## 1. Forums for medieval mereology



One can find discussions of parts and wholes throughout the medieval
philosophical and theological literature. But there are two places
where the student of medieval mereology can reliably look to find
sustained reflections on parts and wholes as such, namely, treatments
of division and the Topics. The main authority on division and the
Topics is the Roman philosopher
 Boethius
 (c. 480-524 C.E.). Boethius is now most famous for his
*Consolation of Philosophy*, but his influence on medieval
philosophy is as much due to his commentaries on Aristotle's
*Categories* and *On Interpretation*, his theological
treatises, and his handbooks on logic (see Chadwick 1981 and Marenbon
2003). Boethius' treatment of division is found in his handbook
*On Division* (*De divisione*). His treatment of the
Topics is found in his handbook *On the Topical Differences*
(*De differentiis topicis*) and in his commentary on
Cicero's treatment of the Topics (*In Ciceronis
Topica*).


### 1.1 Division



The methods of "division" (Greek: *diairesis,*
Latin: *divisio*) and "collection" (Greek:
*sunagoge*, Latin: *collectio*) have their roots in
Plato's later dialogues, and they are common in Ancient
Neoplatonic and Aristotelian treatises on logic. Plato tells us that
collection and division provide us with a way to understand the
relationships between some unity and some plurality (*Phaedrus*
265d-266b, and *Philebus* 16c-17a). Division is a process
whereby any sort of unity is resolved into a plurality. Collection is
the process whereby a plurality is collected into a unity.



For the Neoplatonists, division and collection are first and foremost
applied to genera and species (such as Animal and Human), and
instances of these universals (such as Brownie the donkey and
Socrates). This primary mode of division is often interpreted as a
logical exercise. In particular, it is a method for developing
definitions 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 to
determine the hierarchical relations between a universal and that
which falls under it. But divisions are applied to a variety of other
items. There is no single, canonical list of divisions (for an
overview, see Magee 1998, xxxvii-xlix). The list presented by
Boethius is the one bequeathed to the Latin West (*On Division*
877c-d). Boethius distinguishes between two broad categories:
substantial divisions and accidental divisions. These divisions are
divided 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 term
*locus*. (Note that the translators of Peter of Spain (2014)
have chosen to render it as "place", which is the basic
meaning of the Latin term.) As Stump (1978, Part 2, and 1982) and
Green-Pedersen (1984) have pointed out, the notion of the Topic
evolves over the course of its use in ancient and medieval logic, but
roughly put, medieval logicians think that the study of the Topics
helps one to discover a number of self-evidently true propositions, or
"maximal propositions", that can serve as warrants for
arguments. (See Peter of Spain, *Tractatus* V.3-4 (2014,
198-205).) For example, suppose that someone makes this
inference:



> 
> If Socrates is human, Socrates is animal.
> 



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



> 
> If a species is predicated of something, that species' genus is
> also predicable of that thing.
> 



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



Following a well-established Ancient tradition, the Topic from the
whole 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 the
subtleties of ordinary language. In particular, they are mindful of
the 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 "*totus* *x*" can mean
"all the parts of *x* taken together" or "the
entire *x*" (see
 Section 2
 below). This sensitivity to the distributive sense of
"*totus*" might be what motivates medieval
logicians to add further refinements to the theory of the Topics.
Specifically, the Topic from the whole is routinely divided in a
six-fold manner (see, e.g., Peter of Spain *Tractatus*
V.11-18 (2014, 210-19); following Boethius' terse
remarks in *De topicis differentiis* II, 1189A-C; trans.
Stump 1978, 52-53). In addition to the Topic from the universal
whole and the Topic from the integral whole, there are these
additional 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 considers
propositions where the term is taken universally, such as "Every
*x* is a *y*", or "No *x* is a
*y*". The Topic from the whole in a respect considers a
term in respect to some limiting qualification. So, for example, if
*x* is white *on its surface*, then every part of
*x*'s *surface* is white. The Topic from the whole
in place classifies propositions bounded by the term
"everywhere" or its cognates. So, if water is
*everywhere*, then water is *here* (where
"here" designates a "part in place"). And,
finally, the Topic from the whole in time considers inferences that
one can make from propositions bounded by the term
"always" or "never".



A full account of medieval mereology would consider carefully the
details of all six sub-Topics. But of special interest to us are the
treatments of the Topic from the integral whole and of the Topic from
the integral part. In these discussions, medieval philosophers often
consider whether the traditional maximal propositions associated with
these Topics in fact describe the logical and metaphysical relations
that hold between an integral whole and an integral part.
Specifically, the maximal proposition that applies to integral wholes
is:



> 
> If the whole is, then the part is.
> 



The maximal proposition said to apply to the integral part in relation
to 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 if
Socrates' hand ceases to exist, then Socrates ceases to exist.
As we will see in
 Section 4.2,
 such consequences do not escape the notice of medieval philosophers,
and much of interest regarding the metaphysical implications of the
Topics and their maximal propositions ensues.


## 2. Wholes



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



Closely related to the sense of being complete, "whole"
can have a distributive function and the phrase "*totus x est
y*" ("whole *x* is *y*") can mean
all of *x* is *y*--that is, all the parts of
*x* taken together is *y*. This is called the
term's "categorematic" sense. Medieval logicians
also distinguish a second distributive sense of "whole",
which they call the "syncategorematic" sense of the term
(see, e.g., William of Sherwood *Syncategoremata* [1941, 54];
Ockham *Summa Logicae* II, ch. 6 [*Opera Philosophica*
I, 267-69]; John Buridan *Summulae de Dialectica*
4.3.7-1; Albert of Saxony *Sophismata* 45). When
"*totus x est y*" is interpreted
syncategorematically, it means each and every part (*quaelibet
pars*) of *x* is
 *y*.[1]



Finally, "whole" can signify a thing that is either
composed of some things or divisible into some things. This usually is
the sense of "whole" at work in discussions of collection
and division.



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



First, one should not assume that all wholes are mind-independent
features of the world. Peter Abelard, for instance, argues that
temporal wholes (such as days, weeks, or hours) and universal wholes
are not things (*res*). Yet, provided that we do not reify
these items, Abelard will allow us to treat items like days and hours
as wholes consisting of parts, and he will allow us to talk about
universals and their parts. (On temporal wholes, see
*Dialectica* 554.14-23 and *Logica Ingredientibus*
2, 187.9-14. On universals, see *Logica Ingredientibus*
1, 10.8-16.18 and *Logica 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 their
metaphysical commitments as well as their understanding of what wholes
must 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 species
are not actually parts of genera. These items are merely
"parts", and accordingly their corresponding species and
genera are only "wholes", in a figurative sense. Once this
fact is appreciated, however, Ockham is willing to allow us to speak
of a genus or a species as a "whole of a sort"
(*quoddam totum*), since there is a legitimate sense in which a
species "contains" individuals and a genus
"contains" its species (Ockham *Expositio in librum
Porphyrii de praedicabilibus* ch. 2, [*Opera Philosophica*
II, 54]).



With these caveats in place, we can begin to explore the basic
mereological categories that one finds in medieval treatments of
mereology. As was noted, all manner of items can be wholes, but
generally speaking medieval philosophers believe that this motley
group can be organized under three broad categories: integral wholes,
universal wholes, and potential wholes. It will soon become apparent
that the category of integral wholes is quite broad. Indeed, it is so
inclusive that some philosophers will feel pressure to introduce a
fourth basic category, substantial wholes. But others will suggest
that an integral whole is divisible into two distinct kinds of parts,
quantitative parts and substantial parts. For this reason, I will
consider the issue of substantial wholes and substantial parts in
 Section 3.1.


### 2.1 Integral wholes



Much of what we encounter in the material world is, or can be
considered to be, a whole. Cars, houses, plants, and human beings are
composed 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 one
another, or relatively scattered. Contiguous wholes consist of parts
that are discrete, but spatially close together. Discrete wholes whose
parts are relatively diffuse are "disaggregated" integral
wholes. (For a fairly comprehensive catalog of the kinds of integral
wholes, see Anonymous *Compendium Logicae Porretanum*
III.12.)



Whether an item is a continuous or contiguous whole will depend upon
other metaphysical commitments. Some early medieval philosophers only
have simple artifacts like fences and walls in mind when they talk
about contiguous wholes. But others think that even more complicated
man-made objects, such as wagons, houses, and clocks are contiguous,
and not continuous wholes, because only substances are continuous
wholes. 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 only
God can fuse parts together into a continuous unity. Human
manipulations, no matter how skilled, are only capable of placing
parts in close proximity to one another (*Dialectica*
417.4-37; 419.35-420.6). Drawing upon Aristotle's
reflections on form and matter (especially in his *De Anima*,
*Physics* and *Metaphysics*), Aquinas thinks that only
substances possess the kind of form that can inhere in each and every
part of the whole and thereby make it truly one (*Summa
Theologiae* I, q. 76, art. 8; *Quaestiones de Anima* q. 10;
see also Pasnau 2002, 79-88). An artifact possesses a form that
merely orders and arranges the parts of the whole. We can tell whether
a form imbues its parts in the manner required of a substantial form
by attending to the effect of the form's existence on the
functionality of the parts. The human soul inheres in the hands of a
human being as a substantial form because if the soul were removed,
that part would cease to function as a hand - indeed, Aquinas
insists that this thing would cease to be a hand. By contrast, when
the form of a car is removed from the items that make up the car, many
of these items can still be car-parts. For example, a carburetor can
still be a carburetor; it can be placed in another car and continue
functioning as a
 carburetor.[2]



In addition to artifacts, plants, and animals, some medieval
philosophers expand the class of continuous integral wholes to cover
homogeneous masses, such as some gold or some water, and the class of
disaggregated integral wholes to include scattered mereological sums,
such as the sum of this mountain and this dog. However, one will not
find any medieval philosopher who assents to the modern mereological
thesis of universalism, or what David Lewis calls "unrestricted
fusion" (1991, 74; cf. Simons 1987, 108-12 and the
subsection on unrestricted fusions in the Encyclopedia entry on
 mereology).
 Peter Abelard comes close when he asserts that any two items, even
ones belonging to different categories of being, can constitute a
plurality (*Dialectica* 548.19-22; see also the anonymous
twelfth-century *Introductiones Montanae maiores*, 69rb, where
Abelard's thesis is mocked). But one suspects that even Abelard,
if pressed, might step back from assenting to a full-blown version of
universalism for theological reasons. For in discussions of
God's simplicity, it is routinely noted that God is not
intrinsically composite *and* that He cannot be combined with
anything else *and* that nothing can be combined with Him (see,
e.g., Aquinas *Summa Theologiae* I, q. 3, arts. 7-8).
This strong thesis about the lack of composition in God is maintained
because *being a part of something* implies a certain sort of
potentiality and incompleteness. If anything could be a part of a
whole that also has God as a part, then there is something that God
could be-namely, a part of this composite-that He
currently is not. And even if He currently is a part of the sum of all
things (for instance), given that some of the other parts are
contingent beings, the sum of all things is contingent. Accordingly,
God's parthood is contingent, or has a potentiality. But God is
fully actual and necessary. Hence, God cannot be a part of some
greater
 composite.[3]



As a more general point, it must be remembered that medieval
philosophers are for the most part working within an Aristotelian
framework, and like Aristotle, their paradigmatic examples of unified
composite things are substances. If a thing is not unified by a
substantial form, then that thing has a lesser kind of unity. Humans
and dogs are more unified than houses and wagons. Houses and wagons,
nonetheless, are unified by a form-- namely, an accidental
form--and so they have a unity of a sort. Moreover, this kind
unity will be greater than the unity of a mere collection of things.
Hence, as a general rule, the more gerrymandered a collection of parts
is, the more likely it is that this composite's status as a
whole will be called into question (see, e.g., Aquinas *In
Metaphysica expositio*, Book V, lectio XXI SSSS
1102-4; Jean Buridan *Summulae de Dialectica* 8.1.4).



In addition to material beings, some medieval philosophers allow
non-material items to be integral wholes. For example, Ockham thinks
that complex mental acts can be wholes, and Aquinas insists that
actions such as penance are integral wholes (Ockham *Quaestiones in
physicam* q. 6 [*Opera* VI, 407-10]; *In De
Int.* I, prooemium, 6 [*Opera* II, 354-8]; Aquinas
*Summa Theologiae* III, q. 90, art. 3). Abelard insists that
temporal items are not integral wholes, but he seems to be in the
minority. Non-material integral wholes do not sit easily under either
the continuous or the discrete category, since their parts cannot be
related to one another with respect to location. If asked to pick,
medieval philosophers tend to label temporal wholes and events as
continuous wholes, but they are not continuous in the way that a
bronze rod is continuous. Their parts to not share some spatial
boundary; they come one after another in continuous succession. For
this reason, these wholes are sometimes called
"successive" wholes (Anonymous *Compendium Logiae
Porretanum* III.12). In addition to aggregates of time, processes
(that is, things that take time to unfold) are also sometimes
considered to be successive wholes (see, e.g., Aquinas *Summa
Theologiae* 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 philosophers
insist that the class of integral wholes does not exhaust the domain
of items that can be wholes. In particular, there are two types of
item 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 most
medieval philosophers mark off one special sort of non-material
object, the universal, and treat it as a separate type of whole.



It was noted in
 Section 1.1
 that universals, and especially species and their genera, are related
to one another hierarchically. For instance, the species Human Being
and 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 the
genus Animal. These relations between universals and individuals are
often described in the terms of collection and division. Cicero and
Socrates and all other humans are collected into the species Human
Being, and the species Human Being and Horse and all other species are
collected in the genus Animal. Correlatively, Animal is divided into
its species, and Human Being is divided into individual humans.



This language of collection and division invites medieval philosophers
to call the divisible items wholes, and the products of these
divisions parts, but most medieval philosophers are not tempted to
think that universals are literally *composed* out of
individuals or lesser species. There are noteworthy exceptions. For
example, an anonymous twelfth-century philosopher carefully
articulates and spiritedly defends a version of what is often called
the collection (*collectio*) theory of universals
(Pseudo-Joscelin *De generibus et speciebus* SSSS
84-153 (2014, 158-85)). But, in general, universal wholes
are clearly marked off from other kinds of whole, and their behavior
is thought to be distinct from the behavior of potential and integral
wholes.



There is one significant complication. In his *On Division*,
Boethius distinguishes between the division of a genus and the
division of a whole (877c-d, see also *In Cic. top.*
331.18-19). But then under his treatment of the ways to divide
wholes, he notes that one way to divide a true whole is to divide a
universal into its subordinate individuals (887d). That a universal
can be a true whole will matter below when we try to uncover the
properties that differentiate one kind of part-whole relation from
others (Section
 3.3).


### 2.3 Potential wholes



In addition to integral wholes and universal wholes, Boethius
introduces a third basic type of whole to medieval philosophers: the
whole consisting of "powers" (*potentiae*) or
"virtues" (*virtutes*) (*On Division* 888a).
These wholes are often called "virtual wholes" or
"potential wholes". This article will follow that
practice, but it should be understood that in medieval contexts
"potential" usually signifies something that is in some
respect not fully actualized, and, hence, in some contexts
"potential whole" and "potential part" refer
to things that are possibly, but not actually divided, as when a
continuum with a certain magnitude is potentially divisible but not
actually divided into halves or fourths, or when a substance is said
to be potentially divisible into its elemental constituents (see
Pasnau 2011, 606-629). This standard sense of
"potency" is even at work in the potential wholes that
Boethius is talking about, since the soul's powers are typically
related to the soul's "operations", or activities,
and these activities are clearly not always occurring. A soul might
have the power to see or the power to think, even though it is not
presently seeing or thinking. Nonetheless, these Boethian potential
wholes are actual and actually possess these potential parts; they are
not "potential" in the sense that they are possible things
or things that are merely capable of being divided.



Potential wholes are curious items. They are particulars, not
universals. Yet potential wholes are not composed out of their parts
in the way that, for example, a house is composed out of bricks and
wood. Indeed, the most commonly discussed kind of potential whole is a
soul. But for Aristotelians souls are forms and it is generally held
that forms are mereologically simple. Thus, souls ought to be
mereologically simple. That Boethius, or for that matter, Aristotle
asserts that souls have parts is therefore initially quite puzzling.
(For a list of the places where Aristotle talks about the parts of the
soul as well as a defense of the claim that we should take
Aristotle's use of the term "parts" seriously, see
Johansen 2014.)



Further reflection, however, shows us that a soul (and especially a
human soul) is a complicated thing. It has a variety of capacities,
which seem to be really distinct from one another and possibly from
the soul itself. Some of these capacities even seem to be contrary to
one another. For example, a soul seems to be capable of psychic
conflict. There is a lot of pressure then put on this putatively
simple, yet obviously complicated thing, and therefore it is hardly
surprising that there is a robust medieval debate over the nature of
the parts or powers of the soul (see Perler 2015). Some think that the
pressure is too great to sustain and that one substantially simple
thing cannot be the subject of all these capacities. Two common
reasons are given for the soul's being really divided in a
substantial manner. First, it seems that really distinct operations
must stem from really distinct sources, and since operations are
accidents of substances, this would seem to imply that the operations
of a soul must stem from distinct substantial sources. Second,
according to Aristotle, the rational capacities do not require a
bodily instrument to operate, where the other capacities do. This
leads some medieval Aristotelians to conclude that whereas the
non-rational capacities arise from the matter of the thing, the
rational capacities must come from some external source. This, in
turn, suggests that a human being has at least one substantial form
that is rooted in and arises from the matter, and another substantial
form that comes from an external source. If all of this is right, then
humans at least either have several souls, or one substantially
*composite* 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 thing
can only have one substantial form and therefore only one soul (Pasnau
2002, 126-30). These "unitarians" must come up with
an answer to how the powers do not in fact divide the soul into souls,
especially those unitarians like Aquinas who hold that the powers of
the soul are really distinct from the soul itself. (In fact, according
to King (2008), the dominant view in the medieval period is that the
soul's powers are really distinct from the soul's
substance.) A commonly proposed solution is that the powers
"flow out" from the soul's essence and that the
subordinate powers of the soul are "virtually present" in
the soul (see Shields 2014 for an analysis of Suarez's
version of this solution; also Perler 2015).



In keeping with the introductory nature of his treatise, Boethius
remains silent about many of these issues and instead focuses on the
logical nature of the soul considered as a whole consisting of powers.
Unfortunately, what he says actually makes matters all the more
obscure since he notes that the division of a potential whole
resembles both the division of a genus and the division of a
whole:



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



Some earlier medieval philosophers take Boethius' statement as
an invitation to reduce the soul, and the potential whole in general,
to either a genus or an integral whole. (One of the first attempts to
place the division of soul under the division of the genus is found in
a short letter from a mysterious ninth-century thinker identified only
as "Master L". The letter is preserved in the manuscripts
of St. Gall, and is transcribed by De Rijk (1963, 75-78).)
However, the attempt to reduce soul to either a genus or an integral
whole appears doomed, for potential wholes do not fit well under
either category. Souls are particular, and hence, they cannot be
universals of any kind, let alone genera. On the other hand, by
introducing the potential whole as an *additional* kind of
whole, Boethius appears to be signalling that potential wholes do not
have parts in the same way that other true wholes have parts: Socrates
can be separated into his hands, feet, and so forth; a chemical
mixture can be reduced back into its ingredients. Even a universal
whole can be separated into independently existing parts, namely, the
individuals that are its parts. But a soul is neither constructed out
of its powers, nor does it appear to be separable into freestanding
parts: the powers must be powers *of* a soul.



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



> 
> This is the superior definition of "soul": soul is a
> quality with respect to which a body is made to be alive. It is in
> accord with this definition that "soul" is predicated of
> its parts individually, and thus the power of knowing is a quality
> with respect to which a body is made to be alive, and the power of
> sensing is a quality with respect to which a body is made to be alive.
> This is the inferior definition of "soul": soul is a
> quality constituted out of rationality, out of sensibility, and out of
> vegetability (*vegetabilitate*). And it is with respect to this
> definition that [soul] cannot be predicated of its parts taken
> singularly, since one cannot say that the power of thinking is that
> which is constituted out of rationality, sensibility, and
> vegetability. ((ps?) Abelard, *De divisionibus*
> 194.12-22)
> 



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



In his commentary on Boethius's text, Albert the Great takes a
different approach (*In de Div.* tract. 4, ch. 1; Loe ed.,
75-9). He refers to the view that a potential whole is some sort
of intermediary between a universal and an integral whole, but then he
proceeds to give a different explanation why a potential whole is a
whole: It is a whole because the full and complete list of the
capacities will produce the full and complete account of the soul in
so far as it is the principle of life and self-motion. But it does not
follow from this that any given particular soul will have all of the
powers and capacities on the list. Albert also hastens to add that
while a full list of the capacities of soul gives us the substance of
the principle of life and motion, under this description we are not
attending to a substance in the absolute sense. We, thus, should not
conclude that a soul is divisible in the way that an integral whole is
divisible. That is, these capacities need not answer to little
substances or parts of substances that constitute the soul. To help us
to see his point, Albert offers an example of another potential
whole:



> 
> And again a regime (*regnum*) is a whole of this kind, given
> that it consists of its powers, namely, the king, the prefect, the
> superintendent, 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 is
analyzable and hence consists in smaller roles and their associated
powers. The complete list of the roles in regimes would give the whole
of what it is to be a regime. But like individual souls, a particular
regime will require some of these roles and their attendant powers,
but it need not possess all of them. Moreover, a difference in role
does not imply a difference in subject. The same person could perform
some or even all of these distinct roles.


## 3. Parts



Many kinds of item can be wholes, and many kinds of item can be parts
of 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 part
of that whole. The only clear restriction on what can be a part is
that no part is identical to its whole. In other words, no medieval
philosopher countenances what contemporary mereologists call improper
parts. (On the contemporary notions of *part,* or *improper
part,* and *proper part* consult the entry on contemporary
 mereology,
 and also Simons 1987, 9-11.)



We will first consider the kinds of things that can be parts of
integral 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 of
universals can also be parts of integral wholes. This will prompt us
to consider several criteria that medieval philosophers use to
distinguish universal wholes from integral wholes
 (Section 3.3).
 Finally, we will return to consider potential wholes in light of
these criteria for distinguishing universals from integral wholes
 (Section 3.4).


### 3.1 Parts of integral wholes



Consider the paradigmatic integral whole Socrates. Socrates is
composed 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 ultimately
created by combining the four basic elements, Earth, Air, Fire, and
Water. All these components of Socrates can be considered integral
parts of Socrates.



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



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



Not all medieval philosophers think that the elements are parts of
Socrates. Abelard, for example, believes that the elements are
ingredients, but not every ingredient is a part. Strictly speaking,
only those items that compose some whole and remain in that whole
after composition are parts of the whole (*Dialectica*
575.18-36). Hence, even though the flour is an ingredient of the
bread, the flour is not a part of the bread. The flour has been
altered by a chemical change, and so it does not remain once the bread
is baked. Likewise, while the elements combine into a chemical mixture
that becomes flesh, the earth and water that make up flesh are no
longer present. Only the crumbs and flesh are properly parts of the
bread and Socrates respectively.



Many later medieval thinkers effectively agree with Abelard on this
point. The elements that compose my body only exist in actuality when
my 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. Some
medieval philosophers went so far as to suggest that most or even all
the parts of a substance exist only in potency when the substance is
actual (see Pasnau 2011, 606-29).



Medieval philosophers also like to draw a distinction between the
homogeneous 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. For
example, a hand is composed out of fingers, knuckles, and a palm. It
is also, from another viewpoint, composed out of muscles, skin, and
sinews. No part of the hand is a hand. However, some of the
hand's parts are homogeneous. Muscle, skin, and blood are each
homogeneous, since every bit of muscle is muscle, every bit of blood
is itself blood, and every bit of skin is also skin.



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



Many of the heterogeneous parts of Socrates are best defined in terms
of their function, not their measure. For example, hands are
discriminated from feet based on what functions they perform for an
animal. Many medieval philosophers believe that these functions are
provided either by the form or the soul of the animal. For this
reason, many medieval philosophers call functionally defined parts
"formal parts", or parts *secundum formam*, but in
order to avoid any confusion, I will refer to parts of this sort as
*functional* parts. The fourteenth-century philosopher Walter
Burley tells us that the functional parts "remain the same so
long as the whole remains the same and complete" (*De toto et
parte*, 301). In other words, so long as Socrates' soul
occupies his body, and provided the hand is not cut off,
Socrates' hand remains a hand. Burley contrasts functional parts
with material parts, and he places homogeneous parts such as flesh,
bone, and blood under this category. Socrates' material parts
are in constant flux; Socrates is constantly losing and replacing bits
of flesh and blood.



In addition to the functional parts, Socrates' substantial form
(or forms) and his matter (either his prime matter or his proximate
matter *en masse*) can be considered parts of Socrates. But
here there is some question as to whether form and matter are
*integral* parts of Socrates. What Boethius says is
ambiguous.



> 
> There is also a division of the whole into matter and form. For in one
> manner the statue is constructed out of its parts, and in another
> manner 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 are
not "parts", especially since they are products of a
division. Moreover, if a core meaning of "part of
*x*" is "that out of which *x* comes to
be" (cf. Boethius' *On the Trinity* 2), then the
form and the matter of a thing ought to be the primary parts of that
thing. Bronze without its form is merely bronze, not a statue;
Socrates' matter without Socrates' substantial form is not
a human. Clearly, then, Boethius cannot mean that form and matter are
not parts in any sense.



The task, then, is to determine what kind of parts form and matter
are, and especially what kind of parts substantial form and matter
are. Aquinas routinely distinguishes between the
"quantitative" parts of a thing and the form and the
matter, which he calls "parts of the essence" (*Summa
Theologiae* 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 question
whether the part of an essence is a type of integral part, or whether
it is a distinct kind of part. Other philosophers present a less
ambiguous line: only parts that make up some quantity can be integral
parts (see, e.g., Lambert of Auxerre *Logica* 126; Peter of
Spain *Tractatus* V.14 (2014, 212-4); Albert of Saxony
*Sophismata* 45). Accordingly, many medieval thinkers will
propose that in addition to potential, subjective, and integral parts,
there is a fourth kind of part, which I will call a
*substantial*
 part.[4]



There are other good reasons why medieval philosophers might be
motivated to distinguish the integral parts of a thing from the
substantial parts of the same thing. Firstly, the form does not behave
like 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 a
whole" in each material part of the thing. Integral parts, by
contrast, are thought to be unable to be present in more than one
place at the same time, and for this reason, the integral parts of a
thing are said to sit side-by-side, or "part outside of
part" (*pars extra partem*). Secondly, substantial parts
require something else to perfect them (see Albert of Saxony
*Sophismata* 49). Matter by itself is incomplete; it is only
potentially this or that thing. Matter needs a form in order to
actually be this or that thing. Form, too, is in a sense an incomplete
substance. A dog's form, for instance, is in need of some matter
in order to be the form of this dog. But more fundamentally, given the
Aristotelian critique of Platonic forms-specifically, the
critique of the notion that a form can exist independently of
matter-many medieval thinkers will insist that a form is
incomplete in the sense that it cannot *exist* unless it is
part of a substantial composite. Finally, distinguishing between
substantial parts and integral parts captures an important intuition
about substances: Socrates can lose and gain matter without
compromising his existence. But if Socrates were to lose either his
matter in total or his substantial form, he would cease to exist.



If one draws a sharp distinction between integral parts and
substantial parts, the next natural thought would be to distinguish
between integral and substantial wholes. The trouble is that in many
cases the *same thing* seems to be both an integral whole and a
substantial whole. Socrates, for example, has a quantity, and he is
surely divisible into smaller quantities. It seems to follow that
Socrates is an integral whole. But Socrates is also composed of matter
and substantial form. So, he is a substantial whole. One could reject
the notion that Socrates is an integral whole. But, then, what is it
that has Socrates's size? Socrates' size is supposed to be
an accident of *Socrates*, not of a part of Socrates. It is
perhaps for this reason that some medieval philosophers try a
different approach. As we already noted, Walter Burley draws a
distinction between material parts and parts *secundum formam*
(which we called "functional" parts, above). Related to
this, Burley also draws a distinction between the whole considered in
relation to its matter (*secundum materiam*) and the whole
considered in relation to its form (*secundum formam*) (*De
toto et parte*, 301). Socrates considered formally--i.e. as a
unity of this form with some matter--persists through time and
change. Socrates considered materially is constantly in flux. That is,
Socrates on Monday is not the same whole materially as Socrates on
Friday, because the sum of material parts belonging to Socrates on
Monday is not identical to the sum of material parts on Friday. The
distinction between the whole considered formally and the whole
considered materially will play a role in some medieval theories of
persistence, and we will pursue this use of the distinction in short
order (see
 Section 4.2).
 But, first, we must consider the parts of universals and the special
problems that the parts of universals entail.


### 3.2 Parts of genera and universal wholes



Genera are divisible into their subordinate species, whereas what
Boethius calls "universal wholes" are divisible into their
subordinate individuals. In the medieval tradition that follows
Boethius, 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 of
Animal, and Human Being and Animal are predicable of Socrates. That
is, Socrates is a human being, and Socrates is an animal. However, to
be a true subjective part, not only must the name of the whole be
predicable of the part, the definition of the whole also must be
predicable of the part. A statue of Socrates could in some contexts be
called a human. For example, suppose someone points to the statue and
says, "That's a man.", and then she points to
another statue and says, "And that's a horse.". This
person is saying something intelligible, but the statue of Socrates
cannot be a subjective part of Human Being, since the statue is not a
rational mortal animal; in substance it is some marble shaped like a
human with certain features. Socrates, however, is a subjective part
because 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, in
the same sense that the Black Stallion and Mr. Ed are horses.



Genera and universal wholes both consist of subjective parts, but they
differ with respect to the items that are their subjective parts. This
fact might explain why Boethius places universal wholes under the
broader class of true wholes. Boethius offers four criteria for
distinguishing between genera and true wholes (*On Division*
879b-d):


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



Some of these criteria could be interpreted in a manner such that the
universal consisting of individuals falls under the category of true
wholes. Consider, for example, criterion (a). The genus Animal is
divided by considering what sort (*qualis*) of animal something
is. Human Being is a *rational* animal, Horse is an
*irrational* animal. In contrast, Human Being is not divided
with respect to what sort of human being Socrates is. Socrates and
Cicero are both rational animals. The difference between Socrates and
Cicero is due to the fact that this bit of matter which makes up
Socrates is different from that bit of matter which makes up Cicero. A
difference in matter is typically considered to be a quantitative
difference. Hence, the parts of the universal whole Human Being appear
to be distinguished by quantity rather than quality.



However, some of the other criteria do not clearly mark genera off
from universal wholes. Consider, for example, the second criterion.
Boethius seems to have something like this in mind when he articulates
difference (b): the parts of a genus are dependent upon the genus,
whereas the whole is dependent upon its parts. In other words, if
there are no animals, there can be no dogs or humans. In contrast, the
house depends upon its parts. If you take away the roof and floor, the
house ceases to exist. But universal wholes seem to behave like
genera, not houses. If we annihilate individual humans, we do not
eliminate the universal Human Being. Therefore, difference (b) does
not 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, let
alone whether the universal whole behaves like the genus or the true
whole.



Thus, it is not clear that Boethius's four criteria adequately
separate universals divided into universals from universals divided
into individuals. It may be that the better division is that between
universal wholes and integral wholes, not genera and true wholes, and
indeed many medieval philosophers seem to take this route.



This strategy, however, only provides some relief from the puzzles
that seem to arise from Boethius' four criteria. As we will see
in the next section, there is still much more to be done to secure a
clear and principled distinction between universal wholes and integral
wholes.


### 3.3 Distinguishing universals from integral wholes with respect to their parts



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



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



Peter Abelard reports that there were some medieval philosophers who
drew this very conclusion. Abelard describes and attacks this
"collection theory" of universals in his *Logica
Ingredientibus* (1, 13.18-15.22). For fuller presentations
and 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 thinks
that the collection theory gets the relation of dependence between the
universal and the individuals backwards. According to Abelard, the
collection theory is committed to the view that when Socrates dies,
the universal Human Being is changed, and if one believes, as Abelard
does, that a discrete integral whole is identical to the sum of its
parts, then the Human Being that has Socrates as a part is not
identical to the Human Being without Socrates. There is a new Human
Being. But this, Abelard insists, is contrary to the orthodox
understanding of universals, which states that while the individuals
that fall under a universal are impermanent, the universal itself is
permanent. Indeed, Human Being would exist even if every human being
were annihilated.



Yet, whatever one ultimately makes of the collection theory and
Abelard's counterarguments, this much is clear: the difference
between integral wholes and universal wholes cannot be defined solely
and simply in terms of the kinds of items that are parts of these
wholes. Substances can be parts of both. Rather, the distinction
between universal wholes and integral wholes must be maintained in
terms of more nuanced constraints or properties.



Many medieval philosophers focus on the fact that every part of a
universal whole taken singularly must accept the predication of the
name and the definition of the whole, whereas this is not true of each
and every part of an integral whole (Aquinas *Summa Theol.* III
q. 90, art. 3; Walter Burley *De toto et parte*, 302; and
Buridan *Summulae* 8.1.4). For example, Socrates and Plato are
both parts of the universal whole Human Being, and Human Being is
predicable 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 of *every* part of Human Being. In contrast, an
integral whole is not predicable of its parts taken singularly. That
is, one cannot say that



> 
> This piece of wood is a house.
> 



Integral wholes are only predicable of their parts taken all at
once.



> 
> This wood and this stone and these other parts taken together are a
> house.
> 



This seems to give us a principled way to exclude some items from the
universal whole Human Being and thus to distinguish it from the
integral whole consisting of all human beings. For this seems to be
right: 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, even
though the latter are parts of the integral whole consisting of all
human beings. Socrates and Cicero are parts of Human Being because
Socrates is a human being and Cicero is a human being. The parts of
Socrates and Cicero are not parts of Human Being because
Socrates' hand is not a human being and Cicero's head is
not a human being; whereas they are part of the integral whole
consisting of all the humans, since Transitivity seems to hold for
integral parts: if *x* is a part of *y* and *y*
is a part of *z*, then *x* is also a part of *z*.
(It should be noted, however, that some medieval authors were
suspicious of an unrestricted form Transitivity, even for integral
wholes. See Arlig 2013 for details.) There is also this: while it is
true that some of the parts of the sum of all human beings will accept
the predication of the name and definition of Human Being, if one
examines the matter carefully, it turns out that the definition of
*the sum of all human beings* is not the same as the species
Human 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 are
composed of homogeneous parts. Boethius himself illustrates the
problem with the example of a bronze rod, which he believes is a
homogeneous substance (*On Division* 879d). Recall that if
something is a homogeneous substance, then every part of that thing is
also the same substance. That is, if *y* is a homogeneous whole
and *x* is a part of *y*, the name and definition of
*y* is also predicable of *x*. Every portion of bronze
can take both the name and the definition of bronze, and hence, it
seems that every part of the bronze rod meets the standard of being a
subjective part. Therefore, it seems that the bronze rod is a
universal, which is clearly absurd.



Boethius resolves the puzzle of the rod and its parts by noting that
while it is true that each portion of bronze is bronze, it is not true
that a portion of the original quantity of bronze is that original
quantity of bronze (Boethius *On Division* 880a)
Boethius' solution is often repeated ((ps?) Abelard *De
divisionibus* 169.33-36; Albertus Magnus *In de Div.*
tract. 2, ch. 5 (Loe ed., 35); and Radulphus Brito *In de Top.*
II, q. 9 [1978, 45]; cf. Aquinas *Summa Theologiae* I, q. 3,
art. 7).



The solution points to another difference between integral wholes and
universals. As Abelard puts it, every integral whole "draws
together" (*comprehendere*) some quantity
(*Dialectica*, 546.21-27). The suggestion is that when
some items compose an integral whole, that whole will be measurable
with respect to some quantity or other. The integral whole that is
measured by some quantity need not be composed out of material
elements. Consider the mereological sum of the angel Gabriel and the
angel Michael. If this sum is a whole (and there seems to be no reason
to deny this), it seems it is an integral whole. However, there are
some integral wholes which do not seem to embrace any quantity,
namely, thoughts and actions. So there is need to locate yet another
difference between universal wholes and integral wholes.



The last difference that we will consider is this: Universals, like
Human Being or Horse, are not literally *composed* out of their
species. Integral wholes, such as Socrates or a house, are composed
out of their parts (John Buridan *Summulae de Dialectica*
6.4.4; Ockham *Expositio in librum Porphyrii* 2.16
[*Opera* II, 54]). Composites are dependent upon their
components. Socrates is composed of his body and his soul. If these
components did not exist and combine to form Socrates, Socrates would
not exist. Components are often, but not necessarily, temporally prior
to the whole that they compose. For example, the house is composed by
bits of wood, stone, and iron, and these parts existed prior to the
existence of the house. Universal wholes are neither dependent upon
their parts, nor are the parts of a universal temporally prior to it.
The species existed long before Socrates or Cicero, and will exist
long after Socrates and Cicero. Indeed, Human Being can exist, even if
no individual human being exists.



Unlike the appeal to quantity, the composition criterion can be
applied to thoughts and actions, for a complex thought requires the
simpler concepts that compose it, and penance requires that the
actions 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 to
be "composed" out of a genus and a *differentia*.
These parts are sometimes called "essential" parts of the
universal since the genus and *differentia* together constitute
the essence of the universal (Aquinas *Summa Theologiae* I, q.
8, art. 2, ad 3; and I q. 76, art. 8), although to avoid confusion we
perhaps ought to call them *definitional* parts. The essence of
a universal is usually encoded in its definition. For example, the
definition of Human Being is *rational mortal animal*.
*Animal* is the genus, and rationality and mortality are the
*differentiae*.



Many medieval philosophers try to dampen this criticism by suggesting
that the universal is, strictly speaking, not composite; it merely
mimics composition (again, see Ockham *Expositio in librum
Porphyrii* 2.16). Perhaps this is a viable response, but there is
another problem with the composition requirement: it does not tell us
why all the parts of integral wholes are integral parts. It was
already observed in passing that not all parts of an integral whole
are plausibly components of their wholes. Consider Socrates. The
elements are strictly what compose Socrates. It is only when Socrates
is composed that other parts, such as his hands and feet, come into
existence. Or put another way, it is false to say that one makes a
human being by cobbling together hands, feet, and head. Such a
creature would be Frankenstein's monster, not a human being.



In sum, a number of proposals are offered for how one can distinguish
universal wholes from integral wholes. But perhaps no single proposal
is universally embraced because of the bewildering variety of items
that are integral wholes. An obvious solution would be to reduce the
number of items that can be integral wholes or integral parts. And as
we have seen already, some philosophers in fact do this in certain
cases, as when Abelard eliminates temporal wholes. However, based on
what we presently know, no medieval thinker attempts anything like a
systematic eliminative project in mereology as such. (Thus, while
Ockham, for instance, famously undertakes a comprehensive and severe
eliminative project in ontology, he does not attempt to develop a
systematic theory of what can and what cannot be a part. Rather, for
him, the question whether "part" is being used either
properly or figuratively tends to arise within a specific context and
the resolution often appears to be developed on a case by case
basis.)


### 3.4 Parts of potential wholes



Potential wholes add further complications. Like genera, and unlike
integral wholes, it is usually assumed that potential wholes are not
literally composed out of their parts. Potential wholes, or at least
the subjects underlying them, are items that are fundamentally simple.
But, in another respect, the parts of the potential whole behave like
parts of an integral whole. Different souls can possess only a
selection of the capacities associated with soul, and so in a sense,
there is a construction of a specific soul's total capacity out
of many discrete capacities and powers. And no one power by itself
exhibits or stands in for the whole soul.



The question thus arises as to how one can distinguish powers as parts
from both subjective and integral parts. Here we will look briefly at
only one account. Aquinas separates potential wholes from both
universals and integral wholes by considering two parameters: the
presence of the whole in the part with respect to the whole's
essence, and the presence of the whole in the part with respect to the
whole's power (*Summa Theologiae* I, q. 77, art. 1, ad
1). The universal whole "is present to each of its parts in its
entire essence and power". It is for this reason that each part
of the universal is a subjective part. In contrast, the integral whole
is not in each of its parts either in respect to its entire essence or
in respect to its power. Hence, the integral whole is not predicable
of any one of its parts taken singularly. Finally, the potential whole
is present to each of its parts with respect to its entire essence,
but not with respect to its full power. This is why, even though
one's soul is non-composite and cannot be cut up, the operation
of thinking does not entail along with it the operation of
sensing.


## 4. Mereology and metaphysics



Medieval philosophers are well aware that the study of wholes and
their parts has numerous applications in metaphysics. I will conclude
this survey by examining two applications. First, I will look at how
theorizing about parts and wholes informs medieval reflections on
identity at a time. Second, I will consider how mereology influences
medieval theories of persistence through time and survival through
change.


### 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 and
wholes themselves. If *x* is a quantitative part of *y*,
then *x* is lesser in quantity than *y*. A
donkey's form is not identical to the donkey, because this
donkey is a composite of the donkey's form *and something
else*, namely, the donkey's matter. Socrates' soul is
not identical to the human being who is Socrates, for the human being
is a composite of body and soul. Socrates is not a universal, even
though he is a human being. And the extension of Animal is greater
than the extension of Human Being.



However, there is another question that does divide medieval
philosophers. Consider an integral whole. An integral whole is
composed out of its parts. But is it true that an integral whole is no
thing other than the sum of these parts? Some philosophers, such as
Abelard and Ockham, argue that the whole is no thing other than the
sum of its parts. Others, such as Duns Scotus, argue that the whole is
some thing which is really distinct from the sum of its parts. (On the
debate between Ockham and Scotus, see Cross 1995, Cross 1999, and
Normore 2006.) The former philosophers perhaps base their position on
passages from Boethius' treatments of mereology such as
this:



> 
> Every thing is the same as the whole. For example, Rome is the same as
> that which is the whole citizenry. Each and every thing is also the
> same 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, and
> the 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 often
appeal to Aristotle. For example, an anonymous commentator on
Aristotle's *Sophistical Refutations* argues that
"the five are not the two and the three" on the grounds
that Aristotle has shown in his *Metaphysics* that "the
composite, in general, is something other than its component
parts" (*Quaestiones super Sophisticos Elenchos* q. 831
[1977, 346]; cf. Aristotle *Metaphysics* 8.6,
1045a9-10).



Peter Abelard provides one of the most sophisticated solutions to this
problem. Abelard emphatically asserts that the whole is no thing other
than the sum of its parts (*Dialectica* 344.34-5, and
550.6-7; cf. 560.34-561.2). But this is surely false.
Consider a house. A house is composed out of a specific sum of boards
and bricks. Yet these boards and bricks could be sitting together on
the building site without being arranged as a house. Hence, it is hard
to see how the house is no thing other than the sum of the parts.



Abelard will respond by drawing a distinction between the
*essentia*, or concrete being, of a thing and the
"conditions" (*status*) which a thing may possess
or be in. True, the sum of boards and bricks must possess the
appropriate arrangement in order to be a house (*Logica
Ingredientibus* II, 171.14-17; and *Dialectica*
550.36-551.4). But when the sum takes on a specific structure it
does not take on an extra part. Whatever they are, Abelard makes it
very clear that structures and arrangements like those found in houses
are not *things* (*res*), and only things can be parts.
The imposition of a structure upon the sum of the boards and bricks
does not add a part to the thing that is that sum, and hence, the sum
of the boards and bricks is the same in *essentia* as the
house--they are the very same *thing*. Nevertheless, there
can be different conditions in which this thing is. When the sum of
boards and bricks are arranged in one way, they are in the condition
of being a house. When they are arranged in other ways, they are not
in this condition. In short, to claim that the house is no thing other
than the sum of its parts does not imply that the sum is in the state
of being a house when the sum exists.



Abelard's solution will only work if he can motivate the claim
that a thing taking on a new condition does not make a new thing.
And medieval realists went after such reductivist programs in much the
same way that some contemporary realists do. Specifically, they often
presented truth-maker challenges, or "anti-Razor"
arguments (see Maurer 1984): If at one time this sum of boards and
bricks is not a house and at another time it is a house, then there
must be some item in reality, some thing, that is absent at the
former time and present at the later time. The task, of course, is to
identify the right sort of thing that can be such a difference
maker.



But let us move on to consider another related puzzle about identity
at a time. Abelard's characterization of numerical sameness and
difference seems to give us the tools to answer a common medieval
puzzle which I will call the Problem of the Many Men. Versions of this
problem can be found in a number of medieval works (see, e.g., Abelard
*Theologia Christiana* III, SS 153 (1969, 252); William of
Sherwood *Syncategoremata* (O'Donnell ed., 60); Albert of
Saxony *Quaestiones in Physicam* I, qq. 7-8, and his
*Sophismata* 46, 25va-vb). The puzzle can easily be generated
using a crude understanding of numerical sameness. Assume that
Socrates' body is perfectly intact: he has all his limbs, and
their parts. Now consider every part of Socrates' body except
one finger. Call this whole *W*. *W* is not numerically
the same as Socrates, so it appears that they must be numerically
distinct. Socrates' whole body is imbued with the soul of a man.
But it also happens that *W* is imbued with the soul of a man.
So, there are now two numerically distinct men where it initially
appeared there was one. But it gets worse. Considering the body apart
from one finger was only one of an indefinite number of such
considerations. And by the same reasoning, these other bracketed
wholes composed from Socrates' body are also men. Hence, it is
easy to generate an indefinite number of numerically distinct men
where commonsense tells us that there is only one.



Abelard might appear to be capable of unraveling this puzzle
by employing his distinction between difference in *essentia*
and numerical difference, which he presents most clearly in his
analysis of the Trinity (*Theologia Christiana* III, SS 139
and SS148 (1969, 247 and 250-1); see Brower 2004,
226-34). Abelard's notion of sameness and difference
in *essentia* is informed by his appreciation of the
notion of mereological overlap (see the section on Basic
Principles/Other Mereological Concepts, in the entry on
 mereology).
 His appreciation of mereological overlap in and of itself is a
watershed, 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 in *essentia* as
*y* if and only if every part of *x* is a part of
*y* and every part of *y* is a part of *x*. In
other words, *x* is the same in *essentia* as *y*
if and only if *x* and *y* mereologically coincide. If
*x* is the same in *essentia* as *y*, then
*x* is numerically the same as *y*. However, if
*x* is not the same in *essentia* as *y*, it does
*not* follow that *x* is numerically distinct from
*y*. This is because *x* and *y* could
overlap--that is, share at least one part--even if they do
not coincide. Thus, Abelard's answer to the Problem of
the Many Men is that while there are many overlapping men, each of
which is different in *essentia* from the others, this does not
entail that there are an indefinite number of numerically different
men. One might, however, wonder whether Abelard has completely
dispelled the hint of paradox generated by the initial puzzle. After
all, it would be nice to know *which* of these overlapping, but
non-coincident wholes is Socrates. All that Abelard's
theoretical tools allow us to do is to say that while there is
*this* whole which is imbued by Socrates' soul and there
is *that* whole which is imbued by Socrates' soul, these
wholes are not *numerically* many and, thus, there are not
numerically many men where intuitively we thought there was only one
(see Normore 2006, 749).



Most medieval philosophers tackle the Problem of the Many Men in
another manner. Albert of Saxony, for example, resolves the puzzle by
claiming that nothing which is a part of something else can be a
numerically 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 a
distinct human substance, since it is a part of a human substance,
namely, Socrates. Therefore, even if we grant that there are many
things present, there are not many distinct
*humans* present where we thought that there was only
Socrates. If that appears to be an argument by stipulation, the reader
might find William of Sherwood's elaboration on this strategy
helpful (*Syncategoremata* (O'Donnell ed., 61)). William
concedes that, because Socrates-less-the-finger is perfected by a
human soul, it can *in a certain respect* be said to be a human
being. But in this respect Socrates and Socrates-less-the-finger are
perfected by the *same soul* and if one wants to count humans,
one counts human souls. Thus, Socrates and Socrates-less-the-finger
are not numerically distinct humans. On the other hand, if
Socrates-less-the-finger is considered as a *part* of something
larger that is also perfected by the same soul, then in this respect
Socrates-less-the-finger is numerically different from Socrates, but
it is not a human being. Considered in terms of its being a
*part*, Socrates-less-the-finger is a part of a body of a human
being, and a body of a human being is not a human. Hence, we count too
many humans only if we ignore these different respects under which the
premises of the argument can be made true.


### 4.2 Persistence



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



> 
> 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 whole *x*" can mean the complete *x*.
Accordingly, to say that the whole is destroyed if the part is
destroyed, is merely to say that the whole is incomplete or
"mutilated" if a part is removed. It might also be the
case that "the whole *x*" merely signifies all the
parts of *x* taken together. On this reading, the maximal
proposition would merely imply that, if *x* does not exist,
then the whole consisting of *x* and some other parts does not
exist. 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. (*On
> Division* 879c [Magee 1998, 14]).
> 



Boethius' assertion that the whole consisting of *x*
*and all the rest* will perish when *x* perishes leaves
it open whether the remaining parts are still, say, Socrates or a
house. Naturally, the *creative* parts had to be present in
order 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. If
there had been no nails, wood, or cement, there would have been no
house. Still, commonsense tells us that the creative parts need not
remain present after the house has been created. Some boards and some
nails may be replaced in the house, but this does not compromise the
existence of this house.



Despite the appeal of commonsense, there were some medieval
philosophers who took the maximal proposition to imply that if
Socrates loses a finger, Socrates ceases to exist. At least at one
point in his career, Peter Abelard appears to seriously entertain the
radical 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 of
Abelard's followers, the so-called Nominales, wholeheartedly
embraced the extreme interpretation of the maximal proposition. One of
the positions associated with the Nominales was that "nothing
grows" (*nihil crescit*), and there is a surviving
commentary on the *Categories* with Nominalist features where
the author argues that any addition, removal, or even relocation of a
part compromises the identity of the thing itself (Ebbesen 1999, 397).
This extreme interpretation is derived from the judgment that each
whole is the same thing as a unique set of parts. This house must be
composed out of these nails, these boards, and this cement. If I use
other nails or other boards, I could make a house, but not this very
house. This premise is no doubt controversial, but Abelard and many of
his contemporaries have principled reasons for holding it. Abelard
believes that the ultimate pieces of the universe are tiny,
indivisible bits of matter, souls, and perhaps some forms. The items
that we experience are composites of these elements. Each element is
self-identical. Composite beings are individuated by the elements that
make them up, and in the case of complex, composite beings--such
as artifacts and substances--by the arrangement that these bits
have. Given such a universe, it is quite plausible to assume that the
identity of a composite item is solely determined by its parts. It
would then easily follow that the removal or addition of a composing
element entails that the new whole is not identical to the old whole.
Another whole similar to the original might exist after the
mereological change take place, but strictly speaking the two wholes
are not identical.



Most medieval philosophers are not as extreme as the Nominales. Of
course, there are those parts that are required to keep the whole
intact, but there are also those parts that can be lost without
compromising the integrity of the whole. If we were to cut off
Socrates' head, he would perish. But if we amputate
Socrates' right hand, it seems that Socrates would not cease to
exist. He would merely lack a hand. The parts required to keep a whole
intact are called "principal". Ones that could be lost
without compromising the integrity of the whole were called
"secondary" parts (Anonymous *Introductiones maiores
Montane* 71va-72rb; Albertus Magnus *Commentarii in De
div.* tract. 2, ch. 5 (Loe ed., 33-4); Buridan *Summulae
de Dialectica* 6.4.4). Examples of principal parts would be the
foundation of the house, the heart of a cat, and the brain of
Socrates. Examples of secondary parts are a brick in the house, a
strand of hair belonging to the cat, and a fingernail of Socrates.



In the Twelfth century there was a heated debate about how to
determine whether this or that part is principal (see Pseudo-Joscelin
*De generibus* SSSS 3-20 (2014, 122-29;
Arlig 2013), for as Abelard pointed out the following account is
insufficient:



> 
> *x* is a principal part of *y* if and only if the
> removal of *x* entails the destruction of *y*.
> 



Abelard reminds his audience that even the extremist can accept this
account of principal parts; the only difference is that on his account
*every* part is a principal part (*Dialectica*
549-52).



The best answer that twelfth-century thinkers come up with is that a
principal part of *x* is that which, when removed, compromises
the form of *x* (Pseudo-Joscelin *De generibus*
SSSS 23-6 (2014, 130-31); Arlig 2013,
106-10). Many later medieval philosophers followed suit. At
least in the case of substances, the substantial form could do the
metaphysical work of guaranteeing persistence through time and change
(Pasnau 2011, 689-92). For example, we have already seen that
Burley draws a distinction between the whole *secundum formam*
and the material whole. The whole considered formally persists so long
as the form persists. The whole considered materially is the only
whole compromised by changes in material parts (*De toto et
parte*, 301). Most commonsense objects are identified with a whole
considered with respect to form, and so there would be no reason to
think that these things are substantially compromised by material
changes. (The fate of artifacts, would be different, as their forms
are merely accidental forms. A mereological change in the case of an
artifact would seem to entail that the initial artifact is replaced by
another one. But later medieval philosophers do not seem to be all
that troubled by these implications. Whether *theologians*
could stomach all the consequences that this view of artifacts entails
(for example, for whether something remains ritually pure after it has
been chipped) may be another matter. But this is an area whether
further research is needed.)



This basic Aristotelian picture is complicated, if not outright
altered, by several very influential fourteenth-century thinkers,
including Ockham and Buridan (Normore 2006; Pasnau 2011,
692-702). We will consider Buridan's theory of identity
over time and change (*Quaestiones super De generatione et
corruptione* 1.13; *Quaestiones in Physicam* 1.10; with
English paraphrase in Pluta 2001). According to Buridan, there are
three senses of numerical sameness: (1) *x* is
"totally" the same in number as *y*, (2) *x*
is "partially" the same in number as *y*, and (3)
*x* is numerically the same as *y* in a "less
proper" way. Something is totally the same in number if all its
parts remain the same and it neither acquires nor loses any parts. In
this strictest of senses, no corruptible thing whatsoever persists
through mereological change. Something is partially the same in number
if its "most principal part" remains numerically the same.
This is the sense that allows us to claim that Socrates is numerically
the same man now as that man ten years ago. In particular, it is
Socrates' intellective soul which is the principal part and
guarantor of persistence through change. Finally, something is less
properly the same in number if there is merely a continuous succession
of beings that maintain a similar shape, disposition, and form. This
less proper mode of numerical sameness allows us to claim that the
Nile River here today is numerically the same river as the Nile back
in Herodotus' time. While it is probably not a surprise to learn
that rivers are the same in number only in a less proper sense, it is
striking that Buridan goes on to assert that plants and non-rational
animals 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-rational
creatures do not have the sort of soul that can act as a guarantor of
less proper identity; rather, non-rational souls are themselves
altered when the material parts are altered. Thus, over the span of a
normal lifetime, neither the matter nor the form of a horse or an
orchid remains numerically the same in anything other than a less
proper
 sense.[7]



Buridan's account appears to commit him to the view that the
addition or subtraction of even tiny and seemingly inconsequential
parts can bring about substantial change: strictly speaking we do not
have one horse, but rather a succession of horses unified by three
facts, namely, (1) these successive beings have some of the same
parts, (2) this line of succession is continuous, and (3) each member
in the line of succession is a horse. The picture we get has many
similarities to the accounts of identity that we get in early modern
thinkers like Hobbes and Locke. Thus, it is tempting to think that we
are witnessing in Buridan's work one more way in which the
Aristotelian scholastic edifice is being eroded to the point that it
will quickly crumble in the seventeenth century (see Pasnau 2011, 703
f., and Lagerlund 2012, esp. pp. 481-2).


## 5. Concluding remarks



Medieval philosophers study a variety of wholes and parts, and they
often do so with a remarkable degree of sophistication. To be sure,
some aspects of medieval mereology are idiosyncratic, but many of the
puzzles that medieval philosophers wrestle with are analogous to ones
that interest contemporary students of logic and metaphysics. Medieval
philosophers are particularly attuned to the relationship between
mereology and various problems in metaphysics, and many of their
solutions to puzzles of identity and survival are embraced by
philosophers in other periods. Even the idiosyncratic aspects of
medieval mereology reveal a sophisticated appreciation of three
fundamental questions in mereology, namely, what items are wholes,
what items are parts, and under what conditions is one item a part of
another item. This survey can only hint at the richness of medieval
mereology. In part, this is due to the overwhelming number of medieval
authors who say something or other, somewhere or other, about parts
and wholes. But it is also due to the fact that there are still more
texts to be unearthed, properly edited, and studied with care. Based
on what we have already discovered, we should be confident that we
will find many more interesting reflections on parts and wholes and
that we will uncover further connections between medieval mereology
and more recent theories of parts and wholes.