Three levels of theological models

There are three kinds of metaphysical models of a theological mystery—say, Trinity, Incarnation or Transubstantiation:

• realistic model: a metaphysical story that is meant to be a true account of what makes the mysterious doctrine be true

• potential model: a metaphysical story that is meant to be an epistemically possible account of what makes the mysterious doctrine be true

• analogical model: a story that is meant to be an epistemically possible account of what makes something analogous to the mysterious doctrine be true.

For instance, Aquinas’s accounts of the Trinity, Incarnation and Transubstantiation are realistic models: they are meant to be accounts of what indeed makes the doctrines true. Van Inwagen’s relative identity account of the Trinity or his body-snatching account of the resurrection, on the other hand, are only potential models: van Inwagen does not affirm they are true. And the history of the Church is filled with analogical models.

A crucial test of any of these models is this: Imagine that you believe the story to be true, and see if the traditional things that one says about the mystery (in the case of a realistic or potential model), or analogues of them (in the case of an analogical model), sound like reasonable things to say given what one believes.

For instance, consider a time-travel model of the Incarnation. Alice, currently a successful ultramarathoner and brilliant geologist, will live a long and fruitful life. Near the end of her life, she has lost most of her physical and mental powers, and all her knowledge of geology. She uses a time machine to go back to 2020 when she is in her prime. If we thought this story was true, it would be reasonable to find ourselves saying things like:

• Alice is a successful ultramarathoner and barely able to walk

• Alice understands continental drift and does not not know what magma is

• Alice is young and old

• Alice is in the pink of health and dying.

These things would sound like a contradiction, but the time-travel story shows they are not. However, these claims are also analogous to claims that constitute an especially mysterious part of the mystery of the Incarnation (and I suppose a mysterious part of a mystery is itself a mystery): Christ suffers and is impassible; Christ is omniscient and does not know everything; Christ is timeless and born around 4 BC.

Of course nobody should think that it’s literally true that the Incarnation is to be accounted for in terms of time travel. But what the analogical model does show is that there are contexts in which it is reasonable to describe a non-contradictory reality in terms that are very similar to the apparently contradictory incarnational claims.

Peer disagreement and models of error

You and I are epistemic peers and we calculate a 15% tip on a very expensive restaurant bill for a very large party. As shared background information, add that calculation mistakes for you and me are pretty much random rather than systematic. As I am calculating, I get a nagging feeling of lack of confidence in my calculation, which results in $435.51, and I assign a credence of 0.3 to that being the tip. You then tell me that you you’re not sure what the answer is, but that you assign a credence of 0.2 to its being $435.51.

I now think to myself. No doubt you had a similar kind of nagging lack of confidence to mine, but your confidence in the end was lower. So if all each of us had was their own individual calculation, we’d each have good reason to doubt that the tip is $435.51. But it would be unlikely that we would both make the same kind of mistake, given that our mistakes are random. So, the best explanation of why we both got $435.51 is that we didn’t make a mistake, and I now believe that $435.51 is right. (This story works better with larger numbers, as there are more possible randomly erroneous outputs, which is why the example uses a large bill.)

Hence, your lower reported credence of 0.2 not only did not push me down from my credence of 0.3, but it pushed me all the way up into the belief range.

Here’s the moral of the story: When faced with disagreement, instead of moving closer to the other person’s credence, we should formulate (perhaps implicitly) a model of the sources of error, and apply standard methods of reasoning based on that model and the evidence of the other’s credence. In the case at hand, the model was that error tends to be random, and hence it is very unlikely that an error would result in the particular number that was reported.

Modeling space

The obvious model of a Newtonian space is as the set of all triples (x,y,z) of real-numbered coordinates. But the model does not have isotropy that Newtonian space does. It has privileged directions, such as the x-axis, the y-axis and the z-axis. It has privileged coordinates such as (0,0,0). Of course, physical models generally do have properties that aren't found in what is modeled. If I build a model of the solar system out of fruit, the fact that some of the fruit is sweeter need not model any property of the solar system. If I make a model of an ethanol molecule out of sticks and balls, the balls that represent hydrogen atoms differ in their exact mass, and exhibit scratches, in a way that the hydrogen atoms do not.

Nonetheless, even though this is common to all modeling, there really is something a little unsatisfying when the mathematical model does this. Typically when we mathematically model something, we have to abstract or forget on both sides. On the side of what is modeled, the side of the world, we ignore aspects of the physical structure because otherwise things get too complicated. On the side of the model, we ignore aspects of the mathematical structure because they don't, as far as we know, correspond to anything in the physics. Wouldn't it be nice if we could abstract only on one side, that of the world? But some things that would be nice are not an option.

The above remarks do, I think, make Pythagoreanism less plausible. There seems to be structure in the mathematics that models the world that isn't found in the world. This makes it implausible that the world just is composed of the mathematics.

Properties of the model and the modeled

My apologies for yet another technical post that's just notes-to-self.

Quantum Mechanics models the world using a Hilbert space. I wonder what we can say about just how much of the structure of the model is meant to be found in what is modeled. In contemporary mathematics, I guess ultimately any Hilbert space will be a very complex construction out of the empty set. Yet it seems absurd to think that the low-level details of the set-theoretic implementation (say, different ways of constructing the natural numbers out of the empty set) would reflect differences in the world. There are way too many ways to implement these details.

But there will also be differences at higher levels. For instance, there will be cases where the Hilbert space is L²(X), "the space of square-integrable functions" on some set X. I put that in scare quotes, because that's not what L² is, despite often being described so. Rather, it's the space of equivalence classes of square-integrable functions, where two functions are equivalent provided that the set of points where they differ has measure zero. So now we have a question about the model and the modeled. You could think that different members of an equivalence class correspond to different empirically indistinguishable physical states, and the physics simply makes no prediction as to which of the indistinguishable states is exemplified when. Or you could think that each equivalence class corresponds to a single possible physical state. The latter makes for a theory that is simpler and yet seems to give less understanding. It is simpler because it doesn't posit unexplained differences between states. But it seems to give less understanding, because it means that the wavefunction can no longer be seen as an assignment of values to different points in phase space, but rather a more mysterious kind of entity—one modeled as an equivalence class of such assignments.

There may be a third option: There is a privileged member of each equivalence class, and only the privileged member can be physically actualized. This would give us the best of both worlds. We would have a field over phase-space, and no extra indistinguishable physical possibilities. The lack of linear liftings on L²[0,1] makes it a bit harder to realize this hope than one might have wished, but maybe there is still some hope.