William Angus

Mereology and the Philosophy of Mathematics

Posted on Sunday the 22nd of September, 2024.

What’s Mereology?

Mereology is the Philosophical treatment of parthood. That is to say, it deals with defining what it means, and when one object/thing is part of another. This can be abstracted in many different ways. For example, we can talk about the arm being a part of a body, or we can talk about the mind being part of a person, or even we can talk about love being a part of a relationship.

There are a few basic founding features of parthood. We shall denote this relation as $\leq$ . That is, we write $P \leq Q$ to denote that $P$ is a part of $Q$ . First of all, parthood, at least classical conceived, is a partial order. This is

it is transitive: if $P \leq Q$ and $Q \leq R$ , then $P \leq R$ ;
it is reflexive: $P \leq P$ ; and
it is anti-symmetric: if $P \leq Q$ and $Q \leq R$ , then $P = R$ , in other words, if two objects are parts of each other, they are the same thing.

See the Stanford Article for more information on Mereology. All the examples in the next section can be found in that article.

However, as we shall see, certain “deviant” groups disagree with even those most basic assertions.

Deviant Mereology

“Deviant Mereology” refers to a variation on the above, classical mereology, that denies or alters one of the three properties of parthood that state it is a partial order. These deviations often arise through interesting edge cases that challenge the classical notion of parthood. By exploring these, we can better understand what it means for something to be part of another thing. In particular, various Philosophers have different reasons for denying each one of the above three axioms: transitivity, reflexivity, and anti-symmetry.

One reason to disagree with transitivity of $\leq$ is that one may think that a handle can be part of a door, and the door be part of a house, but the handle not be a part of a house. Although I personally do not agree with such views (I think that different notions of “parthood” are being mixed up), they are certainly held in the literature.

One reason to disagree with the fact that $\leq$ is reflexive is that the following situation seems metaphysically sensical: consider a situation in which a wall is shrunk down to the size of a brick and sent back in time so that it is used to build itself. In this situation one may believe that the wall is, in some sense, part of itself, but not in the trivial sense arising from reflexivity, but rather in a special case complicated by the notion of identity over time. This highlights that self-parthood could be more nuanced them simply being trivially true. This similarly applies to fractals (which are mathematical objects that “contain infinitely many copies of themselves” – the definitions are not exact, even in the fractal literature), which would require further mathematical study to evaluate the plausibility of this response.

One reason to disagree with the fact that $\leq$ is anti-symmetric is that we can easily imagine fictitious cases in which anti-symmetry fails. Consider Borges’s Aleph: the Alpeh is contained within a singular room, and the Aleph contains the entire universe: hence the universe and the Aleph are distinct, but both contain each other. I am sensitive to this view, however one could simply say that fiction has nothing to do with the real notion of parthood, despite the fact that we can conceive of cases where anti-symmetry fails. Although it is certainly true that Philosophers often use though experiments to guide intuition, one may like to say that just because we can conceive of the world being a certain way it does not mean that this should guide our view of the metaphysics of the actual world. But that’s why new supposed counter-examples can be useful. And I will present one in the next section.

As mentioned earlier, mereology is not confined to simple physical objects, but also abstract structures, such as those found in Maths. We can consider what constitutes a part of a specific kind of mathematical structure. This is what we shall do in the case of groups, which leads to a surprising result.

Groups

Let us now move on to what Maths has to do with Mereology, by considering groups.

Groups are ubiquitous across mathematics. Examples of uses are in physics: modelling the fact that a set of symmetries always forms a group, and in cryptography (e.g., RSA). Formally, a group is a (carrier) set $G$ together with a binary operation $\cdot : G \times G \to G$ such that

it is associative: $(x \cdot y) \cdot z = x \cdot (y \cdot z)$ for all $x, y, z \in G$ ;
it has an identified identity element $e \in G$ such that for any $x \in G$ , $e \cdot x = x = x \cdot e$ ; and
each element $x \in G$ has an inverse $x^{-1} \in G$ such that $x \cdot x^{-1} = e = x^{-1} \cdot x$ .

We say that a group is abelian if it is also commutative: that is $x \cdot y = y \cdot x$ for all $x, y \in G$ .

Let $G$ be a group with binary operation $\cdot$ . A subset $H$ of $G$ is a subgroup of $G$ if $H$ together with $\cdot$ also forms a group.

We say that groups $G$ and $H$ , with binary operations $\cdot_G$ and $\cdot_H$ , respectively, are isomorphic if there is a bijection $\phi : G \to H$ such that for all $x, y \in G$ , $\phi(x \cdot_G y) = \phi(x) \cdot_H \phi(y)$ .

It is a surprising fact (at least to me!) that there are two abelian groups $A$ and $B$ with the following properties¹:

$A$ is a subgroup of $B$ ;
$B$ is a subgroup of $A$ ; and
$A$ is not isomorphic to $B$ .

What does this have to do with mereology?

Mathematical Structure and Mereology

It seems to me that a natural notion of parthood for groups is that of “subgroup”, and a natural notion of identity of groups is “isomorphism”. Hence, the above is an example of two abelian groups that are parts of each other, but are not equal, violating the notion of anti-symmetry of parthood.

However, I can also see a case for “subgroup” not being the same as “parthood”, owing to the complexity of the objects, as well as the fact that it is not clear what it means for an abstract structure to have a part; it depends on how one thinks of mathematical structures and the like. That is why I think these two fields of Philosophy may be able to influence each other, at least in this case.

A Final Word on Graphs

I conjecture that in the cases of graphs (as in the most simple definition: a graph is simply a set with a binary relation on it, denoting whether there is an edge between any two points), this result also holds: there are two graphs $G_1$ and $G_2$ , which are both subgraphs of each other, but not isomorphic. I think that in this case, the notions of “subgraph” and “part of” definitely do intersect, and this provides a counter-example to property of anti-symmetry applying to parthood. But again, this needs Philosophical verification from the merging of Mereology and the Philosophy of Mathematics.