NOTE: I put this under draft day, because I'm hopeful someone will read it. I have no problem with its presentation, rough draft or whatever.
I think EA's don't quantify much or very well, but claim to update their knowledge in a quantifiable sense, basically offering a bogus description of what they do when learning, and also providing a cultish social ritual for how they do it, with the occasional addition of increasing their desire for betting behaviors as a part of the exercise of their epistemology, or emphasizing the importance of specific ideas of the future, as needed. I explain this below. Here I am referring to quantifying probabilistic uncertainty, rather than other uses of mathematics.
Perspectives on how or why EA's quantify beliefs and decisions include:
One - The tendency for EA's to "update" or quantify various kinds of beliefs has a structure that I have seen before, in a different group of people. I am familiar with this suspension of disbelief around discussion of mental mechanisms and the commitment to a group dynamic of offering each other communication rituals specific to the in-group during conversation with other members. The communication rituals address and reify a false idea of how a mental mechanism works, and lends an aura to the interaction, as though it is something special, earned, or developed through practice. It's something you find in any in-group or cult. The EA updating ritual is a ritual. You have a discussion. You want to "update, " given that they made you feel convinced. You let them know that you will "update" on the new information. "People updated after event X", "I haven't updated on this in awhile but then I found this information from you", EA's make assertions like that. You know to discuss your updating with people who will understand what you mean, and appreciate it.
Two - There is another perspective on it, the perspective that EA's are sometimes quantifying their feelings of doubt or uncertainty or conviction wrt beliefs. I find this perspective appealing as an interpretation of EA's acting in good faith to "update" or quantify beliefs or decisions. Good introspection will let you notice degrees or shades of feeling level or intensity. Your certainty can indeed increase and decrease. It shouldn't be a surprise to anyone, however, that feelings do not align with our conscious knowledge. Thus our usual awareness of our "irrational" feelings. Measuring feeling intensity is important to make them more accessible to your way of thinking, whatever awareness, insight, or trigger that they represent. Rather than an all-or-nothing phenomenon, feelings become more nuanced and granular, as their level of intensity gets measured and labeled. However, while feelings might tell you something about what you believe in the moment, they don't always tell you what you should believe based on facts or evidence.
CAUTION: Typically, feelings are influenced by motives or beliefs in ways that the person with feelings cannot control when trying to "think clearly". That is a good reason to use external cognitive aids (for example, a written list of evidence or a pros and cons write-up) for some kinds of argument analysis and decision-making because otherwise your feelings would filter out some of what you need to consciously consider.
Three - There's a perspective that EA's are actually betting addicts, and that their ethics and philosophy reflect an unconscious commitment to their risk-seeking behavior. To the extent that that is true, you would expect to see a lot of betting behavior among EA's, and I don't have strong evidence of that. Most EA's seem committed to career interests, not betting. However, the presence of the metaphors of betting in EA epistemics is alarming. Words like "casinos", "cashing out", "odds", show up in most discussions, and serious debates apparently require betting money to resolve. "Updating" from this perspective is like updating a bookie's betting odds, and the EA is the bookie or the bettor.
Four - EDIT: I almost forgot this perspective, though it is one I've held for a while. By the prediction of existential risk with specific probabilities, EA's emphasize the outcome or its contrary. For example, EA's emphasize a path into the future that is ambiguous with respect to the consequences of developing AGI. Supposedly, AGI either threaten our existence or guarantee a bright one for us. There's a chance of each outcome. By selecting a specific probability for one, it or its contrary can be emphasized. Does AGI safety require desperate work or are AGI going to bring prosperity and happiness to everybody? Seems like either way your AGI research job is vital. In the case of Climate Destruction, on the other hand, the consensus among EA's is that its odds of causing an existential crisis are quite low. Instead, EA's emphasize a contrary interpretation of our current path into the future: a status quo approach of economic development and free markets leading to technological utopia, provided AGI safety concerns are solved. Since the estimated probability of a climate apocalypse is not testable, nor requires any real proof, the choice of number simply lends emphasis to a worldview or agenda. In the case of Climate Destruction, a crisis whose presence threatens the status quo of economic, social, and business development, the agenda is to maintain cash flow and the worldview is one that the wealthy hold. The choice of message about Climate Destruction simply reflects that self-serving behavior.
NOTE: EA's do mention tail risks going up, and sometimes mention climate change as a tail risk. Discussing tail risks seems to be another way of claiming that "Yeah, we're on the right path but it's getting more risky to stay on it!" With respect to AGI harming everyone and climate destruction causing an apocalypse, I am scared of both and believe neither's contrary deserves following an ambiguous path towards it. Our current path is the wrong path.
Those are a few critical perspectives on EA updating. Speaking for myself, I prefer a categorical statement of belief followed by learning to constrain the belief to contexts in which it is not contradicted. If there are no such contexts, I move on. There are more and less efficient methods to constrain a belief. However, by "belief", I mean something that I find tolerable to assert as part of a conscious deliberation about what belongs in my ontology, or a conscious investigation of truths about the world. Murkier beliefs that I hold are found somewhere in my unconscious, as intuitions or heuristics or nagging feelings, where they typically remain, surfacing every so often to clear my head or irritate me a bit, lol.
TIP: I have been trying to rid my own self-talk of the cultural tick of referring to possibilities and probabilities when what I mean are plausible ideas (plausibilities, haha) and matches to contexts. I don't necessarily recommend it to you unless you would like some "deprogramming." An alternative term than "context" is "schema", "prototype", or "situation". Contexts have variations or nuances in specific instances, and the more a specific instance varies from some prototypical form, the more the word "ignorance" describes my knowledge about it. "Ignorance", not "uncertainty". If you want to explicitly quantify less, you can try out what I'm doing. For example, you could answer the question, "How should I constrain my belief about this topic?" or the question, "How similar is this context to any that I compare it to? What are the differences?"