Wow, thank you! I need to hear you talk more about the symmetry between this game and some of the structures you mentioned.
Viewing post in Instructions for Solving Mathematical Problems jam comments
That's a BIG topic, but I can point to a few things. Caveat emptor: I have a very strange set of beliefs about philosophy of science that makes some people mad.
We could inspect the idea of "solving" something. It's my impression that mathematicians have a very different approach to what it means to solve things than, say, most scientists do. At least their language and demeanor about it tends to be very different. I think most mathematicians aren't very concerned, for instance, if their proof maps onto the physical world, because math is inherently abstract, etc. (It may have implications for the physical world, but it very quickly becomes science or something other than math--art and occultism spring to my mind, in fact--when you start thinking that way.)
And this process of solution that the game describes, seems very aesthetic to me, especially in the advanced variant. What criteria, what impulse, what set of beliefs inform what processes you use to go about solving something? Similarly, what criteria inform your declaration of the problem being solved/finished? These are, I would argue, essentially aesthetic criteria. I have seen so many mathematicians (my physicist ex among them) who prize elegance and beauty in proofs above all else. I have known mathematicians who have found a perfectly "functional" proof, who then go on off in search of another, more aesthetically pleasing one, or a more interesting one, or some other arbitrary--aesthetic--requirement.
The truth is that there cannot be one, singular set of criteria for proof. Any given paradigm requires input from Outside (there's that word again) against which to judge its own set of assumptions and practices. My normal example is the so-called "Scientific Method," which doesn't functionally exist in any meaningful way. If it did, science wouldn't be able to do anything. (For more on this, I recommend Against Method, Paul Feyerabend. I do not especially recommend Kuhn, but to each their own.) And I argue that the same is true for any functional tradition of knowing (art, occultism, whatever): it does not, cannot exist in a vacuum. All progress happens across worldviews, inter-paradigmatically. There's no objective Aristotelian Platform from which to inspect ourselves and our belief systems, there are only other belief systems. And the belief systems that acknowledge this, the way that art and occultism and some approaches to math and science do, are stronger and more agile and more functional as a result.
And this game implicitly acknowledges that, I think. First, randomizing methodologies is a way of getting Outside (of your personal paradigm, of the prevailing paradigm of your culture of inquiry), of reaching for something beyond the current set of possibilities. And once that becomes a natural thing, you can remove the randomness and allow your burgeoning aesthetic judgment to guide that process. And once you've mastered that, you can apply those techniques and proclivities in other areas (such as solving this game). 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.
So that's not a lot of specifics about parallels to art and occultism, but I think it gestures at a larger point?
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.
I don't have a lot to add to this (without, as you say, responding in the style of the game perhaps), except to say that I think we're in enthusiastic agreement. Nonetheless, some follow-ups:
- I view mathematical realism/non-realism much in the same way I do fictional realism/non-realism. Which is to say: yes to each, at various times, as necessary or expedient.
- If you're into Quine-Duhem, you're gonna love Feyerabend. He gives the strongest possible version of that claim, and Against Method lays out the argument systematically, drawing on as much historiography as theory. It's an underappreciated book, and one that has completely shaped everything I've done since I read it.
- I'm reading the randomness as a "training wheels" method of getting outside, which we can shed as we come to fully embody the skill. Once we lose the need for "artificial" methods like randomness, we have a more direct access to the Otherwise Than Ourselves, hopefully moving beyond even the limitations imposed by the game (tip of the iceberg, after all) and into the world of the empathetic, the subjectivity of the Other. This absolutely requires some facility with breaking down the Outside-Inside dichotomy, which was always false anyway. (Side note: this gets right at a central point of Against Method, that practice/theory or evidence/theory are not separate, and cannot be. We are always doing both at once, as one gesture.)
I know you're busy these days, but if you end up reading Against Method at some point, I'd love to chat about it! ❤