Foundations of Chemistry

, Volume 13, Issue 1, pp 63–76

Mereologies as the grammars of chemical discourses

Authors

    • Georgetown University
  • Jean-Pierre Llored
    • CREA/Ecole Polytechnique
Article

DOI: 10.1007/s10698-011-9103-3

Cite this article as:
Harré, R. & Llored, J. Found Chem (2011) 13: 63. doi:10.1007/s10698-011-9103-3

Abstract

Mereology is the logic of part—whole concepts as they are used in many different contexts. The old chemical metaphysics of atoms and molecules seems to fit classical mereology very well. However, when functional attributes are added to part specifications and quantum mechanical considerations are also added, the rules of classical mereology are breached in chemical discourses. A set theoretical alternative mereology is also found wanting. Molecular orbital theory requires a metaphysics of affordances that also stands outside classical mereology.

Keywords

Part—wholeAtomMoelculeSetIonAffordanceMass substances

Since Robert Boyle’s corpuscularian philosophy, chemistry has been a mereological science. Displacing the metaphysics of `continuous substances’ and `qualities’ as the expression of “principles”’, chemistry has been built on a `part-whole’ metaphysics. The grammar for the use of `part-whole’ concepts is mereology. Taking chemistry to be the science of the transformation of substances by the manipulation of their constituent material parts which are also bits of discrete substances, the elements, this science seems to fit the concepts of classical mereology neatly. The scheme has served as a popular and pedagogical foundation for chemical concepts and explanations in traditional chemical discourse. A sodium atom becoming an ion is included in a sodium carbonate salt structure, but is also a part of the extended material substance, the element sodium.1

However, chemistry has long since ceased to be based on a simple Boylean metaphysics. The way such `components’ as sodium ions are parts of larger chemical entities depends in part in whether those large entities are molecules in the strict sense, that is strongly bound covalent compounds, or molecular ions sometimes called polyatomic ions. Some bonds are partly covalent, Quantum chemistry allows us to determine the relative importance of covalence and iconicity for each chemical bond inside a molecule. Those studies highlight a molecular landscape the nature of which dwells upon local and global contexts. Pure covalent bond is a unifying concept that structures our descriptions. The third possibility is that they are constituents of the chemical elements. Even the third possibility is problematic in that in recent years the ambiguity of the notion of a chemical element has been brought to the attention of philosophers. The totality of the element `iron’ is all the iron there is throughout the universe. This stuff is identified by criteria such as electrical conductivity, specific gravity and so on. Let us call this the `matter’ or `M-sense’ of element. The atoms of iron in compounds like ferrous sulphate are `iron’ in another sense. Let us call this the `Z-sense’. They are identified by different criteria, summed up in the atomic number `Z’. In this paper we want to track the developments in chemistry in relation to the presumptions of a variety of mereologies, grammars of chemical discourses, in relation to a variety of chemical aggregates, depending on various chemical practices.2

Our argument is based on the identification of variations in both of the `poles’ of the Part—Whole relation.

A. Differences in Wholes:
  1. I.

    Dissipative wholes in which material constituents change within a stable structure of processes in contract to wholes in which the parts are material beings self-identical over time.

     
  2. II.

    Structural wholes in which the parts are components of stable structures in contrast to amorphous wholes.

     
B. Differences in Parts:
  1. I.

    Those for which criteria of identity of parts are independent of the wholes of which they are parts.

     
  2. II.

    Those for which criteria of identity of parts are conceptually related to the whole of which they are parts.

     

Instances of chemical discourses in which all four contrasts are salient will be identified and proposals for the mereological principles necessitated will be examined.

We will extract two mereological systems from the literature and show that each has a natural application as a discourse `grammar’ to one of the two main whole-part concepts to be found in chemical thinking but not to the other.

Mereologies as systems of formal rules

The idea that the part-whole relationship was of sufficient importance to warrant a special branch of logic is due to the work of Stanislaus Lesniewski (for Lesniewski’s mereology see Simons 1987: §2.6). Before turning to the recent discussion3 of the details of the idea that the grammar of discourses concerning chemically relevant substances is mereology, that is implies an ontology of wholes consisting of distinguishable parts, which themselves consist of distinguishable parts, it is worth reminding ourselves of the basic principles of general mereology and sketching some of the debates about the way these principles should be deployed.

Classical mereological principles

Two main mereological principles more or less define the system of mereological rules for discontinuous substances and their parts, in which the whole is uniform, and the parts are not differentiated by their roles in an overall and relatively stable structure. We will refer to this system as the C-mereology.

The principle of unique composition

There is a unique being, the sum or `fusion’ of a certain collection of beings, of which every such being is a part and which has no parts other than such a part. So, for example, a certain actual chemical molecule is a unique collection of just these chemical atoms, and only these chemical atoms. We note that the composition of such a collection does not serve to uniquely identify a molecule as a being of certain kind—the properties of molecules include structures as well as components. In practice we need to recognise the difference between `disparate sums’, that is wholes the parts of which instantiate different categories or types and `uniform sums’ in which the parts are all of the same category or type. [Axiom MA3 in Simons (1987)].

The principle of mereological transitivity

If B is a part of A and C is a part of B, then C is a part of A. [Simons (1987), Axiom MA2].

Various exceptions have been offered to this principle. Some turn on the issue of the way a component is a part of the being of which it is a component or part. A gear wheel is a part of a gear box, but is a tooth of that gear wheel a part of the gearbox in the same way? If we include function among the attributes that define how a being is a constituent of another being, that is how it is a part, then clearly a tooth is a part of a gear wheel in a different way from the way a gear wheel is part of a gear box, and transitivity of that part-whole relation fails. Each has a quite different functional relationship to the whole of which it is a part. This observation leads on to the need to formulate a second mereology, one in which the principles include structural–functional relations.

Functional mereological principles

Even though constituents lose their actualised functional attributes when removed from the whole of which they have been parts, they do not cease to exist. Nor do they lose the core attributes that enabled them to count as parts of the relevant whole. In the light of our knowledge of how a component fits into a whole we may want to hold that potential functionality survives some ways of decomposing the original whole. For example setting fire to a chair is a mode of decomposition into parts that does not preserved potential functionality.

Consider the parts of a chair—qua material objects but not identified as beings of certain kinds by the criteria of carpenters. They continue to exist and have all their material attributes, size, shape, weight etc., as bits of wood when the chair is disassembled.

However, they do not preserve their formerly occurrent functional attributes after disassembly—chair parts move from actual to potential functions, e.g., the seat was actually then and there supporting the weight of sitter, but detached from the frame that function is only potential.

This aspect of wholes has been discussed by Rescher and Oppenheim aeons ago (1955). They suggest three conditions on wholes a whole must possess an attribute that is peculiar to it as a whole; the parts of a whole must stand in some special relationship to one another a whole must have a structure.

The above analysis seems to presuppose the concept of an emergent property fully to describe the whole of which functionally specified components are parts. To ascribe a function to a chair leg makes sense only if the assemblage of chair parts has a structure which endows these parts as assembled with certain causal powers, such as the ability to support the weight of a person. Or whatever concept usefully combines material, formal and final causes such as downward causation in some contexts. The aim here is to create an ordering scheme to think about several different relations between wholes and parts.

In general, emergent properties do not satisfy the mereological principle of transitivity.

Mereology of sets, subsets and supersets

Lewis begins his sketch of the basic principles of set theoretical mereology with an example to illustrate the concept of `fusion’ and `sum’. It falls somewhere between the examples of continuous and discontinuous wholes above. `The fusion of all cats is that large, scattered chunk of cat-stuff which is composed of all the cats there are, and nothing else’ (Lewis 1991: 1). Neither past cats nor future cats are parts of the cat-fusion. Simons’s concept of `fusion’ is different from that of Lewis. For Lewis `sum’, that is `all the cats’, is the same as `fusion’. For Simons some bunch of cats taken as a whole is a fusion, though it may not include all the cats. So there may be several cat-fusions. Assuming transitivity Lewis remarks that the parts of cats are also parts of the cat-fusion. This allows Lewis to distinguish the class of cats from the fusion of cats—the mereological attributes of lots of cats from their set theoretical attributes. Since the member of a member of a set is not in general a member of that set, membership is not the same relation as part to whole. However, Lewis does allow that classes do have parts, their subclasses (Lewis 1991: 3). So there is the possibility of a mereologised set theory, or a set-theoretical mereology. We will refer to this system as the S-mereology.

He proposes several mereological principles for sets and their relations to individuals (Lewis, 1991: 7).
  1. 1.

    One class is a part of another if and only if the first is a subclass of the second.

     
  2. 2.

    No class has any part that is not a class.

     
  3. 3.

    Reality divides exhaustively into individuals and classes.

     
  4. 4.

    No class is part of an individual.

     
  5. 5.

    Any fusion of individuals is an individual.

     

It follows from Principles 1 and 2 that fusions of individuals are not classes the membership of which is extensionally equivalent to the individuals that constitute the fusion.

Using the concept of a ‘fusion’ Lewis refines the simple Lesniewskian scheme with alternative axioms for a mereology of sets and subsets (Lewis 1991: 74), in particular adding c below.
  1. a.

    Transitivity: If x is a part of some part of y, then x is a part of y.

     
  2. b.

    Unrestricted Composition: Whenever there are some things, then there exists a fusion of those things.

     
  3. c.

    Uniqueness of Composition: It never happens that the same things have two different fusions.

     

Chemists with stereo-isomers, racemic molecules, carbohydrates, eutectoïds and so on in mind will surely find the principle of uniqueness of composition unintuitive, and inadequate to those rules for chemical part-whole reasoning that are required to accommodate the role of chemical entities in structures, such as atoms in polyatomic ions.

Choosing a mereology for chemical discourses

Chemical discourse is largely based on a distinction between elements, compounds and mixtures. Clearly ion-cores are parts of elements in different way from that in which they are parts of compounds. At this point we need to introduce the distinction between two senses of `element’ first noted by Paneth and much discussed in the literature since, for example by Joseph Earley (2008) and Klaus Ruthenberg (2008). Elements are uniform fusions or sums of atoms all of the same kind—for example the element sodium. This is the most important assumption that sustains all this theoretical framework. Is uniformity of atomic sums a necessary conditions for any mereological models of materials? The trouble is that materials are first and foremost entangled with many others. This is precisely why the concept of element is a unifying metaphysical concept that links mereology to chemical practices of purification (among others).

An atomic constituent of sodium the metal is a part in a different way from that in which a sodium atom is a constituent of molecules or of molecular ions. The former we have called the `M-sense’ of being such and such an element, and the latter the `Z-sense’ of being an atom of that element.

A further complication to a choice of mereology comes about because compounds are disparate aggregates. In general, the constituents of molecules include ion-cores of different elements in the or more exactly their isotopes in the Z-sense. Mixtures are also disparate fusions or sums but the parts are not causally related into relatively permanent structures, nor is there a determinate proportionality among the parts of a mixture. They tend to be parts in the M-sense.

Would there be any virtue in choosing to express the content of chemical knowledge in discourses ordered according to David Lewis’s mereologised set-theory rather than the classical mereology of Lesniewski? Chemical discourse is evidently structured by mereological concepts. This is especially true in current `green’ chemistry because chemists use models of whole/parts interactions to assess molecular eco-toxicology. Mereological questions are more than ever crucial because they can help to suggest new models. But which version of mereology should we prefer? The intuitions behind classical mereology make use of the lowest logical level of beings in the relations of parts and wholes—things.

Two kinds of classical mereological discourses can be distinguished according to whether their several mereological regresses terminate in atoms, in the traditional sense of beings with no proper parts, or do not terminate, every proper part at each level itself having proper parts. A proper part is a part that it not identical to the whole of which it is a part. We will not discuss this distinction further. It becomes relevant in the context of whether chemistry can be reduced to physics. Are the parts of chemical compounds strictly quarks?

As we argued above classical mereology can be extended to include rules for the use of a whole—part relation for contexts in which the parts are functionally distinct relative to the whole of which they form parts. In the absence of the concept of the whole, the shapes of the parts, for example, are mereologically irrelevant to their mereological status. They have no role as parts.

Applications

Which version of mereology should we use to express the structures of molecules and molecular ions—is a molecule a thing of which its parts are also things, a structured collective—or is a molecule a set of which its constituents are subsets? The former would required the classical mereology of Lesniewski, the C-mereology in its functional form. The latter would require one to make sense of the mereologised set theory of Lewis, the S-mereology, in chemical contexts. In that mereology there no provision for distinguishing parts functionally with respect to their roles in the wholes of which they are parts. Is the grammar of chemistry classical mereology or mereologised set theory?

Are chemical elements in the M-sense wholes composed of atoms as parts, or are they sets of singletons, atoms as one membered subsets? Is a sodium atom a part of the element sodium taken in the M-sense in the way that a horse is part of a herd of horses? Very few elements exist as chunks of well-bounded and uniform stuff—diamonds as chunks of carbon or nuggets as chunks of gold perhaps? However, neither diamonds nor nuggets of gold are pure, uniform substances. A teacher might cut a small piece of sodium of the specimen kept in a jar of paraffin to show its reactivity with water. Most of the sodium in our part of the universe exists for us as atoms of sodium in the Z-sense, that is as components of compounds. No doubt there are some lonely singletons floating about in the debris of supernovas.

Do the key chemical concepts of substance, element, molecule, atom, ion and so on fit, at least in part, on to the logic of classes, set theory, or on to the principles of classical mereology. Is the element `sodium’ in the M-sense the set of all sodium atoms, or the mereological fusion of all sodium atoms? What difference would it make which way we jumped to explicate the forms of reasoning available to chemists? Is it a change of ontological foundations or a methodological (epistemological) shift as regards ways of modeling chemical phenomena?

Lewis introduces an ontologically and mereologically significant concept of the `singleton’, the single membered class. Here we have a genuine alternative ontology—are the atomic constituents of molecules and polyatomic ions single member subsets that are subsets of molecular sets?

Mereological rules for continuous substances

In his recent debate with Joseph Earley, Paul Needham (2005) sets out two basic mereological principles for discourses about substances that are considered to be continuous—they have bits but not parts. A lump of gold, not yet wrought into anything shapely, can have bits lopped of it, but they are not parts of the lump in the sense that the legs are parts of the chair. It makes little sense to ask someone to bring them a part of glass of beer, or of the sea. `Bring me some sea water’ does make sense but it would be very odd to say that what is in the bucket is a part of the sea, though it is!. Using water as an example and despite the force of the vernacular use of the word ‘part’, Needham offers the following:
  1. 1.

    The distributive condition: Whatever is a `part’ [sample] `w’, taken from a body of water, `W’ is also water.

     
  2. 2.

    The cumulative condition: If two things, say the contents of a pair of buckets, are water, their sum is water.

     

These principles generalise nicely to other mass substances, such as wood and even fire. This scheme fits the S-mereology. If the parts are subsets of the whole as a set, then each bucket of water is a singleton, that is a single membered set. An empty bucket is a logical possibility in the task that the Sorcerer’s Apprentice was stuck with—a nothing will not do as a `part’ in the C-mereology, but there is no problem with null sets. The empty bucket is, as it were, the intension of the singleton, but it has no extension.

Needham’s principles do very well with the part-whole relation as it appears in the traditional chemistry of elements in the M sense. When we turn to compounds, it is reasonable to consider ions as parts of chemical substances, one molecular cluster of ions is enough to have a substance on hand, but they satisfy neither the distributive nor the cumulative condition above. Any old sodium ion and any old chloride ion as candidate parts of a salt crystal are not salt. Here is the case of the part-whole relation requiring a third ingredient, the right relation between the parts, in order that the criteria of identity for the relevant whole are met. Is the structure a part of the whole? Must the sodium and chloride ions be juxtaposed in the right way in a cubical lattice for there to be salt?

Progression of the mereological principle in chemistry

Earley’s mereological argument

Na+ and Cl ions are not parts of salt lattices after that salt has been dissolved. Being in the solution determines that the solution will afford salt as a mass substance on the carrying out of certain operations on sea water, and not something else. Thus they are at best potential material parts of salt. They are like the parts of chairs in the factory store room. If there are legs, seats and backs there the factory affords chairs if the workmen carry out their instructions.

When it is in the sea, an Na+ is a potentially a part or constituent of a possible salt crystal afforded by a saline solution. When you meld a solution of NaOH (Na+ HO) with a triglyceride (fat or oil), you produce a soap. Na+ is thus a potential part or component of a sodium palmitate which can be a part of a cake of soap. However, it is neither a bit of salt nor a bit of soap. Unlike the legs of chairs, which could hardly be used for any other piece of furniture the ions of chemical elements in the Z-sense are polyfunctional.

Mulliken’s mereological models

The progression of the grammar of chemical discourse concerning molecules and polystomic ions considered with respect to their mereological constituents goes something like this: the classical account of a molecule was of a disparate fusion of ion-cores of different elements in the Z-sense, sustained by their individual combining power.4 About a 100 years ago this modulated into the shared electron theory as the source of bonding, with the perfect octet as the grounding concept. Each electron `orbited’ the nucleus of its own atom since it was a defining constituent of that atom. Thus the `shell’ architecture of sodium with its nuclear cluster of protons and neutrons defined the element in the Z-sense. A molecular carboxyl ion contains carbon in the same way that a bucket of brine contains salt—that is it affords carbon as the result of some quite complex chemical manipulation, just as evaporation forces the bucket of brine to afford salt. However the advent of molecular orbital approximation, such as that of Mulliken (1981), requires a more radical mereological grammar. Even ion-cores lose their thing-like status. According to Mulliken (Ramsay and Hinze 1932: 451) `Attempts to regard a molecule as consisting of specific atoms or ionic units held together by discrete numbers of bonding electrons or electron pairs are regarded [by me] as more or less meaningless’. So, following Mulliken’s thought, there are no atoms in a molecule, for the same reason as there is no salt in the sea.

Electrons as constituents of molecular orbitals as an image of electronic density energy distributions are not related to the nuclei of constituent atoms, but to duplets, triplets etc., the paired or tripled etc., nuclei at the core of the molecule. Formally, molecular orbitals are linear combinations of atomic orbitals, but the atoms that define these wave functions do not actually exist. Mulliken’s view is that they are scaffolding to provide the models needed to arrive at the consequential molecular orbitals and to explain molecular spectra Jean-Pierre Llored (2010). It is the concept of quantum state which becomes the lever of his attribution of electronic configurations to a molecule. This holistic molecular approach makes atomic valency become unnecessary to describe a molecular whole. Mulliken shows for instance how the atom of helium He disappears during the synthesis of the molecule HeH. He also explains how HeH can afford He in certain conditions, just as well as an electrolysis of molten NaCl affords sodium atoms in plenty because NaCl affords the sodium nuclei that are essential to the formation of sodium, atom by atom. There is no sodium in salt. But salt affords sodium.

Does it make any sense at all to ask if elements and their atomic constituents and molecules and their atomic components could be treated as sets? If so there is room for mereological set theory à la Lewis as an alternative grammar for chemicals discourses.

The case for S-mereology

The case for adopting set theory as the Mereology for chemistry begins with the predictions by Odling and Mendeleev of the properties of elements yet to be discovered. At the time of their proposals only the intensions of the set of atoms of eka-iodine was available in chemical discourse. The set had a null extension for the users of the grammar appropriate to the situation as it then stood, since the set had no members, and conceivably might never have any. Obviously there cannot be a fusion or a sum of which there are no parts. To talk of eka-iodine in the grammar of classical Mereology made no sense. It does seem to make sense in a discourse in which the parts of sets are subsets.

Are hydrogen and oxygen atoms (ions) subsets of the water molecule set? Each water molecule would be a subset of the superset, the stuff water. However, what is the intension of the set of which two sets, pair of hydrogen atoms and a singleton oxygen atom are the subsets? Well, it is the properties of whatever it takes to be a subset of the set of water molecules that is the water stuff. The hydrogen atoms (ions) are members of the set of all hydrogen atoms, while the oxygen atom (ion) is a member of the set of all oxygen atoms. Does this have any advantage over the classical mereological grammar?

The case for the C-mereology

The first argument for the C-mereology depends on the possibility of a whole having emergent properties as a result of some structural invariants. A set only accidentally has structural properties because it is a conceptual object. A whole has structural properties because is a material entity, with real relations between its parts. Sets are held together by similarity relations, not by real relations between the parts of wholes such as material connectivity (the parts of a chair) or causality, the parts of a molecule. A set can have only similar members, while a whole can have dissimilar parts. A set is a logical object while a whole is a material object.

The second argument for C-mereology depends on the criteria for class membership that is the intensionality component of the set concept. If H+ and O2− are subsets of the water molecule set what is their common property that makes them members of this set? It can only be that they are constituents of a water molecule. Hence the S-mereology treatment of chemical unity in multiplicity depends on a C-mereological understanding of the relation between atoms (ions) and the molecules of which they are parts.

In the grammar of classical Mereology, the three atoms are the parts of a water molecule which is their (disparate} fusion or sum. The water in the sea is the mereological fusion of certain varieties of hydrogen–oxygen conglomerates as parts. But it is not the sum of these conglomerates, which is a being of much greater dimensions being all the water there is. A bucket of brine as a part of the sea is a fusion of the `water’ ionic conglomerates which are its parts. As Earley has argued the same does not apply to the Na+ and Cl ions in the sea. Here we need to supplement C-mereology with dispositional concepts as illustrated in the simple case of the parts of the chair. The concept of the whole, the chair, cannot be eliminated from the criteria for ascribing dispositional properties to the chair parts. This is indeed a reflection of a general condition for practically oriented cognition.

Mereological distinctions between mixtures and compounds

A mixture is a collectivity or aggregate which includes more than one kind of substance. If we allow `disparate sums’ as a legitimate mereological concept then mixtures are clearly sums or fusions. Mixtures rarely have emergent properties. Their constituents are not causally related nor do they have invariant structures. A sack of sand and cement consists of causally unrelated and independent chunks of sand grains and particles of Portland cement, usually consisting of two kinds of calcium silicate with aluminum and iron oxides among others components. When water is added complex chemical and physical reactions begin and the Portland cement is changed into a paste of mixture ceases to exist. The paste holds together the sand grains and the pieces of rock in the aggregate. The resultant concrete has emergent properties, such as strength in compression, that a scoop of dry sand and cement does not have. A calcium silicate crystal—the strong iconicity of the bond precludes its identification as a molecule—can be a mereological part of a bag of sand and cement, but when that bag of cement becomes a part of a block of concrete of which another though short-lived `part’ was water, the transitivity principle becomes problematic because none of the constituent items in the original mixture have the emergent properties of the mortar.5

Mereological presumptions in the historical analysis of the concept of `molecular orbital’

In proposing a change of terminology from `orbit to `orbital’ Mulliken (1981) reassigns the electro-mechanical model of the atom as a basis for an explanation of atomic spectra to a heuristic role. We can be quite agnostic about whether electrons are little things whizzing round a nuclear `sun’. Mulliken proposal upsets simplistic applications of C- or S-mereological rules to `atomic chemistry though the concept of `atom’ in this framework seems to us still be susceptible to them. However, the role of electrons in binding atomic units into molecules, when interpreted within Mulliken’s molecular orbital theory, upsets the mereology of atoms as well. Or to express this in another way, this theory undercuts the simplistic idea of atoms as simple constituents of molecules. This seems to open the way for a revised mereology of affordances à la Earley. Molecules afford atoms though atoms are not simple molecular constituents.

Just as `electron’ ceases to be literally the name of a moving body, so `atom’ ceases to be literally the name of a molecular constituent. If molecular spectra can be explained by molecular `orbitals’ then the case is more or less made. Mulliken makes use of the concept of `atom’ in two distinct but linked ways. The concept of `atom’ is a conceptual tool which makes it possible to unify multiple relations between empirical data, particularly spectra. Mulliken’s `correlation diagrams bring about a synthesis of the experimental data, chemical representations and certain theoretical models thanks to the mediator concepts of `isolated or unified atoms. These diagrams allow a great number of forecasts not only about the spectral states of the molecules but also as regards their physical properties and to some extent, their chemical reactivity. They make it possible moreover to study the formation of molecules without alluding to a supposed intrinsic valency of the atoms (Llored 2010).

The same account can be given of the role of the concept of `electron’. It plays a heuristic role through the concept of binding capacity of electrons. Mulliken makes use of the process of molecular dynamics to try to rationalize molecular reactivity. The heuristic character of the explanations which he proposes is undeniable but it is not all. Using the `manipulation’ criterion on which to base claims for existence Mulliken describes electrons as if they were existing particles because they can be acted upon by electromagnetic radiation, the interaction having observable consequences (Harré 1996). An electron has a relational capacity to interact with various nuclei in a molecular orbit. The consequences of acting upon electrons are displayed by means of spectroscopy. Mulliken does not believe in electrons because he seeks a theory of the structure of matter but because they can be acted upon electromagnetically. Such effects at the electron level as lengthening of the internuclear distances in a molecule, change of the angles of connection, evolution of energies of dissociation and so on, have spectroscopic effects. Mulliken is looking for whatever causal capacities are at the origin of the molecular phenomena. He tries to quantify the binding capacity of electrons via many spectral studies.

Mulliken believed that this binding capacity is related to the stability and the reactivity of the molecules i.e., on their capacity to act on other molecules. He tried to measure the capacity of electrons to be put in relation with nuclei or electrons of other molecules to produce chemical phenomena. Spectroscopy makes it possible for Mulliken to evaluate this reactive capacity of electrons and to propose ways of envisaging the properties of the molecules starting from analogies between the atomic and molecular spectral states.

The mereological problem posed by molecular orbital theory

Using the expression `diatomic molecule’ for such a thing as a molecule of HCl or H2 suggests that the mereological analysis of these complex entities should lead us to say that the parts of such molecules are hydrogen and chlorine atoms. However, Mulliken’s solution to the problem of how atoms are bound into molecules involves electron orbits that are not centred on the nuclei of the constituent atoms. Instead the one-electron wave function approximation for an electron becomes molecule centred, the paired nuclei serving as the reference for the model interpretation of the new orbital as a linear function of the wave equations for each electron considered with respect to each of the apparently constituent nuclei. If the criterion of identity for an atom or the ionic residue of such an atom, is the composition of the electron shells then these criteria could not be satisfied by the components of a complex molecule. The relevant nuclei form a doublet which, speaking in the accent of Mulliken, are a unit without parts, using the molecular orbital theory of electrons as the criterion for an individual part. A molecule does not have atoms or ions or even the nuclei of ions as its parts. It does have nuclei duplets however, identified as molecular parts with respect to molecular orbitals. Furthermore, new energy levels emerge within the molecular whole that didn’t exist in the previous atoms. Mulliken proposes the concept of electronic promotion to construe molecular electronic configurations fitting with empirical data. A molecular electronic configuration is all but a ‘molecular part’ within this heuristic and holistic approach but a tool for prediction (Llored 2010).

But we can go further and deeper if we consider the use of group theory based on the logic of sets and subsets in such a molecular approach. Mulliken develops the fragment method in 1933, two fragments can interact provided they have the same kind of symmetry and that their energy gap, measured by spectroscopy, is not too high. For the ethylene molecule “C2H4” Mulliken considers two ‘fragments CH2’ and determines suitable molecular orbital by using the irreducible representations of ethylene. He can thus propose a representation of molecular orbital of ethylene by increasing order of energy as well as its correlation diagram thanks to those of the two ‘fragments’. In so doing, he grasps all the characteristics of molecular orbital diagram of the ethylene molecule (Mulliken, 1932). The possibility of an experimental support was all the more important as the nature of the initial fragments can change depending on each specific case. To model the molecule C2H2, Mulliken could just as easily had considered a fragment “C2” and another “H4” of adapted symmetries. The relation between the whole “C2H2” and its ‘fragments’ is of second interest provided that the energy diagram of the “molecular whole” is in agreement with the experiment.

We could express this insight in a mereological principle: Constituent atoms of molecules are not parts of those molecules when we look at the total entity in the light of molecular orbitals. Unlike chair parts which preserve their material properties whether in the chair or on the bench. Nor are they parts in the sense that buckets of water are parts of the ocean.

However, parts of chairs, atoms and the contents of buckets of water are extracted from the wholes of which they are parts by some procedure. Looked at from the point of view of the whole, chairs, molecules and oceans afford things; looked at from the point of view of their constituent parts they are potentialities, not the things that are thereby afforded.

A mereological study must take into account of the interaction of the whole with its environment and should not artificially isolate it from the external world. For even thinking the relation between a molecule and its parts, it is necessary to take account of the capacities of this molecule to act on the external world. This capacity to enter into extra-molecular relations makes it possible to study the chemical properties of atomic aggregates but also accounts for the molecular form i.e., the modeling of the internal relations between the molecule and its “parts”. In addition, the striking analogy between chemical properties and quantum observables is a line of work that could pave the way for a new mereological approach to chemical systems. In the light of our arguments, it seems that Mulliken and Earley offer parallel arguments. Molecules afford atoms in the context of certain manipulations. The material content of a molecule can only be a fusion of atomic potentials, not of atoms. Brine affords salt crystals in the context of certain manipulations. The conclusion from this analysis is that an `atom’ in the molecular orbital framework of concepts is mereologically like a single sodium ionic core in Earley’s sea, that is it affords salt or in appropriate circumstances, soap, as a proper part of a structured whole, or it affords sodium as a widely distributed element.

Continuous and discontinuous substances as wholes

Since the end of the eighteenth century the idea that chemistry is the study of the qualitative transformations of continuous substances has been displaced until recently by the simple atomic hypothesis. The mereology of continuous substances does not fit the logic of classes and their members, set theory, as worked out mereologically, by David Lewis (1991) for example. Taking the sea as a continuous substance, a human choice for some purpose or other, we can say it is a fusion of trillions of buckets full, how many depends on the size of the bucket. It can be considered as a fusion of a dozen or so oceans and seas, or as a fusion of so many drops from an eye dropper and so on. These scale-different fusions illustrate the transitivity of the part-whole relation.

However, atoms of disparate kinds do not make up molecules in the way that members of sets make up sets, though they are the parts of such molecules (neglecting diatomic or polyatomic molecules for the moment). Nor do atoms of the same kind make up elements in that way either, though the stuff, sodium, has sodium atoms as parts, and blocks of it can be kept anhydrously in a vat of paraffin. Horses make up the set of all horses in yet another way, since horses never fuse into larger equine entities in the way that sodium atoms fuse into larger blocks of sodium. So there seem to be two notions of mereological fusion at work in chemistry.

We are forced to conclude therefore that a new set of mereological rules is required for the logic of chemical discourses. It is neither wholly a C- nor wholly an S-mereology. The new mereology requires a revision of the basic principles that are definitive of each of the mereologies set out above.

In classical mereology the Principle of Unique Composition runs up against conclusion that the parts of chemical wholes like molecules and atoms are affordances not themselves concrete entities. However, those same atoms which Mulliken’s approach transforms into affordances are the parts of elements as fusions, that is obey the Unique Composition Principle. It seems to us that Transitivity of the Part-Whole Relation as defined in C-mereology does hold because electron affordances are parts of atoms, and atom affordances are parts of molecules, electron affordances are parts of atoms and so of molecules. That is the conclusion to be drawn from Mulliken’s demonstration of the power of the molecular orbital set.

However, transferring the Principle of the Transitivity of the Part-Whole relation to the S-mereology does appear to be viable. An electron-singleton is a subset of a certain set of electrons, just as a neutron-singleton is a subset of a certain set of neutrons and a positron-singleton is a subset of a certain set of positrons. But could these be subsets of an atom as a set of subatomic objects? The members of a set must share a common intension which is just what the components of an atom do not have. Our conclusion is that the functional version of C-mereology is appropriate to this case, but it cannot be replaced by the S-mereology as the set of rules for this aspect of chemical discourse.

Parts in dissipative structures

A candle flame, such as that which was the subject of Faraday’s famous lecture, is a dissipative structure because it is continuously self-identical as a sum of processes.6 There are many molecular level processes which are parts of the macro process, the flame as a bounded dissipative structure. From this point of view the mereology of Faraday’s candle flame is unproblematic. However, from another point of view, that of ionic cores as constituents of material beings as their sums, C-mereology does not seem to be a good fit. The material constituents of a flame or any other bounded dissipative stricture are continuously changing as more wax molecules interact with more oxygen molecules drawn into the flame. The products, mainly carbon dioxide and water pass out of the flame. Let us call these molecules `fleeting parts’.

This problem has been addressed in the past in other contexts. A rainbow is an optical phenomenon produced by the refraction and internal refection of light from a point source in rain drops. Theodoric of Freiburg in his experimental study of the phenomenon realised that he could treat the rain shower as a dissipative structure, because the rain drops were succeeding one another in the rain shower sufficiently quickly to allow for the stationary array of watery spheres. The rain drops are fleeting parts of a dissipative structure. S-mereology seems to fit the concept of fleeting constituent well. There is a many membered set of raindrops at a certain location in the shower—just as there is a set of oxygen atoms at certain location in the flame. It is this set that is a constituent as a subset of the superset that is the flame.

Chemistry also makes use of `ephemeral’ individuals as parts of wholes. For instance, the swiftly composing and decomposing hydrogen–oxygen structures of which real water is really composed are ephemeral individuals. Water is made up of these beings. As such they are constituents of a certain whole. Here is a mereological set-up for which neither C-mereology nor S-mereology seems well adapted as discourse grammars because they do not integrate the conceptual time-dependence between process of transformation and dissipative structure. And that is why again, the topological chemical quantum turn is of the utmost importance for philosophical enquiries such as this. That is why, too, philosophers need to go back to laboratories of research to grasp what scientists are really doing with their new models and apparatus.

Summary

The long term development of the ontology of chemistry from the days of Robert Boyle, though John Dalton’s mereology to the present day displays a progressive elaboration of the working mereology implicit in chemical discourse. The dual semantics of `sodium’ as an element in the sense of widely distributed metal of a certain electrical conductivity, reactivity and so on, and as the type of an elementary ion of atomic number 11 suggests that both the s-mereology sets and subsets, and the c-mereology sensitive to the distinctive role that part can play in wholes are in use in chemical discourse. However, the ontology implicit in the work of Mulliken suggests yet another option—that the fact that a material set up yields individual items that seem to be parts of the wholes from which they are extracted does not license this inference—products are not necessarily parts—instead they may be given such a status vicariously on the basis of a coherent mereological model.

Footnotes
1

It is worth remembering that Boyle’s famous book—`The Origine of Forms and Qualities’ is based on the principle that different structures manifest different emergent properties, e.g. his discussion of Glauber’s Salt.

 
2

Two reviewers of an early rough draft of this paper drew our attention to the importance of these distinctions in any discussion of chemical mereology.

 
3

The two leading papers are Earley (2005) and Needham (2005).

 
4

Diatomic molecules consist of atoms of the same element in the Z-sense. Perhaps they are like the cattle grid of similar bars in contrast to the farm gate made of parts of different types..

 
5

Sand, as silicon dioxide, does not react chemically with the water and calcium silicate derivatives in concrete. Each silicon ion is surrounded by four oxygens in a repeated tetrahedral layout.

 
6

We are grateful to Joseph Earley for insights into dissipative `entities’.

 

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© Springer Science+Business Media B.V. 2011