Spacetime, Individuation, and Fiber Bundles

How can entities be picked out as individual and distinct entities? Sunny Auyang presents a Kantian model of spacetime as an absolute and indispensable structural scheme we project onto the world to organize it and to pick out individual elements, or events in it. Using fiber bundles she packs together a complex structure of individuating qualitative features that she links to individual points in spacetime.

I’d like to talk again about some stuff I’ve been reading in this book by Sunny Auyang, How is Quantum Field Theory Possible? Specifically in this latest chapter I read on the nature of space or spacetime and the possibility of individuation, individuation being the identification and distinction of entities as separate entities.

Both of these issues have a long history in the history of philosophy but Auyang focuses mostly on the work of the modern period of the last few centuries, most especially on Leibniz, Newton, and Kant. There’s a famous dichotomy or division between the models of space put forward by Leibniz and Newton. And the question there is whether space is an independently existing thing or just a way of conceptualizing the relations between actual entities, like their distances and orientations from each other. So Newton’s view was that space has an independent existence. Even if you took out all other entities in the universe space itself would still be there as its own thing. Also time. So both space and time are “absolute”. But for Leibniz these are relative or relational concepts. Lengths, areas, and volumes are relations between entities but if you take away the entities, the actual things there’s nothing left behind, no empty space. Now I’ve read that those are actually drastic simplifications of their views, which doesn’t surprise me. But regardless of that we can at least have those views in mind to start, with the understanding that they’re traditionally associated with Newton and Leibniz. Auyang actually divides both these views further, so that we have four; two Newton-type views and two Leibniz-type views. And I’ll just introduce those so we can use the descriptive names rather than these two proper names.

On the one side we have the substantival view and the absolute view. Spacetime is substantival if it exists independent of material entities. Spacetime is absolute if its concept is presupposed by the concept of individual entities and things. These are similar but slightly different ideas. Substantivalism is ontological, meaning it actually has to do with being, what is. Absoluteness is conceptual; it pertains to the way concepts fit together and what is necessary for certain concepts to work and be intelligible. These can coincide but they don’t have to. And Auyang is going to argue for a model of spacetime that is absolute but not substantival. So in her view spacetime is not a thing that exists independent of material entities but it is a concept that is required to conceptualize material entities.

On the other side Auyang also distinguishes between the relational view and the structural view. I think this is an even more subtle distinction. The difference between these two is a matter of logical priority, looking at what comes first. So recall that with the relational view the concept of space arises from the relations between entities. Dimensions like length, area, and volume are these relations that we perceive between the entities around us. They’re already there and we perceive them. The structural view is the Kantian view, from Immanuel Kant, that space, and we can say also spacetime, are concepts that we project onto the world to organize it bring structure to it. So we as subjects come first. I’m describing that a little differently than she does in the book but that’s the way it makes most sense for me to think about it. And I think it’s consistent with her view. And between these options Auyang is going to argue for a model of spacetime that is structural rather than relational. So it’s more the Kantian model. So bringing these two together her view of spacetime is absolute and structural. In other words, spacetime is a concept that is required for us to conceptualize material entities, and it is a structure that we project onto the world to organize it and make sense of it.

With that in place let’s get to individuation of entities. How do we say that a thing is the same thing across time, something that we can index or label? And how do we say of a thing that it is this thing and not some other thing? “An entity is an individual that can be singly picked out, referred to, and made into the subject of propositions.” Aristotle said that it incorporates two elements. It’s both a this and a what-it-is. These are the notions of individuality and kind. A specific entity is not only a thing but it is this thing. It’s indexed and labeled. It’s also a certain kind of thing. That doesn’t individuate the single entity from other members of that same kind but it distinguishes that class of entities as a kind. Then within that set of that kind of entity they must be further differentiated and identified as individuals. That gets very complex. Other philosophers instead have also argued for the importance of a cluster-of-qualities notion. An entity is no more than the sum of its qualities. If you get specific enough about your qualities maybe that’s all you need. Every entity has a unique spatio-temporal history at least, even if indistinguishable in all other qualities. At least we may so argue. So some important concepts here are individuality, kind, and qualities. These are ways of individuating.

So we’re going to look at a model of these entities. And the first thing to address is that we’re going to look at this through the lens of quantum field theory rather than classical mechanics. So the primary form of matter, the material entities I’ve been talking about before, shift from discrete mass points in space to continuous fields comprising discrete events. Auyang doesn’t mention this but it reminds me a little bit of Alfred North Whitehead’s process philosophy in which he substituted a substance ontology of things to a process ontology of events. Auyang’s quantum field theory is rather different from that, nevertheless, it was something that came to mind. So anyway, the basic entities we’re going to consider now are events.

A field is a continuous system. “The world of fields is full, in contrast to the mechanistic world, in which particles are separated by empty space.” Every point in a field is assigned a value. So say we have a field, that we’ll call the greek letter ψ, for every point x in that field there will be a value ψ(x). And that field variable ψ(x) doesn’t have to be scalar, i.e. just a number. It can be a vector, tensor, or spinor as well. Actually I’m most accustomed to thinking of field variables as vectors like with a gravitational field or an electric field. So with a gravitational field for instance every point in the field around mass M has a vector oriented toward mass M. And then the magnitude of those vectors varies with the distance from mass M. And that’s just an example, the field variable could be any number of things. And that’s important for individuation because we’re going to want to account for the qualities of an individual event with which we can distinguish it. But also one key idea to keep in mind is that the field variable ψ is indexed to some point x in the field. That’s another method of individuation.

So let’s look at how both qualities and numerical identity get taken up in Auyang’s model. To give a bit of a road map before diving into the details her model will include. She’s going to use 6 major pieces: D, G, π, M, x, and ψ.

D is what’s called the total space.
G is a local symmetry group.
π is a map.
M is a base space.
x is a position in the base space M.
And ψ(x) is an event.

All of this will be put together in a fiber bundle structure. And we’ll get into what all that means in a minute.

First let’s talk about symmetry groups, which will be this G in her model. The concept of the this-something, the individuality of events, is incorporated in field theories through two symmetry groups. Symmetry is a key idea in physics. A related term is invariance, also a very important concept. And it’s basically what it sounds like. It’s some property that doesn’t change. More specifically, we’re interested in the very particular circumstances under which it doesn’t change, called transformations. So you have some object, you transform it in some way – say you rotate it for example – the features that don’t change in that transformation are invariants. And this can tell us important things. The big conservation laws in physics come from invariants as we know from what is called Noether’s Theorem. For example, conservation of energy comes from time invariance. Conservation of momentum comes from translational invariance. Conservation of angular momentum comes from rotational invariance. Very significant. Okay, so backing up again to symmetry groups – that was the whole reason for getting into this. A symmetry group is the group of all transformations under which the object is invariant. Some objects have lots of symmetry – they’ll be invariant under many transformation – others have very little. But the key is that the group of all those transformations where it is invariant – that’s a symmetry group.

The two symmetry groups pertinent to the field theories here are the local symmetry group and the spatio-temporal symmetry group. And these embody different aspects of the individuation of entities. “The idea of kinds is embodied in the local symmetry group, which pertains not to spatio-temporal but to qualitative features. The symmetry group circumscribes a set of possible states and defines a natural kind.” So recall one of the important ideas for identification or individuation was quality. Well the state of an entity covers its qualities. But for localization and identification, its numerical identity, we need a global whole, rather than a local whole, and that is represented by a spatio-temporal symmetry group. “The identities of the events are the invariants in the spatio-temporal symmetry structure.” These two symmetries give us the quality and numerical identity of the entities.

To fit this all together Auyang presents a model for the structure of local symmetries. And she does this using fiber bundles. Fiber bundles are great mathematical tools. The most straightforward way I like to think about fiber bundles is that they are a way to relate single points in some base space to more complex structures in another space. And when I say “space” here these can be abstract spaces, though at least one of these in what follows, the base space, will in fact be a spatio-temporal space. The great thing about this is that it lets us sneak a lot of structure into a spatio-temporal position. And that’s good because we need a lot of structure for these individuating elements. A spatio-temporal position is just one of these individuating elements. We want to bring qualities in there too.

So let’s look at Auyang’s model. This is the featured image for this episode by the way if want to look at it. The objects D, G, and M are differential manifolds, which is basically just a kind of space or surface. These manifolds can be actually spatial or spatio-temporal, which will be the case with our base space M. But they can also be, and often are abstract, which will be the case for our total space D and our local symmetry group G in this model. The first manifold, our total space D, is a set of abstract qualities. So this is where we’re going to get the qualities for our entities from. Then she also has a local symmetry group, G, which is also a manifold. We can label the abstract qualities in D as θ, θ’, and so forth. “At this starting point, both D and G are abstract and meaningless. Our aim is to find the minimal conceptual structure in which we can recognize events as individuals”.

The symmetry group G acts on the total space D and collects subsets of elements in D that are equivalent to each other. Each of these subsets we’ll call a G orbit. The elements in a single G orbit are equivalent to each other. We can start with quality θ and θ’ – those will go into one G orbit. Then we can pick out ξ and ξ’. This divides D up into these G orbit subsets until all elements in D are accounted for. None of resultant G-orbits share common elements. D still has all the same elements as before but they are divided into these subsets. This is quite useful for our purposes of individuation. We have some organization here of all this information.

Next we can take a G orbit and introduce a map π that sends all elements in a G orbit, θ,  θ’, for example, sends all those elements onto a single point x. This point x is on another manifold M, a base space. There’s also an inverse map, π-1, that canonically assigns a unique element x in M to each G orbit in D. M is what’s called a quotient of D by the equivalence relation G. It’s not given in advance but falls out from D. Every spacetime point, x, in the spatio-temporal structure, M, is associated with an event, ψ(x), in the total space D. Speaking of this in terms of set theory, D becomes a set with an indexing set M.

So now we have all the pieces put together: D, G, π, M, x, and ψ. And to review, D is the total space, G is a local symmetry group, π is a map, M is a base space, x is a position in the base space M, and ψ(x) is an event. And what’s the significance of all this in the “real world”, so to speak? M is usually called spacetime and x is a point in spacetime, the spatio-temporal position of an event ψ(x). But the identity of an ψ(x) includes more than just it’s spatio-temporal position, even though it’s indexed to that position. All that extra information is in the total space D. It’s divided up by the local symmetry group G. And then it’s mapped onto the spacetime base space M by the map π. The cool thing about the fiber bundle is that it allows us to cram a lot of information into a single point in spacetime, or at least link it to a lot of extra information.

The main goal that Auyang is working toward with this model is individuation. And to do that she needs enough complexity to carry the kind and quality features of individual entities, as well as spatio-temporal position. What happens in this model is that a spacetime position, x, signifies the identity of an event ψ(x). x uniquely designates ψ(x) and marks it out from others. The symmetry group, G, whose features are typical of all ψ(x), signifies a kind; since it collects those features as group. Then the spatio-temporal structure, M, is a system for identifying individuals in that group. So this “sortal concept that individuates entities in a world involves two operations” that will mark out (1) kinds and (2) numerical identity. First the local symmetry group, G, forms identical equivalence classes of qualities for this notion of kinds. Second the projection map, π, introduces numerical identity for each of these equivalence classes. These together secure the individuality of an event, ψ(x).

One thing we can certainly say about this kind of model is that it is analyzable. Events and spacetime positions are not just given in this view. There’s complex interplay between spacetime positions of events and all the qualities of those events. This is what we get with field theories. Even if we look at the world in the most primitive level, as Auyung says, with field theories, “to articulate even this primitive world requires minimal conceptual structure more complicated than that in many philosophies, which regard sets of entities as given.” So is this necessary, are we just making things more complicated than they need to be? Quoting Auyang again: “Field theories have not added complications, they have made explicit the implicit assumptions taken for granted.” I’m not prepared to defend that point but I’m fine with going along with it for the time being.

To wrap things up let’s look at some ways for thinking about this spatio-temporal structure, M. The complexity of the full conceptual structure of this model (D, M, G, π) is what makes it analyzable and it enables us to examine M’s possible meaning. Auyang characteristically promotes a Kantian take on all this. This is to see M as a “scheme of individuation and identification that we project into the world via the inverse map π-1 and by which we present the world to ourselves as comprising distinct entities.” Recall that in Kant’s thought the world is intelligible to us only because we apply categories of understanding to the raw sense data we bring in, and we use these categories to organize it all and make sense of it. Auyung is saying that this is what M does; this is what the spatio-temporal structure, or our concept of spacetime does.

And this idea of space being what individuates things has a long history. For example, speaking of Kant, in Kant’s philosophy space is what makes identity and difference possible. Hermann Weyl called space the “principium individuationis”, which is really fun to say with the classical Latin pronunciation of the ‘v’. But that’s just this idea we’ve been talking about, individuation, the manner in which a thing is identified as distinguished from other things. Weyl also said space “makes the existence of numerically different things possible which are equal in every respect”. So it’s not just the qualities (non-spatial) that are important. You need space to distinguish entities that are otherwise identical. This doesn’t mean that space is substantival, some independently existing substance. But it is conceptually indispensable. So, say it is something that we bring to the scene, something we impose as an organizing tool. It’s still indispensable for the possibility of individuation. So it’s absolute in that sense.

So to review, I’ll put these in Kantian terms. We start off with what is “out there”, just this pre-conceptualized mass of stuff, our total space D. How is that intelligible? We come at it via a conceptual structure, the mental categories of space and time, or spacetime, M. Then we project these spatial and temporal conceptual categories onto the world using the inverse map π-1. This inverse map is able to pick out individual entities in the total space D that are distinguishable by an organizing operation of the local symmetry group G. The local symmetry group G has divided up the total space D into G-orbits with common elements. Our spatial and temporal categories pick these subsets out as events ψ(x) that are mapped onto spacetime M. And that brings the whole structure together in a way that we can see everything together and pick out individual events as individual elements.

Object-Oriented Theology

Mike and Todd discuss Adam S. Miller’s “Speculative Grace: Bruno Latour and Object-Oriented Theology”, possibly the most rigorous, speculative, and systematic attempt at a professional take on Mormon philosophy ever, that never directly mentions Mormonism. We read between the lines and look at the revolutionary ideas of the Mormon moment in world religious history that are arguably still not fully realized in the ongoing Restoration.

State Spaces, Representations, and Transformations

Sunny Y. Auyang gives a useful model for thinking about the way states and state spaces of objective reality are represented in ways accessible to us and how transformations between a plurality of representations imply not relativism but common states and state spaces that they represent.

I’ve been reading a fascinating book that’s been giving me lots of ideas that I’ve been wanting to talk about. I was thinking to wait until I had finished it but I changed my mind because there are some ideas I want to capture now. It’s one of the books I call my “eye-reading” books because I’m usually listening to a number of audiobooks simultaneously. And I don’t have much time to sit down and actually read a book in the traditional way. But I sometimes save space for one if it looks really interesting and it’s not available in audio. And that applies to this one. The book is How is Quantum Field Theory Possible?, written by Sunny Y. Auyang. I heard about it while listening to another podcast, The Partially Examined Life, which is a philosophy podcast. One of the guys on there, Dylan Casey, mentioned it in their episode on Schopenhauer. It peaked my interest and I knew I had to get it.

The part of the book I want to talk about today is a model she puts together to think about the different ways an objective state can be represented in our scientific theories. To the extent that our scientific models and measurements are conventional what should we think if they represent things differently? Are we condemned to relativism and the arbitrariness of convention? She argues that we are not gives a model that takes things up a level to see different representations from the outside, how they relate to each other through transformations and how they relate to the objective states that they represent. This is necessarily a philosophical project, particularly a question in the philosophy of science. It is to get behind the work of science itself to think about what it is we’re doing when we do science and what it means when we say that things are a certain way and work in a certain way, as described by some theory.

I’d like to give a brief overview of some of those concepts and the vocabulary Auyang uses. And this will just be to get the concepts in our head. John von Neumann had a very funny quip that “in mathematics you don’t understand things. You just get used to them.” Now, I think that’s an overstatement. But in a way I think it’s kind of helpful whenever we’re getting into a discipline that has lots of unfamiliar terms and concepts that can seem really overwhelming. I think it’s helpful to just relax and not worry about fully understanding everything right away. But to take time to just get used to stuff, which takes time. Eventually things will start to come together and make more sense.

So the first idea I want to talk about is a phase space or state space. A phase space is the set of all possible states of a system. That’s very abstract so I’ll start with a concrete example. Say we have a single particle. At any given time this particle has a position in three-dimensional space that we can specify with three numbers along three spatial axes. For example, you could have a north-south axis, an east-west axis, and an elevation axis. You can also add momentum to this. So a particle’s momentum would be its mass multiplied by its velocity. Mass is scalar quantity – it doesn’t have direction – but velocity is a vector, so it does have direction. And in three-dimensions the velocity has three components along the same spatial axes as position. So you can specify the particle’s position and momentum with six numbers: three numbers to give its position and three numbers to give its momentum.

The really cool move from here is that you can then make use of what’s called a phase space. So for a single particle with these six axes we’ve selected this is a six-dimensional space. This is also called a manifold. Don’t worry about trying to visualize a six dimensional space. It’s not necessary. Just go along with the idea that we’re using such a thing. This is an abstract space. It’s not supposed to represent the kind of space we actually live it with length, width, and height. Any point in this six-dimensional space represents a possible state of the particle. You can represent any combination of position and momentum as a point in this phase space. So for example, the 6-tuple in parentheses with the six numbers (0,0,0,0,0,0) represents a state where a particle is at rest and it is sitting at the origin of whatever spatial reference frame we’ve set up. And you can put in any set of numbers to get any possible state of that particle. If we’re looking at multiple states of this particle through time we can think of it tracing out a trajectory in this state space.

Now, here’s where things get crazy. You can add more than one particle to this system. Say we add a second particle. How many dimensions does our phase space have now? It has twelve dimensions because we have axes for the positions and momentum components for both particles in three-dimensional space. And then we’ll have a 12-tuple, twelve numbers in parentheses, to call out the state of the system. And you can add as many particles as you like. For whatever N number of particles we have in our system the phase space will have 6N dimensions. So you can imagine that dimensions will start to pile up very quickly. Let’s say we take a liter of air. That has something on the order of 1022 molecules in it; over a billion billion. The number of dimensions in our phase space for that system will be six times that. Now, in practice we’d never actually specify a state in this kind of system. With gases for instance we don’t worry about what’s going on with every single particle in the system. We use properties like temperature and pressure to generalize the average behavior of all the particles and that’s much, much more practical. But as a conceptual device we can think of this phase space underlying all of that.

In quantum mechanics the state space of a system is called a Hilbert space. So this is the space of all possible states of a quantum system. Then any particular state of the quantum system is represented by a state vector, usually written with the Greek letter phi: |φ⟩. When we run an experiment to get information about the quantum system we look at a particular property that is called an observable. And you can think of an observable as pretty much what it sounds like, i.e. something that can be observed. And this is associated mathematically with an operator. An operator, as the name implies, operates on a function. And there are all kinds of operators. There are operators for position, momentum, total energy, kinetic energy, potential energy, angular momentum, and spin angular momentum. One way to think of this is that with an operator you’re conducting an experiment to measure the value of some property type. Then the result of that experiment is some number. The name for the resulting value is an eigenvalue. So for all those different operators I just listed off they will spit out corresponding eigenvalues. But an eigenvalue is an actual value. So with a kinetic energy operator, for example, your eigenvalue will actually be a number for the value of kinetic energy in some unit for energy, like Joules or whatever your choice of units.

Recall that in our phase space for particles each dimension, and there were many, many dimensions, had an axis in that phase space. In quantum mechanics the state space, the Hilbert space, has a collection of axes that are called a basis. And the basis of a Hilbert space is composed of eigenstates. And we can think of this as the coordinate system, the axes, of the state place of the system. The eigenvalue is what we get when we run an experiment but one of the interesting things about quantum systems is that we don’t always get the same value when we run an experiment, even if we’re applying the same operator to the same system. That’s because a quantum system is a combination (more specifically a linear combination or superposition) of many eigenstates. And each eigenstate has a certain amplitude. As we repeat several measurements of an observable we’ll observe eigenstates with higher amplitudes more often than eigenstates with lower amplitudes. We can actually quantify this. For any given eigenstate the probability that it will be observed with a measurement of an operator is its amplitude squared. So amplitude is a very important property in a system.

So there are many similarities there between the phase space of the system of classical particles and the Hilbert space of a quantum mechanical system. I just wanted to give an overview of those to introduce and talk about the vocabulary in the interest of starting to “get used to it” as von Neumann said, even if that’s a long way from having a comprehensive understanding of it.

Having laid that groundwork down I want to summarize this section of the book where Auyang introduces a model to analyze the relationship between the objective state space of system and its representations in different scientific theories. The objective state space is what is “out there” independent of our observations or awareness of it. The representations are what we interact with. We could definitely invoke Immanuel Kant here with his concepts of the “thing in itself”, that he calls the “noumena”, and the “phenomena” that we experience of it. And Auyang definitely draws on Kant repeatedly in her book.

There’s a figure she refers to over several pages and I’ve posted this on the website. But for those listening on the podcast I’ll try to describe it in a way that hopefully isn’t too difficult to follow. In her diagram she has three boxes. The top box is the state space, “M”. So that’s the set of all possible states of a system. Then in this state space there’s one state, “x”. x is what is objectively out there, independent of our observations and theories of it. But we don’t observe or interact with x directly. What we observe are the representations of x. And those are the lower two boxes.

These lower two boxes are fα(M) and fβ(M). These are the representations of certain properties of state space M. fα and fβ are property types that we could be looking for and then fα(M) and fβ(M) are the possible representations we can find when we run experiments to measure for those properties. Inside each of these lower boxes is a smaller box for the representation of the single objective state x. So these would be fα(x) and fβ(x). These are the definite predicates or values we get from our experiments. These are the things we come into contact with.

To tie this back to quantum mechanics real quick, in the quantum mechanical case the way this general picture would play out is that M would be a Hilbert space, x would be a state vector (x being one state in that state space), fα would be an observable, fα(M) would be a representation, and fα(x) would be an amplitude of x in the basis of the observable fα.

What’s important to understand here is that the values measured by the two representations are not equivalent. Someone trying to get at the objective state x in the objective state space M from fα will see something different than someone trying to get at it from fβ. One will get fα(x) and one will get fβ(x). Which one is right? Well they’re both right. But what does that mean? It depends on how much of this picture we see.

So we’ll look at parts of this model in pieces before we get back to the whole, comprehensive picture. But first I want to make another comparison because this is all quite abstract. I think it’s helpful to compare this to sentences in different languages. Say we have a sentence in English and Spanish. In English we say “The dog eats his food” and in Spanish we say “El perro come su comida”. These are different utterances. They sound very different. But we want to say that they mean roughly the same thing. We can translate between the two languages. And people can respond to the utterances in similar ways that indicate that there is something in common to both. But whatever it is that is common to both is not expressible. We only express the sentences in particular languages. But because they are translatable into each other it makes sense to think that there is some third thing that is the meaning the both share.

OK, so keep that example in mind as we get back to physical theories and the objective states they represent. Looking at our model again say we look only at one of the lower boxes, fα(M). In this picture as far as we’re concerned this is all there is. So one thing to say about this is that the meaning of fα(x) is what Auyang calls “unanalyzable”. And why is that? Well, it’s because fα(x) is “absolute, self-evident, and theory-free”. It’s just given. There is no objective state space M that fα(M) is representing. Rather fα(M) is the immediate bottom level. So there’s nothing here to analyze. We don’t have to think about the process of representation.

OK, well let’s add the second lower box, fβ(M). So now we have just the two lower boxes but still no objective state space M. What do we have now? Well we have plurality. There are multiple representations of the same thing and we don’t have a way of knowing which one is true. And neither can we say that they point to one common thing. So this gets to be a very confusing situation because we have both plurality and unanalyzability. Plurality in that we have two different values representing a state, fα(x) and fβ(x). Unanalyzability because, as with the previous view with only the one box, there’s not objective state space that either of these correspond to. No process of representation to analyze here. What we have are conventions. This is the kind of picture Thomas Kuhn gives in his book The Structure of Scientific Revolutions. And this is a picture of relativism. The conventions are incommensurate and the choice among them is arbitrary. I think it’s fair to say that there’s much that’s unsatisfying with this picture.

Well, now let’s add the top box back in so we have everything. This brings what I’d consider an explanatorily robust conceptual device. As Auyang says, “the introduction of the physical object whose state is x not only adds an element in our conceptual structure; it enriches the elements discussed earlier,” fα(M) and fβ(M). In other words. fα(M) and fβ(M) look a lot different with M than without it. And I’d say they also make a lot more sense.

For one thing, the picture is no longer unanalyzable but analyzable. We understand that there is a process of representation occurring when we collect numerical data from experiments. When we look at property types fα and fβ we understand that these both represent M in different ways. As Auyang says, “Various representations can be drastically different, but they represent the same object.” She gives a concrete example: “The same electromagnetic configuration that is a mess in the Cartesian coordinates can become simplicity itself when represented in the spherical coordinates. However, the two representations are equivalent.” What’s key to understand here, and what makes this third, fuller picture, more powerful and coherent is that there is one objective state space, one object that the various representations point back to. So we circumvent relativism. The picture only looks relativistic when we only have the partial view. But when we see state space M and that fα(M) and fβ(M) map onto it we can appreciate that even though fα(x) and fβ(x) are different, they both correspond to one objective state x.

Another important thing to consider is that there is a transformation between fα(M) and fβ(M) that Auyang calls fβ•fα-1. The transformation is the rule for transforming from representation fα(M) to fβ(M). That there is such a transformation and that it is possible to transform between representations arguably evinces the existence of the objective state space that they represent. As Auyang states: “Since fα to fβ are imbedded in the meaning of [fα(x) to fβ(x)], the transformation fβ•fα-1 connects the two representations in a necessary way dictated by the object x. fβ•fα-1 is a composite map. It not only pairs the two predicates [fα(x) to fβ(x)], it identifies them as representations of the same object x, to which it refers via the individual maps fα-1 and fβ. Since fβ•fα-1 always points to an object x, the representations they connect not only enjoy intersubjective agreement; they are also objectively valid. To use Kant’s words, the representations are no longer connected merely by habit; they are united in the object.” This is related to the example I gave earlier about two sentences in different languages as representations of a common referent.

And I just think that’s a lovely picture. One of my favorite thinking tools is to take things up a level to try and see things that weren’t observable before. Like how you can see more from an airplane than when you’re on the ground. It’s not that the way things are has changed. But we see more of it. And with a more complete pictures the parts that seemed random or even contradictory make more sense and work together as a rational system.

So that’s one bit of the book I wanted to talk about. There are a few other things I’ve read that I want to talk about later too. And I’m only about halfway through. And if it continues to be as jam-packed with insights as it has been up to now I’m sure there will be more I’ll want to talk about.

Hegel Beyond Marxism

Hegel is much more than a precursor to Marx. His philosophy has a lot to offer in many subjects, not least in theology and the philosophy of religion.

I’ve been reading Kantian Reason and Hegelian Spirit: The Idealistic Logic of Modern Theology by Gary Dorrien, a book I sort of found randomly while browsing recently. It scratches an admittedly idiosyncratic intellectual itch, my interest in both theology and nineteenth century cultural and intellectual history. The latter has basically been a thing since my senior year in high school when I was in academic decathlon and we studied Romanticism in the art, literature, and music units. Anyway, I was looking for something on theology and German Idealism and there was this book that is also conveniently available in audio format. Turns out it’s actually really good too, if you’re into that sort of thing. I was listening to it on the way to work today and Dorrien made some comments about Hegel that I really appreciated.

I was first exposed to Hegel in the context of Marx, as an influence of Marx. I learned about the concept of the dialectic first in the form of Marx’s dialectical materialism. And it’s an interesting idea, both the Marxian version and other versions of the dialectic. The basic idea being that there’s a kind of ongoing conversation, effectively, working itself out through history. Ideas, cultures, and systems develop through their interactions and conflicts. For Marx this was fundamentally an economic process with everything else, like religion, being a kind of add-on or “superstructure” on top of that. He saw economic development proceeding through feudalism and capitalism and ultimately to communism. That was his theory of history. And I knew that Hegel had a lot to say about this and that Marx sort of used him as a springboard. And that’s definitely true, Marx was very influenced by Hegel.

But Hegel was a lot more than just a precursor to Marx. And in the nineteenth century I’d say he was a more influential figure. Marx certainly dominates much of the twentieth century. But in the nineteenth century everyone was responding to Hegel. So, why? What was this guy about? Well he wrote about everything. He was a tremendous systems thinker who had a theory of the whole of reality, quite literally. So it’s almost hard to say what his schtick was. His writing is also notoriously difficult so it’s often hard to even understand what he’s talking about. But. His thought definitely includes much more than the elements picked out in the service of Marxism.

And this is something Dorrien emphasized in the parts of his book I was reading today. And he got me nodding along enthusiastically, because he was saying a lot of the stuff I had been thinking recently. I’m actually one of these gluttons for punishment who has read Hegel’s behemoth work The Phenomenology of Spirit in its entirety. I read it this year read in fact. And Dorrien was talking about how much of the Phenomenology is a disorganized mess and I have to agree. That makes it difficult to get through. Still there’s a lot of fascinating stuff there. The part that gets emphasized a lot is the section on the master and slave, which has been incredibly influential. This shows up in a redrawn form in Nietzsche and it definitely has potential political use. Dorrien talks about how interpreters of Hegel, especially those with political interests, have really latched onto this, seeing the Phenomenology through the lens of oppression and liberation. In the twentieth century Alexandre Kojève was a towering interpreter of Hegel, sort of the authority on Hegel. And he approached him through a political lens and a Marxist lens. As a consequence people have tended to downplay the metaphysical and especially the theological side of Hegel. And in my reading of the Phenomenology it’s struck me how metaphysical and theological Hegel was. Those weren’t incidental to Hegel. I’d say there were central. Far from the revolutionary Marxist and atheist interpretations of Hegel he was a profoundly spiritually-interested Christian philosopher. He wasn’t orthodox by any means. Hardly conventional. But Christianity was central to his philosophy, which is one of the reasons I find him so interesting.

Now it’s certainly true that Hegel was a political thinker. Like I said, he wrote about everything. His Philosophy of Right is one of the canonical texts of Western political philosophy. He lived through the Napoleonic wars and the emergence of liberalism. But he had many other interests as well beyond the political. I tend to think the political has a dangerous tendency to metastasize and blind us to many other important features of the world and this is just one example of that. It’s not that the political isn’t important but it’s not everything. So I really appreciated Dorrien talking about this in his book.

Just a few more comments on the Phenomenology. People have struggled to say what this book is about, not least of all Hegel himself. But I’d say it’s about the development of Reason with a capital R, starting from the barest simplicity to the most comprehensive entirety of all reality. And in between those extremes is a systematic study of the way human beings are able to think and reason. So in a way it’s analogous to titles like Locke’s An Essay Concerning Human Understanding and Hume’s An Enquiry Concerning Human Understanding. And of course he, like everyone in his day, was responding to Kant and his Critique of Pure Reason. And I’d say that Hegel, though his language is much more poetic and difficult, is really doing something very similar. Just more epic in scope.

And you can get a pretty good idea of this from the book’s table of contents. The main sections are, in order: consciousness, self-consciousness, reason, spirit, religion, and absolute knowing. And that first section on consciousness is divided into subsections on sense-certainty and perception, so starting with just the most basic animal qualities. Then self-consciousness is more developed then consciousness, reason is more developed than self-consciousness, and so on. That famous master-slave section is actually in the section on self-consciousness. And though it probably does have political associations I’d say it’s really more about the way self-consciousness arises through realization that other people are centers of consciousness like ourselves and that this induces reflection leading to a more recursive self-consciousness, which I think is a very interesting idea, even more so than the political implications. Lots of interesting stuff there.

I’m especially interested in the sections on religion. Hegel has some very interesting ideas that he gets into into the Phenomenology that he develops further in his Lectures on the Philosophy of Religion, which I haven’t had the opportunity to read yet directly, apart from commentaries. But he has an understanding of religion and God in relation to the world progressing through a dialectic, which leads to the religious development we see in history. Something I’m studying in the Bible right now is the way the understanding of God as well as the relationships God has with humanity change throughout history. With my Latter-day Saint background this is also very amenable to our idea of progression, line-upon-line development, and continuing revelation. So Hegel is very interesting for all of that.

So that’s the end of my soapbox on Hegel and why, even if you come across through Marx, like I did, there’s a lot more there. And he’s a very interesting philosopher and, arguably, Christian theologian in his own right.