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.

God In History

The story of God in scripture is one of active and dynamic involvement in history. It is also one of continuous change and progression, both in the relationship between God and humanity and in our understanding of God.

One of the themes I’ve been paying attention to recently in my study of scripture is the way that the relationship between God and humanity changes over time. Both the way we understand God and the way we relate to God. I’m enough of a classical theist to think that there is an aspect to God, probably a very significant aspect, that is constant relative to time. And I say relative to time to admit the possibility, very likely in my view, that these unchanging aspects subsist beyond time, beyond our temporal frame. That being said it’s certainly the case that we experience time and that the relation between us and God and the way we understand God can be different at different times. This is something that the process theologian Charles Hartshorne called attention to in his book Divine Relativity and I also think it’s a comfortably Biblical concept.

There are a few things that motivate my interest in this recently. One is my readings of Hegel that I talked about in my previous episode. Another is my conversation with Alex in the “Conversations Across the Divide of Religious Belief and Unbelief” episode. We talked about progression, both personal and communal, particularly of a religious community. In the Latter-day Saint religion we have this expectation that knowledge progresses, it’s actually something that’s literally an article of faith. And there’s admittedly a tension there, a kind of paradox like those Terryl Givens in his book People of Paradox sees at the heart of Latter-day Saint religious and cultural life. Arguably a productive tension, hopefully much of the time. And it’s not just the Latter-day Saint religion where we see this but I think it’s a good example. We get used to the way things are and the way we understand things, even as we know, at least in principle, that we don’t have the fullness and that there will be change and growth.

I think it’s also important say first that the way we understand things at a later time shouldn’t be a position from which to look down on the ways things were understood at prior times. At least not with any form of malice or contempt, or condescension really. That’s a hard position to strike but I think it should be an aspiration anyway. One of the things that bothers me about theories of religious progress that we see in the nineteenth century in folks like Hegel and in higher criticism with Julius Wellhausen is some unfair treatment of Judaism. I’ve read that some of the higher criticism sort of cast Judaism as a more primitive and unpolished stage of religious development prior to Christianity and I don’t care for that take. And one rebuttal I’d make to that is that Judaism has evolved as well. Quite significantly. It’s not like Jewish thought has remained stagnant for the past two millennia while Christian thought has been evolving. Far from it. In fact I pull a lot of ideas from the great leap in creativity and imagination in Jewish thought from the nineteenth century onward, as in the Jewish Haskalah (השכלה), or Jewish Enlightenment. So when I refer to new perspectives on things that we get from the New Testament scriptures I don’t mean to subordinate Judaism, or the Hebrew Bible for that matter.

So I’ll just rattle off a few examples that I’ve noticed as I’ve been thumbing through the scriptures. One major example is with Moses. Moses is definitely one of these big transition points in scriptures where there is some major development. Certainly the giving of the Law is a major event in Biblical history. Clearly this is something that wasn’t there before and now this group of people is being given. God is establishing a covenant with the people of Israel with a level of detail that wasn’t there before. They are being expected to do things that weren’t expected before, or to not do things that weren’t prohibited before, at least not explicitly. One passage I really like the indicates nicely that Moses is someone special, someone privileged in a notable and new way is in Exodus 6:3 where God tells Moses that he is on what we could call a first-name basis with God in a way that prophets before him where not.

וָאֵרָ֗א אֶל־אַבְרָהָ֛ם אֶל־יִצְחָ֥ק וְאֶֽל־יַעֲקֹ֖ב בְּאֵ֣ל שַׁדָּ֑י וּשְׁמִ֣י יְהוָ֔ה לֹ֥א נֹודַ֖עְתִּי לָהֶֽם׃

“And I appeared unto Abraham unto Isaac and unto Jacob by the name of God Almighty, in Hebrew El Shaddai (אֵ֣ל שַׁדָּ֑י), but by my name YHWH was I not known to them.” So God did not reveal his actual name, YHWH, sometimes said as “Jehovah” in English, he did not reveal his actual to the Patriarchs, Abraham, Isaac, and Jacob. He used a more generic title “El Shaddai” or “God Almighty”, el just being the Hebrew word for a god, lower case g. And the name he reveals is arguably more transcendent and ultimate, the God, upper case G, beyond any particular tribal gods, lower case g. There’s some debate about the philology but I’m inclined to agree with Frank Moore Cross who argued in his book Canaanite Myth and Hebrew Epic that the name YHWH derives from a Canaanite-Proto-Hebrew verb “to be” in its causative imperfect form. So this name in my mind carries more of an association with existence, being itself, which the phrase I AM THAT I AM (ehyeh asher ehyeh, אֶֽהְיֶ֖ה אֲשֶׁ֣ר אֶֽהְיֶ֑ה) can also be understood to evoke.

But there were transition points and developments prior to Moses and the giving of the Law. Another major transition point was the Flood and the institution of what is sometimes called the Noachide Law, which in contrast to the Mosaic Law applies to all humanity rather than just the House of Israel. And this is found in Genesis 9:5-6. “And surely your blood of your lives will I require at the hand of every beast will I require it and at the hand of man at the hand of every man’s brother (ish achiv, אִ֣ישׁ אָחִ֔יו) will I require the life of man (nephesh ha-adam, נֶ֥פֶשׁ הָֽאָדָֽם). Whoso sheddeth man’s blood, by man shall his blood be shed: for in the image of God (be-tzelem elohim, בְּצֶ֣לֶם אֱלֹהִ֔ים) made he man.” So this prohibition against murder is made explicit. Now, arguably it was already implicit. God was none too pleased when Cain killed Abel and seemed to expect that he not have done that. And we could say there was already a natural, unwritten law in effect even before it was codified as positive law. And one reason the Flood was brought in the first place was because of the violence (chamas, חָמָס) of the people. But now it’s really made explicit. No one can pull a George Costanza and say, “Was that wrong? Should I not have done that?” It’s definitely verboten. No excuses.

As another very broad comment about development and progress in Genesis I really like an interpretation I heard from Christin Hayes in her Open Yale course on the Hebrew Bible. I’ll quote a section of her lecture from when she’s just finished talking about Jacob and his wrestle with God:

“With Jacob, who is now Israel, God seems perhaps to finally have found the working relationship with humans that he has been seeking since their creation. God learned immediately after creating this unique being, that he will exercise his free will against God. God saw that he had to limit the life span of humans, or risk creating an enemy that was nearly equal to him. So he casts the humans out of the Garden, blocks access to the tree of life. But humans continue their violent and evil ways, and in desperation, God wipes them out, and starts again. This second creation proves to be not much better. They forget God, they turn to idolatry. God has promised at this point, however, not to destroy all humankind again, so he experiments with a single individual of faith. Abraham’s faith withstands many a trial. He is obedient to God in a way that no one has been up to this point in the narrative, but perhaps ultimately the model of blind obedience is rejected, too. When Abraham prepares to slaughter his own son, perhaps God sees that blind faith can be as destructive and evil as disobedience, so God relinquishes his demand for blind obedience: he stops Abraham himself.”

“The only relationship that will work with humans is perhaps one in which there is a balance between unchecked independence and blind obedience, and God seems to find that relationship with Jacob. And the metaphor for that relationship is a metaphor of struggle, or wrestling. Remember Yisrael means ‘one who wrestles, who struggles with God.’ God and humans lock in an eternal struggle, neither prevailing, yet both forever changed by their encounter with one another.”

I think that’s just a lovely interpretation of the story of the Patriarchs in Genesis and there’s a lot there to ponder. It’s a good metaphor for this subject as a whole, of an evolving relationship between God and human-kind.

One more passage from the Hebrew Bible before moving to the New Testament. This one is from Jeremiah 31:31-33. “Behold, the days come, saith the LORD, that I will make a new covenant (berit chadashah, בְּרִ֥ית חֲדָשָֽׁה) with the house of Israel, and with the house of Judah: not like the covenant which I made with their fathers… I will put My law (torah, תּוֹרָה) within them and on their heart (leb, לֵב) I will write it; and I will be their God, and they shall be My people.” And this is just one example of many such verses where God is making it clear that something new is coming. He is going to do things differently than he had in the past.

And this is definitely a theme that Christianity picks up on, quite understandably, right? And that verse from Jeremiah is quoted in the New Testament. Things definitely change with Jesus. That’s not at all to deny the continuity, of which there is also plenty. But something new is definitely happening here. So that narrative of change and God acting in history is going to play a role and it will also factor into how Christians understand the Hebrew Bible and the things that have changed from what it says.

In fact I’d say it’s precisely those passages that emphasize continuity that also introduce the ideas of change and progression. The Acts of the Apostles has a lot of this. Missionaries like Peter, Stephen, and Phillip are tying Jesus back to the prophecies and the sweeping narrative of Israel. Paul too, certainly. One of the most notable of these is Stephen’s long speech before the Council as he stands in judgment, just prior to his death. That’s in Acts 7. For about 50 verses he reviews the history of Israel with Abraham, Isaac, and Jacob. Joseph in Egypt. Moses leading the people out of Egypt. Then King Solomon and leading up to the time of Jesus. Phillip, when he runs, literally runs, into an Ethiopian man, what does he find? He finds him reading from Isaiah and he explains the scriptures to him and how they prophesy of Jesus.

καὶ ἀρξάμενος ἀπὸ τῆς γραφῆς ταύτης εὐηγγελίσατο αὐτῷ τὸν Ἰησοῦν.

“And beginning from this Scripture he preached Jesus to him.” (Acts 8:35)

One of my favorite examples, and probably the most sophisticated, of the use of the Hebrew scripture in a narrative of God’s evolving relationship with humanity is in the book of Hebrews. And the book of Hebrews could really have its own episode or several episodes. And, let’s be honest it probably will eventually. That something new and different has taken place and is taking place is stated right at the beginning: “God, who at sundry times and in divers manners spake in time past (pálai, πάλαι) unto the fathers by the prophets (en tois prophétais, ἐν τοῖς προφήταις), Hath in these last days (ep eschátou ton hemerón toúton, ἐπ’ ἐσχάτου τῶν ἡμερῶν τούτων) spoken unto us by his Son (en huio, ἐν Υἱῷ), whom he hath appointed heir of all things, by whom also he made the worlds; Who being the brightness of his glory, and the express image of his person, and upholding all things by the word of his power, when he had by himself purged our sins, sat down on the right hand of the Majesty on high; Being made so much better than the angels, as he hath by inheritance obtained a more excellent name than they.” (Hebrew 1:1-4) So, this is how things were done in the times past, but this is how things are being done now, in these last days (ep eschátou ton hemerón toúton, ἐπ’ ἐσχάτου τῶν ἡμερῶν τούτων).

And the author of the Hebrews gets into some very deep analysis of the rituals conducted in the tabernacle as outlined in the Torah and emphasizes the way that it prefigures Christ’s sacrifice. So there’s both continuity and transition. For example in Hebrews 10:1, “For the law (nómos, νόμος) having a shadow (skiá, σκιά) of good things to come, and not the very image of the things, can never with those sacrifices which they offered each year (kat eniautón, κατ’ ἐνιαυτὸν) continually make the comers thereunto perfect.” So there’s a connection there to the past because the law is a shadow (skiá, σκιά) of what is to come. But one of the big ideas in Hebrews is that the practices of the former covenant had to be done repeatedly, each year (kat eniautón, κατ’ ἐνιαυτὸν). And he argues that this need to repeat the practices is a sign of a certain imperfection in them. They weren’t perfect in a way that the new covenant is. “For if that first covenant had been faultless, then should no place have been sought for the second” (Hebrews 8:7). The way things were done formerly, with the high priest of the tabernacle, prefigures the crowning act of perfect sacrifice accomplished in Christ. But Christ, as the new high priest “needeth not daily, as those high priests, to offer up sacrifice, first for his own sins, and then for the people’s: for this he did once, when he offered up himself.” (Hebrews 7:27) This perfect sacrifice didn’t need to be repeated as was needed in former times. “Every priest standeth daily ministering and offering oftentimes the same sacrifices, which can never take away sins: But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God.” (Hebrews 10:11-12) It’s “one and done”, not to describe it too casually, hopefully. The power of Christ’s sacrifice is thoroughly sufficient for all time. And this fulfills the old law, so that many of the things that were required formally don’t need to be done anymore. I think it’s interesting how Jesus also creates this synthesis of continuity and change in his statement that he doesn’t come to destroy (katalúo, καταλύω) the law but to fulfill it (pleróo, πληρόω). It’s a fascinating distinction. (Matthew 5:17).

Two more big examples are circumcision and dietary laws. Circumcision was the distinguishing mark of God’s covenant with his people. But the Christian church determined that it was not necessary for Gentile converts to be circumcised. This was something Paul had to try to persuade people of continually. And Peter has this remarkable vision in Acts 10 where a voice from Heaven actually bids him to eat unclean animals. And Peter impulsively wants to protest. This goes against everything he’s been taught. This seems foundational to his religious identity. “Not so, Lord; for I have never eaten any thing that is common (koinós, κοινός) or unclean (akáthartos, ἀκάθαρτος).” (Acts 10:14) And the voice says to him, “What God hath cleansed, that call not thou common (koinós, κοινός).” (Acts 10:15). It’s indicated here that God has done something that has changed the situation on the ground. And because of God’s actions in history these basic aspects to Peter’s religious identity are going to shift. It’s a fascinating and radical portrayal of process and development.

So what do we make of all this? Well if I’m putting together all these data points and seeing a trend line rather than a flat line it doesn’t seem unreasonable to think that change and process are part of reality and the way God works in history. It seems reasonable to expect that the way things are now in God’s relation to us and the way we understand God now are not necessarily going to be the way things are in the future. Change is likely. And it’s probably a good idea to expect that. Not that we can ever be ready for the unknown since it is, well unknown. But we can at least condition ourselves to be open to change and revelation.