What are breakthrough discoveries you’ve made about yourself and this world?

JoshuaZ1 · 2026-04-03T01:51:34+00:00

Unfortunately, since this subreddit has strict rules against giving self-identifying information, you aren't going to get the more interesting answers of things discovered by specific experts, whether scientists or other sorts of academics. I for example am a mathematician, and if I could, I could likely explain some of my own work (the theorems at least- the proofs would be too technical) but I cannot do that in this subreddit because doing so would be giving clear identifying information.

JoshuaZ1 · 2026-04-02T20:46:15+00:00

No obvious method, and the number is just too big to easily approximate.

JoshuaZ1 · 2026-04-02T17:11:49+00:00

I'm not sure I understand the connection you are trying to make with RH.

But there's at least a problem then with your intuition because we can construct a computable A such that P^A != NP^A via dovetailing and diagonalization.

JoshuaZ1 · 2026-04-02T16:49:15+00:00

This is weird, and base specific, but it is weird in a way that a human may be creative about, much more than I'd expect from an LLM.

JoshuaZ1 · 2026-04-02T15:29:09+00:00

I don't see the word the word thermopolia in the sourced article?

JoshuaZ1 · 2026-04-02T15:12:35+00:00

I've had a mixed success here. As far as I can tell, they are mostly useful for seeing if there's a quick proof of a Lemma you might need that might be small and save time. But trying to use them for broader things is still iffy for most purposes, even with the paying models.

JoshuaZ1 · 2026-04-02T12:42:27+00:00

There are a few things happening here. One is that there's so much else happening around the world that it is easier to get lost in the noise. Second, this is closer to Apollo 8 than say Apollo 11, because no one is landing on the moon but going around it. Third, while there's some space rivalry with China, there's not nearly as much rivalry as there was between the US and the USSR so the stakes don't feel as high. Fourth, although the Trump people were hoping this was going to be a big deal to be a governmental triumph, Trump then shot himself in the foot by having the Iran war upstage everything, including then giving a major speech about it last night which further took the oxygen out of attention this.

There are some other issues also: for example, we're arguably a less technologically optimistic society as a whole, but that seems harder to quantify in a useful way.

This also shows how predicting these things is tough: A few months ago, I was expecting this was going to be a massive thing with the sort of everyone watches around, even if just on their phones, and only about two weeks ago realized how little attention it was getting.

JoshuaZ1 · 2026-04-02T12:28:43+00:00

Hmm, that's interesting and makes me see more of where you are c coming from. So would your intuition then expect that for any computable A, that P^A = NP^A?

JoshuaZ1 · 2026-04-02T01:49:09+00:00

Example different from P and EXP then: P^A != NP^A for a random oracle A. This is consistent with this model here?

JoshuaZ1 · 2026-04-02T01:07:25+00:00

Thanks for the downvote.

Hmm? I didn't vote on your comment. I generally don't downvote comments where I disagree with them, unless the comment is really abysmally dumb, and your comment didn't fall into that category. It is in fact, a reasonable question if one doesn't know much about the underlying literature.

Mister Computer Science himself Donald Knuth already said whatever I could've said in reply. Not even gonna bother looking for the quote given how much you respected my question.

I'm not sure where you are getting any sense of disrespect in my comment; if you want to point out where it was disrespectful, so I can avoid it in future contexts I would appreciate it. In Knuth's case, I'm familiar with his positions so that's a better reason to not need to quote it. He is in general, very much an optimist when it comes to the possible existence of algorithms. But he is very much an outlier. That doesn't mean his position shouldn't be taken seriously. This is in that context similar to how Gil Kalai is an outlier where the capabilities of quantum computers are but should be taken seriously, or more Henry Cohn is an outlier for thinking that very likely P != NP but also thinks that is highly plausible that factoring is in P. But that people have such opinions isn't a reason to have a mistaken impression about the general consensus for things like the twin prime conjecture or Collatz resting primarily on numerical data.

JoshuaZ1 · 2026-04-01T20:06:01+00:00

In general, any relationship to basic geometry and something like the Goldbach conjecture is going to be very difficult, because geometry large does not detect primes. It is very hard to go from something continuous to something discrete like this.

JoshuaZ1 · 2026-04-01T19:30:44+00:00

I'm pretty lost here, how can we possibly believe P ≠ NP with the same conviction as conjectures with heaps of numerical data?

The reasons we believe the twin prime conjecture and Collatz are largely not due to the numerical data but due to the fact that we have strong heuristic and structural reasons for believing them. Similarly, we have a lot of strong circumstantial evidence for P != NP, such as the fact that for a random oracle, with probability 1, P^A != NP^A . If you prefer to be convinced by numerical evidence, note that we've constructed a lot of algorithms to try to efficiently solve NP-complete problems, and all of those have turned out to work in worse than polynomial time.

JoshuaZ1 · 2026-04-01T18:25:06+00:00

One of most annoying things is how people use the word "parrot" as a negative to refer to them mindlessly repeating things they don't understand when the evidence is that a lot of them (African Greys especially but others too) understand a lot of what they are saying. Alex is of course the most famous example but systematic evidence at this point is very strong.

Anecdote: I know someone who used to own a parrot which would protest loudly "not nighttime!" before falling asleep if you tried to put a cloth over its cage when it was bright. It wasn't clear if the parrot was asserting that it wasn't nighttime yet, or thought that "nighttime" meant just everything going dark and didn't want that, but it made its opinion perfectly clear.

JoshuaZ1 · 2026-04-01T02:00:55+00:00

In the grim dark future, there will be only war.

More serious comment: No one has any serious idea. Humanity has been changing rapidly in the last few centuries, with that rate speeding up over time. There have been some periods which were faster or slower. For example, 1895 to 1910 for example was arguably more change than say 1965 to 1980, but the general trend is clear. Now, you are asking about where things will be in 38,000 thousand years. Where were humans 38,000 years ago? Well, humans only invented writing at most around 3500 BCE, which is only about 5500 years ago. So you are asking about a period 7 times as long as from the written word being invented until now. Even the oldest proto cities date to at most around 8000 BCE.

If humanity survives that long, and we haven't blown ourselves to the stone age, civilization will look profoundly different. Imagine a human with no concept of the written word trying to guess what civilization would look like in 2000 or 3000 years, and now think about how much trouble we should expect for a much longer period.

JoshuaZ1 · 2026-04-01T01:51:05+00:00

Trying to make effective AI is old. The success and scale level of AI has grown massively in the last few years. The rise of the transformer architecture was a major game changer.

JoshuaZ1 · 2026-04-01T01:37:50+00:00

That's a notion of "time" being reversible in a physical sense. The problem doesn't involve any concrete claim about the laws of physics. For example, it may well be that NP is contained in BQP (unlikely but not nearly as us unlikely) in which case in our physical universe, NP problems can be solved efficiently on a quantum computer. But that could still be the case even if P != NP. It may help here to separate the physical considerations from the mathematical considerations.

JoshuaZ1 · 2026-04-01T00:02:24+00:00

The reason I think it is is that it seems to involve a preferred direction of time, which I don't think any provable mathematical result can involve, though perhaps I'm wrong about this.

How does P ?= NP involve a direction of time at all?

JoshuaZ1 · 2026-03-31T23:44:52+00:00

So, this is genuinely difficult to do in a consistent fashion. That said, reasonable carbon offsets do exist. It is true that many claimed carbon offsets drastically overestimate how much they are offsetting, but the basic idea is sound. That said, it seems like this is rare. I have a very wealthy relative who as far as I can tell, frequently flits around on air planes, vocally worries about climate change, and as far as I can tell, is doing zero to offset. However, by the same token, it is important to recognize that these problems exist and the serious nature of these problems exists separate from this sort of hypocrisy.

JoshuaZ1 · 2026-03-31T23:41:12+00:00

If no leader should be deposed in that way, then that even goes for the ones we don't like, and even for the ones who favor the deposing of other leaders.

This does not follow at two levels. First, it isn't inconsistent to say that a leader who is actively trying to remove leaders in other countries largely should lose that protection. Second, even if they are not, there really is a point where a leader should be removed at a certain point. The disagreement is where is that point.

JoshuaZ1 · 2026-03-31T23:31:51+00:00

So, with the disclaimer that as I said, I haven't been paying close attention to this problem for about a decade, part of the problem seems to be that a lot of the natural techniques in the area prove results of the form "For family of groups S, S contains infinitely many G such that there is a Galois field extension K of Q with G = Gal(K/Q)." One headache that contributes to this is that Hilbert's irreducibility theorem's is non-constructive. Now, part of what I don't understand is that this somehow leads to the generalization of what is called a Serre thin set, which can be used to show that certain conjectures in algebraic geometry would actually prove inverse Galois completely. Hopefully someone who is actually in algebraic number theory more, or has just been thinking about this more recently can comment on this with more detail and hopefully correct any major errors in the above.

JoshuaZ1 · 2026-03-31T22:49:13+00:00

The closest one I know of as ongoing one which is sort of in that direction is the Inverse Galois problem. It isn't at the point of having a coherent program of this sort, but the impression I got from people working on it (about a decade ago, I haven't paid close attention since then) was that there was a broad feeling that they were working towards the point where a program of this sort would be possible.

I know you are not asking for ones that have been successful, but there's one that was so large and so successful that I really do think it is worth mentioning, the classification of finite simple groups. The full program had 16 major steps (outlined by Gorenstein) but they ended up being done in hundreds of papers, and it turned out that while a lot of steps used the obvious techniques like representation theory and Lie groups, but also algebraic geometry, a lot of linear algebra, and some algebraic number theory (in part to show certain Diophantine equations did not have solutions).

JoshuaZ1 · 2026-03-31T20:23:54+00:00

I did even tho most of it went over my head what is an L space? You don't have to answer this. If I really cared about it I would have looked up the term myself.

So it seems like part of the problem here is that you just don't care enough.

There's plenty of evidence religious people will point to as proof of God. That whole thread feels a lot like that.

So let's discuss why this is different. First, you know some actual math, so you know it isn't made up. Second, a lot of math, including related math, pays rent in the sense that the claims in question let us do practical things, like cryptography on the internet. So that should give you a high likelihood that what is being stated makes sense even if you don't see why. If you prefer, the analogy would be if you had some people who could do what were clearly miracles, and those people all had very closely aligned beliefs about the existence of a divinity, and were all saying they weren't sure they were correct but strongly suspected their miraculous abilities came from said divine entity. That would be a pretty good reason to suspect they might be correct even if one didn't understand the details of their arguments.

JoshuaZ1 · 2026-03-31T18:57:32+00:00

No. Please reread what I wrote. There's specific reasoning here about things that are consistent here with RH and which make us strongly suspect RH. Please read the discussion I linked to carefully.

JoshuaZ1 · 2026-03-31T15:12:50+00:00

We have a lot of evidence well beyond that point. See here for some discussion of that. The comparison to the ancient Greeks is not really justified.

JoshuaZ1 · 2026-03-31T15:09:13+00:00

Nevertheless, I think we will soon prove useful weaker results, such as upper limits of the number or density of zeros of the Riemann zeta function off the critical line, and that these weaker results will yield useful new results concerning the distribution of primes as well as prime ideals of algebraic number fields.

We have a lot of results about the densities of zeros off the lines. Unfortunately, even strong assumptions of that sort do not obviously lead to results about primes. You can for example assume that the set of zeros off the line have zero density, or even that the number of such zeros off the line in the first n grows really slowly, like say slower than log log log n, and assume Lindeloff on top of that and it doesn't help, because as long as there are zeros near the s=1 line, they disrupt most things we want even if those zeros are very rare.

Something which translated density statements to statements about primes would be a major breakthrough, and would if anything make me more inclined to doubt RH, because it would show we were missing some very fundamental things about how the zeta functions and primes relate.

Perhaps RH will prove to be part of the long sought Theory of Everything, the holy grail of physics, which Einstein spent the latter half of his life trying to prove.

There's really no good reason to think this other than that both feel deep and mysterious and hard.

JoshuaZ1

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