It’s been a while since I’ve done an episode for the philosophy of structure series. My interests and reading have taken some other directions over the last few months, dealing especially with theology. But they’ve circled back to the philosophy of structure, by way of theology interestingly enough, particularly in the scholastic theology of Thomas Aquinas. I’ve been wanting to get into some ideas in Aristotelian and Thomistic thought, but in order to do that I feel like there’s some groundwork I want to lay down first, that happens to pass through the philosophy of structure.
This discussion of structure will be a little more general; not as particular as the applications in music and chemistry discussed in previous episodes but more like the first episode on structure as such. But I’ll still refer to some examples in chemistry.
In addition to pre-modern philosophers like Aristotle and Aquinas, some living philosophers I’ve been reading recently are Edward Feser and Kathrin Koslicki. Feser in his books Scholastic Metaphysics: A Contemporary Introduction and Aristotle’s Revenge: The Metaphysical Foundations of Physical and Biological Science. And Koslicki in her book The Structure of Objects. Both Feser and Koslicki argue for a hylomorphic model of material substances. Hylomorphism, from the Greek words ὕλη, hyle, “matter”, and μορφή, morphē, “form”, is the view that physical objects are products of both matter and form. It’s also the view promoted by Aristotle and Aquinas. I take a note from Koslicki and also use the term “structure” to refer to the classical notion of form.
Koslicki gives the following definition of structure:
“Structures are precisely the sorts of entities which make available positions or places for other objects to occupy, provided that these occupants satisfy the type restrictions imposed by the structure on the positions in question; as a result of occupying these positions, the objects in question will exhibit a particular configuration or arrangement imposed on them by the structure.”
We can see the hylomorphic understanding at work here. The formal, structural component of an object is a set of open “slots” that stand in defined relations to each other. Or put another way, it’s the defined relations between these open slots that constitute the structure. But since this is hylomorphism and not just “morphism” these slots need to be filled to produce the substance. That’s the material component.
This is analogous, I think, to functions with variables. Like mathematical functions or functions in a computer program. In a linear function like y=mx+b, where m and b are constants you can see on a graph all the different values of the dependent variable y that correspond to all the different values of the independent variable x. You can substitute any number for x. That’s what makes it a variable. Or in a computer program you can define characters as variables, basically saying, “I don’t want to say exactly what this is now but I want to be able to insert values into it later.” Variables are, by nature, not fixed. They can have different values or be occupied by different objects. The function gives the general structure. But to get the output of a function you need to input specific values or objects for the variables. Similarly, for material substances structures can host various kinds of material components.
Chemical compounds are an instructive example of this kind of structural-material complex. For example, one kind of chemical structure is a tetrahedral molecular geometry, a central atom is located at the center with four substituents that are located at the corners of a tetrahedron. We could think of this as a structure with five open slots, which can be filled with different kinds of atoms. With a carbon in the center slot and hydrogens in each of the four corners the resulting compound is methane. With chlorine in the middle and oxygen in the corners it is perchlorate. With sulfur in the middle and oxygen in the corners it is sulfate. With phosphorus in the middle and oxygen in the corners it is phosphate. In all these cases the molecules have the same geometry and bond angles of 109.5°. It’s also possible to have a third kind of atom in the molecule as with thiazyl trifluoride, in which the center atom is sulfur and corners are occupied by three fluorines and one nitrogen. The takeaway from all this is that with this particular formal component, the structure of the tetrahedral molecular geometry, we have these open slots that can be filled by various kinds of atoms. But it’s the combination of this structure, plus the atoms, i.e. the material component, that makes the physical chemical compound. That’s the hylomorphic description.
Those are examples of a single formal/structural component with varying material elements. But it can also work the other way, with various chemical compounds having common material elements but varying structure.
Koslicki says the following:
“In chemistry, the notion of structure is employed in the following two central ways: the chemical structure of a compound is given by stating (i) the types of constituents of which it consists, i.e. its formula; as well as (ii) the spatial (i.e., geometrical or topological) configuration exhibited by these constituents… the three-dimensional arrangement into which these constituents enter, is equally crucial in characterizing the chemically relevant behavior of a compound. This became apparent in the history of chemistry in connection with the phenomenon or isomers or chiral (“handed”) molecules, compounds which consist of the same constituents, i.e. have the same chemical formula, but whose constituents are differently arranged and which, as a result of this difference in arrangement, behave quite differently in specific circumstances.”
As an example, there are three different chemical compounds with the basic formula C3H4: three carbons and four hydrogens. There’s propadiene, which has a rigid, linear structure with two double bonds and the pairs of hydrogen atoms at each end on planes at right angles to each other. There’s propyne, which has a rigid triple bond and a freely rotating single bond. And there’s cyclopropene, which has a double bond and a ring structure. In this case all the material components are the same but the open slots that they occupy are arranged differently. So the overall structural-material complex is distinct in each case. Again, this is understood most comprehensively from a hylomorphic description.
One reason both material and structural components of a substance are important is that both contribute to its properties. For molecules that have a common structure it makes a difference what atoms fill the available slot. For one thing, different atoms have different masses so a compound with more massive atoms filling those positions is going to be a more massive compound overall. Xenon tetroxide is much more massive than methane, having molecular weights of 195 g/mol and 16 g/mol, respectively, even though they have the same tetrahedral structure. For this same reason xenon tetroxide also has a higher boiling point than methane, 0 and -161.5 degrees Celsius, respectively. Then for isomers, molecules that have the same atoms but different structure, those structural differences can impart important differences in properties. For example, several compounds have the formula C3H6O: three carbons, six hydrogens, and one oxygen. But two are alcohols (allyl alcohol and cyclopropanol), one is an aldehyde (propionaldehyde), one is a ketone (acetone), one is an epoxide (propylene oxide), and one is an ether (methyl vinyl ether). These have boiling points ranging from 6 degrees Celsius for methyl vinyl ether to 101 degrees Celsius for cyclopropanol; quite a range. And all these isomers have slightly different heats of combustion, meaning they release different amounts of heat when they burn.
An important consideration with all of this is that the structural and material components of a substance also have explanatory power regarding the nature of that substance. Not only do we know from experiments that certain substances have different boiling points, heats of combustion, dipole moments, acidities, vapor pressures, and such, though we certainly do obtain such information from experiments. But knowledge about the structural and material components of substances also helps us understand why they have the properties that they do. That’s what we’re really after in science anyway. We’re not just trying to make a giant catalogue of properties. We want to understand the underlying reasons for things.
Beyond particular kinds of substances we’re also interested in the laws of nature that govern the behavior of many or all kinds of substances generally. In a way laws of nature are an even higher order of structure than that of the structural component of substance in the structural-material complex. In the structure of a substance there are open slots that can be filled with material elements that meet the structural requirements. Laws of nature are similarly structural but instead of slots being filled by material elements they are filled by events and conditions. These also behave like functions, associating elements between sets, like inputs to outputs. The outputs are the things that actually occur. But the underlying reasons are represented in the function itself. In the case of physical sciences the natures and propensities of substances are expressed in the laws of nature. It is these laws that dictate what will result from a given set of inputs. We can’t observe these laws directly. We can only observe events. But given large sets of inputs and outputs we can try to figure out what the underlying laws, natures, and propensities must be.
At some point such repeated, higher-order abstraction moves beyond scientific practice itself to something more meta-scientific, reflection on the nature of the physical sciences as such. And this is intrinsically metaphysical. Metaphysics doesn’t replace physics but it can give deeper understanding of it. Many scientists are also metaphysicians, though they may not use that label. Aristotle distinguished between what he called experience and art. I think of these as corresponding to data and theory, physics and metaphysics. Here’s Aristotle in his Metaphysics:
“All men by nature desire to know… But yet we think that knowledge and understanding belong to art rather than to experience, and we suppose artists to be wiser than men of experience (which implies that Wisdom depends in all cases rather on knowledge); and this because the former know the cause, but the latter do not. For men of experience know that the thing is so, but do not know why, while the others know the ‘why’ and the cause. Again, we do not regard any of the senses as Wisdom; yet surely these give the most authoritative knowledge of particulars. But they do not tell us the ‘why’ of anything-e.g. why fire is hot; they only say that it is hot… Since we are seeking this knowledge, we must inquire of what kind are the causes and the principles, the knowledge of which is Wisdom.”
This was Aristotle’s justification for metaphysics. In contrast to this, Auguste Comte, a nineteenth century positivist, saw such metaphysics as something to overcome. He saw history progressing in three stages: theological, metaphysical, and positivistic. Each successive stage would shed the extraneous baggage of the former. In his view, even though educated people of his day had shed the superstitions of religion they still retained ideas of abstractions and invisible forces like gravity and magnetism that looked beyond the bare positive facts of the material world. Comte believed that metaphysics would eventually die out and we’d be left with only empirical data, just what happens, without making any kind of metaphysical inferences about it or even trying to give any kind of explanation for it.
I think Comte got the order of comprehension conceptually backwards, the reason being that structure is ineliminable from an intelligible account of the material world. A more sophisticated form of positivism, one that at least attempted explanation without recourse to metaphysics, reached an apex in the first half of the twentieth century. But it ran into insuperable difficulties, even though it still retains some popular support. But at the end of the day you really can’t have just data by itself. At least not if you’re after a satisfactory account of reality. Underlying, immaterial structures, like laws of nature, are indispensable to make it at all intelligible. Positivism has to give way to metaphysics.
I also happen to think, continuing in the opposite direction as Comte, that further investigation of metaphysics of this kind ultimately demonstrates that certain concepts found in the traditions of religious theology and philosophy are also indispensable to an adequate understanding of reality. My focus with this episode is on the philosophy of structure so I don’t mean to sneak in too much missionary work. But in full disclosure I do in fact think that’s where the logic of all this leads.
Reality is composed of multiple layers of structure. The hylomorphic model of substance is that material substances themselves are most intelligibly understood as structural-material complexes. That’s the best way to think about substances having the kind of properties that they have, with reasons for having the properties they do. Further, physical reality, with its various substances and objects, proceeds according to the natures and propensities inherent in its substances. Physical reality is most intelligibly understood as conforming to certain laws that govern the kinds of events that occur. These laws impose structure on everything around us.