The Trinity, sexual ethics and liberal Christianity

Many Christians deny traditional Christian doctrines regarding sexual ethics while accepting traditional Christian Trinitarian doctrine. This seems to me to be a rationally suspect combination because:

1. The arguments against traditional Christian sexual ethics are weaker than the arguments against the doctrine of the Trinity.

2. A number of the controversial parts of traditional Christian sexual ethics are grounded
   at least as well in Tradition and Scripture as the doctrine of the Trinity is.

Let me offer some backing for claims 1 and 2.

The strongest arguments against traditional Christian sexual ethics are primarily critiques of the arguments for traditional Christian sexual ethics (such as the arguments from the natural law tradition). As such, these arguments do not establish the falsity of traditional Christian sexual ethics, but at best show that it has a weak philosophical foundation. On the other hand, the best arguments against the doctrine of the Trinity come very close to showing that the doctrine of the Trinity taken on its own terms is logically contradictory. The typical Christian theologian is the one who is on the defensive here, offering ways to resolve the apparent contradiction rather than giving rational arguments for the truth of the doctrine.

There are, admittedly, some arguments against traditional Christian sexual ethics on the basis of intuitions widely shared in our society. But we know that these intuitions are very much shaped by a changing culture, insofar as prior to the 20th century, one could run intuition-based arguments for opposite conclusions. Hence, we should not consider the arguments based on current social intuitions to be particularly strong.
But the intuition that there is something contradictory about the doctrine of the Trinity does not seem to be as dependent on changing social intuitions. The merely socially counterintuitive is rationally preferable to the apparently contradictory.

Neither the whole of the doctrine of the Trinity nor the whole of traditional Christian sexual ethics is explicit in Scripture. But particularly controversial portions of each are explicit in Scripture: the Prologue of John tells us that Christ is God, while both Mark and Luke tell us that remarriage after divorce is a form of adultery, and Paul is clear on the wrongfulness of same-sex sexual activity. And the early Christian tradition is at least as clear, and probably more so, sexual ethics as on the doctrine of the Trinity.

I am not saying, of course, that it is not rational accept the doctrine of the Trinity. I think the arguments against the doctrine have successful responses. All I am saying is that traditional Christian sexual ethics fares (even) better.

Possibly giving a finite description of a nonmeasurable set

It is often assumed that one couldn’t finitely specify a nonmeasurable set. In this post I will argue for two theses:

1. It is possible that someone finitely specifies a nonmeasurable set.

2. It is possible that someone finitely specifies a nonmeasurable set and reasonably believes—and maybe even knows—that she is doing so.

Here’s the argument for (1).

Imagine we live an uncountable multiverse where the universes differ with respect to some parameter V such that every possible value of V corresponds to exactly one universe in the multiverse. (Perhaps there is some branching process which generates a universe for every possible value of V.)

Suppose that there is a non-trivial interval L of possible values of V such that all and only the universes with V in L have intelligent life. Suppose that within each universe with V in L there runs a random evolutionary process, and that the evolutionary processes in different universes are causally isolated of each other.

Finally, suppose that for each universe with V in L, the chance that the first instance of intelligent life will be warm-blooded is 1/2.

Now, I claim that for every subset W of L, the following statement is possible:

3. The set W is in fact the set of all the values of V corresponding to universes in which the first instance of intelligent life is warm-blooded.

The reason is that if some subset W of L were not a possible option for the set of all V-values corresponding to the first instance of intelligent life being warm-blooded, then that would require some sort of an interaction or dependency between the evolutionary processes in the different universes that rules out W. But the evolutionary procesess in the different universes are causally isolated.

Now, let W be any nonmeasurable subset of L (I am assuming that there are nonmeasurable sets, say because of the Axiom of Choice). Then since (3) is possible, it follows that it is possible that the finite description “The set of values of V corresponding to universes in which the first instance of intelligent life is warm blooded” describes W, and hence describes a nonmeasurable set. It is also plainly compossible with everything above that somebody in this multiverse in fact makes use of this finite description, and hence (1) is true.

The argument for (2) is more contentious. Enrich the above assumptions with the added possibility that the people in one of the universes have figured out that they live in a multiverse such as above: one parametrized by values of V, with an interval L of intelligent-life-permitting values of V, with random and isolated evolutionary processes, and with the chance of intelligent life being warm-blooded being 1/2 conditionally on V being in L. For instance, the above claims might follow from particularly elegant and well-confirmed laws of nature.

Given that they have figured this out, they can then let “Q” be an abbreviation for “The set of all values of V corresponding to universes wehre the first instance of intelligent life is warm-blooded.” And they can ask themselves: Is Q likely to be measurable or not?

The set Q is a randomly chosen subset of L. On the standard (product measure) understanding of how to probabilistically make sense of this “random choice” of subset, the event of Q being nonmeasurable is itself nonmeasurable (see the Sawin answer here). However, intuitively we would expect Q to be nonmeasurable. Terence Tao shares this intuition (see the paragraph starting “Intuitively”). His reason for the intuition is that if Q were measurable, then by something like the Law of Large Numbers, we would expect the intersection of Q with a subinterval I of L to have a measure equal to half of the measure of I, which would be in tension with the Lebesgue Density Theorem. This reasoning may not be precisifiable mathematically, but it is intuitively compelling. One might also just have a reasonable and direct intuition that the nonmeasurability is the default among subsets, and so a “random subset” is going to be nonmeasurable.

So, the denizens of our multiverse can use these intuitions to reasonably conclude that Q is nonmeasurable. Hence, (2) is true. Can they leverage these intuitions into knowledge? That’s less clear to me, but I can’t rule it out.

Wobbly priors and posteriors

Here’s a problem for Bayesianism and/or our rationality that I am not sure what exactly to do about.

Take a proposition that we are now pretty confident of, but which was highly counterintuitive so our priors were tiny. This will be a case where we were really surprised. Examples:

1. Simultaneity is relative

2. Physical reality is indeterministic.

Let’s say our current level of credence is 0.95, but our priors were 0.001. Now, here is the problem. Currently we (let’s assume) believe the proposition. But if our priors were 0.0001, our credence would have been only 0.65, given the same evidence, and so we wouldn’t believe the claim. (Whatever the cut-off for belief is, it’s clearly higher than 2/3: nobody should believe on tossing a die that they will get 4 or less.)

Here is the problem. It’s really hard for us to tell the difference in counterintuitiveness between 0.001 and 0.0001. Such differences are psychologically wobbly. If we just squint a little differently when looking mentally a priori at (1) and (2), our credence can go up or down by an order of magnitude. And when our priors are even lower, say 0.00001, then an order of magnitude difference in counterintuitiveness is even harder to distinguish—yet an order of magnitude difference in priors is what makes the difference between a believable 0.95 posterior and an unbelievable 0.65 posterior. And yet our posteriors, I assume, don’t wobble between the two.

In other words, the problem is this: it seems that the tiny priors have an order of magnitude wobble, but our moderate posteriors don’t exhibit a correspnding wobble.

If our posteriors were higher, this wouldn’t be a problem. At a posterior of 0.9999, an order of magnitude wobble in priors results in a wobble between 0.9999 and 0.999, and that isn’t very psychologically noticeable (except maybe when we have really high payoffs).

There is a solution to this problem. Perhaps our priors in claims aren’t tiny just because the claims are counterintuitive. It makes perfect sense to have tiny priors for reasons of indifference. My prior in winning a lottery with a million tickets and one winner is about one in a million, but my intuitive wobbliness on the prior is less than an order of magnitude (I might have some uncertainty about whether the lottery is fair, etc.) But mere counterintuitiveness should not lead to such tiny priors. The counterintuitive happens all too often! So, perhaps, our priors in (1) and (2) were, or should have been, more like 0.10. And now perhaps the wobble in the priors will probably be rather less: it might vary between 0.05 and 0.15, which will result in a less noticeable wobble, namely between 0.90 and 0.97.

Simple hypotheses like (1) and (2), thus, will have at worst moderately low priors, even if they are quite counterintuitive.

And here is an interesting corollary. The God hypothesis is a simple hypothesis—it says that there is something that has all perfections. Thus even if it is counterintuitive (as it is to many atheists), it still doesn’t have really tiny priors.

But perhaps we are irrational in not having our posteriors wobble in cases like (1) and (2).

Objection: When we apply our intuitions, we generate posteriors, not priors. So our priors in (1) and (2) can be moderate, maybe even 1/2, but then when we updated on the counterintuitiveness of (1) and (2), we got something small. And then when we updated on the physics data, we got to 0.95.

Response: This objection is based on a merely verbal disagreement. For whatever wobble there is in the priors on the account I gave in the post will correspond to a similar wobble in the counterintuitiveness-based update in the objection.

Intuitive moral knowledge

People intuitively know that stealing is wrong. Maybe stealing is wrong because it violates the social institution of property which is reasonably and appropriately instituted by each community. Maybe stealing is a violation of the natural relation that an agent has to an object upon mixing her labor with it. Maybe stealing violates a divine command. But people's intuitive knowledge that stealing is wrong does not come from their knowledge of such reasons for the wrongness of stealing. So how is it knowledge?

It's not like when the child knows Pythagoras' Theorem to be true but can't prove it. For she knows the theorem to be true because she gets her belief from the testimony of other people who can prove it. But that's not how the knowledge that stealing is wrong works. People can intuitively know that stealing is wrong without their belief having come directly or indirectly from some brilliant philosopher who came up with a good argument for its wrongness.

Perhaps there is some evolutionary story. Communities where there was a widespread belief that stealing is wrong survived and reproduced while those without the belief perished, and there was no knowledge at all the back of the belief formation. However, perhaps, it came to be knowledge, because this evolutionary process was sensitive to moral truth. However, it is dubious that this evolutionary process was sensitive to moral truth as such. It was sensitive to the non-moral needs of the community, and sometimes this led to moral truth and sometimes to moral falsehood (as, for instance, when it led to the conviction that it is right to enslave members of other communities). So if this is the story where the belief came from, it's not a story about knowledge. At best, the intuitive conviction that stealing is wrong, on this story, is a justified true belief, but it's Gettiered.

This, I think, is an interesting puzzle. There is, presumably, a very good reason why stealing is wrong, but the intuitions that we have do not seem to have the right connection to that reason.

Unless, of course, we did ultimately get the knowledge from someone who has a very good argument for the wrongness of stealing. As I noted, it is very implausible that we got it from a human being who had such an argument. But maybe we got it from a Creator who did.

Bigger and smaller infinities

Anecdotal data suggests that a number of people find counterintuitive the Cantorian idea that some infinities are bigger than others.

This is curious. After all, the naive thing to say about the prime numbers and the natural numbers is that

1. while there are infinitely many of both, there are more natural numbers than primes.

For the same reason it is also surely the obvious thing to say that

2. while there are infinitely many of both, there are more real numbers than natural numbers.

So there is nothing counterintuitive about different sizes of infinity. Of course, (1) is false. Our untutored intuitions are wrong about that case. And that fact should make us suspicious whether (2) is true; given that the same intuitions led us astray in the case of (1), we shouldn't trust them much in case (2). However, the fact that (1) is false should not switch (2) from being intuitive to being counterintuitive. Moroever, our reasons for thinking (1) to be false—namely, the proof of the existence of a bijection between the primes and the naturals—don't work for (2).

All in all, rather than taking (2) to show us how counterintuitive infinity is, we should take (2) to vindicate our pretheoretic intuition that cardinality comparisons can take us beyond the finite, even though some of our pretheoretic intuitions as to particular cardinality comparisons are wrong.

Approximation in mathematics

The New York Times has an interesting article arguing that imprecise approximational intuitions--the ability to quickly and roughly reckon things--are crucial to success at abstract mathematics.

To someone whose mathematical work was in real-number based areas--analysis and probability theory--this is very plausible on introspective grounds. But I wonder how true it is in more algebraic fields. I've never been very good at higher algebra--things like the Sylow theorems were very difficult for me (I still passed the algebra comp, but it came noticeably less naturally to me than the analysis comp), perhaps in part because my approximational intuitions were close to useless.

Anecdotal data suggests to me that there are two distinct kinds of mathematical skills. There are the skills involved in analysis, skills tied to problems that are real-number based (complex numbers are real-number based, of course, since C is just the cross product of R with itself), often visualizable, and where approximation and limiting procedures may be relevant. And then there are the skills involved in more algebraic fields, where (as far as I can) approximation gets you nowhere, and while visualization is helpful, the visualization is much more symbolic (visualizing a path of a brownian particle is pretty straightforward, one visualizes quotient groups either explicitly in symbols like "A/H" or perhaps in some strange and highly abstract diagrams). I don't know where to put the combinatorial--it may somewhat straddle the divide (a lot of visualization is involved), but I think is very algebraic in nature.

It is quite possible for a person to be really good at one of these, without being very good at the other. There are fields of mathematics that call upon both sets of skills. And there is an asymmetry: I think the analysis-type skills may be of very little use to mathematicians working in very algebraic areas, but just about every mathematician working in an analysis-type area needs to be able to do algebraic manipulation (though I have a strong preference for proofs in analysis-type fields where the algebraic manipulation is just a way of making precise what is intuitively obvious).

Two kinds of mathematical intuitions

Mathematicians have two kinds of intuition. A speculative intuition occurs when they think about a problem, perhaps think quite a lot, and conclude that the problem has answer A, even though they how no idea how to prove this. "It just looks like A is the answer." I do not know how reliable speculative mathematical intuitions are. I suspect that they are not very reliable. In particular, I think they rarely if ever justify belief. Certainly, I did not acquire belief in an answer on the basis of speculative intuitions when I was a practicing mathematician.

However, there is also such a thing as pedestrian intuition. This tells the mathematician: "Clearly, p." The "clearly" is not speculative. The content of the intuition is not just that p is true but that p can be easily proved from what preceded. John Fournier, my mathematics thesis director, once gave me the following advice on papers submitted for publication: when there are two obvious steps in a row in a proof, you can omit one, but not both.[note 1] When a mathematician sees that something follows, even if she does not actually go through the proof of the fact that it follows, that pedestrian intuition is, I think, very reliable. It may even be that had the mathematician written down the proof, the proof would have contained some minor mistakes. For this intuition does not seem to be based on having the proof in one's mind. Rather, it seems to be a direct non-inferential grasp of the easy provability of p.

One small piece of evidence for the reliability of pedestrian intuition is the incredible reliability of mathematical publications. Errata are extremely rare in mathematical journals.[note 2] I suspect this is not just because of the refereeing process, but because this highly reliable intuition was guiding the mathematician in writing the proof. In fact, I think the epistemic weight of the result proved in a mathematics paper goes beyond the validity of the published proof. The published proof may indeed contain a minor slip here or there. But what makes these slips be minor is precisely that one can intuitively see what should be in their place. My last mathematics paper was published when I was significantly out of practice. It went back and forth between me and the referee several times, and the referee was rightly exasperated by the amount of mistakes in the proofs. However, all the mistakes were easily fixable: the intuition was exactly right, in a pedestrian way, despite the logical gaps in the proofs.

This is surprising. One might think that a proof with logical holes has no value at all—it is like tracing your ancestry to Charlesmagne with only two gaps in the chain (this isn't my comparison). But somehow the reliability of the pedestrian intuition goes beyond the proof written down.

What explains the extremely high reliability of pedestrian intuition in a well-trained mathematician? One possibility is that it is a highly developed pattern-matching skill. In the past we've seen p-type claims following from q-type claims, and we can see that the present case fits into the pattern, and so p follows from q. This explanation fits well with the fact that experience seems important for this kind of intuition. But I am not sure this would be sufficient to give the intuition the kind of reliability it has. Pattern-matching would, I doubt, have the right kind of reliability. In typical cases of writing down a proof of a new result, the case at hand is unlikely to be exactly like past cases.

Or could it be that there is a process involving a mental representation of a proof, but a representation not directly available to consciousness? If so, what is interesting is that this is just as reliable as, or even more reliable than, consciously going through the steps of a proof (in fact, I suspect that the reliability of consciously going through the steps often or always depends on the non-conscious process occuring side-by-side). This is kind of neat and reminds me of the speculations central to Peter Watts' novel Blindsight. Moreover, if this is right, then I think it should challenge internalist epistemologies that require justifications to be conscious. In these mathematical cases, the justification can be made conscious, but the making-conscious does not seem central, since the non-conscious reasoning is more reliable than the conscious reasoning.

It is an interesting question how the two kinds of mathematical intuition connect up with kinds of philosophical intuition. I do find myself with a quite reliable intuition in philosophy akin to the pedestrian sort of mathematical intuition—an intuition as to what conclusions can be made to follow from what kinds of assumptions. In fact, this is probably just the same intuition at work, though I find it is a bit less reliable in philosophy than in mathematics. (I think I have at least three times been significantly deceived by such an intuition, and in a number of other cases have needed to add plausible ancillary assumptions to make an argument go—though on reflection that probably can happen in mathematical cases, too, which slightly weakens what I said in previous paragraphs.)

There is, however, a second kind of intuition: an intuition that pointless torture is wrong, or that we are not identical with our left big toes, or that identity is non-relative, or that the good is to be pursued and the bad avoided, that nothing can be causally prior to itself, or that every contingent truth has an explanation. I am inclined to class this intuition as different from both the pedestrian and the speculative mathematical intuitions. This intuition is of variable strength, unlike pedestrian mathematical intuition which is pretty uniformly very strong. Sometimes, this kind of philosophical intuition gives us certainty, as in the case of the good being to be pursued, and sometimes it merely inclines us in favor of a proposition. The range of strengths here makes it different from speculative mathematical intuition which, I think, never justifies belief, while this kind of philosophical intuition does justify belief.

Or maybe we need to split this second kind of philosophical intuition into two kinds. One kind is speculative, and this is akin to speculative mathematical intuition. I am, let us suppose, inclined to think electrons are not conscious, but this intuition is not sufficient to compel or justify belief. Another kind is self-evidential which presses belief on us, and I suspect justifies it as well. This kind is more like the highly reliable pedestrian mathematical intuition in respect of the way it compels belief (the reliability question is a different matter on which I want to remain silent), but is unlike the mathematical case in that it is substantive and not merely logical in nature.