Mereology
Physical composition: when do objects compose other objects?
Position: Mereological universalism
The composition question for physical objects has been discussed at length in a previous section ("the physical world"). I’ve highlighted the problems inherent to mereological conservatism (chiefly the difficulty to define mind-independent criteria for objecthood). Mereological nihilism is blessed with simplicity, but both its main flavours carry major issues. Pluralist nihilism only recognizes atomic elements of the world, but still needs to allow discourse about ordinary objects and their properties, ultimately looking like covert universalism — a case of having one’s ontological cake and eating it too. Monist nihilism is seductive but a one-object view of the world is a major imposition upon \(\mathfrak{ToP}\). Our modern physics has a more monistic feel than its Newtonian predecessor (think of quantum field theory, or purely geometrical versions of general relativity), but (1) the aggregate of our "best theories" is most definitely not monistic, and (2) we have no idea whether those are even close to some "ultimate truth": for all we know, the next batch of theories might be atomist and Cartesian all over again.
Mereological universalism, the idea that every union of physical entities is an object too, thus seems to be the safest option ontologically. Should monism turn out to be true, or should one of the canonical physical entities be a "big object" (for instance, the whole universe under general relativity), we can still turn universalism inwards and with the right continuity features, slice and dice in every way imaginable, creating proper parts objects in the process (e.g. a segment of spacetime, or the restriction of a wavefunction along some of its variables).
Mental structure: how does macroscopic mentality emerge?
Position: Phenomenal intentionality
The question of physical composition is mirrored in that of mental structure. In exploring such matters however, we’re getting deep into speculative territory. Our discourse about “the concrete” or a hypothetical theory of the “physico-mental” has preserved the illusion of equality between the concrete world \(\aleph\)’s component realms, but the facts tell a different story: even though we may not know the final word about physical laws, our best models tell us something which certainly guides our reflection on physical mereology. About the mental we know very little…
Let us still remind ourselves of some facts, garnered through a previous section (“the mental world”).
The panpsychist credo which we’ve just endorsed has it that mental states are ubiquitous (and not limited to animal and human minds).
The mental realm has its own sub-components in the phenomenal (“what it’s like”) and the intentional (content directed at an object).
Although there is near-universal agreement that the phenomenal and the intentional are conceptually different, their metaphysical relationship is up for grabs. Some would say they are disjunct (realized in strictly distinct mental states), others that they can coexist but nothing more, and finally there are larger groups that prefer to posit that the intentional grounds the phenomenal (intentionalists) or vice versa (phenomenalists). What are the consequences of panpsychist monism in that regard? By adopting it, I’ve already rejected the idea that the mental could just be reduced to the physical. However the difficulties encountered in achieving such reduction are often due to phenomenal attributes like subjectivity or privacy, whilst the intentional is more amenable to functional explanation. The super-computers, “brains-in-a-vat” or “galaxy brains” from classic thought experiments are also more easily conceived in terms of purely intentional states. So the idea of the intentional being at least partly grounded in the physical shouldn’t be dismissed.
On the other hand, the ubiquity of mental states in panpsychism can be more easily envisaged as a ubiquity of qualia (atoms of pure phenomenality). We don’t expect a rock to get started on a memoir anytime soon (“what it’s like to be me”), but seeing its basic qualia as a form of proto-consciousness (being to the consciousness of a sponge what that is to a human’s) can at least be entertained. It is more difficult to picture a rock with intentional states: what would they be directed at and with what content? If qualia are ubiquitous and intentionality seems to be at least partly reducible to the physical (more precisely, it appears to emerge from some complex structures), we could then see the intentional as emerging from the combination of physical and phenomenal states.
I will go one step further and sketch a speculative model as follows:
\(\mathfrak{ToPM}\) will represent the phenomenal via a single ubiquitous “stuff”, which I would call the qualium field \(\mathfrak{Q}\).
\(\mathfrak{Q}\) interacts with physical objects by breaking up into several terms (in a similar way, to paint a picture, that quantum fields do at low energy). Reminding ourselves that \(\aleph\) represents the whole concrete realm and \(\mathbf{1}\) and \(\mathbf{2}\) the physical and mental respectively, we would have it, roughly, that: \[\begin{aligned} [\textrm{Human}\otimes \mathfrak{Q}]_{\aleph} &\approx \textrm{Human}_\mathbf{1} \oplus \mathfrak{Q}^{\textrm{Human}}_{\mathbf{2}} \oplus \mathfrak{I}^{\textrm{Human}}_{\mathbf{2}} \oplus \cdots \\ [\textrm{Computer} \otimes \mathfrak{Q}]_{\aleph} &\approx \textrm{Computer}_\mathbf{1} \oplus \mathfrak{I}^{\textrm{Computer}}_{\mathbf{2}} \oplus \cdots\\ [e^{-} \otimes \mathfrak{Q}]_{\aleph} &\approx e_\mathbf{1} \oplus \cdots \end{aligned}\] So the interaction of a human being’s physical structure with the qualium in \(\aleph\) gives rise as an approximation to mental states that are distinct from that physicality and, again as an approximation, can be decomposed over the phenomenal/intentional divide (with \(\mathfrak{Q}\) for qualia and \(\mathfrak{I}\) for intentions).
A computer’s physical structure allows to convey information in a form that could be considered as intentionally comparable to a mind’s states, although the phenomenal term would be close to that of brute matter and then "near-zero" by any metric.
Simple, unstructured matter, like an electron, also combines with the qualium field but that interaction is fruitless: although qualia are ubiquitous, their "intensity" is near-zero here too, and the lack of complexity also precludes any significant intentionality.
The mental states hereby generated form a class of objects within the theory that can be incorporated into the same mereology as physical objects, thus allowing for the union of mental states together, or of mental and physical entities. Let’s remind ourselves that such unions have ontological existence (hence they can have properties or relations, as we’ll see), but in the general case this doesn’t mean they’re canonical entities of any relevance.
Abstract algebra: how do abstract entities combine?
Position: Abstract combinatorialism
Having just discussed the mereology of physico-mental entities, one could also mull over the existence of such a structure for the abstract world \(\mathbf{3}\). The notion of an algebra of concepts, like so many things, is an old idea that’s been first properly formalized by Leibniz.
The basics are pretty simple: where there are two properties, there should exist an addition and a multiplication of those, defined as \(\forall x ( (B+R)x \leftrightarrow Bx \vee Rx )\) and \(\forall x ( (B \times R)x \leftrightarrow Bx \wedge Rx )\). If \(B\) is "blue" and \(R\) "red", then quite clearly \(B+R\) and \(B \times R\) would be respectively "blue or red" and "blue and red".
In a similar vein we can define inclusion as: \(I < B \leftrightarrow \forall x ( Ix \implies Bx)\) ("indigo" is included in "blue" because anything that’s indigo is also blue). With limited mathematical labor, together with a few other definitions, we can demonstrate that we are really building a Boolean algebra, which, once we remove the zero point, is in fact equivalent to a mereology.
It is not difficult to extend the Leibnizian conceptual algrebra to our abstract realm, give or take a switch to the encoding point of view. This ensures that we have all the predicates and abstracta we need to discuss anything at the macro-level. Remember, physical mereology helps us build macro-objects through a union of simples, e.g. a man. However, it doesn’t help defining any of that union’s properties, since the only concrete predicates we’ve accepted so far are the ones native to \(\mathfrak{ToPM}\), such as "being a photon" or "having mass \(m_e\)". The conceptual algebra allows us to combine those ad infinitum, generating properties of the mereological complexes. Even something as simple (to humans) a "being round" will require a long build-up from wavefunctions to elementary particles, composite like atoms, expected positions, and finally to the shape of polyatomic objects.
With our complete abstract mereology in place, the bare universe borne out of our basic formalism, of mathematical theories, and of \(\mathfrak{ToPM}\), has become considerably richer. We can now harness the original formalism to deal with complex physico-mental entities and abstracta, as well as their properties and relations. Notation-wise, we’ll use \(\odot\) as an all-purpose mereological operator, for brevity (ignoring the profound differences between the physical, mental and abstract mereologies, over which we won’t delve in the present work). The class of mereologically constructed physical or mental objects (via fusion, division, the hypothetical physical-qualium combination...) — the mereological span of \(\aleph\) — can then be denoted as \(\aleph^{\bigodot}\). \[\aleph^{\bigodot} \hat{=} \{ \bigodot x_i, \forall i(x_i \in \aleph) \}\] As to the abstract realm, it can actually be shown that the logical-formal machinery, together with the comprehension principle, ensures that it is, as a whole, mereologically stable, i.e. \[\mathbf{3}^{\bigodot} \cong \mathbf{3}\] However this wouldn’t be the case for sub-spaces of the abstract realm. \(\mathbf{3}_\aleph\) is the raw class of predicates appearing in \(\mathfrak{ToPM}\), but \((\mathbf{3}_\aleph)^{\bigodot}\) is the minimal algebra spanned by those predicates. It contains the class of predicates exemplified by macro-objects, hence \[(\mathbf{3}^{\bigodot})_\aleph \subset (\mathbf{3}_\aleph)^{\bigodot}\]