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.

Star Wars Episode II: Jeff and Todd Strike Back

We talk about George Lucas, his background and influences. Flash Gordon, Akira Kurosawa, Horse Opera and Spaghetti Westerns. The sequel trilogy: overall coherence, the Star Wars tradition of seeing contradictions “from a certain point of view”. The sequel trilogy characters: Rey, Rose, and Vice Admiral Holdo.

A Second Look at Religion

It’s undeniable that living in a secular age brings significant challenges to belief in God and to religious practice. And it’s understandable. Still, there are reasons to give it a second look, or maybe a new first look with a fresh perspective.

I’ve titled this episode “A Second Look at Religion”, which is a kind of working title I’ve been sitting with to express an idea I’ve had bouncing around in my head for a while. The idea being that religion deserves a second look and to express that to people who might doubt it or think otherwise. And why might people think otherwise? Well, I think it’s hard to deny that we live in a secular age. This isn’t new. We’ve been in a secular age for several centuries, though its reach may be expanding and more people may be aware of it. The philosopher Charles Taylor talked about this in his book A Secular Age and referred to this kind of secularism as a condition where religious belief is no longer axiomatic, or just a given. It’s possible to imagine not believing in God. And that’s certainly the world we live in. Even for religious people believing in God is not just a given. We’re at least aware that there are other options. And we know many people who take up the other options. And we can’t realistically just ignore that. Or at least it wouldn’t be very healthy, certainly not healthy for our relationships. We have to find ways of talking to each other even with differing world views.

I think it’s best for people to work to reach each other from both sides. But since I’m a religious person I take up the burden to try and relate to secular people and to translate ideas into terms that make more sense in a secular framework. In a previous episode with Mike on object-oriented theology we talked about this idea of “porting” concepts into different frameworks. I think that’s a fascinating process. And I’m far from knowing how to do that really. I’m in the middle of this, what I expect to be a lifelong process. And I get a lot of inspiration from Paul. Paul was a fascinating cosmopolitan figure, living in a cosmopolitan world. A diaspora Jew from Tarsus, not Judea, well-spoken in Greek and well-versed in Greek thought, rhetoric, and ideas. And focused primarily on a Greek audience. He was able, quite self-consciously, to be different things to different people. He told the Corinthians that to the Jews he became as a Jew, to those under the law as one under the law, to those outside the law as one outside the law (1 Corinthians 9:20-21).

One of my favorite examples of Paul’s Hellenic porting of the God of Israel was on Mars Hill in Athens, where he referenced and appropriated a number of ideas from Greek thinkers and Greek thought (Acts 17:28). For example, he said:

“In him, therefore, we live and move and have our being.” This line is probably associated with Posidonius (135 BC – 51 BC). Also sometimes attributed to Epimenides (7th or 6th century BC).

Paul also quotes this line: “For we too are his offspring,” which he got from Aratus (315 BC/310 BC – 240 BC), a Greek poet from Cilicia who was educated as a Stoic.

I just think this is a fascinating method from Paul. He certainly knew the Hebrew scriptures. And he knew them in the Greek of the Septuagint translation. And he certainly quoted them and interpreted them frequently. But here he went even farther and expressed his ideas in terms that he translated not only linguistically but culturally. I find that wonderfully inspiring.

So anyway, I think the corollary today, to what the Greeks were to Paul, is the secular age. That is the wider world that religion in general and Christianity in particular comes into contact with. And maybe these are incommensurable but I don’t think so.

One of the first things I think I should say as a Christian is, “I get it.” I get the reasons why religion and Christianity can be hard pills to swallow in the modern world. Part of that is because of the supernatural stuff which seems out place in the modern, scientific and technological world. And part of it is a matter of values. The world of the Bible, well really there are several different worlds since it was written over several centuries, but let’s go ahead and generalize to say that many of the values of the Bible are different from the ones we have today. In recent times differences in values regarding LGBT sexual identities are some of the most obvious. So there are many reasons people turn away from God and religion.

But, I think a second look is warranted, maybe after some time away to clear the head and clear the palate. I’m calling it a second look. The Biblical scholar and Christian author Marcus Borg often called it “seeing again for the first time”. I think that’s kind of cool. He had books with titles like Meeting Jesus Again for the First Time and Reading the Bible Again for the First Time. The God We Never Knew follows a similar track. From that last book, The God We Never Knew, he has an idea that’s stuck with me for a long time and that I keep coming back to about something more. This is an idea that there is more to things than is apparent on the surface. And looking closer and deeper is a consummate religious act. Borg says:

“The sacred is ‘right here’ as well as ‘the beyond’ that encompasses everything. This way of thinking about God, I claim, is not only faithful to the biblical and Christian tradition but also makes the most sense of our experience. For there is much in our experience–of nature, human love, mystery, wonder, amazement–that conveys the reality of the sacred, a surpassingly great ‘more’ that we know in exceptional moments. Many of us experience life as permeated and surrounded by a gracious mystery, a surplus of being that transcends understanding, and when we come to know that mystery as God, our faith becomes full of meaning and vitality.”

And let me step back here a second to bring this home to our daily life. I don’t know about you but going about my daily life in the world of work, commuting, paying bills, paying taxes, worrying about the economy, and politics, etc. I sometimes feel a lack and dissatisfaction from all that and wonder, “Is this all there is?” I think one of the first acts of faith is to consider the possibility that it is not. That there is more there. And I like how Borg mentions that this can be both right here and beyond. I personally believe in a more-ness in both, but the right here is probably the more accessible of the two for present purposes, especially from the secular viewpoint.

I was talking with a friend of mine the other day about the importance of having a sense of awe in things. And he’s not a church-goer or religious in the traditional sense but I would say there’s a recovery or reconstitution there of religious activity from a secular direction. When you think about things deeply enough and long enough they can start to seem remarkable.

Borg mentioned a “surplus of being” and I like that too. David Bentley Hart, a Christian scholar of the early church fathers, has this great line that “everything is out of place”. Why is that? There’s this thought experiment from the philosopher Richard Taylor that if you were just wandering in the woods and suddenly came upon a large translucent sphere you’d naturally wonder about that, how it should happen to be there. You wouldn’t just think, well, it’s just one of those things. But in a sense this is true of everything. Life, matter, the universe – it’s a remarkable that any of it is here really. It’s all “out of place”. We just get used to it so we forget to be struck by how remarkable the mere fact of existence, much less our own existence, really is. Recovering that sense of awe at the world around us is a very religious act I think.

In our episode on object-oriented theology we quoted a line from Adam Miller in his book Speculative Grace that, “religion is what breaks our will to go away.” I think that’s great. To go away from what? What’s right here in front of us. To really attend to it. Another quote from that same book: “Religion corrects for our farsightedness. It addresses the invisibility of objects that are commonly too familiar, too available, too immanent to be seen.” An important moment in my religious from a number of years ago, one of these lines that sticks with you, I was attending an event with Marcus Borg – I actually met him and talked with him a little before hand, very nice guy – anyway, someone asked a question to the effect of how to become closer to God or see the hand of God more in the world. And Borg’s answer was to “pay attention”. Just in general, “pay attention”. I think that’s pretty great. Another line I tried to quote in the conversation with Mike but kind of butchered was a line from Hugh B. Brown, a Latter-day Saint leader: “First then we say be aware, for the degree of your awareness will determine the measure of your aliveness.”

This may seem kind of removed from what we think of as religious but I actually think it’s quite relevant and on target to it. But it does seem different certainly. And that’s part of the second look, seeing again for the first time thing. I understand that there are reasons people get disturbed with religion. I know many people who have gone through the New Atheist heyday of the early 2000s with Richard Dawkins, Sam Harris, and company. Or in my own backyard, Latter-day Saints who’ve read the CES Letter, which catalogues a bunch of issues in church history and doctrine. That’s all there. It causes concerns and that’s not insignificant. And this perspective of religion as awe, awareness, more-ness, looking deeper into things doesn’t address those concerns directly. But I think it’s possible and worth considering those aspects separately. Those things that are concerning don’t have to disappear or be resolved. Even if all those things are true I think it doesn’t undermine the core religious truth and religious practice.

And this is something I think I’ll be talking a lot about in the future because it’s both a personal interest and a topic of interest/concern in my social circles. And in the secular world as a whole.

The Tikal-Calakmul Wars

One of the great things about Mesoamerican history is that we know so much about it. Names of places like Tikal, Calakmul, and Copán might inspire images of stone temples with a mysterious, long-lost history. This may have been true one hundred years ago. But today we know a great deal about their history because we can read their writing. We know the names of their rulers. We know important things that they did. In fact, they put a lot of effort into making sure their histories were preserved for posterity. Thanks to these records we have some very exciting and interesting history.

One of the great things about Mesoamerican history is that we know so much about it. Names of places like Tikal, Calakmul, and Copán might inspire images of stone temples with a mysterious, long-lost history. This may have been true one hundred years ago. But today we know a great deal about their history because we can read their writing. We know the names of their rulers. We know important things that they did. In fact, they put a lot of effort into making sure their histories were preserved for posterity. Thanks to these records we have some very exciting and interesting history.

One of the most important things to understand about Mayan politics in the Classic Period was the rivalry between Tikal and Calakmul. The clashes between these two states dominated the region. These are often called the Tikal-Calakmul Wars and they took place between 537 to 744. These two states forged complicated networks of alliances with other states as they competed for dominance. It’s great history, full of violence and intrigue. Other key players in the great wars included Dos Pilas, Copán, Quiriguá, Naranjo, and Caracol. This period of warefare can be divided into three warring periods: The First Tikal-Calakmul War (537–572), The Second Tikal-Calakmul War (650–695), and the Third Tikal-Calakmul War (720–744).

In what follows I will give a historical narrative of events in the Tikal-Calakmul Wars without referring to all the archeological evidence and glyphs that were used to construct it. This kind of detailed background is available in Robert Sharer and Loa Traxler’s excellent book The Ancient Maya.


Before the ascendancy of Calakmul, Tikal had been the dominant power in the region. One of the most important events in the history of Tikal occurred in 378 when it’s ruler, Chak Tok Ich’aak I (Great Jaguar Paw), was killed by foreign invaders. Chak Tok Ich’aak was defeated by one Siyaj K’ak’ (Fireborn) ostensibly under the auspices of Teotichuacan. Siyaj K’ak’ was serving under Spearthrower Owl, possibly the ruler of Teotichuacan. The first ruler of the new dynasty was Yax Nuun Ayiin I. The new dynasty installed in Tikal was thereafter influenced by Teotihuacan culture mixed with Mayan culture. Tikal exercised significant influence over states in the region in ways that would set the stage for the alliances in the Tikal-Calakmul wars. In 426, Tikal installed K’inich Yax K’uk’ Mo’ as the first ruler of Copán. Copán in turn founded the vassal state of Quiriguá in 426. Calakmul was a less important power in the region until the onset of the First Tikal-Calakmul War in 537.

The First Tikal-Calakmul War (537-572)

Calakmul began to wield more influence in the region after Yaxchilan captured the rulers of Bonampak, Lakamtuun, and Calakmul in 537. After its defeat by Yaxchilan, Calakmul retaliated and conquered Yaxchilan. Calakmul then began to show more ambition, conquering other states. In 546 Calakmul defeated Naranjo and installed Aj Wosal as ruler. This gave Calakmul hegemony over Naranjo and extended its sphere of influence. Meanwhile, in Tikal, Wak Chan K’awill oversaw the inauguration of Yajaw Te’ K’inich II in Caracol. Wak Chan K’awill may have been reacting to Calakmul’s growing power by strengthening an alliance with Caracol. In any event a Tikal-Caracol alliance was not to last.

Tikal and Caracol broke whatever alliance they had and the two states went to war with each other in 556. But things were going to get worse for Tikal. In 561, a new ruler named Sky Witness came to power in Calakmul. One year later, in 562, Wak Chan K’awill was captured by Caracol, which had switched alliances and joined Calakmul. Under this new Calakmul-Caracol alliance Wak Chan K’awill was offered as sacrifice and Tikal suffered significant loss of prestige and influence. Tikal did win some less significant battles in the years following, such as a victory over Caracol in 564. But by 572 Tikal was thoroughly defeated and the First Tikal-Calakmul more came to a close. That same year Sky Witness died.

After the First Tikal-Calakmul War, Tikal entered a period of hiatus. During this time Calakmul strengthened its hegemony. In 579 Scroll Serpent came to power in Calakmul. In 611 Scroll Serpent launched a major campaign against Palenque. The reason for this campaign is not known for certain but Palenque may have been a Tikal ally and this may have been part of the larger conflict between the two states. This attack on Palenque was an impressive logistical achievement and a fine demonstration of Calakmul’s power and ambition. It was a long-distance campaign. Palenque lies nearly 300 km from Calakmul and the army needed to cross several rivers, including the Usumacinta, to get there. As an aside, it was during this time that the future king of Palenque, K’inich Janaab Pakal I, was a young boy. K’inich Janaab Pakal I (603-683), or Pacal the Great, would later become one of the most famous rulers in Mesoamerican history. Pacal’s tomb is also one of the most impressive tombs to have been discovered in the Americas.

There was also trouble within the Calakmul alliance. Caracol and Naranjo were both allies of Calakmul but they were also longtime enemies of each other. In 626 Caracol launched two attacks on Naranjo. Caracol’s king, Tajoom Uk’ab’ K’ak’ was dead shortly after in 630, possibly from conflict with Naranjo. In 631 it seems that Calakmul’s ruler, Yuknoom Head, dealt with these inter-alliance conflicts by conquering Naranjo and torturing its ruler.

Meanwhile in Tikal, Ajaw K’inich Muwaan Jol II founded the vassal state of Dos Pilas in 629, installing his own son, B’alaj Chan K’awiil, as its ruler. B’alaj Chan K’awiil was going to have an important part to play in the upcoming resurgence of conflict with Calakmul. Tikal’s influence was beginning to resurface and become a threat to Calakmul, which was now becoming accustomed to its preeminent position.

The Second Tikal-Calakmul War (650–695)

In 636 one of the most important rulers in Maya history came to power in Calakmul. His name was Yuknoom Ch’een II, also known as Yuknoom the Great. He was to reign for 50 years from 636 to 686. In 650 Yuknoom the Great launched an attack on Dos Pilas, Tikal’s recently formed vassal state. This began the Second Tikal-Calakmul War. The ruler of Dos Pilas, B’alaj Chan K’awiil, was the son of the late Tikal ruler Ajaw K’inich Muwaan Jol II. B’alaj Chan K’awiil was forced to flee from Dos Pilas to Aguateca.

In 657 Yuknoom the Great attacked Tikal itself, which was now under the rule of Nuun Ujol Chaak. If Nuun Ujol Chaak was the son of previous ruler, Ajaw K’inich Muwaan Jol II, he would have been the brother of the exiled Dos Pilas ruler B’alaj Chan K’awiil. If these two were indeed brothers they were about to be engaged in a fracticidal conflict of shifting alliances. In any event, Nuun Ujol Chaak was forced to flee Tikal. That same year B’alaj Chan K’awiil accepted Yuknoom the Great as his overlord and Dos Pilas, the vassal state Tikal had founded, moved into the Calakmul alliance.

After submitting to Calakmul, B’alaj Chan K’awiil was able to return to Dos Pilas but 15 years later, in 672, he was forced to flee again. That year Nuun Ujol Chaak attacked Dos Pilas and B’alaj Chan K’awiil fled to Chaak Naah. Seemingly in relentless pursuit, Nuun Ujol Chaak burned down Chaak Naah, forcing B’alaj Chan K’awiil to flee yet again, this time to Hix Witz. In 677 Calakmul defeated Nuun Ujol Chaak at Pulil and B’alaj Chan K’awiil was once again able to return to Dos Pilas.

The conflict between Tikal, Dos Pilas, and Calakmul came to a head in 679, when Dos Pilas and Calakmul finally defeated and killed Nuun Ujol Chaak. The battle was apparently very bloody and celebrated for this fact. In 682 B’alaj Chan K’awiil commissioned inscriptions to commemorate his struggles and ultimate, glorious victory over Tikal. Famously, the inscriptions speak of “pools of blood” and “piles of heads” in the great battle. B’alaj Chan K’awiil spent a lot of effort to assure his legacy, quite successfully it seems. B’alaj Chan K’awiil also strenghtened his position through marriage alliances. He produced his heir, Itzamnaaj Balam, through a wife from the nobility of Itzan. Through his second wife came a daughter, named Lady Six Sky.

Meanwhile, Caracol’s influence began to decline after a defeat at the hands of its old rival, Naranjo, in 680. Caracol’s deposed king, K’ak’ Ujol K’inich II, fled. In 682 the daughter of Dos Pilas ruler B’alaj Chan K’awiil, Lady Six Sky, was chosen as ruler of Naranjo. She proved to be an exceptional leader and appears on several monuments. She was never formally ruler herself but carried out royal functions and seems to have been the de facto ruler. In 688 K’ak’ Tiliw Chan Chaak become official ruler of Naranjo but he was only 5 years old at the time. He was probably Lady Six Sky’s son and she almost certainly acted as regent for quite some time. Lady Six Sky seems to have lead Naranjo in 8 military campaigns in the first 5 years of K’ak’ Tiliw Chan Chaak’s reign, including a victory against Tikal in 695.

Meanwhile, in Tikal a sleeping giant was growing. In 682 Jasaw Chan K’awill I came to power and launched an ambitious cultural revitalization program. This program included the construction of temples and tombs that pointed back to Tikal’s glorious past, before it lost it’s preeminence to Calakmul. It was clear that Jasaw Chan K’awill I’s revitalization not only pointed to a glorious past but looked forward to a similarly glorious future. This was an ominous sign for Calakmul.

In 695 the Second Tikal-Calakmul War came to a climax. Tikal recorded that it successfully confronted Calakmul, “bringing down the flint and shield” of its ruler, Yuknoom Yich’aak K’ak’. In an important symbolic move, in a world rich in symbolism, Tikal captured a huge effigy of one of Calakmul’s patron deities. Jasaw Chan K’awill I celebrated his victory with ceremony and inscriptions. Ever one to connect Tikal’s glory to the past, he chose to hold his victory celebration on the anniversary of the death of Spearthrower Owl, father of the Yax Nuun Ayiin I, who had replaced Chak Tok Ich’aak I as ruler back in 379.

The Third Tikal-Calakmul War (720-744)

The great victory of Tikal over Calakmul in 695 changed the course history in the region. It went a long way to restore Tikal’s prestige and dominance. But there were still some important events in the region following this great battle. The Third Tikal-Calakmul War involved two very key players: Copán and Quiriguá. Recall that before the wars, Tikal had founded Copán in 426, installing as its ruler K’inich Yax K’uk’ Mo’. Copán in turn founded its own vassal state of Quiriguá in 426. This system of relations was critical to what would follow.

In 725 the ruler of Copán, Uaxaclajuun Ub’aah K’awiil (18-Rabbit), installed K’ak’ Tiliw Chan Yopaat as a subordinate ruler in Quiriguá. But by 734 K’ak’ Tiliw Chan Yopaat had declared Quiriguá independent from Copán. Things got even more complicated a couple years later. As a subordinate state to Copán, Quiriguá had been, by extension, subject to Tikal. Quiriguá had been part of a network of alliances tied to Tikal. But in 736 Calakmul’s ruler, Wamaw K’awiil, met with K’ak’ Tiliw Chan Yopaat and forged an alliance. Quiriguá had switched sides.

Quiriguá was a smaller state than Copán and presumably less powerful on its own. But with a superpower like Calakmul backing it, Quiriguá was able to challenge and actually defeat Copán. In 738 K’ak’ Tiliw Chan Yopaat captured his former overlord, Uaxaclajuun Ub’aah K’awiil, and beheaded him. It was a dramatic reversal.

However, Calakmul was ultimately not able to permanently thwart Tikal’s regained dominance. In 743 Tikal conquered El Peru. In 744 also defeated Naranjo and then Calakmul itself ending the Third Tikal-Calakmul War.


The history of both kingdoms was much less glorious following the events of the great battle of 695 and the battles between Copán and Quiriguá. In fact, the entire region seems to have entered into a decline, at least as far as organizational complexity is concerned. The reasons for the decline of the Classic Maya civilization are heavily debated. One theory is that the Tikal-Calakmul Wars took such a toll on the inhabitants of Mesoamerica that it led to a collapse. Other theories include drought, climate change, overpopulation, and mismanagement of natural resources. Popular theories like drought focus on external forces that we can measure and quantify archeologically. But it is also possible that various cultural trends contributed to a change in the structure of the civilization. But a simultaneous decrease in the amount records being kept at this time makes cultural causes difficult to corroborate. Eventually new powers like Uxmal and Chichen Itza came to dominate in Terminal Classic and Postclassic.

Additional Reading

Sharer, Robert J., and Loa P. Traxler. The Ancient Maya. Stanford, CA: Stanford UP, 2006.

Edwin Barnhart. Maya to Aztec: Ancient Mesoamerica Revealed. Chantilly, VA: The Teaching Company, 2015.

Simon Martin, and Nikolai Grube. Chronicle of the Maya Kings and Queens: Deciphering the Dynasties of the Ancient Maya. 2nd ed. New York, NY: Thames and Hudson, 2008.