There's been an ongoing argument about the problem of the ambiguity of BEAF and exactly how crippling it is. Is it reasonable to say that meameamealokkapoowa has size comparable to X? Can we argue that an array has this ordinal growth rate?
More generally this brings us to a debate over what we should do with ambiguous googology. Sbiis proposed the following "grading system" for googology (which I've edited somewhat for clarity, but most words are his):
- Grade A definitions meet any reasonable standard of mathematical rigor. The definitions are one hundred percent precise, and their existence or totality can only be disputed from a metamathematical standpoint.
- Grade B may have a definition which isn't strictly formal, but it is cohesive enough that it makes sense. You can always figure out what to do next, but it isn't known to be total.
- Grade C would be systems lacking a definition clear enough to say that it makes sense. It may be subject to multiple interpretations. It can still be gauged based on theories of what a "reasonable" interpretation is.
- Grade D would be numbers of a definition so vague or strange that it's not even clear what level of recursion we are talking about, or whether it makes sense. None the less, Grade D still at least provides some semblance of an idea.
- Grade F is crank land. Anything and everything goes. It could be complete nonsense and it would still be grade F.
- Grade S is Sam's Number.
The question is: Can we make mathematical claims about googology that isn't grade A?
I'm going to augment the discussion with a bit of an experiment. Below are two boxes that you can edit to argue for your case. Be sure to add your username to show which one you endorse.
Users: Sbiis, Cookiefonster, SuperJedi224
Yes. Even with a grade of B or maybe C, we can make reasonable estimates about the growth rate and recursive level of many googological systems.
Mathematical intuition is indispensable to the creation of googological systems, but this means intuition can often anticipate the strength of a notation even before a complete formalization is in place. Such gauges are essential to googology since we use them all the time to determine whether a given approach is worth pursuing. An informal idea is not necessarily wrong. Consider the development of calculus, which was not fully understood and formalized until much later. Sometimes an idea must be pursued even in the absence of a full understanding, which only comes in exploring the concept intuitively first.
Vague systems are nothing to fear as long as it is recognized as vague. We can maintain a goal of pursuing quality and precision while still making estimates of the potential of "rough ideas".
If we did only accept the utmost formalized stuff, then so many kind-of-good ideas would go down the drain, and that may be pretty upsetting. Viewing math, or anything, as a black-and-white field, prevents openness to further ideas regarding the designations, and that wouldn't be good either.(edit)
Users: Vel!, LittlePeng9
In general, intuition precedes fully structured arguments. There is nothing wrong with broadcasting a half-formed idea, but a half-formed ideas should be treated as opportunities to produce formal mathematics. Dough isn't bread.
The lower grades are most useful to denote works in progress — it is normal for a googological system to undergo a developmental process where it gradually improves in grade. But in these unfinished steps, analysis is futile: undefined mathematics is not mathematics, and in fact it is crankery. Mathematics is a black-and-white field of study, and it doesn't make sense to draw logical conclusions from vague systems. Doing so is unprofessional, wrong, and generally counter to the entire notion of the objectivity of mathematical logic.
Imprecise systems tend to invite opinions and subjective arguments over the power of a system. It's unproductive to argue over whether Bowers meant X or Y -- why can't we simply note that Bowers' writings were ambiguous at that point and get on with our lives? A prerequisite for mathematical proofs is having clearly defined premises, and notations with a grade of B or lower don't satisfy that.
This intolerance for even minor ambiguities is more than just a personal preference — some systems are incredibly sensitive to details. Take, for example, the catastrophic failure of the catching function, which was "evaluated" while glossing over fundamental sequences. A minor ambiguity, you say? Turns out even the smallest change can make an SVO into an epsilon-zero. An analogous (although probably less extreme) case is BEAF itself.
A consequence of pretending that sloppy googology is rigorous: googologists no longer have to put any sort of effort into making precisely defined systems, leading the whole community into a spiral of half-baked systems that, to an outsider, result in total nonsense.(edit)
Use the comments for interactive discussion — the above boxes are for self-contained, complete arguments.