Ask HN: Best resources to gain math intuition? - <AUTHOR> In high school I did the lowest tier maths and then jumped in the deep end by doing a year of electrical engineering. I've now done those harder math classes but I feel like there's holes here and there. I think when I took physics it really brought out these flaws and lack of intuition.
Would anyone have a good resource for building this up?
Thanks <comment><USER4> I feel you need to rethink your goal. it's like you're trying to finding missing chapters from an undefined book.

Try to be around problems that requires a high degree of math skills to solve. let the problems drive you.

intuition is usually built by spending hundreds of hours thinking, eating, and drinking the problems you're truly interested in to arrive to a solution.

If you spend enough time doing this, you'll know exactly what areas you need to inquire about as your understanding to the problem becomes clearer and gaps start to narrow down.

If you don't know what you need to learn you probably haven't spent enough time on the problem or just read the solution which made you feel this way, not realizing that the solution was done by a person that could have spent 100x the time you spent on it and built the intuition you're asking about.

<comment><USER5> There's a loophole. You read about people like Gauss who had insights long before they became math students. There's something to dig there.

<comment><USER6> Gauss definetly used a cheat code. Calculating before starting to talk, fixing errors in payrolls at age of 3... numbers came more naturally to that guy than talking.

If something is to dig here is that the earlier you start the better you will be.

<comment><USER7> > the earlier you start the better you will be

There was a psychology teacher that had this theory, so when he had kids he started teaching them to play Chess as soon as possible and his two daughters became world champions

<comment><USER6> Polgár László was the guy with 3 daughters. And one of the girls became the best women chess player to date.

[https://en.m.wikipedia.org/wiki/Judit_Polg%C3%A1r](https://en.m.wikipedia.org/wiki/Judit_Polg%C3%A1r)

</comment></comment></comment></comment></comment><comment><USER8> Plug, but I have a blog for exactly this. As an EE/physics student you may appreciate this one on Euler's Formula:

[https://betterexplained.com/articles/intuitive- understanding...](https://betterexplained.com/articles/intuitive- understanding-of-eulers-formula/)

<comment><USER9> Kalid , Your post on the magic of "e" still remains etched in my memory. Kudos to what you are doing :-). Nice book, too!

<comment><USER9> For the curious, here's the link: [https://betterexplained.com/articles/an- intuitive-guide-to-e...](https://betterexplained.com/articles/an-intuitive- guide-to-exponential-functions-e/)

</comment></comment></comment><comment><USER10> 3Blue1Brown videos are awesome:

[https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw](https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw)

<comment><USER11> I second this, with emphasis. His videos cover the vast majority of topics at the intersection of math and hn. He has written custom software for the creation of the animations.

It's on another level imo.

<comment><USER12> 3blue1brown is a special kind of awesome.

I'd love to filter the internet by "content that can give you insights right now that would otherwise take years of study in a specific discipline to even know exists".

Another that I saw recently on an HN comment (thanks tptacek) is [https://www.youtube.com/watch?v=nfY0lrdXar8](https://www.youtube.com/watch?v=nfY0lrdXar8)

A quick google for "3blue1brown awesome github" (my usual strategy for finding good similar content) landed me here: [https://github.com/rossant/awesome- math](https://github.com/rossant/awesome-math)

The awesome lists are pretty good starting points for finding good content, but there's a ton of them of variable quality, so you end up with stuff like this: [https://github.com/jonatasbaldin/awesome-awesome- awesome](https://github.com/jonatasbaldin/awesome-awesome-awesome) (which has 68 forks...)

</comment></comment></comment><comment><USER13> My suggestion would be this. Assume that you have your intuition, already, and it is good.

May be it is distinct from others, may be you will not be 'narrowing' to the right answer within seconds -- like many folks who do Olympiads...

Just do basic things, but every other day. Get books/materials that have solutions (not just problems). use those, compare your results, and then try again.

If you feel like you do not understand 'why', you will need a particular subject area. Switch to read about applications of that area, how historically it came about and so.

And then back to problem solving, proofs, and reading other people's papers (when you can..).

It is hard work, but over time you will build up your version of the so called intuition, it will be powerful, you will be able to apply it all over the place.

Also there are a number of math forums where you can reach out, if you are really stuck and cannot figure out how a particular proof, or solution was obtained.

</comment><comment><USER14> I'm interested in building up math intuition too. I'm engaging with math very slowly and just as a hobby, but part of what got me interested was encountering resources that are unreasonably effective at building intuition.

A common theme I have noticed is whatever it is, getting a geometric understanding of it aids intuition significantly. Others have mentioned the 3Blue1Brown videos. They are an excellent example of this.

These HN threads always bear out great resources and I've made note of (and acquired) a few of these, so I'll list them here.

Burn Maths Class - [https://www.goodreads.com/book/show/26195956-burn-math- class](https://www.goodreads.com/book/show/26195956-burn-math-class)

Book of Proof - [https://www.people.vcu.edu/~rhammack/BookOfProof/](https://www.people.vcu.edu/~rhammack/BookOfProof/)

The Topology Of Numbers (number theory)- [http://pi.math.cornell.edu/~hatcher/TN/TNpage.html](http://pi.math.cornell.edu/~hatcher/TN/TNpage.html)

The Evolution of Trust (game theory) - [https://ncase.me/trust/](https://ncase.me/trust/)

Visual Information Theory - [https://colah.github.io/posts/2015-09-Visual- Information/](https://colah.github.io/posts/2015-09-Visual-Information/)

Information Theory For Smarties - [http://tuvalu.santafe.edu/~simon/it.pdf](http://tuvalu.santafe.edu/~simon/it.pdf)

Abstract Algebra - [https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR...](https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK- TP6)

Algebra Cheatsheet - [https://argumatronic.com/posts/2019-06-21-algebra- cheatsheet...](https://argumatronic.com/posts/2019-06-21-algebra- cheatsheet.html)

Control Theory Basics - [https://www.youtube.com/user/ControlLectures/playlists](https://www.youtube.com/user/ControlLectures/playlists)

</comment><comment><USER15> Pure Mathematics for Beginners by Steve Warner runs through the (very) introductory elements of major branches of math [0]. If you want even more leisurely intro to math intuition without worrying about elements of topology/abstract algebra or whatever, Book of Proof by Hammack is great and free [1].

[0] [https://www.amazon.com/Pure-Mathematics-Beginners- Rigorous-I...](https://www.amazon.com/Pure-Mathematics-Beginners-Rigorous- Introduction/dp/0999811754/ref=sr_1_1?keywords=steve+warner&qid=1566861311&s=gateway&sr=8-1)

[1] [https://www.people.vcu.edu/~rhammack/BookOfProof/](https://www.people.vcu.edu/~rhammack/BookOfProof/)

<comment><USER16> Like most people he's probably referring to applied math. It's really what most people outside of graduate students or math majors are interested in.

</comment></comment><comment><USER17> Helps to keep in mind Jeff Raskin's "theorem" \- "Intuitive equals familiar". Although displaced from its original UI context, the reading of "gaining intuition = gaining familiarity" is spot on.

So if you're looking for intuition, you're looking for deep familiarity which means spending the time/effort it takes to get familiar.

Resources don't help much as they're almost always about someone else's "intuition". Struggle that leads into false turns and force you to backtrack help a lot more I think.

An example - some time ago, I told some colleagues that they've been dealing with convolution even before middle school .. which surprised them. Multiplying numbers is the same operation as convolving their digit sequences. But if you didn't fool around enough with multiplication, or try to relate it to algebra, polynomials, and such, this can be hard to see.

Edit:typo fixed

</comment><comment><USER18> If you mean "gaps in my education" or "basic things that I don't quite understand", you could try studying some high-quality texts. Look for books written by extremely smart people who are trying to explain the ideas rather than taking you through the standard topics. Hamming's books on probability and signal processing and Strang's books on linear algebra and applied math come to mind.

Alternatively if you're really interested in intuition, you could also look at the Math Olympiads. Pick a problem, beat your head on it, finally look at the solution, repeat. There are web sites and prep books.

<comment><USER19> Are the Math olympiads similar to competitive coding?

<comment><USER20> Yeah.

At the high school and college level, the Olympiads for math and CS are pretty analogous. But there's really popular semi-formal coding contests which exist outside academia which don't really have a math equivalent.

I'd say math contests are more popular among high schoolers, and semi-formal coding contests more popular among college students.

Art of Problem Solving (AoPS) [[https://artofproblemsolving.com/](https://artofproblemsolving.com/)] is a really good resource, and there's a very healthy online community.

They're also similar in how olympiads are different from the "real thing" (TM).

Academia.SE discussion about this [[https://academia.stackexchange.com/questions/86451/does- the-...](https://academia.stackexchange.com/questions/86451/does-the- international-math-olympiad-help-research-mathematics/86489)]

As someone who did math olympiads in high school, my 2 cents is that they're a fantastic way to learn how to solve and approach problems and gain intuition. And I'd say intuition mainly comes from solving problems.

</comment></comment></comment><comment><USER21> [https://mitpress.mit.edu/books/art-insight-science-and- engin...](https://mitpress.mit.edu/books/art-insight-science-and-engineering) . (free - look for pdf download link )

[https://mitpress.mit.edu/books/street-fighting- mathematics](https://mitpress.mit.edu/books/street-fighting-mathematics) (free - look for pdf download link)

[https://www.betterworldbooks.com/product/detail/How-to- Solve...](https://www.betterworldbooks.com/product/detail/How-to-Solve-It-- A-New-Aspect-of-Mathematical-Method-9780691023564)

</comment><comment><USER22> For a small buffet of abstract math topics (with lots of exposition putting ideas into historical context and an emphasis on building geometric intuition) try Courant & Robbins "What is Mathematics?". Covers number theory, number systems, geometry, topology, optimization, algebra, and knots, among others. You could also play around with branches of math like Knot and Braid Theory, which are interesting in their own right, give insight into lots of different branches of math, and many interesting problems are still accessible to lay people. Number theory is also like this

I think it can also be helpful to learn some things about the history of math and the historical context that different ideas came from. Here's a nice example covering complex numbers [https://www.youtube.com/watch?v=T647CGsuOVU](https://www.youtube.com/watch?v=T647CGsuOVU)

Maybe tangential to your ends, but the Crest of the Peacock is a nice book on non-European mathematical traditions, which provides some insight into how the process of establishing and validating mathematical knowledge works in other cultures.

You might try books written by physicists or that are about mathematical physics (an author to look out for depending on your level is VI Arnold), since arguments will be of a more geometric or physical nature and appeal more to intuition. Stillwell is another author (not a physicist) that tends to write books that give context and geometric intuition

You might like playing around with Pinter & Humphreys for Algebra, or Jänich for Topology (fantastic book for building intuition around topology).

</comment><comment><USER23> Learn linear algebra properly. Like many folks on HN, I really enjoyed Axler's Linear Algebra Done Right, but ymmv. I got a ton out of going through each of the proofs + practice problems and really taking the time to work through the solutions. It wasn't enough for me to just read the text.

There's a linear algebra lurking everywhere in the realm of applied math (some people like to joke that machine learning is really just linear algebra) so it really is worth your time to have a firm understanding of it.

</comment><comment><USER24> It's probably too hard to answer the general question about "how to gain math intuition". Mathematics is just too vast.

However, you have written

> I think when I took physics it really brought out these flaws and lack of > intuition

which suggests you have some good practical experience with physics problem solving which has precipitated a certain feeling that you need to learn more about some kind of math. I would advise that you try to exploit this. In the same breath I want to recognize (as someone who did their BS and MS in physics) that physicists are not always so careful or explicit in how they are doing their mathematics. So learning means eventually going beyond physics sources and into a much wider world of mathematical thought. The particular things that mathematicians care about may or may not be relevant to the problem you are trying to solve in physics, and a good part of developing that intuition is to figure out which particular caveats that a mathematician expounds upon (more often than not, some esoterica about the space(s) that they are working in or the class of isomorphisms under which their results are invariant) matter physically. As you develop and intuition about these things a bonus is that you will be able to skim through mathematics resources much faster.

</comment><comment><USER25>
      You don't understand mathematics,
      you just get used to it - <USER2>

I used to want to _understand_ math (intuition); now I think of it as learning a language: lots of vocab and rules with _some_ pattern. You need to become fluent - _practiced_ , like a cellist or cricketer.

It's also a language when someone is teaching it.

Prior knowledge is assumed; so it's difficult to detect that it is a gap that's making it difficult. When I finally realized my hard-won breakthroughs were about prior material, I dropped back to do that prior level... again and again.

This thorough approach is far too time-consuming for your needs. I really hope you find a quicker way - and please tell me!

</comment><comment><USER26> Also, take a look at A Programmer's Introduction to Mathematics by Jeremy Kun [0].

[0] [https://www.amazon.com/Programmers-Introduction- Mathematics-...](https://www.amazon.com/Programmers-Introduction-Mathematics- Dr-Jeremy/dp/1727125452)

</comment><comment><USER27> "Learn through inundating yourself with examples rather than definitions."

Source: listen from 28:35 to 29:35 [https://overcast.fm/+Soyvpq978/28:35](https://overcast.fm/+Soyvpq978/28:35)

</comment><comment><USER28> The 3 wonderful Roger Nelsen books _Proofs Without Words_ entirely consist of equations & inequalities from various mathematical fields 'proved' in intuitive pictorial form.

[http://gen.lib.rus.ec/search.php?req=proofs+without+words](http://gen.lib.rus.ec/search.php?req=proofs+without+words)

</comment><comment><USER29> Math is big. A good place to start building your intuition may be by learning proofs. I'm currently revisiting math by working through Chartrand [0]. As a non-mathematician, I would recommend this text as a foundation, and a possible bridge to advanced subjects.

[0] Mathematical Proofs: A Transition to Advanced Mathematics [https://g.co/kgs/stSmxJ](https://g.co/kgs/stSmxJ)

</comment><comment><USER30> If you are doing signal processing (Laplace and Fourier transforms especially) and want to gain an intuitive sense of Complex Analysis, I recommend the book Visual Complex Analysis by Tristan Needham.

</comment><comment><USER31> Try finding a cool research paper that contains a lot of math you don't understand. Then spend a few months learning the math in that one paper until you get it. That is a highly motivating and enjoyable way to learn. Far better than textbooks. I did this to learn algebraic topology.

<comment><USER32> I've been trying to penetrate algebraic topology for a bit now (learning barcodes and such). Any suggestions on papers you think are particularly good?

<comment><USER31> I was obsessed with topological data analysis which is basically applied algebraic topology, so I read every paper I could find on TDA. All of them were mostly incomprehensible to me at first, but by working backward, e.g. "What's a homology group?" Oh I need to learn some basic group theory first. Oh what's a Cayley graph? I kept working backward and forward until I could piece it all together. Working backward from a specific goal was very motivating compared to just working through a textbook aimlessly.

</comment></comment></comment><comment><USER33> This isn't an answer, but an observation. I taught an EE course many years ago -- the second semester of electrodynamics. This was after being a college math major, and getting a physics degree.

It doesn't shock me that there are holes. I noticed that some math topics are very important to engineering and physics coursework, but given short shrift in the math department. Examples are the way that complex numbers are used, and specific kinds of differential equations such as the general harmonic oscillator.

My college physics coursework actually had its own "math methods" class, intended to fill some of those gaps, and to get us prepared for the higher level physics courses.

<comment><USER34> You'll find the complementary opinion in Mathematics departments -- a general chagrin about the type of mathematics that they have to teach in their service courses for engineers.

Mathematics is a very general tool. As with any very general tool, a lot of the devil is in the details of how to use it in any particular domain.

For this reason, in-sourcing mathematics service courses is best for everyone. The very best math-adjacent departments in every field tend to do this either directly or indirectly. E.g., in the direct model, many CS departments internalize the Discrete Mathematics course and some combinatorics. And an example of the indirect model is Mathematics departments that hire Math Finance professors to cover the service load for econ/fin/bus depts.

I think this in-sourcing (either directly or indirectly) is best for everyone -- mathematics depts don't do a good job at teaching those service courses and often don't do a great job of it in any case. Unfortunately, most departments don't have the headcount (in students or faculty) for a specialized mathematics curriculum, so they have to share the math faculty with N other majors to predictable effect.

</comment></comment><comment><USER35> I run a 1-on-1 remote mentoring/teaching business where I teach the intuition and reasoning (including some history) behind mathematics, physics, and computer science. If you are interested in a full level overview into higher level mathematics, starting with set theory, reach out to me. We can organize an arrangement that suits your needs, desires, and schedule.

Shoot an email over to firstname.lastname@example.com or text or call +1-718-360-3176 if you are interested or if you have any questions.

This is my print flyer for reference: [https://i.imgur.com/Gdwa7m9.png](https://i.imgur.com/Gdwa7m9.png)

Looking forward to hearing from you.

</comment><comment><USER36> I'm going to answer this assuming you mean gaining number sense, which is something I didn't realize I was missing until I gained it.

I got my undergrad in math and physics. I was good at math. It wasn't until I had been teaching high school for 3-4 years when some gave me a copy of Shortcut Math by Gerard Kelly. After reading it and practicing the techniques, arithmetic made so much sense. I was able to easily add, subtract, and multiply larger numbers in my head.

Interestingly enough, many of the techniques taught in this book are also part of the common core math curriculum. It's a way to help students gain number sense.

</comment><comment><USER37> Related question, anyone know any good resources for learning and developing intuition in Abstract Algebra? Especially ones that explain it well, like _Calculus Made Easy_ did for Calculus. AA is a prereq for cryptography, category theory and and other areas of interest for CompSci.

I asked this a few weeks ago but at an off peak time and not many folks saw the question. But I know HN probably has good recommendations so trying again here. Should still be relevant and helpful to OP.

[https://news.ycombinator.com/item?id=20733422](https://news.ycombinator.com/item?id=20733422)

<comment><USER38> Take a look at "A Book of Abstract Algebra" by Charles Pinter. It's published by Dover, so very affordable. I've seen it criticized as too verbose, but that worked for me.

<comment><USER39> It has been my experience that complaints about a source being "too verbose" rarely come from people learning the subject.

</comment></comment></comment><comment><USER40> I like 'Better Explained'[1]. It specifically focuses on intuitive understanding of mathematics.

A post I like is on adding numbers 1 to 100 [2]. The staple formula is n(n+1)/2, sum of arithmetic progression. How can we intuitively arrive that this formula?

> _Technique 1: Pair Numbers Pairing numbers is a common approach to this > problem. Instead of writing all the numbers in a single column, let's wrap > the numbers around, like this:

1 2 3 4 5

10 9 8 7 6

An interesting pattern emerges: the sum of each column is 11. As the top row increases, the bottom row decreases, so the sum stays the same.

Because 1 is paired with 10 (our n), we can say that each column has (n+1). And how many pairs do we have? Well, we have 2 equal rows, we must have n/2 pairs.

Number of Paris x Sum of Each Pair = (n/2) (n + 1) = n(n+1) / 2_

[1] [https://betterexplained.com/](https://betterexplained.com/)

[2] [https://betterexplained.com/articles/techniques-for- adding-t...](https://betterexplained.com/articles/techniques-for-adding-the- numbers-1-to-100/)

<comment><USER41> This reasoning for adding consecutive numbers is ok, but for some people the geometric reasoning (that you are adding half of a full square plus the diagonal) will be much clearer and intuitive. It also allows a visual generalization to higher dimensions, where the formula is not so evident.

</comment></comment><comment><USER42> Try and understand how you learn. A lot of intuition in technical fields, not just in mathematics, is understanding the material in a way that you understand it.

For me it was trying to frame as much as possible in a geometric lens. Draw pictures of anything and everything as much as possible. Graphs, number lines, 3d animations, whatever it takes.

Also, understand the "bold print" from the "fine print". I mean to say, every theorem will make a general statement, and then have a bunch of conditions where it holds. So worry first about the bold print, then try to understand the fine print.

Much of mathematics is about making the strongest and most general statements in bold print with as little fine print as possible. Basically the less fine print you have, the more important probably your result is.

</comment><comment><USER43> To get math intuition, you need some memorization along with practice. In school, you focus lot on remembering things for exams, that pays off later. As you grow, it becomes difficult to remember mindlessly.

1\. Do exercises from competitive-exam books written for secondary school children.

2\. Try to solve as many as possible exam papers in stipulated time.

This will help you to build intuition. Intuition comes only after doing something for long time (unless you are born-genius). Memorization plays very important role in learning maths, language, music instruments. On second note, don't fear to change resources if you're not comfortable. What I mean is - <USER3> childhood if you like teacher, you excel in subject. So you have to find material which match your temperament. Wish you best luck !

</comment><comment> Iwan-Zotow Polya How to Solve it

</comment><comment><USER44> This really depends on a lot of factors. You might get a lot out if illustrated works, such the _Cartoon Guides ..._ series. You might benefit from kid-friendly, fun books, like _The Number Devil_ or _Alice in Quantum Land._ Or you might benefit enormously from looking for "chapter books" on the subjects -- books filled with paragraphs of actual English instead of endless formulas. This might include something like _A Tour Of The Calculus_ or various things about math history or biographies of mathy people.

It depends on why you seem to be "missing" something and your post doesn't actually give any insight into what seems to be missing for you.

</comment><comment><USER45> Write your own simple 3d fluid or rigid body simulation, and a 3d renderer to display the simulation. (It could be a video game...) Doing that would take you on a tour of all kinds of math and force you to learn it well enough to implement it, linear algebra, geometry, calculus, physics, etc.

For me, math intuition takes practice and a lot of visuals, graphing and diagrams. Some people get it through proofs and equations, but I feel like the way math is taught in school doesn't usually do a great job of developing intuitions. Coding it up forces me to learn it, and playing with it and tweaking a lot along the way helps develop the intuition.

</comment><comment><USER46> Grab a pencil, straightedge, compass, and a stack of paper and do some geometry.

BTW, John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them."

</comment><comment><USER47> R.W. Hamming. _Numerical Methods for Scientists and Engineers_

</comment><comment><USER48> > In high school I did the lowest tier maths and then jumped in the deep end > by doing a year of electrical engineering. I've now done those harder math > classes but I feel like there's holes here and there. I think when I took > physics it really brought out these flaws and lack of intuition.

As a mathematician, I say: what you lack is not math intuition, but math knowledge. Read really hard math textbooks and solve exercises. This is the only way known to me to get better at math.

</comment><comment><USER49> The other answers have suggested a lot of particular references, many of which are excellent.

I'd add, speaking as a math professor: don't worry too much about holes and lack of intuition. The more you learn, the bigger you will feel like your holes are!

Whatever you choose, in my opinion it should involve solving lots of problems. Subject to that, I'd recommend simply that you dive in to any math subject that attracts your attention.

</comment><comment><USER50> 3B1B is the best place to start.

[http://immersivemath.com/ila/index.html](http://immersivemath.com/ila/index.html)

</comment><comment><USER51> I'm building a math practice site covering topics from simple arithmetic to second order differential equations. The idea is that to get better at (or intuit) math you have to practice. I'm trying to incorporate some Skinner Box techniques to make studying easier. I hope it's helpful and I'd love to hear your feedback: [https://studycounts.com](https://studycounts.com)

</comment><comment> eric-brechemier I would recommend brilliant.org. From first-hand experience, this might be exactly what you are looking for: start from the basics and grow into complex matters, by solving one short problem at a time.

I think this is relevant with regards to your perceived lack of intuition: they have a way of making you discover the solution to complex problems by introducing complexity through a succession of simple problems, in a very piecemeal manner.

</comment><comment><USER52> I have found 3Blue1Brown's videos one of the best math sources on youtube. Give his videos a try I think you will find exactly what you are looking for.

</comment><comment><USER53> It's definitely not the best resource, but math intuition is what I'm promoting on [https://wordsandbuttons.online/](https://wordsandbuttons.online/)

I think, with modern tools, you need math intuition more than computational skills. That's why my site is all about interactive experience. There are no proofs or problems but a lot of things to tweak and drag.

</comment><comment><USER54> aside from 3Blue1Brown :

I think category theory will give you an unique point of view of math, also basic logic and philosophy are very important for a good and solid foundation on math intuition.

[https://arxiv.org/abs/1803.05316](https://arxiv.org/abs/1803.05316) along with their youtube lectures.

very basic but refreshing:

[https://www.youtube.com/playlist?list=PL8dPuuaLjXtNgK6MZucdY...](https://www.youtube.com/playlist?list=PL8dPuuaLjXtNgK6MZucdYldNkMybYIHKR)

if you are a programmer or developer:

[https://bartoszmilewski.com/2014/10/28/category-theory- for-p...](https://bartoszmilewski.com/2014/10/28/category-theory-for- programmers-the-preface/) [https://blog.ploeh.dk/2017/10/04/from-design- patterns-to-cat...](https://blog.ploeh.dk/2017/10/04/from-design-patterns-to- category-theory/)

</comment><comment><USER55> I have an employee who just randomly takes courses at the local uni. He's done a basic chip fab course, a stats course, etc.

</comment><comment><USER56> I'm also very curious about this. There seems to be a big gulf separating genuinely math savvy people and people with mediocre understanding of math, despite the resources being available to both. I'm curious if there is a/are common denominator(s) between the math-savvy people, like a habit or a loop that I slept out on.

</comment><comment><USER57> this might be useful to read for some general high level ideas on how to proceed: [http://m.nautil.us/issue/17/big-bangs/how-i-rewired-my- brain...](http://m.nautil.us/issue/17/big-bangs/how-i-rewired-my-brain-to- become-fluent-in-math-rd)

<comment><USER58> Great Article! Thank You!

</comment></comment><comment><USER19> I used to love the abstract symbolic manipulation of pure math, but have not worked with it in many years and find it unintuitive now. I have a harder time trusting the abstractions. I wonder how to get my old sense back - it never, ever came from practice problems for example in the past. I miss proofs

</comment><comment><USER59> A lot of intuition in math is from geometry. High school plane geometry is a good start, but high school solid geometry, where get good tools and intuition for seeing things in 3D instead of just the 2D of plane geometry, is quite a bit better.

Another part of intuition is from Max Zorn (from Zorn's lemma statement of the axiom of choice):

"Be wise, generalize."

E.g., for the set of real numbers R and a positive integer n, a lot that goes on in the n-dimensional vector space R^n is a _generalization_ of what can see in 3D, e.g., from solid geometry.

E.g., in both cases, a biggie is a perpendicular ( _orthogonal_ ) projection and, again, the Pythagorean theorem. E.g., regression in statistics is a perpendicular projection.

Perpendicular (orthogonality) is a biggie and is a major part of, say, Fourier series. I.e., each of the sine/cosine waves used is an orthogonal axis, and to find the corresponding Fourier series coefficient just project onto that axis. The projection is an integral of a product, and that is commonly an _inner product_ which close to just a cosine of an angle as in plane and solid geometry and a perpendicular projection and close to correlation in statistics, etc.

E.g., a huge fraction of applied math is from _analysis_ in pure math, and from G. F. Simmons the two pillars of analysis are "continuity and linearity". Linearity generalizes enormously: The quantum mechanics _super position_ is essentially linearity. Under meager assumptions, differentiation and integration in calculus are linear _operators_. In probability theory, expectation is a linear operator. The wave equation is a linear partial differential equation. Linear programming works on linear equations. Of course, in _linear_ algebra, matrix multiplication is a linear operator. When something is not linear, it may be _locally_ linear which can be enough to get useful results.

For more, a good lesson is to _approximate_ : Commonly we can't get just what we want in just one step but can _iterate_ and _approximate_ as closely as we please. So, can use simple things, sine waves, polynomials, continuous functions, and more, as means of approximation. Such approximation gets us close to more in _continuity_ and, in particular, _completeness_ \-- the real numbers are _complete_ and the rational numbers are not but via iteration can approximate the reals as closely as we please. Then this generalizes: The big point about Hilbert space (as mathematicians but not always physicists define it) is completeness. A joke, partly correct, is that "calculus is the elementary consequences of the completeness property of the real number system". E.g., the integral in calculus (and its better version in measure theory) is defined in terms of an iterative approximation. So, if you are good with sine waves, polynomials, continuous functions, wavelets, and more, then you can iterate and approximate a lot, in many cases, everything there is in that case.

</comment><comment><USER60> What exactly is your goal?

I think that the way to get math intuition is to learn a mathematical language, like Julia, and play around with it. Plot things. Change parameters.

Also learn a theorem prover. Maybe Agda or Coq or Lean.

</comment><comment><USER61> i'm bad at math, lets say that first.

but for basic things, funny enough i feel like some of these math games / apps and math puzzle apps help me recognise patterns in numbers etc. better.

that being said, i hardly use maths ,certainly not advanced maths. but i do feel things like that are somewhat fun way to keep your senses a bit sharper.

in the end, practice makes perfect, but doing maths straight up is boring as hell to me, so i try my hand at little puzzle and maths games / apps to combat the boringness.

</comment><comment><USER62> Do exercises. Do really difficult exercises. Don't cry that there's no solution manual. It will take you many thousands of hours, there is no way around this.

</comment><comment><USER63> I agree with what has been said but what about a book about proof? It may be something to study that will accelerate the mathematics intuition.

</comment><comment><USER64> If you are looking into higher mathematics, e.g group theory, etc, do go over Concepts of Modern Mathematics by Ian Stewart.

</comment><comment><USER65> You may be looking for mathematical philosophy: [https://plato.stanford.edu/entries/philosophy- mathematics/](https://plato.stanford.edu/entries/philosophy-mathematics/)

</comment><comment><USER66> Just go to khan academy and practice your basics and more.

</comment><comment><USER67> This is something that I've struggled with myself so I can only give you advice as I've learned it from my perspective.

Math is a big field so you have to understand what parts you're really interested in and how you want information to be presented so that you'll learn it. For example, I'm interested in computation aspects of mathematics (information theory, computation group theory, abstract algebra, analysis, etc.) and I always prefer a "computer programmers" intuition in how to learn these subjects. That is, understanding how to 'program it', whatever that means for the different subjects I'm interested in.

Here is a small list of intuition I've learned about finding good resources:

* Books, and sometimes textbooks, are still a valuable resource. It's still the case that having a book on a subject that has curated content is better than the random Wikipedia articles or blog posts on the subject. Use Wikipedia, obviously, and look at blog posts, but I search for books in the subject area, especially if it's a field I'm not familiar with.

* When looking for books, prefer books that have "elementary" in the title, as in "elementary introduction". The more "advanced" books are talking about the bounds of research in the area, often fussing over esoteric issues whereas the "elementary" books give the foundation of knowledge in that area.

* Ideally for me, books would have "fundamental algorithms" somewhere in the title, as these books usually are exactly what I need to understand a field.

* When reading, ideally I make sure to do the exercises or run through the proofs myself. Mathematics is not a spectator sport and a large part of it is "learn by doing". Finding good resources so those exercises are meaningful is hard but they still need to be done.

* I often check MathOverflow, MathUndeflow, Physics.Stackexchange, CStheory.Stackexchange and other accompanying sites. There are a surprising number of good answers to questions of the form "what is the motivation behind...". As the subjects get more esoteric, these questions become more infrequent these resources are still invaluable. Asking questions on these sites is also an option and usually helpful.

* In the past I've watched more in depth lectures from mathematicians, either from conferences or from things like OpenCourseWare. There's a lot of 'folklore' wisdom that's embedded with people that sometimes comes out when viewing actual researchers talk about their research that wouldn't otherwise be apparent or emphasized in papers.

* I sometimes visit blogs from mathematicians or about mathematicians.

When I was younger in college, I was fortunate to have a social group of friends who were graduate students and TAs that had an appetite for discovery and teaching. There was a lot of folklore and intuition that was taught which would have been difficult to find otherwise. I think many graduate students in mathematics essentially use their exposure to their advisor, other teachers and other students to build that intuition.

I should also mention that there isn't "one way" to learn about these subjects. I take a computational perspective because that's my preference but I'm fully aware that not everyone thinks that way. Every person has their own perspective on what's fundamental and how they learn and build intuition even if they can be grouped in to rough categorizations (though I'd be hard pressed to quantify those categorizations). I've found the way I learn and optimize for it and I unfortunately have a hard time when information isn't presented in the way I need it to be, at least initially while I'm building intuition and learning a subject for the first time.

I can't find the quote now but there was a mathematician that was talking about Erdos and how Erdos didn't have deep knowledge or at least didn't use "higher mathematics" like Lie theory or other higher abstractions. Yet Erdos was prolific in his sense with his "elementary" methods, probably because he understood his tools and the problems deeply. As an analogy, it'd be like someone who knows assembly well trying to analyze a Haskell script. The Haskell programmer might have intuition from the constructs of that language but someone who knows assembly well understands that each of the abstractions in Haskell must eventually boil down to assembly instructions and can understand it from that perspective.

I also try to employ the "20% effort for 80% gain" rule. There are usually some basic concepts so learning them as fast as possible is the goal. This also allows for maximum gain for effort spent as if the field is interesting, I can dive deeper or move onto another if it's not.

I try to avoid resources that are "TED talk" like, press releases, or other "feel good" resources, like 3Brown1Blue. These are great for being inspired by mathematics (which is important!) but are usually devoid of content. Resources like 3Brown1Blue I find especially pernicious as they couch deep understanding by regurgitating facts without providing any fundamental insight.

I tend to stay away from Springer books as they're usually dense. They might be good for reference but for initial learning I've found them to be pretty bad.

People often say "read the original papers" but I found this to be horrible advice as the original papers often are a very rough 'proto' model of the ideas presented and don't benefit from work that's been done to simplify and extract the important parts of the theory without the cruft. Often times, mathematicians have their own pet notation which further get in the way of understanding. One exception is Shannon's paper on information theory.

In no particular order, here are a list of books I've found exceptional (very much catered to my personal taste):

[0] Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnsen

[1] The Way of Analysis by Strichartz

[2] Introduction to Algorithms by Cormen, Leiserson, Rivest and Stein

[3] Fundamental Problems of Algorithmic Algebra by Yap

[4] Fundamental Algorithms for Permutation Groups by Butler

[5] A Mathematical Theory of Communication by Shannon

[6] Complexity and Criticality by Christensen and Moloney

I have not found what I consider exceptional texts on number theory, Galois theory or cryptography.

Here are some blogs I occasional visit:

[7] [https://rjlipton.wordpress.com/](https://rjlipton.wordpress.com/) \- Godel's Lost Letter and P=NP

[8] [https://terrytao.wordpress.com/](https://terrytao.wordpress.com/) \- Terrence Tao's blog

Here are the SO sites:

[9] [https://mathoverflow.net](https://mathoverflow.net) \- Math overflow

[10] [https://math.stackexchange.com/](https://math.stackexchange.com/) \- Math "underflow"

[11] [https://physics.stackexchange.com/](https://physics.stackexchange.com/) \- Physics SE

[12] [https://cstheory.stackexchange.com/](https://cstheory.stackexchange.com/) \- Theoretical Computer Science SE

Math videos:

[13] [https://www.msri.org/videos/dashboard](https://www.msri.org/videos/dashboard) \- MSRI Videos

</comment><comment><USER66> Just go to "khan academy" and practice your basics and more.</comment>


Rory Marinich: Make Great Shit - <AUTHOR> http://journal.rinich.com/post/216821123/make-great-shit

<comment><USER2> If you haven't read it, I'll save you the time: "Shit shit shit shit shit shit shit." (There is as much value in that one sentence as there is in this guy's entire rambling blog post.)

<comment><USER3> If you're going to criticize what I write, please criticize it rather than just snarking.

<comment><USER4> You deserve the criticism. You didn't "unintentionally" introduce the word many times into your essay. You thought it would be a good idea and banged the reader over the head with it.

"Remind yourself always that your shit is relative. Find ways to expand your definition of shit as much as possible. That way, the first step you take in making your shit will alone surpass the completed shit of the inexperienced."

You can't make that point without using the word "shit" three times? I'm sure it can be better in a dozen ways without ever using the word shit, but you overused "shit" again to keep your theme going.

That is why the original poster is not being snarky. The post literally reads as shit shit shit shit shit, how many times can I say shit in one blog post. The point you are trying to make is completely lost in that.

edit: When you start blaming the reader for comprehension problems, you have failed as communicating. Although I'll keep in mind that you just wrote "it it" and "people're" right after talking about "flows naturally".

<comment><USER3> What can I say? I must read differently than the people here. When I read it it flows naturally; this has been in circulation long enough that I see lots of other people're okay with the style also.

Perhaps it's that I don't skim when I read: I actually digest the words being used. Read aloud, it really doesn't sound like the word is used too much; skimmed, it's all that stands out.

</comment></comment></comment></comment><comment><USER5> Read: "Rory Marinich: Look How Alternative And Modern I Am, I Know Cuss Words"

<comment><USER3> That's cute, coming from somebody so edgy he quotes Banksy in his profile.

<comment><USER5> But honestly, noticeable articles are noticeable for their noticeable ideas, not noticeable words. Noticeable words backfire, turning people off.

Fair point about Banksy, though.

<comment><USER3> This wasn't a good Hacker News article. It wasn't written to be. I've actually got one in the works now that I was going to submit here this week; this was geared towards my usual reader constituency of artsier types.

That comment here was the first time in a week and a half I've seen anybody comment on the word "shit", actually. When I wrote it I expected that might be a turn-off; I guess after it caught on in the blogosphere I got used to people treating it like it wasn't overdosing on the word.

</comment></comment></comment></comment><comment><USER6> That advice was some good shit, maybe great shit.</comment>


I want to build a real-time web application. but... - <AUTHOR>

But I'm limited by my technical skills. I'm well-verse in Javascript, decent in PHP but have zero knowledge in Linux. I tried using APE(ajax push engine) but the learning curve is way too steep. Node.js require linux environment as well. But I really really want to build a web application which i could be proud of, something that is of use to people and maybe to remind myself that i'm not a lousy programmer after all.<p>I need some advise. <comment><USER2> In my opinion you're doing it the wrong way around, it's really hard to just make something if you wanna work with a technology. All in my humble opinion of course.

If you come up with the idea first, like you said one that people can use, you're really more motivated to make it!

If Javascript is a language you feel confident in, I would suggest you do try and take up Node.js. I've played around with it and it makes making real-time web applications a breeze.

You could try making a realtime chat client that leverages websockets and something like Redis. There are probably a ton of those around but you can learn a lot from making one.

There are also a lot of getting started documents for node so zero knowledge of Linux shouldn't be a problem.

hope this helps!

<comment><AUTHOR> thanks bittersweet.. the thing is whether is it possible to avoid linux altogether?</comment></comment>


Entrepreneurs Don't Need Degrees like lawyers and ... - <AUTHOR> http://simplify.tumblr.com/post/81913420/entrepreneurs-dont-need-degrees-like-lawyers-and

<comment><USER2> This is one step away from linking to a tweet. Great quote, but the original article is more fitting for HN.

<comment><AUTHOR> Thanks for the comment. Have posted the original article.</comment></comment>


Wikipedia removing support for old browsers - <AUTHOR> https://en.wikipedia.org/sec-warning <comment><USER2> At the bottom of that page is a link to general browser recommendations: [https://wikitech.wikimedia.org/wiki/HTTPS/Browser_Recommenda...](https://wikitech.wikimedia.org/wiki/HTTPS/Browser_Recommendations)

One piece of advice that I'm glad to see included in that link: "Disable or uninstall any 3rd party "anti-virus" software. Most of them do more harm than good when they interfere with your browser's secure connections."

If you're using Windows, there is little need to use anything other than Defender (built-in). Any 3rd party anti-virus is intercepting your traffic for analysis.

Some 3rd party anti-viruses are even crashing browsers: [https://support.google.com/chrome/thread/17555930](https://support.google.com/chrome/thread/17555930)</comment>


The Windows 3.x Showcase - <AUTHOR> https://archive.org/details/softwarelibrary_win3_showcase&tab=collection <comment><USER2> I was impressed years ago when some bright spark got linux running in a web browser session, and I'm yet again impressed that we've progressed in technology to the point that I can watch a quicktime video in Windows 3.1 in Safari on my laptop. Not to mention a fantastically organised and easy to use collection of historically interesting software, nice!

</comment><comment><USER3> FWIW I still use windows 3.11 on DOS-BOX to view 90s vehicle service manual software.

<comment><USER4> Amazing how many old pieces of manufacturing equipment are still driven by DOS/ Win9x machines.

<comment><USER5> My colleague recently had to troubleshoot an amiga app controlling an industrial press, running in an emulator.

</comment></comment></comment><comment><USER6> I was wondering, could DosBox do networking? If so, you could get windows3 with internet installed on anything that could run Linux and DosBox.

<comment><USER7> There's a patch which adds emulation for the popular NE2000 network card: [http://home.arcor.de/h-a-l-9000/ne2kstuff/db_ne2000.html](http://home.arcor.de/h-a-l-9000/ne2kstuff/db_ne2000.html)

DOS itself had no networking support. I don't think Windows 3.1 had native networking either - you had to install Novell for LAN networking or Trumpet Winsock to use a modem.

<comment><USER8> You're correct that DOS itself (MS-DOS, at least) has no networking support, but you might be surprised to learn that DOSBox _does_ natively have networking support - in more ways than one.

On the DOS side of things, you can configure DOSBox so that one instance acts as an IPX 'server' \- a server that other DOSBox clients can connect to in order to form a virtual IPX network over UDP. Programs running under DOS would then be able to use the IPX network - especially useful for games, as back then IPX was typically the only kind of networking available.

DOSBox also has support for emulating modems, tunneling the communication over TCP. It does this by allowing the Hayes dial commands to accept IP address/port combinations corresponding to the DOSBox client on the other end (which would have been set up as a server), which many programs would then just accept without validating them. In the event that a program _did_ validate them and not let you enter anything except numbers, you can still do it by padding the numbers with zeroes. For example, if you wanted to connect to 192.168.1.45:1234, you could 'dial' 1921680010451234\. Many modem games can then be played this way without having to use any DOS networking support.

<comment><USER9> There was also the port tricks, connect two PCs with a COM or LPT cable and a few games could communicate over them. I recall (but could be wrong) that DOOM 3D could do this. DirectPlay (DirectX networking) also supported this in at least Windows 95. A few VIC 20 games also used this trick.

<comment><USER8> Yep! That was commonly called a "null modem" cable, and DOSBox supports that style of connection too. :)

</comment></comment></comment></comment></comment><comment><USER10> Isn't this risky with regards to copyright laws? This seems to download some of the software (including Win 3.x itslef).

<comment><USER11> I believe there are exemptions for the preservation of copyrighted works. Please don't consider this authoritative or even applicable here, but, as an example: [https://www.copyright.com/Services/copyrightoncampus/basics/...](https://www.copyright.com/Services/copyrightoncampus/basics/fairuse_archive.html)

<comment><USER12> > To qualify for the exception, your library or archive must: [...]

> Produce no more than a single reproduction of a given work.

Copyright law would consider the Internet Archive to be reproducing the software every time someone clicks the link and their browser downloads it, so this wouldn't apply.

</comment></comment></comment><comment><USER13> Wow. I can actually play a .mov video in QuickTime player, in Windows 3.11, emulated in Javascript/DOSBOX. And it actually works.

</comment><comment><USER14> Can anyone tell me if the embedding works for them with Internet Archive stuff like this?

It loads okay at the IA site, but when I use their embed code for this, and the previous times I've tried with their DOS and Arcade games, the games/programs load, but they don't load in the right position in the embed, about 30% too far down, so are essentially unusable.

</comment><comment><USER15> This is amazing, just played Desert Storm: The Air Campaign for a few minutes. The collection of software here is such an important reminder that even seemingly unimportant software has historical context and is worth preserving. The Internet Archive is good people.

</comment><comment><USER16> Wow, I'm very impressed, surprised this works so well. Any chance we can get Exile 3: Ruined World added to this? Probably my favorite Windows 3.1 game and the full version is now available for free from the developer.

[http://www.spiderwebsoftware.com/exile3/winexile3.html](http://www.spiderwebsoftware.com/exile3/winexile3.html)

Actually, if I can do it myself I'll look into it tonight, there's a few titles I could upload like this if possible...

</comment><comment><USER17> Running in a similar manner, I was impressed when I stumbled across [https://win95.ajf.me/](https://win95.ajf.me/) recently.

</comment><comment><USER18> too good for this world, too pure

</comment><comment><USER19> Oh yes, WinTrek, I spent many a relaxing afternoon hunting down Klingons.

</comment><comment><USER20> Awesome, was able to play the tile game Taipei :)</comment>


Interview with MIT Technology Review's Jason Pontin - <AUTHOR> https://chatbotsmagazine.com/mit-technology-reviews-jason-pontin-technologists-consistently-underestimate-the-awfulness-of-926b49b4cc98 <comment><USER2> Not much depth in this interview. Regarding the awfulness of human beings, I suspect this is partly explained by a large intersection between the group of people who have the time and inclination to talk to a chatbot and the people who enjoy trolling.

</comment><comment><USER3> I remember this guy had something to do with harassment of Shanley.</comment>


Ask HN: Cool Demo Project Ideas - <AUTHOR>

The standard (and probably true) wisdom around here is that a good way to get noticed and get the kind of job that most of us would like is to "build something".
What I'm hoping for is ideas for demo projects.  I don't need anything designed to make money or steal anyone's startup idea, just things that are fun to build and show development skills.
Demos you've seen already and liked or that have gotten people job offers are also appreciated. <comment><USER2> Unfortunately topics like this never get much attention. The only thing I can recommend is to find a topic that interests you and then just go find something that sounds neat. It doesn't have to be the next Google and it doesn't even have to make money, just do something.

For example, I've recently become fascinated with data mining. To explore this area and get a basic idea of how it works, I'm playing with the Twitter API and trying to look for neat things that can be done with the data available. One idea I've come up with is an app that recommends people to follow based on the hashtags that you use frequently. It's still a "half baked" idea for now, but maybe something good will come from it.

Good luck.</comment>


The Need for Open Research in Software Security - <AUTHOR> http://breakingbits.net/2015/02/05/open-research-software-security/ <comment><USER2> Sheesh people, this is the second time this week that this topic has come up and no one mentioned the DARPA Cyber Grand Challenge! It was on 60 minutes this weekend for crying out loud. tl;dr DARPA is challenging companies to develop and improve upon the core ideas proven in SAGE.

[https://news.ycombinator.com/item?id=9012051](https://news.ycombinator.com/item?id=9012051)

[http://www.cbsnews.com/news/darpa-dan-kaufman-internet- secur...](http://www.cbsnews.com/news/darpa-dan-kaufman-internet- security-60-minutes/)

Dylan, seeing that you're in NY, I'd be happy to talk this over with you over a few beers at the next NYSEC: [https://twitter.com/nysecsec](https://twitter.com/nysecsec)

<comment><USER3> Wow, this is fantastic. Thanks for the link. I'll be reading up on this now.

Also, I'll take you up on that offer. I've been meaning to make an appearance there for a while.

</comment></comment><comment><USER4> Nice article. The reality is that mainstream IT security is driven by audit and compliance, not by engineers asking how can I break things/how can system X be more secure. Most companies want to be XYZ compliant (and nothing more than that). So they follow check-lists and continually get hacked and really don't care. They'll fire the C level executive responsible and hire another, but nothing really changes. Welcome to modern IT security.

<comment><USER3> Author here. Yes, that's a fair summary of what's happening. There's more too - investing in R&D for security is risky, tedious and sometimes doesn't result in fruitful tools or findings. The companies that can afford to aren't opening up their methods, and the rest can't really afford to develop the novel approaches necessary for real progress.

<comment><USER2> Sorry, but I'm not certain how you make a business model off that idea. To win, you need to ensure that people can build and use these tools to secure the world AND make money off it.

Second, investing in security R&D is not that risky if you know who to partner with. There are many organizations whose sole purpose for existence is to fund foundational->applied R&D across a variety of fields. You seem to have mentioned none of these opportunities in your blog post?

Third, if you're a software developer and you really care, the tools already exist to write secure code. However, there are few to no incentives to use them! I actually disagree somewhat that more R&D is needed on the security side. It's an inaccurate characterization of the state of the art in security and the degree to which people take advantage of it.

For exampleeeee, Mayhem has found and reported hundreds of bugs in Debian and did the work to submit bug reports with reproducible crashes and POCs. I think < 10 of the bugs they reported were fixed. I can't find the link right now, but the exact systems you referenced are being used out in the open to help people and no one is listening.

<comment><USER3> Hey Dan, thanks for responding.

I can't address your first point because I agree with it - part of the reason I'm pessimistic is because there isn't a strong business model.

However, I think that it's hard to know who to partner with. For all the organizations that you can partner with to produce fruitful research, you have many who are trying to make a quick buck on the trendiness of security. We don't have a lot of Mayhems. We have a lot of Acunetixes.

I agree that the methodologies and tools required to write secure code already exist. However, I don't think it's fair to place all the onus on software developers. Even if they avoid C/C++, they still have modern pitfalls in just about any language.

I think more research should be done on security software of course, which is why I wrote this blog post. What my post doesn't address however is that I also believe more research should be done in educating developers. It doesn't matter if all the tools to help them exist if few of them bother to look for them. Research could help in this regard - education.

Finally, I think you're much more familiar with Mayhem than I am, but I call into question the bugs Mayhem found. How many were high severity and/or actually exploitable? (I'm actually asking because I've seen conflicting reports from both Mayhem and Michal Zalewski on the matter).

When I say being used in the open, I mean in a similar way to Project Zero's activities. Mayhem was used once on Debian as far as I'm aware, and to my mind it looks more like publicity than an ongoing effort to find bugs.

<comment><USER2> > Finally, I think you're much more familiar with Mayhem than I am, but I call > into question the bugs Mayhem found. How many were high severity and/or > actually exploitable? (I'm actually asking because I've seen conflicting > reports from both Mayhem and Michal Zalewski on the matter).

Exploitable? Every. Single. One. Each came with a unique PoC demonstrating control of execution.

> However, I think that it's hard to know who to partner with. For all the > organizations that you can partner with to produce fruitful research, you > have many who are trying to make a quick buck on the trendiness of security. > We don't have a lot of Mayhems. We have a lot of Acunetixes.

This sounds like you simply haven't investigated where or how to get money for R&D!

</comment></comment></comment></comment></comment><comment><USER5> We are building an open source symbolic executor based of BARF [1] starting reimplementing pathgrind [2], which it is not working on modern Linux distributions because of the use of SSE2/3 instructions on the current libc versions. After we have a beta, we plan to use it submit bugs on the fuzzing project [3].

[1]: [https://github.com/programa-stic/barf- project](https://github.com/programa-stic/barf-project)

[2]: [https://github.com/codelion/pathgrind](https://github.com/codelion/pathgrind)

[3]: [https://fuzzing-project.org/](https://fuzzing-project.org/)

<comment><AUTHOR> What is your take on [http://blog.regehr.org/archives/1217](http://blog.regehr.org/archives/1217) and [http://blog.regehr.org/archives/1217](http://blog.regehr.org/archives/1217), who seem to be less than totally excited about the output of static analysis tools in terms of bugs found.

</comment></comment><comment><USER6> IMO as long as the research in these tools are being funded by corporate entities (e.g., Microsoft) then there's little hope of any open research.

Fortunately, there's money to be had for open source and research projects that are willing to organize and look elsewhere for some cash. Look at projects like Bro and Suricata -- commercial security tools which are government and educator funded.

<comment><USER7> The problem with open source security tools (e.g., bro, suricata, brakeman) is that they require security expertise to operate, continually. In my experience many small/medium organizations who actually care about security don't have such expertise in-house and can't find good security people to hire. This limits them to buying commercial solutions which (also in my experience) tend to blow.

We need more security engineers, but the problem is I don't even know what that job title requires. The author pokes fun at CISSP, but how else can I figure out if someone is 'good' at security? They are already so rare and mostly employed by google (joke).

<comment><USER3> _> > The author pokes fun at CISSP, but how else can I figure out if someone is 'good' at security?_

I'm the author.

If you are hiring a security consultant for your firm and you know how to judge infosec skill, use a work sample and check references.

If you're hiring a security consultant to perform a penetration test or audit for your (non-infosec) company, hire people who have a healthy mix of the following:

1\. Public, verifiable work (e.g. bug bounties).

2\. Solid references and past experience with clients who themselves understand what to look for in a security consultant. You obviously check these references. Alternatively, a solid reference that the candidate worked at NCC Group, Accuvant, Leviathan, etc.

3\. Research in the field, such as discovering a new class of vulnerability, publishing vulnerabilities in ubiquitous software, etc.

Prioritize #2, because not all adept security folks like to conduct research or participate in publicly verifiable work.

Of the certifications you can have, the Offensive Security[1] certs are pretty rigorous. For example, the OSCP is a good indicator that a candidate knows what they're doing to offensively test a client's network. That's about it. Almost all other certifications are run by people who have, at best, textbook knowledge of information security. People who get the CISSP can probably accurately describe a cross-site scripting attack to you in an interview, but there is no guarantee they can practically find it or defend against it.

The other issue is that while some certifications are good, a lot of folks in infosec just don't care for them. They can find high paying jobs in prestigious companies without a degree _or_ a certification of any kind, so they simply don't bother, even though they could pass it. This means that you can't reliably throw out candidates with no certifications...which circles back to my original recommendation. Work samples, references and public work are the best ways to judge a candidate's talent. I'm directly aware that this system is used at Matasano and Accuvant, and it's likely the norm at the other "quality" security consultancies.

'tptacek would have a lot of great advice to contribute on this matter as well.

[1]: [https://www.offensive-security.com/information-security- cert...](https://www.offensive-security.com/information-security- certifications/)

<comment><USER8> +1 for anything from OffSec I failed their OSCP once and after some reassessing I will not be taking for another year or so. However, there are many companies that list if not require a CISSP and other CISSP holding individuals are hesitant to degrade the efficacy of the cert.

<comment><USER9> Generally: a requirement that candidates hold CISSP is a strong negative signal about the job. This observation would have qualified as "insightful" 10 years ago, but in 2015 it's verging on conventional wisdom.

</comment></comment></comment></comment></comment><comment><USER10> This title is quite broad. There is plenty of open research into software security. It seems you're referring to open development of fuzzing tools. I am not really convinced that any kind of after-the-fact testing is the ultimate mechanism for making security guarantees. It's simply inherently flawed. I'm sure you make a lot of good discoveries with it, but when you're talking about 100% confidence I'd highly doubt it's going to happen. And in security, the difference between 100% soundness versus anything less is huge.

</comment><comment><USER11> I agree that commercialization has perverse incentives...but its a little ironic that we are talking about security research on commercial products. There is an academic side to security research in CoSci, but most security research is basically poking commercial products to see where they fail, and if that failure can be controlled.

That being said, Microsoft should open source MAGE. It's sounds cool.</comment>


Baseline Grid - <AUTHOR> https://observablehq.com/@gianordoli/baseline-grid <comment><USER2> It would be great to have native support in CSS for aligning text and block elements to a baseline grid (without having to fudge it with margins and padding.) I think the latest draft spec for this is CSS Line Grid[1], and it's severely underrated. With people these days doing most of their reading on the web, it's crazy that the web doesn't support what would be a basic layout mode in any desktop publisher.

[1]: [https://drafts.csswg.org/css-line- grid-1/#intro](https://drafts.csswg.org/css-line-grid-1/#intro)

<comment><USER3> _With people these days doing most of their reading on the web, it 's crazy that the web doesn't support what would be a basic layout mode in any desktop publisher._

Except it does, because (as you say) it can be done using current layout technology - what you call "fudging with margin and padding" is what I would call "using margin and padding". No doubt a design app to help work out those values, or some computation using CSS's calc(), or even some progressive enhancement using JS, would be helpful if the design calls for it but designers and developers desire to avoid those using what's there now should not be what drives web standards. "This is not possible with current tech" or "The current tech is inefficient and bad for perf" is what should drive standards, not "I don't like the current tech".

Also, as a bit of a side note, the draft spec you linked to includes the following line: "And careful calculations can be thrown off by user stylesheets." The author believes a layout being altered by the user's browser choices is a bad thing. I disagree with that _so much_. Giving the user the power to completely change their experience of a site is one of the best things browsers do.

</comment></comment><comment><USER4> Nice! I actually just released a Chrome extension that makes it really easy to render a grid–that includes a baseline grid–over any site. I hope others find this as useful as I have.

[https://chrome.google.com/webstore/detail/grids-by-color- set...](https://chrome.google.com/webstore/detail/grids-by-color- sets/ohmjbldcfanjibdmbfpdlffhagjedobb)

<comment><USER5> Thanks, that's actually quite useful for the layout stuff I'm trying to do on a personal project right now :)

</comment></comment><comment><USER6> It's funny to see that all grids fail on the web. Every single example i've checked the grid was 1-2 pixels off. It looked good, but it didn't match.

</comment><comment><USER7> Is it just me or is Observable kind of confusing? I say this as someone who's worked with Jupyter and JS plenty.

</comment><comment><USER8> I'm having a hard time understanding the source code at the bottom. Where are the functions defined? I see a whole bunch of `f`, without a definition.</comment>


Ask HN: Partnership Compensation for Side Project? - <AUTHOR> Does anyone have any suggestions on how I should structure compensation related to a side project I have that makes money?
- I am not incorporated yet. - I do not have immediate plans to raise for it, my main project is something else. - It's profitable, but not really profitable enough to just hire someone. - While it makes more money than it costs, it is not really a significant amount (yet).
The options I was weighing: - Get a cofounder and split it 50/50. - Hire contractors for one-off features (currently trialing this). - Get a partner(s) and split it something like 80/20.
Any thoughts? <comment> apotheosis-neko This depends on the side project and the type of work you want to get done. All tech? Or tech work with all of the other work that goes with a startup?

If you want full support 50/50\. If you plan on doing most things yourself then balance it, 80/20 or 70/30.

If it pays for itself, it would probably be best to keep 100% until the project is more formalized (a company exists). Equity is quite precious and is hard to get back if your teammate does not work out.</comment>


Quake in your browser - <AUTHOR> http://webquake.quaddicted.com/Client/WebQuake.htm <comment><USER2> Quake in the browser via javascript was done 5 years ago using GWT. I recall seeing that port at Google I/O years ago.

[https://code.google.com/p/quake2-gwt- port/](https://code.google.com/p/quake2-gwt-port/)

<comment><USER3> You linked to quake2-gwt-port, which is a Quake 2 port. But this is a Quake 1 JavaScript port called SiPlus' WebQuake.

[https://github.com/SiPlus/WebQuake](https://github.com/SiPlus/WebQuake)

I wonder how a JavaScript port done manually like this would compare to an emscripten port in terms of performance? It was a little choppy on my old laptop that can run Quake 3 decently.

</comment></comment><comment><USER4> ioquake 3 via emscripten: [https://developer.mozilla.org/en/demos/detail/ioquake3js](https://developer.mozilla.org/en/demos/detail/ioquake3js)

I need some help getting MP to work with WebRTC. If you're interested: [https://github.com/klaussilveira/ioquake3.js](https://github.com/klaussilveira/ioquake3.js)

<comment><USER3> How does your port differ from QuakeJS? : [http://www.quakejs.com/](http://www.quakejs.com/)

The source code is at: [https://github.com/inolen/quakejs](https://github.com/inolen/quakejs)

It has working networking through WebSockets. Might want to check that out.

<comment><USER5> I'm not familiar with either implementation, but webrtc (I believe) allows for UDP networking, which is better for FPS games.

</comment></comment></comment><comment><USER6> Oh, this thing is ancient!

Still cool.

[https://github.com/SiPlus/WebQuake](https://github.com/SiPlus/WebQuake)

There is a node.js multiplayer server included in the repo that works half- decent. :)

</comment><comment><USER7> For some reason it triggered vulnerability blocked warning on my device. Looks like one of the WAV files tried to execute code.

Screenshot: [http://d.pr/i/w2A1/2BrF07Eo](http://d.pr/i/w2A1/2BrF07Eo)

<comment><USER8> Woah, what does that mean now? I played through the first level, do I have to worry about an arbitrary code execution exploit having compromised my computer now?

<comment><USER9> The last chunk in the WAV file mentioned has a length of 0xFFFFFFFF, which I don't believe is valid. It's probably just referring to that.

</comment></comment></comment><comment><USER10> I'm on Chrome (on a macbook) and the mouse isn't working... Click works to shoot, but I can't look around using the mouse. Any ideas?

(I'm playing with just the keyboard so far... but I can see the end coming soon.)

<comment><USER11> Try turning on the mouselook. I think the key for that is '\' or something. Check the controls

<comment><USER12> tilda (~) brings up the menu.

should be "+mlook".

</comment></comment></comment><comment><USER13> It actually runs on the iPhone's MobileSafari! (although with quite a bit of stutter and the occasional browser crash)

<comment><USER14> My original Droid ran Quake 3 with no problems at native res at a super solid frame rate. Browser performance has come far, but it's still no match for native from what I've seen.

<comment><USER15> True _but_ web performance is advancing faster than native performance. It's catching up. Give it enough time and the difference won't be big enough to justify the additional cost of developing for multiple platforms when you can get 90% of the performance with a cross-platform code base. Whether that code base is HTML5+WebGL or something more like Unity with WebAssembly is another matter, but I believe, in 5 years or so, writing native code will be not be the norm for mobile apps.

<comment><USER16> Hm. Defying the laws of physics again, are we?

<comment><USER15> I'm not sure how suggesting a uiWebView wrapper might only use 10% of performance compared to native is really 'defying the laws of physics'. The reason it's catching up is because the browser is essentially is moving out of the way.

For example, right now, if I write a GLSL fragment shader and run it in a native app it runs on the GPU exactly how you'd expect. If I then use the same shader in JS using WebGL ... it does _exactly the same thing_ on the GPU. The performance is the same. In a few years the same will be for the rest of the code - it'll be loaded by a browser but it'll be running basically the same instructions on the CPU as a native app.

The browser chrome, security sandboxing, and a few other bits will slow things down a little, but not so much that we'll need to develop native code except for games that really push the limits of the platform they're on.

</comment></comment></comment></comment></comment><comment><USER17> I'd just like to take a moment to appreciate the domain name.

<comment><USER18> Quaddicted is a great site ran by a great Quake archivist/enthusiast (Spirit). He's helped keep back ups of Quake mods and maps and even written a java application to download and install them easily onto your Quake installation from his site. Just don't appreciate the domain name, appreciate the guy too!

</comment></comment><comment><USER19> +mlook doesn't work... any idea why?

<comment><USER20> It didn't work at first for me when I tried it in the very start of the game, but once I went through a portal to enter the 'select your episode' rooms, it started to work.

</comment></comment><comment><USER21> If I used my old PC with the 3DFX card, could I see through the walls to snipe other players?

Ahh, the good ol' days.

<comment><USER22> Voodoo or Voodoo2?

<comment><USER23> Orchid Righteous 3D, the original voodoo add-on card! Remember playing Fatal Racing / Carmageddon with it too!

<comment><USER22> I remember seeing screenshots of GL Quake and QuakeII in PC Mags taken on machines with 3DFX cards, and wanting one badly. However, no way my parents could spend the money on that.

Hat tip for being the elite of the elite PC gamers back in the 90's!

</comment></comment></comment></comment><comment><USER24> The game sounds still give me goosebumps.

<comment><USER25> Yeah I remember this being one of the first games truly to put the sounds deep into my brain, over grinding multiplayer over countless hours.. just learning from the sounds where the enemies were, who took the red armor and where, when the quad appeared ..

Those jump sounds going over the moving platforms in DM2, somebody rocket jumping in DM4 to the top, DM6 rocket camping and somebody coming behind the corner with the grenade launcher ..

Excellent sound design, Trent Reznor and the ID team did a really good job in spatial location via sound. I dont even know many modern games that have reached the same level, modern multiplayer games don't somehow use sound as well as quake always did to tell you what is happening in the environment.

Although not realistic in any way, but it made sense for the game.

</comment></comment><comment><USER26> ~MAP DM4

 _shoots a couple of shotgun blasts_

 _jumps into the lava_

 _dies_

yup, good 'ol quake

<comment><USER27> What a map that was! Used to love getting quad damage on that one and running amok.

<comment><USER26> Still one of the best deathmatch maps ever made! :)

</comment></comment></comment><comment><USER28> worked smoothly on my linux laptop/firefox :) I wonder if quake 3 would work...

</comment><comment><USER16> Awesome! But I was really hoping it was going to be multiplayer. It would be much cooler if you could just hop straight in to shooting at some other people...

<comment><USER29> I believe [http://www.quakejs.com/](http://www.quakejs.com/) has multiplayer.

<comment><USER30> too bad no one's on the servers :/

</comment></comment></comment><comment><USER31> Here is our experiment in online multiplayer shooter in your browser

[http://demo.krie.gg/](http://demo.krie.gg/)

</comment><comment><USER32> Cool as hell, kinda surprised it ran as smooth as it did.

</comment><comment><USER33> technology is so advanced that we can now run 19 year games in our web browsers and be impressed by it

</comment><comment><USER34> TIL it's really hard to play Quake on a Kinesis...!

Great stuff here. Thanks for sharing.

</comment><comment><USER35> Love this blast from the past.

I just spent 20 minutes watching the demo run through :-)

</comment><comment><USER26> Next year is the 20-year anniversary of quake btw! God I'm old

</comment><comment><USER36> Ctrl + W closes the tab while playing. Happened to me a lot :/

</comment><comment><USER37> Wow! It worked super well on Firefox 38.0.5/OS X 10.7.5.

</comment><comment><USER38> Quake was the first game i finished till the end.

</comment><comment><USER39> Awesome

i can hear the zenimax lawyers stampeding now though...

<comment><USER40> This is loading the WAD files from the demo version of Quake, which is fine per the Quake licence.

<comment><USER12> loading screen says "registered version". The shareware/demo did not.

</comment></comment></comment><comment><USER41> IE9 here. I has no quakes in my browser :(

<comment><USER25> Why are you running IE9 ?

<comment><USER42> I second this question, also, why would you expect it to work? I'd far rather they (or anyone) spent their time on clients (browsers) that support features than address issues which could be resolved by someone using a modern browser.

</comment></comment></comment><comment><USER43> Works on Nexus 7 Android 4.4

</comment><comment><USER44> I liked QuakeLive when that was still running. That shit was the bomb.

<comment><USER45> Its still running though a bit dead since StarCraft. They I was very active but they recently dropped Linux support so i cant play anymore.

They apparently did a rerelease on steam where they nerfed the game mechanics. For example strafe jumping is automatic now, causing a lot of rage at old school players.

Would love to see it back on SteamOS and Linux in the future. Such a gold game.

<comment><USER46> > For example strafe jumping is automatic now, causing a lot of rage at old > school players.

No it isn't, you can hold the space bar to jump, that literally doesn't change anything at all.

It's amazing how many retarded rumors sprung out of that release.

</comment></comment></comment><comment><USER47> wow!

Works great on an Acer C720 chromebook. Fantastic!

</comment><comment> anti-shill hardly worked at all on my PC with 3 gig of RAM.</comment>


What bug/time tracking software do you use ? - <AUTHOR>

What bug/time tracking software do you use ? <comment><USER2> We use Flyspray, and it works fine--very lightweight and easy to install anywhere with PHP available (we use it because it integrates relatively easily with the rest of our site).

I've also used Trac and liked it even more. Roundup is also good, if you don't need all of the extra stuff trac provides (wiki, timeline, etc.). If trac hasn't fixed their horribly broken authentication yet, then that would be a strong argument against it, but I can't imagine that it hasn't been fixed by now.

</comment><comment><USER3> We use JIRA from Atlassian and love it.

<http://www.atlassian.com/jira>

<comment><USER4> +1 for Jira.

The Confluence Wiki from Atlassian is also a very nice product.

</comment></comment><comment><USER5> <http://stufftodo.dedasys.com>

It's more of a task list/time tracker. It does the second about as well as is possible without a local client running and is alright at the first.

</comment><comment><AUTHOR> lighthouseapp.com looks like a good hosted solution</comment>


Mode Media shuts down, leaves bloggers unpaid - <AUTHOR> http://www.sfchronicle.com/business/article/Bay-Area-unicorn-Mode-Media-shuts-down-leaves-9228272.php <comment><AUTHOR> As someone who is currently in the Bay Area and whose personal and professional network on Twitter/FB/real-life revolves around the tiny, sad shrinking universe that is the media industry, I'll have to admit that the name "Mode Media" did not register a single clue with me. I've heard vaguely of "Glam Media" and I knew of Ning, which Marc Andreessen co-founded, and which was bought by Glam/Mode for a reported $120M-$150M [0]. This is not to say that Mode and its blog/social network was not _good_ (as plenty of good media companies have gone out of business), but it wasn't talked about much in the editorial sphere -- in contrast to even content-aggregators/farms such as Demand Media, ViralNova, etc. Which is weird, because traditional media is desperate for billion-dollar media success stories.

[0] [https://techcrunch.com/2011/09/20/glam-buys-ning- andreessen-...](https://techcrunch.com/2011/09/20/glam-buys-ning-andreessen- joins-board/)

<comment><USER2> Glam did traffic roll ups with and packaging for publishers -- amongst a bunch of other things. If you were on the revenue or ad-ops side you would know Glam.

Edit, here is some more info: [http://digiday.com/publishers/rise-fall-mode- media/](http://digiday.com/publishers/rise-fall-mode-media/)

</comment></comment><comment><USER3> I fully expect to see more announcements like this in the media industry over the next couple of years specifically for these digital media "startups".

They are raising money to grow their unprofitable digital business with cost structures that will never make sense... it costs them more to create each piece of content then the revenue each one brings in.

IE... the per unit economics don't make sense and they still won't when they scale.

<comment><USER4> Not debating your point, but how come in the old days media outlets were able to become profitable when they had (have) both union costs, and costs related to actually printing and distributing a physical good?

<comment><USER5> Probably because people actually paid for media at the time?

<comment><USER6> But people paying for a newspaper barely covered the cost of printing it, and even then, plenty were taking losses on the cover price.

<comment><USER7> The newspaper was the primary way to get ads into the home. Each city paper had a Google-like position in that city's ad market.

Quality could be higher because ad reach was based in part on subscriber count, not just the attention-grabby-ness of particular headlines.

</comment></comment></comment></comment></comment><comment><USER8> It's disappointing to see a company with such big-name investors, including having Andreessen recently on the board, closing down in a way that screws the employees.

<comment><USER4> And that's why you keep savings, especially if you work at a startup. In California you're only 1 day away from being jobless and only 30 days away from being homeless.

<comment><USER9> Fwiw most leases are a 60-day notice to terminate residency in CA, at least the ones for my tenants- I think that is pretty standard in the California Apartment Association. Also many counties in CA have renter protections in place so only just cause evictions.

But still, it's always good to have 6 months of backup emergency funds!

</comment></comment></comment><comment><USER10> They need to either pay their debts or go bankrupt. Just shutting down with bills unpaid is not an option.

<comment><USER11> If they can't reorg and continue operating, what would a bankruptcy filing accomplish? They're going to default on all their debts either way.

<comment><USER10> There may be some assets to auction off, and creditors are entitled to a piece of that.

<comment><USER12> All of the debt holders will get paid first, which means everyone else will get zero.

<comment><USER13> Don't unpaid wages get paid first?

<comment><USER14> As long as they're bonafide employees and not just independent contractors.

</comment></comment></comment></comment></comment></comment><comment><USER15> I understand there is a difference between cashflow and assets on a balance sheet but how can a company worth $1bn go bankrupt?

Does bankruptcy means something other than insolvent? Is liquidation or receivership a more accurate term?

Otherwise you would think a bank would lend enough cash to keep the lights on.

<comment><USER16> Step 1) Start a company that makes flurbles.

Step 2) I invest in a billionth of your company for $1.

Step 3) Your company is now valued at a billion dollars.

Step 4) Hire an employee

Step 5) Try to make payroll with that $1

Congratulations, your company is now an insolvent unicorn.

The reality is only _slightly_ more complex, but not much.

<comment><USER17> next gen underpants gnome logic

</comment></comment></comment><comment><USER18> It might make sense to force employer's to retain insurance that guarantees compensation to its employees/contractors. Executive compensation should be doled out last but is too often doled out first.

</comment><comment><USER19> Anyone can call themselves a unicorn

<comment><USER20> I thought it was specifically a word for new-ish companies with a $1B+ market cap that hadn't been bought by a larger, older company.

<comment><USER21> $1B is easy.

[https://signalvnoise.com/posts/1941-press- release-37signals-...](https://signalvnoise.com/posts/1941-press- release-37signals-valuation-tops-100-billion-after-bold-vc-investment)</comment></comment></comment>


Pew Study Suggests That Snapchat Only Has 26 Million US Users - <AUTHOR> http://www.forbes.com/sites/jjcolao/2013/10/28/pew-study-suggests-snapchat-has-26-million-u-s-users/

<comment><USER2> "Only"? That word isn't in the original title. In fact, the article states:

> though the study's findings fit other recent indicators pointing to the > app's strength

And

> "Snapchat is really on the rise right now," she says. "It's been on our > radar for some time, but we were pretty impressed with the number."</comment>


Covid-19 Herd Immunity Is Closer Than Antibody Tests Suggest, Say 2 New Studies - <AUTHOR> https://reason.com/2020/07/01/covid-19-herd-immunity-is-much-closer-than-antibody-tests-suggest-say-2-new-studies/ <comment><USER2> If it were the case that one third of the population of London are immune against covonavirus already that would have massive implications on public policy. But we don't really know yet, the science is still shaky on the issue, and we better get it right.

Why then is a think tank, whose purpose is to sell public policy, permitted to discuss science? We know what policy they would like to implement, we don't know if reality agrees, but instead of waiting for better data they are selling their wishes as actual fact. It doesn't work like that in the real world.

<comment><AUTHOR> > but instead of waiting for better data

Do you mean "waiting for peer review" or "waiting for additional replication"?

> they are selling their wishes as actual fact.

"First, a few caveats: Both studies are based on small sample sizes and neither have yet been vetted by peer review."

<comment><USER2> _Do you mean "waiting for peer review" or "waiting for additional replication"?_

That's kind of the same here. Peer review is a very low bar, it just assures that the grossest nonsense doesn't appear in print. The real review happens afterwards when others try to built on a piece of science. Right now we have a really daring hypothesis supported by extremely weak data, and the downside risks are too high to let that guide policy.

We had that before. Anyone remember that paper suggesting that the virus won't spread in a hot climate? The US government liked the idea while Europe went for containment and look where we are.

 _" First, a few caveats: Both studies are based on small sample sizes and neither have yet been vetted by peer review."_

The usual disclaimer for plausible deniability. In Plain English: "Here's crass nonsense that we like because we are paid for it"

</comment></comment></comment><comment><USER3> If that's the case, wouldn't we have had herd immunity in Belarus by now? Unlike even Sweden, Belarus has taken very few official mitigation measures against COVID-19.

We just had large gatherings for July 3 Independence Day, and previously for parades on May 9 as well.

If anybody's going to demonstrate herd immunity first, I'd guess it would be Belarus. But I haven't seen that in practice yet among people I know here.</comment>