User blog comment:Vel!/FGH Gripe/@comment-2033667-20150326185550

@Sbiis This is a very important argument that I've been delaying since it took me a while to figure out how to phrase it.

I'm glad that we agree that at best these are conjectures and not verified mathematical statements, even if they were to be formalized with a real FS system and a definition of the comparability relation. But here's another argument: in my opinion they are just not very interesting conjectures especially in light of the fact that they are unsourced, original work.

I can come up with any old garbage and call it a conjecture. Here's an example for the sake of analysis: "It is conjectured that TREE(3) > BB(10)." If someone were to add this to an article on the wiki without source, I would probably remove it. Why?

I don't think it would be necessarily be uninteresting to see that problem resolved. Hell, if someone proved the inequality true or false, I'd be pretty stoked and it would definitely be wiki-worthy. But a good conjecture must be supported by evidence, such as heuristic arguments or computer searches for counterexamples. But there isn't any such evidence supporting the inequality "TREE(3) > BB(10)." It's just...there. No context, no justification.

This echoes my feelings about these comparability conjectures. I'm bothered by how the only support for them is the fact that they were proposed by people who think they have some intuition for how these functions work. We may very well disagree on how strong the evidence is, since after all the notions of "good conjecture" and "strong evidence" are subjective.

I think a source of confusion in our debate has been the two types of informality at play in these conjectures. The first is a fairly direct kind of informality caused by failure to supply fundamental sequences and a meaning for "comparability." As you have said, this is fairly easily mended, although we're still battling over whether failure to specify FS's is a problem.

The second is a kind of informality that we see in all conjectures. Indeed, a leap of intuition is rather necessary to notice a pattern and propose a conjecture. Of course I have no issues with this line of thought per se, just how it's being applied in this one case. Conjectures in general are not problematic, it's conjectures with no evidence that I want to get rid of.