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«Mereologies as the Grammars of Chemical Discourses ROM HARRÉ AND JEAN-PIERRE LLORED `If you cut a crumb in half do you have two new crumbs or two ...»

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27 August 2009

Mereologies as the Grammars of Chemical Discourses

ROM HARRÉ AND JEAN-PIERRE LLORED

`If you cut a crumb in half do you have two new crumbs or two halves of a crumb?’

John Palmer, quoted in the Sunday Times, 28 June 2009, News Review, p. 16.

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 is a part of a molecule of sodium carbonate and also a part of the extended material substance, the element sodium.

However, chemistry has long since ceased to be based on a simple Boylean metaphysics. From a metaphysical point of view the Boylean picture has been subverted by Earley’s (2004) arguments in favour of a process metaphysical foundation, which is naturally tied in with such notions as causal powers and affordances. From the point of view of working chemists the appearance of molecular orbitals as an account of the binding processes of molecules also subverts important aspects of Boyle’s ontology. 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, taking account of these developments.

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

A. Differences in Wholes:

i. Dissipative wholes in which material constituents change within a stable structure of processes in contrast 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:

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

ii. Those for which criteria of identity 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.

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, 2000: §2.6). Before turning to the recent discussion 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 unstructured. 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 3 instantiate different categories or types and uniform sums in which the parts are all of the same category or type. [Axiom MA3 in Simons (2000)].

There are some difficult questions about how category distinctions are managed in mereology. This reminds one of the legendary Japanese tax collector who charged the poor peasants for three items, the bull, the cow and the pair of cattle. We shall call the addition of a whole level concept to the list of parts the `Japanese tax collector fallacy’.

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 (2000), 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 gear box 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.

–  –  –

then and there supporting the weight of sitter, but detached from the frame that function is only potential. Turning to an essentially Aristotelian view of the scope of usable categories we note that chemical synthesis creates temporary dispositions that allow chemists to lift the reacting systems over the required energy thresholds. As we shall see, Mullikan’s work allows us to maintain the dynamical point of view of the mereology we will be developing in this paper.

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

Set theoretical Mereological Principles.

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 5 He proposes several mereological principles for sets and their relations to individuals (Lewis, 1991: 7). Using the concept of a `fusion’, Lewis refines the simple

Lesniewskian scheme with additional principles for a set theoretical mereology (Lewis, 1991:

74).

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

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

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

Then he sets out the way mereological concepts are to be given a set-theoretical interpretation, that is mapped on to the part – whole distinction.

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

6. No class has any part that is not a class.

7. Reality divides exhaustively into individuals and classes.

8. No class is part of an individual.

9. Any fusion of individuals is an individual.

Chemists with stereo-isomers, carbohydrates and so on in mind will surely find the principle of uniqueness of composition unintuitive, and inadequate to chemical part-whole reasoning.

Lewis’s mereology of sets (to be called the S-mereology) sidelines structure and so already leaves itself open to counter-examples to its principles from chemistry.

Choosing a Mereology for Chemical Discourses.

Chemical discourse is largely based on a distinction, hard won it is true, between elements, compounds and mixtures. It is evidently structured by mereological concepts. Clearly ioncores are parts of elements in different way from that in which they are parts of compounds.

Elements are uniform fusions or sums of nuclei atoms, identified by atomic number, Z.

Compounds are disparate sums, because in general the constituents of molecules include ioncores from different elements or more exactly from their isotopes as well. Mixtures are also disparate fusions or sums but the parts are not causally related into relatively permanent 6 structures, nor is there a determinate proportionality among the parts of a mixture.

But which version of mereology should we prefer? Classical mereology makes use of the lowest level of the current hierarchy of beings in the relations of parts and wholes – things, generally spatio-temporally distinct beings with both criteria of numerical identity and of qualitative identity. Two kinds of classical mereological discourses have been 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 have suggested that classical axiomatic 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, for example a chair, the shapes of the parts, for example, are mereologically irrelevant to their role as parts of that whole. The mereology of functionally defined parts in contrast to an atomic mereology should fit the logic of discourses featuring compounds while atomic mereology should fit the logic of discourses about mixtures.

Moreover, atoms are also parts of distributed elements. Is a sodium atom a part of the element sodium 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? This question is made more complicated by Earley’s argument in favour of a discourse of `ion-cores’ rather than the atoms of Boyle and Dalton.

Faced with this tidy scheme the question for a philosopher of chemistry is whether the key chemical concepts of substance, element, molecule, atom, subatomic particle, field 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’ the set of all sodium atoms, or the mereological fusion of all sodium atoms? Is there a mereologised set-theoretical way of distinguishing mixtures from compounds?

–  –  –

However, Lewis does introduce an ontologically and mereologically significant concept in the `singleton’, the single membered class. Set theory requires the possibility of empty classes that collectively constitute the null class. This comes about because to specify a class one needs an intension, that is the necessary attributes that identify a class member, and an extension, that is the members that meet that criterion. There can be a class intension that nothing in the real world satisfies. Its extension is null.

Here we have a genuine alternative ontology – are the atomic constituents of molecules single member subsets that are parts of molecular sets?



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