Fundamentality

First published Sat Jul 21, 2018; substantive revision Mon Oct 16, 2023

The notion of fundamentality, as it is used in metaphysics, aims tocapture the idea that there is something basic or primitive in theworld. This metaphysical notion is related to the vernacular use of“fundamental”, but philosophers have also put forwardvarious technical definitions of the notion. Among the mostinfluential of these is the definition ofabsolutefundamentality in terms ofontological independence orungroundedness. Accordingly, the notion of fundamentality isoften associated with these two other technical notions, covered underontological dependence andmetaphysical grounding in this encyclopedia.

Why are philosophers interested in fundamentality? One reason comesfrom a certain view of science. It is not uncommon to think thatparticle physics has some special role in our inquiry into thestructure of reality. After all, every material entity is made up offundamental particles. So, one might think that particle physics aimsto describe thefundamental level of reality, which containsthe basic building blocks of nature. We can then employ the notion ofrelative fundamentality, which enables us to express thehierarchical nature of reality disclosed by science, according towhich the facts of biochemistry depend on the facts of elementarychemistry, which depend in turn on the allegedly fundamental facts offundamental particle physics.

The thought that this priority ordering terminates at the fundamentallevel is often expressed with the notion ofwell-foundedness.The view that reality is well-founded in the relevant sense is calledmetaphysical foundationalism, in contrast withmetaphysical infinitism. A further option, which underminesthe priority ordering and suggests that dependence chains can formloops, is calledmetaphysical coherentism.

We can identify two key tasks for the notion of fundamentality. Thefirst is to capture the idea that there is a foundation of being,which consists ofindependent entities. The second is tocapture the idea that the fundamental entities constitute acomplete basis that all else depends on. These tasks arerelated. In fact, the first would seem to require the second, but notthe other way around. We will see that prioritizing one or the otherof these tasks may result in different accounts of fundamentality. Thesecond task may be applied to relative fundamentality and used toexpress the idea that there is a hierarchy of being whereby someentities are more fundamental than others, although strictly speakingthis hierarchical picture is independent of the notion of a completebasis.

This entry will focus on the contemporary discussion, but many of theideas debated in the contemporary literature have been around formillennia. We now have the tools to make these important ideas muchmore precise. Relevant historical issues include ancient atomism(e.g., Leucippus and Democritus, see the separate entry onancient atomism), Aristotle’s many discussions of priority (see, e.g., Peramatzis2011 and the articles in Sirkel & Tahko 2014), Aquinas’sdiscussion of the first cause (see the entry on thecosmological argument and Oberle 2022b), theprinciple of sufficient reason, as discussed by Spinoza and Leibniz, among others, andBolzano’s work as a precursor to some contemporary work onmetaphysical grounding and related notions (see Roski and Schnieder2019 and Schnieder 2020a).

1. Varieties of Fundamentality

There are many senses in which a thing may be said to befundamental—some technical, some relatively intuitive. A verycommon way to think about fundamentality is in terms of independence,whereby for any notion of dependenceD, an entity isD-fundamental if and only if it does notdepend_D on anything else (or on anything else thatdoes not depend on it). This independence-based characterization offundamentality will be discussed in sections1.1 and1.2. There are also other ways to understand fundamentality; these includefundamentality as a complete description of reality (section 1.3) and as primitive (section 1.4). So, there are many ways to understand fundamentality and whetherthere is any single idea of fundamentality that these different waysare trying to capture is a substantial issue. But even if there is nounified sense of fundamentality, one interesting question is whetherthere are any fundamental entities, where “fundamental” isunderstood in one of the different ways that will be specified below.We might also ask whether we need such entities, whether giventheories are committed to their existence, and what the role offundamentalia is in explanation. Before we get started, a fewpreliminary issues need to be mentioned.

For present purposes, we are interested in the notion of a fundamentalthing, or type of thing. The candidates for fundamentalitymay include objects, such as electrons, but they may also includeproperties, or facts. The choice regarding the relevant type of entitymay 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 <electronshave unit negative charge> to be fundamental. Someone who does notwish to ascribe fundamentality to facts (e.g., because they do nothave facts in their ontology), could understand this to be saying thatthe propertyunit negative charge is fundamental andinstantiated in electrons. The translation here is fromelectronshave unit negative charge to the fundamental propertyunitnegative charge that is instantiated in electrons (and onlythem). So, the disagreement between those who consider fundamentalityto concern facts and those who regard objects and properties asfundamentalia may not be as serious as it looks.

One important distinction in the usage of the notion of fundamentalityis worth highlighting: fundamentality as a general metaphysical notionand fundamentality in physics and physical theories (see Morganti2020a, 2020b; Allori 2022). These two ways to use the notion are ofcourse related, but the focus of this entry is on the first, broadernotion of fundamentality. The remit of the latter notion includesquestions such as whether physics is a more fundamental science thanthe 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 andfundamentality in physics can overlap, especially for naturalisticallyinclined philosophers who wish to turn to our best science as a guideto the structure of reality. However, there are at least two importantdifferences in the usage of the notion of fundamentality in these twocontexts. First, when we consider fundamentality in physicaltheories, there is a further question about how we link whatthe theory describes or represents to reality. Second, somediscussions of fundamentality in physics are framed in a moreepistemic fashion. For instance, Crowther (2019) suggests that atheory should be considered fundamental if and only if it possesses aparticular set of epistemic features, such as being predictive at allscales (see Morganti 2020b for discussion).

We should also distinguish between the fundamentality of entitiesbelonging to a certain ontological category and the fundamentality ofthe ontological category itself (see Hakkarainen 2022). It is onething to say that certain properties are fundamental; another to saythat the categoryproperty is a fundamental category. We aremostly concerned with the former issue, but the debate about whichontological categories are fundamental and how many such categoriesthere are has been lively throughout the history of Westernphilosophy. For instance, Aristotle may have thought that substancesare the only category that isseparate from and hence morefundamental than other categories, such as the category universals(AristotlePhysics 185a31–32;Metaphysics1029a27–28; on the status of the category of substance incontemporary metaphysics, see O’Conaill 2022).^[2] Another example is the category ofnatural kinds,which is one of the four fundamental categories in Lowe’s (2006)four-category ontology, and definitive of the view that wemay callnatural kind fundamentalism (Tahko 2021a: Ch. 4.3).Some philosophers (e.g., L. A. Paul 2017) have also defend a viewaccording to which there is just one fundamental category, by way ofcollapsing the distinction between substance and property. On thisview, substances would be mereological complexes of properties andhence not an ontological category in their own right (seeKeinänen & Tahko 2019 for discussion).

Since one key task for the notion of fundamentality is to help usarticulate the view that there is ahierarchical structure toreality, it is usually assumed that the dependence relation we aredealing with must beasymmetric. Typically, the relationwould also be consideredtransitive andirreflexive,hence producing astrict partial ordering. However, each ofthese formal characteristics can in fact be questioned, albeit whetherthis 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 concernsthe hierarchy of fundamentality. Moreover, relative fundamentalitycould arguably be used to define absolute fundamentality (some entityx is absolutely fundamental if and only if it is notrelatively fundamental to any entityy). In contrast,relative fundamentality cannot be defined in terms of absolutefundamentality, so there are reasons to think that we should focus onrelative fundamentality, insofar as the notion makes sense. Giventhis, it is perhaps surprising that there are relatively few explicitaccounts of relative fundamentality in the literature so far, but wewill refer to it where relevant, and there are signs that theliterature is maturing in this regard (see Wilson 2012, 2016; Zylstra2014; Koslicki 2015; Bennett 2017: Ch. 6; deRosset 2017; Correia2021a, 2021b, 2021c; Werner 2021). One complication, as observed byCorreia (2021c), is that the notion of relative fundamentality comesin two varieties:being more fundamental than andbeingas fundamental as. The notion ofmetaphysical groundinghas 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 notionof metaphysical grounding in terms of relative fundamentality. In whatfollows, a variety of grounding-based characterizations offundamentality will be discussed, but this is not the only way tocharacterize fundamentality. In any case, the systematization ofrelative fundamentality is now proceeding at a fast pace, especiallydue to the recent efforts by Correia and Werner, who are partlyreacting to Bennett’s (2017) earlier attempts (see also Shumener2019 and Wilson 2019 on Bennett’s views).

Finally, the notion ofnaturalness and the related notion ofsparseness, familiar especially from Lewis’s work (seeLewis 1986, 2009; Schaffer 2004; Dorr & Hawthorne 2013; McDaniel2013, 2017; Thompson 2016a; Sánchez forthcoming) is sometimesconnected with fundamentality. Sider’s (2011) influential notionofstructure is closely related to the notion of naturalness(cf. Fine 2013, Mathers 2019, Tahko 2020). Perfectly naturalproperties might seem to be good candidates for absolutely fundamentalentities. Naturalness is no doubt a close cousin of fundamentality,but there are some reasons to think that the two notions cannot do thesame jobs (see Bennett 2017: Ch. 5.7). Most strikingly, there may beperfectly natural entities that are dependent in ways that are clearlyruled out by many of the definitions of fundamentality that we willshortly consider.^[4] For further discussion, see the supplement onthe natural/non-natural distinction in the entry onDavid Lewis’s metaphysics.

We should also briefly address another, related notion, familiar fromSider’s work, which links the notions of structure andmetaphysical grounding:purity (Sider 2011: Ch. 7.2).According to the principle of purity, fundamental facts involve onlyfundamental notions. Sider gives the following example, whereC* is a predicate giving a complete microstructural accountof the city of Hamburg (cf. Sider 2011: 108):

Since the notion ofcity is clearly not fundamental, puritydictates that (C) is not fundamental. SinceC* is here takento be a fundamental fact, the status of (C) becomes problematic if weaccept purity – (C) must either be fundamental itself orsomething needs to ground it. If it is fundamental, it violatespurity. But if it is not fundamental, then it needs to be grounded.What could possibly ground (C)? This is the problem ofmeta-ground, as discussed, e.g., by Dasgupta (2014a), Litland(2017, 2018b), and Correia (forthcoming). Unless a satisfactory answercan be given, one must either abandon purity or sever the link betweengrounding 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 labeledAbsolute Independence:

(AI)x isabsolutely independent if and only if, for all metaphysicaldependence relationsD, there is no suchy thatDxy.

What is included inD? We can give an open-ended list ofcandidate relations: grounding, dependence between wholes and theirparts (known as mereological or compositional dependence),realization, existential dependence, essential dependence, and,controversially, even causal dependence (on the link to causation, seeBernstein 2016; Koslicki 2016; Schaffer 2016b; Shaheen 2017). The listis open-ended partly because there is disagreement about what countsas a metaphysical dependence relation in the relevant sense and partlybecause there may be metaphysical dependence relations that we wish torule out. A possible criterion (i.e., the “relevantsense”) to be included in the list is that the dependencerelations in question must have at least some features in common, suchas, perhaps, transitivity. Furthermore, two philosophers may disagree,say, about whether compositional dependence is a genuine metaphysicaldependence relation. So, for these two philosophers,(AI) will produce a different definition of fundamentality, because thescope ofD is different. We will not attempt to give acomplete list or to define all these different kinds of metaphysicaldependence, but for further details on some of them, see the separateentries onontological dependence andmetaphysical grounding.^[5]

What is important is that(AI) is an attempt to define fundamentality in the most general termspossible, using a very broad notion of metaphysical dependence. Thereis, however, at least one kind of dependence that we will likely wishto exclude fromD, namely,modal dependence=_df Necessarily,x exists only ifyexists. The reason for this is simple: nothing will be independent inthe sense of (AI) if modal independence is required (cf. Wang 2016).This is evident if we consider some necessary existents, such asnumbers (assuming that numbers exist necessarily), for it isnecessarily 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 andthe other way around. This obviously generalizes, resulting in noentity whatsoever being “absolutely existentially modallyfree”, 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 isdifficult to find a direct endorsement of (AI) in the literature. Thatbeing said, there are some potential if controversial candidates forabsolutely independent entities, such as God. Perhaps it is alsopossible to think of the universe as a whole as absolutely independentin this sense. This might reflect something like GeorgesLemaître’s theory of the “primeval atom” orthe “Cosmic Egg” hypothesis—an idea now better knownas the Big Bang theory. Lemaître’s hypothesis was that theobserved expansion of the universe may have started from a singlepoint, the primeval atom, which would have contained the entire massof the universe. Now, one may of course postulate that the primevalatom itself could depend on something else, such as God, but thisnevertheless gives us an entirely naturalistically motivated idea ofabsolute independence. The metaphysical position that comes closest tothis idea would be a type ofmonism, perhaps motivated byconsiderations emerging from quantum holism (Calosi 2013; Ney 2015;Ismael & Schaffer 2020; Calosi 2020; Kovacs 2021; and the entry onmonism).

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

Consider symmetry principles in physics. Symmetries can be understoodas a type of invariance under transformation. Some of the simplestcases of symmetries can be found in familiar examples from the naturalworld, such as sunflowers, where we observe both radial and bilateralsymmetry. Symmetry principles play an especially important role infundamental physics, such as in conservation laws: for each conservedquantity, such as energy or linear momentum, there is a correspondingsymmetry property or invariance. This gives rise to a complicatednetwork of dependencies among the variables that feature inconservation 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 somesymmetries do allow violations. However, in the case of symmetriesassociated with conservation laws, Lange (2016) has argued that thereis a stronger type of necessity in effect, stronger than the naturalnecessity usually associated with laws of nature. Whether or not Langeis correct about this, it clearly important to consider thepossibility of non-causal dependencies such as those imposed bysymmetry principles because they may hold between entities that areindependent in some other sense (such as the mereological sense).

Relatedly, Giannotti (2021a) has discussed the possibility offundamental yet mutually grounded entities in more detail,mentioning the case of quarks as well as Leibnizian monads aspotential examples (cf. also Wilson 2016: 192–193; Wilson 2020:293). As Giannotti correctly observes, supposed examples offundamental yet mutually grounded entities are controversial, and heeven 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 mutuallygrounded entities, this does not prevent a grounding-based account offundamentality. However, Giannotti’s solution requires acceptingnon-asymmetric grounding, which itself goes against thegrounding “orthodoxy” (see Raven 2013). Giannotti (2022b)has recently expanded on this and proposed the labels ofradicalbrutalism for the view according to which the fundamental factsare those which arewholly ungrounded andmoderatebrutalism for the view according to which the fundamental factsare those which arepartially ungrounded (see also Bader 2021on bruteness and fundamentality). The upshot of Giannotti’sdiscussion is that the choice between radical and moderate brutalismis currently underdetermined by empirical evidence, leading Giannottito recommendpluralistic brutalism, according to which somefundamental facts are moderately brute whereas others are radicallybrute.

Whichever way we go here, it appears that many intuitive candidatesfor fundamentalia would not be fundamental on the(AI) definition of fundamentality. A key issue, as with Giannotti’sanalysis, concerns the possibility of fundamentalia being mutuallygrounded 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 beeasily revised to accommodate such entities, but this is an issue thatthe proponent of fundamentality should consider further, and there maybe reasons to leave room for self-dependent entities (see Bliss &Priest 2018b).

Secondly, there is a speculative class of entities that would befundamental according to(AI), but are in tension with other plausible theories. These entities aresometimes 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 activerole in the workings of nature”. So, idlers are at leastcausally isolated. However, we can further postulate that“absolute idlers” are absolutely isolated: they depend onnothing, and nothing, at least nothing concrete, depends on them.Absolute idlers might thus be absolutely independent in the senserequired by (AI). Whether there are any such absolute idlers is ofcourse another question. If there are any absolute idlers, they arerather uninteresting entities—indeed, they are presumablycompletely beyond our ken given their isolation. Moreover, for theidea of absolute idlers to have any plausibility, we would surely haveto restrict their isolation in such a way that they may participate inabstract constructions such as sets. This is because classical settheory, and indeed many other theories, entail that all entities haveat least some entities that depend on them. This may be a good reasonto think that no idler could be absolute in the strongest possiblesense.

1.2 Restricted Independence

The second definition of fundamentality to be considered is much moreversatile and weaker than(AI). We call itrestricted independence. This produces arelativized sense of fundamentality, where fundamentality isrelative to a certain variety or varieties of metaphysical dependence.One should not confuse this withrelative fundamentality,which concerns the priority ordering between two (non-fundamental)entities. So, restricted independence starts from the idea that foreach metaphysical dependence relation there is a corresponding notionof fundamentality and we must relativize the notion of fundamentalityaccordingly:

(RI)x isrestrictedly independent if and only if there is one or moremetaphysical dependence relation(s) \(D_{1}\), \(D_{2}\) …\(D_{N}\) such that there is noy such that \(D_{1xy}\) or\(D_{2xy}\) or … or \(D_{Nxy}\).

(RI) is restricted to concern one or more specific kinds ofmetaphysical dependence, because it is plausible that somemetaphysical dependence relations, such as modal dependence (as we sawinsection 1.1), would immediately rule out fundamental entities. Notice that (RI)allows the possibility of including several dependence relations, butwe 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 thatare considered relevant to fundamentality, so here two philosopherscould disagree about which dependence relations to include in thatsubset. Variations of this conception of fundamentality can be foundthroughout the literature and it is probably as close to a standardconception 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 differencesbetween the various accounts. Variations of (RI) are standard enoughto have been picked up by most of those critical towards thisconception 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 fundamentalreality needs arelational underpinning (on the notion of arelational underpinning, compare Fine 2001: 25). In other words,whatever fundamentality amounts to, it must be the case that one (orseveral) of the various relations of metaphysical dependence may beused to define fundamentality. But since (RI) leaves completely openwhich of the metaphysical dependence relations are in fact relevant tofundamentality, we should mention some examples. At one point in theearly 2000s, it may have been the case that most philosophers usingthe notion of fundamentality had in mind what we might callmereological fundamentality whereby the relevant kind ofdependence 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 acertain property, may depend on the properties and relations had byits parts. (2010: 183; see also Markosian 2005; Thalos 2010, 2013)

However, even though mereological dependence is still sometimesconsidered to be relevant to fundamentality, it is becoming lesscommon to view it as theonly relevant kind of dependence(Wilsch 2016; Bennett 2017: 8–9). Be that as it may, there iscertainly an esteemed history for this type of idea, given that(mereological) atomism of the type defended already by ancientphilosophers such as Leucippus and Democritus would appear to be aninstance of the view (see the separate entry onancient atomism for details, and compare with Schnieder 2020a, where Bolzano’sversion of atomism is discussed).

It is important to see that the thesis that some things aremereologically fundamental does not entail a commitment to atomism.Here it may be helpful to introduce the distinction between (priority)pluralism andmonism. We have already seen that amonist might be attracted to a view about fundamentality where thereis just one fundamental entity, such as the cosmos or universe as awhole (Schaffer 2010a is a good exposition and defense of thisSpinoza-inspired view; see also Newlands 2010). This view does notdirectly entail anything about the relative fundamentality relationsamong the non-fundamental, but it enables us to better understand thatthe converse of mereological dependence, as Kim defines it, mighthold: instead of wholes depending on their parts, the parts coulddepend on the whole. This contrasts with the more familiar idea, oftenassociated with but not entailed by atomism, that mereologicaldependence must run from the larger to the smaller, mereological atomsbeing the fundamental entities. So, the choice concerning thedirection of the relevant dependence relation is often reflected bythe choice between priority pluralism and monism, although strictlyspeaking 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 dependencerelations is relevant to fundamentality, may disagree on the directionof the relevant dependence relation(s).^[8]

Moving on to a different subspecies of(RI), we see a clear link between grounding and fundamentality, wheregrounding is understood as expressing a non-causal connection betweentwo things. For example, a certain act might be considered evilbecause it causes harm. The “because” in this statementdoes not express a causal link; instead, it tells us whatgrounds the evilness of the act. Similarly, one might thinkthat mental states hold in virtue of neurophysiological states or thata substance is prior to its tropes. The notions of “in virtueof” and “prior to” in these cases may be understoodin terms of grounding. What is likely to be the most commoncontemporary understanding of (RI) is that the most important (if notthe only) relation of metaphysical dependence that is relevant tofundamentality is grounding. Note, however, that grounding as wellcould be understood as a family of dependence relations (Trogdon2013), and there is an on-going discussion about the relationshipbetween grounding and ontological dependence more generally (e.g.,Schnieder 2020b; Rydéhn 2021). Furthermore, since ourdiscussion of fundamentality has focused onentities, it isworth noting that bridging fundamentality and grounding suggests acommitment toentity grounding, whereby all sorts of entitiescan stand in grounding relations. However, some prefer to restrictgrounding tofacts only, and this would accordingly require asomewhat different formulation of the relevant fundamentality claims(see the separate entry onmetaphysical grounding and especially the section onregimentation for further details).

On the grounding-based characterization of fundamentality, thefundamentalia areungrounded entities: everything is eitherungrounded or ultimately grounded in the fundamental, ungroundedentities (Schaffer 2009: 353; Audi 2012: 710; Dasgupta 2014a: 536;Raven 2016: 613). For instance, Audi (2012: 710) explicitlydistinguishes between theexplanatorily fundamental and thecompositionally fundamental where the first is associatedwith grounding and the second with mereological dependence. As Audicorrectly notes, one would think that often these two notions offundamentality would overlap. But recall that we’ve justobserved that the converse of mereological dependence, running fromthe smaller to the larger, could hold. For the priority monist, thisdependence relation running from the smaller to the larger isgrounding, so it would also produce a different understanding offundamentality.

If explanatory fundamentality and compositional fundamentality aregenuinely two different notions corresponding to two independentnotions of fundamentality, then(RI) does allow the overlap of explanatory and compositionalfundamentality. But there could certainly be other considerations thatcount against this. For instance, if (RI) is understood to mean thatto be fundamental is just to be at the termination point of somedependence relation, then the debate between monism and pluralismcould just be a debate about the direction of compositionalfundamentality, where compositional fundamentality terminates in themereological atoms and its converse terminates in the whole cosmos. Inother words, this could be understood as a debate about whetherexplanatory fundamentality aligns with compositional fundamentality orits converse. Whichever way we go here, we would seem to end upaligning some notions of fundamentality and misaligning others.

An important question arises: should we indeed postulate severalindependent notions of fundamentality, relativized to each of themetaphysical dependence relations, or should we aim to define only onesense of fundamentality, either to be defined in terms of just onerelation of metaphysical dependence or in terms of some privilegedproper subset of these relations?

Given that there are good reasons to exclude some notions ofdependence, such as modal dependence and perhaps causal dependence, itwould appear that there is an additional question about why only somenotions of dependence are such that we want to define a correspondingnotion of fundamentality for them. There is an obvious challenge forthe view that we should postulate several relativized notions offundamentality. The challenge is simply that the notion offundamentality would seem to do little in addition to the variousrelativized notions of (in)dependence. Indeed, this is likely to causeconfusion, because the notion of fundamentality is sometimes used inthe literature without any mention of which relativized sense ofindependence is in question. Moreover, since different types ofmetaphysical dependence have different formal properties (e.g., someare strict partial orders, but some might be symmetric, reflexive ornon-transitive) and they can perhaps even run in different directions,it is difficult to see what couldunify the different notionsof fundamentality, i.e., the proper subset of those dependencerelations that we consider to be relevant for fundamentality.^[9]

Those who think that grounding is strongly unified often appeal to itsformal 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, itshould at least fit these properties. Relevant formal propertiesinclude the following: grounding is a strict partial order,non-monotonic in the sense that we cannot add arbitrarygrounds and expect that grounding still holds (i.e., ifA isgrounded inB, then it does not follow thatA isgrounded inB andC), and it is thought that groundsmetaphysically necessitate what they ground (although see Leuenberger2014; Skiles 2015 against necessitation). In contrast, Bennett’sbuilding relations (seenote 10) do not share all their formal properties. She holds thatthey’re all antisymmetric and irreflexive, but not necessarilytransitive (Bennett 2017: 46). Another example of this type of“multi-dimensional” approach to fundamentality can befound in Koslicki’s (2012, 2015, 2016) work.^[10]

It’s important to see that on a multi-dimensional view wherebythe various dependence relations relevant to fundamentality might evenrun in different directions, one must be very careful indeed tospecify which relativized notion of fundamentality one has in mind. Ifit turns out that some entity is independent in all the relativizedsenses of independence relevant to fundamentality, then it would seemthat we are back to absolute independence(AI) (or independence “full stop”, as Bennett 2017: 106 callsit).

Here we would do well to systematically distinguish between thoseproponents of(RI) who hold that only one metaphysical dependence relation, e.g.,mereological dependence or a strongly unified notion of grounding, isrelevant to fundamentality and those who think that a proper subset ofthese relations is relevant to fundamentality. Sometimes the labelsmonism andpluralism are used to distinguish betweenthe singular fundamentality view and the multi-dimensional view, butin the interest of clarity we should introduce different terms, as wehave already used these labels for another purpose.^[11] So, let us use the labels (RI-one) and (RI-many) to distinguishbetween those who think that there is only one notion offundamentality and those who think that there are several. Typically,there is an easy translation between these views. Let us say that aproponent of (RI-one) thinks that only mereological dependence isrelevant to fundamentality. Well, a proponent of (RI-many), providedthat mereological dependence is one of the relations they considerrelevant to fundamentality, can simply translate the (RI-one) notionof fundamentality into their (RI-many) notion of mereologicalfundamentality. Hence the disagreement is that from the (RI-one) pointof view, there is a single notion of fundamentality, and they wouldthink that (RI-many) mistakenly holds other notions of fundamentalityto be genuine, whereas from the (RI-many) point of view, there aremany relativized notions of fundamentality, and (RI-one) mistakenlypicks 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 allproponents of(RI): is there a relativized notion of fundamentality foreverymetaphysical dependence relation, or only some of them? Further, ifonly a proper subset of metaphysical dependence relations are relevantto fundamentality, then what makes these dependence relations relevantin this way? A proponent of (RI) may refuse to answer this question,since it would be an additional commitment to privilege somemetaphysical dependence relations in this way. However, given that onemajor task for the notion of fundamentality is often thought to berelated to the idea of the hierarchical structure of reality, it seemsthat we may have to accept only a proper subset of metaphysicaldependence relations as being relevant for fundamentality if this taskis to be preserved (of course, it could also be abandoned!). Forinstance, perhaps only those metaphysical dependence relations thatare asymmetric and transitive would qualify. Still, there may also bedependence relations that are relevant to fundamentality that fail tobe transitive. At this point, it could be helpful to move on to thethird potential definition of fundamentality, as something along thelines of this definition is often used to characterize fundamentalitydefined in terms of (RI) as well.

1.3 Complete Minimal Basis

The conception of fundamentality to be considered in this section isoften used to explicate the second key task for the notion offundamentality, that of the fundamental entities acting as the basicbuilding blocks of reality. According to this approach, thefundamental entities determine everything else. By giving a completelist of the fundamental entities, we can provide a minimal completedescription of reality. So, we are now shifting our focus from thatwhich everything else depends on to that which, as it were, supportseverything else. This idea is often invoked to characterizefundamentality, 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 ofacomplete minimal basis:

(CMB)x isfundamental if and only ifx belongs to a plurality ofentitiesX andX forms a minimal complete basis thatdetermines everything else. The complete basis is minimal if no propersub-plurality of the entities belonging toX iscomplete.

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 fundamentaliamay 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 thanmere necessitation or supervenience, although the idea is closelyrelated 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 moveon to a more general discussion about(CMB).

(CMB) is compatible both with monism and the idea that reality mayhave anirreducibly plural underpinning whereby ifXforms a complete minimal basis, no proper sub-plurality ofXwill be complete.^[15]
We have included aminimality condition in the definitionof (CMB). This is an important addition, because otherwise we couldtake the plurality of all the entities in the world and call itcomplete, given that this plurality as well would include all thefundamental entities. So, according to (CMB), the complete minimalbasis must include all and only the fundamental entities.
There is an open question concerninguniqueness. Couldthere be several distinct pluralities that are minimal and complete?In other words, could there be distinct minimal pluralities that areeach complete and hence capable of determining everything else?Bennett (2017: 112ff.) leaves the possibility of distinct minimalcomplete pluralities open but makes some use of the notion of aunique minimally complete plurality, whereas Tahko (2018)speculates about the possibility of several “ontologicallyminimal descriptions”, dropping the requirement of uniqueness.^[16]
It is, in principle, possible to define relativized versions of(CMB), just as we did insection 1.2 with(RI). So, one could distinguish betweenabsolute completeness andrestricted completeness. It is easy to see how the restrictednotion of completeness is supposed to work. For instance, themereologically minimal complete basis is the minimal completeplurality of mereological elements that determine everythingmereological. This could leave out something determined, say, by apotentialnon-mereological notion of composition or parthood,such as Armstrong’s notion of states of affairs (see Armstrong1997: 118–127). However, taking this route does not only invitethe challenges observed with regard to (RI), but may also diminish thepotential initial attractiveness of (CMB) (Bennett 2017: 110). So, wewill 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 acomplete minimal description of reality. This may be compared toSchaffer’s “fundamental supervenience base”,Jenkins’s “by appeal to which all the rest can beexplained”, Raven’s “ineliminability”, andTahko’s “ontological minimality”; see also Lewis(1986: 60). On this reading, less emphasis is put on the active roleof the fundamentalia, the focus being instead on making sure toinclude a basis for everything in reality.

To have the broadest possible appeal, a characterisation offundamentality should ideally be applicable in different scenariosregarding what the world is like. One such scenario is a “flatworld”, a world where everything is independent, either in thesense of(AI) or (RI), and nothing is built, to use Bennett’s notion (Bennett2017: 123–124). In a flat world, nothing determines anythingelse in the sense of (CMB), since nothing depends on anything else.Initially, it may seem that (CMB) is not plausible in this type ofscenario, since everything is included in the unique minimallycomplete plurality. However, as noted in the previous paragraph, thedefinition of (CMB) does not require any “active”contribution from the fundamentalia; it’s not crucial that thefundamentalia do any determining. So, the upshot is that, in flatworld, everything is fundamental both in the sense of (AI) or (RI) aswell 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, theproponent of this more “active” version, (CMB+), requiressomething more. Driven by the flat world example, one might then thinkthat(AI) and(RI) are prior to (CMB+). According to (CMB+), a flat world—ratherthan being a world where everything is fundamental—would be aworld where nothing is fundamental (or derivative), since nothing is“actively” determining anything. A world which contains nosuch structure would then turn out to be a world which contains nopriority, hence no fundamentality (cf. Wilson 2016: 199).

Another difference between(CMB) and(RI) concerns the possibility ofmetaphysical coherentism, whereloops of dependence are possible (see Bliss 2014, 2018; Barnes 2018;Morganti 2018, 2019; Nolan 2018; Thompson 2018; Swiderskiforthcoming). (CMB) can accommodate the possibility of metaphysicalcoherentism, but (RI) would seem incompatible with it. The primaryreason for this is that metaphysical coherentism, as it is usuallyunderstood, violates asymmetry and hence drops the idea that thestructure of reality is hierarchical. While this requires abandoningone of the key tasks that the notion of fundamentality is oftenthought to have, it does help in accommodating the idea that we shouldadopt a more holistic approach whereby entities can be mutuallyrelated and form loops or cycles. Sometimes an analogy fromepistemology is offered, as these loops of dependence could resemble aQuineanweb of belief, so that each entity depends on one ormore other entities. Perhaps the most plausible example of thepossibility of loops of dependence comes fromontic structural realism (OSR). (OSR) suggests that objects may be reduced to—or moremoderately, are ontologically on a par rather than priorto—relational structures. If (OSR) is true, we might have torevise our views about what the fundamentalia could be, as they mightbe relations rather than objects (for discussion on OSR andfundamentality, see Wolff 2012; McKenzie 2014; Morganti 2018, 2019;Tahko 2018). What exactly happens to (CMB) on this type of viewdepends on the details of the coherentist framework, but onepossibility—on a moderate understanding of (OSR)—is thatthe fundamentalia would include mutually dependent relations andobjects, which then determine everything else. Metaphysicalcoherentism is beginning to receive increasing attention. Forinstance, Calosi and Morganti (2021) have recently put forward acoherentist account of quantum entanglement, and Swiderski(forthcoming) has made an important effort to systematize theliterature by defining four different conceptions of coherentism,namely (we refer the reader to the original paper for illustrativediagrams):

Holism: For anyx and anyy,x(partially) groundsy andy (partially) groundsx (ibid., p. 5).
Insularism: For anyx, there is someysuch thaty groundsx (andx ≠y ) and somez such thatx does not groundz (andx ≠z ). In addition, allgrounding relations are regarded as symmetric (ibid., p. 10). Thisview suggests that there are more than one networks of symmetric orcoherent grounding, where no grounding between these different insularclasses of facts takes place, but we have maximal coherence withineach class.
Hierarchism: For anyx, there are somey’s such that each of they’s groundsx andx grounds each of they’s, andeither (i) there are somez’s (distinct from they’s) such that each of thez’s groundsall the otherz’s, and thez’s groundthey’s, or (ii) there are somew’s(distinct from they’s) such that each of thew’s grounds all the otherw’s, and they’s ground thew’s (ibid., p. 17). Thisview suggests that there is a hierarchy of facts that mutually groundeach other which form alevel, and any level has got either(i) a level below it or (ii) a level above it. (Swiderski alsospecifies two further varieties ofhierachism depending onwhether one wishes to include or rule out infinite descent.)
Rebarism: For anyx, the followingexclusive disjunction is true:either (i) there aresomey’s such that each of they’sgroundsx andx grounds each of they’s, or (ii) there are somez’s suchthat each of thez’s grounds all the otherz’s, and thez’s groundx(ibid., p. 21). This view suggests that only the fundamental levelinvolves symmetric or coherent grounding, while everything else (onhigher levels) stands in asymmetric grounding relations, generatingthe familiar hierarchical picture of reality.

While Swiderski does not explicitly defend any of these versions ofcoherentism, his discussion does point out some crucial decisionpoints. Most importantly, Swiderski demonstrates that while holism andinsularism are not compatible with asymmetric grounding, hiearchismand rebarism do allow for (limited cases of) asymmetric grounding.However, all four views retain a commitment to what Swiderski (ibid.,p. 4) calls theCoherentist Canon: “(i) For anyx, there is somey such thaty groundsx, and (ii) there is somez and somew suchthatz (perhaps indirectly) groundsw and viceversa”. This also suggests a commitment to the idea of mutualgrounding (cf. Giannotti 2021a), which was briefly discussed insection 1.1.

Finally, a specific form of metaphysical infinitism, where the samestructure repeats infinitely, to be discussed in more detail insection 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 definefundamentality. But we may be able to characterize it, and it is to beexpected that(RI) and(CMB) are likely candidates in this regard. This type of view may be whatFine (2001: 26) gestures towards, when he remarks that it is theworld’s intrinsic structure that is fundamental. One way todevelop the idea is by defining absolute fundamentality in terms ofrelative fundamentality and introducing a primitive notion of“Reality” as it is in itself (Fine 2001; see also Fine2009). Note however that Fine proposes this notion in connection tothe debate between realism and anti-realism, whereby“Reality” may be understood as objectivity. So, it is notentirely clear that we can understand fundamentality in terms of thisnotion.

More generally, we may think that the absolute notion of fundamentalreality is not in need of arelational underpinning (Fine2001; Wilson 2014: 561). This is in contrast to the variouscomparative or relational notions of dependence that we have beendiscussing (see also Fine 2015). Note that primitivism must also becontrasted with(CMB) insofar as it is understood as a definition of fundamentality, eventhough it is not uncommon to see primitivist characterizations offundamentality that resemble (CMB). For instance, a primitivist whoaccepts Fine’s notion of “Reality” would like todistinguish between what is Real in itself and what may be true (i.e.,objective) even if it does not concern the way that things arefundamentally. As Fine puts it,

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

The idea behind the primitivist view is actually very simple and itmay be prudent to try to capture it without the controversial notionof “Reality”. One way to do so would be to understand theentities in the fundamental base to be basic in the sense that theyplay a role analogous to axioms in a theory, or that the fundamentalentities are all that God had to bring about to make the world (Wilson2014: 560; 2016; see also Dorr 2005). We can find several occurrencesof this type of heuristic in the literature on fundamentality,e.g.,

The primary is (as it were) all God would need to create. (Schaffer2009: 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 termsby saying that all God had to do in order to create the world was fixthe fundamental facts. (Glazier 2016: 35)

To drive the idea home, consider the following passage fromKripke:

Suppose we imagine God creating the world; what does He need to do tomake the identity of heat and molecular motion obtain? Here it wouldseem that all He needs to do is to create the heat, that is, themolecular motion itself. If the air molecules on this earth aresufficiently agitated, if there is a burning fire, then the earth willbe hot even if there are no observers to see it. God created light(and thus created streams of photons, according to present scientificdoctrine) before He created human and animal observers; and the samepresumably holds for heat. How then does it appear to us that theidentity of molecular motion with heat is a substantive scientificfact, that the mere creation of molecular motion still leaves God withthe additional task of making molecular motion into heat? This feelingis indeed illusory, but whatis a substantive task for theDeity is the task of making molecular motion felt as heat. To do thisHe must create some sentient beings to insure that the molecularmotion produces the sensationS in them. Only after he hasdone this will there be beings who can learn that the sentence“Heat is the motion of molecules” expresses anaposteriori truth in precisely the same way that we do. (Kripke1980: 153)

Now, Kripke is of course not trying to put forward an account offundamentality here, but rather to specify that the qualitativeexperience of feeling heat is something additional to molecularmotion. The idea, however, is strikingly close to the primitivistaccount of fundamentality, whereby the important thing is to find asufficiently rich basis for everything in the world—this is ofcourse 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, onemight wonder how it is that she can decide on what the fundamentalentities are. In other words, what is our epistemic access to issuesconcerning fundamentality? A possible answer is that we proceed thesame way as we do with other primitives in metaphysics. That is, byasking how the view thatx is fundamental fits in our overalltheory. How does it fare with regard to our other commitments? Theseare issues where theoretical virtues such as simplicity andexplanatory power might be employed (for a related discussion, seeSchaffer 2014). However, this is not the place to tackle theadmittedly difficult epistemic questions surrounding metaphysicaltheories in general, or the criteria that are to be employed intheory-choice. The primitivist about fundamentality may have one moreprimitive in their overall theory, and this, of course, calls forjustification. But conceiving of the fundamental as primitive is nomore mysterious than conceiving of, say, naturalness or grounding asprimitive. This is not to say that postulating naturalness orgrounding as primitives wouldn’t also require some justificationthough.

The primitivist about fundamentality does, however, face somechallenges that are specific to fundamentality. Some of thesechallenges have been raised by Schaffer (2016a), targetingWilson’s proposal. A potentially helpful clarification fromSchaffer is the following: there is something primitive in each of thetheories on offer, but the primitivist about fundamentality takesbeing-absolutely-fundamental as that primitive, whereas atleast 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 theprimitive (Schaffer 2016a: 157). As Schaffer notes, one may also takeboth to be primitive, as Fine perhaps does. Schaffer argues that whileabsolute fundamentality can be quite easily defined in terms ofgrounding, as we’ve seen with (RI), it may not be so easy todefine grounding in terms of absolute fundamentality. As observedearlier, there have also been recent attempts to define grounding interms of relative fundamentality (Correia 2018), as well as attemptsto define notions of relative fundamentality in terms of grounding(Correia 2021b, 2021c; Werner 2021), but none of these options aredirectly available to the primitivist.

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

One possibility is to try fixing the direction of priority even in theabsence of an absolutely fundamental level (Wilson 2016: 196ff.).Drawing on a suggestion from Montero (2006: 179), Wilson proposes thatanalogously to an infinite sequence of numbers such as 1/2, 1/3, 1/4… being still “bounded below” by zero, there couldbe an infinite descent of fundamental entities that approaches a limitwhereby the limit acts as the fundamental level even though it isnever reached. Related ideas may be found elsewhere in the literature(Tahko 2014; Morganti 2015; Raven 2016). One difficulty with thisproposal is that applying the notion of a limit assumes that we canassign a numerical measure to the descending entities that approachthis limit. So, we would need to be able to construct the hierarchicalstructure of relative fundamentality in such a way that this numericalmeasure applies to it. Moreover, even if approaching a limit does notnecessarily need to be constructed numerically, the notion of a limititself requires postulating the (fundamental) limit, in this casezero. While zero is not part of the sequence, it does seem to be partof the ontology.

Another possible strategy to address this challenge to primitivism isto argue that fixing the fundamental is not (always) enough to fix thepriority ordering. Rather, we should pay close attention to thevarious dependence relations that interest us, such as the parthoodrelation, and assess the nature of the non-fundamental entities thatthese relations relate. Importantly, we might get different answersdepending on which view about these natures is correct (see Wilson2016: 200ff.).

This concludes our discussion of the different ways to understandabsolute fundamentality. We will now move on to discuss a variety ofimportant 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 keytasks for the notion of fundamentality is to capture the idea thatthere is a foundation of being and that everything else depends on thefundamental entities. This idea about fundamentality is oftenexpressed in terms ofwell-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 ofwell-foundedness itself is sometimes used without much furtherqualification, and there are many instances where another term is usedinstead for what is clearly the same general idea (e.g., Lowe 1998:158; R. Cameron 2008; Paseau 2010; Rosen 2010). A common formulationof this general idea is as follows: a priority/dependence chain iswell-founded if and only if itterminates, i.e., has an endconstituted by one or more entities that do not depend on any otherentity. But not all the above-mentioned authors would be happy withthis particular formulation, and it turns out that sometimesphilosophers may even have talked past each other due to havingslightly different formulations of well-foundedness in mind.Fortunately, the literature has matured, and we now have a number ofmuch more precise accounts of the various potential formulations ofwell-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 withthe hope of making the metaphysical idea more precise. A simpleformulation of set-theoretic well-foundedness can be found in Cotnoirand Bacon (2012: 187):

An order < on a domain is said to bewellfounded if everynonempty subset of that domain has a <-minimal element.

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

The set-theoretic formulation of well-foundedness is the sense inwhich, e.g., Fine (2010: 100) appears to use the notion. Applied tochains of grounding, the proposal is that well-foundedness would ruleout infinite, non-terminating grounding chains and entail that“the grounds of any truth that is grounded will ‘bottomout’ in truths that are ungrounded”. We can see slightlydifferent 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-theoreticnotion of well-foundedness is that it is a straightforward applicationof the standard mathematical definition of a well-founded relation(this is also recognized by Morganti 2015: 556fn2). The only problemis that the standard understanding of well-foundedness may be toostrict for the task at hand. Many authors have considered weakenedversions of the well-foundedness requirement, which may be moresuitable for expressing the desired limits on infinite groundingchains (R. Cameron 2008: 4; Trogdon 2013: 108; Leuenberger 2014:170–171). So, to summarize, it might at first seem intuitive tointerpret well-foundedness simply as a ban on infinite chains andcycles of grounding, in accordance with set-theoreticwell-foundedness. But this would seem to go against the intuition thatthere are cases of infinite chaining which are acceptable for thefoundationalist metaphysician (R. Cameron 2008; Bliss 2013). This hasrecently been made explicit by Dixon (2016) and Rabin & Rabern (2016).^[20]

To get us started on the idea that foundationalist metaphysicianscould accept violations of well-foundedness, consider an importantobservation by Bliss (2013: 416), which also comes up, e.g., in Rabin& Rabern (2016: 362). She distinguishes between finite andwell-founded grounding chains where a finite grounding chain not onlyterminates in something fundamental, but is also such that we canreach the fundamental entities in a finite number of steps fromanywhere in the chain. According to this approach, a well-foundedgrounding chain is indeed one that is grounded in somethingfundamental, but it may itself be infinitely long. So, on thisterminology, finite grounding chains are always well-founded, but awell-founded grounding chain could be infinite. But note that thisalready violates set-theoretic well-foundedness as defined above. Todemonstrate this, consider an infinite chain of dependence \(f <\ldots d_{3} < d_{2} < d_{1}\), where the chain of dependententities \(d_{n}\) terminates in some minimal elementf. Now,if we take a subset of that chain of dependence without the minimalelementf, then we are left with a chain that lacks a<-minimal element, hence violating the set-theoretic definition ofwell-foundedness.

To get clearer on what is at issue here, let us introduce anothernotion, being “bounded from below” or “having alower bound”. We can say that an order < on a domain isbounded 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 ismore 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 lowerbound 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 isset-theoretically well-founded and bounded from below. We’vejust seen that some infinitely descending chains of dependence may bebounded from below yet fail to be set-theoretically well-founded. AsRabin & Rabern (2016: 360) put it, an infinitely descending chainmay or may not have a “limit”, where limit is the greatestlower bound of a set. In the previous example, 3 is the greatest lowerbound of the interval [3, 4, 5]. So, one might suggest that therelevant sense of metaphysical well-foundedness is captured either bythis idea of being bounded from below or by having a limit, a greatestlower bound. But it turns out that even these may be too strong. Thisis the key insight in the recent work of Dixon and Rabin & Rabern.If this is right, then we need to have a sense of ametaphysicalfoundation that is not only compatible with infinite chains ofdependence, but also with infinite chains that do not have a lowerbound (i.e., “unbounded” chains).^[21]

It might be helpful to present a somewhat simplified example of a casewhere well-foundedness and being bounded from below come apart. Let ususe a variation of Trogdon’s (2018b) reconstruction of anexample by Rabin & Rabern (2016: 361; this is an example of whatDixon 2016: 448 calls a “fully pedestalled chain”).Consider a spherical region of spaceS, which we divide up insuch a way that each of the proper sub-regions ofS has aproper sub-region, and each of those sub-regions in turn has a propersub-region. In principle, we can continue this process infinitely.Now, let us assume that each region of space derives its being in partfrom (or is partially grounded in) its sub-regions. Here we would seemto have an infinite regress where, to use the familiar phrase, beingis forever deferred, never achieved. But suppose in addition thatthere are spatial points and that these points are fundamental. Thenwe might like to say thatS and indeed each of its infinitelymany proper sub-regions fully derives its being from (or is fullygrounded in) spatial points. In this case, the infinite regress of thesub-regions would appear to be innocuous since each of them is afterall fully accounted for by the fundamental spatial points. Still, thecase for a sense of fundamentality that is compatible with at leastsome kind of non-well-founded infinite descent or other seems to havebeen made sufficiently clear in recent work. Metaphysicalfoundationalism does not seem to be properly captured by the idea ofset-theoretic well-foundedness. So, what is it about?

We can find numerous formally precise attempts to capture the relevantsense of a metaphysical foundation in recent work. Dixon’s(2016: 446) preferred understanding (which he calls “fullfoundations”) suggests that every non-fundamental fact is fullygrounded by some fundamental fact(s). Rabin & Rabern (2016: 363)attempt to capture the idea of a metaphysical foundation with thephrase “having a foundation”, whereby a groundingstructure has a foundation if and only if there is some set of factsthat (i) together ground all the derivative facts and (ii) arethemselves ungrounded. Raven suggests (2016: 612) that metaphysicalfoundationalism could be understood in terms ofineliminability (to be discussed insection 3). Tahko (2014: 263) tries to analyze metaphysical foundationalism interms of an “ontological” sense of well-foundedness, whichrequires that an ontologically well-founded chain terminates in afundamental supervenience base. Trogdon (2018b) follows the recentsuggestions for a weaker understanding of metaphysical foundationalismand defines it as the view that, necessarily, any non-fundamentalentity is fully grounded by fundamental entities. Even though theterminology varies, it’s quite clear that there are weakersenses of metaphysical foundationalism than the one defined in termsof set-theoretic well-foundedness. We arrive at the followingdefinition ofmetaphysical foundationalism:

(MF) Everynon-fundamental entity is dependent_{D₁,D₂ … D_N} on some fundamentalentity or entities that fully account for itsbeing/reality.

This definition of metaphysical foundationalism is somewhat vaguesince it attempts to capture the idea for all the different varietiesof fundamentality that we have discussed, but it may be supplementedwith 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 ofdependence, exactly as with the schemas proposed in(AI) and(RI) insection 1. Furthermore, we need to understand “fully account for”correspondingly, for instance, in the case of grounding it should beparsed as “fully grounds” and in the case of compositionaldependence it should be parsed as “fully composes” (albeitin some cases it may not be entirely clear that a full/partialdistinction properly applies). Now that we have a working notion ofmetaphysical foundationalism, we may proceed to discuss arguments infavor and against this view.

3. Metaphysical Foundationalism

Metaphysical foundationalism is the view that reality has afoundation—that there is a “fundamental level”, in asense that needs to be specified. The most common way to specify theidea of having a foundation is in terms of well-foundedness, but as wehave seen insection 2, set-theoretical well-foundedness may be too strong to capturemetaphysical foundationalism. Metaphysical foundationalism comes in avariety of strengths, depending on how much the well-foundednessrequirement is weakened. It seems reasonable to say that until quiterecently metaphysical foundationalism was the default position (R.Cameron 2008; Schaffer 2009; 2010a; Bennett 2011a). The relevantintuition is often captured with the much-cited phrase that without afoundation, “a ground of being”, “[b]eing would beinfinitely deferred, never achieved” (Schaffer 2009: 376; 2010a,62).

Sometimes the foundationalist intuition is explicitly tied tocomposition (i.e., mereological dependence) and the (im)possibility ofgunk, namely, the idea that everything has a proper part:“the anti-gunk worry is that composition could never have gotoff the ground” (R. Cameron 2008: 6). The worry is that complexobjects are not possible in gunky worlds, so given that there iscomposition, there must be a foundation. However, we’ve seenthat fundamentality need not be tied to compositional/mereologicaldependence. Moreover, others (McKenzie 2011; Bliss 2013; Tahko 2014;Morganti 2014, 2018; Bohn 2018; Trogdon 2018b) have been suspicious ofthe driving intuition behind this sense of metaphysicalfoundationalism, and now even some of those who have earlier defendedmetaphysical foundationalism opt for agnosticism on the question(Bennett 2017: 120ff; Rosen 2010: 116). Indeed, there is, perhaps, nowa consensus that it is very difficult to come up with a properargument in favor of metaphysical foundationalism, something thatwould go beyond the just stated intuition. The idea has a statuscloser to a type of metaphysicalaxiom orlaw asBohn (2018) puts it (see Morganti 2018 for a good overview). Thisconclusion seems even more warranted given that defining metaphysicalfoundationalism simply in terms of set-theoretical well-foundednessturned out to be too strict. To establish a clearer sense ofmetaphysical foundationalism which is in accordance with the mostrecent literature, we shall understand “metaphysicalfoundationalism” as it was defined insection 2 (MF) (see Oberle 2022a for an overview of the most recentdebate).

Can we be clearer on the core idea of metaphysical foundationalism? Ifwe can, then perhaps potential arguments for the view would also bemore easily available. One interesting attempt to do so is Raven’s.^[22] Raven’s version of metaphysical foundationalism relies on thenotions of “eliminability” and“ineliminability” whereby an entity is eliminable ifreality is described no worse without mentioning it and ineliminableif reality cannot be completely described without mentioning it. Todemonstrate this, let us take advantage of Raven’s ownterminology (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 ofsome fact about it, a last occurrence of it after which the entitynever recurs. So, for an entity to persist is for some facts about itto be unbounded. Importantly, there are two ways to be unbounded:being ungrounded or having grounds but forever recurring. The firsttype of persistence is the familiar sense of relative independence(RI) as defined insection 1.2. But the second type of persistence, where an entity forever recurs inthe chain of dependence, is novel.^[23]

A potentially helpful way to illustrate the options available to usand clarify the relationship between metaphysical foundationalism andinfinite descent is Morganti’s (2015: 562) “emergencemodel of being”. This model may be contrasted with the“transmission model”, which is, perhaps, what themuch-quoted phrase from Schaffer that we led with reflects. Accordingto the transmission model, the ungrounded entities are the“ground of being”. But the emergence model suggests thatsomething may act as a foundation in the absence of ungroundedentities; the infinite “starts playing an active role, andprogress is made as the chain lengthens” (Morganti 2015: 562).The emergence model draws inspiration from an analogy withepistemology where the “emergence of justification” froman infinite chain of reasons has recently been an area of activeresearch (Klein 2007; Peijnenburg & Atkinson 2013). Thus, the coreof the emergence model seems to be that there is no privilegedfundamental level of reality that serves as the basis for being.Instead, we should understand being more holistically, as it were, andexplore the idea that it may emerge gradually. Let us call thisemergentist infinitism. On the face of it, emergentistinfinitism looks like the denial of(MF) and hence metaphysical foundationalism. However, there is some roomfor interpretation here, because the holistic model suggests that thewhole infinite chain could perhaps be considered to ground its“parts”.

We might compare this line of thinking to Leibniz’sprinciple of sufficient reason (PSR), which states that forevery entity that exists, there is an explanation or reason for itsexistence (Della Rocca 2010; Guigon 2015; Dasgupta 2016; Amijee 2020)and the separate entry onprinciple of sufficient reason). In the contemporary literature, we may compare (PSR) with theinheritance 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 withepistemology. Both with epistemic infinitism and epistemiccoherentism, it is typical to consider foundational justification tobe impossible: all possible cases of epistemic justificationmust conform to the infinitis/coherentist picture. It is notimmediately clear whether the corresponding metaphysical views(metaphysical infinitism andmetaphysicalcoherentism) would need to hold by necessity in a similar way. Infact, some of the arguments in the literature suggest that this maynot be the case, e.g., where thepossibility of differenttypes of infinite decent is considered (cf. Tahko 2014). If thestructure of reality is contingent in this regard then there may be aprincipled difference between the epistemic and the metaphysicalcases.

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

4. Metaphysical Infinitism

To endorse metaphysical infinitism is to reject metaphysicalfoundationalism(MF). But as we’ve seen, the sense of metaphysical foundationalismdefined at the end ofsection 2 does not require accepting strong, set-theoretic well-foundedness,and hence it is compatible with at least some types of infinitedescent. Accordingly, metaphysical infinitism is a somewhat strongerview than it may first have seemed.

Using the technical notion of being bounded from below or having alower bound introduced insection 2, we can start with the simple idea that, for a given notion ofdependence, a chain has a lower bound only if there is an element thatevery element in the chain depends on (Rabin & Rabern 2016: 366).As we saw earlier, infinitely descending chains of dependence may havea lower bound, that is, terminate in an independent element that mayor may not be part of the chain itself, yet fail to beset-theoretically well-founded. But there is an even weaker conditionthan having a lower bound that can satisfy(MF), namely, Rabin & Rabern’s (2016: 363) “having afoundation” or Dixon’s (2016: 446) equivalent “fullfoundations”. Both of these are based on the idea of having aninfinitely large foundation. An example of this type of foundation canbe constructed with the help of infinitary disjunctions, as both Rabin& Rabern and Dixon demonstrate (Litland 2016b discusses theproblems associated with this and constructs further examples). Insuch a case, (MF) is satisfied because every element depends on someindependent element, despite there being no lower bound. There is nolower bound because the chain does not terminate if the foundation isinfinitely large. In contrast, if the foundation is finite, then thereis a lower bound and hence the weaker requirement of (MF) is alsoentailed. In decreasing order of strength, we have the requirement ofset-theoretical well-foundedness, having a lower bound, and having afoundation or full foundations. We defined (MF) in terms of theweakest of these three requirements insection 2, but a proponent of metaphysical foundationalism may of course make astronger requirement as well, such as having a lower bound.

In this section, we are interested in the possibility of the type ofstrong metaphysical infinitism that denies all of the threerequirements. The denial of(MF) entails that there are at least some entities that arenon-fundamental yet do not depend for their being on any fundamentalentity or entities. There may be several ways in which this couldhappen, but the most extreme possibility is a type ofinfinitecomplexity whereby there is an infinite descent of differentkinds of entities, each depending on entities further down the chainbut never terminating and never being “fully accountedfor”. We might think of this in terms of a violation of theprinciple of sufficient reason (PSR), at least if (PSR) is thought torequire that we must reach an ultimate reason rather than just areason at each layer below the previous one. Accordingly, infinitecomplexity entails a lack of structure with the type of explanatoryimport that (PSR) requires. This type of view would be more radicalthan the various, potentially innocuous types of infinite descentoutlined so far. The view might strike many as implausible, at leastwhen 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 ofinfinite repetition? This type of idea has been discussed under thelabel ofboring infinite descent (Schaffer 2003: 505, 510;Tahko 2014). A boring or repetitive structure entails that somewheredown the chain of dependence we stop encountering novel types ofentities or novel structure. The boring part of the structure thatrepeats infinitely could be of any length as long as it starts aneweventually. A description of the repetitive part only needs to besupplemented with an instruction to continue as before. Forexample:

The world stands on four elephants, the four elephants stand on aturtle, the turtle stands on two camels, the camels stand on fourelephants, the four elephants stand on a turtle … and repeat adinfinitum. (Tahko 2014: 261)

The idea is that the boring structure, whatever shape it may take, canbe fully described in terms of the mentioned entities (or types ofentities, and perhaps the “standing” relations betweenthem): four elephants, a turtle, and two camels. It has been suggestedthat this produces a “minimal” description of reality, butit is debatable whether it satisfies(MF) or is a case of strong metaphysical infinitism (Raven 2016; Tahko2018).

It seems that this type of infinite descent is less radical thaninfinite complexity, but some open questions remain. For instance, isthe type of infinite regress at hand innocuous or vicious? Can wecapture the difference between metaphysical foundationalism andmetaphysical infinitism in terms of the non-viciousness or viciousnessof the regress (Nolan 2001; Bliss 2013; R. Cameron 2022; Oberle2022b)? We will leave these issues open, but to improve on the toyexample, we might briefly discuss a more concrete case. Sometimes theNobel Prize winner Hans Dehmelt’s (1989) model is mentioned as apotential example (Schaffer 2003, Morganti 2014, Tahko 2014). Dehmeltspeculated that there could be a quark/lepton-like substructure beyondthe known level of quarks, based on the model of the triton, thenucleus of hydrogen’s radioactive isotope tritium:

I propose to extend the triton substructure scheme to an infinitenumber of layers. Below the four layers listed above [up tosubquarks], they contain higher order \(d_N\) subquarks, with \(N = 5\rightarrow \infty\). In each layer the particles are not identicalbut resemble each other in the same way as quarks and leptons do, withmasses varying as much as a factor 108. In an infinite regression tosimpler particles of ever increasing mass, they asymptoticallyapproach Dirac point particles. (Dehmelt 1989: 8618)

Up to \(N = 3\), the level of electrons, Dehmelt’s model ismotivated 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 agenuine case of boring infinite descent, given that the regressappears to terminate in Dirac point particles. What matters here iswhether these Dirac point particles are treated as genuine entities,but there is no suggestion in Dehmelt’s model that this shouldbe the case. Rather, as the above quote suggests, the infiniteregression of non-point particles asymptotically approaches the ideallimit of Dirac point particles. This is a type of mathematicalabstraction not unlike the one we find in the case of gunky space,where regions asymptotically approach “points” (seeZimmerman 1996). Just like Dirac point particles in Dehmelt’smodel, 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 derivetheir “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 inarithmetic), not like a diachronic, dynamic physical relation (like inthermodynamics, or action theory).

The thought here is that the transmission model illegitimately assumesa dynamical “starting point” to any grounding chain. If weabandon the transmission model and the dynamical view, we arrive atthe idea ofindefinitely descending ground whereby all factshave a ground and hence there are no fundamental facts (Bohn 2018).This violates even the weakest of the three requirements that ametaphysical foundationalist might impose, so it is a strict denial of(MF). Note, however, that this concerns only the precisification of (MF) interms of grounding. So, this type of metaphysical infinitism targetsthe conception of fundamentality as ungroundedness (as specified interms of(RI) insection 1.2).

If(MF) does not require the transmission model, the choice betweenfoundationalism and infinitism is difficult. There is, perhaps, onetest case: the possibility ofgunk (everything has a properpart),junk (everything is a proper part), andhunk(everything both is and has a proper part). There is an on-goingdebate about the modal status of these scenarios, and one might appealto their possibility in defense of indefinitely descending ground (asBohn 2018 does).^[26] However, these scenarios of course concern only the relation ofproper parthood (mereological dependence), and hence even ifmereological infinitism were coherent, there may be other notions ofdependence for which the corresponding notion of infinitism isincoherent.

This concludes our discussion of metaphysical infinitism andfundamentality. This survey has focused on a subset of the growingliterature on fundamentality with the aim of clarifying the variousterminological issues surrounding the key notions and identifying somecommon themes. The ideas behind these notions are old as are theintuitions associated with the various arguments that we’ve seenfor and against metaphysical foundationalism and infinitism. Recenthelpful efforts to systematize some of the central ideas, especiallythat of well-foundedness, have made it much easier to haveconstructive discussions about the topic of fundamentality. It can beexpected that further attempts to formulate arguments for and againstdifferent strengths of metaphysical foundationalism and infinitismwill emerge in the near future. The literature is also paying moreattention toward the notions of relative fundamentality andmetaphysical coherentism. Finally, there is an increasing amount ofliterature that applies more technical notions of fundamentality tovarious other debates in metaphysics: e.g., Morganti (2019) on onticstructural realism, Calosi and Morganti (2021) on quantumentanglement, Giannotti (2021b) on powers, Scarpati (2021) onhaecceitism, Tahko (2021) on logical realism, Allori (2022) on quantumtheory, Hamri (2022) on causation, Rabin (2022) on the physicalismdebate, and Spencer (2022) on relativity.

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Acknowledgments

The author would like to thank Umut Baysan, Karen Bennett, Einar Bohn,Christina Conroy, Martin Glazier, David Mark Kovacs, Jon Litland,Matteo Morganti, Donnchadh O’Conaill, Jan Plate, Gabriel Rabin,Mike Raven, Kelly Trogdon, Jessica Wilson, and Justin Zylstra for manyhelpful comments and suggestions. Special thanks to two anonymousreferees for SEP for very detailed comments.

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