Your analogy is off. With number theory, even if no one had an external use for it, progress was (is) made, new theorems were proved, so time was not wasted in that regard. The problem with the Collatz conjecture is whether the effort spent on it is actually generating any insights into the problem.
But there is 'progress'. Looking at the Wikipedia page:
"The proof of the conjecture can indirectly be done by proving the following:
- no infinite divergent trajectory occurs
- no cycle occurs
thus all numbers have a trajectory down to 1.
In 1977, R. Steiner, and in 2000 and 2002, J. Simons and B. de Weger (based on Steiner's work), proved the nonexistence of certain types of cycles."
I am placing 'progress' in quotes because one cannot measure progress in maths. Before one has a proof, we cannot know whether existing approaches are true dead ends or whether they just need that one extra insight.
Not really. This is about being able to access the specific papers written; it is not about being able to read an aggregate of the research without necessarily being able to look at the original source (which is what Wikipedia is).
Indeed, Wikipedia's particular style won't work for science.
The shame of modern science is that what might work is so simple to build, because it is already built. Research papers already form a web: They all have references to prior research papers, which in turn have references of their own. Moreover, the community of researchers and their students would quickly and happily turn all those links into clickable ones -- if only paywalls didn't prevent them. [1]
It's no coincidence that even the simplest of web pages is already perfectly suited for scientific research: Tim Berners-Lee built HTML and HTTP for that purpose. But now science lags behind, say, LOLcats or the Twilight books when it comes to online discussion, and I assert that it's not merely because science is esoteric [2]: It's because scientific publishing is suffocated by rent-seekers.
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[1] Actually, my understanding is that cross-journal hyperlinks are slowly evolving within academia: Once you're inside the university library's firewall you can make actual working hyperlinks that are only, say, 65% less efficient than the average Wikipedia link. This is great for those who are currently inside the ivory tower. Not so great for me and the other 99.9% of humanity.
[2] What I wonder is the extent to which science remains esoteric because it has always required so much training and money to read the literature that the literature has never needed to evolve actual readability. Why bother learning to explain your work to a popular audience when your writing will only ever appear in journals that are only visible to your colleagues?
Historically, we've relied on great science journalism to make up the difference. (Gods bless you, Isaac Asimov.) But great science journalism, like all journalism, is rare and getting rarer.
Important to whom? What people find remarkable and notable varies wildly. What may be notable to one group (say the residents of X-town may find the mayoral election notable) might not be to another (Y-town doesn't care).
Something remarkable or notable may happen every day, but it's not necessarily the case that something remarkable or notable to a specific person happens every day.
I'm curious to see the smooth infinitesimal analysis approach to complex analysis. Using the limit definition (the dreaded epsilon-deltas), you get the correct definition of continuity (for the usual topology).
How (if at all) do you do the same thing with dz's?
