Theory of Constructed Emotion

Jared and Todd discuss the theory of constructed emotion and the work of Lisa Feldman Barrett. In contrast to a location-based understanding, in which discrete emotion categories consistently and specifically correspond to distinct brain regions, constructionism proposes that such emotion categories are constructed of more general brain networks not specific to those categories. We discuss degeneracy, the difference between affect and emotion, different forms of dualism, affinities to the philosophies of Kant and Wittgenstein, and the ways concepts and words have meaning.

Slavoj Žižek and the Sublime Object

Slavoj Žižek is a uniquely entertaining and engaging philosopher with an animated oratorical style. But this can often make it difficult to follow his train of thought and pick out key ideas from it. This episode gives an overview of his most important book, The Sublime Object of Ideology, which presents many of the ideas persisting in all his work. In particular, this overview focuses on Žižek’s theory of the Sublime. Žižek’s ideas bring him into close contact with the German philosophers Immanuel Kant and G.W.F. Hegel.

If we were to put philosophers on a scale from dry, boring presentation styles on one end to engaging, entertaining presentation styles on the other, Slovenian philosopher Slavoj Žižek would be off the chart with his exuberant and oratorical approach. Half of his philosophy seems to consist of jokes. He has a charming (or maybe frustrating) tendency to evade direct argument and debate, opting instead for a kind of friendly banter that opens up a space for people to make up their own minds. For all that, it can be difficult to follow his train of thought or identify some kind of thesis from it all. That difficulty would certainly not be unique among philosophers, especially those philosophers with whom Žižek tends to engage. For example, most of Žižek’s philosophy is a kind of interpretation of G.W.F. Hegel through the ideas of Jacque Lacan. Hegel and Lacan may be two of the most difficult philosophers in the entire tradition of European Continental philosophy. So it’s not surprising that Žižek would be somewhat difficult to follow as well. Be that as it may, Žižek does have some core ideas that recur in his books and talks. Many of these core ideas are present in what has been called his magnum opus, his 1989 book, The Sublime Object of Ideology.

I haven’t found much out there by way of detailed summaries for this book, so after I read it I thought I’d make one myself. I won’t give a summary of all the ideas in the book. There are a lot of them. Instead I’d like to focus on the subject I found most interesting. And that is Žižek’s theory of the Sublime. Žižek is perhaps best known for his purported and self-attributed communism. A provocative claim, to be sure. And that provocation might even be its most significant aspect. Žižek’s communism and leftist politics are interesting features of his thought but for the purposes of this summary I’m not going to focus as much on the political aspects of the book, in spite of the word “ideology” being right there in the title. Instead I want to focus on the concept of the Sublime and look at how Žižek fits into the history of the philosophical treatment of the Sublime.

I’ll introduce his basic position here first and then fill in the details and background information to explain it. Žižek says that there are certain objects that we elevate and believe have or act as if they had greater qualities behind their surface level. But, there is actually nothing greater behind this surface level. Nevertheless, for Žižek there is a real experience of sublimity. It is not of some greater thing, in or behind the object. Rather we experience the Sublime in the very process of discovering that there is nothing, no greater thing behind the object. This realization is Žižek’s sublime object, the realization that there is nothing being concealed.

This will all require some explanation and background. First let’s look at the concept of the Sublime and its philosophical history.

Žižek’s theory of the Sublime is in dialogue with the aesthetic theory of Immanuel Kant. In Kant’s thought aesthetic responses include the Beautiful and the Sublime. The Beautiful is well-ordered and harmonious. The Sublime is awesome and even frightening. The Sublime is found in those things that transcend our ability to grasp them; powers of nature like thunder and lightning. Žižek’s underlying theory of the Sublime is quite different from Kant’s but it also has a quality of not being directly approachable. He says in The Sublime Object of Ideology, that “The sublime object is an object which cannot be approached too closely”. But for Žižek the reason for this is very different from Kant’s. For Žižek, “if we get too near it, it loses its sublime features and becomes an ordinary vulgar object”. Kant’s sublime object would not be something that loses its sublimity. But for Žižek the object of the Sublime is not anything special in itself. And this is an important difference. For Žižek it’s not the object itself that is special. The object is elevated to a special position through a formal fiction. And the sublimity is found in the very process of recognizing that there is nothing greater behind the ordinary, vulgar object.

What are some examples of this kind of formal fiction? Žižek finds theoretical similarities of this process in the work of Marx and Freud. Both Marx and Freud theorize that certain formal processes endow ordinary objects with fictionally extraordinary qualities. This is also called a fetishization. For Marx this happens with the commodity and for Freud this happens with dreams. In both cases people act as if the commodity or the dream have some greater significance. They don’t. But the process of acting as if they did still has real effects in the social structures of which they are a part. Fiat currency is a pretty clear case of this. Money only has value because people act as if it did. And so, in a way, it does. Žižek says:

“There is a fundamental homology between the interpretative procedure of Marx and Freud – more precisely, between their analysis of commodity and of dreams. In both cases the point is to avoid the properly fetishistic fascination of the ‘content’ supposedly hidden behind the form: the ‘secret’ to be unveiled through analysis is not the content hidden by the form (the form of commodities, the form of dreams) but, on the contrary, the ‘secret’ of this form itself.”

Žižek says the key is not the content, like what a dream means for instance. The key is that there is a formal process of apparent concealment in the first place. We come to understand this process not by figuring out what is being concealed but by realizing that the only thing being concealed is that nothing is concealed. There’s a pattern that will recur here, the kind of pattern to which Žižek will apply this sort of deflation. There’s an object and a presumed concealed object behind it. For all such patterns Žižek will locate the Sublime in the process of dispelling the notion of concealment to reveal that there is nothing being concealed. 

Another example where Žižek applies this pattern is to Kant’s philosophy of phenomena and noumena. That is (1) things as we conceive of them and (2) what we posit to be the inaccessible things in themselves. In Kant’s philosophy we understand the world around us using mental structures called categories that organize the raw perceptions of our senses. We take raw sense data and impose our categories onto them like unity, plurality, causality, dependence, necessity, contingency, etc. We use these categories to order our sense perceptions and make them intelligible. But what’s crucial for Kant is that the product of any such mental construction is just that: a construction. It is not the thing in itself. This is the Kantian distinction between noumena and phenomena. A noumenon is the “thing in itself” as it is, independent of our conception of it. A phenomenon is the thing as it is to us after it has passed through our mental processing. For Kant the phenomenon is not the same thing as the noumenon. In fact, the noumenon, the thing in itself, is forever inaccessible to us.

Žižek compares this to there being something hidden behind a curtain. Sort of like in the Wizard of Oz. The phenomenon is what we see, but we imagine that there’s something deeper. Something behind the curtain. Žižek calls this “the Kantian gap dividing forever the Thing from the world of phenomena”. And naturally, Žižek is going to challenge this idea. For Žižek the Sublime arises in the realization that there is nothing behind the curtain: “The illusion that there is something hidden behind the curtain is thus a reflexive one: what is hidden behind the appearance is the possibility of this very illusion – behind the curtain is the fact that the subject thinks something must be behind it.” What the Kantian gap conceals is “a foreboding that perhaps this Thing is itself nothing but a lack”. For Žižek this introduces a paradox. For Kant the noumenon is forever inaccessible. But for Žižek (and for Hegel) it’s this very failure that enables us to access – not the thing in itself, since that’s a fiction – but the Sublime: “The paradox of the Sublime is as follows: in principle, the gap separating phenomenal, empirical objects of experience from the Thing-in-itself is insurmountable – that is, no empirical object, no representation (Vorstellung) of it can adequately present (darstellen) the Thing (the suprasensible Idea); but the Sublime is an object in which we can experience this very impossibility, this permanent failure of the representation to reach after the Thing.”

Žižek traces this insight back to Hegel and sees it as a Hegelian corrective to Kant. A few illustrative quotes on this point:

“Hegel’s reproach to Kant… is… that it is Kant himself who still remains a prisoner of the field of representation… Kant still presupposes that the Thing-in-itself exists as something positively given beyond the field of representation, of phenomenality… Hegel’s position is, in contrast, that there is nothing beyond phenomenality, beyond the field of representation.”

One of the most important ideas in the interpretation of Hegel’s philosophy is of the dialectic. A dialectic is, in its most basic form, a dialogue, a conversation in which ideas are exchanged. Often the ideas being exchanged are in tension and the hope is that in this interchange of competing ideas there can be some kind of productive insight, that this dialectic will be philosophically generative. This was an idea going around among German philosophers at the time. We also see this in Johann Fichte. The pattern is sometimes expressed as a triad of a thesis, a competing antithesis, and a resulting synthesis. We also find a materialist, economic version of this in Marx.

Žižek has a somewhat different and interesting perspective on the Hegelian dialectic, one that he understands to be more true to Hegel’s own. For Žižek the result of the dialectic is fundamentally a change in perspective that reveals the Nothingness, that dispels the illusion of concealment. It’s the pattern we’ve been seeing where there’s an object, a supposed concealed object, and then a revelation that there’s nothing being concealed. Žižek also sees this in the Hegelian dialectic.

“The only philosophical counterpart here is Hegelian dialectics: at the very beginning of his Logic, Being and Nothingness are not complementary, neither is Hegel’s point that each of them obtains its identity through its difference from the other. The point is that Being in itself, when we try to grasp it “as it is’, in its pure abstraction and indeterminacy, without further specification, reveals itself to be Nothingness.”

I’ve gone far too long talking about Žižek without sharing any of his jokes so I’ll share one here that he uses to highlight this idea:

“Here we have a kind of dialogic economy: we articulate a proposition denying the subject, our attempt fails, we experience the absolute contradiction, the extreme negative relationship between the subject and the predicate – and this absolute discordance is the subject as absolute negativity. It is like a well-known Soviet joke about Rabinovitch, a Jew who wants to emigrate. The bureaucrat at the emigration office asks him why; Rabinovitch answers: ‘There are two reasons why. The first is that I’m afraid that in the Soviet Union the Communists will lose power, there will be a counter-revolution and the new power will put all the blame for the Communist crimes on us, Jews – there will again be anti­-Jewish pogroms… ‘But’, interrupts the bureaucrat, ‘this is pure nonsense, nothing can change in the Soviet Union, the power of the Communists will last forever!’ ‘Well,’ responds Rabinovitch calmly, ‘that’s my second reason’. The logic is the same here as in the Hegelian proposition ‘the spirit is a bone’: the very failure of the first reading gives us the true meaning… The Rabinovitch joke also exemplifies the logic of the ill-famed Hegelian triad: if the first reason for emigrating is the ‘thesis’ and the bureaucrat’s objection the ‘anti-thesis’, then the ‘synthesis’ is not any kind of return to the thesis, some kind of healing of the wound made by the anti-thesis – the ‘synthesis’ is exactly the same as the ‘anti-thesis’, the only difference lies in a certain change of perspective.”

So is the problem that the Communists will lose power or that they won’t? That, for Rabinovitch, is precisely the problem. It doesn’t matter. It’s bad either way. This is a humorous, sort of parabolic illustration of the kind of dialectic where Žižek sees discord as fundamental; “absolute discordance” as he calls it. Early in the book Žižek proposes that Hegelian dialectics don’t magically resolve antagonisms. Instead, Žižek asks: “What if, for Hegel, the point, precisely, is to not ‘resolve’ antagonisms ‘in reality’, but simply to enact a parallax shift by means of which antagonisms are recognized ‘as such’ and thereby perceived in their ‘positive’ role?”

This kind of recognition of the fundamentality of discordance is instrumental in Hegel’s system for how the mind progresses to higher levels of knowledge. Arriving at such discordance is not a failure but a successful insight into the nature of things. “We pass from Understanding to Reason not when this analysis, or tearing apart, is overcome in a synthesis that brings us back to the wealth of reality, but when this power of ‘tearing apart’ is displaced from being ‘merely in our mind’ into things themselves, as their inherent power of negativity.” This “tearing apart” is intrinsic to the nature of things. Later in the book Žižek says in a gesture of “decisive emphasis” that “it is not only that the appearance, the fissure between appearance and essence, is a fissure internal to the essence itself; the crucial point is that, inversely, ‘essence’ itself is nothing but the self rupture, the self fissure of the appearance.” Contradiction is “an internal condition of every identity.”

What’s crucial is that in this realization of negativity, negativity itself takes on a positive role. When an illusion is dispelled and we realize that there’s nothing behind it this is a negativity. But it’s not just a negativity. There’s a positive aspect to it in the realization of the negativity, in seeing the negativity as such, where before it was unrealized and unnoticed. 

“So ‘we’ (who have already ‘gone through the fantasy’) can see that there is nothing where the

consciousness thought that it saw something, but our knowledge is already mediated by this ‘illusion’ in so far as it aims at the empty space which makes the illusion possible. In other words, if we subtract from the illusion the illusion itself (its positive content) what remains is not simply nothing but a determinate nothing, the void in the structure which opened the space for the ‘illusion’. To ‘unmask the illusion’ does not mean that ‘there is nothing to see behind it’: what we must be able to see is precisely this nothing as such – beyond the phenomena, there is nothing but this nothing itself, ‘nothing’ which is the subject.”

This idea of a “determinate nothing” is interesting. It reminds me of another joke that Žižek often tells, not in this book but elsewhere. “‘Waiter! A cup of coffee without cream, please!’ ‘I’m sorry, sir, we have no cream, only milk, so can it be a coffee without milk?’” There’s a difference to us between just simply nothing and a determinate nothing. The determinate nothing is the nothing where before, because of the illusion, there was thought to be something.

Žižek believes that this kind of stripping away toward Nothingness is instrumental for Hegel. Hegel’s most famous and arguably most important book was his Phenomenology of Mind. The Phenomenology is a challenging and intimidating book, but I might summarize it as a philosophy of the way that the mind comes to understand things. This is basically the project Hegel outlines in his Preface to the book. Process is fundamental for all of this. Hegel outlines how the mind progresses along successive stages toward fuller understanding.

Žižek challenges one interpretation of Hegel in which he is understood to say that the process of progressive understanding is one of continual accretion, constantly taking on more and more information. “Is not the Hegelian Idea effectively a voracious devourer which ‘swallows up’ every object it comes upon?” Žižek says No! That this is not a correct understanding of Hegel. “The standard reading constructs the Hegelian absolute Substance-Subject as thoroughly constipated [classic Žižek right there] – retaining within itself the swallowed content.” Continuing with the digestive metaphor Žižek says the Hegelian system is a much more healthy process of the usual consumption and excretion, integrating and using conceptual nutrients for sustenance and development, then discarding what’s left over. He calls this process one of “notional deployment”. The idea is not just to accumulate and retain everything, but rather to process and use. “The Idea, in its resolve/decision, ‘freely releases itself’ into Nature, lets Nature go, leaves it off, discards it, pushes it away from itself, and thus liberates it.”

Thus for Žižek abstraction, which is a crucial process of understanding, is also a process of subtraction. “For Hegel the true problem is… the fact that, when we observe a thing, we see too much in it, we fall under the spell of the wealth of empirical detail which prevents us from clearly perceiving the notional determination which forms the core of the thing.” We might think of the difference between someone enjoying nature while out on a hike and a biologist seeking to identify some property of a particular species in that setting. The hiker wants to experience the totality without reduction. But the biologist can’t do that. The biologist needs to isolate, dissect, and analyze.

What is the endpoint of the process of abstraction and subtraction? Žižek says: “If we make an abstraction, if we subtract all the richness of the different modes of subjectivation, all the fullness of experience present in the way the individuals are ‘living’ their subject-positions, what remains is an empty place which was filled out with this richness; this original void, this lack of symbolic structure, is the subject, the subject of the signifier. The subject is therefore to be strictly opposed to the effect of subjectivation: what the subjectivation masks is not a pre- or trans-subjective process of writing but a lack in the structure, a lack which is the subject.”

So here again we see Nothingness behind objects. The idea being that this would be the eventual result of repeated abstraction. But this idea that the end point of the process of abstraction is a void is not nihilistic. At least I don’t understand it to be nihilistic. We’re often interested in more general principles than in the particulars to which they refer. Scientists use data sets to find general patterns. The patterns, as in the form of equations, strip away many of the particulars and abstract from them. If there is some Theory of Everything it will certainly be highly abstract and several levels removed from almost all the particulars that populate our immediate reality. This is just a consequence of the kind of process that abstraction is.

That being said however, non-abstracted, un-subtracted particulars are also useful. What are we to do with the Sublime object in those settings where we have grasped the negativity behind the illusion? Interestingly enough, Žižek says we ought to “come to terms with it”.

“The point is not just that we must unmask the structural mechanism which is producing the effect of subject as ideological misrecognition, but that we must at the same time fully acknowledge this misrecognition as unavoidable – that is, we must accept a certain delusion as a condition of our historical activity, of assuming a role as agent of the historical process… There is no solution, no escape from it; the thing to do is not to ‘overcome’, to ‘abolish’ it, but to come to terms with it, to learn to recognize it in its terrifying dimension and then, on the basis of this fundamental recognition, to try to articulate a modus vivendi with it.”

Why is that? Why find some way to come to terms with the fundamental discordance? The reason is that the alternative is unacceptable. And here we see some of political implications of Žižek’s ideas:

“All ‘culture’ is in a way a reaction-formation, an attempt to limit, canalize – to cultivate this imbalance, this traumatic kernel, this radical antagonism through which man cuts his umbilical cord with nature, with animal homeostasis. It is not only that the aim is no longer to abolish this drive antagonism, but the aspiration to abolish it is precisely the source of totalitarian temptation: the greatest mass murders and holocausts have always been perpetrated in the name of man as harmonious being, of a New Man without antagonistic tension…”

“We have the same logic with democracy: it is – to use the worn-out phrase attributed to Churchill – the worst of all possible systems; the only problem is that there is no other which would be better. That is to say, democracy always entails the possibility of corruption, of the rule of dull mediocrity, the only problem is that every attempt to elude this inherent risk and to restore ‘real’ democracy necessarily brings about its opposite – it ends in the abolition of democracy itself.”

There’s a bit of similarity here to Richard Rorty’s liberal ironism. One of Žižek’s favorite phrases is: “They know very well what they are doing, but still, they are doing it.” Whether it’s in the canalizing aspects of culture or in a reverence for democracy we persist in acting out our cultural and democratic roles as if the ideologies underlying them were not discordant and illusory. Far worse would be to try to iron out the wrinkles, as totalitarian regimes strive to do.

I’ll wrap all this up with two parting thoughts. Or rather, two parting questions. First is: How accurate is Žižek’s interpretation of Hegel? I guess that presumes that there are accurate and inaccurate interpretations of Hegel, something that the logical positivists might have taken issue with. But whatever. Hegel is one of those philosophers who has so many commentators he almost lives entirely through them. The Hegel of Alexandre Kojève, for example, being one very significant incarnation of Hegel. The second question is what to make of Žižek’s ideas as such. In particular the idea we experience the Sublime in the very process of discovering that there is nothing, no greater thing behind the object; the realization that there is nothing being concealed. In how many domains does this idea apply? Are all apparent concealments false concealments? Are there any true “things in themselves” that exist in reality, even if beyond our reach? I’m inclined to think that there are.

So, if for whatever reason you found yourself interested in Žižek and wondering what some of his ideas were about I hope this was useful to you and not too far off base. Thanks for listening.

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.