Countless mathematical systems might be devised that are not at all interesting or useful. Some, maybe just a few, might be really useful. But the vast majority will inevitably be nonsensical, contradictory, useless, etc.

Some people might claim that 2+2=5. Maybe most people will assert ridiculous crap like "the ratio of a circle's circumference to its diameter is exactly 3".

Right, because math has reference to reality, so some mathematical systems are worthless or, for the purpose of practical application, false, and some are valuable and help us to determine real things.

Is logical/mathematical intuition the same kind of feeling?

I can't rule out these being grouped together, but I think that logical intuition is much more innate. Regardless, the same kind of feeling as what? As a mathematical system hypothetically would be (based on how you mischaracterized me above)? I think that's an opaque psychological question and I don't see it as especially relevant.

Does that imply that mathematical claims are purely nonsensical meaningless feelings?

No, of course not. I wouldn't describe them as feelings in the first place, but beyond that I wouldn't call them meaningless because we can observe the utility of using math to solve practical problems.

Let's keep it simple and consider the value of these systems to be purely in their practical usefulness. The systems in which "2+2=4" consistently prove more useful, so there's a clear reason to prefer those.

Right

Are there no moral assertions that consistently prove more useful than their negations? Is it not substantionally more useful to assume that "murdering your children for fun is wrong" than to assume the opposite?

If we need to tell children that Santa is watching them in order to get them to behave, then we can call that useful, but that does not thereby make Santa real as anything but an arbitrary social construct. Likewise, "good" and "bad" aren't made real beyond their existence as arbitrary social constructions by the fact that you can make useful social norms using those concepts.

Often times mathematical discoveries are the result of some sort of logical leap, a feeling that there is some kind of underlying, unifying pattern. Is this intuitive feeling so different from the one that leads you to conclude that it is wrong to murder your children for fun?

Yes, it is different, because you can work backward from a correct logical leap (or forward with the new aim in mind, depending) and have an actual proof of this idea being correct. You cannot do this with moral intuition, you just have your feeling and it can never advance beyond that in terms of establishing the reality of your belief.

This is not to say that that intuition is generally correct, whether it be mathematical or moral. And it would be wise to seek confirmation or proof.

And of course most peoples' intuition is wrong on a variety of topics, from probability to justice.

Most people would get the Monty Hall problem wrong. Most people would give a nonsensical account of morality. I think we can agree the first case is simply a matter of education and investigation.

If people were given sufficient means and opportunity, I think they'd achieve consensus on the underlying principles of probability. Can you imagine a future in which people might achieve consensus on principles of morality?

It's possible for there to be moral consensus in the future, but it would have nothing to do with a process similar to consensus in math, it would be basically the same as a consensus in aesthetic sensibilities, but with governments having a much stronger incentive to promote it. I can't really say how likely or unlikely it is, though I am inclined to say that there will probably never be a positive consensus because I believe the negative position -- that morality is imaginary and other things should inform our actions -- is the much more useful position. I'm not saying everyone will agree with me, there probably won't be a negative consensus either, I just think it's more likely than a positive consensus.

If you look back on the comment I made to the other person, you'll notice that I said that politics as people commonly think of it needs to be separated into political (moral) values and political-economic analysis of real systems. I believe that one of the greatest strengths of Marxism as a political theory is that it rejects moral values and argues in terms of material benefit and personal agency on the basis that they are things people subjectively desire as a product of the way people are (the dreaded "human nature"), with this being a necessary element in the class conflict that has driven history. I want a society that is oriented toward human flourishing, and I think it's much more realistic to make the argument that socialism can benefit people materially and help them to have agency in their life rather than tell them that this or that thing is "good" or "bad," whatever those terms even mean, because they can arbitrarily reject that such a thing is true and there's nothing to prevent them from doing so. With Marxism's claims about reality, it is there own desire to benefit that would prevent it.

It reminds me of why Geoff Waite said "Nietzsche is the only position outside of socialism," because it's perfectly possible and indeed appealing to some people to take up a highly aesthetic form of ethics that rejects these ideas of material benefit and turns one's life into an art performance for the sake of a subjective sense of personal satisfaction, even if it is immensely harmful even to themselves from our perspective. Morally, there is nothing you can say to this person, because they've chosen their moral system and it's just different from yours. If you can make an appeal, I think the only option actually is Marxism (which would make this sort of a rejection of Waite's thesis), by trying to reason with them that their attitude is pathological and they can be happier if they embrace materialist analysis and pro-sociality. Ultimately though, and this is why I'm bringing this up, you can't really tell them they are wrong because values have no independent reality, believing in natural rights is not comparable to believing in the sun, the only thing that you can do is appeal to their subjective sensibilities -- the same general paradigm that got them in this mess -- and give them something that appeals to them more.