most mathematicians aren't very concerned with whether their proof maps onto the real world...
I'd agree. The point of math is that it is an a priori science, as one of my teachers called it-- that is, hypothetically you could "do" mathematics by some standard without any experience of being in the world. However, some mathematicians believe that it's an empirical science or, alternatively/also, that while problems could be solved by Bertrand Russell's Dasein-less robot, to "do math" is to be able to present, communicate, and understand a proof's structure intuitively. I lean more toward that last one (could you tell) and I think it bleeds into this game. Along with, of course, a surprisingly noncontroversial belief that "math is art".
re: Outsideness in criteria for proof
I'll have to read that book. What you describe seems like a really, really strong brand of the Quine-Duhem thesis, and I dig it. At the same time, there's a fun tension between "beliefs can't entail beliefs" and "phil math can suggest aesthetic beliefs and vice versa". Not a contradiction, I don't think, but a tension that might actually be where the fun in this game lies.
First, randomizing methodologies is a way of getting Outside
I'm glad you said this, because I think you're right. The other justification I found (which is not at odds with this one, of course) is that any of the six rolls are a valid course of action for getting somewhere. This is good for two reasons. The first is that the goal you set is not necessarily the most interesting part of the problem-- and, if we are interested in mathematics as art, our goal is to invent/discover/notice/formalize beauty, wherever we find it. The second is that, unlike in science or philosophy, in mathematics beliefs directly entail other beliefs-- specifically, all beliefs in a given worldview are entailed by its axioms. In other words, if you keep walking in the forest, you'll eventually get everywhere, not just somewhere.
I'm interested in your belief that (if I'm reading right) removing the randomness preserves the Outsideness. I mean, obviously I wrote the game in a way that implies it, and it's a pretty good manifesto bullet point for the jam. But, writing this, I think that the advanced variants are advanced because they require inside-Outside synthesis, which is rather hard considering we don't even know what that means. I mean, how are you supposed to know when there's another way, or whether there really is a meaningful iceberg? And technically, looking back, I'm uncertain about the meaning of Whenever you cannot solve a problem and Whenever you can solve a problem as joint triggers. The obvious interpretation is "oh, so like, whenever you want" but I think it requires synthesis.
Side note! So now we've written, like, three study guides of text about the study guide, which to be honest is a design philosophy I hold dear-- Whenever it is possible, answer a text with its own form. That is, to respond to poetry, write a poem; to state something about games, write a game; and so on, and so forth. This isn't always possible or even advisable (or even a posteriori)-- using only that rule disallows most metaphors, for example. But this game believes a lot of things, and the point is that I don't believe they can be summarized without creating a game. (Not that this posting isn't valuable, of course-- it's obviously pretty informative, and besides summary isn't the only thing to glean from a text.)
All of this points to an approach to knowledge acquisition that's open and syncretic, that allows input from other, inherently incommensurable ways of knowing.
I think so. The fact that you're explicitly told to view the game as "the tip of an iceberg" or whatever you roll probably seals this pretty nicely.