimal basis should be understood as giving a
complete minimal description of reality. This may be compared to
Schaffer's "fundamental supervenience base",
Jenkins's "by appeal to which all the rest can be
explained", Raven's "ineliminability", and
Tahko's "ontological minimality"; see also Lewis
(1986: 60). On this reading, less emphasis is put on the active role
of the fundamentalia, the focus being instead on making sure to
include a basis for everything in reality.



To have the broadest possible appeal, a characterisation of
fundamentality should ideally be applicable in different scenarios
regarding what the world is like. One such scenario is a "flat
world", a world where everything is independent, either in the
sense of
 (AI)
 or (RI), and nothing is built, to use Bennett's notion (Bennett
2017: 123-124). In a flat world, nothing determines anything
else in the sense of (CMB), since nothing depends on anything else.
Initially, it may seem that (CMB) is not plausible in this type of
scenario, since everything is included in the unique minimally
complete plurality. However, as noted in the previous paragraph, the
definition of (CMB) does not require any "active"
contribution from the fundamentalia; it's not crucial that the
fundamentalia do any determining. So, the upshot is that, in flat
world, everything is fundamental both in the sense of (AI) or (RI) as
well as in the sense of (CMB).



Accordingly, it would be an additional commitment to
 (CMB)
 to require that the fundamentalia must "actively"
determine something in order to count among the fundamentalia, call it
(CMB+). So, while (CMB) is vacuously true in a flat world, the
proponent of this more "active" version, (CMB+), requires
something more. Driven by the flat world example, one might then think
that
 (AI)
 and
 (RI)
 are prior to (CMB+). According to (CMB+), a flat world--rather
than being a world where everything is fundamental--would be a
world where nothing is fundamental (or derivative), since nothing is
"actively" determining anything. A world which contains no
such structure would then turn out to be a world which contains no
priority, hence no fundamentality (cf. Wilson 2016: 199).



Another difference between
 (CMB)
 and
 (RI)
 concerns the possibility of *metaphysical coherentism*, where
loops of dependence are possible (see Bliss 2014, 2018; Barnes 2018;
Morganti 2018, 2019; Nolan 2018; Thompson 2018; Swiderski
forthcoming). (CMB) can accommodate the possibility of metaphysical
coherentism, but (RI) would seem incompatible with it. The primary
reason for this is that metaphysical coherentism, as it is usually
understood, violates asymmetry and hence drops the idea that the
structure of reality is hierarchical. While this requires abandoning
one of the key tasks that the notion of fundamentality is often
thought to have, it does help in accommodating the idea that we should
adopt a more holistic approach whereby entities can be mutually
related and form loops or cycles. Sometimes an analogy from
epistemology is offered, as these loops of dependence could resemble a
Quinean *web of belief*, so that each entity depends on one or
more other entities. Perhaps the most plausible example of the
possibility of loops of dependence comes from
 ontic structural realism
 (OSR). (OSR) suggests that objects may be reduced to--or more
moderately, are ontologically on a par rather than prior
to--relational structures. If (OSR) is true, we might have to
revise our views about what the fundamentalia could be, as they might
be relations rather than objects (for discussion on OSR and
fundamentality, see Wolff 2012; McKenzie 2014; Morganti 2018, 2019;
Tahko 2018). What exactly happens to (CMB) on this type of view
depends on the details of the coherentist framework, but one
possibility--on a moderate understanding of (OSR)--is that
the fundamentalia would include mutually dependent relations and
objects, which then determine everything else. Metaphysical
coherentism is beginning to receive increasing attention. For
instance, Calosi and Morganti (2021) have recently put forward a
coherentist account of quantum entanglement, and Swiderski
(forthcoming) has made an important effort to systematize the
literature by defining four different conceptions of coherentism,
namely (we refer the reader to the original paper for illustrative
diagrams):


* *Holism*: For any *x* and any *y*, *x*
(partially) grounds *y* and *y* (partially) grounds
*x* (ibid., p. 5).
* *Insularism*: For any *x*, there is some *y*
such that *y* grounds *x* (and *x* [?]
*y* ) and some *z* such that *x* does not ground
*z* (and *x* [?] *z* ). In addition, all
grounding relations are regarded as symmetric (ibid., p. 10). This
view suggests that there are more than one networks of symmetric or
coherent grounding, where no grounding between these different insular
classes of facts takes place, but we have maximal coherence within
each class.
* *Hierarchism*: For any *x*, there are some
*y*'s such that each of the *y*'s grounds
*x* and *x* grounds each of the *y*'s, and
either (i) there are some *z*'s (distinct from the
*y*'s) such that each of the *z*'s grounds
all the other *z*'s, and the *z*'s ground
the *y*'s, or (ii) there are some *w*'s
(distinct from the *y*'s) such that each of the
*w*'s grounds all the other *w*'s, and the
*y*'s ground the *w*'s (ibid., p. 17). This
view suggests that there is a hierarchy of facts that mutually ground
each other which form a *level*, and any level has got either
(i) a level below it or (ii) a level above it. (Swiderski also
specifies two further varieties of *hierachism* depending on
whether one wishes to include or rule out infinite descent.)
* *Rebarism*: For any *x*, the following
*exclusive* disjunction is true: *either* (i) there are
some *y*'s such that each of the *y*'s
grounds *x* and *x* grounds each of the
*y*'s, or (ii) there are some *z*'s such
that each of the *z*'s grounds all the other
*z*'s, and the *z*'s ground *x*
(ibid., p. 21). This view suggests that only the fundamental level
involves symmetric or coherent grounding, while everything else (on
higher levels) stands in asymmetric grounding relations, generating
the familiar hierarchical picture of reality.



While Swiderski does not explicitly defend any of these versions of
coherentism, his discussion does point out some crucial decision
points. Most importantly, Swiderski demonstrates that while holism and
insularism are not compatible with asymmetric grounding, hiearchism
and rebarism do allow for (limited cases of) asymmetric grounding.
However, all four views retain a commitment to what Swiderski (ibid.,
p. 4) calls the *Coherentist Canon*: "(i) For any
*x*, there is some *y* such that *y* grounds
*x*, and (ii) there is some *z* and some *w* such
that *z* (perhaps indirectly) grounds *w* and vice
versa". This also suggests a commitment to the idea of mutual
grounding (cf. Giannotti 2021a), which was briefly discussed in
 section 1.1.



Finally, a specific form of metaphysical infinitism, where the same
structure repeats infinitely, to be discussed in more detail in
 section 4,
 could motivate
 (CMB)
 (Bliss 2013; Tahko 2014; Morganti 2014, 2018). On this type of view,
it could turn out that nothing is independent in the sense of
 (RI),
 but a sense of completeness could nevertheless be retained.


### 1.4 Primitivism



According to primitivism about the fundamental, we cannot define
fundamentality. But we may be able to characterize it, and it is to be
expected that
 (RI)
 and
 (CMB)
 are likely candidates in this regard. This type of view may be what
Fine (2001: 26) gestures towards, when he remarks that it is the
world's intrinsic structure that is fundamental. One way to
develop the idea is by defining absolute fundamentality in terms of
relative fundamentality and introducing a primitive notion of
"Reality" as it is in itself (Fine 2001; see also Fine
2009). Note however that Fine proposes this notion in connection to
the debate between realism and anti-realism, whereby
"Reality" may be understood as objectivity. So, it is not
entirely clear that we can understand fundamentality in terms of this
notion.



More generally, we may think that the absolute notion of fundamental
reality is not in need of a *relational underpinning* (Fine
2001; Wilson 2014: 561). This is in contrast to the various
comparative or relational notions of dependence that we have been
discussing (see also Fine 2015). Note that primitivism must also be
contrasted with
 (CMB)
 insofar as it is understood as a definition of fundamentality, even
though it is not uncommon to see primitivist characterizations of
fundamentality that resemble (CMB). For instance, a primitivist who
accepts Fine's notion of "Reality" would like to
distinguish between what is Real in itself and what may be true (i.e.,
objective) even if it does not concern the way that things are
fundamentally. As Fine puts it,



> 
> 
> even though two nations may be at war, we may deny that this is how
> things really or fundamentally are because the entities in question,
> the nations, and the relationship between them, are no part of Reality
> as it [is] in itself. (Fine 2001: 26)
> 
> 
> 



The idea behind the primitivist view is actually very simple and it
may be prudent to try to capture it without the controversial notion
of "Reality". One way to do so would be to understand the
entities in the fundamental base to be basic in the sense that they
play a role analogous to axioms in a theory, or that the fundamental
entities are all that God had to bring about to make the world (Wilson
2014: 560; 2016; see also Dorr 2005). We can find several occurrences
of this type of heuristic in the literature on fundamentality,
e.g.,



> 
> 
> The primary is (as it were) all God would need to create. (Schaffer
> 2009: 351)
> 
> 
> 
> all God had to do when making the world was fix the qualitative facts;
> (Dasgupta 2014b: 14)
> 
> 
> 
> We often explain the notion of fundamental reality in intuitive terms
> by saying that all God had to do in order to create the world was fix
> the fundamental facts. (Glazier 2016: 35)
> 
> 
> 



To drive the idea home, consider the following passage from
Kripke:



> 
> 
> Suppose we imagine God creating the world; what does He need to do to
> make the identity of heat and molecular motion obtain? Here it would
> seem that all He needs to do is to create the heat, that is, the
> molecular motion itself. If the air molecules on this earth are
> sufficiently agitated, if there is a burning fire, then the earth will
> be hot even if there are no observers to see it. God created light
> (and thus created streams of photons, according to present scientific
> doctrine) before He created human and animal observers; and the same
> presumably holds for heat. How then does it appear to us that the
> identity of molecular motion with heat is a substantive scientific
> fact, that the mere creation of molecular motion still leaves God with
> the additional task of making molecular motion into heat? This feeling
> is indeed illusory, but what *is* a substantive task for the
> Deity is the task of making molecular motion felt as heat. To do this
> He must create some sentient beings to insure that the molecular
> motion produces the sensation *S* in them. Only after he has
> done this will there be beings who can learn that the sentence
> "Heat is the motion of molecules" expresses an *a
> posteriori* truth in precisely the same way that we do. (Kripke
> 1980: 153)
> 
> 
> 



Now, Kripke is of course not trying to put forward an account of
fundamentality here, but rather to specify that the qualitative
experience of feeling heat is something additional to molecular
motion. The idea, however, is strikingly close to the primitivist
account of fundamentality, whereby the important thing is to find a
sufficiently rich basis for everything in the world--this is of
course reminiscent of
 (CMB),
 but here the heuristic is simply used to characterize fundamentality,
not to define it.



Since the primitivist thinks that we cannot define fundamentality, one
might wonder how it is that she can decide on what the fundamental
entities are. In other words, what is our epistemic access to issues
concerning fundamentality? A possible answer is that we proceed the
same way as we do with other primitives in metaphysics. That is, by
asking how the view that *x* is fundamental fits in our overall
theory. How does it fare with regard to our other commitments? These
are issues where theoretical virtues such as simplicity and
explanatory power might be employed (for a related discussion, see
Schaffer 2014). However, this is not the place to tackle the
admittedly difficult epistemic questions surrounding metaphysical
theories in general, or the criteria that are to be employed in
theory-choice. The primitivist about fundamentality may have one more
primitive in their overall theory, and this, of course, calls for
justification. But conceiving of the fundamental as primitive is no
more mysterious than conceiving of, say, naturalness or grounding as
primitive. This is not to say that postulating naturalness or
grounding as primitives wouldn't also require some justification
though.



The primitivist about fundamentality does, however, face some
challenges that are specific to fundamentality. Some of these
challenges have been raised by Schaffer (2016a), targeting
Wilson's proposal. A potentially helpful clarification from
Schaffer is the following: there is something primitive in each of the
theories on offer, but the primitivist about fundamentality takes
*being-absolutely-fundamental* as that primitive, whereas at
least certain proponents of the relative independence
 (RI)
 view (such as Schaffer himself and, e.g., Rosen 2010) take the
"linking notion" of grounding,
*being-relatively-more-fundamental-than-and-linked-to*, as the
primitive (Schaffer 2016a: 157). As Schaffer notes, one may also take
both to be primitive, as Fine perhaps does. Schaffer argues that while
absolute fundamentality can be quite easily defined in terms of
grounding, as we've seen with (RI), it may not be so easy to
define grounding in terms of absolute fundamentality. As observed
earlier, there have also been recent attempts to define grounding in
terms of relative fundamentality (Correia 2018), as well as attempts
to define notions of relative fundamentality in terms of grounding
(Correia 2021b, 2021c; Werner 2021), but none of these options are
directly available to the primitivist.



A further issue arises if there is no fundamental level at all. In
this scenario, the primitivist has no way to account for relative
fundamentality, and hence one of the two key tasks for fundamentality
would be lost. In contrast, the friend of
 (RI)
 could still construct a priority ordering using her favorite notion
of dependence, since no absolutely fundamental level is needed to get
the ordering started (Schaffer 2016a,
 158).[18]
 How can the primitivist reply?



One possibility is to try fixing the direction of priority even in the
absence of an absolutely fundamental level (Wilson 2016: 196ff.).
Drawing on a suggestion from Montero (2006: 179), Wilson proposes that
analogously to an infinite sequence of numbers such as 1/2, 1/3, 1/4
... being still "bounded below" by zero, there could
be an infinite descent of fundamental entities that approaches a limit
whereby the limit acts as the fundamental level even though it is
never reached. Related ideas may be found elsewhere in the literature
(Tahko 2014; Morganti 2015; Raven 2016). One difficulty with this
proposal is that applying the notion of a limit assumes that we can
assign a numerical measure to the descending entities that approach
this limit. So, we would need to be able to construct the hierarchical
structure of relative fundamentality in such a way that this numerical
measure applies to it. Moreover, even if approaching a limit does not
necessarily need to be constructed numerically, the notion of a limit
itself requires postulating the (fundamental) limit, in this case
zero. While zero is not part of the sequence, it does seem to be part
of the ontology.



Another possible strategy to address this challenge to primitivism is
to argue that fixing the fundamental is not (always) enough to fix the
priority ordering. Rather, we should pay close attention to the
various dependence relations that interest us, such as the parthood
relation, and assess the nature of the non-fundamental entities that
these relations relate. Importantly, we might get different answers
depending on which view about these natures is correct (see Wilson
2016: 200ff.).



This concludes our discussion of the different ways to understand
absolute fundamentality. We will now move on to discuss a variety of
important views that are often expressed in terms of fundamentality,
putting the notion into use.


## 2. Well-Foundedness



As we noted in the very beginning of this entry, one of the two key
tasks for the notion of fundamentality is to capture the idea that
there is a foundation of being and that everything else depends on the
fundamental entities. This idea about fundamentality is often
expressed in terms of *well-foundedness* (Morganti 2009: 272;
Orilia 2009: 333; Fine 2010: 100; Schaffer 2010a: 37; Bennett 2011a:
30; Bliss 2013: 416; Trogdon 2013: 108; Tahko 2014: 260; Raven 2016:
614; Bohn 2018; Jago 2018; Pearson forthcoming). But the notion of
well-foundedness itself is sometimes used without much further
qualification, and there are many instances where another term is used
instead for what is clearly the same general idea (e.g., Lowe 1998:
158; R. Cameron 2008; Paseau 2010; Rosen 2010). A common formulation
of this general idea is as follows: a priority/dependence chain is
well-founded if and only if it *terminates*, i.e., has an end
constituted by one or more entities that do not depend on any other
entity. But not all the above-mentioned authors would be happy with
this particular formulation, and it turns out that sometimes
philosophers may even have talked past each other due to having
slightly different formulations of well-foundedness in mind.
Fortunately, the literature has matured, and we now have a number of
much more precise accounts of the various potential formulations of
well-foundedness (see especially Dixon 2016; Rabin & Rabern 2016;
Litland 2016b; Wigglesworth 2018).



Let's start with the origin of the term
"well-foundedness". The term is familiar from mathematics,
especially set theory; no doubt it was adopted from set theory with
the hope of making the metaphysical idea more precise. A simple
formulation of set-theoretic well-foundedness can be found in Cotnoir
and Bacon (2012: 187):



> 
> 
> An order < on a domain is said to be *wellfounded* if every
> nonempty subset of that domain has a <-minimal element.
> 
> 
> 



The first thing to notice here is that well-foundedness is relativized
to a giver order, a given relation of dependence, just like
 (RI)
 in
 section 1.2.
 So, strictly speaking, we ought to specify which relation we have in
mind and consider the various complications that this introduces,
which we discussed in
 section 1.
 However, much of the literature on well-foundedness (although not all
of it) focuses on grounding, and for purposes of exposition it is
easiest to focus on this literature, also assuming that grounding
establishes an absolute order of priority.



The set-theoretic formulation of well-foundedness is the sense in
which, e.g., Fine (2010: 100) appears to use the notion. Applied to
chains of grounding, the proposal is that well-foundedness would rule
out infinite, non-terminating grounding chains and entail that
"the grounds of any truth that is grounded will 'bottom
out' in truths that are ungrounded". We can see slightly
different but equivalent formulations in Schaffer (2010a, 37), Bennett
(2011a, 30), Trogdon (2013: 108), Tahko (2014: 260), Dixon (2016:
452), and Jago
 (2018).[19]
 As Dixon (ibid.) puts it, the attraction of using the set-theoretic
notion of well-foundedness is that it is a straightforward application
of the standard mathematical definition of a well-founded relation
(this is also recognized by Morganti 2015: 556fn2). The only problem
is that the standard understanding of well-foundedness may be too
strict for the task at hand. Many authors have considered weakened
versions of the well-foundedness requirement, which may be more
suitable for expressing the desired limits on infinite grounding
chains (R. Cameron 2008: 4; Trogdon 2013: 108; Leuenberger 2014:
170-171). So, to summarize, it might at first seem intuitive to
interpret well-foundedness simply as a ban on infinite chains and
cycles of grounding, in accordance with set-theoretic
well-foundedness. But this would seem to go against the intuition that
there are cases of infinite chaining which are acceptable for the
foundationalist metaphysician (R. Cameron 2008; Bliss 2013). This has
recently been made explicit by Dixon (2016) and Rabin & Rabern
 (2016).[20]



To get us started on the idea that foundationalist metaphysicians
could accept violations of well-foundedness, consider an important
observation by Bliss (2013: 416), which also comes up, e.g., in Rabin
& Rabern (2016: 362). She distinguishes between finite and
well-founded grounding chains where a finite grounding chain not only
terminates in something fundamental, but is also such that we can
reach the fundamental entities in a finite number of steps from
anywhere in the chain. According to this approach, a well-founded
grounding chain is indeed one that is grounded in something
fundamental, but it may itself be infinitely long. So, on this
terminology, finite grounding chains are always well-founded, but a
well-founded grounding chain could be infinite. But note that this
already violates set-theoretic well-foundedness as defined above. To
demonstrate this, consider an infinite chain of dependence \(f <
\ldots d\_{3} < d\_{2} < d\_{1}\), where the chain of dependent
entities \(d\_{n}\) terminates in some minimal element *f*. Now,
if we take a subset of that chain of dependence without the minimal
element *f*, then we are left with a chain that lacks a
<-minimal element, hence violating the set-theoretic definition of
well-foundedness.



To get clearer on what is at issue here, let us introduce another
notion, being "bounded from below" or "having a
lower bound". We can say that an order < on a domain is
bounded from below if any subset of that domain has a lower bound.
More precisely, a lower bound of a given set is any element that is
more minimal or equal to all of the elements of the set. For example,
1, 2, and 3 are all lower bounds of the interval [3, 4, 5]. A lower
bound of a set does not need to be an element of the set itself.
Consequently, the chain \(f < \ldots d\_{3} < d\_{2} < d\_{1}\),
is bounded from below. More generally, any finite chain is
set-theoretically well-founded and bounded from below. We've
just seen that some infinitely descending chains of dependence may be
bounded from below yet fail to be set-theoretically well-founded. As
Rabin & Rabern (2016: 360) put it, an infinitely descending chain
may or may not have a "limit", where limit is the greatest
lower bound of a set. In the previous example, 3 is the greatest lower
bound of the interval [3, 4, 5]. So, one might suggest that the
relevant sense of metaphysical well-foundedness is captured either by
this idea of being bounded from below or by having a limit, a greatest
lower bound. But it turns out that even these may be too strong. This
is the key insight in the recent work of Dixon and Rabin & Rabern.
If this is right, then we need to have a sense of a *metaphysical
foundation* that is not only compatible with infinite chains of
dependence, but also with infinite chains that do not have a lower
bound (i.e., "unbounded"
 chains).[21]



It might be helpful to present a somewhat simplified example of a case
where well-foundedness and being bounded from below come apart. Let us
use a variation of Trogdon's (2018b) reconstruction of an
example by Rabin & Rabern (2016: 361; this is an example of what
Dixon 2016: 448 calls a "fully pedestalled chain").
Consider a spherical region of space *S*, which we divide up in
such a way that each of the proper sub-regions of *S* has a
proper sub-region, and each of those sub-regions in turn has a proper
sub-region. In principle, we can continue this process infinitely.
Now, let us assume that each region of space derives its being in part
from (or is partially grounded in) its sub-regions. Here we would seem
to have an infinite regress where, to use the familiar phrase, being
is forever deferred, never achieved. But suppose in addition that
there are spatial points and that these points are fundamental. Then
we might like to say that *S* and indeed each of its infinitely
many proper sub-regions fully derives its being from (or is fully
grounded in) spatial points. In this case, the infinite regress of the
sub-regions would appear to be innocuous since each of them is after
all fully accounted for by the fundamental spatial points. Still, the
case for a sense of fundamentality that is compatible with at least
some kind of non-well-founded infinite descent or other seems to have
been made sufficiently clear in recent work. Metaphysical
foundationalism does not seem to be properly captured by the idea of
set-theoretic well-foundedness. So, what is it about?



We can find numerous formally precise attempts to capture the relevant
sense of a metaphysical foundation in recent work. Dixon's
(2016: 446) preferred understanding (which he calls "full
foundations") suggests that every non-fundamental fact is fully
grounded by some fundamental fact(s). Rabin & Rabern (2016: 363)
attempt to capture the idea of a metaphysical foundation with the
phrase "having a foundation", whereby a grounding
structure has a foundation if and only if there is some set of facts
that (i) together ground all the derivative facts and (ii) are
themselves ungrounded. Raven suggests (2016: 612) that metaphysical
foundationalism could be understood in terms of
*ineliminability* (to be discussed in
 section 3).
 Tahko (2014: 263) tries to analyze metaphysical foundationalism in
terms of an "ontological" sense of well-foundedness, which
requires that an ontologically well-founded chain terminates in a
fundamental supervenience base. Trogdon (2018b) follows the recent
suggestions for a weaker understanding of metaphysical foundationalism
and defines it as the view that, necessarily, any non-fundamental
entity is fully grounded by fundamental entities. Even though the
terminology varies, it's quite clear that there are weaker
senses of metaphysical foundationalism than the one defined in terms
of set-theoretic well-foundedness. We arrive at the following
definition of *metaphysical foundationalism*:


* (MF) Every
non-fundamental entity is dependent*D1,
D2 ... DN* on some fundamental
entity or entities that fully account for its
being/reality.



This definition of metaphysical foundationalism is somewhat vague
since it attempts to capture the idea for all the different varieties
of fundamentality that we have discussed, but it may be supplemented
with an appropriate restriction on the type of entities it concerns
(e.g., facts) and the subscripts \(D\_{1}\), \(D\_{2}\) ...
\(D\_{N}\) may be replaced with one's preferred kind or kinds of
dependence, exactly as with the schemas proposed in
 (AI)
 and
 (RI)
 in
 section 1.
 Furthermore, we need to understand "fully account for"
correspondingly, for instance, in the case of grounding it should be
parsed as "fully grounds" and in the case of compositional
dependence it should be parsed as "fully composes" (albeit
in some cases it may not be entirely clear that a full/partial
distinction properly applies). Now that we have a working notion of
metaphysical foundationalism, we may proceed to discuss arguments in
favor and against this view.


## 3. Metaphysical Foundationalism



Metaphysical foundationalism is the view that reality has a
foundation--that there is a "fundamental level", in a
sense that needs to be specified. The most common way to specify the
idea of having a foundation is in terms of well-foundedness, but as we
have seen in
 section 2,
 set-theoretical well-foundedness may be too strong to capture
metaphysical foundationalism. Metaphysical foundationalism comes in a
variety of strengths, depending on how much the well-foundedness
requirement is weakened. It seems reasonable to say that until quite
recently metaphysical foundationalism was the default position (R.
Cameron 2008; Schaffer 2009; 2010a; Bennett 2011a). The relevant
intuition is often captured with the much-cited phrase that without a
foundation, "a ground of being", "[b]eing would be
infinitely deferred, never achieved" (Schaffer 2009: 376; 2010a,
62).



Sometimes the foundationalist intuition is explicitly tied to
composition (i.e., mereological dependence) and the (im)possibility of
*gunk*, namely, the idea that everything has a proper part:
"the anti-gunk worry is that composition could never have got
off the ground" (R. Cameron 2008: 6). The worry is that complex
objects are not possible in gunky worlds, so given that there is
composition, there must be a foundation. However, we've seen
that fundamentality need not be tied to compositional/mereological
dependence. Moreover, others (McKenzie 2011; Bliss 2013; Tahko 2014;
Morganti 2014, 2018; Bohn 2018; Trogdon 2018b) have been suspicious of
the driving intuition behind this sense of metaphysical
foundationalism, and now even some of those who have earlier defended
metaphysical foundationalism opt for agnosticism on the question
(Bennett 2017: 120ff; Rosen 2010: 116). Indeed, there is, perhaps, now
a consensus that it is very difficult to come up with a proper
argument in favor of metaphysical foundationalism, something that
would go beyond the just stated intuition. The idea has a status
closer to a type of metaphysical *axiom* or *law* as
Bohn (2018) puts it (see Morganti 2018 for a good overview). This
conclusion seems even more warranted given that defining metaphysical
foundationalism simply in terms of set-theoretical well-foundedness
turned out to be too strict. To establish a clearer sense of
metaphysical foundationalism which is in accordance with the most
recent literature, we shall understand "metaphysical
foundationalism" as it was defined in
 section 2
(MF) (see Oberle 2022a for an overview of the most recent
debate).



Can we be clearer on the core idea of metaphysical foundationalism? If
we can, then perhaps potential arguments for the view would also be
more easily available. One interesting attempt to do so is
 Raven's.[22]
 Raven's version of metaphysical foundationalism relies on the
notions of "eliminability" and
"ineliminability" whereby an entity is eliminable if
reality is described no worse without mentioning it and ineliminable
if reality cannot be completely described without mentioning it. To
demonstrate this, let us take advantage of Raven's own
terminology (2016: 614-5). Ineliminable entities
"persist" in some fact about them, whereas eliminables
"disappear" from all facts about them. For an entity to
"disappear" is for there to be a bound in the grounds of
some fact about it, a last occurrence of it after which the entity
never recurs. So, for an entity to persist is for some facts about it
to be unbounded. Importantly, there are two ways to be unbounded:
being ungrounded or having grounds but forever recurring. The first
type of persistence is the familiar sense of relative independence
 (RI)
 as defined in
 section 1.2.
 But the second type of persistence, where an entity forever recurs in
the chain of dependence, is
 novel.[23]



A potentially helpful way to illustrate the options available to us
and clarify the relationship between metaphysical foundationalism and
infinite descent is Morganti's (2015: 562) "emergence
model of being". This model may be contrasted with the
"transmission model", which is, perhaps, what the
much-quoted phrase from Schaffer that we led with reflects. According
to the transmission model, the ungrounded entities are the
"ground of being". But the emergence model suggests that
something may act as a foundation in the absence of ungrounded
entities; the infinite "starts playing an active role, and
progress is made as the chain lengthens" (Morganti 2015: 562).
The emergence model draws inspiration from an analogy with
epistemology where the "emergence of justification" from
an infinite chain of reasons has recently been an area of active
research (Klein 2007; Peijnenburg & Atkinson 2013). Thus, the core
of the emergence model seems to be that there is no privileged
fundamental level of reality that serves as the basis for being.
Instead, we should understand being more holistically, as it were, and
explore the idea that it may emerge gradually. Let us call this
*emergentist infinitism*. On the face of it, emergentist
infinitism looks like the denial of
 (MF)
 and hence metaphysical foundationalism. However, there is some room
for interpretation here, because the holistic model suggests that the
whole infinite chain could perhaps be considered to ground its
"parts".



We might compare this line of thinking to Leibniz's
*principle of sufficient reason* (PSR), which states that for
every entity that exists, there is an explanation or reason for its
existence (Della Rocca 2010; Guigon 2015; Dasgupta 2016; Amijee 2020)
and the separate entry on
 principle of sufficient reason).
 In the contemporary literature, we may compare (PSR) with the
inheritance principle discussed in Schaffer (2016b) and Trogdon
(2018b). An open question, although not one that we will pursue here,
is whether emergentist infinitism is compatible with (PSR).



We should clarify one further issue regarding the analogy with
epistemology. Both with epistemic infinitism and epistemic
coherentism, it is typical to consider foundational justification to
be impossible: all possible cases of epistemic justification
*must* conform to the infinitis/coherentist picture. It is not
immediately clear whether the corresponding metaphysical views
(*metaphysical infinitism* and *metaphysical
coherentism*) would need to hold by necessity in a similar way. In
fact, some of the arguments in the literature suggest that this may
not be the case, e.g., where the *possibility* of different
types of infinite decent is considered (cf. Tahko 2014). If the
structure of reality is contingent in this regard then there may be a
principled difference between the epistemic and the metaphysical
cases. 



To conclude this section, it should be noted that not all the views
discussed above have been put forward under the label of metaphysical
foundationalism. But once it is made clear that the foundationalist
idea is not tied to the strong, set-theoretic sense of
well-foundedness, the requirement for a foundation is much weaker than
it may once have seemed. If we broaden the scope of metaphysical
foundationalism accordingly, is there still an interesting sense of
metaphysical infinitism to be discussed?


## 4. Metaphysical Infinitism



To endorse metaphysical infinitism is to reject metaphysical
foundationalism
 (MF).
 But as we've seen, the sense of metaphysical foundationalism
defined at the end of
 section 2
 does not require accepting strong, set-theoretic well-foundedness,
and hence it is compatible with at least some types of infinite
descent. Accordingly, metaphysical infinitism is a somewhat stronger
view than it may first have seemed.



Using the technical notion of being bounded from below or having a
lower bound introduced in
 section 2,
 we can start with the simple idea that, for a given notion of
dependence, a chain has a lower bound only if there is an element that
every element in the chain depends on (Rabin & Rabern 2016: 366).
As we saw earlier, infinitely descending chains of dependence may have
a lower bound, that is, terminate in an independent element that may
or may not be part of the chain itself, yet fail to be
set-theoretically well-founded. But there is an even weaker condition
than having a lower bound that can satisfy
 (MF),
 namely, Rabin & Rabern's (2016: 363) "having a
foundation" or Dixon's (2016: 446) equivalent "full
foundations". Both of these are based on the idea of having an
infinitely large foundation. An example of this type of foundation can
be constructed with the help of infinitary disjunctions, as both Rabin
& Rabern and Dixon demonstrate (Litland 2016b discusses the
problems associated with this and constructs further examples). In
such a case, (MF) is satisfied because every element depends on some
independent element, despite there being no lower bound. There is no
lower bound because the chain does not terminate if the foundation is
infinitely large. In contrast, if the foundation is finite, then there
is a lower bound and hence the weaker requirement of (MF) is also
entailed. In decreasing order of strength, we have the requirement of
set-theoretical well-foundedness, having a lower bound, and having a
foundation or full foundations. We defined (MF) in terms of the
weakest of these three requirements in
 section 2,
 but a proponent of metaphysical foundationalism may of course make a
stronger requirement as well, such as having a lower bound.



In this section, we are interested in the possibility of the type of
strong metaphysical infinitism that denies all of the three
requirements. The denial of
 (MF)
 entails that there are at least some entities that are
non-fundamental yet do not depend for their being on any fundamental
entity or entities. There may be several ways in which this could
happen, but the most extreme possibility is a type of *infinite
complexity* whereby there is an infinite descent of different
kinds of entities, each depending on entities further down the chain
but never terminating and never being "fully accounted
for". We might think of this in terms of a violation of the
principle of sufficient reason (PSR), at least if (PSR) is thought to
require that we must reach an ultimate reason rather than just a
reason at each layer below the previous one. Accordingly, infinite
complexity entails a lack of structure with the type of explanatory
import that (PSR) requires. This type of view would be more radical
than the various, potentially innocuous types of infinite descent
outlined so far. The view might strike many as implausible, at least
when it comes to the actual
 world.[24]



Infinite complexity is a strong version of metaphysical infinitism,
but what if, instead of infinite complexity, we had some kind of
infinite repetition? This type of idea has been discussed under the
label of *boring infinite descent* (Schaffer 2003: 505, 510;
Tahko 2014). A boring or repetitive structure entails that somewhere
down the chain of dependence we stop encountering novel types of
entities or novel structure. The boring part of the structure that
repeats infinitely could be of any length as long as it starts anew
eventually. A description of the repetitive part only needs to be
supplemented with an instruction to continue as before. For
example:



> 
> 
> The world stands on four elephants, the four elephants stand on a
> turtle, the turtle stands on two camels, the camels stand on four
> elephants, the four elephants stand on a turtle ... and repeat ad
> infinitum. (Tahko 2014: 261)
> 
> 
> 



The idea is that the boring structure, whatever shape it may take, can
be fully described in terms of the mentioned entities (or types of
entities, and perhaps the "standing" relations between
them): four elephants, a turtle, and two camels. It has been suggested
that this produces a "minimal" description of reality, but
it is debatable whether it satisfies
 (MF)
 or is a case of strong metaphysical infinitism (Raven 2016; Tahko
2018).



It seems that this type of infinite descent is less radical than
infinite complexity, but some open questions remain. For instance, is
the type of infinite regress at hand innocuous or vicious? Can we
capture the difference between metaphysical foundationalism and
metaphysical infinitism in terms of the non-viciousness or viciousness
of the regress (Nolan 2001; Bliss 2013; R. Cameron 2022; Oberle
2022b)? We will leave these issues open, but to improve on the toy
example, we might briefly discuss a more concrete case. Sometimes the
Nobel Prize winner Hans Dehmelt's (1989) model is mentioned as a
potential example (Schaffer 2003, Morganti 2014, Tahko 2014). Dehmelt
speculated that there could be a quark/lepton-like substructure beyond
the known level of quarks, based on the model of the triton, the
nucleus of hydrogen's radioactive isotope tritium:



> 
> 
> I propose to extend the triton substructure scheme to an infinite
> number of layers. Below the four layers listed above [up to
> subquarks], they contain higher order \(d\_N\) subquarks, with \(N = 5
> \rightarrow \infty\). In each layer the particles are not identical
> but resemble each other in the same way as quarks and leptons do, with
> masses varying as much as a factor 108. In an infinite regression to
> simpler particles of ever increasing mass, they asymptotically
> approach Dirac point particles. (Dehmelt 1989: 8618)
> 
> 
> 



Up to \(N = 3\), the level of electrons, Dehmelt's model is
motivated by current physics, but it is speculative from \(N = 4\)
onwards, where electron substructure is postulated. However,
it's not entirely clear whether Dehmelt's model is a
genuine case of boring infinite descent, given that the regress
appears to terminate in Dirac point particles. What matters here is
whether these Dirac point particles are treated as genuine entities,
but there is no suggestion in Dehmelt's model that this should
be the case. Rather, as the above quote suggests, the infinite
regression of non-point particles asymptotically approaches the ideal
limit of Dirac point particles. This is a type of mathematical
abstraction not unlike the one we find in the case of gunky space,
where regions asymptotically approach "points" (see
Zimmerman 1996). Just like Dirac point particles in Dehmelt's
model, the points in the gunky space are not considered real entities;
they are merely a mathematical
 abstraction.[25]



A final argument to be considered in this section challenges the
"transmission model", whereby derivative entities derive
their "being" from the fundamental entities. Bohn (2018:
170) makes a helpful observation, aimed against the transmission model
(applied to grounding):



> 
> 
> Grounding is like a synchronic, static mathematical relation (like in
> arithmetic), not like a diachronic, dynamic physical relation (like in
> thermodynamics, or action theory).
> 
> 
> 



The thought here is that the transmission model illegitimately assumes
a dynamical "starting point" to any grounding chain. If we
abandon the transmission model and the dynamical view, we arrive at
the idea of *indefinitely descending ground* whereby all facts
have a ground and hence there are no fundamental facts (Bohn 2018).
This violates even the weakest of the three requirements that a
metaphysical foundationalist might impose, so it is a strict denial of
 (MF).
 Note, however, that this concerns only the precisification of (MF) in
terms of grounding. So, this type of metaphysical infinitism targets
the conception of fundamentality as ungroundedness (as specified in
terms of
 (RI)
 in
 section 1.2).



If
 (MF)
 does not require the transmission model, the choice between
foundationalism and infinitism is difficult. There is, perhaps, one
test case: the possibility of *gunk* (everything has a proper
part), *junk* (everything is a proper part), and *hunk*
(everything both is and has a proper part). There is an on-going
debate about the modal status of these scenarios, and one might appeal
to their possibility in defense of indefinitely descending ground (as
Bohn 2018
 does).[26]
 However, these scenarios of course concern only the relation of
proper parthood (mereological dependence), and hence even if
mereological infinitism were coherent, there may be other notions of
dependence for which the corresponding notion of infinitism is
incoherent.



This concludes our discussion of metaphysical infinitism and
fundamentality. This survey has focused on a subset of the growing
literature on fundamentality with the aim of clarifying the various
terminological issues surrounding the key notions and identifying some
common themes. The ideas behind these notions are old as are the
intuitions associated with the various arguments that we've seen
for and against metaphysical foundationalism and infinitism. Recent
helpful efforts to systematize some of the central ideas, especially
that of well-foundedness, have made it much easier to have
constructive discussions about the topic of fundamentality. It can be
expected that further attempts to formulate arguments for and against
different strengths of metaphysical foundationalism and infinitism
will emerge in the near future. The literature is also paying more
attention toward the notions of relative fundamentality and
metaphysical coherentism. Finally, there is an increasing amount of
literature that applies more technical notions of fundamentality to
various other debates in metaphysics: e.g., Morganti (2019) on ontic
structural realism, Calosi and Morganti (2021) on quantum
entanglement, Giannotti (2021b) on powers, Scarpati (2021) on
haecceitism, Tahko (2021) on logical realism, Allori (2022) on quantum
theory, Hamri (2022) on causation, Rabin (2022) on the physicalism
debate, and Spencer (2022) on relativity.