The recent book Mathematica by David Bessis attempts to describe this exact topic. One core theme of the book is that math is done by building mental models and using our intuitions that come from. Formalism is then used to shape and refine these models. By spending time developing these models, insights eventually become obvious. These mental models are the essence of mathematics, not theorems. The author explicitly uses neural networks as an example to describe how this negative feedback could work to make these changes to our mental models happen in our brains.
The core premise of the book is to describe how mathematicians work and think and to show that this is a process that everyone can do (although some will be better at it than others). It includes interesting accounts of Grothendieck, Bill Thurston, and Descartes as well as from the author's own research career at Yale and École normale supérieure. The book is targeted at the general reader and at times reads a little like a self-help book, especially in the first third or so. However, I found it to be an enjoyable and fascinating read. It provoked a lot of interesting questions about the nature of learning and provided a framework to begin to answer them (e.g. "How can I have proved something and yet feel no understanding of it?", "How can some people solve problems orders of magnitude faster than other smart people, as if they don't even have to think about it?", "Why do I sometimes watch a presentation on a new topic, follow every step, and come away feeling like I've learned nothing?"(* see except below)). I don't think I'm doing it justice here, so I'll stop by saying I highly recommend it based on your comment.
_______
* I'll use this as an excuse to provide a related excerpt featuring Fields Medalist and Abel Prize winner Jean-Pierre Serre:
One day, I had to give a lecture at the Chevalley Seminar, a group theory seminar in Paris. I didn't have substantial new results to announce, but it was an opportunity to make a presentation even simpler than usual. [...] A couple of minutes before the talk was to start, Serre came in and sat in the second row. I was honored to have him in the audience, but I let him know right off that the presentation might not be very interesting to him. It was intended for a general audience and I was going to be explaining very basic things.
What I didn't tell him, of course, was that his presence was intimidating. Still, I didn't want to raise the level of my talk only to keep him interested. I just kept an eye out to see if he'd taken off his glasses, which would mean he was getting bored and had stopped listening. No worries there—he kept his glasses on till the end.
I gave my presentation as I would have without him there, speaking to the entire audience, especially the students seated in the back, whom I was pleased to see listening and looking like they understood.
It was a normal presentation, fairly successful, not very deep but well prepared, clear, and intelligible. At the end of the seminar, Serre came up to me and said—and here I quote verbatim: "You'll have to explain that to me again, because I didn't understand anything."
That's a true story, and it plunged me into a state of profound perplexity.
Apparently, Serre wasn't using the verb to understand the way most people use it. The concepts and reasonings of my talk couldn't really have caused him any difficulty. I'm sure he wanted to say that he understood what I had explained, but he hadn't understood why what I had explained was true.
There are two levels of understanding. The first level consists of following the reasoning step by step and accepting that it's correct. Accepting is not the same as understanding. The second level is real understanding. It requires seeing where the reasoning comes from and why it's natural.
In thinking again about Serre's comment, I realized that my presentation had too many “miracles,” too many arbitrary choices, too many things that worked without my really knowing why. Serre was right; it was incomprehensible. His feedback helped me become aware of a number of very big holes in my understanding of the objects and situations I was working on at the time. In the years that followed, research into explanations for these various miracles allowed me to fill in some of the holes and achieve some of the most important results of my career. (However, some of the miracles remain unexplained to this day.)
But the most troubling aspect was the abruptness, the frankness with which Serre had overplayed his own incomprehension.
My intuition is that some people are able to develop abstractions on top of abstractions (compression is a key part of intelligence) that allow them to traverse the search space much, much faster than without these, and with enough repetition this becomes routine, even faster in the brain.
I don't think we have a good theory of how this works, or at least I haven't come across it.
Neural network guided search is somewhat similar but I think we are missing several key pieces
Adult Swim's 404 page always has a dark, sometimes surreal, shaggy-dog story that ends with a reference to the page being 404.[1]
The page seems to return the same story each time you access it (at least on the same day). I'm not sure when they change from one story to another. The author has posted some of the other stories on other sites.[2][3][4][5] I still vividly recall reading this one in particular (although this reproduction is missing the bolding of the text in the second to last paragraph).[6]
To provide a bit of context, this article is about Timothy Burke. His most famous work is the compilation video of news anchors on Sinclair-owned local news stations reading the same script about how media bias is "extremely dangerous to our democracy.”[1] He is also known for breaking the infamous Manti Te'o fake girlfriend story.[2] Additionally, he is noted for having a complicated computing setup to monitor, archive, and disseminate videos from dozens of live sources simultaneously. Some of these details appeared in a profile by the New York Times [3] and also appear in this article.
Burke often shares short clips of offbeat moments from sports and news channels. The article states that he has recently shared behind-the-scenes clips of Tucker Carlson on his former Fox show; it appears likely that this raid was related to determining the origin of those clips.
The amazing thing is even the writers didn’t know that episode was going to end that way until the night before they shot the ending. Here’s Larry David and Jerry Seinfeld talking about it: https://youtu.be/uZPSO4yte8k
The prefix notation may be throwing you off or perhaps you have mistaken the function `bottles` for the string "bottles".
The code `(bottles i)` in the loop is calling the function `bottles` (defined at the top) with the argument `i`. The value of `(bottles 99)` is `(string 99 " bottles")` which evaluates to "99 bottles", as expected.
This image is even more worthless than it seems. The post on Wikimedia is an original work. Its description states: "As the percentage of graduates increases the minimum IQ to include at least that percentage of graduates inherently decreases. Since 2000 the intelligence required to be a college graduate has been less than the intelligence required to graduate from high school in 1940, based on a standard distribution."
It seems the author took the the percentage of the population that graduated high school/college each year and then found the corresponding percentile on an IQ bell curve and used those as the y-values. This methodology only makes sense if you assume that high school/college graduates are exactly the highest IQ population and that everyone who does not graduate isn't intelligent enough to do so. This chart also almost certainly doesn't normalize IQ over time, even though IQ is constantly redefined so that 100 is average while raw intelligence scores have increased over time [1].
What this chart actually shows is the highest possible IQ of the graduate with the lowest IQ in a given year, a statistic that seems to have dubious value.
You’re right it is worse. It looks like they assumed college-going students would approximate the standard distribution of scores for the population as a whole, which… no. An awful assumption for an activity where academic ability is a primary gatekeeper at the same time that they’re attempting to apply a metric of IQ essentially as a proxy for academic ability/knowledge/whatever. (“Whatever” because the entire concept of intelligence is filled with varying definitions)
I agree; this story is apocryphal. I was annoyed enough that I did some research into the story. Via this Atlantic article[1], I found the original Valleywag post[2] containing the letter from the employee. It's worth noting that the letter from the employee is pure text (i.e. not a scan of a letter), is only 170 words long, and has a couple of misspellings. The Valleywag post provides a paraphrasing of what the employee told them, followed by what seems to be the entire body of the email that they received.
The Musk detective work is described by Vance in a footnote:
“Musk would later discover the identity of this employee in an ingenious way. He copied the text of the letter into a Word document, checked the size of the file, sent it to a printer, and looked over the logs of printer activity to find one of the same size. He could then trace that back to the person who had printed the original file. The employee wrote a letter of apology and resigned.”
This leaves me a with more than a few questions:
1. Why would this email/letter have been printed out? It's short, informal, and did not seem to contain any corporate materials that would require access from a work computer. Surely, this would be better sent as an email from a personal computer? All I can think of is that the employee did print some kind of private company information (perhaps as proof?) to send to Valleywag. But wouldn't this mean that the print-job sizes wouldn't match since there would be printed materials not made public? It seems beyond insane to physically mail a tip to a gossip site that was built around emailed tips.
2. If this was sent as a physical letter, why would the quote in the article contain the typos? Why would they even take the time to type up the entire letter when the article summarizes every single point that the Tesla employee mentioned? Shouldn't they have taken some care to not verbatim reproduce text that could have gotten their source into trouble? I will say that the minimal journalistic standards employed by former Gawker-network sites provide convenient explanations to these questions, so these aren't particularly damning.
3. Is this really all the text that was sent to Valleywag? The quoted part of the letter provides no salutation or signature. Sending this text exactly as quoted as a physical letter seems bizarre, even for an anonymous tip.
4. Would the sizes of the files sent to the printer even match up considering the document metadata? This actually seems somewhat plausible.
5. Would the print-job really be the best way to figure out who the leaker was? In October 2008, before the letter was written, Tesla only had 363 employees[3] and may have laid off a few dozen of them before this letter was written. This employee claims to have joined in 2004. A Wired article from 2006[4] mentions a meeting of 30 Tesla employees and board members in December 2004. It seems like there are a very small number of people who could have been the potentially leaker. How many of those people were using the corporate printers the day after the mentioned all-hands meeting to print a single page document?
--------
I imagine that this story was told to present Musk as smarter than everyone else while also threatening disloyalty, which seems to be a frequent Musk bugbear. The bit about the caught employee writing a letter of apology and then resigning (amidst large-scale layoffs at Tesla in 2008!) also seems a little too cute, in a chain-email-atheist-professor-humiliated-by-freshman-Albert-Einstein kind of way.
This discussion should link to the Reuters article[1] it quotes from rather than a site whose Google search meta description reads “Trump News, and Breaking News Updated 24/7.”
The core premise of the book is to describe how mathematicians work and think and to show that this is a process that everyone can do (although some will be better at it than others). It includes interesting accounts of Grothendieck, Bill Thurston, and Descartes as well as from the author's own research career at Yale and École normale supérieure. The book is targeted at the general reader and at times reads a little like a self-help book, especially in the first third or so. However, I found it to be an enjoyable and fascinating read. It provoked a lot of interesting questions about the nature of learning and provided a framework to begin to answer them (e.g. "How can I have proved something and yet feel no understanding of it?", "How can some people solve problems orders of magnitude faster than other smart people, as if they don't even have to think about it?", "Why do I sometimes watch a presentation on a new topic, follow every step, and come away feeling like I've learned nothing?"(* see except below)). I don't think I'm doing it justice here, so I'll stop by saying I highly recommend it based on your comment.
_______
* I'll use this as an excuse to provide a related excerpt featuring Fields Medalist and Abel Prize winner Jean-Pierre Serre:
One day, I had to give a lecture at the Chevalley Seminar, a group theory seminar in Paris. I didn't have substantial new results to announce, but it was an opportunity to make a presentation even simpler than usual. [...] A couple of minutes before the talk was to start, Serre came in and sat in the second row. I was honored to have him in the audience, but I let him know right off that the presentation might not be very interesting to him. It was intended for a general audience and I was going to be explaining very basic things.
What I didn't tell him, of course, was that his presence was intimidating. Still, I didn't want to raise the level of my talk only to keep him interested. I just kept an eye out to see if he'd taken off his glasses, which would mean he was getting bored and had stopped listening. No worries there—he kept his glasses on till the end.
I gave my presentation as I would have without him there, speaking to the entire audience, especially the students seated in the back, whom I was pleased to see listening and looking like they understood. It was a normal presentation, fairly successful, not very deep but well prepared, clear, and intelligible. At the end of the seminar, Serre came up to me and said—and here I quote verbatim: "You'll have to explain that to me again, because I didn't understand anything."
That's a true story, and it plunged me into a state of profound perplexity.
Apparently, Serre wasn't using the verb to understand the way most people use it. The concepts and reasonings of my talk couldn't really have caused him any difficulty. I'm sure he wanted to say that he understood what I had explained, but he hadn't understood why what I had explained was true.
There are two levels of understanding. The first level consists of following the reasoning step by step and accepting that it's correct. Accepting is not the same as understanding. The second level is real understanding. It requires seeing where the reasoning comes from and why it's natural.
In thinking again about Serre's comment, I realized that my presentation had too many “miracles,” too many arbitrary choices, too many things that worked without my really knowing why. Serre was right; it was incomprehensible. His feedback helped me become aware of a number of very big holes in my understanding of the objects and situations I was working on at the time. In the years that followed, research into explanations for these various miracles allowed me to fill in some of the holes and achieve some of the most important results of my career. (However, some of the miracles remain unexplained to this day.)
But the most troubling aspect was the abruptness, the frankness with which Serre had overplayed his own incomprehension.