The threefold whole

In his Metaphysics Δ Aristotle says there are two senses of the term “whole”:

Whole means that from which none of the things of which it is said to consist by nature are missing; and that which contains the things contained in such a way that they form one thing.

The first sense corresponds to our usage of the word when we say things like, “he managed to eat the whole sandwich” and “she read the whole book in one day.” The second sense corresponds to what we refer to when we speak of general part-whole relations, for instance when we say that my arms and legs are part of my body. This second sense is what we’re interested in here. Aristotle further divides this into two kinds:

But this occurs in two ways: either inasmuch as each is the one in question, or inasmuch as one thing is constituted of them.

These are two very different kinds of whole. The second kind is perhaps the one we’re most familiar with: bodies are constituted by organs, tables are constituted by legs and tops, computers are constituted by transistors and other electronics. This kind is referred to as integral, so that integral wholes are constituted by integral parts. We might not think to talk about the first kind as a whole, but it does fit one sense of the general definition. It’s a whole in the sense that a universal applies to (and thereby “contains”) all the particulars that instantiate it: humanness contains all individual humans, treeness contains all individual trees, and so on. This kind is referred to as universal, so that universal wholes apply to universal parts.

Aristotle construes the difference between these two kinds of whole in terms of how the parts are made “one” in different senses. Integral parts come together to form one individual which we call the whole. We refer to this as numerical unity. Universal parts are each themselves an individual which instantiate a common universal. We refer to this as specific unity.

Later the Scholastics discovered a third kind, which they called potential. How potential relates to integral and universal depends on how you analyse the differences between the kinds. Aquinas, for instance, analysed them in terms of the presence of a whole in its parts, which in turn correlates to how truly the whole can be predicated of its parts. This led him to placing the potential as midway between the integral and universal:

… the universal whole is in each part according to its entire essence and power; as animal in a man and in a horse; and therefore it is properly predicated of each part. But the integral whole is not in each part, neither according to its whole essence, nor according to its whole power. Therefore in no way can it be predicated of each part; yet in a way it is predicated, though improperly, of all the parts together; as if we were to say that the wall, roof, and foundations are a house. But the potential whole is in each part according to its whole essence, not, however, according to its whole power. Therefore in a way it can be predicated of each part, but not so properly as the universal whole. (ST I, Q77, A1, ad1)

Intrinsicality

My preferred analysis is in terms of the intrinsicality of the potency and act by which the parts of a whole are distinguished and unified respectively. For the remainder of this post we will unpack this, and reflect on how the different kinds relate to one another on this account.

Now, any material being is a mixture act and potency (or, equivalently, actualities and potentials). By this we mean that it has capacities for various states or behaviours, some of which are realised. We call these capacities potentials, and insofar as a potential is realised we call it an actuality or an actualised potential. For example a coffee cup has potentials for being various temperatures, a person has potentials for being various levels of educated in some subject, and a squirrel has potentials for jumping and running. That last example indicates that potentials aren’t always potentials for static states, but can also be potentials for dynamic activities. So also actualities can be static or dynamic, depending on the kind of potential they’re the actualisation of.

These two things, namely (1) the distinction between act and potency and (2) the realisation that individuals are mixtures of various acts and potencies, enable us account for very fundamental features of reality like change and multiplicity. We’ve spoken about change before, but it’s worth saying something about multiplicity here. Parmenides famously held that multiplicity is impossible since if A and B have being, then the only thing that can distinguish them is non-being, which is nothing. But if nothing distinguishes them then they are not distinguished, and therefore they are identical. Thus everything is one, a unity without multiplicity. His mistake was failing to realise (as we have) that being is divided into act and potency, and that beings are mixtures of these two principles. Two things can be unified by being actual in the same way, but diversified (or multiplied) by this common actuality resulting from the actualisation of distinct potencies. So you and I can be unified in our both being educated, but diversified by the fact that my being educated is the actualisation of my potency for being educated and your being educated is the actualisation of your distinct potency for being educated. So long as we properly divide being into act and potency, then, we can affirm both unity and multiplicity.[1]

So that’s act and potency, next we turn to intrinsicality. Intuitively, to be intrinsic to something is to be wholly contained within it. Slightly more formally, A’s being B is intrinsic to A relative to some C insofar as A’s being B doesn’t depend on C. Alice’s being educated is intrinsic to her relative to Bob’s being educated, for example, because it does not depend on Bob’s being educated. Intrinsicality is, naturally enough, contrasted with extrinsicality. In a water molecule, the hydrogen’s bonding to the oxygen is extrinsic insofar as it depends on the cooperation of the water molecule.

It’s clear enough that the primary sense in which we talk about the acts and potencies of something is as intrinsic acts and potencies, since these are what constitute the being of that thing. In order to outline all three kinds of whole, however, we will need to expand our focus to secondary senses. That being said, when considering something in terms of an act and potency at least one of these must be intrinsic to that thing, since if this weren’t the case, no sense could be made of our considering that thing rather than something else.

In general a whole, in the sense we’re interested, is “a unity of ordered parts.”[2] Parts, of themselves, are diverse and are brought together into a unity through an ordering of some kind, like an arrangement or structure or process. Now, since act unifies and potency diversifies, it follows that a whole arises through the actualisation of the potencies by which the parts are distinguished from one another. So for each part we can talk about the actualisation that unifies it with the other parts, and potency that distinguishes it from the other parts.

This allows us to state our taxonomy of the kinds of whole. For any part, either this unifying actualisation is intrinsic to the part or it is not. If it is extrinsic then, as we said above, the diversifying potency must be intrinsic to the part. If the actualisation is intrinsic, then either the potency is also intrinsic or it is not. An integral whole arises when we have an extrinsic act and intrinsic potency, a universal whole arises when we have an intrinsic act and intrinsic potency, and a potential whole arises when we have an intrinsic act and an extrinsic potency.

Breakdown of the three kinds of whole
Breakdown of the three kinds of whole

Integral wholes

All of this is rather abstract, and some examples might help for clarity. Starting with integral wholes we’ve already seen an example: a water molecule made up of hydrogen and oxygen molecules. Each of the parts has an intrinsic potential to be bonded with the others. There is one bond which actualises all of these distinct potencies resulting in one water molecule, and so this actualisation is extrinsic to the parts. Second, there’s a simple wooden table made up of a tabletop on four legs. Here each of the five pieces have potencies for being structured in various ways, and the binding of them together into the table is an actualisation of these potencies. And finally, there’s a living animal. What the parts are here is not totally obvious; they might be the various organs, the interconnected organic systems, or the cells, bones, and other organic materials. Whatever they end up being, the point of interest is that the extrinsic actualisation here is a dynamic process involving the parts, rather than the static structure of the table. This process is what constitutes the difference between a living animal on the one hand, and a corpse on the other.[3]

With these three examples in hand, we can introduce some technical vocabulary. In an integral whole call the extrinsic actualisation the configuration, and call a part with the configuration abstracted away an element. The element is that in which the intrinsic potency inheres. If we consider a hydrogen molecule while abstracting away whether it is free or bound in some other molecule, then we’re considering the hydrogen molecule element. When we consider a free-hydrogen-molecule or a water-bound-hydrogen-molecule, then we’re considering the element together with a configuration.

Universal wholes

Moving on to universal wholes, let’s consider the example of the wooden table and how it differs depending on which kind of whole we’re considering. The integral whole in this case is the table itself, with the integral parts being the tabletop and legs. The universal whole, on the other hand, is tableness and the universal part of this whole is the individual table (that is, the particular instantiating tableness). Each table — each universal part — will have its own intrinsic actualisation that accounts for its being a table as opposed to something else. This actualisation is common to all tables (it is in virtue of this that we call them tables in the first place), but it is not some numerically one thing. Rather, each has their own instance of this actualisation, each being actualised in the same way.

Again we can introduce some technical vocabulary. Well actually, we can re-introduce some technical vocabulary first introduced by Aristotle. The common actualisation intrinsic to each universal part is called the form, and when we abstract away the form of a part we’re left with its matter. Of itself matter is indeterminate between a number of alternatives, and form is the determination to one of these. (Put in terms of act and potency, of itself matter has potencies for alternatives, and form actualises one of these potencies.) The difference with integral wholes may now be apparent: with integral wholes the elements are the individual pieces of wood, but with universal wholes the matter is the wood itself. After all, if we have a table of wood and we abstract away the table bit all we have left is the of wood bit.

Because much of modern science has focused on integral wholes, we as moderns will always be tempted to confuse form and matter for configuration and elements.[4] We’ve already seen the difference with the wooden table: the elements are the pieces while the matter is the wood. With the living animal the elements are often said to be the cells, and so the configuration would be the organising process of those cells.[5] For universal wholes, however, the matter of a living thing is called its body and the form of a living is called its soul.[6] Considered broadly, there are three classes of living things: plants, animals, and humans. The soul of a plant makes it vegetative, the soul of an animal makes it sentient, and the soul of a human makes it rational.[7] If we abstract away the particular soul of a living thing, then all we know is that it is living; and this matter we call a body. The lesson here is that form and matter carve up the world very differently from configuration and element.

One more example should do to get this point across: consider the case where my hand moves into your face. The motion of my hand alone is indeterminate between me attacking you, and me reaching to get something and hitting you by mistake. The form that determines which of these is the case is my intention. Together the motion (as matter) and the intention (as form) constitute my action. The configuration of my action, by contrast, would presumably pick out how I hit you with my hand, like the path my hand took through the air. This something very different from the intention of the action.

Potential wholes

Finally, potential wholes. Of the three kinds this is the most foreign to us, and it is also arguably the most fundamental. The key here is this: in both integral and universal wholes we have cases where a single act can actualise multiple potencies at once. This is clear enough in integral wholes, but it can also apply with universal wholes: an animal’s soul actualises potencies for walking, grasping, flexing, seeing, smelling, touching, and so on. Now, whenever a single act involves the actualisation of a number of potencies, we can distinguish between sub-acts of that act. If some act A is involves the actualisation of potencies P, Q, and R, then we can consider the sub-acts of A as the actualisation of P and the actualisation of Q and the actualisation of R. The potential whole is the act, and the potential parts are these sub-acts which are distinguish by extrinsic the potencies found in the elements.

Notice the difference here: the parts do not have potencies, but are just sub-acts we differentiate by reference to extrinsic potencies. Consider the water molecule again as an integral whole, so that we have a configuration of elements. Each part is the result of an element being actualised with the configuration, and so each part includes some potency inside it. The whole water molecule includes both potency (from the elements) and act (from the configuration). But now abstract away the elements so that all you’re left with is the configuration itself. This doesn’t include a potency; it is just an act. And when we sub-divide this configuration into sub-configurations (each the actualisation of a different element), these are also just acts: the configurings of the hydrogen molecules and the configuring of the oxygen molecule. Potency plays a role is distinguishing the sub-acts from one another, but the potencies are extrinsic to these sub-acts.

Something similar happens in the case of a form informing matter. For each distinct potency actualised by the form, we can discern a sub-act which is that form considered with respect to that extrinsic potency. The potential parts of a human soul are roughly the various powers it gives a human: vegetative powers like digestion, animal powers like walking and seeing, and rational powers like abstraction and judgement.[8]

So far we’ve illustrated potential wholes by reusing examples from integral and universal wholes. This is partly because we want to show the sense in which potential wholes are most fundamental, but also because it helps us gain some initial intuitions. There are other examples of potential wholes, two of which we’ll go through now. First, communities are potential wholes. This is true in general, but focus on one for now: an orchestra playing a piece of music. The playing is the result of a co-ordinated effort from all the members of the orchestra, and is a single activity of the orchestra. We can consider the sub-activities of this activity as the playing of the individual members, and these would be the potential parts of the playing of the orchestra as a whole.

Second, there are what we might call “composite actions” like faith. At its most general level, faith is thinking with assent. “Thinking” involves having intellectual confidence in something, less than certitude.[9] “Assent” picks out the mood of the thinking: that which I think I also desire. So thinking uses the intellect and assenting uses the will, but these are being used together in one and the same act, which we call faith. So then the act of faith is a potential whole with the potential parts of thinking and assenting, each distinguished by the rational faculty they are the use of.

With both integral and universal wholes we introduced technical vocabulary to capture the specific kind of act and potency at play in each case (configuration-element and form-matter). With potential wholes, however, the act in view seems to be as varied as actuality in general. As such, it seems the best we can do is distinguish between super-act and sub-act, where the super-act is the potential whole and the sub-act is the potential part. Depending on which kind of act we’re considering we’ll restrict the vocabulary, and we’ll usually drop the “super-” bit from the whole. We’ve been doing this all already: configurations and sub-configurations, activities and sub-activities, actions and sub-actions. We also sometimes spoke about the potential parts by using a proxy, as when we used powers as a proxy for sub-forms of an animal soul.

Conclusion

Aristotle discovered two kinds of whole: integral and universal. The Scholastics discovered a third, the potential whole, and extended Aristotle’s analysis of wholes in terms of predication. We saw an example of this in Aquinas, and in that case potential wholes fell between the other two kinds. With the present analysis in terms of intrinsicality there doesn’t seem to be a linear way of ordering the different kinds, although their relations are captured well in the diagram we saw earlier.

Notes

  1. One might wonder if we haven’t just pushed the question about what multiplicity is back a step, since multiplicity of things arises from multiplicity of potencies. But this misses the point since we’re not trying to give an analysis of multiplicity, but rather trying to account for the reality of multiplicity with our principles. Because Parmenides had just being and non-being he could not account for multiplicity. But because we have divided being into being-in-potency and being-in-act, we are thereby able to account for it.
  2. See Svoboda’s Thomas Aquinas on Whole and Part.
  3. Rob Koons discusses in some detail how this process interacts with the parts in his Stalwart vs. Faint-Hearted Hylomorphism. David Oderberg argues in his Synthetic Life and the Bruteness of Immanent Causation the process of life is one involving immanent causation.
  4. Even Eleonore Stump, who is a very careful expositor of Aquinas, falls into this trap. I made the same mistake in an earlier post.
  5. While it is common to refer to the elements of an organism as a cell, this is technically wrong. But the details are not particularly important to our present point.
  6. See Mike Flynn’s blogpost series In Search of Psyche (introduction, part 1, part 2, part 3, and part 4).
  7. This is a technical term: any animal we take to be rational is a human. See David Oderberg’s Can There Be a Superhuman Species? for a related discussion.
  8. We say they are “roughly” the powers, since strictly they are the vehicles of the powers. Every power is grounded in a particular intrinsic actualisation, which we call the vehicle of that power. But such technicality is not necessary here.
  9. As Aquinas said, “[Thinking] is more strictly taken for that consideration of the intellect, which is accompanied by some kind of inquiry, and which precedes the intellect’s arrival at the stage of perfection that comes with the certitude of sight.” (ST II-II, Q2, A1, corp)

The metaphysics of gender

I recently listened to this talk by John Finley titled The Metaphysics of Gender: A Thomistic Approach. Below are my notes of this. I skip the introductory remarks and follow the four-section division of the talk. Note that by “gender” here we do not mean the psychological or social construct introduced by modern feminists. Rather, by “gender” we mean the biological distinction between male and female. Some have come to refer to this as “sex” but in the introduction John notes that both terms have ambiguity and so he just picked one. By-and-large parentheses represent my own thoughts, but this is not always the case. And finally, the times for each of the sections are written next to each of their headings.

Aquinas’s position (8:56-20:30)

A man is a male human being and a woman is a female human being. Male and female are distinguished by their mode of generation: the male is that which can generate in another, while the female is that which can generate in itself. Whatever meanings man and woman could have, they need be connected to these meanings.

So, then, what is the connection between male or female on the one hand, and being human on the other? It does not affect that one has a human nature: one’s gender does not elevate or detract from one’s being a human being. Perhaps, it’s better to say that gender affects how one participates in human nature. “It might be better to say that men and women share human nature equally but differently, according to their respective generative abilities. In an analogous way, being blue-eyed and being brown-eyed pertain equally but differently to the human power of vision.” Nevertheless, gender must be a more significant personal attribute than eye-color, since it involves distinct organs, activities, and purposes. It is also more uniform than other less significant attributes, which appear more sporadically throughout the human population.

Thomas has two classifications of accidents: (1) a logical classification (in terms of genus, species, etc.) in The Disputed Questions on the Soul and (2) a metaphysical classification (as arising from form and matter) in On Being and Essence:

On the logical classification there are three sorts of accidents: proper accidents (eg. risibility in humans) result from the principles of the species and so characterise all members, inseparable accidents (eg. masculine and feminine) result from the principles of the individual through permanent causation and so characterise that member in a lasting fashion, and separable accidents (eg. sitting and walking) flow from the principles of the individual through temporary causation and so only characterise that member at particular times. The main focus here is the inseparable accidents, however it’s not clear what other examples of such accidents there are. Aquinas gives examples like eye color, bone structure, and natural temperament, but as noted above these seem less significant than gender. A question arises as to which principles of the individual (soul, or body, or both) bring these accidents about. This is addressed by his metaphysical classification in On Being and Essence.

Regarding the metaphysical classification, we note that the whole substance is the true subject of all accidents, but since humans are composed of two principles (form and matter) certain accidents flow more from form and others more from matter. Thomas describes four kinds of accident (two following from form primarily, and two following from matter primarily). First, of those following from form, rational activities — understanding and willing — occur entirely in the soul and have no share in matter (though there is a measure of dependence on the physical sense organs). “Other accidents following from form, like sensation, do have a share in matter since they properly reside in the composite substance. The soul, that is, originates powers of sensation but it can’t sense on its own.” “Moving downward, accidents following from matter will always have some relation to form since matter on its own is pure potency, uncharacterized by any feature.” So, in the third case, some accidents following from matter relate to a particular kind of form. For Aquinas, masculine and feminine are accidents that follow from matter but precisely in relation to an animal form. So when the animal dies, and the animal form is separated from the body, it is no longer gendered in a univocal way. Finally, “other accidents following from matter relate to a more general form, as one’s skin color occurs through matter’s relation to the form of some elemental mixture. The color thus remains even after the person has died.”

Combining the two accounts, Aquinas takes gender to be an inseparable accident following from one’s matter in direct relation to one’s substantial form as an animal. This helps us distinguish it from other inseparable accidents, as they would follow from one’s matter in direct relation to some form other than one’s animal substantial form. It seems that gender is the only example of this special class of inseparable accident we have, and so it is in this sense a metaphysically unique feature.

“Now, if being male or female relates necessarily to the form of an animal why does Thomas assigns gender’s origin to matter?” He gives two reasons:

First, for both Aristotle and Thomas, the male and female roles in generation are active and passive respectively. The male semen contains the formal principle of generation whereas the female seminal fluid contains the material principle, such that when the two come together a human is generated. Insofar as every act of generation is directed toward producing one’s likeness and since the male is more active is the generative act, the act naturally tends toward a male offspring, and a female results from an accidental alteration in the male semen. Since gender is determined by the manner in which the seminal matter has been affected, it is seen to follow from matter as opposed to form. Aquinas agrees that one’s reproductive power — as all powers — arise because of the soul, but the difference in gender is owed to a defect in the matter of the female (since the male, insofar as he is more active, has the more reproductive power more perfectly).

Second, for both Aristotle and Thomas since form is what makes matter to be a certain kind or species, a difference in form must result in a difference in species. Thus differences applying to individuals of the same species must be differences originating from matter.

Note that genders origination from matter does not mean that it has no bearing on the soul. “While the soul in its own right is not gendered, just as the soul on its own possesses no sensation, presumably the soul of a male can be derivatively considered a male soul and the same in the case of the female, since the soul’s identity is marked by it’s being the soul of a male or female body. One’s gender then, as following from the principles of the individual, characterizes the person as a whole.”

Brief evaluation of Thomas’s account (20:31-23:15)

Thomas’s logical classification of gender as an inseparable accident makes sense insofar as gender doesn’t apply to the species as a whole, but individual members. “Moreover, current biology’s understanding of genetic systems, chromosomal patterns, gonadal structures, and sexual organs affirms that the principles of the individual exercise permanent causation in their originating one gender or another.” In spite of this, the fact that gender seems to be in a class of its own — separate from other accidents — calls for further inquiry. And this inquiry would have to focus on Aquinas’s metaphysical account of gender arising from matter in relation to a specific form.

It’s not totally clear what it means for an accident to follow from the matter in relation to a specific form. If this is taken to mean simply that the accident flows from the principles of the individual as such, then it is well-taken since evidently, one gender is not a characteristic of the species. This would still leave open, however, which of the individual’s principles is at work here (soul, matter, or both). But Aquinas, in saying that the female gender arises from an accidental alteration of the semen, answers this second question. “That is, he holds not just that gender stems from the principles of the individual, but also that being male or female stems concretely from the side of one’s matter, rather than one’s substantial form or soul.”

Now, current biology, of course, has shown that the female reproductive abilities are not imperfect versions of the male ones. Man and woman, respectively, do not supply the active formal principle of generation and the passive material principle of generation. That a man’s production of semen and a woman’s ovulation each supply distinct elements of the offspring’s genetic material reveals that, in this capacity, the two are co-contributors to the offspring. Since man and woman do not relate generatively as perfect to imperfect it is not the case that any given act of generation seeks the male. As contemporary science shows, the male and female are equally intended at the biological level. So Thomas’s empirical reason for attributing gender to matter — the first reason I mentioned earlier — is no longer tenable.

This leaves us with the question of whether the second reason given still works. Is it true that gender must arise from the matter and not the form because the form cannot account for something that arises from the individual?

A revised account (23:16-36:42)

The aim here is to argue that the Thomistic principles suggest that gender flows more from substantial form than from matter, that is more from the soul than the body.

As both Aristotle and Aquinas saw, male and female are of a different category to black and white. The former are tied up with the essential teleos of the human being and contain the substance’s essence within their definitions, whereas the latter are not and do not. “The presence of an organ indicates a particular configuration of matter for the sake of one of the soul’s powers, which in turn flows from the essence of the soul. The soul itself arranges material structures as organs so that they might fittingly serve as means through which the soul’s various powers can operate effectively.” As Thomas says in The Disputed Questions on the Soul, “the soul constitutes diverse parts in the body even as it fits them for diverse operations.”

To unpack this we might say that like the vegetative powers the reproductive powers slowly manifest as the being matures, and as the soul actualizes and shapes the individual it constitutes these powers in particular organs within the body. Just like the sensory powers, if the soul were to leave the body so too would the generative powers. Unlike the sensory powers, however, not all humans share the same set of generative powers (instead we have something like a 50/50 split across the population).

The generative powers of man and woman should be considered, strictly speaking, co-generative, since they possess a two-fold formal object distinguished hierarchically. As “generative” they possess the same ultimate object, namely procreation of another human being. While as “co-” their proximate objects differ by way of offering distinct sexual organs and activities yet in relation with each other. The ultimate object of the co-generative powers points to the unity of nature shared by man and woman since another of the same species, whether male or female, is generated. The proximate object of the co-generative powers points to the distinction within human nature as found in either man or woman, albeit only at the level of the reproductive capacities.

Since the reproductive powers are two distinct co-generative — as opposed to one at varying levels of perfection — it seems clear that they must be accounted for by the substantial form of the individual. Of course, since the generative powers intrinsically depend on organs they would this should not be thought of as an attempt to separate the soul from the body, but rather to highlight the soul’s role in constituting the powers in the body. Thus we provisionally include gender in those accidents that stem from the soul and have a share in matter, as with the senses. In order to develop this account further, we address three objections.

The first objection is that modern biology seems to support Aquinas’s position that gender is better attributed to matter than soul. This is because modern biology teaches us that gender is intimately connected with various genetic networks, especially the chromosomal patterns XY or XX found in the zygote. But this does not so much entail that gender differentiation arises from matter primarily as show us more clearly how intimately related substantial form and matter relate to one another in the constitution of a human being. Any becoming of a substance requires appropriately disposed matter; after all, the being is generated by the actualization of potencies in the matter. But it is the resultant form (the actuality) that primarily characterizes the being the is generated.

The second objection comes from the second argument given by Aquinas above, that difference in form constitutes difference in species. Since men and women clearly share the same species, their difference must, therefore, arise from matter. Moreover, the notion of an individual brings forth — for Thomists at least — thoughts of matter insofar as it is the principle of individuation. But we must make a distinction between a universal form and a particular form. Aquinas grants that when a soul is commensurated to a particular body (that is, when they mutually limit one another so as to constitute an individual) in a sense it takes on additional characteristics, an obvious example being individuation even after separation from the body at death. It is inevitable that gender is of the form, since matter does not configure itself into particular organs (being indeterminate between any such configurations) it must be the soul that does so in and through matter “for the sake of the particular powers that work through those organs.”

The position I have argued affirms the notion that particular souls are essentially commensurated to particular bodies, but claims that within this commensuration gender begins at the level of the soul and is received into the corresponding matter accordingly designated by the genetic pattern.

As to the concern about this introducing a distinction between two species of human, we can say two things. Rather than being an additional power that future determines the essence of the individual, gender concerns the maintenance of the essence that the other powers constitute. “As oriented towards the species itself, [the generative powers] cannot in themselves constitute new species.” Second, as noted above gender is a co-generative power which differentiates it from the other powers given by the soul insofar as they are independent in some sense. They exclude each other in definition (“four-legged” excludes “winged”) or in fact (“scaled” excludes “feathered”). Gender’s nature, however, presupposes “one like itself” and so depends on and includes its contrary both in fact and in definition. Male is defined in terms of female and vice versa through the co-generative relation. The reproductive powers are not merely distinct as one sense is distinct from another, but as mutually dependent powers contributing to a single action (ie. generation). They are not to be understood as characterizing distinct species, then, but rather as integral parts of the same species considered at the reproductive level. (This is a consequence of us being social animals: humans are not wholly intelligible in terms of an individual, but require that that individual be understood in the context of some community. This reoccurs again at the higher level with powers that enable us to rationally cooperate, which are a consequence of us being political animals.)

The third objection takes issue with the description of co-generative powers. Why could we not accept that there is one generative power manifested in different ways, depending on the body to which the soul is united? This would entail that gender differentiation stems from matter as opposed to form. Note that this is much like Aquinas’s view insofar as he sees one power actualized to differing levels of perfection. Now in some sense, the objector is right, namely insofar as both generative powers have the same ultimate object. Because of this, they can be naturally grouped together, just as the various sensory powers can be naturally grouped together. But insofar as the generative powers have distinct proximate objects (their organs and activities), they can be distinguished. Interestingly, even in the woman, we see multiple generative powers in a single being: powers for generation, support, and nourishment of the offspring all of which are required for procreation (since the ultimate object of generative powers is a human and not merely a clump of flesh). Since there are really distinct generative powers, their distinction must arise from the substantial form and not the matter.

In order to affirm that a numerically single (that is, really identical) power to be differentiated only by matter, we would need to accept Aquinas’s account which, as we’ve seen, is falsified by modern biology. Otherwise, we’d need a “generic power” had by both male and female, which would need to be an abstract power or a power that includes both. But the first alternative is incoherent in Thomistic metaphysics (and even in much of modern metaphysics), and the second would involve an entire set of the person’s powers being denied and frustrated merely in virtue of them being an individual human. This “opposes Thomas’s thought and the majority of human experience.”

Being male or female, therefore, follows principally from one’s soul in relation to that soul’s correspondingly disposed matter.

Three ramifications (36:43-43:41)

The first concerns “gender’s status in relation to the person.” Gender is closely related to the person but is different from other such attributes. Other attributes (like free will, reason, soul, body, growth, and sensation) are understood when the human essence is abstracted from individuals and reproduction is like this. But it differs that when considered in itself the essence includes both male and female, but when it comes to exist there is a split into the co-generative powers. “The human essence in itself includes male and female; only a consideration of that essence as actually existent entails male or female.”

Turn, then, back to the metaphysical classification given above. We’ve seen that reproduction, like sensation, falls into the second category of those accidents which follow from form that have a share in matter. But given the differences between reproduction and sensation, there must be a real distinction within this category. The difference is between those accidents which flow from the nature itself, and those accidents which flow from the nature as it exists in this or that individual.

And in this sense, one’s gender is not as close to one’s fundamental humanity as are the other powers of the soul. Being man or woman — you might say — is more proper to the human individual than to the human individual. As Thomas would put it being gendered at all is proper to human nature, but being a man or a woman is proper to this instance of human nature, this soul and this matter.

All of the other accidents that flow primarily from the soul characterize the whole species, and so we call them the proper accidents (or properties), like sensation and risibility. But gender differs from the other individual accidents insofar as it characterizes one’s structure, abilities, and purpose. Insofar as the gender so characterises an individual we might say that it is “the primary attribute of the existing person”, not as something that constitutes the person (since this is given by the soul and matter), but as that which is most truly proper to individual person (so, in this sense, it’s like a property at the individual level).

The second concerns “gender’s status to the human essence or nature.” Man and woman are not distinct species of human nature, but nor are they merely individuals of human nature. It is good, therefore, to introduce some notions that can describe the genders with regards to their human nature. Man and woman are principles of the nature, they’re parts of it, they are ways of it existing or ways of a soul incarnating in a body, and they are relational as mutually fulfilling complements. “Thomas compares male and female to odd and even in the numerical realm.” But even this misses out the relational nature of humanity.

The third concerns “gender in its specifically human meaning as the intersection of eros and generation.” A slight modification of the Aristotelian definition of male and female is, “the male is what co-generates in another, the female is what co-generates in itself.” There’s nothing peculiar to humans here; we are gendered because we are animal. But human gender has richer meaning than non-human gender insofar as the procreative activity is integrally marked by rational choice.

By nature the generative act is a human act, and not just the act of a human. Thus, what is distinctively human in gender comes to light most manifestly in the “co-” dimension of the co-generative relationship to the extent that deliberation, choice, and love are integral moments within human sexual activity, which thus transcends merely instinctual limitations.

The distinctive human dimension of all this is one of the reasons that it is considered problematic if the human generative act occurs without proper mutual consent, since it “presents a co-generative act with the co- aspect as distinctively human. Since the entire act is co-generative, if one aspect lacks distinctively human structure, so does the whole.” That the co- aspect is human and therefore higher than mere biological generation, it elevates the generative aspect which is primarily animal. The biological tendency becomes subsumed into a conscious intention in love.

Further, as Thomas points out, generating another like oneself in the case of a human involves continued rational and affective dimensions beyond those of the sexual sphere, since the mature human only comes to be after an extensive period of support, nourishment, training, education, and love.

Beyond metaphysics (43:42-46:55)

Here we comment on some things beyond the metaphysical question but which depends on the metaphysical answer, namely issues in the psychological, social, and ethical realms. There are two putative objections that might be raised from modern concerns against the claim that gender stems from the soul. First is the issue of sex-reassignment surgery, second is the reality of intersex persons.

With regards to the first, if in fact sex-reassignment surgery actually changed one’s sex/gender then it would constitute a concern. However, even if such surgery can change the outward appearance of an organ it nonetheless leaves the patient sterile. So rather than say that one’s gender has changed it is more accurate to say that it has to some degree been lost (or blocked).

With regards to the second, just as with sensation defects and abnormalities are possible so too with gender. This arises from the fact that gender (like the senses) arises from the soul working in and through matter. “Aside from the assistance of medical technologies in such cases, it’s crucial to recall that one’s gender, though integral to the person, is neither the defining nor the most important aspect of the person.” To quote Thomas on the place of gender in human life:

Among animals there is a vital activity nobler than generation to which their life is principally directed. Therefore the masculine sex is not in continual union with the feminine in perfect animals, but only at the time of coition, so that we may consider that through coition male and female are made one. But [humans are] further ordered to a nobler vital activity, which is to understand. Therefore there had to be a greater reason for the distinction of these two forces in [them], so that the female should be produced separately from the male and yet they might be fleshly joined as one for the work of generation.

In commenting on this, John closes with:

The ultimate telos of a human being involving the flourishing of a life suffused with knowledge and love reminds us that relationality and fruitfulness occur in realms higher than the physical. If, with Aristophanes in the Symposium, one were tempted to picture the human being simply as a longing half, the passage just quoted offers a larger view. In his own way, Thomas calls to mind Socrates’ and Diotima’s assent to the beautiful.

Common goods

I had originally intended to tie up the thoughts begun in previous posts on natural and moral goodnesssubstantial activitiesbasic goods, and virtual existence, but it has since occurred to me that this would be too ambitious for a single blog post. So, I’ll attempt to approach the topic in installments as I find the time. Those previous discussions are important for the direction I want to go, since we will be using much of the terminology and conclusions there. As such I strongly recommend reading them if you haven’t done so, and perhaps even rereading them if you haven’t done so for a while. In this post we will be introducing the notion of common goods, which will be much of our focus hereon out.

In general something is good to the extent that it realises its end. This is what Aquinas meant when he said that the “good has the nature of an end” (ST Q94 A2 corp). We’re most familiar with ends as intended by rational beings, but these are just a small number of the ends we’re considering. Non-rational animals act for particular ends too, of course. Beyond this the development process of living things is directed toward the end of healthy adulthood. And we’ve seen every substance is in some sense directed toward its characteristic behaviours given by its nature. (Besides the posts linked above, I also discussed this in section 2.2 here.)

Since goods and ends are so linked, a common good is therefore the realisation of a common end. And since common ends belong to communities or societies, it follows that common goods are the goods of these communities. But what is a community? It turns out the answer isn’t a simple matter: there are alternatives and each putative answer gives a slightly different notion of what the commonness of common goods involves. For the remainder of this post we will be unpacking all of this, with the help of our foregoing discussions.

In our discussion on virtual existence we outlined the three ways parts relate to their wholes: (1) parts which are actually present in their aggregate, (2) parts considered in themselves which are virtually present in their substance, and (3) parts considered as parts which are actually present in their substance (in the sense that they derive their being from the substance itself, and this substance is actually present). In (1) the parts each maintain their individual ends, and the end of the aggregate is merely the sum of the ends of its parts. Substances, on the other hand, have ends intrinsic to themselves. In (2) the end of the substance “overrides” the ends the parts would otherwise have in isolation, and in (3) the parts have the same end as the substance because they share in its being and nature.

In our discussion on substantial and aggregate activities, we noted that there is an analogous sense in which activities can be understood as substances or aggregates. And everything we’ve said about wholes equally applies to activities. For instance, we can also speak of virtual existence in the context of substantial activities. We introduce the idea by applying our hylomorphic analysis of virtual existence to a concrete example. Imagine we’re considering an orchestra playing a piece of music, and imagine we zoom in on one of the violinist’s playing. Recall that an action can be analysed hylomorphically, with the matter being the movement and the form being the intention. And recall that the virtual existence of parts in themselves involves retaining the matter while “filling in” (through intellectual activity) a form the part would have in isolation from the whole. What do we get in the case of our imagined example? Well, the intention of the violinist considered as a part of the orchestra is to play with piece together with the rest of the orchestra members. An intention that we might fill in would be the violinist practicing the piece by themselves. In this way actions can exist virtually in the substantial activities they belong to.

Now, are communities to be understood as wholes, or activities, or some combination of the two? It doesn’t seem correct to identify the community with the activity because the parts of the activity are the individual actions whereas the parts of the community are the individuals themselves. At the same time it seems mistaken to completely divorce a community from its activity. The same group of humans could be an orchestra and a soccer team, for instance, but surely the orchestra is distinct from the soccer team? Put another when we consider the members of the orchestra we consider them as musicians, but when we consider the members of the soccer team we consider them as soccer players.

As such, it seems to me that we should consider communities in terms of both wholes and activities. Again, hylomorphism gives us a natural way of doing so: when considering a group of individuals it is their activity that determines what community they are. That is, the group is an otherwise indeterminate substratum and the activity is what determines them to being this or that community. That is, the group is the matter and the activity is the form of the community.

So communities represent a third category which is a hylomorphic combination of the first two. And just as there are three ways for parts to relate to their wholes, and three analogous ways for actions to relate to their activities, so there are three analogous ways for individuals to relate to their communities. How should we understand these in terms of the wholes and activities that make up the communities? With regards to matter (the whole), it seems intuitive that the underlying whole of a community will always be some kind of aggregate of individuals, each of which will be substances in their own right. With regards to form (the activity) we have three options: (1) an aggregate activity in which the individual actions actually exist, (2) a substantial activity in which the individual actions virtually exist, and (3) a substantial activity in which the individual actions actually exist. Each of these would translate to a different kind of community. In (1) the community is merely the aggregate of the individuals, and its end is the sum of the disparate ends of these individuals. In this case, the only things that can truly be called a substance are the individual substances. In (2) we see the reverse of this: the individuals are the parts of the community considered in themselves, and as such their individual ends will be “overridden” by the ends of the substantial community. (3) represents somewhat of a middle ground, and will be of much interest to us. Here the individuals are parts of the substantial community, but not in such a way that they have their ends overridden. This is because their actions are all directed toward the common end of the community.

At least two of these views already have names: (1) is called atomic individualism and (2) is called organic collectivism. Matthew O’Brien and Robert Koons introduce them as follows:

In attending to social nature, the ethically minded metaphysician must avoid both the Scylla of atomistic individualism and the Charybdis of organic collectivism. The attempt to navigate successfully the narrow strait between them has been a recurring theme in Western metaphysics, from the time of Plato to the present. The organic collectivist holds that the most fundamentally real things (the “substances”) are complete and sovereign human societies; on this view, typified by Jean Jacques Rousseau, for example, individual human beings are merely cells of the social organism, with a nature, an identity, and an existence wholly dependent on that of the whole. In contrast, the atomistic individualist, such as Ayn Rand, holds that individual human beings are the substances, with societies as mere aggregations or “heaps” (to use Aristotle’s expression)….

For organic, collectivist pictures of human life, the good of individual human beings carries no weight, since, strictly speaking, there is no such thing as an individual: the good of the society as a whole is everything. For atomic individualists, the ‘common good’ consists of nothing but the sum of measures of the individual welfare of participants.

Their article doesn’t work from the exactly same distinctions we’ve made, but it’s clear from the quoted passage that for the organic collectivist the community’s being a substance in some way “overrides” the individuals that are part of it. That is, the community is a substance at the expense of the individuals, which corresponds with what we’ve said of (2). I don’t know of a name for (3), so for the sake our discussion here we will refer to it as unitivism.

So we have outlined the three views of (1) atomic individualism, (2) organic collectivism, and (3) untivism. Each gives us a different picture of what makes a community, as well as a different understanding of the commonness of common goods. It is this that we must unpack to adequately answer the question at hand.

Let’s start with atomic individualism. On this view the community is merely the sum of its individuals, and therefore so is its end, and thus the common good is also understood as an aggregate of individual goods. A good is common, in this sense, by virtue of being predicated of the many individuals of in the community. So, for instance, health or wealth would be common goods since it is good for each individual to be healthy and sufficiently wealthy. And the health of the community, for instance, would be the aggregate of the health of the individuals. Common goods, in this sense, are contrasted with singular goods in that to be common to be predicated of many whereas to be singular is to be predicated of one. So, we speak of the health of the community as opposed to the health of this or that individual.

Next consider organic collectivism. On this view the community is a substance at the expense of the individuals. Since it is a substance it has its own end, and this is what the common good would be. Since the individuals exist only virtually in the community, this common good overrides their individual goods. An example comes from some socialist economic theories, where individuals are to give up their individual right to private property in order to be part of the political community. So we find that common goods, in this sense, are contrasted with individual goods. The common good, in our example, being the common property which is contrary to the private property of individuals, or what we might call “individual property”.

Finally there’s unitivism. The unitivist agrees with organic collectivist that the community is a kind of substance, but disagrees that this comes in such a way as to override the individuals. We achieve this by noting that the realisation of the common end toward which all the members work together is a good for each member, and it is on account of their shared intention toward this end that they are considered a substantial community in the first place. Moreover the unitivist agrees with the atomic individualist that the goods of the community are the goods of the individuals, but disagrees that these goods are merely shared by virtue of predication and aggregation. We achieve this by noting that the common end is numerically the same for all the individuals, and its realisation is a single good shared by the individuals of the community without thereby being diminished. Consider, for instance, that the piece played by the orchestra is one and the same piece played by each of the musicians, a victory in war is one and the same victory for the entire nation, and so on. To use some Thomistic jargon the common good is a universal cause not a universal predicate. The common good, in this sense, is contrasted with private goods in that to be common is to be shareable with thereby being diminished and to be private is either to be unshareable or always diminished when shared.

Perhaps we should spend some more time unpacking this distinction between common and private goods. First some examples. We mentioned the playing of the piece for the orchestra and the victory in war for the winning nation are both common goods. Other examples are manifold, so long as we can identify the aggregate wholes engaging in substantial activities for common ends: victory in a sports game is a common good for the winning team, financial success is a common good for many companies, the picking up of a car by two friends is a common good for them. A previously mentioned example of a private good was food, for “if there is a loaf of bread between me and someone else, the more the I eat the less there is for the other person to eat.” Two other examples of private goods would be the two goods listed as common by the atomic individualist: health and wealth. While many individuals have health (on account of which it is a common predicate), they do not all share in one and the same health. Wealth is more or less a generalisation of food, in that the more money I give you the less I have for myself. Of course, private property would also be a private good.

Second, we note that in most (if not all) communities there will be certain private goods the members need in order to participate in and enjoy the common goods of that community. This often involves some form of equipment and training, but can also include other things. We will have cause to speak about this in more in later posts. We note this here because it reminds us that while common goods and private goods are contraries conceptually, they needn’t be (and often aren’t) contraries in practice.

Third, what we mean by activity should be construed quite broadly so as to apply to every kind of community we might consider. Indeed, once we do this we begin to see hierarchies of communities form. For instance, a soccer team participates in a soccer game, which itself is part of a larger tournament, which is run by the local soccer league, which is part of the national soccer league. The soccer team’s activity is also more than this or that game, but rather includes all their games as well as their practicing, recruiting, purchasing of equipment, and so on. The hierarchy of communities entails that when communities are parts of bigger ones, they can have private goods themselves. For example, playing a soccer game is a common good for both teams, but victory is private to one of the teams. That same victory, however, is common to the members of the winning team. So whether a good should be characterised as common or private depends on the community and individuals in focus.

Fourth, an important qualification: while common goods can be shared without thereby being diminished it doesn’t follow that sharing always leaves them undiminished. For instance, orchestras are limited in their size because once they get too big they become unmanageable. The same goes for political communities and friendships and presumably any community. Furthermore, including bad musicians in an orchestra might also diminish the end insofar as those musicians get in the way of the orchestra performing well. But in these cases it is not the sharing per se that is diminishing the good, but rather the sharing with too many people or sharing with bad musicians. With private goods, no matter how you share you will always diminish your ends.

Now, all three accounts of common goods can and do occur in reality. Of the three, however, it seems that the unitivist’s notion is most relevant to the study of the good of humans in social or political contexts. That we seek to study human goods means we are not primarily interested in goods that by their very nature occur at the expense of the human individuals. And that we seek to study human goods in social and political contexts means we are not primarily interested in goods that are mere aggregations of individual goods.

Uninstantiatables in Aristotelian Mathematics

Any successful Aristotelian foundations of mathematics needs to account for mathematical objects that are uninstantiated and even uninstantiatable. Examples include (1) positive whole (or “natural”) numbers larger than the number of objects in reality, (2) negative numbers, and (3) infinities.

Uninstantiated natural numbers

As the Aristotelian sees things, we abstract quantity and structure from reality, isolate certain aspects of these (which we call axioms), and extend these abstracted notions beyond our experience. Call these three stages abstraction, isolation, and extension respectively. Even though we can technically distinguish between isolation and extension, in practice these two steps occur together in the same cognitive action. We’ll use the term synthesis to refer to the activity involving isolation and extension. These activities of abstraction and synthesis are not unique to mathematics: we use them all the time. Once we have an concept of a horse and the concept of blackness, for instance, we can consider the combination of these two concepts without having ever seen a black horse. True, the Aristotelian says that “whatever is in the intellect was first in the senses”, but this mustn’t be taken to mean that a concept can exist in the intellect only if it was sensed. Rather it should be taken to mean that sensation provides the raw data from which concepts are abstracted. This is consist with some concepts being synthesized from others.

Once we understand this, then, the problem of uninstantiated whole numbers seems to disappear. Initially we come to see the concept of quantity by considering the relation from an aggregate to a unit. For instance, we consider the relation between a specific aggregate of apples and the unit apple. If we have six apples and six oranges, then the aggregate of apples is related to the unit apple in precisely the same way that the aggregate of oranges is related to the unit orange.[1] It is on account of this that we say that the two aggregates are of the same size. We can label all the various aggregate sizes: 1, 2, 3, 4, 5, … We can also see that all aggregates of size 3 contain aggregates of size 2, all aggregates of size 2 contain aggregates of size 1, and so on. Thus we come to see that there is an ordering amoung these numbers. We can also see that an aggregate of size 3 together with an aggregate of size 2 makes an aggregate of size 5, an 2 aggregates of size 3 together make an aggregate of size 6. Thus we come to understand addition and multiplication, and similarly with subtraction and division (restricting ourselves to just the natural numbers for the time being).

Depending on which mathematician you talk to, 0 will sometimes be considered a natural number and other times not. Typically we will use whatever is convenient at the time. We could get 0 by considering an empty aggregate’s relation to any unit, or by considering a non-empty aggregate’s relation to a unit not contained in that unit (the relation of 6 apples to the unit orange).

At this point we will have experienced a number of aggregates, but there will inevitably be aggregates of sizes that are impossible for us to experience (either because of cognitive limitations or limitations on the number of things in reality). As we saw earlier, however, this does not stop us from having concepts of such aggregates. Through a (usually complicated and messy) combination of abstraction and synthesis we can come to consider any and all natural numbers.

Negative numbers

What about negative numbers? At this point we move from talking about natural numbers to talking about integers, which are whole numbers that are either positive or negative or zero. We might be tempted to try and extend our work above to negative numbers in straightforward way. After all, surely all natural numbers are also integers? Well, kind of.

We said above that natural numbers are relations between aggregates and units. Integers, on the other hand, are relations of difference between two aggregates. Let’s return to our apples and oranges. Say we have 10 apples and 6 oranges. One of the relations between these two aggregates is that if I take away 4 apples from the former, then I will have two aggregates of the same size. More precisely, there will be in a one-to-one correspondence between apples and oranges such that every fruit is matched to some other fruit. This same relation holds between an aggregate of 11 apples and 7 oranges, 9 apples and 5 oranges, and so on. This relation (or any relation co-extensive with it) is the negative integer 4. Now imagine I had it the other way around: 6 apples and 10 oranges, 8 apples and 12 oranges, and so on. These are related in a way inverse to negative 4, since now in order to make the former equal size to the latter we’d need to add 4 apples. This relation is the positive integer 4.

This parallels what we do when constructing the integers out of the natural numbers in first year mathematics courses. Starting with the Peano axioms we get the natural numbers. Then we build the integers up from pairs of natural numbers, where the pair (a,b) intuitively represents the difference between a and b.[2]

Just as we came to understand ordering, addition, subtraction, multiplication, and division with the natural numbers, so we can with the integers. Assume you have two integers x and y. As we have seen, each integer is a relation between two natural numbers, so let x be the relation from a to b andy be the relation from c to d (where a, b, c, and d are all natural numbers), written as x = (a, b) and y = (c, d) respectively. Again, as we have seen, an integer can be a relation between more than one pair of natural numbers, as when the integer -4 holds between 5 and 1, 6 and 2, 7 and 3, and so on. Using this fact we can align a and c, by which I mean the following: because of how the natural numbers are ordered either a > c, a < c, or a = c. If a > c then a – c is a natural number and y = (c + a – c, d + a – c) = (a, d + a -c). If a < c then we do this the other way and get x = (a + c – a, b + c – a) = (c, b + c – a). And finally, if a = c then we needn’t change anything. At the end of this alignment we will have three variables e, f, and g such that x = (e, f) and y = (e, g). Given this alignment, we say that the ordering between x and yis the same as the ordering between f and g. The intuition behind this is as follows: if both x and yrepresent adding or removing a certain amount from an aggregate of size e, then the ordering of the two integers is the same as the ordering between these two results.

Next consider addition. Once again assume we have two integers x = (a, b) and y = (c, d). This time, however, align b and c to give us x = (f, e) and y = (e, g). Then x + y = (f, g). The intuition here is that the addition of two integers is the same as applying the one to the result of the other.

I will leave subtraction, multiplication, and division as an exercise to the reader. Each time you will extend the respective operation from the natural numbers. There is, however, a new operation that arises with integers which we might call “additive inversion”: a is the additive inverse of b if and only ifa = -b. This is fairly simple to get using the notion of relations: for any integer x, x = (a, b) if and only if -x = (b, a).

In summary then, integers are understood as relation of differences between aggregates, and so negative numbers do not pose much of a problem for the Aristotelian. As before, through a combination of abstraction and synthesis we can come to consider any and all integers, even those we haven’t (or couldn’t have) experienced.

Infinities

As you might expect, when we start talking about infinites we need to get more abstract and precise in our approach. One of the hallmarks of modern mathematics is that we seek a universal foundation for the things we study. Often this is some form of set theory, but in the past century we’ve also seen that categories, topoi, types, and others can serve as a foundation equally as well. For any of these foundations, the Aristotelian can do something similar to what we were doing above for numbers. For the sake of simplicity here we’ll just use sets as our foundation, and not worry too much about their details. I will also treat 0 as a natural number here, which is an inconsequential philosophically but helps with presentation. (If you’d prefer to not think of 0 as a natural number, then you can assume we’re talking about non-negative integers.)

We’ve previously explained that when the mathematician speaks of “defining” things in terms of sets, what he really does is establish what we called a “correspondence of aspect using analogy”. This involves “encoding” those things in terms of sets such that the relevant aspects of the things are captured from the perspective of the set. So, for instance, say we wanted to study ordering amoung the natural numbers. We can do this from the perspective of sets by considering the following “definition”:

  1. Let 0 be defined as ∅, the empty set.
  2. Let any natural number n be defined as {0, 1, 2, 3, … n-1}, the set of all previously defined natural numbers.

When writing this definition out verbosely, we’ll get the following:

  • 0 = ∅
  • 1 = {0} = {∅}
  • 2 = {0, 1} = {∅, {∅}}
  • 3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}}

From this perspective, one number is less than another number when the former is contained in the latter. That is, “1 < 20” is the same as saying that “1 ∈ 20”. This way, we can study the ordering amoung the natural numbers quite easily from the perspective of sets.

Notice that this definition only captures those aspects we want to study about numbers. If we wanted to study a different aspect, we might need a different set-theoretic definition of numbers. On the other hand, two different definitions might be equivalent for the purposes of studying a given aspect. Consider another putative set-theoretic definition of natural numbers:

  1. Let 0 be defined as ∅, the empty set.
  2. Let any natural number n be defined as {*n-1*}.

So, verbosely, this looks as follows:

  • 0 = ∅
  • 1 = {∅}
  • 2 = {{∅}}
  • 3 = {{{∅}}}

Using this definition it would be a lot more difficult to define what “1 < 20” means in terms of sets, but it would be equally as easy to define what “19 is immediately before 20” means as the first definition (namely, “19 ∈ 20”).

The point of all of this, for our purposes, is that not all definitions are equal, and it is this very fact that we exploit when studying infinities. We will focus on two “types” of infinity: cardinal infinities and ordinal infinites. In case you didn’t know there are an infinite number of each of these infinites. (Just let that sink in.)

Now natural numbers have a bunch of aspects, and we study different infinites by focusing on one of these to the exclusion of the others. This restriction effectively enables us to go beyond the finiteness of numbers. Depending on which restriction we make, we get a different type of infinity.

The aspects of numbers include quantity, matching, and ordering. Now both matching and ordering are more fundamental than quantity. This video gives a good explanation of why matching is more fundamental, but basically the idea is that I can know facts about matching or ordering without knowing the quantities involved. For instance, I can know that there are as many people as there are chairs in the room without knowing how many there are of either, and I can know that you finished the race before me without knowing our respective places.

Cardinal infinities

When we choose to focus on the matching aspect of numbers we study cardinal infinities. These are the infinites marked by the Hebrew letter ℵ (aleph). If we have two sets X and Y, there are three possibilities for matching:

  1. We can pair elements of X and Y such that every element in X is paired with exactly one element in Y, and there are no elements in Y left over. For finite sets this occurs when the two sets are the same size.
  2. No matter how we pair the elements one-to-one, there will always be some elements in Y left over. For finite sets this occurs when Y is bigger than X.
  3. No matter how we pair the elements one-to-one, we will never be able to pair every element in X. For finite sets this occurs when X is bigger than Y.

If we just focus on matching we can talk of the “size” of infinite sets, in terms similar to those just listed, but we must avoid thinking that we’ll get exactly the same kinds of results as in the finite cases. In finite cases sizes link to quantities, and it is exactly this link that we remove in order to study infinities. For instance, we can match each natural number to an even number such than none are left over, and so there are “as many” natural numbers as there are even numbers. The cardinal infinites represent the various “infinite sizes” that we could have. ℵ0 is the “size” of the natural numbers and any set for which we can give pair with the natural numbers with no left overs on either side. Thus, ℵ0 is also the size of the even numbers. When speaking precisely, we might say that infinite sets don’t have “size”, but rather cardinality. Cardinality is a notion that captures “matchability” or “pairability”. In finite cases, size and cardinality are the same. Of course, we rarely speak so precisely, and happily use the words interchangeably for infinite cases too.

An early result in set theory from Georg Cantor is that for any set (finite or infinite), the set of all subsets of that set will always be a bigger cardinality than that original set. This means that there are bigger infinities than ℵ0. One case he proved in particular was that no matter how you match up the natural numbers with the real numbers (points on the continuum, or numbers that can be represented with decimal expansions), there will always be some real numbers left over. So if we have a set of cardinality ℵ0, we say that the cardinality of the set of all subsets of that set is ℵ1, and the cardinality of the set of all subsets of that set is ℵ2, and so on.

Notice how the Aristotelian has no problems with any of this, for all we’ve done is the same thing we’ve been doing all along: abstraction and synthesis. In this case we’ve abstracted the notion of matching and synthesised the general notion of cardinality.

Ordinal infinities

We do something similar with ordinal infinites, which focus on the aspect of order. Imagine we went with the first set-theoretic definition of natural numbers given above. What number would set of all natural numbers represent? Presumably none of them, since no natural number is such that all natural numbers is less than it. But from the perspective of order, it would represent what we’d informally take to be the infinite-th position in a list. Just as before we have a general notion of ordinalwhich, when finite, agrees with the usual meaning of position or index, but which can also be used of infinite positions. And just as before we have a specific letter for ordinal infinities: the Greek symbol ω (omega). The first ordinal infinity is ω0, and using our first set-theoretic definition we have that ω0 = {0, 1, 2, 3, …}.

At this point we can see an interesting difference between the two different set-theoretic definitions we gave above: only the former is capable of capturing ω0. We can understand this from two perspectives. Formally, from a set-theoretic perspective the axiom of foundation prohibits infinitely nested sets, and this is exactly what we’d need if we were to give the definition of ω0 on the second account. Informally, from an intuitive perspective because ω0 is the infinite-th position there cannot be a natural number that is immediately before it. But this second definition effectively encodes the natural numbers in terms of the natural number immediately before them (n is defined solely in terms of n-1).

For the Aristotelian, this serves to show that what we can synthesise depends on how we abstract.

Now, just like the cardinals, there is more than one ordinal infinity. Unlike the cardinals, the next ordinal after ω0 is ω0+1 = {0, 1, 2, 3, …, ω0}.[3] Then it’s ω0+2, ω0+3, …, ω1 (=ω0+ω0), ω1+1, and so on.

Again the Aristotelian has no problems with any of this. In this case we’ve abstracted the notion of order and synthesised the general notion of ordinality.

Notes

  1. Readers will note that this establishes an analogy of proper proportionality of the form “apple aggregate : apple :: orange aggregate : orange”.
  2. We later take equivalence classes of these pairs, which corresponds to the idea that the same difference relation that holds between 6 and 10 also holds between 7 and 11, 8 and 12, and so on.
  3. With the cardinals, ℵ0+1=ℵ0. For instance, if we have some set {a, 0, 1, 2, 3, 4, …} which is cardinality ℵ0+1, then we can create a paring from {0, 1, 2, 3, 4, …} to it as follows: 0 → a, 1 → 0, 2 → 1, 3 → 2, … Thus, given how cardinals are defined, {a, 0, 1, 2, 3, 4, …} also has cardinality ℵ0.

How Aristotle starts the Nicomachean Ethics

In the opening passage of the Nicomachean Ethics Aristotle seeks to pick out the specific subject of his study for the remainder of the book. His discussion is often misunderstood, but a good understanding of it will serve us well in understanding the study of ethics. We will consider the passage bit by bit with comments and clarifications as we go along, doing our best to read it according to the principle of charity.

The good has the nature of an end

Every art and every inquiry, and similarly every action and choice, is thought to aim at some good; and for this reason the good has rightly been declared to be that at which all things aim. (emphasis added)

Contrary to what some people think, Aristotle is not committing a quantifier shift fallacy here. Rather, he’s picking out some determinable, the good, which is common to all things done for some end. Let’s unpack this.

In general, something is less determinate (and therefore more indeterminate) if it is vaguer or less specific. So, for instance, red is less determinate that scarlet. Furthermore, determinateness comes in degrees: red is less determinate than scarlet, and coloured is less determinate than red. We use the term “determinable” to refer to some partially indeterminate feature which can be determined in some way. So, coloured is a determinable which red determines and red is a determinable which scarlet determines.

When two things resemble one another it is on account of them sharing some determinable feature which they each determine in some way: a scarlet thing resembles a crimson thing in that they are both red (that is, they share the determinable red), and both resemble a green thing in that they are all coloured things (that is, they share the determinable coloured). Just as determination comes in degrees, so too does resemblance: the scarlet and crimson things resemble each other at more levels of determination that the scarlet and green things. Speaking discretely, the scarlet and crimson things resemble each other as red and as coloured, whereas the scarlet and green thing only resemble each other as coloured.

In this opening passage Aristotle seeks to narrow the focus of his study by picking out the determinable that all desired things share, according to which they resemble each other as desired or as aimed at in some activity. He notes that whenever we desire or aim at something it is because of some good in it, and therefore the good is rightly declared to be this determinable he’s looking for. Now, just as what makes something one colour as opposed to another will depend on the particular way in which the determinable coloured has been determined, so too the reason why this or that thing is desirable or aimed at will depend on the particular thing in view. Good ice-cream and good vacations are desirable for different reasons, and so determinethe good in different ways, but they resemble each other in that they are pursued.

Later treatments would make explicit a question which, as far as I can tell, Aristotle leaves implicit or thinks obvious: is something good because I desire it, or do I desire it because it’s good? It cannot be the former, since I often desire things I later realise were in fact bad for me.

For haven’t we all had the experience of wanting something which we ourselves then admitted was not good? I wanted that last drink at the party, but afterwards I admit that it was not good for me. I wanted to drive 100 mph down the winding road, but later, on my hospital bed, I admit that it was not good. If wanting something made it good, then my wanting the last drink would have made it good for me. (Edmund Waldstein, The Good, the Highest Good, and the Common Good, thesis 2)

It must be, therefore, that I desire something because it seems good to me in some way. That is, the goodness I perceive causes the desire in me. Of course this perception may be incorrect, but the point remains that it because of the good I perceive in something (correctly or incorrectly) that attracts me to it as something worth pursuing.

Returning to our passage, Aristotle is noting here that in general the good “has the nature of an end” (cf. ST I-II Q9 A1 corp). An end is “that for the sake of which something is done” and a means is “that which is done for the sake of something”. These are complementary notions such that whenever we have one we also have the other. We see Aristotle make this same connection in the Physics where he lays out his four kinds of causes. In the passage he identifies ends as what later would be called final causes:

Then there are things which are causes in the sense that they are the ends of the other things, and are the good for which they are done. Without quibbling about whether it is an actual good or an apparent good, that at which other things are aimed — that is, their end — tends to be what is best. (Aristotle, Physics II.3 195a23-25)

Note that the Aristotelian “cause” is much broader than the modern’s “cause”. The modern usage most closely approximates the Aristotelian efficient cause. For those unfamiliar with the Aristotelian usage, perhaps “four kinds of explanations” is more helpful for conveying what he’s getting at.

Note also that the phrase “tends to be what is best” is just there to explain how ends and goods relate: something is good to the extent that it fulfills its end, and so to achieve its end in full is best (that is, most good). This is all he means.[1]

In summary, then, this first passage involves distilling this determinable the good, which is what accounts for the resemblance between things as desired in some activity. It picks out something as an end or that for the sake of which the activity is done.

Ends are better than means

But a certain difference is found among ends; some are activities, others are products apart from the activities that produce them. Where there are ends apart from the actions, it is the nature of the product to be better than the activities.

Now, as there are many actions, arts, and sciences, their ends also are many; the end of the medical art is health, that of shipbuilding a vessel, that of strategy victory, that of economics wealth. But where such arts fall under a single capacity — as bridle-making and the other arts concerned with the equipment of horses fall under the art of riding, and this and every military action under strategy, in the same way other arts fall under yet others — in all of these the ends of master arts are to be preferred to all the subordinate ends; for it is for the sake of the former that the latter are pursued. It makes no difference whether the activities themselves are then ends of the actions, or something else apart from the activities, as in the case of the sciences just mentioned.

Aristotle here makes distinctions regarding how activities and ends relate to one another. First, either the end and activity are the same or they are distinct. An orchestra playing a piece is an example of the former, since the performance is both the end and the activity. A carpenter making a chair is an example of the latter, since there’s a real distinction between the production of the chair (activity) and the chair (end). We must note that by “product” we don’t only mean physical objects that result from some activity, as the chair results from the carpentry. Rather, we mean any outcome which is distinct from the activity that brings it about. So, winning a sports game is the end and product pursued when playing the game.

Second, an activity can be made up of other activities. In this case, we might say that the subsuming activity is superordinate (or “master”), and the subsumed activities are subordinate. Subordinate activities are parts of superordinate activities.

Both distinctions show us different ways in which ends and means might arise and relate. Sometimes the end and the means are really the same thing, as when an activity is the end we desire. In this case, the distinction we impose is merely conceptual. Other times they are really distinct, as when an activity produces something external to it. Moreover, when one activity is subordinate to another the former is done for the sake of the latter, and so the former relates to the latter as a means to an end.

Twice in this passage he picks cases where there is a real distinction between ends and means (the product and activity, and the superordinate and subordinate activities), and notes that the end is always better than the means. This is true because “it is for the sake of the former that the latter are pursued”. That is, the end is more truly the thing desired, whereas the means is desired only in a derivative way. The end is desired through the means.

To make this more precise we need distinguish between the thing itself on the one hand and the thing desired on the other. Now, a thing is desired to the extent that it — and not something else — fulfills that desire, and so the desire for the thing itself is proportional to how closely it relates to the thing desired.[2] Thus, if just this or that feature of the thing is desired, then that feature is more desired that the thing itself. For instance, if I buy a torch because I want the lightbulb inside of it, then I desire the lightbulb more than I desire the torch. Conversely, if the thing itself is just one feature of what is desired, then the greater whole will be more desired than the thing itself. For instance, if I desire a violin performance because I desire an orchestra performance, then I desire the orchestra performance more than the violin performance.

Applying this to the cases Aristotle mentions, we can see why his claims are true. First, there’s the case when an activity is desired for the sake of some product really distinct from it. Here the activity is desired because of one of its features, namely the ability to bring about the desired product, and so the acquisition of the product is desired more than the activity itself. Second, there’s case of a subordinate activity being desired for the sake of some superordinate activity. Here the subordinate activity is desired because it is part of the superordinate activity, and so the superordinate activity desired more than the subordinate one.

We might also arrive at this conclusion from a slightly different angle. We’ve seen that goodness has the nature of an end. Thus to be better (that is, more good) is to be more of an end. Now, something is more of an end if it is closer to some final or ultimate end, as riding is closer to strategy than bridle-making.[3] But if A is a means to B, then B is closer to some final end, and is therefore more of an end, and therefore better.

At this point we must make two clarifications.

First, people sometimes mistakenly interpret Aristotle as assuming that bridle-making is only ever done for the sake of strategy. The passage does not require that we interpret him this way, and given his historical context he surely knew that bridle-making could also be done for the sake of other things, like recreation or sport. What he’s doing here is picking one of these instances as a concrete example of how activities might relate to one another such that some subsume others. If you prefer you could use an example where bridle-making is subsumed under some activity other than strategy, but the point would remain the same.

Second, to say that that ends are always better than means is not to say that things that are ends are always better than things that are means. Rather, we’re claiming that things considered as ends are always better than things considered as means. Part of the import of the first passage is that whenever we consider something better than another thing, it must be with respect to some end. But the complexity of human desires means that the same activity might be desirable for more than one reason, and therefore on account of more than one end. Imagine, as an example, that our friends have come together to study as a group. On the one hand, this might be desirable because studying produces knowledge. On the other, we might desire it because we enjoy spending time with our friends. The conclusion here, and Aristotle’s point, is that knowledge is better than studying considered as a means to knowledge. We’re saying nothing about the relationship between knowledge and studying considered as a part of spending time with friends.

In general, the claim that ends are always better than means is not the same as the claim that if B is ever a means to A, then A is always better than B. Rather, it is that claim that whenever and insofar as B is a means to A, A is better than it.

In summary, then, this passage notes certain helpful distinctions regarding activities and ends. Sometimes the activity and the end are the same, and sometimes they are distinct. In the latter case, the end is better than the activity. Activities themselves can often be divided into sub-activities (called subordinate activities), and in these cases the superordinate activities are better than the subordinate activities.

The chief good has the nature of a last end

If, then, there is some end of the things we do, which we desire for its own sake (everything else being desired for the sake of this), and if we do not choose everything for the sake of something else (for at that rate the process would go on to infinity, so that our desire would be empty and vain), clearly this must be the good and the chief good.

This passage has caused much confusion for readers, and it certainly would have been better if Aristotle had spent some more space clarifying his meaning here. Some have thought that by “chief good” Aristotle is picking out some particular final end of all human life. I’m inclined to think that he only introduces such a notion later in the first book, and even then with more nuance than some commentators would grant him. Alas, we will have to leave that for a future post.

But if he is not talking about a specific end, then what could he be talking about? Good question. In the first passage Aristotle arrived at this determinable the good which, having the nature of an end, is that for the sake of which everything is desired. Now, as with all determinables, we can determine this in various ways to various levels of specificity. So we can talk about the good, the good thing, the good artist, the good musician, the good violinist, the good first violinist, and so on. This parallels how we can talk about the coloured thing, the red thing, the scarlet thing, and so on. But notice how we could determine things differently, so that instead of determining coloured to red we could determine coloured to brightly coloured. This kind of “alternative determination” is what Aristotle is doing here in this third passage.[4] We’ll first discuss the structure, and then explain his defense.

The first part of the sentence reintroduces the notion already discussed in the first passage — this determinable the good — which he refers to as the “end of the things we do, which we desire for its own sake”. Now in the second passage you’ll recall he discussed ends and means and how they arise in various general ways, and he noted that the ends are always better than their means. Now consider some particular case where A is desired for the sake of B, B is desired for the sake of C, and so on, but where this chain comes to some final end Z. In this case Z is different from all the other members in the chain in that it is not desired for the sake of something else, or in other words it does not derive its desirability from another as a means derives its desirability from its end. This is the property Aristotle wishes to use in his alternative determination of the good, and it gives us this determinable the chief good. Now the chief good is still fairly indeterminate, and there is nothing in the notion itself that requires that it pick out one particular good in all cases. Depending on how we determine it we will get different goods: the chief medical good is health, the chief economic good is wealth, and the chief military good is victory. The important thing here is that the chief military good is not bridle-making, since the latter is an activity subordinated under the activity of strategy and as such derives its desirability from that superordinate activity.

So, if we were to repeat the passage, taking out the parentheses and highlighting corresponding determinables and their names, we would get the following:

If, then, there is some end of the things we do, which we desire for its own sake… and if we do not choose everything for the sake of something else… clearly this must be the good andthe chief good.

This, then, is the structure of the passage. But you’ll notice that Aristotle thinks that all chains of desire must end in some or other chief end. He summarises the reason for this in the second pair of parentheses when he says that if a specific chain didn’t come to an end then “the process would go on to infinity, so that our desire would be empty and vain”. Someone unfamiliar with the distinctions and arguments introduced by him in his other works like the Physics and Metaphysics can be forgiven for missing that he is just summarising and applying these here, as opposed to working them out from the start again. For the sake of clarity we will expand his summary slightly.

In those works Aristotle makes use of a general distinction between what would later become called per se causal chains and per accidens causal chains. These days they are also sometimes called essentially ordered causal chains and accidentally ordered causal chains respectively. The defining characteristic of a per se causal chain is that each member in the chain acts only insofar as it is acted upon, so that it derives its power to act from some other member in the chain. The standard example of such a chain is that of a stick which pushes a rock, which it does only insofar as it is pushed by me. On the other hand members in a per accidens causal chain do not depend on each other in this way. Here the standard example is that while my father depends on my grandfather for his coming to be, it is not the case that my father begets me only insofar as my grandfather begets him. (As a reminder note that while I’m using efficient causal chains as illustrative examples, the term “cause” here is used in the broader Aristotelian sense and not in the limited modern sense.)

Now, any particular per se causal chain requires an ultimate cause, by which we mean something with underived causal power in the relevant sense. This ultimate cause is also sometimes called a “first” cause, but when using this term we must remember that we aren’t concerned with something first in the sense of being earlier than all the other cause, but rather something being independent of the other causes and on which they depend. Indeed, when considering chains of final causes, this “first” cause is actually the last end.[5]

The reason why per se chains need ultimate causes is because each intermediate cause merely propagates the causal power it derives from another, so that unless there’s some originating cause there would be no causal power to propagate in the first place. Now because people are prone to misunderstand what’s being said we note that this point isn’t primarily concerned with the number of causes, but their kind — namely, that they are derivative causes. For example, it doesn’t matter how many water pipes you have, they will never by themselves be able to direct a flow of water unless something puts water into the system. Similarly, if everything in a collection can push only insofar as it is itself pushed, then that collection cannot by itself push anything.

Aristotle is here applying this to the notion of final causes: if everything in a collection produces desire in me only insofar as it derives that desirability from another, then that collection cannot by itself produce desire in me. What we need is something which is desired for its own sake and not for the sake of another, failing which the chain would have no power to produce desire in me (or, as Aristotle says, “our desire would be empty and vain”). This ultimate final cause, or ultimate end, would then be an example of a chief good.

So far we have left one thing in the passage unexplained: if the point about per se chains isn’t primarily about the number of causes, then why is Aristotle concerned that “the process would go on to infinity“? We can take Aristotle’s words in two ways, each of which complements the other. First, it might be that he’s using the term to pick out the notion of an infinite regress in the sense that there is no ultimate cause. This is sometimes how the term is used these days, and in this sense of the term it is consistent with there being infinitely many intermediate causes between the ultimate cause and the final effect (assuming such a thing is coherent). Second, while the point about per se chains isn’t primarily about quantity, it has a secondary consequence about quantity. It follows that the chain must be finite from the facts that (1) for each cause there is a next member of the chain (that which it causes), (2) there is a first member (the ultimate cause), and (3) there is a last member (the final effect).[6]

In summary, then, this last passage combines the insights from the first two and, by way of “alternative determination”, picks out the primary focus of the rest of the rest of his study, namely this determinable the chief good. This is just the beginning, however, and in a later post we will discuss the narrowing of his focus to the particularly human chief good that occurs later in the book.

Related resources

The biggest influence on the approach I followed here was David Oderberg, particularly his papers On an Alleged Fallacy in Aristotle and The Content and Structure of the Good. I quoted Edmund Waldstein’s The Good, the Highest Good, and the Common Good, and I highly recommend reading that too.

The notion of a determinable and it’s distinction from a cause is critically important for precision of thought. Ronald McArthur’s paper Universal in praedicando, universal in causando is an invaluable resource for understanding the distinction between predication (which corresponds to our determinables) and causation. I highly recommend it.

For more information on per se vs per accidens causal chains I recommend Edward Feser’s blogposts Cross on Scotus on causal series and Edwards on infinite causal seriesm Caleb Cohoe’s paper There Must Be a First, and Gaven Kerr’s paper Essentially Ordered Series Reconsidered. Cohoe’s paper is the one where I realised that additional reasons need to be given for thinking that per se causal chains are finite.

Notes

  1. In general something fails to fulfill its end only to the extent that it is prevented in some way, due to either internal defect or external interference. For instance, a carpenter might aim at making a chair which can hold people up and not fall down easily, and will only fail to achieve this if impeded by something internal (like lack of ability) or something external (like bad materials). Unless such interference occurs the carpenter will achieve their end in full, which is the best result. We’ve mentioned before that classical thinkers like Aristotle realised that there is a broad sense in which all things are orientated to certain ends given by their natures. And in the Physics he has this more general notion in mind. For instance, the development process of a dog is directed toward the growth of four legs with which the dog can walk, and only fails to achieve this when the dog has some kind of genetic defect or has some external blocker is present, like an accident or lack of food. Again, unless it is interfered with the process will achieve it’s end in full, which is the best result.
  2. In coming up with this phrasing I thought of a number of alternatives, which I include here for posterity: “the thing exhausts the desire without excess”, “the thing itself, and not some part or some greater whole, is what’s desired”, “the thing itself is desired, and not some part thereof or some whole of which it is part”, “the thing is desired neither merely in part or as a part”, “the whole of the thing is the whole of what’s desired”, “the thing is all and only what is desired”, “the thing in reality matches the thing desired”, “a thing is desired to the extent that it matches the object of that desire”, “the thing satisfies the desire without excess or deficit”, “of the thing itself and the thing desired, neither is a part of the other, but they agree completely”, “the thing itself, and not something more or less, fulfills the desire”.
  3. Shortly we will defend the claim that every chain of ends must have some final, ultimate, or “chief” end. However, this is not required for this point. If we have some infinite series of ends, then to say that B is more of an end than A it is sufficient that it be closer to some end C which is further along the chain.
  4. The rough idea is as follows: when we determine some determinable we contract or qualify it in some way. Now we come to know determinables through abstracting away these qualifications, and so the most “natural” way of determining them is by the same road we took to get there in the first place. But nothing in this constrains us to this, and we are free to qualify or contract in a different way to how we first arrived at the determinable. This different way is what I’m calling an “alternative determinations”.
  5. I’m reminded of the saying of Christ that “the last will be first, and the first last.” (Matt 20:16)
  6. While Aristotle wouldn’t have expressed it in these terms exactly, it seems from his other works that he understood the principles at play here. The three facts combined mean that the causal chain is one-to-one mappable onto a bounded contiguous range of natural numbers, which is only possible if the chain is finite.

From morality to nature and back again

Below is a talk I recently gave at a local apologetics meet-up. The goal was to introduce and partially defend natural law theory to a group of fellow-Protestants who, as far as I was aware, had not engaged extensively with natural law theory before. The talk was recorded in various parts, with video coming in the second part. At the end there is a collection of resources for those interested in some further reading.

In our previous meeting I got the impression that my views on morality as an Aristotelian and Thomist are particularly different from the views of many of you here, as well as Protestants more generally these days. I have two goals here tonight. The first is to introduce and partially defend the views I’ve come to hold on these issues, and the second is to explain how these relate particularly to Protestant approaches to Scripture and modern uses of the moral argument for God’s existence.

We’ll be concerning ourselves primarily with issues of meta-ethics, which is that subfield of ethics that concerns itself with (1) what we mean by certain terms like “good”, “moral”, “virtue”, “justice”, “ought”, as well as (2) how such things are grounded, by which we mean giving an account of what makes things good, moral, virtuous, etc. There are roughly two meta-ethical theories I want to talk about:

  1. Divine command theory, which I imagine is the view many of us here hold.
  2. Natural law theory, which is the view I want to recommend as best.

Now, before we start I should note that these two theories have very different approaches in terms of how they are developed. Like most modern meta-ethical theories, the divine command theory we’ll be talking about takes the term “moral” as picking out some special or mysterious class of facts that need to be defined and grounded by the theory. On the other hand, for the classical natural law theory we’ll be talking about, the term “moral” doesn’t pick out any particularly special, and is defined before we even start the theory. The focus of natural law theory is instead the notion of “goodness”.

This is noteworthy because we’re going to use the word “moral” in both senses tonight, and it can get confusing unless you keep this difference in mind.

1. Essentialist Divine Command Theory

I imagine the divine command theory that is most commonly held here is the so-called essentialist divine command theory defended by people like William Lane Craig and Robert Adams.[1] We can sketch the rough outlines of the theory in about 7 points:

  1. “Moral” picks out those fact which are most fundamental and important. If our government commands us to do something immoral, for example, we still have a duty to refrain from listening to them, since our moral duties are more important than our duties to our government.
  2. We divide moral facts into moral values and moral duties. Moral value refers to the worth or goodness of something. Moral duties refer to the moral obligations or prohibitions that apply to us, what we ought and ought not do, rights and wrongs.
  3. Moral value is ultimately grounded in God’s nature or essence, in the sense that he is the paradigm of moral goodness. Because God is a person, persons are morally valuable. Because God is loving, love is morally good. And so on.
  4. Moral duties are grounded in God’s commands to us, which are given explicitly through revelation or implicitly through conscience. The idea here is that in general duties arise from commands from qualified authorities. For example, when a policeman commands me to do something I have a legal duty to do that, since policemen are qualified legal authorities. God, being the paradigm of moral goodness, is uniquely qualified to be a perfect moral authority, and so his commands constitute moral duties.
  5. God’s commands, and therefore our duties, are not arbitrary because they are based on God’s unchanging nature, which we said in (3) is the paradigm of moral goodness. Nor are they based on something external or “bigger” than him because his nature is something internal to him.
  6. Moral virtues are those habits that dispose us to doing good and right things as they are grounded in God’s nature and commands.
  7. Because moral duties arise from God in this way, it seems that so must our personal motivations for obeying them. In a Christian context this would mean that the reason we follow God’s commands is out (1) love for God and desire to be with him, and (2) fear of just punishment.

2. Thomistic Natural Law Theory

We move now to natural law theory. The particular brand of natural law that I’m interested in here is the one from by Thomas Aquinas, who himself was developing the natural law theory of Aristotle.

2.1. Morality is about practical reason

Now, as I said, as a classical theory we have the term “moral” defined upfront: “moral” picks out things relating to the will, and therefore also our actions. For example, classically we can by divide reason into speculative reason and practical reason. Speculative reason relates to our intellect and has to do with applying reason to further expand our understanding of reality. The habits that lead to good speculative reasoning are called the intellectual virtues. Practical reason relates to our will and has to do with applying reason to govern how we will and act. So habits that lead to good practical reasoning are called moral virtues.

So, while we might have inherited the word “moral” from Aristotle, it no longer has the same meaning. Classically, it did not denote some special or fundamental class of value of duty, it wasnot connected with the will of God in such a way that he could be said to be a lawgiver, and it does not carry the psychological weight of being bound by some law. In his Ethics Aristotle discusses both moral and intellectual virtues, with neither being more important than the other.[2] The reason for this is that both moral and intellectual virtues part of being a good human.

As we said, the best starting point would be how classical natural law understands the notion of “goodness”.

2.2. Good has the nature of an end

Aquinas said that in general the good “has the nature of an end”[3] and we’ll use this as our starting point. In a way, though, our modern ears aren’t prepared for this definition, because we’ve been taught to think of conscious deliberation whenever we think of something working for an end. But for Aristotle and Aquinas our consciousness is just a special case of the goal-directedness that exists throughout nature. For them, everything that exists has tendencies toward certain ends determined by its nature.

The thought is roughly as follows: at every level things exhibit certain natural regularities or tendencies toward certain effects. We see this in living things, like how hearts regularly pump blood, or how dogs regularly grow up to have four legs so they can walk, or how seeds regularly grow into trees. We also see this in non-living things, like how matches tend to combust when struck, the moon tends to orbit the earth, salt regularly dissolves in water, rocks regularly fall to the ground, and so on. In each case we have something consistently producing its specific effect unless its prevented from doing so in some way.

And notice that each regularity involves the production a specific effect rather than something else or nothing at all. Matches produce fire as opposed to producing ice or nothing at all. Seeds grow into trees and not into rocks. Salt dissolves as opposed to combusting. Rocks fall as opposed to exploding. And the same goes for all the numerous regularities that exist throughout the universe. But, that things consistently work to produce their specific effects seems to make sense only if “there is something in them that is directed at or points to specifically those outcomes rather than any others”.[4]

So, in some broad sense hearts are directed at pumping blood, the development process of dogs works to produce an organism that walks on four legs, matches are directed at combusting when struck, salt is directed to dissolving in water, and so on. At the end of the day we find that everything that exhibits some form of natural regularity must be directed by its nature towards that behaviour as kind of end or goal. This is the kind of “teleology” that Aristotle and Aquinas have in mind when they talk about goal-directedness in nature, which by-and-large isn’t due to the conscious deliberation of the things themselves. Of course, working this out completely requires a fairly lengthy side-track into metaphysics and philosophy of nature, but hopefully the examples I gave will give you enough of an intuition.

Now, let’s go back to what Aquinas was saying about good having the nature of an end. What he’s getting at is that whenever we talk about an end we can also talk about goodness: something is good to the extent it fulfills its end and bad or defective to the extent that it fails to fulfill its end. If I’m playing a sports match, for instance, then my actions are good for me to the extent that they help me win the game. On the other hand, losing the game would be bad for me, and could happen because I played badly or because my opponent played better than me. A chair is good to the extent that it realizes the carpenter’s end of making something that holds people up and doesn’t fall over. And a music performance is good the extent that it achieves the orchestra’s end of playing the piece.

2.3. Natural goodness

So we have that (1) everything is in some sense directed toward certain ends by their nature and (2) whenever somethings works for an end we have a measure of goodness for that thing. This gives us a very general sense of goodness that applies to almost everything. Because this notion of goodness is so closely linked with the natures of things we can call it “natural goodness”.

It might sound odd, but this natural goodness is in some sense both relative and objective. It is relative because what is good for you is dependent on the kind of thing that you are. If you had had a different nature, then different things would be good for you. It’s bad for cats to have two legs, but it is good for humans to have two legs. A good match causes fire when used, and a good fire extinguisher stops fire when used. However it’s still objective because at the more fundamental levels you don’t decide your own nature, and cannot change it.

Now, this natural goodness serves as the springboard for all ethical reasoning in natural law theory. The basic idea is that because we can study our human nature through various empirical methods and philosophical reasoning, we can also come to a better understanding on how to live well as humans.

2.4. Accountability, duty, and authority

While we can’t go through all the details here, what I would like to do is give you a rough idea of how on natural law theory we can move from this natural goodness to thinks like moral accountability, duties, and authority.

Accountability, it seems to me, is ambiguous between two things, which we’ll take in turn: responsibility and punishment. We noted earlier that moral virtues are a special case of virtues in general, and I think something similar happens when you consider moral responsibility and responsibility in general. In general, being responsible for an action means that that action was up to you. And people typically that one’s responsibility is in some way proportional to one’s knowledge, or at least one’s capacity for knowledge. The idea here is that an action is up to you only to the extent that you understand what you’re doing. So we generally hold adults more responsible for their actions than children, who we hold more responsible for their actions than our pets, who we hold more responsible for their actions than this or that rock.

Now, humans have been traditionally been called rational animals. We don’t mean by this that humans are always perfectly rational: they’re not. Roughly, what makes animals rational is their ability to grasp and be conscious of universal concepts that particular things fall under. So there’s the particular human called Socrates, and there’s the universal concept of humanness which Socrates, Plato, and all other humans fall under. All animals are conscious in some way of particular things, but rational are those animals which are also conscious universal concepts. Now, this ability to understand universal concepts means we have the ability to understand the natural goodness and evil, that we were talking about earlier, both for ourselves and for others, as well as the ability to choose to pursue or avoid this goodness. This additional understanding about our actions results in us being held more responsible for them, and this additional layer or responsibility is what we mean by “moral” responsibility. At the end of the day, we say that an action is morally good or evil to the extent that the end or means willed in that action are naturally good or evil.

For example, if due to genetic defect or accident I have only one leg this is bad for me but I am not responsible it. In this case we have a natural evil without a moral evil. On the other hand, if I cut my own leg off then this is an evil for which I am responsible. In this case we have a natural evil with a moral dimension, since the natural evil is the product of my will.

As for punishment, one way it arises is as follows: humans are not merely rational animals but also political animals, by which we mean that it is natural and good for us to be part of various communities like families, sports teams, companies, friendships, and states. When a part is a detriment to the good of the whole, it is good for that part to be removed from that whole or to otherwise incur some debt so as to restore the good of the whole.[5, 6] For example, if my hand has gangrene it is good for me to cut it off. This removal or debt will be punishment, and if properly administered it will have to be done according to the principle of retributive justice.[7]

What about duties? On divine command theory we have divine legal duties which arise from God’s commands to us. And although it’s not as big a focus in natural law, we can also say something about duties. We’ve seen that our nature sets certain ends for us, and to the extent that an action contributes to our fulfillment of these ends it is good. This gives us the fact that, if I will the good, then I ought act so as to fulfill my natural ends. But if we think about it, in general we act for something because we will it, and we will it because it seems good to us in some way. “The mugger who admits that robbery is evil nevertheless takes his victim’s wallet because he thinks it would be good to have money to pay for his drugs.”[8] What this means, however, is that we always will what seems good to us, even if sometimes we incorrectly prioritize some goods over others. Combining this with our earlier fact we get to the following conclusion:

  1. If I will the good, then I ought act so as to fulfill my natural ends.
  2. I do will the good.
  3. Therefore, I ought act so as to fulfill my natural ends.[9]

After some reflection on our natures this will result in various duties such as “I ought not steal”, “I ought not murder”, “I ought honor my parents”, and so on. But what kind of duty is this? It’s certainly not a legal duty that we get from divine command theory, since it doesn’t arise from any command. We might call it a rational or a natural duty since it arises out of our natural capacity for practical reason. It serves to show us that we should be interested in what is naturally good for us.

And finally, what of authority? Here we combine some of the points we’ve already made. The idea here is that someone has authority over me if they are in charge of my good, since I ought seek my good, and therefore I ought listen to their commands. Different people will have authority over different areas of my life and to different degrees depending on their position and qualification, and in each case something like this idea applies.

2.5. The four laws

Now, is there any place for a divine legislator on natural law theory? This is one of the main areas where Aristotle and Aquinas differ. For Aristotle, God is not a divine legislator and the only place he takes in the ethics is as the object of our highest end which is philosophical contemplation about him. Aquinas, however, thinks Aristotle made a mistake here. In unpacking what he thinks is the correct view, Aquinas explains that there are ultimately four kinds of law:

  1. There’s the eternal law, which embodies God’s knowledge of all the various natures of things he could have created, and so what would have been good for them.
  2. There’s the natural law, which is what we’ve been speaking about here. For humans this forms the foundation for all our practical reasoning. It tells us what it means to act well as the kinds of things that we are. It’s called “natural” law because all of this derives from our natures.
  3. There’s human law, which are laws promulgated by a human legislator in charge of a community. Natural law is often very vague and general and it’s application in particular cases requires careful consideration by wise people. “[H]uman law is essential for living the good life because it makes the general precepts of the natural law more specific.”[10] Human law is authoritative because it’s based on natural law.
  4. Finally there’s the divine law, which are laws promulgated by God, the divine legislator. This law most closely represents that law that we think of in divine command theory, and they are the laws that are proclaimed through some form of revelation.

So there is a place for divine law, but it’s embedded in this bigger theory of ethics. Ultimately I think every intuition we have explained in divine command theory can be relocated somewhere in natural law theory, with a richer foundation, since natural law gives us accounts of things like authority, responsibility, and so on.

3. Modern Protestant objections

So with that overview of natural law theory, let’s talk briefly about it means for Protestantism. I think a lot of Protestants these days are quite resistant to the idea that moral prescriptions or substantial moral knowledge might come from somewhere outside of scripture. I say “these days”, because neither the church historically nor the reformers themselves had a problem with natural law theory. John Calvin, for example, said the following in his Institutes:

It is a fact that the law of God which we call the moral law is nothing else than a testimony of the natural law and of that conscience which God has engraved upon the minds of men.[11]

I think our modern hesitance arises from a combination of two things. On the one hand there’s been an increasing loss of acquaintance with natural law thinking in the past few hundred years, because of what I take to be certain philosophical errors of the early moderns like Descartes and Locke. Recently we’ve started correcting these errors, but our culture as a whole has lost its grip on this kind of thinking. And when we consider certain doctrines like original sin and sola scriptura against this backdrop they might seem to be at odds with what I’ve been saying.

So consider original sin, which says that our natures have been disordered, which in turn undermines our ability for unaided reason and therefore the moral conclusions we draw from it. But there’s nothing in this that contradicts what I’ve been saying. The claim that we can come to know ethical truths through philosophical reflection does not require that we be infallible in our conclusions. All that follows from our fallibility is that our understanding of ourselves, like our understanding of any part of nature, needs to be a community effort that spans many generations and societies. And the same thing can be said of our understanding of scripture itself. To quote John Goyette:

The collective effort required for the development of the arts and sciences is, for Aquinas, one of the reasons why man is a political animal. But the same is true of human law: it a collective effort requiring experience and time, and the wisdom of the wise. Just as men perfect the arts and sciences as part of a community, so do men perfect their knowledge of the natural moral law by participating in the [political community].[10]

What about the doctrine of sola scriptura, or “scripture alone”? There seem to be a number of slightly different of ways of formulating the doctrine, [12] but if it’s to be consistent with scripture it can’t claim that scripture is the only source of moral knowledge, for two reasons. First, because scripture itself references other sources like conscience. One of the clearest places where we see this is in Paul’s letter to the Romans where he talks about the Gentiles and he says that even though they haven’t been given the law through revelation, “they show that the requirements of the law are written on their hearts, their consciences also bearing witness.”[13]

The second reason is because scripture must presuppose some knowledge of the world, and this knowledge includes some things pertaining to morality. J. Budziszewski gives the following example:

Consider for example the prologue to the Ten Commandments, where God reminds the Hebrew people of their indebtedness to Him: “And God spoke all these words, saying, ‘I am the LORD your God, who brought you out of the land of Egypt, out of the house of bondage. You shall have no other gods before me ….'” How is it that the people of Israel, before the proclamation of the law, already know the law of gratitude? The answer is that the basics of natural law are already impressed upon the innermost design of the created moral intellect. We know a part of God’s will for us even before receiving it in words.[14]

3.1. The role scripture

I suppose we might wonder what does scripture adds if we can to know moral conclusions apart from it. There are a number of things we can say here.[15]

  1. In general there are things about God and ourselves that we can’t know through unaided reason and scripture is needed for these. Things like God’s triune nature or his dealings in human history, particularly what we call redemptive history, what will happen after we die, that marriage is a symbol for Christ and the church, and so on.
  2. Because of God’s revelation to us through scripture and through Jesus we are able know God personally, which wouldn’t be possible otherwise, since friendship requires communication between friends. As Jesus says in John’s gospel, “No longer do I call you servants, for the servant does not know what his master is doing; but I have called you friends, for all that I have heard from my Father I have made known to you.”[16]
  3. Revelation of God’s commands serves to introduce divine law and duties, which we wouldn’t have otherwise.
  4. Revelation about morality serves as a guide and summary of natural law. We’ve already seen that it can be difficult to work out the details of natural law, and besides that not everyone has the gifts or time to work them out. So through revelation God enables more people to know and will the good.

4. Apologetics

As we close I want to say a few things about what natural law means for apologetics today.

4.1. The moral argument

Like most arguments for God’s existence the “moral argument” is really a family of arguments. The one most heard today is formulated as follows:

  1. If God does not exist, then objective moral values and duties do not exist.
  2. Objective moral values and duties do exist.
  3. Therefore, God exists.

The question I want to address is how natural law effects the prospects for a moral argument like this.

Now, Aquinas gave arguments for God’s existence in various places throughout his writings, although most famous are the so-called “five ways” he lays out in the Summa Theologica. As far as I can tell Aquinas never gave a moral argument. I think the reason for this is that from a natural law perspective morality is not some special part of reality that calls out for an explanation, but is rather the result of the combination of otherwise non-moral features of reality: (1) the goal-directedness we see throughout nature and (2) the wills of rational beings.

The closest thing Aquinas gives to a moral argument is his fifth way, which is a teleological argument.[17] I should note, though, that the teleology Aquinas has in mind is different from the kinds of teleology we see in modern arguments for God’s existence.[18] He’s not concerned with the complexity of living things or the fine-tuning of the universe, for instance, but rather the goal-directedness we spoke about earlier, which is required by the various regularities that exist at all levels of nature both complex and simple.

Now, the question arises of how something can be directed toward and end. It’s clear how this happens with intelligent beings, since there the end in some sense existing in the intellect of that being, and so it can guide the actions of that being. But with non-intelligent things, since they lack an intellect, their ends can’t influence them in the same way. So it seems that non-intelligent things must be directed toward their ends by something with intelligence. And in fact, we could see why this is the case if we spent some time analyzing the notion of intelligence, but we don’t have time for that now. This is how Aquinas summarizes what we’ve been saying in his fifth way:

Now whatever lacks intelligence cannot tend towards an end, unless it be directed by some being endowed with knowledge and intelligence; as the arrow is shot to its mark by the archer. Therefore some intelligent being exits by whom all natural things are directed to their end; and this being we call God.[19]

Of course there is still lots to be unpacked. After he gives his five ways, Aquinas spends some time explaining why this is the being we call God. He argues for God’s oneness, goodness, omnipotence, omniscience, simplicity, eternity, and a host of other divine attributes, but we just don’t have time to give and defend those arguments now.

Coming back to the moral argument. I think technically we can still use it as I formulated it, but we must recognise that it is partly dependent upon something like the fifth way for its soundness. At the end of the day I think much moral debate can be had without reference to God, since it is based on what is knowable about our nature. But ultimately I think any viable ethics depends on God, including natural law.

4.2. Cultural apologetics

Finally, from the perspective of cultural apologetics natural law serves as a common ground for Christians and non-Christians to discuss ethical issues, since particular moral conclusions do not depend on whether one thinks God exists or not. For example, atheist philosopher Phillipa Foot has said that,

… the Summa Theologica is one of the best sources we have for moral philosophy, and moreover that St. Thomas’s ethical writings are as useful to the atheist as to the Catholic or other Christian believer.[20]

In the Western world it’s becoming increasingly important that we be able to defend the value of human life, and of family life, and particularly the rights of children. Our culture truly is a “culture of death”, in which people think it’s OK to kill innocent human beings so long as they’re young enough or helpless enough, and more generally in which we ignore the rights of children so that adults can do what they want. All too often these days I see people object to things like these on so-called “religious grounds” and then get ignored because the secular world doesn’t share their religious convictions. But there are good arguments wholly apart from any religious confession, and these need to be the primary go-to point for us.

A secondary point is that as we show the reasonableness of so-called “traditional” moral conclusions, we also show in part the reasonableness of the Christian worldview. In this way natural law can help show our culture that Christianity is an intellectually viable worldview, which is something they’ve forgotten amongst all the hype with the New Atheists.

Other resources

For more on natural law, Feser’s blogpost Whose nature? Which law? which goes into more detail about this technical word “natural”. There’s also his article Natural Law, Natural Rights, and Private Property, which has a short introduction to natural law as well as an example application of it to property rights. If you’re interested in structure of the approach many natural law theorists take when unpacking the specifics of natural law, see my blogpost Goods, basic goods, and facultiesand David Oderberg’s paper The Structure and Content of the Good.

On the topic of original sin, there’s J. Budziszewski’s three-part blogpost series Natural Law and Original Sin (part 1, part 2, part 3).

On the relationship between God and natural law see Edward Feser’s blogposts Natural law or supernatural law? and Does morality depend on God? While the fifth way is a sound argument for God’s existence, I tend to prefer the second way. See Edward Feser’s An Aristotelian Proof for the Existence of God for a good talk on this, as well as a taste of how we might go about arguing why we call this being God. If you’re interested in the fifth way, I recommend his paper Between Aristotle and William Paley: Aquinas’s Fifth Way.

Finally, one of the things that cam up in the question time was the notion of divine simplicity. William Lane Craig on divine simplicity is a blogpost by Edward Feser where he discusses some contemporary objections, and On Three Problems of Divine Simplicity is a paper by Alexander Pruss doing likewise.

For more, see the notes below, as well as the long list of categorised resources over at my blog.

Notes

  1. See, for instance, Robert Adams’s Finite and Infinite Goods. In contrast to essentialist versions of divine command theory there are the voluntarist versions like the one put forward by Ockham, which place both values and duties at God’s commands. I’ve also discussed what I call derivative divine command theory, in which duties are prior to values.
  2. I mean more important in the sense of needing to be studied. He thinks that the intellectual virtues are better than the moral virtues, since the highest end of man (or, to use modern terminology, man’s superordinate basic good) is philosophical contemplation of God.
  3. ST I-II Q9 A1 corp.
  4. Edward Feser, Scholastic Metaphysics.
  5. “Now every part is directed to the whole, as imperfect to perfect, wherefore every part is naturally for the sake of the whole. For this reason we observe that if the health of the whole body demands the excision of a member, through its being decayed or infectious to the other members, it will be both praiseworthy and advantageous to have it cut away. Now every individual person is compared to the whole community, as part to whole. Therefore if a man be dangerous and infectious to the community, on account of some sin, it is praiseworthy and advantageous that he be killed in order to safeguard the common good…” (ST II-II Q64 A2 corp)
  6. “… whatever rises up against an order, is put down by that order or by the principle thereof. And because sin is an inordinate act, it is evident that whoever sins, commits an offense against an order: wherefore he is put down, in consequence, by that same order, which repression is punishment.” (ST I-II Q87 A1 corp) This is a more general version of what was said in [5]. I’ve briefly discussed this comment elsewhere.
  7. The argument, very briefly, is as follows: in order for the good of the whole to be best upheld, punishment ought only be of guilty people, ought be proportional to the crime, and ought be equal (ie. like punishment for like crimes). There are at most four putative theories of justice: deterrence, correction, preventative, and retributive. Only the last safeguards all three of these conditions. This is not to say that punishment couldn’t also include deterrence, correction, and prevention, but it must minimally be based on the principal of retribution. A supporting argument is that only retributive justice sees the agent as a human, and is therefore the only theory that affords them proper respect. Deterrence sees only a behaviour, correction only a patient, and prevention only a future threat.
  8. Edward Feser, Classical Natural Law Theory, Property Rights, and Taxation.
  9. Compare this argument to the following: (1) If I will to draw a straight line, then I ought use a ruler, (2) I will to draw a straight line, (3) Therefore, I ought use a ruler. (3) is consistent with me not realising that rulers are the best way of drawing straight lines. Similarly, that I will the good is consistent with me not having a perfect grasp of what that involves. And even once I realise it involves acting so as to fulfill my natural ends, I still won’t have a perfect grasp of what such fulfillment involves.
  10. John Goyette, On the Transcendence of the Common Good
  11. John Calvin, Institutes of the Christian Religion, IV. XX. 15
  12. This arises because we need to find a formulation that is not the Roman Catholic doctrine ofprima scriptura but at the same time doesn’t lead to self-defeat.
  13. Romans 2:15, New International Version.
  14. J. Budziszewski, Does Sola Scriptura Mean “No Natural Law”?
  15. I’m particularly fond of what John O’Callaghan says about the general relationship between theology and philosophy here. A noteworthy quote is: “Theology doesn’t take place in a vacuum just because it something heard from the mouth of God… and so we need to understand what’s presupposed to being able to hear what is being preached to us or what is being revealed to us, and then a systematic reflection upon it. Theology shouldn’t take place in a vacuum.”
  16. John 15:15, English Standard Version.
  17. For a lengthy and substantive defense of Aquinas’s fifth way see Edward Feser’s Between Aristotle and William Paley: Aquinas’ Fifth Way.
  18. See Edward Feser’s Teleology: A Shopper’s Guide.
  19. ST I Q2 A3 corp.
  20. Phillipa Foot, Virtues and Vices.

Lonergan on Aquinas on Causation

Below is an excerpt from Bernard Lonergan’s incredible book Grace and Freedom, discussing Thomas Aquinas’s views on causation and how they relate to Aristotle’s views on the topic. Except for the term “actio” I’ve replaced Latin phrases with their English translations in square brackets.

Causation is the common feature of both operation and cooperation; its nature is of fundamental importance in this inquiry. But if St Thomas certainly disagreed with Hume, who held causation to be purely subjective, it is less clear what object he considered to constitute the objective reference of the proposition “A causes B.” Was causation for him something in between A and B? Or was it simply the relation of dependence of B on A? Or was it some entity added to A as actually causing? Let us take each of these three views in turn.

As to the first view, that causation is in between cause and effect, St Thomas constantly and explicitly denied it in the case of divine activity. Avicennist biology had distinguished between a [a moving power commanding (something)] and a [a motive power effecting (something)], and St Albert had drawn a parallel distinction between the [divine created power] and a [divine uncreated power]. But St Thomas, while he used the biological opinion at least in his commentary on the Sentences, always asserted that God was his own virtue, operated without any mediating virtue, indeed operated [by the immediacy of power]. The matter is less clear with regard to causation by creatures. Even in later works there is a variety of expressions which appear to imply something in between agent and recipient. Still, it should seem that these are but modes of expression or of conception; for what is in between, if it is something, must be either substance or accident; but causation as such can hardly be another substance; and if it were an accident, it would have to be either the miracle of an accident without a subject, or else, what St Thomas denied, an accident in transit from one subject to another.

On the second view, causation is simply the relation of dependence in the effect with respect to the cause. This is the Aristotelian position presented in the Physics and explained by St Thomas as follows. First of all, this analysis prescinded from the case of the mover being moved accidentally; for instance, a terrestrial body acts through contact and cannot touch without being touched; but this does not prove that the cause as cause undergoes change but only that the terrestrial body as cause does so. In the second place, it was argued that the emergence of a motion or change involved the actuation of both the active potency of the cause and the passive potency of the effect. In the third, place the thesis was stated: one and the same act actuates both potencies, and this act is the motion produced in the object moved. Fourthly, there came the ground of this position: if causation, actio, were an entity inherent in the cause, then, since it is a motion, it would follow either that “[every moving thing is moved],” or else that motion inheres in a subject without the subject being moved; but the latter is contradictory, and the former would preclude the idea of an immovable mover; therefore, causation is not inherent in the cause but in the effect. Finally, the objective difference between action and passion was explained: both are really identical with the motion of the recipient; they differ notionally, for action is this motion as from the cause, [movement of this as from this], while passion is the same motion as inhering in the effect, [movement of this as in this].

It would seem that St Thomas accepted this Aristotelian analysis as true and did not merely study it as a detached and indifferent commentator. Not only did he repeat the same exposition in commenting the parallel passage in the Metaphysics, while in the De anima he argued that sound and hearing, instances of action and passion, must be one and the same reality, else every mover would be moved; but in works that are entirely his own the same view at least occasionally turns up. In the Summa theologiae the definition of actual grace appeals to the third book of the Physics for the doctrine that “[an act of a mover is a movement in the thing moved]”; the analysis of the idea of creation was based upon the Aristotelian identification of action and passion with motion; and the fact that this identification involved no confusion of action with passion was adduced to solve the object against the Blessed Trinity, namely, that since the divine Persons were identical with the divine substance they must be identical with one another. Still, this is not the whole story. In his commentary on the Sentences St Thomas brushed aside the notion that action and passion were on and the same reality, while in the parallel passage in the Summa theologiae a solution is found that does not compromise the authority of Aristotle. This difference involves a change attitude, prior to the Pars prima and perhaps posterior to the De potentia, raising the question of the initial Thomist view.

In earlier works, then, the theory of causation seems to have been worked out on the analogy of the familiar distinction between the [being towards] and the [being in] of the relation. In action one has to distinguish between a formal content described as [from an agent] or [as proceeding from an agent to another], and on the other hand, a reality, substantial or accidental, termed the [principle of action] or the [cause of action] or even loosely actio. This terminology is to be found no less in the commentary on the Sentences than in the De potentia, but at least in the latter work it also is quite clear that the formal content is no more than a notional entity. In the two passages quoted below, the reader will be able to verify the following six propositions: (A) change from rest to activity is change in an improper and metaphorical sense; (B) the reverse change from activity to rest takes place without any real change in the agent; (C) when the agent is acting there is no composition of agent and action; (D) what remains unchanged is the [principle] or [cause of action]; (E) what comes and goes without changing the agent is the formal content, [from an agent]; (F) the analysis holds even in the case of a created agent such as fire.

And so a relation is something inhering (in a subject), though that does not result from the mere fact that it is a relation; as action, too, from the fact that it is action, is considered as from an agent, but as an accident it is considered as in the acting subject. And therefore, there is nothing to prevent an accident of this kind (B) from ceasing to be without (involving) a change of that (subject) in which it is, because its being is not realized insofar as it is in that subject, but insofar as it passes on to another; with the removal of that (passing on), the being of this accident is removed (E) in what regards the act but remains (D) in what regards the cause; as is the case also when, with the removal of the material (to be heated), the heating (F) is removed, though the cause of heating remains (De potentia, q. 7, a. 9, ad 7m)

But that which is attributed to something as proceeding from it to something else does not enter into composition with it, as (C) neither does action (enter into composition) with the agent… without any change in that which is related to another, a relation can cease to be through the change alone of the other; as also is clear about action (B), that there is no movement as regards action except metaphorically and improperly; as we say that (A) one passing from leisure to act is changed; which would not be the case if relation or action signified something remaining in the subject (Ibid. a. 8 c.)

If our interpretation of these passages is correct, then at least in the De potentia St Thomas had arrived at a theory of action that was in essential agreement with Aristotle’s. Evidently the two terminologies differ completely: on the Aristotelian view action is a relation of dependence in the effect; on the Thomist view action is a formal content attributed to the cause as causing. But these differences only serve to emphasize the fundamental identity of the two positions: both philosophers keenly realized that causation must not be thought to involve any real change in the cause as cause; Aristotle, because he conceived action as a motion, placed it in the effect; St Thomas, who conceived it simply as a formal content, was able to place it in the cause; but though they proceed by different routes, both arrive at the same goal, namely, that the objective difference between [to be able to act] and [to actually act] is attained without any change emerging in the cause as such.

This real agreement in terminological difference solves the problem of St Thomas’s thought on causation. John of St Thomas listed the passages in which action is placed, now in the agent and now in the recipient; from this he drew the conclusion that action, according to St Thomas, was inchoatively in the agent and perfectively in the recipient. But in point of fact St Thomas simply had two ways of saying that action involved no new entity in the agent; and so far was he from differing really from Aristotle that he seems to have been quite unaware of even his terminological departure from the Aristotelian position. This latter fact not only solves Cajetan’s perplexity over the apparent divergence between the commentary on the Physics and regular Thomist usage but also provides the most conclusive evidence against such as position of Billuart’s that a real distinction in the agent between [power to act] and [the act itself] is one of the pillars of Thomist thought.

Virtual existence

It might not seem like it, but a proper understanding of virtual existence can be significantly helpful when trying to understand the structure of human communities. To this end, I’d like to spend some time thinking about this puzzling notion here.

Substances and aggregates again

You’ll recall that, in our discussions about substantial activities, we spent a fair amount of time introducing the notion of substance. There we said that a substance is something which has intrinsic directedness towards an end, or equivalently something which has intrinsic causal powers.

The guiding intuition here is that when a substance does something it is the substance itself doing it, as opposed to its parts or something external. So, the doing is intrinsic to the substance as opposed to its parts or something outside. Now, we use the term characteristic behaviours to pick out what something does always, or for the most part, given the kind of thing that it is; that is, how it behaves so long as it’s not being “blocked” in some way. Plants and animals grow to become healthy adults unless prevented by genetic defect or environmental factors, hydrogen combusts under certain circumstances unless prevented, the phosphorous in a match head will ignite when struck unless prevented, and so on.[1] We mentioned before that the only way such characteristic behaviours can be made intelligible is in terms of that thing’s being directed toward that behaviour by virtue of the kind of thing that it is. And this directedness, you’ll recall, needn’t be the result of conscious deliberation.

Putting this all together we get that if the doing of something always or for the most part is intrinsic to a thing, then so is the directedness towards this behaviour. Similarly, since a thing can’t do something without the power to do it, if the doing of something is intrinsic to a thing then so are the causal powers needed to do it. It is roughly along these lines that we (following Aristotle and the Scholastics) come to understand substances in the terms mentioned above. For a longer discussion of this, as well as responses to objections, I suggest you read Edward Feser’s Scholastic Metaphysics.

In contrast to substances, aggregates are those things which have only extrinsic directedness or causal powers. That is, an aggregate’s causal powers are reducible to the sum of the causal powers of its parts and what is imposed on it from outside.

A pile of rocks would be an obvious example of an aggregate. Its power to hold something 2 meters above the ground is merely the sum of the individual rocks that make it up. Above Aristotle used an example of a bed, which is merely an aggregate of the materials (wood and metal) that make it up.

Some aggregates, because of their complexity, are less obviously aggregates. Examples of these are things like watches and computers. A watch’s power for time-telling is imposed on it by us, and its power for the circular motion of its hands is merely the sum of the powers of its parts such as the conduction of electricity and so on. Similarly for a computer or a calculator.

The pile of rocks would be a table to the extent that it is intended as such by the individual or community that has access to it. Here it would be the sum of the parts together with an outside intention that make the pile of rocks a table.

With the watch there is nothing intrinsic to it or its parts that enables it to tell the time. This is something derived from us on the outside, as interpreters of the mechanical symbols we used in the watch’s construction. If you took us out of the equation, all that would remain are cogs, electrically stimulated, moving other cogs and pieces of metal at a fairly constant rate.

I suspect this is less clear to many of us in the case of computers. But this is more a function of our ignorance about how computers work than anything substantial about computers themselves. In this case various electrical components alternate their charges by interacting with one another, typically terminating in patterns of colours on a screen or sounds from a speaker. Sure, we’ve managed to do this faster and with smaller components, but there is nothing of significant difference (at least not for our purposes) between this and purely mechanical computers. We impose meaning on these patterns of colour and sound, and thereby impose on the computer the ability to compute things that are not intrinsic to the metal or electrical currents themselves. This is not unlike we impose material symbols and utterances with meaning in written and spoken communication.

Now, between substances and aggregates the substances are more ontologically fundamental. Or, as it has been put, substances are the most fundamentally real things. Of course, both aggregates and substances depend on their parts, but (1) aggregates are always made up of substances, and (2) with substances there is also a sense in which the parts depend on the whole. A full examination of (1) will require a deeper understanding of per se causal chains than we have space here to discuss so, as before, we’ll put this off until another time. We will spend the rest of our space here attempting to make inroads to understanding (2). Throughout these attempts we will being using the insight that what a thing is (its nature) is closely tied up with its directedness, characteristic behaviours, and causal powers. Indeed, we said last time and have noted elsewhere, that a thing’s nature just is what it is directed towards.

The actual existence of parts in aggregates

Now, an aggregate’s causal powers and directedness are by definition not intrinsic, but rather extrinsically derived from its parts and from outside. Thus we find that aggregates don’t really have a nature or existence over and above the substances that make them up and the ends imposed on them from outside. That is, their nature and existence are wholly reducible to extrinsic sources, and it is therefore by reference to these extrinsic sources that these aggregates are intelligible.

Furthermore, the substances that make up (or impose on) an aggregate retain their intrinsic directedness, characteristic behaviours, and causal powers. We use the term “actual existence” to refer to the way in which these substances exist, and say that they are “actually present” in (or around) the aggregate.

This is consistent with saying that, by virtue of being part of an aggregate these substances have their characteristic behaviours and causal powers influenced by one another. In this case they don’t take on new behaviours or powers, but rather have their behaviours and powers redirected through interaction with one another. Think, for instance, of how cogs influence each other in mechanical clocks or how pipes redirect the flow of water.

The virtual existence of parts in substances

What can we say, then, about parts of substances? A substance does have a nature and existence over and above its parts and outside imposition. It’s nature and existence are not wholly reducible to extrinsic sources, and it is therefore to some extent intelligible apart from these extrinsic sources. At this point we must tread carefully, for it is easy to misunderstand what is being said. In the interest of clarity, then, we make a distinction: a part of a substance can be considered in two ways, either (1) in itself or (2) as a part. In the former case we consider the part in isolation from the substance it belongs to, and in the latter case we consider the part in the context of the substance it belongs to.

We can illustrate this distinction with some examples. A previously mentioned example of a substance was a water molecule. We pointed out that “water boils at 100°C while hydrogen, considered in itself, boils at -252.9°C and oxygen, considered in itself, boils at -183°C.” On the other hand, hydrogen and oxygen, considered as a parts of water, boil at 100°C. In some ways this is obvious, for the continued existence of the water depends on the continued existence of the hydrogen and oxygen that make it up, and since water doesn’t combust when it comes into contact with fire it follows that neither do its parts. On the other hand, I’m sure that for many of us hearing something like “the boiling point of hydrogen is 100°C” causes somewhat of a knee-jerk reaction. In a way this highlights the point of the distinction. Presumably we have such knee-jerk reactions because of what we learnt in chemistry class. But chemistry, like many sciences, seeks primarily to understand the essential features of the objects it studies, and therefore typically studies these objects in isolation from outside influence. They therefore have little to say about these objects when considered as part of another. So such knee-jerk reactions are neither surprising nor hindering to our discussion.

Another example of a substance previously mentioned was an animal. For each of the organs an animal has we can consider it in itself or as a part. In this case we typically have the reverse intuitions as above: with the molecules we are accustomed to thinking about them in themselves, but with organs we are accustomed to thinking about them as parts. Consider, for instance, the claims “hydrogen boils at -252.9°C” and “the heart pumps blood”. The former tells us how hydrogen behaves in isolation from outside influence, and the latter tells us what the heart does in the context of the rest of the body. Now, considered in itself, “an organ is merely a clump of flesh which decomposes if left to its own devices.” Indeed, don’t we see this all the time with severed limbs and corpses? On the other hand, considered as parts organs “are each capable of their individual functions in the body (walking, grasping, thinking, sensing, pumping blood, and so on) and they are all capable of participating in the life of the animal”, where by “life” we mean the ability of a thing to “produce, conserve and repair its proper functioning as the kind of thing it is”. For example, compare what happens when you cut a severed hand (or some other non-living thing) and when you cut a living thing. The latter will repair itself to some extent whereas the former will do nothing.

You’ll notice from the examples given that the characteristic behaviours, directedness, and causal powers of things can differ quite significantly depending on whether we’re considering them in isolation and or as parts. Therefore, so do their natures: a clump of flesh is a significantly different kind of thing to a heart pumping blood, a free hydrogen atom is a different kind of thing to a hydrogen atom in a water molecule, and so on. An implication of this is that the things considered in themselves are not actually present in the substances they belong to, at least not in the same sense that the things considered as parts are. In these cases, we say that the things in themselves are “virtually” present in the wholes they belong to.

What of the parts considered as parts? They are directed by the nature of the substance and derive their causal powers from the substance. I don’t mean that the substance is something separable from its parts, for of course it is constituted by them. Rather, it is on account of the parts being configured together so as to produce a whole which is capable of more than the mere sum of its parts, each considered in itself, that the parts cease to behave like they would in isolation and take on a new nature grounded in the overall configuration itself. It is because of this configuration of the substance that the parts have different causal powers, behaviours, and directedness. It is in this sense that the parts all together take on the nature of the substance and share in it’s existence. And it is this sense that parts depend on the substance they belong to.

A hylomorphic account of virtual existence

Let’s summarise what we’ve said so far. In aggregates the parts considered in themselves actually exist, since they continue doing what they do in isolation. And although they continue behaving as they would in isolation, they can nonetheless have this behaviour redirected by the other parts of the aggregate. Finally, in aggregates there isn’t really a distinction between the parts considered in themselves and the parts considered as parts.

In substances the parts considered in themselves virtually exist, since they do not continue doing what they do in isolation. By virtue of how they are configured in the substance, they behave in a new ways which share in the existence of the substance. Finally, in substances there is a distinction between the parts considered in themselves and the parts considered as parts.

Now, we might be tempted to see the word “virtual” and think that what’s being claimed is that the parts of substances, considered in themselves, do not make any causal contribution to the substance they belong to. On the contrary, Scholastics call this “merely logical” existence and distinguish it from virtual existence. Something has merely logical existence when its existence is wholly dependent upon intellectual activity. Examples would be fictional stories, imaginary friends, hallucinations, and dreams. On the other side of the spectrum are substances, which, as we’ve been saying, have actual existence. In this context this entails that they “fully” exist independently of intellectual activity. Virtual existence sort of stands in between these two opposites: to some extent they have mind-independent existence, and to some extent they are dependent upon intellectual activity. This may sound slightly strange, but this conclusion is implicit in what we’ve been saying. Their partial dependence on intellectual activity derives from the fact that while they are part of a substance they do not behave as they do in isolation, and so it requires intellectual activity to “fill in” what’s currently absent. Their partial independence of intellectual activity derives from the fact that the parts could not be configured so as to constitute the substance unless they were as they are in themselves. For instance, it is precisely because of how hydrogen and oxygen molecules are in themselves, that they can come together to form a water molecule.

We can shed some light on this somewhat strange property of virtual existence by means of the Aristotelian theory of hylomorphism.[2] According to hylomorphism every material thing is composed of “form” and “matter”, where by matter we mean some otherwise indeterminate substratum and by form we mean the configuration of the matter that determines it to this rather than that. So stated, hylomorphism is completely general, and we illustrate it with three very different cases.

First, there’s the sense in which we’re talking about material substances like trees, dogs, humans, water molecules, wood, and so on. Our matter is the “stuff” we’re all made out of, and our forms are the configurations of this matter into the various kinds of material things there are. We humans have our matter configured in a way quite differently from how the matter in the tree outside or in my pet cat is configured. It is on account of these different forms that we have our distinctive behaviours, directedness, and causal powers, and on account of which we are called humans, trees, and cats.

Second, there are things like written or spoken sentences or pieces of music. With the sentence the matter would be the words or letters and the form would be the syntax together with some kind of “semantic coherence” (since syntax alone isn’t enough). With the music we have something similar, but I suspect there syntax is enough.

Third, there are actions. Here, I think, is where we begin to see the generality of the form-matter distinction. Consider the motion of my hand into your face. This movement itself is indeterminate between at least two possibilities: either I am punching you or I am doing something else and have hit you by mistake. As such, the movement is the matter of my action. What is the form? Surely it’s my intention. If I intend to hurt you then the action is me punching you, otherwise it is a mistake. So, while the form-matter distinction primarily applies to material substances, at the end of the day it goes far beyond this to almost everything.[3]

Now, using the form-matter distinction we can say the following. Something merely logically exists if it has neither form nor matter in reality, but is only understood or imagined in such terms. Something actually exists if it is constituted by the composition of form and matter which are intrinsic to it; that is, the thing is made up of its own form and matter. And something virtually exists if it is constituted by the composition of intrinsic matter and extrinsic form; that is, the thing contributes its matter to the form of something else, or equivalently the thing’s matter is informed by something else.

This captures, in the technical jargon of hylomorphism, what we were talking about earlier with the parts making contribution to the configuration of the substance and thereby participating in its existence. It also enables us to make sense of the partial dependence of virtually existing things on intellectual activity: such activity is necessary to “fill in” the missing intrinsic form, but not the matter.

Notes

  1. We’ve spoken about these kinds of generalisations before, at which point we called them “Aristotelian categoricals”. John Haldane, in his talk Aquinas and Realism, calls them “generics”. David Oderberg, in his paper Essence and Properties, calls them “properties”, along with Scholastics more generally.
  2. We won’t be able to do complete justice to the theory now, so if you’re interested in more than what I have to say here I recommend all the resources I’ve listed on my resources page under the sections “Hylomorphic dualism” and “Hylomorphism in general”.
  3. One might be tempted to equate form and structure. While in some cases these are the same (music pieces, for example), this is not true in general. See David Oderberg’s paper Is Form Structure? for a more detailed discussion of this.

Actualisation of potentiality as such

While we’re on the topic of confusing things Aquinas said, we can talk about his analysis of change, which he in turn gets from Aristotle.

We’ve noted before that the first step in analysing change is the realisation that it involves the actualisation of a potential:

When a hot cup of coffee gets cold, for example, what is happening is that the cup’s potential for the being cold is actualised by the coldness in the surrounding air… When I pick the cup off the ground and place it on the desk, I am actualising the cup’s potential to be a meter above the ground

But, as we noted, not all such actualisation of a potential involves change. The thing that sets change apart from other actualisations of potentials is that it involves the movement from potential to actual. It is on account of this that the ancients and Scholastic happily used the words “motion” and “change” somewhat interchangeably.

Now, while calling change the movement from potential to actual serves as a helpful start it is by no means the end of a satisfactory analysis. At the end of the day we want to know what this movement consists in, and we want it terms as basic as possible. This is where the confusing phrase from Aquinas comes in, for he says that “motion is the act of that which is in potentiality, as such.”[1] In this phrase Aquinas is abbreviating a slightly-less-confusing phrase from Aristotle who says that “change is the actuality of that which exists potentially, in so far as it is potentially this actuality.”[2]

To see what these two are getting at, return to the example of the cup’s resting on the table a meter above the ground. At any given moment, there are two senses in which this potential of the cup’s might be being actualised: first, by the cup actually resting on the table a meter above the ground and second, by me currently being in the process of picking the cup off the ground and placing it on the table. We might put it like this, given that I’ve started this process I’m eitherfinished it (the first case) or I’m still doing it (the second case). In both cases the cup’s potential for resting on the table a meter above the ground is being actualised, but only in the second case is this actualisation an instance of movement. In the first case the cup is sitting on the table a meter above the ground; in the second case it’s not there yet, but it’s on it’s way there. Put (rather verbosely) in terms of act and potency, in the first case the cup’s potential for resting on the table is being actualised and the cup is actually resting on the table, whereas in the second case the cup’s potential for resting on the table is being actualised and the cup is merely potentially resting on the table.

More generally (and symbolically), if we’re considering some object X that has some potential for P currently being actualised, then either X is actually P or X is potentially P. In the former case there is no movement toward P, since X is already P. In the latter case there is movement towards P, since the only way X can have this potential currently being actualised and not be there yet is if X is on its way to P. In the above example X is the cup, and P is “resting on the table a meter above the ground”.

Perhaps this diagram will help you, but if it doesn’t just ignore it. The arrow represents the motion of X to P. Notice how X’s potential for P is being actualised both when X is actually P and when X is potentially P. As we’ve been saying this is that the latter case is when X is moving toward P.

This, then, is what Aquinas and Aristotle are getting at: an actualisation of a potential is movement when, and only when, the thing being actualised is still potentially at its end. Or, more succinctly, movement is the actualisation of a potential while it is still potential.

Notes

  1. Summa Contra Gentiles Ch 13
  2. Physics 3.1 201a10-12

Analogy at the foundations of mathematics

Consider the Benacerraf identification problem in philosophy of maths: there are multiple different ways of “defining” natural numbers in terms of sets, so there is no way of determining which definition is the “correct” one. This is not just a problem about natural numbers but they’re a useful notion to introduce the problem with. In fact, it’s not even just a problem with sets, since there are even multiple foundations from which to choose to define the natural numbers (I’m thinking about type theory and category theory here, considered as foundations).

Now, I’m broadly Aristotelian about mathematical objects: we abstract quantity and structure from reality, isolate certain aspects of these (which we call axioms), and extend these abstracted notions beyond our experience. One of these abstraction-and-extensions is the natural numbers. Presumably something similar could be said for things like sets, groups, rings, topological spaces, metric spaces, the real numbers, the rational numbers, and so on. So, as far as I’m concerned (as are some other non-Aristotelians, I’m sure), the natural numbers are intelligible wholly apart from any set-theoretic “definition” of them.

So what do we do when we “define” natural numbers (or anything) in terms of sets? For starters, I don’t think we define them at all. Rather we characterize numbers in terms of sets (although we nonetheless talk about this in terms of defining or constructing). But what does that mean?

Consider a concrete example: characterize 0 as ∅ (the empty set), 1 as {∅} = {0}, 2 as {∅, {∅}} = {0, 1}, 3 as {0, 1, 2}, and more generally n = {0, 1, 2, …, n-1} for any natural number n. For the sake of brevity, call these sets which we characterize the natural numbers with the natural sets. Now the natural sets have certain features or aspects, partly due to being sets and partly due to their relations with one another (relations they might not share with non-natural sets). Similarly, the natural numbers have their own features or aspects. There are aspects of the natural numbers that aren’t aspects of the natural sets, and there are aspects of the natural sets which aren’t aspects of the natural numbers. The important thing is that there is a correspondence between some of the natural number aspects and some of the natural set aspects. Call this the correspondence of aspect. The extent and nature of the correspondence will depend entirely on which sets we choose and how we choose to relate them to the numbers. Our considered characterization, for example, quite naturally captures the notion of order between natural numbers. That is, n < k (considered as numbers) if and only if n ∈ k (considered as sets).

So far so good, but we can push this further. What does correspondence of aspect consist in? There’s a mathematical way to make this precise: isomorphism. But we’re talking about the very interpretation of mathematics itself, so that won’t work.

Perhaps we could look to metaphysics, particularly Scholastic metaphysics. Here we find the notion of analogy, and it turns out to be quite useful. Much could be said about this, but roughly analogical predicates fall between strictly univocal and strictly equivocal predicates. They are not used in exactly the same sense when predicated, but they are nonetheless related. An example is the notion “seeing” when we see a table (say) and when we see how a conclusion follows from the premises of an argument. Clearly we don’t “see” in exactly the same way in both cases (like the “ball” in “rugby ball” and “soccer ball“), but yet the two seeings aren’t completely unrelated either (like the “ball” in “matric ball” and “rugby ball“). We say that my seeing the table is analogous to my seeing the conclusion. Much can be said about analogy, but for our purposes it will suffice to note that this particular kind of analogy is called analogy of proper proportionality. In these cases we have a relation between relations: our eyes are to the table (relation 1) as our intellect is to the conclusion (relation 2). We write this as “eyes : table :: intellect : conclusion”.

Ok, now come back to our correspondence of aspect. It seems to me that this consists in analogy of proper proportionality (and perhaps sometimes in other kinds of analogy, but I don’t see where they’d apply at this point so I’m sticking with proper proportionality for now). For instance, the characterization we’re considering captures that each natural set is to ∈ as each natural number is to <. If you feel like conveying this in a particularly obfuscated way for just 0, then you could write “∅ : ∈ :: 0 : <“.

So, in summary, we don’t define numbers in terms of sets (strictly speaking). We establish a correspondence of aspect using analogy between sets and numbers, and study those aspects from the perspective of sets. More generally, when we try use sets as a foundation what we’re doing is finding correspondence of aspect using analogy between various structures or quantities and sets and studying them from that perspective. And depending on the correspondence and sets in view, we might end up studying slightly different collections of aspects of these quantities and structures. Although I suspect for many purposes the differences will be irrelevant.