Probably the most familiar ontology in my typology, and the most natural to English speakers, is a substance-ontology. The idea is that being has a basic structural dichotomy, noun-like substances, and predicates - things said of substances. My coffee cup is a substance, and it has properties like being mostly empty of coffee, or being black and silver; locations - on my desk, in Indiana; states - not in motion relative to my desk, legally owned by me; relations - smaller than a breadbox, larger than a coffee bean, and so on. On this picture the world is basically divided into a variety of things, and ways for things to be. Aristotle, one of the great examples of a substance-ontologist, says it well:
"Thus everything except primary substances is either predicated of primary substances, or is present in them, and if these last did not exist, it would be impossible for anything else to exist." (Categories 2b: 4-5, McKeon's translation)
Substance-ontologies are divided into several kinds, thing-ontologies, stuff-ontologies, bundle-ontologies, lattice-ontologies, and maybe some others.
In a thing-ontology, the world is primarily made up of things and ways for things to be. That is - substances are like count nouns: chairs, humans, coins, lizards, trees, and so on. Things carry individuation criteria with them in their own being. One lizard and another lizard are two separate things, and things are importantly countable, and individuatable.
In a stuff-ontology, substance works more like mass-nouns: water, bronze, money, justice, thought, etc. Water is not countable or individuatable, it's just too liquid to be discrete. We can say "three cups of water" but "three waters" sounds like a grammar error. Its not that stuff can't be individuated or counted, its just that it doesn't come that way already, you need a measure or an individuation criteria to do it, because that isn't inherent to its own nature.
A thing-ontologist might be very comfortable saying that my coffee cup is a substance; a stuff-ontologist might say that the metal and plastic in my cup are genuine substances and that my coffee cup is made out of them. Things are often called objects, stuff is often called material, or by Aristotle's term, hyle. In either case, properties are going to be ways for substances to be, rather than themselves beings (and it can get tricky to differentiate properties from relations, states, locations, and other ways for substances to be).
Stuff-ontologies go back at least to Anaxagoras among the pre-Socratic Greek thinkers, and even earlier arche theorists like Thales look pretty stuff-like.Ancient Greek atomism, such as Democritus, is a great example of a thing-ontology and atomism was present in ancient India too, under both Carvakan and Vaisheshikan styles. It seems that Aristotle's own position on things vs. stuff is too nuanced to fall snugly in either camp, but there are certainly interpreters who read him as a thorough stuff-ontologist. Some later Aristotelians are basically thing-ontologists with only a few nods to stuff, though. The Stoics are basically thing-ontologists or Monists; the world is divided into things which exist, and other ways of being which subsist. By the 1600s the West had swung back mostly to thing-ontologies, and the rise of atomic theory certainly supported this trend. Descartes, Berkeley, and Fichte are all basically thing-ontologists.
Bundle theory, which is well-developed in Hume, and (disputably) already present in Locke presents us with a somewhat tricky case, and my current temptation is to lump it as 3rd kind of substance-ontology distinct from both object based thing-ontologies, and material based stuff-ontologies. The idea here is that properties (or at least some primary properties) are genuine ontologically primary beings, but that substances only have a kind of derivative being as the locus of a collection of properties. As Locke says "... a philosopher ... whatever substantial forms he may talk of, has no other idea of those substances than what is framed by a collection of those simple ideas which are found in them ..." It is still the case for this picture that the world at its most basic levels is divided into things, and ways for things to be; but here the things are but placeholders for habitual patterns of collections of properties. Similarly, it is possible that what I will later describe as lattice-ontologies are really best thought of as a subvariety of substance-ontology similar to Bundle theory where substances are constituted by relations rather than by properties. Or perhaps both should be seen as distinct from substance ontologies, because they privilege properties or relations over substances.
Patrick Suppes in Probabilistic Metaphysics argues that old Aristotelian stuff-ontology makes better sense of 20th century physics than many other pictures. But even fairly recently, thing-ontologies have been tempting to high-end philosophers in the West. Frege's logic assumes two kinds of beings, objects and functions, which fit the basic substance division, of things and ways for things to be. Substance-ontologies make a lot of sense for speakers of a subject-predicate language, and my guess is that most speakers of Western languages use common-sense substance-ontologies of some kind unless they get exposed to a lot of philosophy. It is still common to refer to substance/property ontologies as "traditional" metaphysics.
The most famous of what I'm calling a type-ontology (in the West) is Plato's theory of the forms. The idea is that forms, or types, or categories are the most fundamental layer of being: human, red, just, circle. The goal is to efface the noun-like/predicate-like distinction, and imagine the important beings as able to stand on either side of it. A type or form is amphibiously noun-like and adjective-like. We can say of a rock that it is circular, or of a circle that it is red. The form circle can be said of other beings, or other beings can be said of it. This seems grammatically counter-intuitive in Germanic languages, but in Greek and in Romance languages it is common to have terms that are amphibiously adjectives or nouns. "Alba" can mean "white" the adjective or "white thing" with equal ease in Latin. In the Platonic theory of forms, what really ultimately exists, are the forms: the white, the circular, the red, the just, the good, and so on. Or perhaps we might name them whiteness, circularity, redness, justice, goodness, and so on. My coffee cup will then be a derivative being, dependent for its being on the many forms in which it participates, cupness, circularity, blackness, particularity, and so on.
It would be tempting to think of a form or type, say the red or redness, as the set of all red things, as if types were built out of tokens, but that would be to use a thing-ontology instead of a type-ontology. In a thing-ontology there are things which have the property red, and then another thing, a set, which contains all red things. But, for a type-ontology, red things are only red because they participate in the form of the red; the form is fundamental and the things only derivative. Tokens are built out of types. The red is the being it is on its own, and causes red things to be red, or well if not "cause" then the form "constitutes" red things as red, or even better "red things depend in their being red on the being of the red."
Thus, instead of a fundamental divide between beings and ways for beings to be - type-ontologies have every being also being a way for other beings to be, the cosmos is an array of combinations of beings.
Type-ontologies also do a good job with category hierarchies, because types participate in other types. The human is a variety of the mammal, which is a variety of the vertebrate, which is a variety of the animal, the living, the physical and the existent. Type-ontology is a kind of anti-atomism, instead of the world being built up from tiny basic things, into larger and larger collections of things which are themselves things of a different kind, it imagines the whole of being being sub-divided into types, and sub-types, and sub-sub-types, until reaching the level of particularity.
Properties exist in their own right on this picture, but there is no real dividing line between properties and things, things are just particular properties.
In the West, Pythagoras develops a type-ontology by extending Anaximander's thought that the unlimited is the arche, by asserting that the unlimited and the limited are jointly the arche, and that number and ratio are the primary layer of being. Plato modifies the role of the limited, to get a less numerical picture of form. Type-ontologies of Pythagorean, Platonic, or Neo-Platonic styles are interacting with substance-ontologies in the West until modern philosophy when they fall out of favor.
I can't think of any Indian type-ontologies off-hand. But normal Chinese thought is type-ontological at least until philosophers get involved. Confucius appears to think type-ontologically, without highlighting this fact. Logician Kung-sun Lung appears to have directly advocated this kind of picture in ancient China, perhaps even against the thought of logician Hui Shi.
In much the same way that substance-ontologies just seem like normal common sense for languages that have rigid subject-predicate distinctions, type-ontologies just seem like common sense for languages that conflate nouns and adjectives, like Chinese. Even after philosophers start proposing other options in China, many Chinese philosophers retain allegiance to type-ontologies until the 20th century, (and probably do unconsciously still when they aren't being careful).
Another thoroughly ancient approach to ontology, is to insist that being is united and singular, that there is really only one being, which might be called Being, or The One, or The All, or Nature, or the Cosmos, or Brahman, or God, or Prakriti, etc. If being is radically one, then it doesn't really matter "one what?" Monists can say that the one being is the only substance, or the over-type, of the total event, or the one great fact, or the field of all being, or whatever. Monism becomes a kind of end-point for all other ontologies. However, everything else, becomes a secondary existence, of non-ultimate existence of some kind. Everything else that is, is a way for the one being to be. As Spinoza puts it in Ethics, Prop 6:
"the modes of any given attribute are caused by God, in so far as he is considered through the attribute of which they are modes, not so far as he is considered through any other attribute."
Everything that is, is a part of God, and is caused by God, as a mode of the being of God. But this is so, of people, and circles, and redness, and events, and so on. One danger is that our categories of being will be swamped by the single distinction of ultimate being vs. everything else, and so any Monist that wants to do metaphysics (like Spinoza or Plotinus) will need to try to rescue differences within the "everything else" category to some extent. In this account, properties will be non-fundamental beings, modes of the being of the one true being at best, and there isn't much reason to hope for properties to be very distinguishable from relations, or events, or things, unless the particular Monist-ontologist does a lot of work.
Monist-ontologies are very old in India. They go back at least to Parmenides in Greece. Both in the West and in India they received both religious and naturalist elaborations. Plotinus and some later Christians and Muslims, think of the one true being in God-like terms, as does Sankara (opening them to charges of pantheism). Thinkers like Parmenides or Sartre think of the one being in fairly naturalistic terms (opening them to the charge of atheism). Thinkers like Spinoza, the Stoics, Einstein, or the Samkhya-darsana are willing to admit that the one being is an appropriate target of both religious and naturalist impulses. It is worth mentioning, that four-dimensionalist pictures of the cosmos even from physicists often wind up sliding into Monism if they aren't trying hard to resist it. After all they make the whole of time look like what is, and all events, facts, things, properties, etc., look merely like modes of being of the whole of the cosmos. Monism has been popular in India in some times, but it is at least present if not popular, in almost all times and places, because it is a sort of end-point for any other ontology.
Another classic strategy on fundamental ontology is to give up, refuse to answer, or retreat to perplexity, what the Greeks call an aporia. As Plato says in the Sophist:
"For manifestly you have long been aware of what you mean when you use the expression 'being.' We, however, who used to think we understood it have now become perplexed." (The Sophist, 244a)
The Skeptics suspended judgment on the question of the most fundamental being as they did on other topics. Skepticism of the fundamental nature of being was present in ancient India too. In ancient China, Zhuangzi argues that fundamental ontology is the wrong project to pursue throughout chapter two of the book named after him. Some ancient and medieval aporetic-ontologists were not skeptics, and were willing to make lots of claims on many topics, but as one approached the fundamental layer of being, they retreated into negative theologies, apophatic theologies, and other elaborate devices to avoid logizing the ontos. Pseudo-Dionysius and Maimonides are good examples. Kant develops elaborate metaphysics, but his price for doing so is giving up on the question of fundamental ontology, even claiming that "it is precisely in knowing its limits that philosophy exists."
Even Heidegger is in this camp. His famous Being and Time, raises the question of the meaning of being, and seeks to once again tackle a question he feels has been covered over and forgotten by the philosophical tradition, the question of the most fundamental meaning of being. He argues that our being, which he calls da-sein, is the being to be interrogated to pursue the question of the meaning of being, and that "thus, fundamental ontology, from which alone all other ontologies can originate, must be sought in the existential analysis of da-sein" (Being and Time, p. 11).
But Heidegger sees his task has having two parts each with three divisions. Part two would be "destructuring of the history of ontology on the guideline of the problem of temporality (p. 35)." He never wrote part two of Being and Time, but didn't really need to for his project to work. Part one was "The interpretation of Da-sein on the basis of temporality and the explication of time as the transcendental horizon of the question of being (p. 35)." And it was to have three divisions,
1) the preparatory fundamental analysis of Da-sein,
2) Da-sein and temporality, and
3) Time and Being.
So the idea was we try to understand our being in division 1, use that to understand temporality in division 2, and use this to understand the meaning of being (as opposed to just our being) in division 3 (and then argue against three rival pictures in Part 2). But Heidegger never wrote Division 3! Thus Heidegger's REAL answer to the question of the meaning of being, is to work diligently on the project for 400 pages, and then give up 2/3rds of the way to the answer! He just never found a way to make his mediations on the relation between our being and temporality shed any light on being in general. It is possible, but controversial, that even the later works of Wittgenstein are ultimately aporetic with regard to fundamental ontology.
2.5 Factor-Ontologies or Dharma-Ontologies
The Buddhist tradition develops a different picture of fundamental ontology early on, traditionally called dharma theory, or Abhidharma theory. Within a few centuries of Buddha's teaching a series of texts called the Abhidharmas (roughly, "advanced teachings") arises. In the words of Paul Williams, a scholar of Buddhism, they are "lists which enumerate with maximum possible exactitude what is actually occurring in a particular psychological or physical situation spoken of in the Sutras or in life generally. The lists are lists of what is seen to be the case by one who sees things the way they really are (Buddhist Thought, 2000, p. 88)."
We have a reduction program in which normal and unusual "occasions" are to be analyzed into their components, the skandas or "heaps", which are in turn to be analyzed further until we reach the ultimate level of being. Those elements which remain even at the ultimate level are called dharmas, a Sanskrit word that elsewhere means everything from "thing" to "truth" to "teaching" to "factor" to "topic of discussion." So for example, "eating a strawberry" consists of some volitions or habits (samkhara), consciousnesses (samjna), sense-consciousnesses (vijnana), feelings (vedana), and physical forms (rupa).
The vijnana will further divide into particular strawberry-taste-consciousnesses, strawberry-mouthfeel-consciousnesses, and so on. The material form of the strawberry will divide into solidity, fluidity, energy, and motion, and perhaps some other physical factors dependent on these. But the particular solidity of the strawberry cannot be analyzed any further. Solidity is simply one of the basic factors in physical occasions.
Dharmas are primary existents (dravyasat) and are what all other existents are composed of. They have "own-being" (svabhava), which is how they differ from secondary existents, like chairs, or persons which do not have own-being. Each dharma has an "own-mark" (svalaksana), which is how it differs from every other dharma. A dharma is a genuine factor in the ultimate ontological analysis of complex occurrences. Later Buddhist schools disagree with each other over exactly how many kinds of dharma there are, how to categorize them, what the best way to analyze particular occurrences into dharmas is, and what dharmas themselves are like.
In the Sautrantikan school, dharmas look a lot like short events, or even what Dretske calls event-aspects, and they exist only in the present. For them what it is to be is to exert activity, and only present factors do that. Whereas in the Sarvastivadin (literally "everything exists") school, dharmas in the past and future do not exist in quite the same way that present dharmas do, but they do nonetheless still exist. Sarvastivadins do, however, admit that present dharmas, and not past or future ones, have a sakaritra, or "characteristic activity," which is the present form of its own-mark, or characteristic feature. Many of these schools die out; only Theravada, retains this picture of Buddhism today.
Where does this leave us for properties? Well some things that look like objects do wind up being dharmas: space, greed, pleasure, attention, and so on. But other dharmas look more like properties, red, for example, is a particular visual-sense-consciousness, or in most cases a particular visual-sense-contact, but it is a real factor in the ultimate analysis of occasions. Contrariwise, many properties will wind up being secondary existents, rather than dharmas; personhood is a real existent, but it is a secondary one, not a dharma at all (for most schools of Buddhism).
What is ultimately real of beings on this picture, comes in many varieties, and these varieties combine to give us the complex occasions we encounter. Factor-ontology winds up being a kind of logical atomism, that isn't much like physical atomism, or any of the metaphysical pictures of the West, except perhaps the event-aspect metaphysics that is beginning to be explored now.
Somewhere around 0 BCE, a newer form of Buddhism develops which calls itself Mahayana, or the great vehicle. One of its chief characteristics is to reject the dharma-ontology discussed above and replace it with something called prajnaparamita - "the perfection of wisdom." At least early on, the point does not seem to be that dharma-metaphysics is wholesale wrong, but rather that it, too, is not the most deep or fundamental truth of things; that it is a metaphysics but not really an ontology.
Exactly what was supposed to replace the dharma picture as our understanding of the deepest reality of things, was taken even by the Mahayanists to be a mysterious, difficult picture to express. See my longer work "Logical Atoms and the Astasahasrika Prajnaparamita Sutra," for a fuller discussion. But the basic story goes like this.
Give up all talk of dharmas, or factors, or things, or properties or occasions, none of that is ultimate. Being is "empty" of all that. Things just are the way they are. "Thusness" or "suchness" is the main feature of genuine being. Being is "just like this." If we are to speak with precision (such as in a merit dedication formula) we say "let the merit, such as it is, generated donating this book, understood as the Buddhas understand it, to this monastery, understood as the Buddhas understand, be dedicated to the leading all beings, taken as they are seen by the Buddha, to ultimate enlightenment, as it is understood by the most holy ones."
The danger is that we will falsely reify such concepts as merit, book, monastery, all beings, or enlightenment. Dharma metaphysics allows us to avoid reifying books, or selves, or monasteries, but at the cost of reifying the ultimate factors, space, sense-consciousness, etc.
Prajnaparamita metaphysics says avoid reifying anything by your verbal expressions, let being be as it is. The solution is to repeat, "such as they are," or "such as they are understood by the Buddhas." The ontological picture is not intended to amount to Monism or nihilism, but to walk a middle path between falsely reifying what does not exist, and denying the existence of what does. It is almost as if the ancient Buddhists agreed with Wittgenstein's first claim, "the world is all that is the case" but disagreed with his second claim "the world is the totality of facts, not of things" instead thinking "the world is not the totality of facts, or things, or properties, or factors, or dharmas, or activities; it simply is as it is."
Prajnaparamita-ontologies do not reject properties or things entirely, they are among those things that exist conventionally, but they are simply not present at the deepest level of reality. For these pictures, it is often appropriate to speak of properties or things or factors, when our goal is not to speak about ultimate reality, or not to speak with aching precision, but instead to converse with people on day to day matters or to introduce them gradually to the deeper aspects of reality.
Tu Shun lived from 557-640 CE in China, and was the first Patriarch of the Hua-Yen or Flower Garland school of Chinese Buddhism, which focused on the interpretation of the Sanskrit Buddhist text the Avatamsaka Sutra (which existed by 420 CE and parts or all of it are probably earlier than that). Tu Shun thinks he is just interpreting the Avatamsaka Sutra, and that may be so, or perhaps his commentary is improving or systematizing the ideas of the Sutra. His school teaches that we come to understand reality in a series of stages.
Tu Shun's succinct formulation is: 'First, one in one. Second, all in one. Third, one in all. Fourth, all in all.'" (Buddha Boogie, "The Tautological Paradigm", pg 305).
Object/property metaphysics, Monism, and the prajnaparamita picture, are each correct stages of understanding, but none are the ultimate layer of being, which is the "all in all." The Avatamsaka Sutra uses the metaphor of the Jeweled Net of Indra (the king of the gods). Imagine a vast net (that is a lattice) with a jewel at each juncture. Each jewel reflects every other jewel in the net, so that every jewel stands in "relation" or "connection" with every other jewel, so that any change in any jewel is reflected in some way in every jewel. Now let each jewel represent an individual life form, cell, atom or unit of consciousness. For the Avatamsaka Sutra, and Tu Shun reality at its most basic is a vast array of inter-relationships. Individual objects are constituted by their pattern in the whole, and indeed their distinctness and independent existence is fairly suspect. They are at all only because of their network of interrelationships to all other beings.
In Bundle theory, we saw a picture where substance is dependent in its being on primary properties which are ontologically fundamental. A lattice-ontology imagines relations, rather than properties or objects to be fundamental, and properties and objects to be derivative upon the relations.
Interestingly, Leibniz in the West advances almost exactly the same picture in his Monadology. One of the principle features of monads is that each reflects the entire universe. In Leibniz, though, monads are thought of as "substantial forms of being." So probably it is best to think of him as a substance-ontologist, who is trying to resist both thing and stuff pictures. It is only if one goes one step further and says that interrelations are the fundamental beings, and substances are merely modes of interrelation that the ontology starts looking distinct from substance-ontology. My guess is that this kind of picture is going to become slowly more popular. Substance-ontologies work great for mechanical views of the world, but Vitalist and ecological views tend to fit better with process-ontologies or lattice-ontologies. Also from a modern mathematical perspective it is very natural to think of properties as unary relations, and to think of k-ary relations as interpretations of k-ary predicates, so it is fairly easy to think of properties and predicates as degenerate cases of relations.
If we think of substances and properties as nouns and adjectives, then it is natural to look for a metaphysical category for verbs or adverbs. Western traditional metaphysics has a lot of trouble with verbs and verbal phrases. It tends to assimilate, adjectival phrases, prepositional phrases, and verbal phrases all to the broader metaphysical category of predicates, and then have troubles distinguishing them again.
In Aristotle, for example, action is categorically parallel to quantity, quality, relation, place, position, state, time, and even affection. How exactly verbs are supposed to work is a source of controversy and confusion in India, too. But you could, in theory, make verbs or adverbs the primary layer of being, and try to make noun or adjective-like modes of being look derivative. Modernly, this is the goal of Process-ontologies. It has been suggested that some parts of ancient Chinese thought look a lot like process-ontologies or precursors of them, especially reflection on the Yi Jing (the Classic on Changes), or Yin-Yangist reflection on the wu hsing (5 elemental processes).
But even trickier is the thorny question of how to interpret early philosophical Taoist thought, such as the Daodejing (the Classic on the Way and the Power) attributed to Laozi. Here the fundamental ontological category is the Dao or Tao or "Way." It is the origin of heaven and earth, and the mother of all things, but is not itself very thing-like. The very first line of the text (in its modern versions) is "tao k'o tao fei ch'ang tao" or "The 'way' that 'can be' 'wayed/walked' 'cannot be' the 'unchanging' 'way.'" The word "dao" appears 3 times in 6 words: as a noun, a verb and a modified noun. It is a path and a walking of a path, a river's course and its coursing of its course. And changing is essential to it. Exactly what this text is trying to say is a subject of much contention and interpretation, but it is pretty clearly rejecting substance and type ontologies.
Maybe it too is advancing a process-ontology of some kind. Maybe it is essentially a kind of verbal-monism, in which all things are modes of the way. But this can be interpreted as a kind of adverbialism where all things other than the way are at root adverbs, ways for the way to way. One of my old colleagues liked to put it like this: The world is appearing to me treely. The world is happening stop-the-war-in-Iraqly. The world is appearing to me confusedly. The world is happening driving-down-the-roadly.
If adverbs are the fundament of being, then processes would merely be constituted by the more fundamental ways to process, and objects and properties would be merely odd ways of trying to express adverbial realities. A property like being red, winds up being a way of happening redly. I'm not entirely convinced that anyone, even the Taoists, have actually advanced an adverbial-ontology, and its not clear to me what you buy yourself ontologically by making adverbs fundamental, but its possible that this is what some Taoists are trying to do, and even if they aren't it's a logically possible form of ontology.
There is no real requirement to say that at the basic level reality is of one grammatical category. Even traditional western ontology usually acknowledges both substances and predicates as jointly fundamental. You could, in principle, hold that objects and factors are both fundamental, or that processes and types are, or that the one substance (Nature) and variety of ways for the subject to behave are jointly fundamental. You could even say that all these ontological categories are among the ways for being to be, and plenty more besides, and that forms of being are not typically built up out of more basic forms of being, but rather that there are lots of varieties of primitive or fundamental being. But this is not usually popular.
There is some kind of powerful reductionist impulse in the history of world ontologies, whereby most ontologists seem to want to reduce other categories of being to some one or two fundamental ones. Perhaps this is because people are reductionists at heart, or because one's native language has a kind of trump, or maybe it is because Unificationists are right about how explanation works.
Medieval Neo-Confucians in China, (Like Zhu Xi) interpret the Yi Jing as describing two distinct but inter-related layers of reality "the tao of every class of things" and "the tao of the transformation of all things." Maybe Zhu Xi is best thought of as hybrid-ontologists who admit the fundamental reality of both process and type. Maybe. Similarly, contemporary western thinkers Casati and Varzi grant that you could think of events and objects as equally ontologically fundamental (in their Stanford Encyclopedia of Philosophy article "Events"), but they don't give any examples of thinkers who actually do. Hybrid-ontologies seem to be theoretically possible, but simply not very tempting, both anciently and contemporarily.
2.10 Hamiltonian mechanics, Field-Ontologies, Group-Ontologies and Others
By the 1800s Western mathematicians are inventing ontological categories left and right, but not generally arguing for them as being fundamental. In 1788, LeGrange had found a way to mathematically reformulate Newton's mechanics, so that the results were all the same, but the equations were (sometimes) much easier. Legrangian mechanics did not and does not seem to have altered anything important at the ontological level. William Rowan Hamilton invents quaternions in 1843, and then some of his allies such as Peter Tait argue for restating the claims of physics in terms of them. Maxwell's famous equations are stated in quaternion notation in 1865 in his own research, but the quaternion notation isn't really that far from our normal intuitions about objects as occupying space and time. Indeed, a big chunk of Hamilton's goal is to show why the one dimension of time seems to work so differently than the three dimensions of space.
Hamilton's ontology leaves objects and properties intact, and merely tries to situate them in a 4-space in which three of the dimensions work differently than the fourth. In a Hamiltonian formulation of a classical mechanical system we have three basic ingredients, states (which in many ways are algebraic precursors of Wittgenstein's states of affairs), observables (which work just like properties), and dynamics (by which we can represent conjugate momenta, via only 1st order differential equations.) Hamilton is trying to re-understand space and time, and he doesn't think he is shaking object-property metaphysics at all. But in order to make equations for figuring out object-property physics easier, he is taking short-cuts through quite different ontological categories, such as "states" and "dynamics" which are probably setting the stage for later fact-ontologies and process-ontologies.
But by the late 1800s other mathematical physicists are moving to a more radical position, that fields, especially vector-fields or later tensor-fields or spinor-fields, are fundamental. A vector-field is an assignment of a vector (a quantity + direction) to every point of an N-space of some kind. So, for example, the gravitational field of a single large mass (say the earth), would be a collection of force vectors pointing towards the center of the earth, with magnitudes depending on (roughly) the inverse square of the distance to the center of the mass. The gravitational field of a two mass system, (say the earth and moon) is more complex, but still quite mathematically tractable.
Newton imagines gravitational attraction as a thing, namely a "force" with properties like magnitude and direction. You could think of a field as an infinite array of possible forces depending on one's location. But it is more natural to think of a field as a single entity, but one that isn't very thing like. It spreads over a whole space and can be fully co-spacious with other fields. It is natural to think of the two-body gravitational field as the product of two distinct one-body fields.
During the 1800s fields are becoming more and more prominent in physics. Electricity, magnetism, gravitation, hydrodynamics, and meteorology all start using them. They begin as notational devices, but increasingly become ontological categories. Mathematical physicists like Gibbs, Heaviside, and Lord Kelvin, argued for physical theories to be expressed in field notations. Gibbs, for example, believed that mathematics IS a language, and thus that the job of the theoretical mathematical physicist is to improve the language in which we can express physical truths. Thus moving to vector-field notations amounted in Gibb's mind to linguistic-ontological reform in how we think and speak about the fundamental ontology of physical entities. The Gibb-Heaviside field picture of physics is powerful, flexible and extremely abstract, relying (intentionally) heavily on high end algebra and analysis. It is intended to be as general to matter as possible and to work well on many scales and contexts, rather than to focus on matter's structure at very-small scales, or very large ones. I'm inclined to think it is a departure from traditional object/property metaphysics (fields aren't really either) and from natural language (preferring the artificial language of algebraic analysis). The Gibb-Heaviside picture is still used a lot, especially for physics and engineering at super-atomic scales, but it's rare to find folk who think it is fundamental anymore. It is certainly also possible to use field notations in physics without particularly subscribing to a field-ontology. Einstein, for example, described his theories as "A true triumph of the methods of the general differential calculus founded by Gauss, Riemann, Christoffel, Ricci, ... p. 626." Thus, for him, the fields used in differential calculus were methodological innovations, not insight into how things are.
Field-ontology isn't the only attempt to make 19th century mathematical entities fundamental constituents of being. My old differential equations teacher, Pimon Ajanapon, an ex-Buddhist monk and mathematician, was trying to build an ontology in which sets would be the only fundamental beings, but as far as I know he never got very far.
More importantly, Max Klein began a project in algebra called the Erlangen Program which wanted to use group theory to look at invariants over transformations. After the success of the Lorenz transformations, this started looking like it might be the path to fundamental physical ontology, and motivated a lot of the work of the 1930s and 1940s.
The Weyl-Fock-London gauge theory was popularized by Pauli in the 1940s. By 1947 Hungarian physicist Cornelius Lanczos is arguing that all physical laws are self-adjoint variational principles, in effect that gauge groups are the presence of physical laws at the level of fundamental ontology. Yang and Mills explored non-abelian gauge symmetry groups in the 1950s and later models, such as the Standard Model and string theory continue to have a hefty gauge group component today. If I am understanding correctly, and that's a big if, particles (that is objects with properties using the traditional ontology), fields (now often tensor or spinor fields, rather than the more basic vector fields), and gauge groups are all used in many formulations of physical theories. The Standard Model, for example, is expressed in gauge group notation, but can derive particle-notations and field-notations with a fair bit of ease.
It seems that some physicists think of the particles as fundamental and the fields and gauge groups as notational schemes to make the math nice, whereas others think of the fields, or gauge-groups as fundamental. On the other hand, "the Standard Model" is probably not all the way to fundamental yet. The question is, if it could be extended to a model that included gravity as well as electromagnetism, strong nuclear, and weak nuclear interactions, would it rely on particles, fields, gauge-groups, all three or something else? Gauge groups are, or perhaps are close to being, sub-species of the forms that Plato was on about, they are invariants over local and global transformations. It is possible that gauge group ontologies are really best thought of as a contemporary sub-variety of type-ontology, but I'm suspending judgment on that at the moment.
There is an active contemporary group called the IIFB, the International Institute of Field-Being, it was formed in 1996 and holds symposia and meetings in conjunction with the American Philosophical Association. Its goal is the study of non-substantialist ontologies in both the East and West, and has roots in American study of Chinese thought and Chinese study of American thought. While fields are part of Lik Kuen Tong's ontology, the IIFB are really pretty clearly what I'm calling process-ontologists. See http://www.iifb.org/. They categorize all ontologies into two kinds, substantialist and non-substantialist, which seems to me to be running together a lot of alternatives, but gives a feel for where the current thinking on typologies of fundamental ontology is prior to this paper.
Continued in "A Taxonomy of Fundamental Ontologies, Part 3"