The term “mereology” is sometimes used to refer to any oneof the formal languages that describe the part-to-whole relation. Inthis article, I will use the term more broadly to refer to anytheoretical study (formal or informal) of parts, wholes, and therelations (logical or metaphysical) that obtain between them. In whatfollows I will survey some of the ways that philosophers in themedieval Latin West thought about parts and wholes. (While there isvery little contemporary scholarship on medieval Arabic, Hebrew, orByzantine mereology (although now see Thom 2019), many of the Latinnotions and distinctions surveyed in this article surely havecorrelates in these traditions.) The article will highlight many ofthe key medieval mereological concepts and principles, and it willoutline some of the fundamental issues that confront mereologists inthe Middle Ages. Specific philosophers and their doctrines will beused to illustrate some of these concepts, principles, and puzzles.Many of these concepts and principles may seem strange to the modernstudent of parts and wholes, but behind this alien veneer one will seethat medieval mereologists share many of our concerns about wholes,their parts, and the metaphysical implications of mereology.

• 1. Forums for medieval mereology
  • 1.1 Division
  • 1.2 The Topics
• 2. Wholes
  • 2.1 Integral wholes
  • 2.2 Universal wholes
  • 2.3 Potential wholes
• 3. Parts
  • 3.1 Parts of integral wholes
  • 3.2 Parts of genera and universal wholes
  • 3.3 Distinguishing universals from integral wholes with respect to their parts
  • 3.4 Parts of potential wholes
• 4. Mereology and metaphysics
  • 4.1 Identity
  • 4.2 Persistence
• 5. Concluding remarks
• Bibliography
  • Ancient and Medieval Texts Cited
  • Select Secondary Sources
• Academic Tools
• Other Internet Resources
• Related Entries

1. Forums for medieval mereology

One can find discussions of parts and wholes throughout the medievalphilosophical and theological literature. But there are two placeswhere the student of medieval mereology can reliably look to findsustained reflections on parts and wholes as such, namely, treatmentsof division and the Topics. The main authority on division and theTopics is the Roman philosopherBoethius (c. 480–524 C.E.). Boethius is now most famous for hisConsolation of Philosophy, but his influence on medievalphilosophy is as much due to his commentaries on Aristotle’sCategories andOn Interpretation, his theologicaltreatises, and his handbooks on logic (see Chadwick 1981 and Marenbon2003). Boethius’ treatment of division is found in his handbookOn Division (De divisione). His treatment of theTopics is found in his handbookOn the Topical Differences(De differentiis topicis) and in his commentary onCicero’s treatment of the Topics (In CiceronisTopica).

1.1 Division

The methods of “division” (Greek:diairesis,Latin:divisio) and “collection” (Greek:sunagoge, Latin:collectio) have their roots inPlato’s later dialogues, and they are common in AncientNeoplatonic and Aristotelian treatises on logic. Plato tells us thatcollection and division provide us with a way to understand therelationships between some unity and some plurality (Phaedrus265d–266b, andPhilebus 16c–17a). Division is a processwhereby any sort of unity is resolved into a plurality. Collection isthe process whereby a plurality is collected into a unity.

For the Neoplatonists, division and collection are first and foremostapplied to genera and species (such as Animal and Human), andinstances of these universals (such as Brownie the donkey andSocrates). This primary mode of division is often interpreted as alogical exercise. In particular, it is a method for developingdefinitions of things, which can then be used in demonstrations.Collection is construed as a method for classification.

The primary purpose of division for Late Ancient philosophers is todetermine the hierarchical relations between a universal and thatwhich falls under it. But divisions are applied to a variety of otheritems. There is no single, canonical list of divisions (for anoverview, see Magee 1998, xxxvii–xlix). The list presented byBoethius is the one bequeathed to the Latin West (On Division877c–d). Boethius distinguishes between two broad categories:substantial divisions and accidental divisions. These divisions aredivided further. Of the substantial divisions there are:

1. The division of the genus into its species.
2. The division of the whole into parts.
3. The division of a word into its meanings.

Of the accidental divisions there are:

4. The division of a subject into accidents.
5. The division of accidents into subjects.
6. The division of accidents into accidents.

The most important material for our purposes is Boethius’treatment of the first and second substantial modes of division.

1.2 The Topics

“Topic” is the standard translation for the Latin termlocus. (Note that the translators of Peter of Spain (2014)have chosen to render it as “place”, which is the basicmeaning of the Latin term.) As Stump (1978, Part 2, and 1982) andGreen-Pedersen (1984) have pointed out, the notion of the Topicevolves over the course of its use in ancient and medieval logic, butroughly put, medieval logicians think that the study of the Topicshelps one to discover a number of self-evidently true propositions, or“maximal propositions”, that can serve as warrants forarguments. (See Peter of Spain,Tractatus V.3–4 (2014,198–205).) For example, suppose that someone makes thisinference:

If Socrates is human, Socrates is animal.

Students of the Topics claim that this inference is warranted by thefollowing maximal proposition:

If a species is predicated of something, that species’ genus isalso predicable of that thing.

The student of medieval mereology will be extremely interested in themaximal propositions presented in treatments of the Topic from thewhole and the Topic from the part.

Following a well-established Ancient tradition, the Topic from thewhole is usually divided into two sub-Topics:

1. The Topic from the universal whole.
2. The Topic from the integral whole.

From the beginning, the best medieval philosophers are aware of thesubtleties of ordinary language. In particular, they are mindful ofthe distributive function of the adjectival term “whole”(totus/tota/totum). That is, the term“totus” can act like a universal quantifier and,hence, the phrase “totusx” can mean“all the parts ofx taken together” or “theentirex” (seeSection 2 below). This sensitivity to the distributive sense of“totus” might be what motivates medievallogicians to add further refinements to the theory of the Topics.Specifically, the Topic from the whole is routinely divided in asix-fold manner (see, e.g., Peter of SpainTractatusV.11–18 (2014, 210–19); following Boethius’ terseremarks inDe topicis differentiis II, 1189A–C; trans.Stump 1978, 52–53). In addition to the Topic from the universalwhole and the Topic from the integral whole, there are theseadditional Topics:

3. The Topic from the whole in quantity.
4. The Topic from the whole in a respect (in modo).
5. The Topic from the whole in place.
6. The Topic from the whole in time.

The Topic from the whole in quantity classifies and considerspropositions where the term is taken universally, such as “Everyx is ay”, or “Nox is ay”. The Topic from the whole in a respect considers aterm in respect to some limiting qualification. So, for example, ifx is whiteon its surface, then every part ofx’ssurface is white. The Topic from the wholein place classifies propositions bounded by the term“everywhere” or its cognates. So, if water iseverywhere, then water ishere (where“here” designates a “part in place”). And,finally, the Topic from the whole in time considers inferences thatone can make from propositions bounded by the term“always” or “never”.

A full account of medieval mereology would consider carefully thedetails of all six sub-Topics. But of special interest to us are thetreatments of the Topic from the integral whole and of the Topic fromthe integral part. In these discussions, medieval philosophers oftenconsider whether the traditional maximal propositions associated withthese Topics in fact describe the logical and metaphysical relationsthat hold between an integral whole and an integral part.Specifically, the maximal proposition that applies to integral wholesis:

If the whole is, then the part is.

The maximal proposition said to apply to the integral part in relationto its whole is:

If the part is not, the whole is not.

These maximal propositions are quite startling. They seem to entail,for example, that if Socrates exists, then his hand must exist, and ifSocrates’ hand ceases to exist, then Socrates ceases to exist.As we will see inSection 4.2, such consequences do not escape the notice of medieval philosophers,and much of interest regarding the metaphysical implications of theTopics and their maximal propositions ensues.

2. Wholes

Sometimes the phrase “totus x (est)” can mean“x is whole”, in the sense thatx iscomplete or not lacking anything. This is a sense of“whole” that Aristotle identifies in hisMetaphysics and that he contrasts with the notion of being“mutilated” (kolobon) (Metaphysics 5.26and 5.27, respectively).

Closely related to the sense of being complete, “whole”can have a distributive function and the phrase “totus x esty” (“wholex isy”) can meanall ofx isy—that is, all the parts ofx taken together isy. This is called theterm’s “categorematic” sense. Medieval logiciansalso distinguish a second distributive sense of “whole”,which they call the “syncategorematic” sense of the term(see, e.g., William of SherwoodSyncategoremata [1941, 54];OckhamSumma Logicae II, ch. 6 [Opera PhilosophicaI, 267–69]; John BuridanSummulae de Dialectica4.3.7–1; Albert of SaxonySophismata 45). When“totus x est y” is interpretedsyncategorematically, it means each and every part (quaelibetpars) ofx isy.^[1]

Finally, “whole” can signify a thing that is eithercomposed of some things or divisible into some things. This usually isthe sense of “whole” at work in discussions of collectionand division.

The ancient practice of collection and division, and especially theproclivity to call both the products of collection and the things tobe divided wholes, has a lasting influence on medieval mereology. Formedieval philosophers a variety of items can be wholes. Universals,concepts, material objects, masses (such as water or gold), souls, andtime can all be wholes—and this is only to mention some of themore common items studied by medieval philosophers. Thus, in general,anything that is composed out of other items or that can be dividedinto other items is a whole. But there are two importantqualifications.

First, one should not assume that all wholes are mind-independentfeatures of the world. Peter Abelard, for instance, argues thattemporal wholes (such as days, weeks, or hours) and universal wholesare not things (res). Yet, provided that we do not reifythese items, Abelard will allow us to treat items like days and hoursas wholes consisting of parts, and he will allow us to talk aboutuniversals and their parts. (On temporal wholes, seeDialectica 554.14–23 andLogica Ingredientibus2, 187.9–14. On universals, seeLogica Ingredientibus1, 10.8–16.18 andLogica Nostrorum 515.10–522.9.For a helpful overview of Abelard’s anti-realist metaphysics,consult King 2004.)

Second, some medieval philosophers, again motivated by theirmetaphysical commitments as well as their understanding of what wholesmust be, will prefer to distinguish between “true”, or“proper”, wholes and quasi-wholes. Ockham, for example,insists that individuals are not actually parts of species and speciesare not actually parts of genera. These items are merely“parts”, and accordingly their corresponding species andgenera are only “wholes”, in a figurative sense. Once thisfact is appreciated, however, Ockham is willing to allow us to speakof a genus or a species as a “whole of a sort”(quoddam totum), since there is a legitimate sense in which aspecies “contains” individuals and a genus“contains” its species (OckhamExpositio in librumPorphyrii de praedicabilibus ch. 2, [Opera PhilosophicaII, 54]).

With these caveats in place, we can begin to explore the basicmereological categories that one finds in medieval treatments ofmereology. As was noted, all manner of items can be wholes, butgenerally speaking medieval philosophers believe that this motleygroup can be organized under three broad categories: integral wholes,universal wholes, and potential wholes. It will soon become apparentthat the category of integral wholes is quite broad. Indeed, it is soinclusive that some philosophers will feel pressure to introduce afourth basic category, substantial wholes. But others will suggestthat an integral whole is divisible into two distinct kinds of parts,quantitative parts and substantial parts. For this reason, I willconsider the issue of substantial wholes and substantial parts inSection 3.1.

2.1 Integral wholes

Much of what we encounter in the material world is, or can beconsidered to be, a whole. Cars, houses, plants, and human beings arecomposed out of bits of metal, plastic, cellulose, or flesh and bone.These items are not only divisible into the components just mentioned;they are divisible into other parts such as carburetors, doors,leaves, and hands.

Medieval philosophers consider the wholes mentioned above to be either“continuous” or “contiguous” integral wholes,where a contiguous whole is a type of “discrete” whole.Continuous wholes are wholes whose parts share a common boundary.Discrete wholes are wholes whose parts do not share a common boundary.The parts of discrete integral wholes can be either close to oneanother, or relatively scattered. Contiguous wholes consist of partsthat are discrete, but spatially close together. Discrete wholes whoseparts are relatively diffuse are “disaggregated” integralwholes. (For a fairly comprehensive catalog of the kinds of integralwholes, see AnonymousCompendium Logicae PorretanumIII.12.)

Whether an item is a continuous or contiguous whole will depend uponother metaphysical commitments. Some early medieval philosophers onlyhave simple artifacts like fences and walls in mind when they talkabout contiguous wholes. But others think that even more complicatedman-made objects, such as wagons, houses, and clocks are contiguous,and not continuous wholes, because only substances are continuouswholes. Abelard and Aquinas both hold the latter position, and thus,that only individual donkeys, palm trees, human beings, and the like,are continuous wholes. Abelard thinks that this is true because onlyGod can fuse parts together into a continuous unity. Humanmanipulations, no matter how skilled, are only capable of placingparts in close proximity to one another (Dialectica417.4–37; 419.35–420.6). Drawing upon Aristotle’sreflections on form and matter (especially in hisDe Anima,Physics andMetaphysics), Aquinas thinks that onlysubstances possess the kind of form that can inhere in each and everypart of the whole and thereby make it truly one (SummaTheologiae I, q. 76, art. 8;Quaestiones de Anima q. 10;see also Pasnau 2002, 79–88). An artifact possesses a form thatmerely orders and arranges the parts of the whole. We can tell whethera form imbues its parts in the manner required of a substantial formby attending to the effect of the form’s existence on thefunctionality of the parts. The human soul inheres in the hands of ahuman being as a substantial form because if the soul were removed,that part would cease to function as a hand – indeed, Aquinasinsists that this thing would cease to be a hand. By contrast, whenthe form of a car is removed from the items that make up the car, manyof these items can still be car-parts. For example, a carburetor canstill be a carburetor; it can be placed in another car and continuefunctioning as a carburetor.^[2]

In addition to artifacts, plants, and animals, some medievalphilosophers expand the class of continuous integral wholes to coverhomogeneous masses, such as some gold or some water, and the class ofdisaggregated integral wholes to include scattered mereological sums,such as the sum of this mountain and this dog. However, one will notfind any medieval philosopher who assents to the modern mereologicalthesis of universalism, or what David Lewis calls “unrestrictedfusion” (1991, 74; cf. Simons 1987, 108–12 and thesubsection on unrestricted fusions in the Encyclopedia entry onmereology). Peter Abelard comes close when he asserts that any two items, evenones belonging to different categories of being, can constitute aplurality (Dialectica 548.19–22; see also the anonymoustwelfth-centuryIntroductiones Montanae maiores, 69rb, whereAbelard’s thesis is mocked). But one suspects that even Abelard,if pressed, might step back from assenting to a full-blown version ofuniversalism for theological reasons. For in discussions ofGod’s simplicity, it is routinely noted that God is notintrinsically compositeand that He cannot be combined withanything elseand that nothing can be combined with Him (see,e.g., AquinasSumma Theologiae I, q. 3, arts. 7–8).This strong thesis about the lack of composition in God is maintainedbecausebeing a part of something implies a certain sort ofpotentiality and incompleteness. If anything could be a part of awhole that also has God as a part, then there is something that Godcould be–namely, a part of this composite–that Hecurrently is not. And even if He currently is a part of the sum of allthings (for instance), given that some of the other parts arecontingent beings, the sum of all things is contingent. Accordingly,God’s parthood is contingent, or has a potentiality. But God isfully actual and necessary. Hence, God cannot be a part of somegreater composite.^[3]

As a more general point, it must be remembered that medievalphilosophers are for the most part working within an Aristotelianframework, and like Aristotle, their paradigmatic examples of unifiedcomposite things are substances. If a thing is not unified by asubstantial form, then that thing has a lesser kind of unity. Humansand dogs are more unified than houses and wagons. Houses and wagons,nonetheless, are unified by a form— namely, an accidentalform—and so they have a unity of a sort. Moreover, this kindunity will be greater than the unity of a mere collection of things.Hence, as a general rule, the more gerrymandered a collection of partsis, the more likely it is that this composite’s status as awhole will be called into question (see, e.g., AquinasInMetaphysica expositio, Book V, lectio XXI §§1102–4; Jean BuridanSummulae de Dialectica 8.1.4).

In addition to material beings, some medieval philosophers allownon-material items to be integral wholes. For example, Ockham thinksthat complex mental acts can be wholes, and Aquinas insists thatactions such as penance are integral wholes (OckhamQuaestiones inphysicam q. 6 [Opera VI, 407–10];In DeInt. I, prooemium, 6 [Opera II, 354–8]; AquinasSumma Theologiae III, q. 90, art. 3). Abelard insists thattemporal items are not integral wholes, but he seems to be in theminority. Non-material integral wholes do not sit easily under eitherthe continuous or the discrete category, since their parts cannot berelated to one another with respect to location. If asked to pick,medieval philosophers tend to label temporal wholes and events ascontinuous wholes, but they are not continuous in the way that abronze rod is continuous. Their parts to not share some spatialboundary; they come one after another in continuous succession. Forthis reason, these wholes are sometimes called“successive” wholes (AnonymousCompendium LogiaePorretanum III.12). In addition to aggregates of time, processes(that is, things that take time to unfold) are also sometimesconsidered to be successive wholes (see, e.g., AquinasSummaTheologiae III, q. 90, art. 3, ad 3).

In short, an astounding variety of items can be integral wholes. Yet,no matter how large this category becomes, most medieval philosophersinsist that the class of integral wholes does not exhaust the domainof items that can be wholes. In particular, there are two types ofitem that require their own category: universals and the so-called“potential wholes”.

2.2 Universal wholes

Many non-material items are considered to be integral wholes. But mostmedieval philosophers mark off one special sort of non-materialobject, the universal, and treat it as a separate type of whole.

It was noted inSection 1.1 that universals, and especially species and their genera, are relatedto one another hierarchically. For instance, the species Human Beingand the species Horse both fall under the genus Animal. Additionally,individuals are related hierarchically to their species and genera.Hence, Socrates and Cicero fall under the species Human Being and thegenus Animal. These relations between universals and individuals areoften described in the terms of collection and division. Cicero andSocrates and all other humans are collected into the species HumanBeing, and the species Human Being and Horse and all other species arecollected in the genus Animal. Correlatively, Animal is divided intoits species, and Human Being is divided into individual humans.

This language of collection and division invites medieval philosophersto call the divisible items wholes, and the products of thesedivisions parts, but most medieval philosophers are not tempted tothink that universals are literallycomposed out ofindividuals or lesser species. There are noteworthy exceptions. Forexample, an anonymous twelfth-century philosopher carefullyarticulates and spiritedly defends a version of what is often calledthe collection (collectio) theory of universals(Pseudo-JoscelinDe generibus et speciebus §§84–153 (2014, 158–85)). But, in general, universal wholesare clearly marked off from other kinds of whole, and their behavioris thought to be distinct from the behavior of potential and integralwholes.

There is one significant complication. In hisOn Division,Boethius distinguishes between the division of a genus and thedivision of a whole (877c–d, see alsoIn Cic. top.331.18–19). But then under his treatment of the ways to dividewholes, he notes that one way to divide a true whole is to divide auniversal into its subordinate individuals (887d). That a universalcan be a true whole will matter below when we try to uncover theproperties that differentiate one kind of part-whole relation fromothers (Section3.3).

2.3 Potential wholes

In addition to integral wholes and universal wholes, Boethiusintroduces a third basic type of whole to medieval philosophers: thewhole consisting of “powers” (potentiae) or“virtues” (virtutes) (On Division 888a).These wholes are often called “virtual wholes” or“potential wholes”. This article will follow thatpractice, but it should be understood that in medieval contexts“potential” usually signifies something that is in somerespect not fully actualized, and, hence, in some contexts“potential whole” and “potential part” referto things that are possibly, but not actually divided, as when acontinuum with a certain magnitude is potentially divisible but notactually divided into halves or fourths, or when a substance is saidto be potentially divisible into its elemental constituents (seePasnau 2011, 606–629). This standard sense of“potency” is even at work in the potential wholes thatBoethius is talking about, since the soul’s powers are typicallyrelated to the soul’s “operations”, or activities,and these activities are clearly not always occurring. A soul mighthave the power to see or the power to think, even though it is notpresently seeing or thinking. Nonetheless, these Boethian potentialwholes are actual and actually possess these potential parts; they arenot “potential” in the sense that they are possible thingsor things that are merely capable of being divided.

Potential wholes are curious items. They are particulars, notuniversals. Yet potential wholes are not composed out of their partsin the way that, for example, a house is composed out of bricks andwood. Indeed, the most commonly discussed kind of potential whole is asoul. But for Aristotelians souls are forms and it is generally heldthat forms are mereologically simple. Thus, souls ought to bemereologically simple. That Boethius, or for that matter, Aristotleasserts that souls have parts is therefore initially quite puzzling.(For a list of the places where Aristotle talks about the parts of thesoul as well as a defense of the claim that we should takeAristotle’s use of the term “parts” seriously, seeJohansen 2014.)

Further reflection, however, shows us that a soul (and especially ahuman soul) is a complicated thing. It has a variety of capacities,which seem to be really distinct from one another and possibly fromthe soul itself. Some of these capacities even seem to be contrary toone another. For example, a soul seems to be capable of psychicconflict. There is a lot of pressure then put on this putativelysimple, yet obviously complicated thing, and therefore it is hardlysurprising that there is a robust medieval debate over the nature ofthe parts or powers of the soul (see Perler 2015). Some think that thepressure is too great to sustain and that one substantially simplething cannot be the subject of all these capacities. Two commonreasons are given for the soul’s being really divided in asubstantial manner. First, it seems that really distinct operationsmust stem from really distinct sources, and since operations areaccidents of substances, this would seem to imply that the operationsof a soul must stem from distinct substantial sources. Second,according to Aristotle, the rational capacities do not require abodily instrument to operate, where the other capacities do. Thisleads some medieval Aristotelians to conclude that whereas thenon-rational capacities arise from the matter of the thing, therational capacities must come from some external source. This, inturn, suggests that a human being has at least one substantial formthat is rooted in and arises from the matter, and another substantialform that comes from an external source. If all of this is right, thenhumans at least either have several souls, or one substantiallycomposite soul, where the parts are viewed as“incomplete” substances that combine to form one“perfected” soul (see Pasnau 1997; Perler 2014). Others,and most famously Aquinas and his followers, hold that a living thingcan only have one substantial form and therefore only one soul (Pasnau2002, 126–30). These “unitarians” must come up withan answer to how the powers do not in fact divide the soul into souls,especially those unitarians like Aquinas who hold that the powers ofthe soul are really distinct from the soul itself. (In fact, accordingto King (2008), the dominant view in the medieval period is that thesoul’s powers are really distinct from the soul’ssubstance.) A commonly proposed solution is that the powers“flow out” from the soul’s essence and that thesubordinate powers of the soul are “virtually present” inthe soul (see Shields 2014 for an analysis of Suárez’sversion of this solution; also Perler 2015).

In keeping with the introductory nature of his treatise, Boethiusremains silent about many of these issues and instead focuses on thelogical nature of the soul considered as a whole consisting of powers.Unfortunately, what he says actually makes matters all the moreobscure since he notes that the division of a potential wholeresembles both the division of a genus and the division of awhole:

For in that each and every part of it entails the predicate“soul” it is brought into connection with the division ofthe genus, each and every species of which necessarily entails thegenus itself. On the other hand, in that not every soul is composed ofall parts but each one is composed differently, in this it isnecessarily brought into connection with the nature of a whole.(On Division 888c–d [trans. Magee 1998, 41])

Some earlier medieval philosophers take Boethius’ statement asan invitation to reduce the soul, and the potential whole in general,to either a genus or an integral whole. (One of the first attempts toplace the division of soul under the division of the genus is found ina short letter from a mysterious ninth-century thinker identified onlyas “Master L”. The letter is preserved in the manuscriptsof St. Gall, and is transcribed by De Rijk (1963, 75–78).)However, the attempt to reduce soul to either a genus or an integralwhole appears doomed, for potential wholes do not fit well undereither category. Souls are particular, and hence, they cannot beuniversals of any kind, let alone genera. On the other hand, byintroducing the potential whole as anadditional kind ofwhole, Boethius appears to be signalling that potential wholes do nothave parts in the same way that other true wholes have parts: Socratescan be separated into his hands, feet, and so forth; a chemicalmixture can be reduced back into its ingredients. Even a universalwhole can be separated into independently existing parts, namely, theindividuals that are its parts. But a soul is neither constructed outof its powers, nor does it appear to be separable into freestandingparts: the powers must be powersof a soul.

We begin to see some progress in the twelfth century. For example, ina treatise that has been attributed to the young Peter Abelard welearn that there are two definitions of soul, a “superior”one in virtue of which the soul “has an affinity with auniversal whole”, and an “inferior” definition invirtue of which soul has an affinity with an integral whole:

This is the superior definition of “soul”: soul is aquality with respect to which a body is made to be alive. It is inaccord with this definition that “soul” is predicated ofits parts individually, and thus the power of knowing is a qualitywith respect to which a body is made to be alive, and the power ofsensing is a quality with respect to which a body is made to be alive.This is the inferior definition of “soul”: soul is aquality constituted out of rationality, out of sensibility, and out ofvegetability (vegetabilitate). And it is with respect to thisdefinition that [soul] cannot be predicated of its parts takensingularly, since one cannot say that the power of thinking is thatwhich is constituted out of rationality, sensibility, andvegetability. ((ps?) Abelard,De divisionibus194.12–22)

This solution helps to explain the strange predicative behavior ofsoul and its powers. It also, interestingly, provides us with asolution to what a potential part is: The soul turns out to be aquality composed in a fairly familiar sense of“composition” out of other qualities. The commentator thusattempts to explain the tricky notion of potential-parthood in termsof more understandable relations, namely, the predication relation andthe relation of an integral part to its whole.

In his commentary on Boethius’s text, Albert the Great takes adifferent approach (In de Div. tract. 4, ch. 1; Loe ed.,75–9). He refers to the view that a potential whole is some sortof intermediary between a universal and an integral whole, but then heproceeds to give a different explanation why a potential whole is awhole: It is a whole because the full and complete list of thecapacities will produce the full and complete account of the soul inso far as it is the principle of life and self-motion. But it does notfollow from this that any given particular soul will have all of thepowers and capacities on the list. Albert also hastens to add thatwhile a full list of the capacities of soul gives us the substance ofthe principle of life and motion, under this description we are notattending to a substance in the absolute sense. We, thus, should notconclude that a soul is divisible in the way that an integral whole isdivisible. That is, these capacities need not answer to littlesubstances or parts of substances that constitute the soul. To help usto see his point, Albert offers an example of another potentialwhole:

And again a regime (regnum) is a whole of this kind, giventhat it consists of its powers, namely, the king, the prefect, thesuperintendent, and others of this sort. (In de Div. tract.4, ch. 1; Loe ed., 75)

Albert is thinking of the regime in terms of an act or role, which isanalyzable and hence consists in smaller roles and their associatedpowers. The complete list of the roles in regimes would give the wholeof what it is to be a regime. But like individual souls, a particularregime will require some of these roles and their attendant powers,but it need not possess all of them. Moreover, a difference in roledoes not imply a difference in subject. The same person could performsome or even all of these distinct roles.

3. Parts

Many kinds of item can be wholes, and many kinds of item can be partsof these wholes. In general, any item that composes a whole is a part,and any item that is a product of a division of some whole is a partof that whole. The only clear restriction on what can be a part isthat no part is identical to its whole. In other words, no medievalphilosopher countenances what contemporary mereologists call improperparts. (On the contemporary notions ofpart, orimproperpart, andproper part consult the entry on contemporarymereology, and also Simons 1987, 9–11.)

We will first consider the kinds of things that can be parts ofintegral wholes (Section 3.1). We will then turn to the parts of universals (Section 3.2). As it will turn out, some of the things that can be parts ofuniversals can also be parts of integral wholes. This will prompt usto consider several criteria that medieval philosophers use todistinguish universal wholes from integral wholes (Section 3.3). Finally, we will return to consider potential wholes in light ofthese criteria for distinguishing universals from integral wholes (Section 3.4).

3.1 Parts of integral wholes

Consider the paradigmatic integral whole Socrates. Socrates iscomposed out of a soul and a body. His body is composed out of flesh,bone, and blood. And the flesh, bone, and blood in turn are ultimatelycreated by combining the four basic elements, Earth, Air, Fire, andWater. All these components of Socrates can be considered integralparts of Socrates.

Socrates is divisible into a number of other parts. We can cutSocrates in half, and thereby create the top half of Socrates and thebottom half of Socrates. We can also divide Socrates into his hands,feet, torso, heart, and so forth. All these products of the divisionsof Socrates can be considered integral parts of Socrates as well.

Medieval philosophers separate this plethora of integral parts into anumber of distinct categories. The top and bottom halves of Socratesare often called quantitative parts, since they divide Socrates solelywith respect to a quantity, or measure. Flesh, bone, and blood, aswell as the elements that compose these components also are oftencalled quantitative parts, for they comprise Socrates’ matter,and matter is often associated with quantity.

Not all medieval philosophers think that the elements are parts ofSocrates. Abelard, for example, believes that the elements areingredients, but not every ingredient is a part. Strictly speaking,only those items that compose some whole and remain in that wholeafter composition are parts of the whole (Dialectica575.18–36). Hence, even though the flour is an ingredient of thebread, the flour is not a part of the bread. The flour has beenaltered by a chemical change, and so it does not remain once the breadis baked. Likewise, while the elements combine into a chemical mixturethat becomes flesh, the earth and water that make up flesh are nolonger present. Only the crumbs and flesh are properly parts of thebread and Socrates respectively.

Many later medieval thinkers effectively agree with Abelard on thispoint. The elements that compose my body only exist in actuality whenmy body has been dissolved back into elemental matter (see Ward 2014,125–44). While I am alive, these elements only exist“virtually” or “potentially” in me. Somemedieval philosophers went so far as to suggest that most or even allthe parts of a substance exist only in potency when the substance isactual (see Pasnau 2011, 606–29).

Medieval philosophers also like to draw a distinction between thehomogeneous and heterogeneous parts of Socrates’ body.Heterogeneous parts are such that, if they are themselves divided,their constituents are not of the same type as the original. Forexample, a hand is composed out of fingers, knuckles, and a palm. Itis also, from another viewpoint, composed out of muscles, skin, andsinews. No part of the hand is a hand. However, some of thehand’s parts are homogeneous. Muscle, skin, and blood are eachhomogeneous, since every bit of muscle is muscle, every bit of bloodis itself blood, and every bit of skin is also skin.

The distinction between heterogeneous and homogeneous parts isbequeathed to medieval philosophers by Aristotle and Boethius.Aristotle imposes a loose hierarchy on these types of parts, claimingthat the heterogeneous parts are composed out of homogeneous parts(History of Animals 486a13–14). This in turn suggeststhat the division of a whole into its parts is best initiated bydividing it into its heterogeneous parts, and only then into itshomogeneous parts. Boethius is less explicit, suggesting that theremay be many equally acceptable ways to begin to divide up a thing intoits parts (On Division 888a–b).

Many of the heterogeneous parts of Socrates are best defined in termsof their function, not their measure. For example, hands arediscriminated from feet based on what functions they perform for ananimal. Many medieval philosophers believe that these functions areprovided either by the form or the soul of the animal. For thisreason, many medieval philosophers call functionally defined parts“formal parts”, or partssecundum formam, but inorder to avoid any confusion, I will refer to parts of this sort asfunctional parts. The fourteenth-century philosopher WalterBurley tells us that the functional parts “remain the same solong as the whole remains the same and complete” (De toto etparte, 301). In other words, so long as Socrates’ souloccupies his body, and provided the hand is not cut off,Socrates’ hand remains a hand. Burley contrasts functional partswith material parts, and he places homogeneous parts such as flesh,bone, and blood under this category. Socrates’ material partsare in constant flux; Socrates is constantly losing and replacing bitsof flesh and blood.

In addition to the functional parts, Socrates’ substantial form(or forms) and his matter (either his prime matter or his proximatematteren masse) can be considered parts of Socrates. Buthere there is some question as to whether form and matter areintegral parts of Socrates. What Boethius says isambiguous.

There is also a division of the whole into matter and form. For in onemanner the statue is constructed out of its parts, and in anothermanner out of matter and form—i.e. out of bronze and its shape(species). (On Division 888b [Magee 1998, 40])

It is striking to learn that the form and the matter of the statue arenot “parts”, especially since they are products of adivision. Moreover, if a core meaning of “part ofx” is “that out of whichx comes tobe” (cf. Boethius’On the Trinity 2), then theform and the matter of a thing ought to be the primary parts of thatthing. Bronze without its form is merely bronze, not a statue;Socrates’ matter without Socrates’ substantial form is nota human. Clearly, then, Boethius cannot mean that form and matter arenot parts in any sense.

The task, then, is to determine what kind of parts form and matterare, and especially what kind of parts substantial form and matterare. Aquinas routinely distinguishes between the“quantitative” parts of a thing and the form and thematter, which he calls “parts of the essence” (SummaTheologiae I, q. 8, a. 2, ad 3; I, q. 76, art. 8; III, q. 90, a.2). However, he does not provide a clear answer to the questionwhether the part of an essence is a type of integral part, or whetherit is a distinct kind of part. Other philosophers present a lessambiguous line: only parts that make up some quantity can be integralparts (see, e.g., Lambert of AuxerreLogica 126; Peter ofSpainTractatus V.14 (2014, 212–4); Albert of SaxonySophismata 45). Accordingly, many medieval thinkers willpropose that in addition to potential, subjective, and integral parts,there is a fourth kind of part, which I will call asubstantial part.^[4]

There are other good reasons why medieval philosophers might bemotivated to distinguish the integral parts of a thing from thesubstantial parts of the same thing. Firstly, the form does not behavelike the integral parts of the thing, since for many Aristotelians(and especially, as we saw above, for Aquinas and other unitarians)the form has the unique capacity to be present “as awhole” in each material part of the thing. Integral parts, bycontrast, are thought to be unable to be present in more than oneplace at the same time, and for this reason, the integral parts of athing are said to sit side-by-side, or “part outside ofpart” (pars extra partem). Secondly, substantial partsrequire something else to perfect them (see Albert of SaxonySophismata 49). Matter by itself is incomplete; it is onlypotentially this or that thing. Matter needs a form in order toactually be this or that thing. Form, too, is in a sense an incompletesubstance. A dog’s form, for instance, is in need of some matterin order to be the form of this dog. But more fundamentally, given theAristotelian critique of Platonic forms–specifically, thecritique of the notion that a form can exist independently ofmatter–many medieval thinkers will insist that a form isincomplete in the sense that it cannotexist unless it ispart of a substantial composite. Finally, distinguishing betweensubstantial parts and integral parts captures an important intuitionabout substances: Socrates can lose and gain matter withoutcompromising his existence. But if Socrates were to lose either hismatter in total or his substantial form, he would cease to exist.

If one draws a sharp distinction between integral parts andsubstantial parts, the next natural thought would be to distinguishbetween integral and substantial wholes. The trouble is that in manycases thesame thing seems to be both an integral whole and asubstantial whole. Socrates, for example, has a quantity, and he issurely divisible into smaller quantities. It seems to follow thatSocrates is an integral whole. But Socrates is also composed of matterand substantial form. So, he is a substantial whole. One could rejectthe notion that Socrates is an integral whole. But, then, what is itthat has Socrates’s size? Socrates’ size is supposed to bean accident ofSocrates, not of a part of Socrates. It isperhaps for this reason that some medieval philosophers try adifferent approach. As we already noted, Walter Burley draws adistinction between material parts and partssecundum formam(which we called “functional” parts, above). Related tothis, Burley also draws a distinction between the whole considered inrelation to its matter (secundum materiam) and the wholeconsidered in relation to its form (secundum formam) (Detoto et parte, 301). Socrates considered formally—i.e. as aunity of this form with some matter—persists through time andchange. Socrates considered materially is constantly in flux. That is,Socrates on Monday is not the same whole materially as Socrates onFriday, because the sum of material parts belonging to Socrates onMonday is not identical to the sum of material parts on Friday. Thedistinction between the whole considered formally and the wholeconsidered materially will play a role in some medieval theories ofpersistence, and we will pursue this use of the distinction in shortorder (seeSection 4.2). But, first, we must consider the parts of universals and the specialproblems that the parts of universals entail.

3.2 Parts of genera and universal wholes

Genera are divisible into their subordinate species, whereas whatBoethius calls “universal wholes” are divisible into theirsubordinate individuals. In the medieval tradition that followsBoethius, these subordinate items are said to be“subjective” parts of the universal whole or the genus,since the part is a subject and the whole is predicable of the part.Socrates is a subjective part of Human Being and a subjective part ofAnimal, and Human Being and Animal are predicable of Socrates. Thatis, Socrates is a human being, and Socrates is an animal. However, tobe a true subjective part, not only must the name of the whole bepredicable of the part, the definition of the whole also must bepredicable of the part. A statue of Socrates could in some contexts becalled a human. For example, suppose someone points to the statue andsays, “That’s a man.”, and then she points toanother statue and says, “And that’s a horse.”. Thisperson is saying something intelligible, but the statue of Socratescannot be a subjective part of Human Being, since the statue is not arational mortal animal; in substance it is some marble shaped like ahuman with certain features. Socrates, however, is a subjective partbecause he is a human being and a rational mortal animal. Likewise,the statue of a horse is not a horse in the proper sense, that is, inthe same sense that the Black Stallion and Mr. Ed are horses.

Genera and universal wholes both consist of subjective parts, but theydiffer with respect to the items that are their subjective parts. Thisfact might explain why Boethius places universal wholes under thebroader class of true wholes. Boethius offers four criteria fordistinguishing between genera and true wholes (On Division879b–d):

1. The genus is divided by means of a qualitative difference, whereasthe whole is divided by means of a quantitative difference.
2. The genus is naturally prior to its species, whereas the whole isnaturally posterior to its parts.
3. The genus is the matter for its species, whereas the parts are thematter for the whole.
4. The species is always the same thing as its genus, while the partis sometimes not the same thing as its whole.

Some of these criteria could be interpreted in a manner such that theuniversal consisting of individuals falls under the category of truewholes. Consider, for example, criterion (a). The genus Animal isdivided by considering what sort (qualis) of animal somethingis. Human Being is arational animal, Horse is anirrational animal. In contrast, Human Being is not dividedwith respect to what sort of human being Socrates is. Socrates andCicero are both rational animals. The difference between Socrates andCicero is due to the fact that this bit of matter which makes upSocrates is different from that bit of matter which makes up Cicero. Adifference in matter is typically considered to be a quantitativedifference. Hence, the parts of the universal whole Human Being appearto be distinguished by quantity rather than quality.

However, some of the other criteria do not clearly mark genera offfrom universal wholes. Consider, for example, the second criterion.Boethius seems to have something like this in mind when he articulatesdifference (b): the parts of a genus are dependent upon the genus,whereas the whole is dependent upon its parts. In other words, ifthere are no animals, there can be no dogs or humans. In contrast, thehouse depends upon its parts. If you take away the roof and floor, thehouse ceases to exist. But universal wholes seem to behave likegenera, not houses. If we annihilate individual humans, we do noteliminate the universal Human Being. Therefore, difference (b) doesnot cleanly demarcate genera from universal wholes.

The third and fourth criteria present their own special problems,since it is far from clear how to interpret these differences, letalone whether the universal whole behaves like the genus or the truewhole.

Thus, it is not clear that Boethius’s four criteria adequatelyseparate universals divided into universals from universals dividedinto individuals. It may be that the better division is that betweenuniversal wholes and integral wholes, not genera and true wholes, andindeed many medieval philosophers seem to take this route.

This strategy, however, only provides some relief from the puzzlesthat seem to arise from Boethius’ four criteria. As we will seein the next section, there is still much more to be done to secure aclear and principled distinction between universal wholes and integralwholes.

3.3 Distinguishing universals from integral wholes with respect to their parts

Boethius and the majority of his readers assume that universal wholesand integral wholes are irreducibly distinct kinds of whole. But thereare some questions, many raised by minority parties among those whostudied Boethius, about whether a clear, principled distinctionbetween these two kinds of whole can be maintained.

To see how these questions can be raised, consider one particularintegral whole that Boethius himself seems to countenance, namely, theintegral whole consisting of all the human beings on the planet(On Division 888c). Granted, this is a very large and diffusediscrete whole. But if we allow crowds and flocks to be integralwholes, there seems to be no principled reason to reject the existenceof the sum of all humans. (And, again, medieval logicians would havehad the authority of Boethius as support.) At the same time each ofthese human beings is a subjective part of the universal Human Being.If wholes are distinguished by the type of parts that they have, itseems that the universal Human Being just is the integral wholecomposed of all human beings.

Peter Abelard reports that there were some medieval philosophers whodrew this very conclusion. Abelard describes and attacks this“collection theory” of universals in hisLogicaIngredientibus (1, 13.18–15.22). For fuller presentationsand evaluations of Abelard’s critique, one should consult Henry(1984, 235–59), Freddosso (1978), and Tweedale (1976,113–15). Here I will give only one objection: Abelard thinksthat the collection theory gets the relation of dependence between theuniversal and the individuals backwards. According to Abelard, thecollection theory is committed to the view that when Socrates dies,the universal Human Being is changed, and if one believes, as Abelarddoes, that a discrete integral whole is identical to the sum of itsparts, then the Human Being that has Socrates as a part is notidentical to the Human Being without Socrates. There is a new HumanBeing. But this, Abelard insists, is contrary to the orthodoxunderstanding of universals, which states that while the individualsthat fall under a universal are impermanent, the universal itself ispermanent. Indeed, Human Being would exist even if every human beingwere annihilated.

Yet, whatever one ultimately makes of the collection theory andAbelard’s counterarguments, this much is clear: the differencebetween integral wholes and universal wholes cannot be defined solelyand simply in terms of the kinds of items that are parts of thesewholes. Substances can be parts of both. Rather, the distinctionbetween universal wholes and integral wholes must be maintained interms of more nuanced constraints or properties.

Many medieval philosophers focus on the fact that every part of auniversal whole taken singularly must accept the predication of thename and the definition of the whole, whereas this is not true of eachand every part of an integral whole (AquinasSumma Theol. IIIq. 90, art. 3; Walter BurleyDe toto et parte, 302; andBuridanSummulae 8.1.4). For example, Socrates and Plato areboth parts of the universal whole Human Being, and Human Being ispredicable of both Socrates and Plato. That is,

Socrates is a human being, and Socrates is a rational, mortal animal.

and

Plato is a human being, and Plato is a rational, mortal animal.

This is true ofevery part of Human Being. In contrast, anintegral whole is not predicable of its parts taken singularly. Thatis, one cannot say that

This piece of wood is a house.

Integral wholes are only predicable of their parts taken all atonce.

This wood and this stone and these other parts taken together are ahouse.

This seems to give us a principled way to exclude some items from theuniversal whole Human Being and thus to distinguish it from theintegral whole consisting of all human beings. For this seems to beright: only whole human beings are parts of Human Being. That is,Socrates and Cicero are parts of Human Being, but Socrates’finger and Cicero’s head are not parts of Human Being, eventhough the latter are parts of the integral whole consisting of allhuman beings. Socrates and Cicero are parts of Human Being becauseSocrates is a human being and Cicero is a human being. The parts ofSocrates and Cicero are not parts of Human Being becauseSocrates’ hand is not a human being and Cicero’s head isnot a human being; whereas they are part of the integral wholeconsisting of all the humans, since Transitivity seems to hold forintegral parts: ifx is a part ofy andyis a part ofz, thenx is also a part ofz.(It should be noted, however, that some medieval authors weresuspicious of an unrestricted form Transitivity, even for integralwholes. See Arlig 2013 for details.) There is also this: while it istrue that some of the parts of the sum of all human beings will acceptthe predication of the name and definition of Human Being, if oneexamines the matter carefully, it turns out that the definition ofthe sum of all human beings is not the same as the speciesHuman Being, for the sum of all human beings is not itself a rational,mortal animal.

There is a complication that is raised by considering wholes that arecomposed of homogeneous parts. Boethius himself illustrates theproblem with the example of a bronze rod, which he believes is ahomogeneous substance (On Division 879d). Recall that ifsomething is a homogeneous substance, then every part of that thing isalso the same substance. That is, ify is a homogeneous wholeandx is a part ofy, the name and definition ofy is also predicable ofx. Every portion of bronzecan take both the name and the definition of bronze, and hence, itseems that every part of the bronze rod meets the standard of being asubjective part. Therefore, it seems that the bronze rod is auniversal, which is clearly absurd.

Boethius resolves the puzzle of the rod and its parts by noting thatwhile it is true that each portion of bronze is bronze, it is not truethat a portion of the original quantity of bronze is that originalquantity of bronze (BoethiusOn Division 880a)Boethius’ solution is often repeated ((ps?) AbelardDedivisionibus 169.33–36; Albertus MagnusIn de Div.tract. 2, ch. 5 (Loe ed., 35); and Radulphus BritoIn de Top.II, q. 9 [1978, 45]; cf. AquinasSumma Theologiae I, q. 3,art. 7).

The solution points to another difference between integral wholes anduniversals. As Abelard puts it, every integral whole “drawstogether” (comprehendere) some quantity(Dialectica, 546.21–27). The suggestion is that whensome items compose an integral whole, that whole will be measurablewith respect to some quantity or other. The integral whole that ismeasured by some quantity need not be composed out of materialelements. Consider the mereological sum of the angel Gabriel and theangel Michael. If this sum is a whole (and there seems to be no reasonto deny this), it seems it is an integral whole. However, there aresome integral wholes which do not seem to embrace any quantity,namely, thoughts and actions. So there is need to locate yet anotherdifference between universal wholes and integral wholes.

The last difference that we will consider is this: Universals, likeHuman Being or Horse, are not literallycomposed out of theirspecies. Integral wholes, such as Socrates or a house, are composedout of their parts (John BuridanSummulae de Dialectica6.4.4; OckhamExpositio in librum Porphyrii 2.16[Opera II, 54]). Composites are dependent upon theircomponents. Socrates is composed of his body and his soul. If thesecomponents did not exist and combine to form Socrates, Socrates wouldnot exist. Components are often, but not necessarily, temporally priorto the whole that they compose. For example, the house is composed bybits of wood, stone, and iron, and these parts existed prior to theexistence of the house. Universal wholes are neither dependent upontheir parts, nor are the parts of a universal temporally prior to it.The species existed long before Socrates or Cicero, and will existlong after Socrates and Cicero. Indeed, Human Being can exist, even ifno individual human being exists.

Unlike the appeal to quantity, the composition criterion can beapplied to thoughts and actions, for a complex thought requires thesimpler concepts that compose it, and penance requires that theactions that constitute penance occur.

On the other hand, it appears that a universal is also a composite.Every universal except for the most general of genera, can be said tobe “composed” out of a genus and adifferentia.These parts are sometimes called “essential” parts of theuniversal since the genus anddifferentia together constitutethe essence of the universal (AquinasSumma Theologiae I, q.8, art. 2, ad 3; and I q. 76, art. 8), although to avoid confusion weperhaps ought to call themdefinitional parts. The essence ofa universal is usually encoded in its definition. For example, thedefinition of Human Being isrational mortal animal.Animal is the genus, and rationality and mortality are thedifferentiae.

Many medieval philosophers try to dampen this criticism by suggestingthat the universal is, strictly speaking, not composite; it merelymimics composition (again, see OckhamExpositio in librumPorphyrii 2.16). Perhaps this is a viable response, but there isanother problem with the composition requirement: it does not tell uswhy all the parts of integral wholes are integral parts. It wasalready observed in passing that not all parts of an integral wholeare plausibly components of their wholes. Consider Socrates. Theelements are strictly what compose Socrates. It is only when Socratesis composed that other parts, such as his hands and feet, come intoexistence. Or put another way, it is false to say that one makes ahuman being by cobbling together hands, feet, and head. Such acreature would be Frankenstein’s monster, not a human being.

In sum, a number of proposals are offered for how one can distinguishuniversal wholes from integral wholes. But perhaps no single proposalis universally embraced because of the bewildering variety of itemsthat are integral wholes. An obvious solution would be to reduce thenumber of items that can be integral wholes or integral parts. And aswe have seen already, some philosophers in fact do this in certaincases, as when Abelard eliminates temporal wholes. However, based onwhat we presently know, no medieval thinker attempts anything like asystematic eliminative project in mereology as such. (Thus, whileOckham, for instance, famously undertakes a comprehensive and severeeliminative project in ontology, he does not attempt to develop asystematic theory of what can and what cannot be a part. Rather, forhim, the question whether “part” is being used eitherproperly or figuratively tends to arise within a specific context andthe resolution often appears to be developed on a case by casebasis.)

3.4 Parts of potential wholes

Potential wholes add further complications. Like genera, and unlikeintegral wholes, it is usually assumed that potential wholes are notliterally composed out of their parts. Potential wholes, or at leastthe subjects underlying them, are items that are fundamentally simple.But, in another respect, the parts of the potential whole behave likeparts of an integral whole. Different souls can possess only aselection of the capacities associated with soul, and so in a sense,there is a construction of a specific soul’s total capacity outof many discrete capacities and powers. And no one power by itselfexhibits or stands in for the whole soul.

The question thus arises as to how one can distinguish powers as partsfrom both subjective and integral parts. Here we will look briefly atonly one account. Aquinas separates potential wholes from bothuniversals and integral wholes by considering two parameters: thepresence of the whole in the part with respect to the whole’sessence, and the presence of the whole in the part with respect to thewhole’s power (Summa Theologiae I, q. 77, art. 1, ad1). The universal whole “is present to each of its parts in itsentire essence and power”. It is for this reason that each partof the universal is a subjective part. In contrast, the integral wholeis not in each of its parts either in respect to its entire essence orin respect to its power. Hence, the integral whole is not predicableof any one of its parts taken singularly. Finally, the potential wholeis present to each of its parts with respect to its entire essence,but not with respect to its full power. This is why, even thoughone’s soul is non-composite and cannot be cut up, the operationof thinking does not entail along with it the operation ofsensing.

4. Mereology and metaphysics

Medieval philosophers are well aware that the study of wholes andtheir parts has numerous applications in metaphysics. I will concludethis survey by examining two applications. First, I will look at howtheorizing about parts and wholes informs medieval reflections onidentity at a time. Second, I will consider how mereology influencesmedieval theories of persistence through time and survival throughchange.

4.1 Identity

Medieval philosophers think that no part is identical to its whole.^[5] The reasons why this is true are as varied as the types of parts andwholes themselves. Ifx is a quantitative part ofy,thenx is lesser in quantity thany. Adonkey’s form is not identical to the donkey, because thisdonkey is a composite of the donkey’s formand somethingelse, namely, the donkey’s matter. Socrates’ soul isnot identical to the human being who is Socrates, for the human beingis a composite of body and soul. Socrates is not a universal, eventhough he is a human being. And the extension of Animal is greaterthan the extension of Human Being.

However, there is another question that does divide medievalphilosophers. Consider an integral whole. An integral whole iscomposed out of its parts. But is it true that an integral whole is nothing other than the sum of these parts? Some philosophers, such asAbelard and Ockham, argue that the whole is no thing other than thesum of its parts. Others, such as Duns Scotus, argue that the whole issome thing which is really distinct from the sum of its parts. (On thedebate between Ockham and Scotus, see Cross 1995, Cross 1999, andNormore 2006.) The former philosophers perhaps base their position onpassages from Boethius’ treatments of mereology such asthis:

Every thing is the same as the whole. For example, Rome is the same asthat which is the whole citizenry. Each and every thing is also thesame as all its parts when they are gathered together into a unity.For example, a man is the same as the head, throat, belly, feet, andthe rest of the parts gathered together and conjoined into a unity.(In Cic. top. 285.24–28)

Those who claim that the whole is not identical to its parts oftenappeal to Aristotle. For example, an anonymous commentator onAristotle’sSophistical Refutations argues that“the five are not the two and the three” on the groundsthat Aristotle has shown in hisMetaphysics that “thecomposite, in general, is something other than its componentparts” (Quaestiones super Sophisticos Elenchos q. 831[1977, 346]; cf. AristotleMetaphysics 8.6,1045a9–10).

Peter Abelard provides one of the most sophisticated solutions to thisproblem. Abelard emphatically asserts that the whole is no thing otherthan the sum of its parts (Dialectica 344.34–5, and550.6–7; cf. 560.34–561.2). But this is surely false.Consider a house. A house is composed out of a specific sum of boardsand bricks. Yet these boards and bricks could be sitting together onthe building site without being arranged as a house. Hence, it is hardto see how the house is no thing other than the sum of the parts.

Abelard will respond by drawing a distinction between theessentia, or concrete being, of a thing and the“conditions” (status) which a thing may possessor be in. True, the sum of boards and bricks must possess theappropriate arrangement in order to be a house (LogicaIngredientibus II, 171.14–17; andDialectica550.36–551.4). But when the sum takes on a specific structure itdoes not take on an extra part. Whatever they are, Abelard makes itvery clear that structures and arrangements like those found in housesare notthings (res), and only things can be parts.The imposition of a structure upon the sum of the boards and bricksdoes not add a part to the thing that is that sum, and hence, the sumof the boards and bricks is the same inessentia as thehouse—they are the very samething. Nevertheless, therecan be different conditions in which this thing is. When the sum ofboards and bricks are arranged in one way, they are in the conditionof being a house. When they are arranged in other ways, they are notin this condition. In short, to claim that the house is no thing otherthan the sum of its parts does not imply that the sum is in the stateof being a house when the sum exists.

Abelard’s solution will only work if he can motivate the claimthat a thing taking on a new condition does not make a new thing.And medieval realists went after such reductivist programs in much thesame way that some contemporary realists do. Specifically, they oftenpresented truth-maker challenges, or “anti-Razor”arguments (see Maurer 1984): If at one time this sum of boards andbricks is not a house and at another time it is a house, then theremust be some item in reality, some thing, that is absent at theformer time and present at the later time. The task, of course, is toidentify the right sort of thing that can be such a differencemaker.

But let us move on to consider another related puzzle about identityat a time. Abelard’s characterization of numerical sameness anddifference seems to give us the tools to answer a common medievalpuzzle which I will call the Problem of the Many Men. Versions of thisproblem can be found in a number of medieval works (see, e.g., AbelardTheologia Christiana III, § 153 (1969, 252); William ofSherwoodSyncategoremata (O’Donnell ed., 60); Albert ofSaxonyQuaestiones in Physicam I, qq. 7–8, and hisSophismata 46, 25va–vb). The puzzle can easily be generatedusing a crude understanding of numerical sameness. Assume thatSocrates’ body is perfectly intact: he has all his limbs, andtheir parts. Now consider every part of Socrates’ body exceptone finger. Call this wholeW.W is not numericallythe same as Socrates, so it appears that they must be numericallydistinct. Socrates’ whole body is imbued with the soul of a man.But it also happens thatW is imbued with the soul of a man.So, there are now two numerically distinct men where it initiallyappeared there was one. But it gets worse. Considering the body apartfrom one finger was only one of an indefinite number of suchconsiderations. And by the same reasoning, these other bracketedwholes composed from Socrates’ body are also men. Hence, it iseasy to generate an indefinite number of numerically distinct menwhere commonsense tells us that there is only one.

Abelard might appear to be capable of unraveling this puzzleby employing his distinction between difference inessentiaand numerical difference, which he presents most clearly in hisanalysis of the Trinity (Theologia Christiana III, § 139and §148 (1969, 247 and 250–1); see Brower 2004,226–34). Abelard’s notion of sameness and differenceinessentia is informed by his appreciation of thenotion of mereological overlap (see the section on BasicPrinciples/Other Mereological Concepts, in the entry onmereology). His appreciation of mereological overlap in and of itself is awatershed, since he is one of the first (and perhaps one of the few)medieval philosophers to clearly understand this phenomenon.^[6] According to Abelard,x is the same inessentia asy if and only if every part ofx is a part ofy and every part ofy is a part ofx. Inother words,x is the same inessentia asyif and only ifx andy mereologically coincide. Ifx is the same inessentia asy, thenx is numerically the same asy. However, ifx is not the same inessentia asy, it doesnot follow thatx is numerically distinct fromy. This is becausex andy couldoverlap—that is, share at least one part—even if they donot coincide. Thus, Abelard’s answer to the Problem ofthe Many Men is that while there are many overlapping men, each ofwhich is different inessentia from the others, this does notentail that there are an indefinite number of numerically differentmen. One might, however, wonder whether Abelard has completelydispelled the hint of paradox generated by the initial puzzle. Afterall, it would be nice to knowwhich of these overlapping, butnon-coincident wholes is Socrates. All that Abelard’stheoretical tools allow us to do is to say that while there isthis whole which is imbued by Socrates’ soul and thereisthat whole which is imbued by Socrates’ soul, thesewholes are notnumerically many and, thus, there are notnumerically many men where intuitively we thought there was only one(see Normore 2006, 749).

Most medieval philosophers tackle the Problem of the Many Men inanother manner. Albert of Saxony, for example, resolves the puzzle byclaiming that nothing which is a part of something else can be anumerically distinct existing being (Quaes. in Arist.Physicam I q. 8, 131–32;Sophismata 46, 25vb; cf.Fitzgerald 2009). Socrates’ body less the finger is not adistinct human substance, since it is a part of a human substance,namely, Socrates. Therefore, even if we grant that there are manythings present, there are not many distincthumans present where we thought that there was onlySocrates. If that appears to be an argument by stipulation, the readermight find William of Sherwood’s elaboration on this strategyhelpful (Syncategoremata (O’Donnell ed., 61)). Williamconcedes that, because Socrates-less-the-finger is perfected by ahuman soul, it canin a certain respect be said to be a humanbeing. But in this respect Socrates and Socrates-less-the-finger areperfected by thesame soul and if one wants to count humans,one counts human souls. Thus, Socrates and Socrates-less-the-fingerare not numerically distinct humans. On the other hand, ifSocrates-less-the-finger is considered as apart of somethinglarger that is also perfected by the same soul, then in this respectSocrates-less-the-finger is numerically different from Socrates, butit is not a human being. Considered in terms of its being apart, Socrates-less-the-finger is a part of a body of a humanbeing, and a body of a human being is not a human. Hence, we count toomany humans only if we ignore these different respects under which thepremises of the argument can be made true.

4.2 Persistence

Medieval philosophers also worry about the identity of objects overtime and through change. Medieval examinations of identity over time,or persistence, are often occasioned by reflection on the maximalproposition associated with the Topic from the integral whole, whichstates:

If the whole is, the part is.

This maximal proposition, however, implies

If the part is not, the whole is not.

But this seems to entail that if Socrates’ hand does not exist,Socrates does not exist.

There is an innocent interpretation of this Topical maxim. Recall that“the wholex” can mean the completex.Accordingly, to say that the whole is destroyed if the part isdestroyed, is merely to say that the whole is incomplete or“mutilated” if a part is removed. It might also be thecase that “the wholex” merely signifies all theparts ofx taken together. On this reading, the maximalproposition would merely imply that, ifx does not exist,then the whole consisting ofx and some other parts does notexist. And, indeed, Boethius seems to mean only this:

For if a part of the whole perishes, the whole—namely, the[whole] whose one part was destroyed—will not exist. (OnDivision 879c [Magee 1998, 14]).

Boethius’ assertion that the whole consisting ofxand all the rest will perish whenx perishes leavesit open whether the remaining parts are still, say, Socrates or ahouse. Naturally, thecreative parts had to be present inorder to create Socrates or the house. For example, this sum of wood,cement, and nails is the sum of the creative parts of a house. Ifthere had been no nails, wood, or cement, there would have been nohouse. Still, commonsense tells us that the creative parts need notremain present after the house has been created. Some boards and somenails may be replaced in the house, but this does not compromise theexistence of this house.

Despite the appeal of commonsense, there were some medievalphilosophers who took the maximal proposition to imply that ifSocrates loses a finger, Socrates ceases to exist. At least at onepoint in his career, Peter Abelard appears to seriously entertain theradical reading of the maximal proposition (Henry 1991, 92–139).Abelard himself might have eventually backed away from this view(Martin 1998), but there are reports that suggest some ofAbelard’s followers, the so-called Nominales, wholeheartedlyembraced the extreme interpretation of the maximal proposition. One ofthe positions associated with the Nominales was that “nothinggrows” (nihil crescit), and there is a survivingcommentary on theCategories with Nominalist features wherethe author argues that any addition, removal, or even relocation of apart compromises the identity of the thing itself (Ebbesen 1999, 397).This extreme interpretation is derived from the judgment that eachwhole is the same thing as a unique set of parts. This house must becomposed out of these nails, these boards, and this cement. If I useother nails or other boards, I could make a house, but not this veryhouse. This premise is no doubt controversial, but Abelard and many ofhis contemporaries have principled reasons for holding it. Abelardbelieves that the ultimate pieces of the universe are tiny,indivisible bits of matter, souls, and perhaps some forms. The itemsthat we experience are composites of these elements. Each element isself-identical. Composite beings are individuated by the elements thatmake them up, and in the case of complex, composite beings—suchas artifacts and substances—by the arrangement that these bitshave. Given such a universe, it is quite plausible to assume that theidentity of a composite item is solely determined by its parts. Itwould then easily follow that the removal or addition of a composingelement entails that the new whole is not identical to the old whole.Another whole similar to the original might exist after themereological change take place, but strictly speaking the two wholesare not identical.

Most medieval philosophers are not as extreme as the Nominales. Ofcourse, there are those parts that are required to keep the wholeintact, but there are also those parts that can be lost withoutcompromising the integrity of the whole. If we were to cut offSocrates’ head, he would perish. But if we amputateSocrates’ right hand, it seems that Socrates would not cease toexist. He would merely lack a hand. The parts required to keep a wholeintact are called “principal”. Ones that could be lostwithout compromising the integrity of the whole were called“secondary” parts (AnonymousIntroductiones maioresMontane 71va–72rb; Albertus MagnusCommentarii in Dediv. tract. 2, ch. 5 (Loe ed., 33–4); BuridanSummulaede Dialectica 6.4.4). Examples of principal parts would be thefoundation of the house, the heart of a cat, and the brain ofSocrates. Examples of secondary parts are a brick in the house, astrand of hair belonging to the cat, and a fingernail of Socrates.

In the Twelfth century there was a heated debate about how todetermine whether this or that part is principal (see Pseudo-JoscelinDe generibus §§ 3–20 (2014, 122–29;Arlig 2013), for as Abelard pointed out the following account isinsufficient:

x is a principal part ofy if and only if theremoval ofx entails the destruction ofy.

Abelard reminds his audience that even the extremist can accept thisaccount of principal parts; the only difference is that on his accountevery part is a principal part (Dialectica549–52).

The best answer that twelfth-century thinkers come up with is that aprincipal part ofx is that which, when removed, compromisesthe form ofx (Pseudo-JoscelinDe generibus§§ 23–6 (2014, 130–31); Arlig 2013,106–10). Many later medieval philosophers followed suit. Atleast in the case of substances, the substantial form could do themetaphysical work of guaranteeing persistence through time and change(Pasnau 2011, 689–92). For example, we have already seen thatBurley draws a distinction between the wholesecundum formamand the material whole. The whole considered formally persists so longas the form persists. The whole considered materially is the onlywhole compromised by changes in material parts (De toto etparte, 301). Most commonsense objects are identified with a wholeconsidered with respect to form, and so there would be no reason tothink that these things are substantially compromised by materialchanges. (The fate of artifacts, would be different, as their formsare merely accidental forms. A mereological change in the case of anartifact would seem to entail that the initial artifact is replaced byanother one. But later medieval philosophers do not seem to be allthat troubled by these implications. Whethertheologianscould stomach all the consequences that this view of artifacts entails(for example, for whether something remains ritually pure after it hasbeen chipped) may be another matter. But this is an area whetherfurther research is needed.)

This basic Aristotelian picture is complicated, if not outrightaltered, by several very influential fourteenth-century thinkers,including Ockham and Buridan (Normore 2006; Pasnau 2011,692–702). We will consider Buridan’s theory of identityover time and change (Quaestiones super De generatione etcorruptione 1.13;Quaestiones in Physicam 1.10; withEnglish paraphrase in Pluta 2001). According to Buridan, there arethree senses of numerical sameness: (1)x is“totally” the same in number asy, (2)xis “partially” the same in number asy, and (3)x is numerically the same asy in a “lessproper” way. Something is totally the same in number if all itsparts remain the same and it neither acquires nor loses any parts. Inthis strictest of senses, no corruptible thing whatsoever persiststhrough mereological change. Something is partially the same in numberif its “most principal part” remains numerically the same.This is the sense that allows us to claim that Socrates is numericallythe same man now as that man ten years ago. In particular, it isSocrates’ intellective soul which is the principal part andguarantor of persistence through change. Finally, something is lessproperly the same in number if there is merely a continuous successionof beings that maintain a similar shape, disposition, and form. Thisless proper mode of numerical sameness allows us to claim that theNile River here today is numerically the same river as the Nile backin Herodotus’ time. While it is probably not a surprise to learnthat rivers are the same in number only in a less proper sense, it isstriking that Buridan goes on to assert that plants and non-rationalanimals too can only be numerically the same in this less proper sense(Quaestiones super De gen. 1.13 (2010, 114–15)).Buridan’s reason for thinking this is that non-rationalcreatures do not have the sort of soul that can act as a guarantor ofless proper identity; rather, non-rational souls are themselvesaltered when the material parts are altered. Thus, over the span of anormal lifetime, neither the matter nor the form of a horse or anorchid remains numerically the same in anything other than a lessproper sense.^[7]

Buridan’s account appears to commit him to the view that theaddition or subtraction of even tiny and seemingly inconsequentialparts can bring about substantial change: strictly speaking we do nothave one horse, but rather a succession of horses unified by threefacts, namely, (1) these successive beings have some of the sameparts, (2) this line of succession is continuous, and (3) each memberin the line of succession is a horse. The picture we get has manysimilarities to the accounts of identity that we get in early modernthinkers like Hobbes and Locke. Thus, it is tempting to think that weare witnessing in Buridan’s work one more way in which theAristotelian scholastic edifice is being eroded to the point that itwill quickly crumble in the seventeenth century (see Pasnau 2011, 703f., and Lagerlund 2012, esp. pp. 481–2).

5. Concluding remarks

Medieval philosophers study a variety of wholes and parts, and theyoften do so with a remarkable degree of sophistication. To be sure,some aspects of medieval mereology are idiosyncratic, but many of thepuzzles that medieval philosophers wrestle with are analogous to onesthat interest contemporary students of logic and metaphysics. Medievalphilosophers are particularly attuned to the relationship betweenmereology and various problems in metaphysics, and many of theirsolutions to puzzles of identity and survival are embraced byphilosophers in other periods. Even the idiosyncratic aspects ofmedieval mereology reveal a sophisticated appreciation of threefundamental questions in mereology, namely, what items are wholes,what items are parts, and under what conditions is one item a part ofanother item. This survey can only hint at the richness of medievalmereology. In part, this is due to the overwhelming number of medievalauthors who say something or other, somewhere or other, about partsand wholes. But it is also due to the fact that there are still moretexts to be unearthed, properly edited, and studied with care. Basedon what we have already discovered, we should be confident that wewill find many more interesting reflections on parts and wholes andthat we will uncover further connections between medieval mereologyand more recent theories of parts and wholes.

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• Mediaeval Logic and Philosophy. Downloadable texts, links to other sites.
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Related Entries

Abelard [Abailard], Peter |Aquinas, Thomas |Boethius, Anicius Manlius Severinus |Buridan, John [Jean] |Burley [Burleigh], Walter |identity |identity: over time |medieval philosophy |mereology |Radulphus Brito