As an exercise on programming Android, I've made an app to log predictions you make and keep score of your results. Like PredictionBook, but taking more of a personal daily exercise feel, in line with this post.

The "statistics" right now are only a score I copied from the old Credence calibration game, and a calibration bar chart.

I'm hoping for suggestionss for features and criticism on the app design.

Here's the link for the apk (v0.4), and here's the source code repository. You can download it at Google Play Store.

Pending/Possible/Requested Features:

• Set check-in dates for predictions
• Tags (and stats by tag)
• Stats by timeframe
• Beeminder integration
• Trivia questions you can answer if you don't have any personal prediction to make
• Ring pie chart to choose probability

Edit:

2015-08-26 - Fixed bug that broke on Android 5.0.2 (thanks Bobertron)

2015-08-28 - Change layout for landscape mode, and add a better icon

2015-08-31 -

• Daily notifications
• Buttons at the expanded-item-layout (ht dutchie)
• Show points won/lost in the snackbar when a prediction is answered
• Translation to portuguese

The excellent Scott Aaronson has posted on his blog a version of a talk he recently gave at SPARC, about Aumann's agreement theorem and related topics. I think a substantial fraction of LW readers would enjoy it. As well as stating Aumann's theorem and explaining why it's true, the article discusses other instances where the idea of "common knowledge" (the assumption that does a lot of the work in the AAT) is important, and offers some interesting thoughts on the practical applicability (if any) of the AAT.

(Possibly relevant: an earlier LW discussion of AAT.)

Here's a 32-minute presentation I made to provide an introduction to some of the core LessWrong concepts for a general audience:

You Are A Brain [YouTube]

You Are a Brain [Google Slides] - public domain

I already posted this here in 2009 and some commenters asked for a video, so I immediately recorded one six years later. This time the audience isn't teens from my former youth group, it's employees who work at my software company where we have a seminar series on Thursday afternoons.

I'm David. I'm reading through the books in the MIRI research guide and will write a review for each as I finish them. By way of inspiration from how Nate did it.

Naive Set Theory

Halmos Naive Set Theory is a classic and dense little book on axiomatic set theory, from a "naive" perspective.

Which is to say, the book won't dig to the depths of formality or philosophy, it focuses on getting you productive with set theory. The point is to give someone who wants to dig into advanced mathematics a foundation in set theory, as set theory is a fundamental tool used in a lot of mathematics.

Summary

Is it a good book? Yes.

Would I recommend it as a starting point, if you would like to learn set theory? No. The book has a terse presentation which makes it tough to digest if you aren't already familiar with propositional logic, perhaps set theory to some extent already and a bit of advanced mathematics in general. There are plenty of other books that can get you started there.

If you do have a somewhat fitting background, I think this should be a very competent pick to deepen your understanding of set theory. The author shows you the nuts and bolts of set theory and doesn't waste any time doing it.

Perspective of this review

I will first refer you to Nate's review, which I found to be a lucid take on it. I don't want to be redundant and repeat the good points made there, so I want to focus this review on the perspective of someone with a bit weaker background in math, and try to give some help to prospective readers with parts I found tricky in the book.

What is my perspective? While I've always had a knack for math, I only read about 2 months of mathematics at introductory university level, and not including discrete mathematics. I do have a thorough background in software development.

Set theory has eluded me. I've only picked up fragments. It's seemed very fundamental but school never gave me a good opportunity to learn it. I've wanted to understand it, which made it a joy to add Naive Set Theory to the top of my reading list.

How I read Naive Set Theory

Starting on Naive Set Theory, I quickly realized I wanted more meat to the explanations. What is this concept used for? How does it fit in to the larger subject of mathematics? What the heck is the author expressing here?

I supplemented heavily with wikipedia, math.stackexchange and other websites. Sometimes, I read other sources even before reading the chapter in the book. At two points, I laid down the book in order to finish two other books. The first was Gödel's Proof, which handed me some friendly examples of propositional logic. I had started reading it on the side when I realized it was contextually useful. The second was Concepts of Modern Mathematics, which gave me much of the larger mathematical context that Naive Set Theory didn't.

Consequently, while reading Naive Set Theory, I spent at least as much time reading other sources!

A bit into the book, I started struggling with the exercises. It simply felt like I hadn't been given all the tools to attempt the task. So, I concluded I needed a better introduction to mathematical proofs, ordered some books on the subject, and postponed investing into the exercises in Naive Set Theory until I had gotten that introduction.

Chapters

In general, if the book doesn't offer you enough explanation on a subject, search the Internet. Wikipedia has numerous competent articles, math.stackexchange is overflowing with content and there's plenty additional sources available on the net. If you get stuck, do try playing around with examples of sets on paper or in a text file. That's universal advice for math.

I'll follow with some key points and some highlights of things that tripped me up while reading the book.

Axiom of extension

The axiom of extension tells us how to distinguish between sets: Sets are the same if they contain the same elements. Different if they do not.

Axiom of specification

The axiom of specification allows you to create subsets by using conditions. This is pretty much what is done every time set builder notation is employed.

Puzzled by the bit about Russell's paradox at the end of the chapter? http://math.stackexchange.com/questions/651637/russells-paradox-in-naive-set-theory-by-paul-halmos

Unordered pairs

The axiom of pairs allows one to create a new set that contains the two original sets.

Unions and intersections

The axiom of unions allows one to create a new set that contains all the members of the original sets.

Complements and powers

The axiom of powers allows one to, out of one set, create a set containing all the different possible subsets of the original set.

Getting tripped up about the "for some" and "for every" notation used by Halmos? Welcome to the club:
http://math.stackexchange.com/questions/887363/axiom-of-unions-and-its-use-of-the-existential-quantifier
http://math.stackexchange.com/questions/1368073/order-of-evaluation-in-conditions-in-set-theory

Using natural language rather than logical notation is commmon practice in mathematical textbooks. You'd better get used to it:
http://math.stackexchange.com/questions/1368531/why-there-is-no-sign-of-logic-symbols-in-mathematical-texts

The existential quantifiers tripped me up a bit before I absorbed it. In math, you can freely express something like "Out of all possible x ever, give me the set of x that fulfill this condition". In programming languages, you tend to have to be much more... specific, in your statements.

Ordered pairs

Cartesian products are used to represent plenty of mathematical concepts, notably coordinate systems.

Relations

Equivalence relations and equivalence classes are important concepts in mathematics.

Functions

Halmos is using some dated terminology and is in my eyes a bit inconsistent here. In modern usage, we have: injective, surjective, bijective and functions that are none of these. Bijective is the combination of being both injective and surjective. Replace Halmos' "onto" with surjective, "one-to-one" with injective, and "one-to-one correspondence" with bijective.

He also confused me with his explanation of "characteristic function" - you might want to check another source there.

Families

This chapter tripped me up heavily because Halmos mixed in three things at the same time on page 36: 1. A confusing way of talking about sets. 2. Convoluted proof. 3. n-ary cartesian product.

Families are an alternative way of talking about sets. An indexed family is a set, with an index and a function in the background. A family of sets means a collection of sets, with an index and a function in the background. For Halmos build-up to n-ary cartesian products, the deal seems to be that he teases out order without explicitly using ordered pairs. Golf clap. Try this one for the math.se treatment: http://math.stackexchange.com/questions/312098/cartesian-products-and-families

Inverses and composites

The inverses Halmos defines here are more general than the inverse functions described on wikipedia. Halmos' inverses work even when the functions are not bijective.

Numbers

The axiom of infinity states that there is a set of the natural numbers.

The Peano axioms

The peano axioms can be modeled on the the set-theoretic axioms. The recursion theorem guarantees that recursive functions exist.

Arithmetic

The principle of mathematical induction is put to heavy use in order to define arithmetic.

Order

Partial orders, total orders, well orders -- are powerful mathematical concepts and are used extensively.

Some help on the way:
http://math.stackexchange.com/questions/1047409/sole-minimal-element-why-not-also-the-minimum
http://math.stackexchange.com/questions/367583/example-of-partial-order-thats-not-a-total-order-and-why
http://math.stackexchange.com/questions/225808/is-my-understanding-of-antisymmetric-and-symmetric-relations-correct
http://math.stackexchange.com/questions/160451/difference-between-supremum-and-maximum

Also, keep in mind that infinite sets like subsets of w can muck up expectations about order. For example, a totally ordered set can have multiple elements without a predecessor.

Axiom of choice

The axiom of choice lets you, from any collection of non-empty sets, select an element from every set in the collection. The axiom is necessary to do these kind of "choices" with infinite sets. In finite cases, one can construct functions for the job using the other axioms. Though, the axiom of choice often makes the job easier in finite cases so it is used where it isn't necessary.

Zorn's lemma

Zorn's lemma is used in similar ways to the axiom of choice - making infinite many choices at once - which perhaps is not very strange considering ZL and AC have been proven to be equivalent.

robot-dreams offers some help in following the massive proof in the book.

Well ordering

A well-ordered set is a totally ordered set with the extra condition that every non-empty subset of it has a smallest element. This extra condition is useful when working with infinite sets.

The principle of transfinite induction means that if the presence of all strict predecessors of an element always implies the presence of the element itself, then the set must contain everything. Why does this matter? It means you can make conclusions about infinite sets beyond w, where mathematical induction isn't sufficient.

Transfinite recursion

Transfinite recursion is an analogue to the ordinary recursion theorem, in a similar way that transfinite induction is an analogue to mathematical induction - recursive functions for infinite sets beyond w.

In modern lingo, what Halmos calls a "similarity" is an "order isomorphism".

Ordinal numbers

The axiom of substitution is called the axiom (schema) of replacement in modern use. It's used for extending counting beyond w.

Sets of ordinal numbers

The counting theorem states that each well ordered set is order isomorphic to a unique ordinal number.

Ordinal arithmetic

The misbehavior of commutativity in arithmetic with ordinals tells us a natural fact about ordinals: if you tack on an element in the beginning, the result will be order isomorphic to what it is without that element. If you tack on an element at the end, the set now has a last element and is thus not order isomorphic to what you started with.

The Schröder-Bernstein theorem

The Schröder-Bernstein theorem states that if X dominates Y, and Y dominates X, then X ~ Y (X and Y are equivalent).

Countable sets

Cantor's theorem states that every set always has a smaller cardinal number than the cardinal number of its power set.

Cardinal arithmetic

Read this chapter after Cardinal numbers.

Cardinal arithmetic is an arithmetic where just about all the standard operators do nothing (beyond the finite cases).

Cardinal numbers

Read this chapter before Cardinal arithmetic.

The continuum hypothesis asserts that there is no cardinal number between that of the natural numbers and that of the reals. The generalized continuum hypothesis asserts that, for all cardinal numbers including aleph-0 and beyond aleph-0, the next cardinal number in the sequence is the power set of the previous one.

Concluding reflections

I am at the same time humbled by the subject and empowered by what I've learned in this episode. Mathematics is a truly vast and deep field. To build a solid foundation in proofs, I will now go through one or two books about mathematical proofs. I may return to Naive Set Theory after that. If anyone is interested, I could post my impressions of other mathematical books I read.

I think Naive Set Theory wasn't the optimal book for me at the stage I was. And I think Naive Set Theory probably should be replaced by another introductory book on set theory in the MIRI research guide. But that's a small complaint on an excellent document.

If you seek to get into a new field, know the prerequisites. Build your knowledge in solid steps. Which I guess, sometimes requires that you do test your limits to find out where you really are.

The next book I start on from the research guide is bound to be Computability and Logic.

I wonder if anyone has thought about setting up an online community dedicated to peer-to-peer tutoring.  The idea is that if I want to learn "Differential Geometry" and know "Python programming", and you want to learn "Python programming" and know "Differential geometry," then we can agree to tutor each other online.  The features of the community would be to support peer-to-peer tutoring by:

• Facilitating matchups between compatible tutors
• Allowing for more than two people to participate in a tutoring arrangement
• Providing reputation-based incentives to honor tutoring agreements and putting effort into tutoring
• Allowing other members to "sit in" on tutoring sessions, if they are made public
• Allowing the option to record tutoring sessions
• Providing members with access to such recorded sessions and "course materials"
• Providing a forum to arrange other events

One fallacy that I see frequently in works of popular science -- and also here on LessWrong -- is the belief that we have strong evidence of the way things were in pre-history, particularly when one is giving evidence that we can explain various aspects of our culture, psychology, or personal experience because we evolved in a certain way. Moreover, it is held implicit that because we have this 'strong evidence', it must be relevant to the topic at hand. While it is true that the environment did effect our evolution and thus the way we are today, evolution and anthropology of pre-historic societies is emphasized to a much greater extent than rational thought would indicate is appropriate. 

As a matter of course, you should remember these points whenever you hear a claim about prehistory:

• Most of what we know (or guess) is based on less data than you would expect, and the publish or perish mentality is alive and well in the field of anthropology.

• Most of the information is limited and technical, which means that anyone writing for a popular audience will have strong motivation to generalize and simplify.

• It has been found time and time again that for any statement that we can make about human culture and behavior that there is (or was) a society somewhere that will serve as a counterexample. 

• Very rarely do anthropologists or members of related fields have finely tuned critical thinking skills or a strong background on the philosophy of science, and are highly motivated to come up with interpretations of results that match their previous theories and expectations. 

Results that you should have reasonable levels of confidence in should be framed in generalities, not absolutes. E.g., "The great majority of human cultures that we have observed have distinct and strong religious traditions", and not "humans evolved to have religion". It may be true that we have areas in our brain that evolved not only 'consistent with holding religion', but actually evolved 'specifically for the purpose of experiencing religion'... but it would be very hard to prove this second statement, and anyone who makes it should be highly suspect. 

Perhaps more importantly, these statements are almost always a red herring. It may make you feel better that humans evolved to be violent, to fit in with the tribe, to eat meat, to be spiritual, to die at the age of thirty.... But rarely do we see these claims in a context where the stated purpose is to make you feel better. Instead they are couched in language indicating that they are making a normative statement -- that this is the way things in some way should be. (This is specifically the argumentum ad antiquitatem or appeal to tradition, and should not be confused with the historical fallacy, but it is certainly a fallacy). 

It is fine to identify, for example, that your fear of flying has a evolutionary basis. However, it is foolish to therefore refuse to fly because it is unnatural, or to undertake gene therapy to correct the fear. Whether or not the explanation is valid, it is not meaningful. 

Obviously, this doesn't mean that we shouldn't study evolution or the effects evolution has on behavior. However, any time you hear someone refer to this information in order to support any argument outside the fields of biology or anthropology, you should look carefully at why they are taking the time to distract you from the practical implications of the matter under discussion. 

I continually train my ten-year-old son’s working memory, and urge parents of other young children to do likewise.  While I have succeeded in at least temporarily improving his working memory, I accept that this change might not be permanent and could end a few months after he stops training.  But I also believe that while his working memory is boosted so too is his learning capacity.    

I have a horrible working memory that greatly hindered my academic achievement.  I was so bad at spelling that they stopped counting it against me in school.  In technical classes I had trouble remembering what variables stood for.  My son, in contrast, has a fantastic memory.  He twice won his school’s spelling bee, and just recently I wrote twenty symbols (letters, numbers, and shapes) in rows of five.  After a few minutes he memorized the symbols and then (without looking) repeated them forward, backwards, forwards, and then by columns.    

My son and I have been learning different programming languages through Codecademy.  While I struggle to remember the required syntax of different languages, he quickly gets this and can focus on higher level understanding.  When we do math learning together his strong working memory also lets him concentrate on higher order issues then remembering the details of the problem and the relevant formulas.     

You can easily train a child’s working memory.  It requires just a few minutes of time a day, can be very low tech or done on a computer, can be optimized for your child to get him in flow, and easily lends itself to a reward system.  Here is some of the training we have done:     

• I write down a sequence and have him repeat it.
• I say a sequence and have him repeat it.
• He repeats the sequence backwards.
• He repeats the sequence with slight changes such as adding one to each number and “subtracting” one from each letter.
• He repeats while doing some task like touching his head every time he says an even number and touching his knee every time he says an odd one.
• Before repeating a memorized sequence he must play repeat after me where I say a random string.
• I draw a picture and have him redraw it.
• He plays N-back games.
• He does mental math requiring keeping track of numbers (i.e. 42 times 37).
• I assign numerical values to letters and ask him math operation questions (i.e. A*B+C).        

The key is to keep changing how you train your kid so you have more hope of improving general working memory rather than the very specific task you are doing.  So, for example, if you say a sequence and have your kid repeat it back to you, vary the speed at which you talk on different days and don’t just use one class of symbols in your exercises.

tl;dr: If you struggle with motivational problems, it's likely that the problem is not intrinsic to you, but instead that you haven't yet found work that you find very interesting.  

How I discovered how to do great work

Last winter I did something that I had never done before. I spent ~1500 hours working on genuinely original scientific research.

I had done research for my PhD in pure math, but faced squarely, the problems that I worked on were of very little interest to anyone outside of the fields, and I was not very engaged with my research. Pure math is very heavily stacked with talent, and the low hanging fruit has been plucked, so unless you're one of the most talented people in the world, your prospects for doing anything other than derivative work are very poor. 

What I did last fall was entirely different. As I trained to be a data scientist, I found that there's far more low hanging fruit in the field than there is in pure math, and found myself working on novel problems that are of broad interest almost immediately.

Having very high intrinsic motivation made a huge difference. I found myself spending all waking hours (~90 hours / week) working on it obsessively, almost involuntarily. Once I emerged, I realized that what I had done over the past ~3.5 months was far more significant than all of the other work that I had done over the span of my entire life combined. I was astonished to find myself having ascended to the pantheon of those who have made major contributions to human knowledge, something that I hadn't imagined possible in my wildest dreams.

The problem isn't "laziness"

Many of the most interesting people who I know are achieving at a level far below their potential. They often have major procrastination problems, and believe this to correspond to them having a character flaw of "laziness". I've become convinced that these people's problems don't come from them being insufficiently disciplined.

Their problems come from them spending their time trying to do work that they find boring. If you find your work boring, it's very likely that you should be doing something else.

References

My position is not unique to me: it's common to extremely high functioning people.

[1] Steve Jobs created Apple, which owns ~0.1%+ of the world's wealth. In his 2005 Stanford commencement address he said:

I'm convinced that the only thing that kept me going was that I loved what I did. You've got to find what you love. And that is as true for your work as it is for your lovers. Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work. And the only way to do great work is to love what you do. If you haven't found it yet, keep looking. Don't settle. As with all matters of the heart, you'll know when you find it. And, like any great relationship, it just gets better and better as the years roll on. So keep looking until you find it. Don't settle. 

[2] Bill Thurston is one of the greatest mathematicians of the 20th century. He formulated the geometrization conjecture, which subsumes the 100 year old Poincare conjecture, considered one of the ~7 most important unsolved problems. He describes his own character as follows:

My attention is more inward than that of most people: it can be resistant to being captured and directed externally. Exercises like these mathematics lessons were excruciatingly boring and painful (whether or not I had "mastered the material"). I used to think my wandering attention and difficulty in completing assignments was a defect, but now I realize my "laziness" is a feature, not a bug. Human society wouldn't function well if everyone were like me, but society is better with everyone not being alike.

[3] Scott Alexander / Yvain is widely regarded as a great writer. Political celebrity Ezra Klein characterized his blog as fantastic. Scott wrote:

On the other hand, I know people who want to get good at writing, and make a mighty resolution to write two hundred words a day every day, and then after the first week they find it’s too annoying and give up. These people think I’m amazing, and why shouldn’t they? I’ve written a few hundred to a few thousand words pretty much every day for the past ten years.

But as I’ve said before, this has taken exactly zero willpower. It’s more that I can’t stop even if I want to. Part of that is probably that when I write, I feel really good about having expressed exactly what it was I meant to say. Lots of people read it, they comment, they praise me, I feel good, I’m encouraged to keep writing, and it’s exactly the same virtuous cycle as my brother got from his piano practice.

[4] Paul Graham is the co-founder of Y-Combinator, a seed funder with a portfolio of combined value exceeding $30 billion (with investees including Dropbox, AirBnB and Stripe). In What You'll Wish You Had Known he wrote

One of the most dangerous illusions you get from school is the idea that doing great things requires a lot of discipline. Most subjects are taught in such a boring way that it's only by discipline that you can flog yourself through them.

Now I know a number of people who do great work, and it's the same with all of them. They have little discipline. They're all terrible procrastinators and find it almost impossible to make themselves do anything they're not interested in. One still hasn't sent out his half of the thank-you notes from his wedding, four years ago. Another has 26,000 emails in her inbox.

I'm not saying you can get away with zero self-discipline. You probably need about the amount you need to go running. [...] But once they get started, interest takes over, and discipline is no longer necessary.

Do you think Shakespeare was gritting his teeth and diligently trying to write Great Literature? Of course not. He was having fun. That's why he's so good.

Cross Posted at the EA Forum

At Event Horizon (a Rationalist/Effective Altruist house in Berkeley) my roommates yesterday were worried about Slate Star Codex. Their worries also apply to the Effective Altruism Forum, so I'll extend them. 

The Problem:

Lesswrong was for many years the gravitational center for young rationalists worldwide, and it permits posting by new users, so good new ideas had a strong incentive to emerge.

With the rise of Slate Star Codex, the incentive for new users to post content on Lesswrong went down. Posting at Slate Star Codex is not open, so potentially great bloggers are not incentivized to come up with their ideas, but only to comment on the ones there. 

The Effective Altruism forum doesn't have that particular problem. It is however more constrained in terms of what can be posted there. It is after all supposed to be about Effective Altruism. 

We thus have three different strong attractors for the large community of people who enjoy reading blog posts online and are nearby in idea space. 

Possible Solutions: 

(EDIT: By possible solutions I merely mean to say "these are some bad solutions I came up with in 5 minutes, and the reason I'm posting them here is because if I post bad solutions, other people will be incentivized to post better solutions)

If Slate Star Codex became an open blog like Lesswrong, more people would consider transitioning from passive lurkers to actual posters. 

If the Effective Altruism Forum got as many readers as Lesswrong, there could be two gravity centers at the same time. 

If the moderation and self selection of Main was changed into something that attracts those who have been on LW for a long time, and discussion was changed to something like Newcomers discussion, LW could go back to being the main space, with a two tier system (maybe one modulated by karma as well). 

The Past:

In the past there was Overcoming Bias, and Lesswrong in part became a stronger attractor because it was more open. Eventually lesswrongers migrated from Main to Discussion, and from there to Slate Star Codex, 80k blog, Effective Altruism forum, back to Overcoming Bias, and Wait But Why. 

It is possible that Lesswrong had simply exerted it's capacity. 

It is possible that a new higher tier league was needed to keep post quality high.

A Suggestion: 

I suggest two things should be preserved:

Interesting content being created by those with more experience and knowledge who have interacted in this memespace for longer (part of why Slate Star Codex is powerful), and 

The opportunity (and total absence of trivial inconveniences) for new people to try creating their own new posts. 

If these two properties are kept, there is a lot of value to be gained by everyone. 

The Status Quo: 

I feel like we are living in a very suboptimal blogosphere. On LW, Discussion is more read than Main, which means what is being promoted to Main is not attractive to the people who are actually reading Lesswrong. The top tier quality for actually read posting is dominated by one individual (a great one, but still), disincentivizing high quality posts by other high quality people. The EA Forum has high quality posts that go unread because it isn't the center of attention.