What is it like to understand advanced mathematics?

-3 morganism 14 August 2016 11:14PM

a Quora answer that is pretty interesting.

"But the quantitative and logical techniques you sharpen as a mathematician allow you to take many shortcuts that make learning other fields easier"

https://www.quora.com/What-is-it-like-to-understand-advanced-mathematics/answers/873950?srid=p6KQ&share=1

and another

Intellectuals Are Freaks

the pool from which scholars, writers, and policy experts is drawn is already a small one. It is even more exclusive in practice, because the children of the rich and affluent are over-represented among those who go to college.

http://www.newgeography.com/content/005352-intellectuals-are-freaks/

 

he-he

http://www.joelkotkin.com/content/001258-today%E2%80%99s-tech-oligarchs-are-worse-robber-barons

 

 

Making Less Wrong Great Again

-16 PeerGynt 01 June 2016 04:34AM

Trump

 

 

 

 

Please post other Making Less Wrong Great Again memes in the comments

 

It's a fact: male and female brains are different

3 araneae 07 October 2010 08:15PM

In Which I Present The Opposing Side's Hypothesis and Falsify It

This post is in part in response to a New Scientist article/book review "Fighting back against neurosexism."  And the tagline is "Are differences between men and women hard-wired in the brain? Two new books argue that there's no solid scientific evidence for this popular notion."  

Full disclosure here: I haven't read the books, although I do have a B.S. in neurobiology. But you don't even need to understand anything about neurobiology to falslify their most basic hypothesis: that male and female brains have no hardwired behavioral differences.  

And it's easy to falsify: if male and female brains were the same, all humans would be completely bisexual.  If it's true that female brains, on average, prefer to fuck, date, and marry men, and male brains, on average, prefer to fuck, date, and marry women, then male and female brains are in fact different.

continue reading »

Beyond Statistics 101

19 JonahSinick 26 June 2015 10:24AM

Is statistics beyond introductory statistics important for general reasoning?

Ideas such as regression to the mean, that correlation does not imply causation and base rate fallacy are very important for reasoning about the world in general. One gets these from a deep understanding of statistics 101, and the basics of the Bayesian statistical paradigm. Up until one year ago, I was under the impression that more advanced statistics is technical elaboration that doesn't offer major additional insights  into thinking about the world in general.

Nothing could be further from the truth: ideas from advanced statistics are essential for reasoning about the world, even on a day-to-day level. In hindsight my prior belief seems very naive – as far as I can tell, my only reason for holding it is that I hadn't heard anyone say otherwise. But I hadn't actually looked advanced statistics to see whether or not my impression was justified :D.

Since then, I've learned some advanced statistics and machine learning, and the ideas that I've learned have radically altered my worldview. The "official" prerequisites for this material are calculus, differential multivariable calculus, and linear algebra. But one doesn't actually need to have detailed knowledge of these to understand ideas from advanced statistics well enough to benefit from them. The problem is pedagogical: I need to figure out how how to communicate them in an accessible way.

Advanced statistics enables one to reach nonobvious conclusions

To give a bird's eye view of the perspective that I've arrived at, in practice, the ideas from "basic" statistics are generally useful primarily for disproving hypotheses. This pushes in the direction of a state of radical agnosticism: the idea that one can't really know anything for sure about lots of important questions. More advanced statistics enables one to become justifiably confident in nonobvious conclusions, often even in the absence of formal evidence coming from the standard scientific practice.

IQ research and PCA as a case study

In the early 20th century, the psychologist and statistician Charles Spearman discovered the the g-factor, which is what IQ tests are designed to measure. The g-factor is one of the most powerful constructs that's come out of psychology research. There are many factors that played a role in enabling Bill Gates ability to save perhaps millions of lives, but one of the most salient factors is his IQ being in the top ~1% of his class at Harvard. IQ research helped the Gates Foundation to recognize iodine supplementation as a nutritional intervention that would improve socioeconomic prospects for children in the developing world.

The work of Spearman and his successors on IQ constitute one of the pinnacles of achievement in the social sciences. But while Spearman's discovery of IQ was a great discovery, it wasn't his greatest discovery. His greatest discovery was a discovery about how to do social science research. He pioneered the use of factor analysis, a close relative of principal component analysis (PCA).

The philosophy of dimensionality reduction

PCA is a dimensionality reduction method. Real world data often has the surprising property of "dimensionality reduction":  a small number of latent variables explain a large fraction of the variance in data.

This is related to the effectiveness of Occam's razor: it turns out to be possible to describe a surprisingly large amount of what we see around us in terms of a small number of variables. Only, the variables that explain a lot usually aren't the variables that are immediately visibleinstead they're hidden from us, and in order to model reality, we need to discover them, which is the function that PCA serves. The small number of variables that drive a large fraction of variance in data can be thought of as a sort of "backbone" of the data. That enables one to understand the data at a "macro /  big picture / structural" level.

This is a very long story that will take a long time to flesh out, and doing so is one of my main goals. 

Autism, or early isolation?

17 JonahSinick 17 June 2015 08:52AM

I've often heard LWers describe themselves as having autism, or Asperger's Syndrome (which is no longer considered a valid construct, and was removed from the Diagnostic and Statistical Manual of Mental Disorders two years ago.) This is given as an explanation for various forms of social dysfunction. The suggestion is that such people have a genetic disorder.

I've come to think that the issues are seldom genetic in origin. There's a simpler explanation. LWers are often intellectually gifted. This is conducive to early isolation. In The Outsiders Grady Towers writes:

The single greatest adjustment problem faced by the intellectually gifted, however, is their tendency to become isolated from the rest of humanity. Hollingworth points out that the exceptionally gifted do not deliberately choose isolation, but are forced into it against their wills. These children are not unfriendly or ungregarious by nature. Typically they strive to play with others but their efforts are defeated by the difficulties of the case... Other children do not share their interests, their vocabulary, or their desire to organize activities. [...] Forms of solitary play develop, and these, becoming fixed as habits, may explain the fact that many highly intellectual adults are shy, ungregarious, and unmindful of human relationships, or even misanthropic and uncomfortable in ordinary social intercourse.

Most people pick up a huge amount of tacit social knowledge as children and adolescents, through very frequent interaction with many peers. This is often not true of intellectually gifted people, who usually grew up in relative isolation on account of lack of peers who shared their interests.

They often have the chance to meet others similar to themselves later on in life. One might think that this would resolve the issue. But in many cases intellectually gifted people simply never learn how beneficial it can be to interact with others. For example, the great mathematician Robert Langlands wrote:

Bochner pointed out my existence to Selberg and he invited me over to speak with him at the Institute. I have known Selberg for more than 40 years. We are on cordial terms and our offices have been essentially adjacent for more than 20 years.This is nevertheless the only mathematical conversation I ever had with him. It was a revelation.

At first blush, this seems very strange: much of Langlands' work involves generalizations of Selberg's trace formula. It seems obvious that it would be fruitful for Langlands to have spoken with Selberg about math more than once, especially given that the one conversation that he had was very fruitful! But if one thinks about what their early life experiences must have been like, as a couple of the most brilliant people in the world, it sort of makes sense: they plausibly had essentially nobody to talk to about their interests for many years, and if you go for many years without having substantive conversations with people, you might never get into the habit.

When intellectually gifted people do interact, one often sees cultural clashes, because such people created their own cultures as a substitute for usual cultural acclimation, and share no common background culture. From the inside, one sees other intellectually gifted people, recognizes that they're very odd by mainstream standards, and thinks "these people are freaks!" But at the same time, the people who one sees as freaks see one in the same light, and one is often blind to how unusual one's own behavior is, only in different ways. Thus, one gets trainwreck scenarios, as when I inadvertently offended dozens of people when I made strong criticisms of MIRI and Eliezer back in 2010, just after I joined the LW community.

Grady Towers concludes the essay by writing:

The tragedy is that none of the super high IQ societies created thus far have been able to meet those needs, and the reason for this is simple. None of these groups is willing to acknowledge or come to terms with the fact that much of their membership belong to the psychological walking wounded. This alone is enough to explain the constant schisms that develop, the frequent vendettas, and the mediocre level of their publications. But those are not immutable facts; they can be changed. And the first step in doing so is to see ourselves as we are.

The value of learning mathematical proof

3 JonahSinick 02 June 2015 03:15AM

The social justice movement espouses the notion that people who are privileged are often unfairly judgmental of those who were less privileged. Until recently, what they said didn't resonate with me. I knew that I had major advantages out of virtue of having been born a white, middle class male. But I recently realized that there were other privileges that I hadn't acknowledged as having benefited enormously from.  In particular, I had the unusual experience of growing up with a very intellectually curious father, which gave me a huge head start in intellectual development.

I used to get annoyed when LWers misread my posts in ways that they wouldn't have if they had been reading more carefully. I conceptualized such commenters as being undisciplined, and being unwilling to do the work necessary to maintain high epistemic standards. I now see that my reading was in many cases uncharitable, analogous to many of my teachers having misread my learning disability as reflecting laziness. Many of my readers have probably never had the opportunity to learn how to read really carefully.

How did I myself learn? I don't remember in detail, but the one factor that seems most significant is my study of the mathematical subject of real analysis. A number of strongest thinkers who I know characterized the experience as a turning point in their development as well. It's the subject where one goes through rigorous proofs of the theorems of calculus. 

Consider the extreme value theorem

If a real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain a maximum and a minimum, each at least once. 

The theorem may seem obvious, but almost no undergraduate math majors would be able to come up with a logically impeccable proof from scratch. This ties in with why I almost never try to present rigorous arguments. If it's not clear to you that it might be very difficult to construct a rigorous proof of the extreme value theorem, you'd probably benefit intellectually from more exposure to mathematical proof. The experience of seeing how difficult it can be to offer rigorous proofs of even relatively simple statements trains one to read very carefully, and not make any unwarranted assumptions.

If you've studied calculus, haven't yet had the experience of proving theorems from first principles beyond high school geometry, and would are interested, I would recommend:

The last of these books is great for developing a sense for how superficially plausible statements are often false. 

Social class amongst the intellectually gifted

6 JonahSinick 02 June 2015 11:02PM

Something that I've come to realize is that as a practical matter, intellectually gifted people who haven't developed very strong ability in a quantitative subject tend to be at a major disadvantage relative to those who have. The quantitative subjects that I have in mind as "quantitative subjects" are primarily math, physics, theoretical computer science and statistics, though others such as electrical engineering may qualify. [1]

This point is usually masked over by the fact that people who don't have very strong technical ability are often reasonable functional by the standards of mainstream society, and don't realize how far they're falling short of their genetic potential. They tend to have jobs that don't fully use their strengths, and experience cognitive dissonance around being aware on some level of far they are from utilizing their core competencies. 

Consider the following:

  • The Google co-founders met as computer science graduate students at Stanford. Sergei Brin double majored in math and physics and was an NSF graduate fellow. He comes from a mathematical family: his father was a math professor at University of Maryland. 
  • Bill Gates took Math 55 as a freshman at Harvard, which is the class designed for the most mathematically talented students at Harvard. During his sophomore year he did research which he later published a paper on with well known theoretical computer science professor Christos Papadimitriou.
  • James Simons comes close to being the only elite mathematician to leave academia for the business world. He founded the hedge fund Renaissance Technologies and made ~$12.5 billion.
  • Charles Munger, the Vice-Chairman of Berkshire Hathaway (net worth ~$1.3 billion) often quotes the maxim of the 19th century mathematician Carl Gustav Jacob Jacobi Invert, Always Invert, and characterizes him using that concept to solve difficult business problems 

I can't give a brief justification for this, but I have good reason to believe that the ~10000x+ differential in net worth comes in large part  from the people having had unusually good opportunities to conducive to becoming very technically proficient, that resulted in them developing transferable reasoning abilities and having had an intellectually elite peer group to learn from.

I know a fair number of brilliant people who didn't have such advantages. The situation actually seems to me like one in which amongst intellectually gifted people, there's an "upperclass" of people who had opportunities to develop very strong technical ability and an "underclass" of people who who could have developed them under more favorable environmental circumstances, but haven't. Many intellectually gifted people who didn't have the chance to develop the abilities mistakenly believe that they lack the innate ability to do so. And people who did have the opportunities to develop them often look down on those who didn't, unaware of how much of their own relative success is due to having had environmental advantages earlier in their lives.


[1] James Miller points out that graduates of elite law schools may have analogous advantages – that's a population that I haven't had exposure to. 

Gettier walks into a bar, um, barrista

-2 HalMorris 30 April 2015 09:26PM

Gettier walks up to the counter. Before he can order, the Barrista confuses him for a regular and chirps “I know what you want.” By coincidence, Gettier ends up with exactly the drink he desired. (from Alvin Goldman, Epistemologist Extraordinaire)

Harry Yudkowsky and the Methods of Postrationality: Chapter One: Em Dashes Colons and Ellipses, Littérateurs Go Wild

-7 Will_Newsome 06 July 2014 09:34AM

 

"If you give George Lukács any taste at all, immediately become the Deathstar." — Old Klingon Proverb

 

There was no nice way to put it: Harry James Potter-Yudkowsky was half Potter, half Yudkowsky. Harry just didn’t fit in. It wasn't that he lacked humanity. It was just that no one else knew (P)Many_Worlds, (P)singularity, or (P)their_special_insight_into_the_true_beautiful_Bayesian_fractally_recursive_nature_of_reality. Other people were rolesand how shall an actor, an agent, relate to those who are merely what they are, merely their roles? Merely their roles, without pretext or irony? How shall the PC fuck with the NPCs? Harry James Potter-Yudkowsky oft asked himself this question, but his 11-year-old mind lacked the g to grasp the answer. For if you are to draw any moral from this tale, godforsaken readers, the moral you must draw is this: P!=NP.

 

One night Harry Potter-Yudkowsky was outside, pretending to be Keats, staring at the stars and the incomprehensibly vast distances between them, pondering his own infinite significance in the face of such an overwhelming sea of stupidity, when an owl dropped a letter directly on his head, winking slyly. “You’re a wizard,” said the letter, while the owl watched, increasingly gloatingly, “and we strongly suggest you attend our school, which goes by the name Hogwarts. 'Because we’re sexy and you know it.’”

 

Harry pondered this for five seconds. “Curse the stars!, literally curse them!, Abra Kadabra!, for I must admit what I always knew in my heart to be true,” lamented Harry. “This is fanfic.”

 

“Meh.”

 

And so, as they'd been furiously engaged in for months, the divers models of Harry Potter-Yudkowsky gathered dust. In layman’s terms...

 

Harry didn’t update at all.

 

Harry: 1

Author:  0

 

 

(To be fair, the author was drunk.)

 

Next chapter: "Analyzing the Fuck out of an Owl"

...

Criticism appreciated.