Icy0

I was going to write a post disagreeing with this on the basis of the fact that the reader lacks the background information the LLM has. For example, if I were to prompt "explain the proof of quadratic reciprocity using Gauss sums" most readers would need the entire LLM's answer (and much more, probably) and not just the prompt.

But then I realized that the reader can prompt the LLM with the same prompt for the same or equivalent expanded text. Most people don't do this as it's extra effort, but it's interesting to imagine a world where this is the default way of engagement with a text, assumed by both writers and readers alike.

I'm not getting any processing errors. Seems to render fine on both Chromium and Firefox.

Love how Notepad has syntax highlighting for many languages and even has linting for JSON, HTML, and JS :)

My take is the fall of the current order of scientific institutions was already happening well before Trump's actions this year. Increased academic fraud, the reproducibility crisis, increase in people prioritizing career growth over pursuit of knowledge, overzealous publish-or-perish culture, administrative bloat, and so on.

Can we save it? I want to believe it, but I'm more and more of the mind that we need to create a new scientific institution to replace the old one, whatever that means.

Grothendieck's disgust of the mathematics community also reached a breaking point in 1970!

> Not sure what'll fix it though. Perhaps efforts to promote good science as opposed to a great one like accepting publications for failed attempts (michaelson morley style), replication results of earlier works

Too often we try to solve social problems by "adding" something, whether it be adding an incentive or adding a program. I think to really solve the problem of publish or perish mentality, we first need to understand the root cause or causes of this mentality, then work to remove them. What I'm seeing here is humans being shepherded by enormous economic and social pressure to engage in selfish behavior for survival and/or social acceptance. Adding an incentive or a program therefore ultimately does not work because it does nothing about the fact that the humans are still largely enslaved by the aforementioned pressure. So, we must remove the pressure. Remove the pressure, remove the selfish behavior. But how to remove the pressure?

I’ve come to simply realize the problem isn’t AIs, it’s humans.

More specifically, humans who feel the need to make a quick buck.

Actually, the problem isn’t those humans, it’s the societal structure creating incentives and pressures that lead to a large percentage of humans feeling like they need to make a quick buck. Whether that quick buck be for cost of living or for social status gain.

I happened to notice the exact same thing when playing with this puzzle in Lean!

You can solve this by working backwards from the outside in, replacing a (b x) with c a b x wherever you can (not inside out, or else you get into an infinite loop!)

A cute alternative expression to solve the curried composition puzzle is c c c c c, just 5 c's in a row :)

Finally, Lean is a great language to do these puzzles in. See the following code:

  /-- Compose! -/
  def c (g : β → γ) (f : α → β) := g ∘ f
  /-
  ?f (?g (?h ?x))
  -/
  #reduce c (c c) c ?f ?g ?h ?x
  /-
  ?f (?g (?h ?x))
  -/
  #reduce c c c c c ?f ?g ?h ?x

To be clear, I meant that the conversation (the argument) would be a better experience for everyone involved, not the argument itself being somehow strengthened by it.

Nevertheless, I'll try to recall how I arrived at this conclusion. The biggest influence was seeing for all my life, the left and right in America each claim the other is stupid, irrational, and un-convinceable by rational argument. When I was 16 it really did seem to me that the right was the irrational side here, but seeing everything come full circle with what's happening on the left these days, I realized it's kind of weird and unlikely that half the population, divided on a political line, is actually worse at thinking than the other half.

(See the bottom of this thread for a vivid example.)

> when the opposite side cannot be convinced by rationality, which is most of the time

Which possibility is the "failure" possibility here, that the opposite side gets convinced, or that the opposite side doesn't get convinced? I'd argue the opposite side getting convinced is the failure. This one singular exchange somehow managed to convince him of your viewpoint amidst his entire lifetime of experiences that led him to conclude the opposite.

I think it's the old rationalist way of thinking to think "If your mind isn't changed by a syllogistically correct argument, you must not be arguing in good faith (or otherwise can't be convinced by rationality)."

Everyone arrives at conclusions about various topics from the data they obtain throughout their lives through a continuous complex inference process they cannot communicate. Included in this process is a judgment of which sources are credible and which are less credible. Your friend is more credible than the guy who is showing you a study that contradicts your belief, for the purpose of changing your mind. (There are lots of studies out there so it is not hard to find one that supports your argument.) Trying to distill this complex inference process into a linear argument will necessarily lead to an inaccurate representation of how they arrived at the belief.

Doesn't mean everyone's right, though. Governments, cults, and friend groups are very good at shaping the data an individual receives. Even in these cases, you can see how based on the limited information that they do receive, it is quite reasonable from that internal perspective to believe what they do believe.

Arguments would be better if by "argument" we mean some effort, no matter how difficult, to communicate how you arrived at the beliefs you have, rather than a "well" crafted linear rational argument.

> That's an interesting point. Ideally students would have both. My impression is that the latter is far less trainable, and the best you can do is go through enough worked examples, spread out so that every problem in the space of expected learning is within a reasonably small distance to some worked example.

I simply mean that researcher team A will claim a positive result for method A because their test tested task fluency, while team B will claim a positive result for method B because their test tested ability to wade through new and confusing territory. (btw, I think "generalization ability" is an unhelpful term here. The flip side to task fluency I think more of as debugging, or turning confusing situations into unconfusing situations.)

> If you have time, would you mind elaborating a bit more on this?

I don't know what good theoretical underpinnings for human learning looks like (I'm not a time traveler), but to make an analogy imagine chemistry before the discovery of the periodic table, specifically how off-the-mark both sides of arguments in chemistry must have been back then.

> My impression is that general problem-solving training falls into the category of lack of adequate theoretical underpinnings, but I doubt that's what you mean to refer to with this point.

By the way, I see problem solving as a goal, not as a theory. If your study measures mathematical knowledge without problem solving, your tests will look like standardized tests given to high school students in the USA. The optimal way to teach a class for those tests will then be in the style of "When good teaching leads to bad results" that Alan Schoenfeld wrote about in regards to NYC geometry teachers.

I did! On MESE first, then on Hacker News.

Usually when there's a replication crisis, people talk about perverse incentives and p-hacking. But there's 2 things I want to mention that people don't talk as much about:

- Lack of adequate theoretical underpinnings.

- In the case of math education, we need to watch out for the differences in what researchers mean by "math proficiency." Is it fluency with tasks, or is it ability to make some progress on problems not similar to worked examples?

Nice to see a response from you!

I have read the rest of the argument. However, my take upon reading it is that this is just one more contribution in a back-and-forth argument about every aspect that has been studied in math education. Despite the fact that this was published in 2010, the landscape in 2024 very much points to "it's unclear" as the answer to "is [anything] effective?", at least for me, unfortunately.

You seem to have linked a collection of general research on teaching and learning, which I am aware of exists. I'm talking about randomized controlled trials, where you assign a group of students to receive the intervention and another group to not receive it, and if it's single- or double-blinded, without them and/or the researchers being aware of which group they are in. Even writing this brings up logistical questions about how you might get a reliable research result doing this for teaching (instead of, say, medicine, where it's easy to fool a patient into thinking a placebo is the drug).

> There is no body of research based on randomized, controlled experiments indicating that such teaching leads to better problem solving.

I'm sorry but one don't exactly come across randomized controlled experiments in teaching very often... not to even mention ones that are well designed... so this isn't saying much.

I tend to agree with this on discord, as I really dislike dms about not-directly-personal things when they can just share it in the server.

In real life though, my severe congenital hearing impairment means communication ability is stunted for all but one-on-one meetings. Cursed...

I love this! And even more is true -- you can read off the Euler characteristic from adding up how many fractions of a circle are lost over all the points.

For the cube, at each vertex you've lost a quarter circle, and there are 8 vertices -- hence the Euler characteristic of a cube is 2.

For the two-disk model of the sphere, a similar thing should be true, I think, but I haven't worked it out in detail -- the integral of "circles lost" over the sphere (the support of this integral is the shared boundary of the disks) should be 2 as well.

This is the Gauss-Bonnet theorem.

That's a good response! I had to spend some time to work out what goes wrong here. But I figured it out.

Parallel transport is broken in your model of the sphere. Take this example. Take a vector pointing north in circle 1, and send through the north portal. It should be pointing south when it gets to the other circle. Fine -- north in circle 1 corresponds to south in circle 2.

Now send the north-pointing vector east instead. It is going to point north in circle 2.

So the north vector changes direction depending on which portal it goes through. So parallel transport changes directions. Hence your model of the sphere is not flat.

You're right that there are no (smooth) flat embeddings of a torus into 3-space.

To understand how a torus can be flat, it's best to replace the idea of folding with the idea of placing portals on edges. Start with a square and put portals between the north and south edges and between the left and right edges. Intuitively this is flat, and this intuition does indeed capture the mathematical notion that a torus is flat.