On Ontology, Existence, and the Objects of Reality
Thus far in these essays, I have argued from my metaphilosophy to my general philosophy of commensurablism, which is any philosophy that is neither dogmatic nor cynical, and neither transcendent nor relativist.
Then I explored the implications of commensurablism on the philosophy of language, including both logic and rhetoric.
In this essay I will now start to explore its implications on the specific subtopics of philosophy concerning reality and knowledge, beginning with ontology.
Ontology (from the Greek word ontos, meaning "being") is the study of being, as in existence, or reality. It is about the kinds of things that exist, and what it is to exist, or to be real. As such it is the core field of the slightly wider field of metaphysics, and often simply what it meant by the latter term. I think of it as the study of the objects of reality, in that it is about the things that are real, the things being described by descriptive speech-acts. I would also characterize it as being about the criteria by which we judge such descriptive speech to be correct, inasmuch as descriptive speech makes claims about what states of affairs are real, and ontology is about what it is for some state of affairs to be real.
Philosophers commonly distinguish between two different kinds of existence: concrete and abstract. Concrete existence is the kind of existence that familiar physical things like rocks, trees, tables, and chairs have, though it is not conceptually limited to physical things, as some philosophers hold that there are non-physical things that concretely exist. Abstract existence is the kind of existence that mathematical objects like numbers might have; as well as what philosophers call "universals", like redness, roundness, or bigness; and fictional things, like the One Ring from the Lord of the Rings books; if any of those can be said to exist at all, which is a topic of philosophical contention.
I do not draw a sharp division between all things that exist into one of these two categories, but rather hold them to overlap significantly, with the vast majority of existing things being technically abstract objects posited as explanations of the concrete world that we most directly experience. Those are instrumental abstractions, and I will explain soon how I hold that most of the things we most commonly consider real, like those rocks, trees, tables, and chairs, are to some extent abstractions, that are nevertheless real inasmuch as they explain more concrete phenomena.
In the remainder of this essay, I will first discuss concrete existence, and then proceed to discuss instrumental and purely abstract existence.
On Concrete Existence
As should be expected from the positions already argued for in my previous essays on commensurablism, and especially against relativism and against transcendentalism, my general position on the nature of reality is empirical realism. To recap the argument for that position:
- The task of philosophy is to find a way of discerning correct answers from incorrect answers to questions of any kind, whether about what is real or true or existent, or about what is moral or good or valuable.
- The position that there is such a thing as a correct opinion, in a sense beyond mere subjective agreement, is to be called "universalism", and its negation "relativism".
- If we assume relativism rather than universalism, then in case there does happen to be such a thing as the correct opinion after all, we will never find it, because we never even attempt to answer what it might be, and we will remain incorrect forever.
- Therefore to successfully do philosophy at all we must at least tacitly assume universalism, rejecting relativism.
- Universalism about reality or existence is to be called "realism": holding that some things are actually really existent, not just merely perceived or believed.
- Therefore to successfully do philosophy about reality or existence (ontology) we must at least tacitly assume realism.
- The position that there is always a question as to which opinion, and whether or to what extent any opinion, is correct, is to be called "criticism", and its negation "dogmatism".
- If we assume dogmatism rather than criticism, then in case our opinions do happen to be incorrect after all, we will never find out, because we never question them, and we will remain incorrect forever.
- Therefore to successfully do philosophy at all we must at least tacitly assume criticism, rejecting dogmatism.
- The position that any contest of opinion is to be settled by comparing and measuring the candidates against the common scale of the experiential phenomena accessible by everyone, and opinions that cannot be thus tested are thereby disqualified, is to be called "phenomenalism", and its negation "transcendentalism".
- Phenomenalism is entailed by criticism: if we are going to hold every opinion open to question, we have to consider only opinions that would make some experiential, phenomenal difference, where we could somehow tell if they were correct or incorrect.
- Therefore, since to successfully do philosophy at all we must at least tacitly assume criticism, rejecting dogmatism, we must likewise at least tacitly assume phenomenalism, rejecting transcendentalism.
- Phenomenalism about reality or existence is to be called "empiricism": appealing to sense experiences for descriptive justification.
- Therefore to successfully do philosophy about reality or existence (ontology) we must at least tacitly assume empiricism.
- Therefore to successfully do ontology we must at least tacitly assume empirical realism.
That is to say, to hold that there definitely is a universal reality, as opposed to any kind of relativism, which hold that what is real is relative to someone's beliefs or perceptions, or else (as I consider equivalent to those) that nothing is actually real at all. But to also hold that the content of that reality is entirely empirical in nature, that there is nothing real that is in principle beyond all observation, that if something exists, there will be some noticeable difference in the reality that we experience compared to what we would experience if it did not exist, and the whole of that thing's existence is the observable differences in reality it makes.
This empirical realism might well also be called physicalist phenomenalism, in that it holds that only physical phenomena exist – which is to say, things that are observable (phenomena) in a universal (physical) way accessible to all observers and not mere figments of any one person's imagination. This kind of view traces back to at least John Stuart Mill, who held the
permanent possibilities of experience to constitute the entirety of an object's existence.
This is a kind of monism, holding that there is one kind of stuff that exists that all the many things in reality are made up of, in contrast with pluralist ontologies that hold that there are multiple fundamentally different kinds of stuff, especially with dualism as espoused by the likes of Rene Descartes which holds that there are wholly different mental and physical kinds of stuff. It is not quite the usual monism held contrary to that dualism, namely materialism, though as described above it is definitely physicalist; nor is it quite the other usual kind of monism, idealism, though as described above it is definitely phenomenalist.
Neither is it quite neutral monism in the usual sense, as espoused by the likes of Baruch Spinoza, as that holds that there is one kind of stuff that has both mental and material properties; whereas I hold, as will be elaborated by the end of this essay, that there are not so much different kinds of properties, much less different kinds of stuff, as there are what could crudely be called mental and material ways of looking at the same properties and the same objects, that are essentially both mind-like and matter-like in different ways, that distinction no longer really properly applying when we get down to the details.
On the Indubitability of Reality
I would say that the most concrete things that exist are, as Alfred North Whitehead called them,
occasions of experience. These are the things of which we have the most direct, unmediated awareness, and the only things of which we can have no doubt.
Rene Descartes famously attempted to systematically doubt everything he could, including the reliability of experiences of the world, and consequently of the existence of any physical things in particular; which he then took, I think a step too far, as doubting whether anything at all physical existed, but I will return to that in a moment. He found that the only thing he could not possibly doubt was the occurrence of his own doubting, and consequently, his own existence as some kind of thinking thing that is capable of doubting.
But other philosophers such as Pierre Gassendi and Georg Lichtenberg have in the years since argued, as I agree, that the existence of oneself is not strictly warranted by the kind of systemic doubt Descartes engaged in; instead, all that is truly indubitable is that thinking occurs, or at least, that some kind of cognitive or mental activity occurs. I prefer to use the word "thought" in a more narrow sense than merely any mental activity, as I've touched upon in my previous essay on language, so what I would say is all that survives such a Cartesian attempt at universal doubt is experience: one cannot doubt that an experience of doubt is being had, and so that some kind of experience is being had.
But I then say that the concept of an experience is inherently a relational one: someone has an experience of something. An experience being had by nobody is an experience not being had at all, and an experience being had of nothing is again an experience not being had at all. This indubitable experience thus immediately gives justification to the notion of both a self, which is whoever the someone having the experience is, and also a world, which is whatever the something being experienced is. One may yet have no idea what the nature of oneself or the world is, in any detail at all, but one can no more doubt that oneself exists to have an experience than that experience is happening, and more still than that, one cannot doubt that something is being experienced, and whatever that something is, in its entirety, that is what one calls the world.
It is of course possible that oneself is the world, that self and world are the same thing. But then one just has solipsism, which is trivial: even if the world just is oneself, there is still a divide between the parts of it that one has direct knowledge and control over, and parts that are beyond ones knowledge and control. Even if that whole world-self is all there is, there is still the same practical reason to investigate into and act upon the parts of it that seem 'other' exactly as one would if it were in fact other. In the end, identifying the world with oneself (divided into a known and controlled part and a largely unknown and uncontrolled part) is no different than identifying oneself (which one knows and controls) as just a part of the world (the rest of which is largely unknown and uncontrolled), which is uncontroversial.
So from the moment we are aware of any experience at all, we can conclude that there is some world or another being experienced, and we can then attend to the particulars of those experiences to suss out the particular nature of that world. The particular occasions of experience are thus the most fundamentally concrete parts of the world, and everything else that we postulate the existence of, including things as elementary as matter, is some abstraction that's only real inasmuch as postulating its existence helps explain the particular occasions of experience that we have.
On the Web of Reality
George Berkeley famously said that
to be is to be perceived, and as I've already detailed in my previous essay against relativism, I don't agree with that entirely, in part because I take perception to be a narrower concept than experience in a broader sense, and because I don't think it is the actual act of being experienced per se that constitutes something's existence, but rather the potential to be experienced. I would instead say not that to be is to be perceived, or that to be is to be experienced, but that to be is to be experienceable.
And I find this adage to combine in very interesting ways with two other famous philosophical adages: Socrates said that
to do is to be, meaning that anything that does something necessarily exists; and more poignantly, Jean-Paul Sartre said that
to be is to do, meaning that what something is is defined by what that something does. Being, existence, can be reduced to the potential for or habit of some set of behaviors: things are, or at least are defined by, what they do, or at least what they tend to do. To combine this with my adaptation of Berkeley's adage, we get concepts like "to do is to be experienced", "to be experienced is to do", "to be done unto is to experience", and "to experience is to be done unto".
This paints experience and behavior as two sides of the same coin, opposite perspectives on the same one thing: an interaction. Our experience of a thing is that thing's behavior upon us. For example, an object is red inasmuch as it appears red, and it appears red inasmuch as it emits light toward us in certain frequencies and not others: the emission of the right frequencies of light, a behavior in a very broad sense, constitutes the property of redness.
Every other property of an object is likewise defined by what it does, perhaps in response to something that we must do first: an object's color may be relative to what frequencies of light we shine on it (e.g. something that is red under white light may be black under blue light), the shape of the object as felt by touch is defined by where it pushes back on our nerves when we press them into it, and many other more subtle properties of things discovered by experiments are defined by what that thing does when we do something to it.
We can thus define all objects by their function from their experiences to their behaviors: what they do in response to what it done to them. The specifics of that function, a mathematical concept mapping inputs to outputs, defines the abstract object that is held to be responsible for the concrete experiences we have. Every object's behavior upon other objects constitutes an aspect of those other objects' experience, and every object's experience is composed of the behaviors of the rest of the world upon it. All of reality can then be seen as a web of these interactions, the interactions themselves being the most concrete constituents of that reality, with the vertices of that web constituting the more abstract objects, in the usual sense, of that reality.
We each find ourselves to be one complex object in that web, and the things we have the most direct, unmediated awareness of are those interactions between our own constituent parts, and between ourselves and the nearest other vertices in that web, those interactions constituting our experience of the world, and also our behavior upon the world. By identifying the patterns in those experiences, we can begin to build an idea of what the rest of the world beyond that is like, inferring the existence and function of other nodes beyond the ones we are directly connected to by their influence in the patterns of behavior of (and thus our experience of) those nearest nodes.
On Physics and Ontology
This work is where philosophy ends, as far as investigating reality goes at least, and the physical sciences take over, postulating the existence of abstract objects with functions that would give rise to the concrete experiences we have of the world. Early physics began by identifying the behavior of large complex objects, and the different kinds of stuff that they are made of, "elements" like "earth", "water", and "air". But in time it has found those all to be made of many kinds of smaller particles of a similar nature to each other, molecules, interacting with each other in different ways.
Those many diverse molecules have in turn been found to all be composed of a more limited set of still smaller particles, atoms; and those in turn of an even smaller set of smaller particles still, electrons and nucleons like protons and neutrons; the latter in turn made up of triplets of two still smaller and more fundamental particles called up and down quarks. And I believe that contemporary physics has come far enough along, dug deep enough into the constituent particles of reality, that it has now identified as its most fundamental particles objects that are literally identifiable with the very "occasions of experience" that make up the web of reality described in my ontology above.
For clearest illustration, consider the experience of vision, which is now understood to be mediated by particles of light called photons. Whenever we see anything, all we're actually seeing in the most technical sense is the photons that hit our eyes; the objects we see, in the casual sense most people mean, are only inferred from the patterns in those photons hitting our eyes. Because of the distortion of space and time relative to motion, from the frame of reference of any given photon, the distance that it travels between whatever emitted it and your eye is zero, and the journey takes no time at all; from the photon's perspective, it exists only at a point and only for an instant, the whole of its being constituted entirely by the interaction between whatever emitted it and your eye.
Contemporary theories of physics hold that fundamentally, all of the most fundamental particles are essentially like photons in that way, all naturally moving at the speed of light, and so finding themselves, in their own frames of reference, to exist for but an instant at the point where two other objects interact, and their own existence consisting entirely of that interaction between them. It is only the aggregate patterns of interactions between these particles that gives rise to the appearance of the conventional, slower-moving particles out of which all of the aforementioned structures arise, up to the macroscopic scale we're familiar with.
For instance, electrons as we commonly understand them are understood to be an aggregate pattern of two different light-like particles, each similar to an electron and to each other but differing from each other in a property called spin, neither of which is able to travel any measurably large distance in space without immediately interacting with something called the Higgs field. The Higgs field absorbs that particle, and immediately emits another identical to it other than having opposite spin, only for that to be immediately reabsorbed and a particle like the first one emitted again, the overall pattern of those two kinds of particles, oscillating between each other immeasurably quickly as they interact with the Higgs field, constituting the particle that we conventionally think of as an electron.
Those light-like fundamental particles, that I think are identifiable with the interactions or "occasions of experience" that constitute the web of reality as described here in my ontology, thus make up, in a sense, the electrons and quarks that make up the atoms that make up the molecules that make up all of the matter that makes up the entire world, including people like you and me.
On Abstract Existence
Though philosophers broadly agree on the general notion that objects are abstract to the extent that they are not concrete, there is not a clear consensus on what it is that defines the distinction between concrete and abstract existence. I gestured at the distinction in the introduction to this essay through example: abstracts objects are things like numbers and other mathematical objects, or universals properties that can have many particular instances, or invented things like any given story or game (but not any particular instance thereof), or the elements of such a story or game. But that doesn't give a definition of what makes those things abstract and not concrete.
One common type of definition is a negative one, as in, concrete things are those things that are located in space and time, while abstract things are not located in space and time, and so not the sort of things we can interact with like we would paradigmatic concrete things like rocks, trees, tables, and chairs; not even mental events like thoughts and feelings that still occur at a particular place and time and can both be caused by concrete experiences of, and cause concrete behaviors by, the people who have them. I think that that is a helpful description for better understanding the distinction, but does not itself constitute a full definition of the distinction.
The definition that I offer for concrete things is that they are things that are real in the sense described in the section above on concrete existence, things that we think exist because we can interact with them, similar to the more common definition above. But on my definition abstract things aren't merely things that aren't concrete like that, but rather they have a defining feature of their own in common that concrete things lack: they are in some sense defined into existence, or equivalently discovered through definition, the distinction between creation and discovery not properly applying to abstract objects, as discussed at the end of my earlier essay on rhetoric and the arts.
As mentioned at the end of my earlier essay on logic and mathematics, a mathematical object is defined by fiat as whatever obeys some specified rules, and then the logical implications of that definition, and the relations of those kinds of objects to each other, are explored in the working practice of mathematics. Universal properties similarly exist inasmuch as we can define what we mean by them, what the common feature between many particulars is. Fictional objects are likewise defined into existence by the works of fiction. The stories themselves, or similar inventions like games, are also defined into existence by their authors: rewriting part of my own copy of The Silmarillion doesn't change that story in the abstract, it merely creates a derivative story of which I have the only copy.
In this way, as gestured at in the section on concrete existence above, abstract objects also play a role in structuring our experience of the concrete world, as all of our beliefs and theories about what is real beyond the immediate occasions of experience we are directly interacting with are suppositions about something or another out there behaving according to some pattern of rules that makes our experience of it be the way that it is.
On Indispensable Abstractions
Some of these abstract things are so fundamental that we could scarcely conceive of any intelligent beings comprehending reality without the use of them. Immanuel Kant called these kinds of things, things we cannot exactly observe but which we cannot help but use to structure the things that we do observe, "categories". The ones that I will describe here are not exactly the ones that he describes, though there is significant overlap.
On Qualitative and Quantitative Identity
The first thing we need to do to structure our experiences is to identify patterns in them. To do that, we need a pair of concepts that I call "quality" and "quantity", which allow us to think of there being several things that are nevertheless the same, without them being just one thing: they can be qualitatively the same, while being quantitatively different.
Any two electrons, for instance, are identical inasmuch as they are indistinguishable from each other, because every electron is alike, but they are nevertheless two separate electrons, not one electron. In contrast, the fictional character Clark Kent is, in his fictional universe, identical to the character of Superman in a quantitative way, not just a qualitative way: though they seem vastly different to casual observers, they are in fact the same single person.
If two people are said to drive "the same car", there are two things that that might mean: it could mean that they drive qualitatively identical cars (or as close to it as realistically possible, e.g. the same year, make, and model), or it could mean that they drive the same, single, quantitatively identical car, one car shared between both of them.
With these concepts of quality and quantity, we can describe patterns in our experience as quantitatively different instances or tokens of qualitatively the same tropes or types. Out of this arises the notion of several different things being members of the same set of things ("qualities" as I mean them here mapping roughly to the mathematical concept of "classes", an abstraction away from sets, and "quantities" as I mean them here mapping roughly to the mathematical concept of "cardinality", an abstraction away from the measure of a set or class). And with that can be conducted all of the construction of increasingly complex abstract objects built from sets as detailed in my previous essay on logic and mathematics.
On Spatial, Temporal, and Modal Dimensions
Then to further structure those patterns, we need conceptual spaces in which to figuratively lay out those instances of those tropes, in which to cluster them together and separate them apart. The most elementary of those conceptual spaces, I hold, is what mathematicians and physicists call a phase space or configuration space, which is an entirely abstract, imaginary kind of space wherein each point represents one way the system under consideration could be, a kind of abstract space of possibilities, wherein the potential changes of our experiences can be structured.
If we then identify patterns, trends, in the movement of our experience through that abstract space of possibilities, we have constructed the concept of time in its usual linear sense, with one direction in the configuration space being designated the past and another designated the future. I hold that time is best conceived of literally as a line through an abstract configuration space like this, with other "possible worlds", other possible configurations of the world, being literally, ontologically the same kind of thing as other times, other times being merely a special subset of other possible worlds in which the present, or the actual world, can be found.
My conception of possible worlds, each being an instantaneous possible configuration of the universe, is different from the usual kind promoted by supporters of modal realism – this view that other possible worlds really exist – like David Lewis, who hold that other possible worlds each contain within them a whole temporal history changing from past through present to future.
Other philosophers, such as logician Saul Kripke, seem to take possible worlds to be instantaneous configurations of the universe like I do, and speak of things being possible, necessary, etc, "from" one world or another, rather than in absolute terms; something being possible in some world if it occurs in any other world "accessible" from that world, rather than just if it occurs in any possible world at all.
I would interpret that, on my model, as being equivalent to the temporal relationship between possible worlds: a world that is "accessible" from another is a possible future of that other world, and things that lie in possible futures of a given world are in a sense relatively possible from that world. But we can still speak meaningfully about things being possible in the absolute sense of occurring in any possible world at all, irrelative to any particular world from which that world is accessible.
With a concept of time established, we can then construct a concept of space, that being the time that it takes a change to one part of our experience to affect another part of our experience. This does not depend on any particular claims of contemporary physics about the speed of light being constant or anything like that: any arbitrary speed could be picked, even if the speed of light were not constant, by which to derive distances from durations, and to construct a concept of space from a concept of time. This is actually quite commonly done in casual speech: places may be said to be hours away by foot or minutes away by car, their distances given as the time it takes to travel there at a given speed (implied by the mode of transit).
On Spatial, Temporal, and Modal Bundles
Within these spaces, we can then separate bundles of experiences from each other into different objects, giving rise to concepts such as substance, which as outlined in my previous essay against transcendentalism is not directly observable and cannot really be said to exist unto itself, but is a useful concept for structuring bundles of experiences in space.
Patterns in our experience bundled together in time give rise to the concept of causation, which as David Hume famously argued also cannot be directly observed, only patterns of
constant conjunction. I agree with him about that as much as I agree with the likes of George Berkeley about substance, in that neither can really be directly known or said to really exist in any independent way, but they are nevertheless indispensable concepts in structuring the patterns of our experiences in space and time.
(These notions of substance and causation also map neatly onto the notions of mass and energy, inasmuch as mass can be thought of as the amount of substance in an object, and energy thought of as the capacity to cause changes. Coupled with that association, the notion that to be is to do, as discussed in the section on concrete existence above, seems to me a vague predecessor to the notion of mass-energy equivalence).
Lastly, the concept of a kind of modal identity is useful for structuring ideas about counterfactual scenarios, bundling things together across possible worlds in a way broader than mere temporal causation. For instance we might want to say that had I made different choices in the past I would find myself in different circumstances in the present, and yet that counterfactual me in some other possible present is still nevertheless me in some sense, just as much as the past version of me that we both have in common is also me in some sense, connected to this present me by a chain of causation. I am presently what that past me became, in this timeline; and some other possible me is what that past me could have become, had things unfolded differently.
On Emergent Instrumental Abstract Existence
With those basic, most indispensable abstractions, we are able to structure our concrete experiences into a basic physical model of the world. But there are many other kinds of things that we ordinarily talk about that are more sophisticated than just simple inert masses causing each other to do things simple things like billiard balls colliding. There are all manner of different chemical substances that interact with each other in complicated ways, forming and transforming all the different rocks, liquids, and gasses of our planet. There are living organisms that interact with each other and the rest of the world in even more complicated ecosystems. And of course there are thinking beings like ourselves and our dizzyingly complex social systems.
It is often supposed that some of these kinds of things are reducible to other kinds of things, or conversely, that some of them emerge from other kinds of things. Sometimes those are taken to be opposing ideas, emergentists opposing reductionism for claiming that psychology is "nothing but" biology, which is "nothing but" chemistry, which is "nothing but" physics; or conversely, reducionists opposing emergentism for claiming that wholly new psychological, biological, and chemical properties arise "like magic" from the underlying, ultimately physical reality. I find that conflict unnecessary, because both of the opponents targeted by those arguments can be in the wrong simultaneously, while other forms of both reductionism and emergentism remain salvagable.
When it comes to emergentism, there are already two well-defined kinds that allow us to make such a distinction. The kind I am not against is called "weak" emergentism. That merely holds that there are useful aggregate properties to speak of at some larger scales of abstraction, that ignore irrelevant details at smaller scales of abstraction; but it doesn't hold that anything genuinely new starts to happen when the larger-scale systems are constructed out of smaller parts. On the other hand, "strong" emergentism, which I am against, holds some wholes to be truly greater than the sums of their parts, and thus that when certain things are arranged in certain ways, wholly new properties apply to the whole that are not mere aggregates or composites of the properties of the parts.
A useful example to illustrate this difference is temperature, which is a weakly emergent property of the motion of molecules in a substance: if you modeled the motion of all the molecules in a material, you would end up modeling something that exhibited temperature "for free", without adding any new laws to the model; though if that was the scale you were interested in, you could usefully model just that aggregate property of temperature instead and ignore the details of the motion of individual molecules. In contrast, if you could not possibly model temperature without explicitly adding rules that assign such a property to aggregates of molecules, even though no particular molecules had that property, then temperature would be a strongly emergent property.
We do not yet have a complete account of all physical sciences that perfectly reduces them all down to fundamental physics, but the question at hand is whether it is reasonable to expect that that can eventually be done, or if instead it is just in principle not possible. Just as in the pragmatic principles underlying my entire philosophy, I think it is merely giving up to assume that it cannot be done, rather than to assume that it can in principle and we just haven't done it yet. To say that emergence does happen but it cannot possibly be explained exactly when or why (for that would be a reduction) is essentially to invoke magic, running counter to my argument against supernaturalism in my previous essay against transcendentalism.
On the other hand, I am not here endorsing what Daniel Dennett calls "greedy reductionism", which I would instead term "strong reductionism" in contrast to the "weak reductionism" that I do endorse (as does Dennett), in parallel to "weak" and "strong" emergence. Of the temperature example above, weak reductionism would only says that temperature is understandable in terms of something more fundamental than itself; but strong reductionism would say that, because temperature is thus reducible, there is in fact no such thing as temperature. Such strong reductionism would thus be a form of nihilism, which I have already argued against in my previous essay against relativism.
The remaining mix of both weak emergence and weak reduction to the exclusion of either strong reduction or strong emergence is simply about having no discontinuities in our understanding: having every account of everything transition seamlessly into each other with no sudden new fundamental laws of nature added anywhere. It's about being able in principle to zoom into the details of something on one level of abstraction and see a complex arrangements of things on another lower level, or zoom out and ignore the details on the smaller scale to see new structures on a larger scale, understanding how all of these different kinds of objects at different scales relate to each other.
I suspect that we most commonly perceive the stack of emerging or reducible layers of abstraction from the top down. As thinking, social beings, we perceive first a social world of other minds like ourselves. Then we see, underlying that, an ecosystem of living things, not all of which have minds quite like ours. And then, only later, we start to understand those living things in terms of the same microscopic chemical processes that govern the transformations of non-living things. And only later still do we begin to understand those chemical processes in terms of elementary physical interactions, and dig deeper and deeper into more fundamental understandings of such physics.
In the models of such deep physics, the instrumental abstractions employed begin to get very abstract indeed, and we begin to model the most fundamental interactions of all – which I earlier equated to those "occasions of experience" that are the most concrete elements of reality on this account – as the inputs and outputs of complex mathematical functions.
On Pure Mathematical Abstract Existence
At the end of my earlier essay on logic and mathematics, I gave an account of how from nothing but otherwise empty sets, one of the simplest kinds of abstract objects, we can construct all variety of numbers, and from sets of numbers all variety of geometric objects. In this section, I will now illustrate that such construction can even build up to a physical reality like ours, and argue for a position on the real existence of even pure, non-instrumental abstract objects from there.
Geometric things like lines and planes are examples of the more general type of geometric object called a space. Spaces can be very different in nature depending on exactly how they are constructed, but a space that locally resembles the usual kind of straight and flat spaces we intuitively speak of (called Euclidian spaces) is an object called a manifold, and such a space that, like the real number line and the complex number plane, is continuous in the way required to do calculus on it, is called a differentiable manifold. Such a differentiable manifold is basically just a slight generalization of the usual kind of flat, continuous space we intuitively think of space as being, and it, as shown, can be built entirely out of sets of sets of ultimately empty sets.
Meanwhile, a special type of set defined such that any two elements in it can be combined through some operation to produce a third element of it, in a way obeying a few rules that I won't detail here, constitutes a mathematical object called a group. A differentiable manifold, being ultimately a kind of set, can also be a group, if it follows the rules that define a group, and when it does, that is called a Lie group.
Also meanwhile, another special kind of set whose members can be sorted into a two-dimensional array constitutes a mathematical object called a matrix, which can be treated in many ways like a fancy kind of number that can be added, multiplied, etc. A square matrix (one with its dimensions being of equal length) of complex numbers that obeys some other rules that I once again won't detail here is called a unitary matrix. Matrices can be the "numbers" that make up a geometric space, including a differentiable manifold, including a Lie group, and when a Lie group is made of unitary matrices, it constitutes a unitary group.
And lastly, a unitary group that obeys another rule I won't bother detailing here is called a special unitary group. This makes a special unitary group essentially a space of the kind we would intuitively expect a space to be like – locally flat-ish, smooth and continuous, etc – but where every point in that space is a particular kind of square matrix of complex numbers, that all obey certain rules under certain operations on them, with different kinds of special unitary groups being made of matrices of different sizes.
I have hastily recounted here the construction of this specific and complicated mathematical object, the special unitary group, out of bare, empty sets, because that special unitary group is considered by contemporary theories of physics to be the fundamental kind of thing that the most elementary physical objects, quantum fields, are literally made of. Excitations of those quantum fields, which is to say particular states of those special unitary groups, constitute the fundamental particles of physics, which combine to make atoms, molecules, stars, planets, living cells, and organisms, including us, so in a very distant way we can be said to be made of empty sets.
(And as all of the truth functions, and so all the set operations, and all the other functions built out of set operations, can be built out of just conegation, and the objects they act upon are built up out of empty sets, everything can in a sense be said to be "made out of negations of nothing").
In the same way that when, in my earlier essay on logic and mathematics, we constructed a series of sets that behave exactly like the natural numbers and so are indistinguishable and thus identical to them, so too can we construct complicated mathematical objects like this that behave indistinguishably from the fundamental constituents of reality and so are, for all intents and purposes, identical to them. And it is not a special feature of contemporary physics that says reality is made of mathematical objects; rather, it is a general feature of mathematics that whatever we find things in reality to be doing, we can always invent a mathematical structure that behaves exactly, indistinguishably like that, and so say that the things in reality are identical to that mathematical structure.
If we should find tomorrow that our contemporary theories of physics are wrong, it could not possibly prove that those features of reality are not identical to some mathematical structure or another; only that they are not identical to the structures we thought they were identical to, and we need to better figure out which of the infinite possible structures we could come up with it is identical to. We just need to identify the rules that reality is obeying, and then define mathematical objects by their obedience to those same rules. It may be hard to identify what those rules are, but as previously described in my essay against relativism, we can never conclusively say that reality simply does not obey rules, only that we have not figured out what rules it obeys, yet.
The mathematics is essentially just describing reality, and whatever reality should be like, we can always come up with some way of describing it. One may be tempted to say that that does not make the description identical to reality itself, as in Alfred Korzybski's adage "the map is not the territory". In general that adage is true, and we should not arrogantly hold our current descriptions of reality to be certainly identical to reality itself. But a perfectly detailed, perfectly accurate map of any territory at 1:1 scale is just an exact replica of that territory, and so is itself a territory in its own right, indistinguishable from the original.
And likewise, whatever the perfectly detailed, perfectly accurate mathematical model of reality should turn out to be, that mathematical model is a reality: the features of it that are perfectly detailed, perfectly accurate models of people like us would find themselves experiencing it as their reality exactly like we experience our reality. Mathematics "merely models" reality in that we don't know exactly what reality is like and we're trying to make a map of it. But whatever model it is that would perfectly map reality in every detail, that would be identical to reality itself. We just don't know what model that is.
There necessarily must be some rigorous formal (i.e. mathematical) system or another that would be a perfect description of reality. The alternative to reality being describable by a formal language would be either that some phenomenon occurs, and we are somehow unable to even speak about it; or that we can speak about it, but only in vague poetic language using words and grammar that are not well-defined.
I struggle to imagine any possible phenomenon that could cause either of those problems. In fact, it seems to me that such a phenomenon is, in principle, literally unimaginable: I cannot picture in my head some definite image of something happening, yet at the same time not be able to describe it, as rigorously as I should feel like, not even by inventing new terminology if I need to. At best, I can just kind of... not really definitely imagine anything in particular.
All of this is building up to me addressing the central question in the philosophy of mathematics, which is about the existence of purely abstract objects, like numbers and everything else that I've just been discussing. There are two main answers to that question, and some positions intermediate to the two, but I want to offer a position that I consider to be off of that spectrum entirely.
One of the usual two positions is platonism, sometimes called either platonic realism or platonic idealism, which holds that abstract objects, or as Plato called them "forms" or "ideas", are real in the same sense that concrete objects, like rocks and trees and tables and chairs, are real; but that they don't exist in our space and time, and instead live in some separate, spaceless, timeless realm, from which they are somehow connected with the things in our realm that "partake" of them, in the way that a triangular rock "partakes of the form of the triangle".
It is held by platonists that the existence, in some way, of these abstract objects is necessary in order for mathematical and other abstract statements that seem nominally to be about them to be true: for instance, the Pythagorean theorem, which describes the relations of the legs of a right triangle to the length of its hypotenuse, is not made true by the existence of any particular triangular objects, but rather by facts about the form of triangles generally, even if no concrete triangular objects existed at all.
I am not very amenable to this position at all, holding it to fall heavily afoul of the principles I laid out in my previous essay against transcendentalism.
The second of the usual two positions is called nominalism, which holds that abstract objects are merely empty names, that do not refer to real things that exist at all, and are just names for the similar properties of, and collections of, particular concrete objects. I am much more amenable to that position generally, but there is still something unsavory in how it effectively declares that e.g. "numbers don't really exist", that they are just things we've made up, in a way that threatens to fall into relativism.
I think that a kind of existence can, despite my objections to transcendentalism, nevertheless be applied to abstract objects after all; a kind of existence abstracted away from the more familiar phenomenal notion of concrete existence. This view avoids the troubles of transcendentalism because the underlying objection to transcendent things is their demand that we take someone’s word for it why one arbitrary transcendent thing exists but not others. The view I am about to lay out, on the other hand, aims to eliminating a problem of arbitrariness: to dissolve the outstanding question of why the concrete, phenomenal world is this way, fitting the patterns of the mathematical objects that it does, rather some others instead.
In the most restricted sense, one could say "only what I am experiencing right here right now exists". Everything else that we talk about existing is some degree of inference and abstraction away from that. There is a position in the philosophy of time, called presentism, that holds that only the present exists, neither the past nor the future. I agree with them to the extent that in a sense only the present exists: only the present presently exists, right now.
But a part of what I'm experiencing right now in the present is memory, from which I infer (automatically, intuitively, without thinking about it) the existence of other times, having an experience of moving between different times, from those remembered past times and toward projected future times, and there is a perfectly serviceable sense in which I can say that those other times "exist" in a timeless sense of the word: they don't exist now, presently, for sure, but they still exist at other times.
And in that "movie", so to speak, of my past, present, and future experiences that I have now inferred, I have the experience of seeming to move around different places, so I further infer that other places exist too, besides just the here that I am experiencing now. Like with presentism, only the place I am at exists here, but those other places can still reasonably be said to exist elsewhere.
In this way, a spatiotemporal kind of existence is already abstracted away from the more primitive kind of existence relevant to my local, present experiences. But beyond that, some philosophers such as David Lewis hold, and I agree, that other possible worlds, like the kind that we use to make sense of talk of alethic modalities like necessity and possibility, really exist, and aren't just useful fictions, even though they don't actually exist, because "actual" is an indexical term like "present" or "local": it refers to things relative to the person using the word. Just as other times don't presently exist but are still real in a more abstract sense, so too, on this account, other possible words don't actually exist, because "actually" means "in the possible world I am a part of", but they are nevertheless still real in a still more abstract sense.
As already explained in the section on indispensable abstractions above, I view possible worlds as the same kind of thing as other times, and this relationship makes for a much more intuitive argument for modal realism than usual. If we're already conceding that other times are as real as the present, and then we also run with the usual intuition that there is not only one definite future, but multiple possible futures, those together entail that there are multiple possible futures that are all equally as real as the present. And since that was thus in the past too, there are also multiple possible presents, all equally as real as this, actual, present. And so on back forever, possible futures of distant pasts spreading out to any way things could possibly have ended up: all those possible ways the world could have been as real as the way the world actually is.
Likewise, to finally get on to my point about the existence of mathematical objects: since we can in principle equate our concrete universe with some mathematical structure or another, and that mathematical structure definitely concretely exists (because it just is the concrete universe), we can say of all other mathematical structures, i.e. abstract objects, that while they don't concretely exist – because "concretely" is indexical, like "actually", it means "as a part of the mathematical structure that is our universe" – they can nevertheless be reasonably called "real" in some even broader sense, the most abstract sense possible: they abstractly exist.
So, why does a universe like this one exist instead of another one? What a presumptuous question! The others ones all exist too – they're just not this one that we're a part of. So there is no unanswerable question about why the universe is like this and not like that: this universe may be like this, but another one is like that, and it's equally real, it's just not the one we're part of. It's only being like this, with us in it and such, that makes this universe this one.
This position is held by physicist Max Tegmark, and he calls it the "ultimate ensemble"; it is more broadly called the mathematical universe hypothesis, or mathematicism, and it has precursors tracing back to the Pythagorean philosophers of ancient Greece.
This kind of existence for abstract objects does not run afoul of my position against transcendentalism the way that platonism does, because the abstract objects don't exist in some wholly different kind of way separate from the kind of concrete objects that we can empirically observe. They are just the loosest part of the broader framework of explanation for our empirical observations.
We cannot directly observe other times or places, only the local present, but postulating the existence of other times and places helps to explain the patterns in our local, present experiences. Those other times and places aren't held to be discontinuous or of a completely different nature than the local present, they are just postulated extensions of the here and now. Likewise, I hold, with postulating other possible worlds, continuous with the one we find ourselves in and of same nature as it; and also likewise with other abstract objects besides whichever one is identical with the concrete universe, continuous with it and of the same nature as it.
This view of the relation between the concrete and abstract also bears a similarity to what Immanuel Kant called the phenomenal and the noumenal, where on his account we cannot ever have direct experiential contact with noumena, but instead only project our ideas about them behind the world of phenomena that we experience, much like how on my account the purely abstract has no direct influence on the concrete world we experience, and we can only project our ideas of abstract objects behind that concrete world in an attempt to understand and explain it.
On Impossible Objects and Worlds
There are some few kinds of abstractions that I would not go so far as to grant the existence of, because they are impossible. That is to say, I hold that there do not exist any such things as impossible objects, or impossible worlds. Those are merely things that are gestured at by incomplete fragments of representation, which fragments, if expanded upon and connected into a unified whole, would contain contradictions, and so in fact could not be expanded upon and connected into a unified whole.
Impossible worlds, for instance, would be the worlds depicted in works of fiction that make contrary claims about such worlds. All works of fiction are by necessity incomplete depictions of the worlds that they depict, but those worlds could in some cases be possible worlds, if those partial depictions were coherent and self-consistent, such that they could in principle be matched up to a complete description of a possible world without any disagreement. But because works of fiction are thus incomplete, and thus fragmented, depictions of worlds, it is possible that one fragmentary depiction might match a possible world, and another fragment might match another world, but there is no way that those could be the same world, and so no one single possible world could possibly match the overall depiction of the work of fiction.
Impossible objects, similarly, are objects like the Penrose Staircase, or the Devil's Tuning Fork. These are what seem to be three-dimensional objects depicted in two-dimensional art, but no such three dimensional object could possibly have the appearance depicted in the art. In the case of the Penrose Staircase, what is depicted appears to be a spiral staircase that one could walk continually upward around, only to inexplicably end up back on the step one started on, without ever descending downward. The Devil's Tuning fork, similarly, appears on one end to be the base of a two-pronged tuning fork, yet by the other end inexplicably has three prongs, without any prongs ever splitting or diverging from parallel lines. These are, like impossible worlds, representations that fragmentarily match possible things, but whose fragments cannot be united into a single representation of any one possible things.
Continue to the next essay, On the Mind, Consciousness, and the Subjects of Reality.