A utility maximizing Bayesian says "what course of action will give me the highest expected return given the probability distribution I have for all my hypotheses?"

That matches my own description of what the brain does. I wrote briefly:

your actions are a product of a combination of your probability assignments and your value assignments, the costs and benefits.

which I explain elsewhere in more detail, and which matches your description of the utility maximizing Bayesian. It is the combination of your probability assignments and your value assignments which produces your expected return for each course of action you might take.

Does that help?

Depends what you mean. You are agreeing with my account, with the exception that you are saying that this describes a "utility maximizing Bayesian", and I am saying that it describes any brain (more or less). That is, I think that brains work more or less in accordance with Bayesian principles, at least in certain areas. I can't think that the brain's calculation is tremendously precise, but I expect that it good enough for survival.

Here's a simple idea: everything we do is an action. To speak is to do something. Therefore speech is an action. Speech is declaration of belief. So declaration of belief is an action.

Now, let us consider what CuSithBell says:

Your actual belief in different things does not, so far as I can tell, depend on how useful it is to act as if those things are true. How you act in response to your beliefs does.

So, he agrees that how you act depends on utility. But, contrary to what he appears to believe, to declare a belief is to act - the action is linguistic. Therefore how you declare your beliefs depends on utility - that is, on the utility of making that declaration.

The utility of a declaration depends on its context, on how the declaration is used. And declarations are used. We make assertions, draw inferences, and consequently, act. So our actions depend on our statements. So our statements must be adjusted to the actions that depend on them. If someone is considering a highly risky undertaking, then we will avoid making assertions of belief unless our probability assignments are very high.

Maybe people have noticed this. People adjusting their statements, even retracting certain assertions of belief, once they discover that those statements are going to be put to a more risky use than they had thought. Maybe they have noticed it and believed it to be an inconsistency? No - it's not an inconsistency. It's a natural consequence of the process by which we decide where the threshold is. Here's a bit of dialog:

Bob: There are no such thing as ghosts.

Max: Let's stay in this haunted house overnight.

Bob: Forget it!

Max: Why not?

Bob: Ghosts!

For one purpose (which involves no personal downside), Bob declares a disbelief in ghosts. For another purpose (which involves a significant personal downside if he's wrong), Bob revises his statement. Here's another one:

Bob: Bullets please. My revolver is empty.

Max: How do you know?

Bob: How do you think I know?

Max: Point it at your head and pull the trigger.

Bob: No!

Max: Why not?

Bob: Why do you think?

For one purpose (getting bullets), the downside is small, so Bob has no trouble saying that he knows his revolver is empty. For the other purpose, the downside is enormous, so Bob does not say that he knows it's empty.

Ah, I think I see the problem. It seems that you acting under the assumption that conscious declaration of being "convinced" should cause you to act like the claim in question has probability 1. Thus, one shouldn't say one is "convinced" unless one has a lot of evidence. May I suggest that you are possibly confusing cognitive biases with epistemology?

Not at all. In fact I pointed out that my account of being "convinced" is continuous with Pascal's Wager, and Pascal argued in favor of believing on the basis of close to zero probability. As the Stanford Encyclopedia introduces the wager:

“Pascal's Wager” is the name given to an argument due to Blaise Pascal for believing, or for at least taking steps to believe, in God.

Everyone is familiar with it of course. I only quote the Stanford to point out that it was in fact about "believing". And of course nobody gets into heaven without believing. So Pascal wasn't talking about merely making a bet without an accompanying belief. He was talking about, must have been talking about, belief, must have been saying you should believe in God even though there is no evidence of God.

I would hesitantly suggest that for most questions if one can't conceive easily of what such evidence would look like then one probably hasn't thought much about the matter.

The issue is two-fold: whether mathematicians are less interested in elementary proofs than before, and if they are, why. So, how would you go about checking to see whether mathematicians are less interested in elementary proofs? What if they do fewer elementary proofs? But it might be because there aren't elementary proofs to do. So you would need to deal with that possibility. How would you do that? Would you survey mathematicians? But the survey would give little confidence to someone who suspect mathematicians of being less interested.

As part of the reason "why", one possible answer is, "because elementary proofs aren't that important, really." I mean, it might be the right thing. How would I know whether it was the right thing? I'm not sure. I'm not sure that it's not a matter of preference. Well, maybe elementary proofs have a better track record of not ultimately being overturned. How would we check that? Sounds hard.

So, say math had some terribly strong political bias, what would we expect? Do we see that? Do we not see it?

Well, as I recall, his actual claim was that liberalism causes mathematicians to evade accountability, and part of that evasion is abandoning the search for elementary proofs. So one question to ask is whether liberalism causes a person to evade accountability. There is a lot about liberalism that can arguably be connected to evasion of personal accountability. The specific question is whether liberalism would cause mathematicians to evade mathematical accountability - that is, accountability in accordance with traditional standards of mathematics. If so, this would be part of a more general tendency of liberal academics, liberal thinkers, to seek to avoid personal accountability.

In order to answer this I really think we need to come up with an account of what, exactly, liberalism is. A lot of people have put a lot of work into coming up with an account of what liberalism is, and each person comes up with a different account. For example there is Thomas Sowell's account of liberals in his Conflict of Visions.

What, exactly, liberalism is, would greatly affect the answer to the question of whether liberalism accounts for the avoidance (if it exists) of personal accountability.

I will go ahead and give you just one, highly speculative, account of liberalism and its effect on academia. Here goes. Liberalism is the ideology of a certain class of people, and the ideology grows in part out of the class. We can think of it as a religion, which is somewhat adapted to the people it occurs in, just as Islam is (presumably) somewhat adapted to the Middle East, and so on. Among other things, liberalism extols bureaucracy, such as by preferring regulation of the marketplace, which is rule by bureaucrats over the economy. This is in part connected to the fact that liberalism is the ideology of bureaucrats. However, internally, bureaucracy grows in accordance with a logic that is connected to the evasion of personal responsibility by bureaucrats. If somebody does something foolish and gets smacked for it, the bureaucratic response is to establish strict rules to which all must adhere. Now the next time something foolish is done, the person can say, "I'm following the rules", which he is. It is the rules which are foolish. But the rules aren't any person. They can't be smacked. Voila - evasion of personal responsibility. This is just one tiny example.

So, to recap, liberalism is the ideology of bureaucracy, and extols bureaucracy, and bureaucracy is in no small part built around the ideal of the avoidance of personal responsibility. One is, of course, still accountable in some way - but the nature of the accountability is radically different. One is now accountable for following the intricate rules of the bureaucracy to the letter. One is not personally accountable for the real-world disasters that are produced by bureaucracy which has gone on too long.

The liberal mindset, then, is the bureaucratic mindset, and the bureaucratic mindset revolves around the evasion of personal accountability, at least has a strong element of evasion.

Now we get to the universities. The public universities are already part of the state. The professors work for the state. They are bureaucratized. What about private universities? They are also largely connected with the state, especially insofar as professors get grants from the state. Long story short, academic science has turned into a vast bureaucracy, scientists have turned into bureaucrats. Scientific method has been replaced by such things as "peer review", which is a highly bureaucratized review by anonymous (and therefore unaccountable) peers. Except that the peers are accountable - though not to the truth. They are accountable to each other and to the writers they are reviewing, much as individual departments within a vast bureaucracy are filled with people who are accountable - to each other. What we get is massive amounts of groupthink, echo chamber, nobody wanting to rock the boat, same as we get in bureaucracy.

So now we get to mathematicians.

Within a bureaucracy, your position is safe and your work is easy. There are rules, probably intricate rules, but as long as you follow the rules, and as long as you're a team player, you can survive. You don't actually have to produce anything valuable. The rules are originally intended to guide the production of valuable goods, but in the end, just as industries capture their regulatory authority, so do bureaucrats capture the rules they work under. So they push a lot of paper but accomplish nothing.

I mean, here's a prediction from this theory: we should see a lot of trivial papers published, papers that don't really advance the field in any significant way but merely add to the count of papers published.

And in fact this is what we see. So the theory is confirmed! Not so fast - I already knew about the academic paper situation, so maybe I concocted a theory that was consistent with this.