Custom Processor Speeds Up Robot Motion Planning - <AUTHOR> http://spectrum.ieee.org/automaton/robotics/robotics-software/custom-processor-speeds-up-robot-motion-planning-by-factor-of-1000 <comment><USER2> Here's my interpretation of the result (posted in the hope that it will aid others):

The authors are interested in solving motion planning problems with both fixed and dynamic obstacles. They do this using a combination of offline pre- processing and online search.

During pre-processing a general-purpose PRM (read: state-space graph) is constructed using only information about the movement capabilities of the robot and the location of fixed obstacles in the robot's environment.

At run-time the location of dynamic obstacles is detected and all edges from the PRM which would result in a collision with these dynamic obstacles are pruned away. The remaining problem is easy: just find a shortest path in the remaining graph, from the start location to the goal position.

Anyway, it's this online collision checking operation which they implement in and parallelise with custom hardware.

Neat.

I wonder if in their experiments with software-only planners they also made a distinction between off-line preprocessing and online search? The paper doesn't seem to say. I would hope the comparison is apples-to-apples.

<comment><USER2> If I've said something wrong in the above I would appreciate a comment rather than an anonymous down vote.

</comment></comment><comment><USER3> I think we'll look back at movies like "The Terminator" and its slow lurching robots with a wry sense of humour. Instead the future is going to give us robots that can move so fast that they're just visual blurs.

 _As part of Mr. Lee 's good neighbor policy, all Rat Things are programmed never to break the sound barrier in a populated area. But Fido's in too much of a hurry to worry about the good neighbor policy. Jack the sound barrier. Bring the noise._

</comment><comment><USER4> The slowness of this arm makes it hard/impossible to highlight planning speed.

<comment><USER5> This is incorrect. There is usually a huge delay before any motion happens. The speed of that motion is independent of the planning speed. If you have to _replan_ during a move then yes the two can become intermingled, but if you watch most ROS based planning today what you see is a good 30 seconds of compute before the robot moves at all. Sometimes it can be multiple minutes!

</comment></comment><comment><USER6> I would like to know what about motion planning makes it more suitable for a FPGA than a GPU.

Wasn't able find any scholarly papers on this particular project searching the various references. Can any supply some more information?

<comment><USER7> It's very easy to go from FPGA "source code" to a foundry and get a chip with exactly the same thing implemented, and at a cheap price if your scale is large enough.

<comment><USER8> I said the same thing a year ago when I had FPGA synthesized verilog and decided to go get a real chip made. Closing timing along with optimizing for PPA for an ASIC vs an FPGA is a royal PITA. And that's just on the front end design work... A core on a FPGA gives you no real idea when it comes to placement or all of the other hurdles of the backend flow of a real ASIC!

We ended up re doing everything (which overall was a good thing, as we made a lot better decisions from the beginning by redesigning), but that's show biz.

<comment><USER9> This used to be Altera's "Hardcopy" advantage: an easy gate-array version of their FPGAs could (in theory) be made because: 1. the gate array uses the same gates as the FPGA, just the interconnect is customized. 2. Use Primetime compatible timing constraints for the FPGA design to begin with.

<comment><USER8> While obviously better than just the FPGA, it is still very inefficient compared to a real ASIC. Altera and Xilinx both stopped offering their programs in this area due to the fact that it as it is not really cost effective to have the same transistor level mask set and then have to buy all new masks on top once you get to sub 28nm processes.

As for primetime, we use Synopsys Synplify Pro for FPGA synthesis, and while it does a better job than Altera/Xilinx's tools, it does no where near as well (and works very differently) compared ot physical aware synthesis from Design Compiler or RC/Genus.

<comment><USER9> Primetime is just the timing checker.. where the syntax for sdc files originated I think.

We found that Synplify Pro is not quite as good as the vendor tools (xst/quartus) for FPGA synthesis, but I've not compared them recently.

Actually there used to be a version of DC for FPGAs, but it was not good at all. I think it was not as prepared to duplicate logic or flops as compared with the FPGA specific tools.

</comment></comment></comment></comment></comment></comment><comment><USER10> The Boston Dynamics robots seem to have realtime motion planning that works in unfamiliar environments, which seems to make pre-processing impossible.

Would anyone with experience in robotics hazard a guess on how they implement that?

</comment><comment><USER11> This could be used in reverse as a robot trainer for fencers by setting it to find a way to make a collision with objects instead of avoiding them.

<comment><USER12> This is a much easier problem to solve. There are certainly different ways in which objects can collide though, so that would make the problem more interesting and complex.</comment></comment>


I'm crazy: I ask job candidates to do homework - <AUTHOR> https://no-kill-switch.ghost.io/i-am-crazy-i-ask-job-candidates-to-do-homework/ <comment><USER2> Or some candidates go through the easy route and privately fork an existing solution from GitHub which they heavily modify it to submit that as their own to go to the onsite interviews.

"Take home projects" are not enough to identify this so-called "talent" that you are searching for. Just ask all the candidates if they have any significant open source projects or patches sent to relevant high-profile projects and you have raised the barrier to entry to weed out hello-world, junior, mid type candidates.

So yes, You're crazy.

</comment><comment><USER3> It's good for other employers that companies do this sort of thing.

Few people would be bothered doing up-front tests.</comment>


How Parents and Community Groups Use Data - <AUTHOR> http://mobility-labs.com/2015/how-parents-community-groups-use-data

<comment><USER2> sounds like a good reason for homeschooling [http://eus.sagepub.com/content/39/2/264.short](http://eus.sagepub.com/content/39/2/264.short)

<comment><AUTHOR> Homeschooling is great but a lot to ask of parents. My mom managed to stay very active in my education despite being a single mother working all the time. A few of my friends mothers were the same. I think if they would have been given the chance to understand the status of our entire class(where we were deficient or doing well), they may have organized in a similar way.</comment></comment>


Computing the maximum salary - <AUTHOR> http://mathoverflow.net/questions/6217/computing-the-maximum-salary

<comment><USER2> I was reading over the first response, and the details of the (second, correct) n-private sum computation, and I noticed they're omitting a small but important detail. The protocol has you distribute various numbers, to various people, the sum of which is your salary, and claims that no one gains any information about your salary. If the numbers you distribute are all positive, then when you give someone the number X they gain the information that your salary is greater than X, so the claim is incorrect. This problem can be removed by allowing the numbers you distribute to be negative, because then you could distribute a number higher than your salary. The ability to distribute negative numbers is critical to the n-private property of the protocol.

<comment><USER3> In the first version, the first person adds their salary, modulo 10 _N_ B, so you don't get any information.

In the second version, you are correct, with a sign error. You are presenting an _upper_ bound on your salary, which is probably not a serious problem. That said, you do need negative numbers to truly conceal all the information about the initiator's salary.</comment></comment>


GUI interface using visual basic to track the killers IP address CSI - <AUTHOR> https://www.youtube.com/watch?v=hkDD03yeLnU

<comment><AUTHOR> I could just use the command line, but let's face it a black screen and some text doesn't make for very good TV.

I think I would use something with better OpenGL support so I could do something more like the Jurassic park "this is unix" interface for tracking the killer.</comment>


Pernosco System Debuginfo - <AUTHOR> https://pernos.co/about/system-debuginfo/ <comment><AUTHOR> Yihaa! I guess this will provide the equivalent of a Windows Symbol Server but for Linux. Retrieving debuginfo and sources för Ubuntu is a mess. I've heard that Feodora is better but they're both leaps behind Windows in this regard. Bruce Dawson has sön a blog series about setting up a symbol server för Linux. It was fiddly.

<comment><AUTHOR> Here's one of Bruce Dawsons articles about debug symbols. It contain links to the rest of the articles in the series. I hadn't realized how much progress Feodora has made on providing a symbol server where packages can be downloaded based on buildid.

<comment><AUTHOR> [https://randomascii.wordpress.com/2013/02/20/symbols-on- linu...](https://randomascii.wordpress.com/2013/02/20/symbols-on-linux-part- three-linux-versus-windows/)</comment></comment></comment>


New Intel Microcodes for Windows 10 Released to Fix CPU Bugs - <AUTHOR> https://www.bleepingcomputer.com/news/microsoft/new-intel-microcodes-for-windows-10-released-to-fix-cpu-bugs/ <comment><USER2> > _Intel Microcode updates are not installed via Windows Update and must be > installed manually._

What does this mean? That only those who know that they need them should apply them? What should I do (as a software developer) with my machine?

<comment><USER3> A microcode update is a binary blob loaded to CPU on boot.

Modern Intel CPUs itself are sort of a runtime environment that has its language and program RAM. By placing said blob onto BIOS(UEFI firmware) ROM, CPU can find it, fetch it and load onto internal mini-RAM during power-up.

The structure of BIOS/firmware ROM differs among board manufacturers and it cannot reliably be altered from Windows by Microsoft(its all the same in actuality though). Thus the microcode update blob must be integrated into and installed by manufacturer or as per their instructions.

Personally I don't care. I only care about speed. I want the latest AGESA, but I opt Meltdown patches be left off.

<comment><USER4> The OS can update the Intel microcode at runtime just fine as well, and e.g. Linux and Windows offer such updates (like the one described by this very article). So BIOS/UEFI update/involvement is not needed.

</comment></comment></comment><comment><USER5> Cloudfare wants me to solve a capcha to access the site (firefox). Amazingly, if I turn off ublock origin, the capcha solves itself.

<comment><USER6> It's regrettable that this is current state. Google equates a login and recent activity with them as being a "legit person". Anything short of that means you're a malicious bot. The traffic dominance of Cloudflare and Google deserves some real independent scrutiny.

<comment> yjftsjthsd-h That's an interesting thought: If Cloudflare didn't exclusively use Google to determine if you're a human, I wouldn't mind them nearly as much.

<comment><USER7> they don't. they also have their own cloudflare "token" thing. can't remember the name of it right now.

found it: privacy pass

<comment><USER8> Being asked to install arbitrary code to run in browser extension privileged context is perhaps one of the only things they could have done worse than google's captcha.

<comment><USER9> For what it's worth, the extension is open source: [https://github.com/privacypass/challenge-bypass- extension](https://github.com/privacypass/challenge-bypass-extension)

<comment><USER10> And does not work almost always.

</comment></comment></comment></comment></comment></comment></comment><comment><USER11> I'll await some benchmarks before installing these. The last ones caused a 14% performance hit.

<comment><USER12> I could be totally wrong, but this was part of my justification for buying our new servers with AMD Epyc chips and not Intel. Mainly that the numbers made more sense, but also recent vulnerabilities and mediation issues.

</comment></comment><comment><USER13> Will these at some point be available through Windows update? If not then how does Microsoft expect the vast majority of non-tech savvy people to deploy these protections?

<comment><USER14> Yes.

</comment></comment><comment><USER15> [https://support.microsoft.com/en- us/help/4497165/kb4497165-i...](https://support.microsoft.com/en- us/help/4497165/kb4497165-intel-microcode-updates) has the full list of CPUs supported by the update.

</comment><comment><USER16> ...AMD seems hurt a lot less than intel by these exploits so far, and are also releasing some really cool hardware. It might be good to buy some stock and see how the next few years go?

<comment><USER17> IIRC (not a down and dirty systems guy, not familiar with parlance) AMD CPUs weren't speculatively executing transitions to lower rings, which was the crux of Meltdown (as opposed to Spectre).

The big thing with AMD from my outsiders' perspective was their gambit on chiplets, which improved yields at 7nm quite immensely and could be done by Intel at 10/5nm down the road.

In terms of money though the thing to look at is power consumption, which dominates mobile and server spaces. Zen2 is a marvel because of its efficiency, and should start to appear among the big purchasers for cloud services in the coming months.

</comment></comment><comment><USER18> Can I install these from Linux?

<comment><USER19> It should make it to your distro in the form of an intel-ucode package or similar name.

</comment></comment><comment><USER20> Does this disable any features?

<comment><USER21> Is speed a feature?</comment></comment>


Ask HN: How do you make logging consistent - <AUTHOR> This is a software design question: I have different components which do similar operations (similar, not identical) and I want to normalize the logs that each component emits so I can make diagnosing issues easier. What approach should I take?
One thing I have thought about is using an error code for a specific known fault and then emit a message based on the error code (think M$ error codes) - same message parameters for all components but the values differ.
I'm more interested in hearing how this has been addressed by HNers who have had to build this functionality. If there are references, please share. <comment><USER2> For inspiration, look at how the syslog api works. You want to make each log message a 3-tuple (timestamp, sender, message-text) and log that somewhere. Be it a csv-file, database table, stderr or whatever. If you are thinking about complicating it more, e.g by introducing additional fields to the 3-tuple, then remember that the logs doesn't have to be 100% perfect. As long as they are consistent enough for the user who is grepping them.

Logging error codes is not a good idea since they add a layer of indirection. The text "file not found" is always clearer than "error code: 412".

Another thing to keep in mind is to make your log messages distinct. If a user reports the error "Connection reset by peer" and there are 28 places in your code which can cause that message, then that is harder to debug than if the error only can be logged from exactly one location.

</comment><comment><USER3> If it were my problem, I'd probably start by normalizing the log files _after_ the log files were produced. Which is to say, I'd start by writing a program to normalize the log files rather than modifying the programs producing the log files.

The reasons I'd approach it that way is it would decouple the process of designing a new log format from the process of refactoring all the code generating the logs. At the point where I knew what was worth doing and could come up with a plan for doing it before touching all the components. Then I could implement changes as it made sense rather than all at once.

Good luck.

</comment><comment><USER4> We log every request by simply shipping JSON to a hosted elasticsearch cluster, which comes with Kibana to search and visualize all requests. We can log whatever we want, make a graph of whatever we want, and find whatever we want. We have full control yet don't worry about managing anything. We just use it when debugging errors, and to alert us of errors.

</comment><comment><USER5> Don't use error codes. Use structured messages, such that their serialization to JSON is easy. Then you can just put in any relevant fields and be done with that.

</comment><comment><USER6> Logstash is your friend.</comment>


3 SSL vulnerability discovered in 3 months, anything wrong?  - <AUTHOR>

-Apple's Goto Fail -GnuTLS certificate issue - OpenSSL Heartbleed
It may be a coincidence but they all have something in common:
- affects SSL. - 'simple' coding error. - had been there for a long time. - flaw affects the fundamental feature of SSL. Affects certificates specifically while not needing anything at all in the third.
It may just be paranoia but there are very disturbing flaws.
OR
Maybe people have a very bad coding and 'auditing' habit.
At an age where there are so many privacy and surveillance issues, it gets worrisome. <comment><USER2> _Gozer the Traveler. He will come in one of the pre-chosen forms. During the rectification of the Vuldrini, the traveler came as a large and moving Torg! Then, during the third reconciliation of the last of the McKetrick supplicants, they chose a new form for him: that of a giant Slor! Many Shuvs and Zuuls knew what it was to be roasted in the depths of the Slor that day, I can tell you!_

</comment><comment><USER3> Security vulnerabilities are often found in clusters, whether because of heightened attention or simply because coincidence & cognitive bias.

<comment><USER4> Exactly. Similar thing happened to Rails last year. One big vulnerability discovery led to a couple more being discovered soon afterwards.</comment></comment>


Google keyword advertising is waste of money, says eBay report - <AUTHOR> http://www.guardian.co.uk/technology/2013/mar/13/google-keyword-advertising-wastes-money-ebay

<comment><USER2> the beef is against buying your brand keywords. I've always found brand keywords to be the best CPA though - much lower than non-brand. plus, you can cookie returning users and take them out of your conversion calc to see exactly how many new customers you are getting and what it is returning you. maybe it is just that google ads are expensive and competitive and there needs to be better alternatives to reaching people. I've yet to have great success with FB but still working it cause I want alternatives.

<comment><AUTHOR> I've heard Facebook is cheaper but haven't seen a comparison on effectiveness yet. We're actually about to try a Facebook campaign. Would love to hear what you're trying and whether it works.</comment></comment>


HackerNews for iPhone out now - <AUTHOR> https://itunes.apple.com/us/app/hackernews-for-iphone/id642041103?ls=1&mt=8

<comment><USER2> Congratulations on the App Store release. I'm assuming that we're sharing a certain amount of vitriol regarding provisioning and code signing.

A few ideas that might be of your interest:

* Automatic text shrinking in UILabel is likely the numero uno bad feature put in place to encourage bad interface design. It makes font sizes inconsistent, it destroys visual rhythm, and it jags the lines. Try to implement variable height table views and be typographically consistent.

* UIActionSheet on iPhone should be presented from the bottom-most edge of the screen, not any random view, and not from the bottom of a view above the toolbar.

* The Comments view is using the grouped style with one cell per section. This is not a good idea because it simply introduces much more visual noise.

* Readability integration, read-later services and social sharing are all present in news:yc. They are not actually selling points when everybody can get it done practically in no time.

<comment><AUTHOR> Hey Evadne, The text shrinking was actually requested by 20 different beta testers, even though I initially hated the idea, and am looking at ways of improving it.

The UIActionSheet should come from the bottom, it's a bug I need to fix (that randomly came about as I deployed)

I accept the point on comments, I'm currently redesigning them for the next version.

As for readability/social etc, it's a useful feature, so why not include it as a bonus?

</comment></comment><comment><USER3> Could you try and sell it a bit more energetically to us? In what grounds is it better than the myriad of other clients that came before? How is it better than news:yc app, which has been recognized as one of the best around and Open Source?

<comment><AUTHOR> It's a personal preference thing, personally, I felt that News:YC was kinda bloated when all I really wanted it for was reading. My app was built to be nice and simple, with a core feature set, News:YC does have benefits for some people, like the ability to post, not everyone wants to though.

</comment></comment><comment><AUTHOR> HackerNews for iPhone is a new client for HackerNews, with readability integration, read-later services and social sharing. At 69p (99 cents), it's less than your morning coffee, and you deserve that right?

</comment><comment><USER4> Which api do you use to get this data? I'm curious about building an app (for learning purposes)

<comment><AUTHOR> I built a custom one in NodeJS as the existing ones aren't reliable enough. This is hosted on heroku, so I can scale as required.

<comment><USER4> Are you scraping the site to build it?</comment></comment></comment>


The evolution of computer languages  - <AUTHOR> http://www.extremetech.com/computing/91572-the-evolution-of-computer-languages-infographic

<comment><USER2> Previous submissions:

<http://news.ycombinator.com/item?id=2815112>

<http://news.ycombinator.com/item?id=2816073>

<http://news.ycombinator.com/item?id=2816862>

It's not highly regarded.

Just for fun, here's an alternative chart. It doesn't show language evolution, but it's a fun take on motivation - every language tries to fix something:

<http://news.ycombinator.com/item?id=2816526>

</comment><comment><USER3> This is a wretchedly bad infographic. It has terrible lacunae, incorrect facts, and the overall design implies a teleology that is utterly fantastic, at best.

There's a great graphic in the evolution and cross-pollination of computer languages struggling to get out; this one (<http://www.levenez.com/lang/>) is a lot better than the original article, although it severely lacks in aesthetic appeal and functional usability.</comment>


Is a BS Degree really worth it? - <AUTHOR> http://thedailyscrum.net/2012/12/03/bsing-about-bs-degrees/

<comment><USER2> Been thinking about this a lot lately. Opted for a 2-year program (Associate rather than Bachelor), because when I was 18 I figured I already knew how to code and didn't need to spend a lot of time or money to going to college to prove it. I'm still not sure if this was the right decision. Not much college debt, but there are a few doors that are closed to me.

Now that I do more interviewing I've also noticed a low correlation in candidates between education level and programming competency. It's important to note that this is /after/ they've passed a technical phone screen though. I'm sure for the general population the numbers are much different. I think all my experience is reflecting is that if you know your stuff, it doesn't make a huge difference how you learned it. I'm also wary of some personal bias. :)</comment>


Continuous Deployment: just ship your code to customers as often as possible - <AUTHOR> http://timothyfitz.wordpress.com/2009/02/08/continuous-deployment/

<comment><USER2> _> Continuously deploy. Every commit should be instantly deployed to production._

ah... no. Maybe this is just viable for a single developer, as a substitution for continuous integration. But with multiple developers checking in, on a complex system your site will be down. A lot.

If you're frequently having problems where new deployments are bringing down the site, then doing that more often isn't the answer. As un-fun as it may be, you should be looking at how you could have caught the problem _before_ it went live, whether that entails more testing, more robust design or something else.

<comment><USER3> Agree'd. It seems like the author either has no customers or very understanding customers. Pushing every commit to production is just non-sense by any metric. What about commits that muck with the data model where bugs might potentially destroying production data? What about complex developments that consist of more than one commit?

This almost reads like a link-bait. I somehow doubt the author really believes what he's writing there.

<comment><USER4> Well, it's certainly not "no customers"... Timothy is referring to IMVU. Whether the customers are understanding or not is another issue. However, when we implemented this "cluster immune system", we had outages all of the time. But every single time, we took steps to prevent that class of failure from happening again, and now we deploy code to the cluster twenty times a day.

<comment><USER3> Well, I'm not against frequent deploys. I just found the article a bit light- hearted in tone - as if no testing at all was going on.

No matter how many classes of failures you have ironed out, in any reasonable complex system you will still regularly have regressions that are not caught by a quick "two eyes" check.

</comment></comment></comment></comment><comment><USER5> I wonder if you could do some automated A/B system here, and instead of every commit, just do it once a day: Every night, update one set of servers with the latest stable branch code. Route N% of your user to those servers. If faults start showing up, push N to 0 and notify developers. Otherwise, slowly move N towards 100 throughout the day. By end of the day you're either fully deployed, or rolled back.

(I already see faults with this...it's presented here as a thought experiment, not advice)

<comment><USER6> What faults?

I ask because I swear A/B and multivariatate tests have been around my head a lot lately, and when I finished reading the article, the first thing I thought was: Why not just deploy to 1% of the users and see if it works?

Then I thought about how hard would be to manage multiple versions of the same software, specially data, amongst different user. Certain features presented to the 1% might be incompatible with the other 99%. But that's a technical problem. Very hard to solve, but manageable. Then I imagined somekind of framework that would make communication between different versions of the data floating around easier, with "how to transform" data from-and-to version 1.1 and 1.2 easily.

Anyway, I am really curious, because it sounds like a good solution :)

<comment><USER7> As far as data transofmration goes you have two options:

1\. Something akin to ActiveMigration is RubyOnRails world. This allows going back and forth different versions of your data's schema.

2\. Use a more open data scheme such as Google App Engine uses where adding/removing properties to an object is not as disruptive compared to SQL- based solutions.

<comment><AUTHOR> It turns out the system works fine with SQL based alters. We do have to do real work to deploy expensive alters (apply them to standbys, fail over, repeat, or worse) but in general it's cheap to change schemas.

Unfortunately, it's very manually intensive to roll back schema changes, so it's one of the few places where we put old school process in place (a DBA who reviews all schema changes prior to deployment)

<comment><USER7> Did you look at ActiveMigration? Even though I never used ruby/ror I found it a very sensible approach to thinking about schema evolution.

<comment><AUTHOR> ActiveMigration really solves a different problem. Our problem is that adding indexes or altering popular tables is impossible to do on a live and in production database. To get those changes out we have to go through quite a bit of extra work. It's really a MySQL limitation, not a process problem.</comment></comment></comment></comment></comment></comment>


If It Were Up To Me, The iPad Would Have A Touch Sensitive Case  - <AUTHOR> http://www.techcrunch.com/2010/01/26/ipad-touch-sensitive-case/ <comment><USER2> I hope they got millions for that Blackberry AD at Techcrunch.com/welcome // the 'skip this ad' button doesn't work in fact it just reloads the ad (chrome 4.)

<on topic> If the device is a good size, why not have a keyboard (touch keyboard) on the back with a phantom (see thru) screen showing your current finger position /on the bottom 30% of the screen? ~ sounds far fetched though.</comment>


A Mathematician's Lament (2002) [pdf] - <AUTHOR> http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf <comment><USER4> "There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could all be having so much more fun."

As a Mathematician I've always identified as a creative person, yet struggle to convince "artsy" persons that what I do is creative. As soon as I utter the word math they convulse and shutter as if they were afflicted with a PTSD flashback. I try explaining what the idea involves and draw a diagram or three, but all they do is just nod and say "yeah, yup, okay.." Are the ideas of math that inaccessible to the general population when compared to a work of Art?

<comment><USER5> I think that language (at least English) fails us here. The word "creative" can be used to mean both "relating to or involving the imagination or original ideas, especially in the production of an artistic work" [1] or "resulting from originality of thought, expression, etc.; imaginative" [2]. It seems that most people identify the word more strongly with the artistic sense of the word, in which creativity is a proxy for a kind of self-expression that is not bound by any rules, logic, or structure. In doing so, they seem to mistake one of the more visible manifestations of creativity with its essence, which really lies in the "originality of thought" and "imagination" of the creative person.

If we had a specific, unique, and widely used word for the "artsy" free- expression type of creativity, I think the confusion many people express when you try to convince them that mathematics is a creative endeavour would be greatly diminished.

[1] [http://oxforddictionaries.com/us/definition/american_english...](http://oxforddictionaries.com/us/definition/american_english/creative)

[2] [http://dictionary.reference.com/browse/creative](http://dictionary.reference.com/browse/creative)

<comment><USER6> > creativity is a proxy for a kind of self-expression that is not bound by any > rules, logic, or structure

I do not find this to be the case. Art is full of structure, rules and logic. When artists "break" the rules, it usually means they been able to operate using _underlying_ rules, and understand well the rules they break.

<comment><USER7> Or they're creating new rules.

<comment><USER6> Yes, indeed. For instance, the creation of bebop in jazz. Of course Bird could only do this by a comprehensive understanding of the existing rules and seeing deeper.

</comment></comment></comment></comment></comment><comment><USER8> _" but later in college when they finally get to hear all this stuff, they'll really appreciate all the work they did in high school."_

So painfully spot-on. My mathematical education was horrible. Meanwhile I had been writing code since I was a little kid. It wasn't until I was an adult that I realized how much math I had been learning while programming. And worse, that I had been completely miseducated about what math actually is.

<comment><USER9> Interestingly, the converse is also true.

I was always into computers and tech but never really programming.

I was a math major in college, however, and after graduating I started programming. The transition was almost seamless, I picked up programming really quickly, it was surprising to me how much the "ways of thinking" are alike.

<comment><USER7> One of the key mistakes of educational strategy is dividing up knowledge into a bunch of subjects like "Language", "Math", and so on. You can find linkages in just about everything; this siloing really only serves to make people think they're particularly good at one thing and not another, when really it's a preference for one perspective over another.

<comment><USER10> So the pragmatists question is then naturally, how do you teach it? How do you shard the curriculum if not by subject? It seems a bit overbearing to ask each teacher to have the proper renaissance man's education of knowing a bit of everything.

<comment><USER7> To me, a teacher's proper role isn't having knowledge and dumping it into someone's brain. It's knowing _where to find_ that knowledge. My three Rs are "reading, research, and reflection": a teacher's job is to (1) provide useful material to consume, via lecture or homework or whatnot, (2) point towards larger resources for further exploration, and (3) guide thought processes to make useful conclusions.

A teacher's job is not to teach. It's to provide a space in which a student can learn. A focus is useful for this, but the focus doesn't need to be an abstract subject. It's a MacGuffin; it can be anything.

</comment></comment></comment></comment></comment><comment><USER11> I read this a couple weeks ago and decided to try and find a math book that incorporates some history in it. I found _Journey through Genius: The Great Theorems of Mathematics_ with some great reviews.

I am working on an iPad app that pairs up people to mentor each other through books like this. It has video chat and a shared whiteboard, so it is ideally suited for discussing math. If anyone is interested in reading the book with someone, email me at firstname.lastname@example.com. I could really use some beta testers for the app!

If you have already read it, you could still mentor someone in it to review the material again.

<comment><USER12> To add to the suggestions here, "Number" by Tobias Dantzig is absolutely a wonderful history of mathematics. I mean, shoot, it's actually got a quote from Einstein:

"This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. If people know how to treasure the truly good, this book will attain a lasting place in the literature of the world. The evolution of mathematical thought from the earliest times to the latest constructions is presented here with admirable consistency and originality and in a wonderfully lively style."

—Albert Einstein

[http://www.amazon.com/Number-Language-Science-Tobias- Dantzig...](http://www.amazon.com/Number-Language-Science-Tobias- Dantzig/dp/0452288118)

<comment><USER13> if you are in a hurry:

[http://www.engineering108.com/Data/Engineering/Maths/Number_...](http://www.engineering108.com/Data/Engineering/Maths/Number_the_language_of_science_by_Joseph- Mazur_and_Barry-Mazur.pdf)

</comment></comment></comment><comment><USER14> Still haven't finished reading this and all the painful memories came back from middle school. Not math (although it too wasn't a fun experience) but art. For four years we had a teacher that thought music and visual arts by dictating. For 45 minutes we would write down everything word for word. Then after a few weeks she would ask us broad questions and we'd have to recite everything back to her. I still remember (I'm almost 30 now) rote learning from my notebook, walking around the house, repeating those stupid sentences in my head, memorising descriptions of works of art I've never seen, painting techniques I've never witnessed, music I've never heard. Wow, just talking about it makes me angry, makes me want to to choke the bitch's neck. Sadly, that's the state of schools in Croatia, people equate schools with rote learning and memorising. You have to do it to get into a good university and get a good job. Who will create those jobs? No wonder the country is in the shitter.

<comment><USER14> oops, meant taught instead of thought.

</comment></comment><comment><USER15> Love this essay. I read it years ago when my brother was working with Paul Lockhart, who deeply influenced him as a math teacher.

My brother and his wife have since started an organization called Math For Love ([http://www.mathforlove.com](http://www.mathforlove.com)) focused on changing the way math is taught. They run workshops for teachers and provide great material for students.

If you're in Seattle and interesting in pedagogy and math, you should check them out.

</comment><comment><USER16> I love this article, but: what can a practicing math teacher take away from it? How can you apply this stuff if you still have to teach a standard curriculum?

I'm really asking -- my friend is about to start as a high-school math teacher.

I guess the first recommendation would be: motivate every new technique by starting with one or more problems that the technique helps to solve. (Here "problems" is meant in the Lockhart sense -- real puzzles, not exercises.)

But how often are "techniques" actually taught in high school math, especially algebra and precalculus? A lot of high school math consists of digesting new definitions, or the generalization of old definitions. A fair amount of it consists of learning theorems that go unproven, or that are proven (by the teacher) too quickly for students to understand where they come from -- and in general it isn't satisfying to solve a puzzle with a theorem that one doesn't actually understand.

On top of that... students have to spend time with problems before they become genuinely interested in their solutions, so progress would be slower with this method. It's not clear that you could teach a whole year's curriculum in one year like this. (And if you fail to do that you'll eventually get fired.)

Any insight? I believe that it's possible to teach math, even standard high school curriculum, in such a way that students are at all times intrinsically interested in what's presented. But it would be awfully hard to do at scale, at the standard pace, as a high school teacher would have to. How might a teacher start in that direction?

<comment><USER17> I co-founded Dev Bootcamp and while I was still there one of my not-so-secret missions was to make mathematics less alienating. I only say that because _it was incredibly difficult_ , even in an environment where I had complete autonomy and authority to make whatever curricular and pedagogical decisions I wanted. The problem becomes combinatorially more complex in a public school where teachers have much less autonomy, have to teach to a common set of state-wide standards, and have students of varying levels of interest.

Here are my scattered thoughts, though. I'm going to try to not suggest a pie- in-the-sky solution like "new curriculum!"

First, I majored in mathematics at the University of Chicago, but I hate, hate, _hated_ mathematics in high school. Take something you'd see in Algebra II like matrix multiplication, matrix inverses, and solving systems of linear equations. You're presented with these things called matrices and taught a bunch of rules. Where did these rules come from? Why are we calling this "multiplication" when it doesn't look or act anything like multiplication?

And sure, I see that when I go through the steps you tell me to go through like a monkey I get an answer that works, but how do we know there aren't more correct answers? How did anyone even come up with these steps in the first place? It's not like someone sat down and tried a trillion random combinations of symbols and steps until one of them happened to work.

Augh. In that world the only recourse for students is to memorize, usually just enough to do the homework or pass the test, and then promptly forget. The only experience they associate with math is the utterly humiliating feeling of being terrible at it.

So, I think that's one of the root problems. People remember what they feel and most people remember feeling stupid, humiliated, and possibly ashamed when it comes to mathematics. It's only a matter of time before that becomes part of their identity. "Oh, I'm terrible at math. Oh, I'm not smart enough to do math." and so on.

If I were a HS math teacher my top priority would be to watch out for when those counterproductive, self-defeating beliefs were forming and do whatever I could to preempt them.

Second, I think the way math is taught is overly symbolic. What most non- mathematicians don't realize is that when most mathematicians look at a set of abstract symbols they don't "see" the symbols _per se_ , they see what those symbols are meant to represent. They freely move between a geometric and algebraic picture of the world, but the algebraic picture is usually incredibly compressed.

I think the key thing is not to pick a side -- algebra vs. geometry -- but to show the relationship between the two. Geometric objects admit a symbolic representation and vice versa.

Third, students have this idea that math is all about being "right" or "wrong", that it's "black" or "white", that there's some universe of Proper Math that is insisting on certain rules for no rhyme or reason

Here's a silly but illustrative example that I think students would cover in 6th or 7th grade: order of operations.

Hey class! Look at this expression: 4 _5+6. What does it equal?

A bad teacher says "It's 26 and any other answer is wrong." An ok teacher says, "Remember the order of operations. If we apply those rules we get 26, so that's the right answer."

A great teacher shows their students that some things are necessarily true and other things are definitionally (or conventionally) true. This teacher would do something more like...

Who got 26? Who got 44? Students who said the answer was 26, how did the students who got 44 arrive at their answer? Students who said the answer was 44, how did the students who said 26 arrive at their answer? Neither of you are wrong _per se _. We could have chosen to live in either world, but we have to choose one consistent set of rules.

These rules lead us to 26. If we chose the other set of rules, we'd get at 44. We only do this because we don't want to have to write down parentheses all the time, but without them it's unclear what order we're supposed to apply + and _. So we need to agree on a set of rules so that two people looking at the same expression both understand how to make sense of it.

It's like traffic laws. There's nothing stopping people from driving on the left side of the road. In fact, there are countries where everyone does drive on the left side of the road. The important thing is that everyone agrees on a convention -- left-side or right-side. It works as long as everyone agrees and breaks if people don't.

I could go on, but I'll stop here. Like I said, these are my scattered thoughts. :)

<comment><USER18> > when most mathematicians look at a set of abstract symbols they don't "see" > the symbols per se, they see what those symbols are meant to represent.

Might we benefit from a different set of symbols that actually convey the geometrical meaning behind them? If instead of π, we used a glyph that shows a circle over a diameter, instead of x for a variable, we show an empty rectangle that shows that it's a placeholder for a value?

> Students who said the answer was 44, how did the students who said 26 arrive > at their answer?

I came across a great example of this approach in Chess: The Complete Self- Tutor by Edward Lasker. Instead of just showing the right answer for a chess puzzle, he tells you what you did right in your answer and what you missed in getting an even better answer. This was a printed book that was completely interactive.

</comment></comment></comment><comment><USER19> _" Michelangelo decorated a ceiling, but I'm sure he had loftier things on his mind."_

That's almost Douglas Adams levels of dry wit.

This is a fascinating article, as someone who's never really contemplated the playfulness of maths....I mean, for sure the wonder of maths or the power of maths - I've just never really put the pieces together to link that back to my grounding in maths, beyond the practical, functional stuff I incorporate into my daily life, without considering it's mathiness. Very interesting.

</comment><comment><USER20> This is intensely thought-provoking and beautifully well-written. It's not directly about hacking, but it's the type of treasure that hackers love to stumble upon.

It's worth noting the times when HN really delivers. I doubt I'd have come across this anywhere else.

<comment><USER21> >It's worth noting the times when HN really delivers. I doubt I'd have come across this anywhere else.

At risk of getting to meta, this piece seems to pop up everywhere. I don't mean that in a bad way, but this is at least the fifth time over the course of many years that I have seen this piece pop up on completely unrelated sites.

<comment><USER22> Not to mention it's been here on HN at least 8 times, probably 10 or more. I've stopped bothering to direct people to previous conversations - they just say the same things over and over again anyway.

</comment></comment></comment><comment><USER23> I thought I had been taking math my whole life until I finally took a math class.

<comment><USER24> Back when I was in academia, one of my favorite student evaluations from an abstract algebra class had the following comment (paraphrased from memory):

"This course was like climbing Mt. Everest. It's difficult, and sometimes slow going, but at the end, it's breathtaking how much you've learned and accomplished."

</comment></comment><comment><USER25> The article resonated with me on some level, because it does take a long time to learn how to actually do math. If you are at the point of just doing algebraic manipulations on equations to try to figure something out, you've lost the battle (as opposed to using algebraic manipulations to _encode_ your thoughts, and work out the details).

On the other hand, I think everybody really did see the beauty in geometry. Yes, the initial manipulation to prove something about symmetric angles is a bit silly, but you aren't being taught symmetric angles here, you are being taught how to do a geometric proof. Which is not easy to learn, so you start with something super simple and obvious. I didn't observe anyone in my classes (well this was back in 1982, so memory is a challenge here) confused about that point. And, as soon as we learned it, the requirement for formalism was dropped. It was the same in algebra. In the first weeks you weren't allowed to go directly from x + 3 = 1 to x = -2. You had to do something like x + 3 - 3 = 1 - 3; x + 0 = 1 - 3; x = 1 - 3; x = -2; with all the rules that you are using written out. Annoying yes, once you grasp it, but once you proved you grasped it that was the end of that, and we never had to do it again.

But, I had good teachers that always tried to explain the 'why' of what we were doing, and did not make us engage in pointless formalisms. But you need to understand terminology like ABC in geometry; when you get to tough problems you'll be using it. So, learn it with the easy problems.

<comment><USER21> My best math teachers never made us engage formalisations. What they would do is manipulate us into having an arguement, or construct an apparent contradiction. We then discussed the problem until we all agree (often without the Teacher talking). The inevitable result of this is that everyone learned formalisation: because it makes for really convinving arguements.

</comment></comment><comment><USER26> I hated math until I got to calculus. I never knew why, but I might as well quite this mathematician... They took an exciting topic that interested me as a youth and killed it with rote repetition.

The question I struggle with is... How do you really get by without math as a mandatory topic? How can you teach science? How can you teach someone to balance a checkbook? The current system is awful, but do we push the subject to electives as a result?

<comment><USER27> > The question I struggle with is... How do you really get by without math as > a mandatory topic?

That's the wrong question, because "mandatory learning" is a contradiction in terms. It very clearly doesn't work. Beyond the most primitive pavlovian conditioning, you can't force people to learn if they don't want to. Intrinsic motivation is the dominating factor in how well a person can master an intellectually demanding task.

The right question is, how do you inspire people to want to learn?

<comment><USER26> True enough. And that's a tough question too. I waited from 4th grade to calculus to get inspired again. And then I think it was the material more than the teaching method.

</comment></comment></comment><comment><USER28> The author of this article teaches a course on coursera that I can't recommend enough, he put so much effort into that course. [https://www.coursera.org/course/maththink](https://www.coursera.org/course/maththink) As you might have guessed its not really about 'math' in traditional sense.

</comment><comment><USER29> This article just blew my mind. I feel little silly, especially since I majored in my math. I have never seen these types of ideas expressed so clearly and powerfully.

He is 100% right. We should not be teaching the formulaic rules of basic arithmetic until high school, and only then as a part of life skills class. It should be taught as the art form that it is.

</comment><comment><USER30> As an erstwhile math major (I couldn't hack the honors basic algebra class - the difference between a euclidean domain and a principal ideal domain got too confusing; but I rocked proofs in analysis) I have to say that the author is confusing Mathematics (which is an art) and Arithmetic (which is a skill). Part of what make the opening farce absurd is that musical skills and painting are not terribly necessary as a matter of life and death, or even to a certain degree, quality of life or death; whereas understanding sums and compounding processes ARE.

While SALVIATI is completely correct in his analysis of the situation, he offers no solution, and I identify with SIMPLICIO more.

Perhaps the problem is that in our schools we conflate arithmetic with mathematics. Surely, they are related, but perhaps they need to be delineated and the difference understood.

<comment><USER7> > whereas understanding sums and compounding processes ARE [terribly necessary > as a matter of life and death, or even to a certain degree, quality of life > or death].

That's not actually true.

<comment><USER30> in an ideal world, I would agree with you. We do however live in the real world.

<comment><USER7> Please give one example applicable to a reasonable majority of human beings on the planet at this current time where "understanding sums and compounding processes" are a matter of life and death.

<comment><USER31> It seems probable that the subprime mortgage crisis was a contributing factor in a number of deaths, via suicide, stress-induced illness, or, with the help of alcohol, violent or vehicular incidents.

<comment><USER7> But

(1) The subprime mortgage crisis could not have been averted by a larger percentage of the population understanding sums and compounding processes.

(2) The advent of civilization was a contributing factor to everything that's happened in the last several thousand years, including the paper cut I just got.

<comment><USER31> you understand that interest-bearing loans are a compound process, right - <USER2> that an understanding thereof might be of relevance to someone entering into a loan agreement which they probably, if they really understood it, were not going to be able to repay...

But let's not only blame the borrowers of subprime loans, let's also ask whether the banks didn't entirely understand the risk models of the derivatives they were compiling out of subprime mortgages because many of their senior managers also didn't understand compounding processes, or, possibly, sums...

<comment><USER7> > an understanding thereof might be of relevance to someone entering into a > loan agreement which they probably, if they really understood it, were not > going to be able to repay...

Oh, it's relevant. It's sort of like how being able to load and fire a gun is relevant to the decision whether or not to commit suicide. Sure, it can modify one of the many, many details, but some people just make a noose and hang themselves.

Keep in mind, this is your argument: if more people in the world understood sums and compounding processes, this would have prevented the subprime mortgage crisis and thus the many deaths that were inspired by the resultant fallout. There is no possibility that anything else caused the crisis, and no possibility that anyone is at fault for these deaths other than parents and teachers.

This claim, if true, actually _absolves_ the lenders that you talk about below, because they can be held responsible only for _their own_ understanding of sums and compounding processes.

Amusingly, if you accept your argument, you can also make an interesting inverse version. The fact that lenders understood sums and compounding processes led to their employment at unscrupulous institutions which then mandated their sign-off on high-risk mortgages, which then caused the deaths of all those people.

Math, apparently, kills.

</comment></comment></comment></comment></comment></comment></comment></comment><comment><USER32> I've read this article several times at this point (it does tend to pop up everywhere) and it resonates with me but I'm not sure what to do about it.

I really want to experience the kind of math the author writes about; can anyone recommend a place to start as someone who has only ever done "fake" high school math? I'm in college now and I'm halfway through a computer science degree; I've tried a few times to break into theoretical math classes but I've found the bar for entry pretty high (especially when I only have room for one or two courses), with most classes and even peers asking for years of experience and "mathematical maturity." Have any of you ever succeeded in learning some math outside formal curricula?

<comment><USER33> Hey Imartel -- I've been in a similar situation. I think my first starting point was "Mathematics and the Imagination" or "Gödel's Proof." Mathematics and the Imagination is a good high level overview, and would provide some foundational notions that'll reappear repeatedly -- but, it won't given you any practice in mathematical methods. For that I would recommend "What is Mathematics?" by Courant and Robbins. Can be pretty challenging, but you'll actually get somewhere if you put effort into it. If you haven't had much experience with proofs, it's worth focusing briefly on the process of proving explicitly as a preliminary. That's actually a good thing to do with another person or in a class -- can be pretty difficult to get some of the subtleties involved, and to know when you've done things correctly or not.

I mention "Gödel's Proof" because it's the first thing I came across that informed me I am in fact interested in mathematical systems. I'm more interested in architectures than problem solving though. If you suspect that might be the case for yourself, might check it out -- it's about 100 pages.

<comment><USER34> Courant and Robbins is really great. I'd recommend checking it out after the Lockhart book (or together with it), as it's more textbookish; there's more danger of bouncing off.

</comment></comment></comment><comment><USER21> During high school, at the end of the year everyone in my Calculus class had to make a math presentation (to go along with a paper). During lunch one day we stood with our posters to explain our project the the students that choice/were encouraged to attend. I explained my topic (re-ordering conditionally convergent series) to a group of students. They seemed to understand what I was saying, showed awe at some of the key insights/tricks, and even interupted me to excitadly finish the arguement. Then, when I was done, they pointed to the equations and asked me to explain them. My response was that those equations are exactly what I had just said, and I repeated the arguement while showing them where it is in the equations.

</comment><comment><USER35> Richard P. Feynman

"Physics is like sex: sure, it may give some practical results, but that's not why we do it."

</comment><comment><USER36> We really need to teach people _how_ to teach induction, which is only done right when you put quotes around your Boolean statements; the "implies" symbol gets jumbled up with everything else otherwise, and not using it at all is passing up on a great tool. One can do simple proofs-by-induction without a single English word, completely symbolically, and have it be understood easily, if one uses quotes and correct LaTeX formatting (or good handwriting)

Induction doesn't just involve numbers and equality signs, it involves _statements_ with variables inside of them, and non-programmers need to be made well-aware of this (and taught Boolean logic early, PLEASE)

<comment><USER37> There was a time people were taught logic and geometry and math and history all at the same time. Also, philosophy.

The reductionist approach of our education is it's major failure.

</comment></comment><comment><USER38> A classic issue that mathematicians, philosophers, and even computer "scientists" have is the idea that they can somehow reason their way to the truth. Oh, he makes a good argument. But the greeks made great arguments about how the sun goes around the earth.

I could say that this or that argument is flawed. But really, the only valid arguments are data. His data is severely lacking, and many modern methods of teaching are much more supported by experimentation. Logic is a tool for finding logically consistent imaginary realities. Science is a tool for finding out about this reality.

</comment><comment><USER39> I had to overcome my school-age brainwashing in order to enjoy math. One book that helped me was "Who is Fourier? A Mathematical Adventure" ([http://www.amazon.com/ho-Fourier-Mathematical-Adventure- Edit...](http://www.amazon.com/ho-Fourier-Mathematical-Adventure- Edition/dp/0964350432)). I highly recommend it for those looking to find enjoyment in math.

</comment><comment><USER28> Attempts to present mathematics as relevant to daily life inevitably appear forced and contrived: "You see kids, if you know algebra then you can figure out how old Maria is if we know thatshe istwo years older than twice her age seven years ago!" (Asif anyone would ever have access to that ridiculous kind of information, and not her age.)

I wince every time I see these examples in my niece's textbook. So ridiculous.

</comment><comment><USER38> For music, the formal curriculum does involve memorizing songs and scales and whatnot. The problem is that the best way to start a subject is to fuck around with it (playing random songs you like on your guitar, drawing cool shapes and noticing patterns in them, etc), but our schools are ill equipped for such things, and enabling this style is difficult and incredibly expensive.

</comment><comment><USER37> Reflects my experience. During school I had interest and good grades in all classes, including physics and geometry (those are separate from math in Brazil), the only exception being math (which covers algebra, polynomials, etc.). How a kid can get an A in physics and a D in math is beyond my comprehension, but I did. I only started appreciating math much later in life.

</comment><comment><USER40> _" The saddest part of all this "reform" are the attempts to "make math interesting" and "relevant to kids' lives.""_

+1

Also s/math/science/ +1

</comment><comment><USER41> In contrast, my 5- and 3-year-olds are enjoying DragonBox on the iPad, not knowing they're learning Algebra.

</comment><comment><USER42> This is how I feel about doing most coding tutorials versus the MIT intro to CS through python, or project euler.

When coding is presented as: \- here is a problem \- how would you solve this problem? \- here are some hints to get you started

it is incredibly fun for me. when it is presented as:

\- follow along \- look what you did!

it can be a bit dry.

<comment><USER34> Hi whiddershins, here's a little present for you then: [https://github.com/darius/regexercise](https://github.com/darius/regexercise)

(It's not quite finished and I'd love to get feedback.)

</comment></comment><comment><USER43> Please read the article with a critical eye, some of it is complete non-sense, for example:

 _CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned._

"Mathematics of motion", which makes it sound so simple, has in fact perplexed philosophers and mathematicians for centuries and continues to perplex a great many people even today, consider for example the Zeno paradox:

[http://en.wikipedia.org/wiki/Zeno%27s_paradoxes](http://en.wikipedia.org/wiki/Zeno%27s_paradoxes)

The ideas of Newton and Leibniz were hardly simple, they had some valid intuitions and managed to do formal manipulations that led to correct results, but in their day it was impossible to at all logically understand why what they are doing works, and not for some god knows how complicated things, but even for most elementary ones. You don't even have to go back to writings of Newton or Leibniz, just have a look at a 19th century textbook of calculus to see how noticeably strange and illogical the exposition of the subject was even then, with "infinitely small quantities" and an air of mysticism about it:

[http://archive.org/stream/elementsofdiffer00woolrich#page/n1...](http://archive.org/stream/elementsofdiffer00woolrich#page/n17/mode/2up)

This kind of approach simply doesn't make sense, even though it happens to apparently produce correct results sometimes. Now, "function-based approach" is a weird phrase, but I guess he means the common modern exposition of elementary calculus using limits. This however wasn't developed in response to "various analytic crises". The only explanation of this statement I see is that he knows history of mathematics poorly and confuses the latter developments by Lebesgue, Jordan etc. that led to what we now call real analysis (inspired by considerations of nowhere continuous functions, continuous but nowhere differentiable functions etc.) with the earlier and more general lack of any decent understanding of how calculus works at all that was solved by Cauchy, Weierstrass and others. It is their introduction of what the author considers "unnecessary formalism" that made us finally really understand "mathematics of motion" and satisfactorily resolved things like the before-mentioned Zeno's paradox.

If it is only motivated appropriately, the concept of a limit is actually very interesting and powerful. There is a ladder of granularity with which you can treat computational problems, with the most elementary approach being always trying to get the exact answer. However, the class of problems that can be solved this way is very narrow. You can jump over this severe restriction by getting a bound, with inequalities for example, or you could try to get an equality in the limit (when n approaches y, the sought thing x approaches w*z). Unfortunately in school people almost exclusively learn to look for the exact answer, while in mathematics proper and in real world it is much more common to look for approximations and limiting behaviour. Furthermore, since the limit concept so powerfully extends the range of problems for which we are able to state anything interesting, there are lots of mathematical disciplines that rely on it to a great extent, for example probability theory (laws of large numbers, central limit theorem, ...). You won't understand almost any higher mathematics without learning limits first!

One can get an excellent and well motivated introduction to reasonably rigorous calculus using limits in Courant's "What is mathematics?" in less than a 100 pages, up to the point of understanding basic differentiation and integration, the exponential function, power series etc. The problem is not the formalism, but the teachers who can't motivate the material well enough both mathematically and physically and students who are not always mature enough to put in the amount of work necessary to understand calculus, which for most of them will be by far the most difficult thing they ever attempted to learn.

<comment><USER44> The point he's making is against unnecessary rigorization of introductory calculus and I think you are getting a bit too hung up on the "function-based approach". I repeat - introductory calculus. There's lot of time and space to make things more rigorous in a class like Analysis.

When I help students with calculus most of them have no trouble with the ideas but the implementation that they are required to perform. I spend a lot of time getting them to understand the simple particulars. There are ways of teaching calculus that would dispense with some of the more rigorous aspects and make it a much more bearable experience for most.

<comment><USER43> Introductory calculus classes are hardly ever rigorous. The only formalism you see is functions and limits, and that's a very useful one, far from unnecessary, it might just not always be motivated appropriately by poor teachers who themselves have little understanding of its usefulness. Your interpretation is also very far from what he has actually written. I edited my parent comment to make what I mean more clear.

<comment><USER45> it is true that rigour is eventually needed, but the principia, laplace's celestial mechanics and countless other works (heard of euler?) were all published before cauchy and weirstrass. all of the wonderful work in elliptic functions by gauss, abel and jacobi was done before rigour was en vogue. euclidean and non-euclidean geometries both flourished wonderfully w/o modern rigour (when was hilbert's book on geometry published?)

you're also wrong about your examples. it wasn't nowhere differentiable functions, but fourier series that motivated lebesgue. that's what the author is referring to regarding analytical traps ie, monotone convergence.

it is also quite a leap to assert the arithmetical definition of limits solves the zeno paradox!!! i few of my colleague's might disagree with you.

don't led your initial fascination with rigour (it can be addicting) get in the way of your intuition. rigour is necessary, but it comes after - <USER3> to the chagrin of some. look at the teaching of modern algebra. pure abstraction and rigour, with complete detachment of all the wonderful ideas, and experiences that gave it rise.

up with triangles i say :)

<comment><USER43> _it is true that rigour is eventually needed, but the principia, laplace 's celestial mechanics and countless other works (heard of euler?) were all published before cauchy and weirstrass. all of the wonderful work in elliptic functions by gauss, abel and jacobi was done before rigour was en vogue. euclidean and non-euclidean geometries both flourished wonderfully w/o modern rigour (when was hilbert's book on geometry published?)_

Now any hard-working student of university calculus can without problems understand and reproduce the results of those treatises. I think this would not be possible without the modern systematic methods, including the epsilon- delta approach.

 _you 're also wrong about your examples. it wasn't nowhere differentiable functions, but fourier series that motivated lebesgue. that's what the author is referring to regarding analytical traps ie, monotone convergence._

I mentioned fourier in another comments. Those weird functions were postulated in the discussion that arised somewhere in the same time period. I don't know what the author is referring to, because he is irritatingly vague, especially for a mathematician.

 _it is also quite a leap to assert the arithmetical definition of limits solves the zeno paradox!!! i few of my colleague 's might disagree with you._

If you are familiar with the concept of a limit, you can notice that Zeno considers a limiting process of two related quantities, time and the difference in the position of Achilles and the Tortoise. Since in this limiting process time get arbitrarly close to some definite value (the meeting time of Achilles and Tortoise), but never gets equal to it, it stops being so surprising that the distance between them never reaches zero, though it gets arbitrarly close. The formalism clarifies what seems parodoxical when described in natural language. I find this quite convincing, and I never found a better explanation, altough I know philosophers still dispute this.

I never said rigour takes precedence over intuition. I just think it's the inherent difficulty of calculus that stops students from understanding it, and not the epsilon-delta stuff.</comment></comment></comment></comment></comment>


Sisters' diary of their quest to make their oat cuisine a hit - <AUTHOR> http://www.dailymail.co.uk/femail/article-1141409/Porridge-pot-sisters-hilarious-diary-quest-make-oat-cuisine-hit.html#

<comment><USER2> I'm very surprised that they were allowed to run a food production business (initially) from a home kitchen. That would never fly in the USA.

<comment><USER3> The USA is almost schizophrenic about food.

On the one hand, god forbid anything is unpasteurized, on the other - all the corn syrup you can handle.

On a somewhat related subject, I know this is a story about a startup, but does it feel like someone's testing what stories will _crack_ HN?

<comment><USER4> What's wrong with corn syrup?

<comment><USER3> You are being taxed to fund the production of a much more expensive sugar substitute.

High tariffs on cane sugar are there to protect corn farmers, but the wast majority of businesses enjoy no such protections.

Because some things like hard candy just can not be made with corn syrup, some hard candy manufactures have moved factories to Mexico.

Note that it wasn't labor costs that made them move, it's the fact that imported sugar has tariffs, but manufactured hard candy - 100% sugar, does not.

This costs the US productive manufacturing jobs.

And lastly, in some people's subjective experience many things made with cane sugar just taste better.

Many people who have tasted both US and Mexican or Canadian coke or pepsi claim this, but again that's subjective.

</comment></comment></comment></comment><comment><USER5> Hey, I knew Abi a few years ago! So weird to read about her here. Sounds like a great business and lessons everyone can learn on the role luck can play in your success or failure. Not sure it's a typical HN story though.</comment>