User blog comment:Alemagno12/BM2 has a problem/@comment-30754445-20180724073311/@comment-30754445-20180725130923

@Ecl1psed276

"You seem to think that even a very good and detailed analysis can be completely useless, just because it's informal"

No.

What I am saying is that an analysis which doesn't bring any arguments to support its claims, is not "good analysis" and therefore it is completely useless.

"And again, stop imposing an 'analysis must be formal' rule on us."

I'm not imposing anything. You can do whatever you want. Just don't expect me to take your analyses seriously, when you're giving zero evidence for it being correct.

@Alemagno12

"He means to make a formal proof (not just a table of ordinals) that (0,0)(1,1)(2,2) = BHO. This can be done by proving that each standard BM2 expression below (0,0)(1,1)(2,2) corresponds to an ordinal below the BHO."

Precisely.

And since writing ordinals up to the BHO is pretty straight forward, this shouldn't be too difficult to anyone who is well-versed in BM2. Any takers?

@P進大好きbot

"Anyway, writing a sufficiently big table is just an evidence, but not a formal proof."

Since the world of numbers (and ordinals) is infinite, writing (finite) big tables isn't even evidence... unless you also have some kind of theoretical justification for assuming that your sample is large enough / covers all the bases.

For example:

"The Riemannian hypothesis has a table with \(10^{13}\) lines of evidence, but has not verified yet."

10 trillion zeros is indeed compelling evidence. But in the days we only knew of 10 million zeros, people were much more skeptical.

So what's difference? Isn't 10 million pieces of evidence enough? Well, as it turns out, there are deep mathematical reasons for the first few dozen million zeros to fall on the critical line. So no, in this case, 10 million pieces of evidence isn't impressive at all.

But 10 trillion zeros, in this specific case, is convincing. Not because of the sheer number (10 trillion) but because of how this number compares to our theoretical understanding of the situation. And this is one of the reasons that virtually all mathematicians today accept the Riemann Hypothesis as true, even though they don't yet have a proof.

On the other hand, there are cases where even 10 trillion bits of evidence aren't compelling. Take for example Gauss' conjecture that Li(n)>pi(n) for every n. Today we know that:

(1) This inequality holds for all numbers less than 1E27

(2) Gauss' conjecture is false, with the first counter-example occuring at some point before 1.4E316.

So that's a whopping octillion bits of evidence for a conjecture that's false. Worse: the difference Li(n)-pi(n) actually grows larger and larger for n<1E27. Seems like overwhelming evidence... but of-course, there are perfectly good number-theoretical reasons for this to happen.