16 ms·

I no longer understand my PhD dissertation

682 points by refrigerator 11y ago

hnarayanan 11y ago
As a PhD in spplied math, I must say I concur wholeheartedly with the author. The true value of a PhD in a quantitative field is less about specific domain knowledge, and more in the set of general problem solving skills you pick up.
- forrestthewoods 11y ago
  This raises an interesting question. Is the PhD process the best way to acquire those general problem solving skills? Or is there a better way to learn them?
  - Silhouette 11y ago
    From my experience of both academia and life in general, I'd say traditional tertiary education -- meaning lectures and research work from undergraduate upwards in a university setting -- is actually quite a poor way to learn anything. I'd say the ideal way to develop knowledge, understanding and skill in almost any field is a combination of systematic practice and receiving personalised guidance and training from someone who thoroughly understands the field itself to at least the level you are trying to reach and is able to share that understanding effectively based on your current level of understanding. Sadly, this is usually hopelessly unrealistic, because there are nowhere near enough suitable trainers around to give everyone close to 1:1 training in that format. But the further we drift from it, the more impersonal and generic training becomes, the more isolated individual practice becomes, the less immediate and detailed feedback becomes, the less effective the training regime as a whole will be. Given that neither undergraduate-style mass lectures nor postgraduate-style research are particularly efficient at conveying useful information, guiding practice, or promoting rapid and actionable feedback, I personally don't rate either particularly highly on an idealised scale. Perhaps a more practically useful question would be whether there are ways to improve university-level training that are realistic given the time and money constraints and, beyond a certain point, the lack of many if any people who actually are more knowledgeable or skillful in increasingly specialised fields than the research student who is dedicated to exploring them.
    - vidarh 11y ago
      Both the CS and maths undergrad courses at my uni were structured with voluntary lectures coupled with compulsory group study with 10-15 students led by a post-grad TA for many of the larger courses. I skipped most of the lectures, and focused on the group study, and it was far more rewarding.
      - Silhouette 11y ago
        That sounds like an all-too-familiar story. On the subject of lectures, it does slightly surprise me that in 2016 we still have researchers with neither much interest in teaching nor the presentation skills to do it well being asked/compelled to deliver undergraduate lecture courses at individual universities. You'd think with the easy access to video presentations and supporting materials now offered by the Internet, universities might have collaborated by now to build the personal elements of tuition around video lectures given by academics who do have the interest and are gifted presenters. On the other hand, I suppose that would expose how little personal attention many students actually receive in return for the fees and debts they take on, and universities don't want to encourage potential students to question how much real value they provide. Surely it would be more reliable and efficient as an education method if they focused their efforts on small group tuition and individual guidance, though.
  - zzleeper 11y ago
    I was thinking the same thing.. 5-7 years is an insane amount of time (and money) to spend just to get better general problem solving skills.
  - brohee 11y ago
    Well the PhD process lets you experience failure without putting much more than your self esteem in danger...
  - lmm 11y ago
    An interesting question indeed, but too broad to make much progress on. If you have ideas for a) an incremental change to the PhD process that would improve this or b) an alternative that might work better then please do pursue them. (Remember that the credential aspect is important as well as the actual learning).
  - hnarayanan 11y ago
    In general (and in my opinion), this is not the best way to acquire such skills. But it is a wonderful way to spend 5–7 years in a relatively stress-free environment while you can exercise your brain, explore different avenues, and pick up some had skills and stick–to–itiveness along the way.
jedberg 11y ago
> Beyond scarcely stretching the boundaries of obscure mathematical knowledge, what tangible difference has a PhD made to my life? The same thing a bachelors degree does for everyone else. You've proven that you can start, stick with, and complete a task that takes multiple years and a complicated set of steps.
- tluyben2 11y ago
  Which feels like a 'life is suffering' weirdness. I did it and never needed my diplomas for anything. Maybe I benefited somehow but I was a coder and entrepreneur before that; I had a software company before uni. It was over 20 years ago and in hindsight I find it pretty pointless and a waste of time. Maybe I became a better problem solver on some level but unless you are going into research or are not a self starter I would not recommend it.
  - eru 11y ago
    At least it keeps you from having regrets about `should have done a degree'.
  - vidarh 11y ago
    I went into taking my MSc explicitly to be able to add the letters to my resume - I started not long after the dot-com bubble burst, as a precaution. I don't regret it; I learned some interesting things during my thesis (the rest of it was regurgitating stuff I already knew; but it was distance learning so I didn't have to put in much effort), but similar experience - I don't really use it much. If I'd done it full time, it would have been a tremendous waste of time, though. But a lot of the reason for this experience for me at least was that I started uni after having spent about 15 years learning to program already, and by the time I picked up again and did my masters, I'd had another 10 years of commercial software development experience. These things are not really geared for people like us that came to them with a lot of pre-existing knowledge, but at people like my class-mates first time around that had hardly touched a computer before, and that did need a lot of hard work to come out with a good understanding of the problems. As a hiring manager this is why I rarely care whether someone has a degree or not if they can demonstrate experience. And on the other side of the table, I only took that degree because in the UK there are still sufficiently many employers that have an obsession with degrees regardless of experience...
bitL 11y ago
So, you basically ended up being better coming up with new algorithms?
amelius 11y ago
This is what happens to most programmers as well, when they try to read code that they wrote a while ago.
pmarreck 11y ago
Math seems to have a very ephemeral lifetime in the brain. I skipped a year of college once, and when I returned I realized I had to basically abandon any major with a math requirement, because I had seemingly forgotten everything. I'm currently struggling with an online Machine Learning class (the Coursera one... at the tender age of 43), and I can only take it (so far, at least... just failed my first quiz... fortunately I can review and re-take) because I was rather obsessed with matrices, oh, about 28 years ago. "You mean I can rotate things in an n-dimensional space without using trig?"
- baby 11y ago
  > because I had seemingly forgotten everything graduated in math, let me tell you this is how all my fellow students felt like every semester in every class.
sbardle 11y ago
A PhD isn't so much a test of intelligence as it is of perseverance.
- analog31 11y ago
  I think it's hard to generalize about PhDs because of the huge diversity of experiences. A PhD student should have a lot of freedom to define for themselves what they get out of their education. They are responsible adults and if they wanted a "marketable skill," they would have finished with a BS or MS. Predictably, the flexibility of PhD education doesn't always happen, and even when it does, it's both a blessing and a curse.
nanis 11y ago
First, I am not sure Functional Analysis is as obscure as some other areas. But, second, this just shows, once again, that one ought never to use "clearly," "obviously" etc in proofs. It is the same principle as writing programs so they are easier for the next programmer to read. That person may be you.
- rafekett 11y ago
  it also leads to proof by intimidation which often masks errors.
- vidarh 11y ago
  I don't read maths papers, but in CS papers too, words like "clearly" and "obviously" etc. serve as red flags to me that screams "hand-waving or big gaps coming up". Often it's understandable short-cuts, but often it also turns out that the author has left out very substantial chunks of knowledge, or sometimes clearly don't understand why they got the results they did. In CS papers there's an additional red flag: Maths. Outside of a few maths heavy areas of CS where it is justified, if a CS paper is full of equations, it's a good sign they'll have glossed over a lot of essential information, such as parameters that often turns out to be essential to be able to replicate their results. Not always, but often enough for me to be vary. I'm guessing it is because in the instances that include pseudo-code or working code, it is instantly obvious that something is missing, both to the author and to reviewers, but when it's obscured in equations it takes more effort to identify the same flaw because so many steps are often legitimately left out because of conventions that it's non-trivial for someone not steeped in the same notation to determine which bits should be defined and which bits are not necessary. I'm sure most of the time it's not intentional. But taking that shortcut seems to make it a lot easier to forget which additional information is actually necessary. And the irony is that I've seen plenty of example where the equations have taken up just as much space as pseudo-code or even working-but-naive implementations would have taken.
  - reikonomusha 11y ago
    Wait, let me get this straight. It's a red flag when a computer science paper has math in it? Computer science (in the asymptotic limit) is math. And the papers ideally should read like math papers. Otherwise it's not CS.
chx 11y ago
> Mathematics is an excellent proxy for problem-solving I went to a special, maths focused high school class and this rings true on that lower level too. I am a reasonably successful programmer/architect today and I have -- repeatedly -- attributed my successful attitude toward solving my problems to the 1200 or so problems we solved during those four years. Our maths education was literally nothing else but solving one problem after the other.
closure 11y ago
This does not surprise me in the least. Math was always extremely easy for me growing up. Up through my first differential equations class I found almost everything trivial to learn (the one exception is that I always found proving things difficult). I made the mistake of minoring in math and that slowly killed my enjoyment of it. Once I got to differential geometry and advanced matrix theory it all just became too abstract and I just wanted to get away from it. For several years after college I would routinely pull my advanced calculus text out and do problems "for fun". After a while I stopped doing that. Within a few years of no longer being exposed to math, I found it all incredibly foreign and challenging, to the point where I would say I have a bit of an aversion/phobia to it. I'm trying to reverse that now by tackling a topic I'm interested in but have previously avoided due to the math-heavy nature of it - type theory. Hopefully I can find the joy in math again through this. I think my point is that you can lose competence in math very very quickly through lack of constant exposure. The same is probably true of programming but I hope to never end up in that position.
- pmarreck 11y ago
  > type theory Not sure if related to your username, but I have a book on it (that was recommended to me here, in fact) on this topic, which I was curious about due to its implications in the programming field
  - closure 11y ago
    TAPL by Pierce is as far as I can tell the book everyone recommends, but if you have another you think is good I'm interested.
- zarkov99 11y ago
  The same is indeed of true of programming, at least in my experience. I took a management/consulting detour that lasted about 18 months and getting back into programming was quite though. I could not focus, could not hold the problem in my head, could not go from design concepts to implementation details. It took me a solid 6 months to get back to my former level.
- 11y ago
jholman 11y ago
All of these arguments are arguments for replacing the mathematics curriculum with video gaming. Games require generalized problem solving (arguably better-generalized than math, and arguably better-transferrable to other domains). Games build character: grit and tenacity, cautious optimism, etc blah blah etc. And games are fun (for many more people than find math fun). Guess math teachers should start learning to play League of Legends and Pokemon. Alternatively, I guess we need better reasons than those to teach a subject.
- deleted 11y ago
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- YeGoblynQueenne 11y ago
  Don't laugh it off. Personally, I learned to code by building Magic: the Gathering decks and playing at tournaments at high school. I learned things about resource management, reducing solutions, integrating disparate components into a functional system and so on, and even a bit of probabilities along the way. Not to mention what it did for my ability to concentrate and analyse an adversarial situation. If you think about it, a lot of education is really a kind of game and games themselves are often educational, usually by accident. Frex, I think a lot of people would recognise the value of teaching kids to play chess in order to improve their concentration and problem-solving skills. Well, why not more modern board games?
- j2kun 11y ago
  Video games are designed for grinding or following a fixed story, and your creativity is completely limited by the possibilities programmed for you. The exceptions to this (minecrafters designing Turing machines, hacking pokemon) are an insignificant fraction that are entirely discovered and engineered by those who understand mathematics and computer science and apply it to the video game. Why do you think the kinds of problem solving in video games is "better generalized" than math?
csense 11y ago
TLDR: The author independently re-discovered what you may know as Old Code Syndrome. I think that's because mathematical papers place too much value on terseness and abstraction over exposition and intuition. This guy's basically in the position of a fairly new developer who's just been asked to do non-trivial update of his own code for the first time. All those clever one-liners he put into his code made him feel smart and got the job done at the time. But he's now beginning to realize that if he keeps doing that, he's going to be cursed by his future self when he pulls up the code a few months later (never mind five years!) and has zero memory of how it actually works. I'm not intending to disparage the author; I've been there, and if you've been a software developer for a while you've likely been there too. Any decent programmer with enough experience will tell you the fix is to add some comments (more expository text than "it is obvious that..." or "the reader will quickly see..."), unit tests (concrete examples of abstract concepts), give variables and procedures descriptive names (The Wave Decomposition Lemma instead of Lemma 4.16), etc.
- bbcbasic 11y ago
  Kind of old code syndrome. I would say it is more like changing from Java to Ruby on Rails full time, and looking at your old Java code which is now a rusty language, with rusty memories of the class libraries and frameworks you used. Even with beautiful code it will be a struggle.
- gjulianm 11y ago
  > I think that's because mathematical papers place too much value on terseness and abstraction over exposition and intuition. If it only happened on mathematical papers... It is all over the place, in books, classes, etc. Few professors tell you what are they doing and why; they just throw formulas, theorems and symbols at you. Intuition is not given its necessary importance, first because it's hard to grasp some concepts, and second because sometimes there's no easy intution behind them unless you have deeper knowledge in the subject. Regarding "unit tests", or descriptive names... That's very difficult once you get into advanced math. Examples are probably going to be trivial or too contrived to be useful. And if you have to give a name to a theorem or lemma, you'll end up describing its full contents.
MicroBerto 11y ago
My response is that for every 100 of these types of papers, one of them may prove to be pivotal or inspirational in something truly groundbreaking and functionally useful. For this reason, I am all for 100 different people spending their time doing things like this, because eventually one of them will make an impact that is greater than 100x the efforts of 100 normal men. It's just a different kind of "brick in the wall" - only the diamonds in the rough can turn out to be hugely important for something else in the future.
- mfn 11y ago
  Great point. I think this applies to scientific research in general, which is why the constant emphasis on only funding research with clear and immediate economic payoff seems a bit shortsighted. In reality, chances are that most research won't lead to anything significant - but the 1% that does will have outsized impact that will more than pay for the rest. And we don't know which 1% this will be in advance.
  - dnautics 11y ago
    Unfortunately, you can very easily tell that what is excused as basic science is often just 'solutions looking for a problem'. Most "basic science" these days is just professors looking to get paid with no accountability. Although there have been a few Fouriers across history, the most compelling brilliant scientists had one foot in applied science working on theory in the spare time. Euler, Gauss, Faraday, Langmuir. Most of the best basic science had goals "to explain something pressing" (which is not the same as "do something because there will be no payoff") anyways, like Planck, Einstein, Peter Mitchell. There's something to be gained about cutting your teeth on problems with results, instead of just lollygagging about in theoryland. Feynman talked about this in an exercise where he described the motion of spinning discs - a very "applied" problem, and remarked that later those insights proved useful in a separate, unrelated problem.
- pluteoid 11y ago
  Exactly. I have the broadly same issue with my doctorate: I know essentially what I showed, but I only ever really completely understood my thesis's more abstruse passages around the time I was writing them (and being examined on them). But I don't find this surprising or concerning. The overriding motivation for doing a PhD was to try to contribute some original knowledge and insights to a field I find interesting. All this article talks about is, what did my PhD do for me personally, and tangentially how can the PhD experience help to train the mathematical brain. To me those are secondary issues, my reward is that my thesis, although hardly earth-shattering, gets checked out of the library every so often, and I get emails from interested and interesting people asking me about it.
chipsy 11y ago
It speaks to how finite we are around "knowledge." At the moment we reach understanding, we experience a sophomoric feeling of confidence. But as it fades farther and farther from our working memory, we become less fluent and more hesitant. The emerging pattern becomes one of "I can understand these interesting concepts, but it takes a lot of work and they don't last, so I have to choose to understand practically and situationally." And then in the end our bodies and personalities turn out to control our minds more than we might want to believe, as we turn away from one problem and towards a different one on some whim, never able to view the whole. As I recognize this more in myself, I am more inclined to become a bit of a librarian and develop better methods of personal note-taking and information retrieval, so that I lose less each time my mind flutters. At the moment that's turned into a fascination with mind maps - every time I need to critically-think through a problem I start mapping it. In the future I might look into ways of searching through those maps.
- timroy 11y ago
  You might also check out spaced repetition software. I've started putting anything I want to remember for the long haul into org-drill, an SRS package for Emacs, though Anki and Mnemosyne are more well known. I even schedule articles that I want to return to.
smonff 11y ago
I experiment the same problem with "trivial regular expressions" in two yeats old Perl programs.
eccstartup 11y ago
Functional Analysis or Nonlinear Functional Analysis?
analog31 11y ago
My PhD is in physics, from 20+ years ago, and I would not be able to explain or defend it today without studying it for a while. I've even forgotten the language (Pascal) that I wrote my experimental control and analysis code in. My experiment formed the basis of a fairly productive new research program for my thesis advisor, so at least it lived on in somebody's brain, but not in mine. ;-)
ck2 11y ago
Reminds me of this (well without the forgetting part, but I do that with old code all the time) http://matt.might.net/articles/phd-school-in-pictures/ http://matt.might.net/articles/phd-school-in-pictures/
tokenadult 11y ago
An interesting read. But I think the author should have explicitly written out the point he is really making: you can't be too careful about making your writing clear, even to yourself. I recall reading (I'd point to the book with a link if I could remember in what book I read this) that mathematicians who occasionally write expository articles on mathematics for the general public are often told by their professional colleagues, fellow research mathematicians, "Hey, I really liked your article [name of popular article] and I got a lot out of reading it." The book claimed that if mathematicians made a conscious effort to write understandably to members of the general public, their mathematics research would have more influence on other research mathematicians. That sounds like an important experiment to try for an early-career mathematician. More generally, in the very excellent book The Sense of Style: The Thinking Person's Guide to Writing in the 21st Century,[1] author and researcher Steven Pinker makes the point that the hardest thing for any writer to do is to avoid the "curse of knowledge," assuming that readers know what you know as they read your writing. It's HARD to write about something you know well without skipping lots of steps in reasoning and details of the topic that are unknown to most of your readers. This is one of the best reasons for any writer to submit manuscripts to an editor (or a set of friends, as Paul Graham does) before publishing. And, yes, if you think what I wrote above is unclear, as I fear it is, please let me know what's confusing about what I wrote. I'd be glad to hear your suggestions of how to make my main point more clear. I'm trying to say that anyone who writes anything has to put extra effort into making his point clear. [1] http://www.amazon.com/gp/product/B00INIYG74/ http://www.amazon.com/gp/product/B00INIYG74/
- tonyedgecombe 11y ago
  It's almost like you want the source code as well as the compiled program.
- Houshalter 11y ago
  This is a really common experience. See Explainers Shoot High, Aim Low!: http://lesswrong.com/lw/kh/explainers_shoot_high_aim_low/ http://lesswrong.com/lw/kh/explainers_shoot_high_aim_low/ >We miss the mark by several major grades of expertise. Aiming for outside academics gets you an article that will be popular among specialists in your field. Aiming at grade school (admittedly, naively so) will hit undergraduates. This is not because your audience is more stupid than you think, but because your words are far less helpful than you think. You're way way overshooting the target. Aim several major gradations lower, and you may hit your mark. This phenomenon has a name, the Illusion of Transparency: http://lesswrong.com/lw/ke/illusion_of_transparency_why_no_one_understands/ http://lesswrong.com/lw/ke/illusion_of_transparency_why_no_o... Also see some of the other posts there on the issue, it's quite interesting.
shalmanese 11y ago
I'm truly shocked by the multiple people in the thread who claim that Math knowledge can be completely erased through as little as a year of non-practice. For me, Math has always resembled riding a bike more than anything else. Sure, the first few moments, the path is a bit overgrown and all the weeds need to be cleared off but it was always significantly easier revisiting a topic than understanding it for the first time. For those who forget so quickly, I wonder if you felt like you truly understood it in the first place?
- marcosdumay 11y ago
  I've recently taken a calculus problem (multidimensional real optimization) I guess for the first time since I was an student. While your description has some merit, there's a huge amount of trivia in the format of "I can do X, I just have to do Y first", "X has no known solution, try something else", and "operation X is very useful, try it". That goes away, and everything gets way harder.
  - markus2012 11y ago
    I've always felt that if I had done all of my calculus with Mathematica I would have left college with an excellent grasp on how to use higher level functions provided by Mathematica that would have largely abstracted away all of this. Of course, the higher level functions might get covered in cobwebs - but I suspect not the same way; I would have kept these higher level skills up to date because: - I recently went though a couple of books on Bayes and computer vision. I would have used Mathematica - refreshing my memory. - I sometimes need to do some stats / analysis - Refresh... - I recently picked up a Student's guide to Maxwell's equations - Refresh... - I need to help my children with Calculus... If I had been using a high level tool my whole life I think I actually would make use of calculus and other mathematics. I'm curious if anyone else feels the same.
    - pjscott 11y ago
      That's roughly what I did, and it worked out about as well you predict. As soon as I understood what was going on with some kind of math, I'd use whatever tools I had to automate it: calculators at first, then computer algebra systems, numpy, whatever. No regrets; it was a great time-saver with very few drawbacks, and I made good educational use of the time it freed up.
bbcbasic 11y ago
It is not like riding a bike! I am reading up on stats after 15 years away from the subject and even the very basic stuff I have forgotten. Although the 'muscle memory' is there so that perhaps it is a bit easier then when totally new. What I also find is I am more interested in the application/intuition behind something now rather than the mechanics of the formulas. Maybe that has to do with a different aim i.e. usefulness vs. passing an exam..
ikeboy 11y ago
> I have attempted to deliver [these lectures] in a spirit that should be recommended to all students embarking on the writing of their PhD theses: imagine that you are explaining your ideas to your former smart, but ignorant, self, at the beginning of your studies! -Richard Feynman
- j2kun 11y ago
  After spending the last five years trying to explain math to a general technical audience, I'd agree with this, but with one minor change: s/smart/dumb :)
  - repsilat 11y ago
    I find it gets harder and harder to do as you progressively become more affected by the topic. I like to think I'm pretty good at explaining concepts in physics to lay audiences, but the more real physics I do the more I think like a physicist, and the less I can see what an explanation looks like to the layman. Quantum mechanics is the worst. I dislike a lot of the "popular science" language and analogues used to describe it, but the real (academic) pedagogical material is completely inappropriate for regular people. I'm worried that I'll be an inscrutable physicist before I grok it well enough to explain to a highschooler, though.
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admirethemeyer 11y ago
I had several exceptional Math teachers throughout my education, but the piece of advice that stuck with me the most is: "If you're not sure what you want to do with your life, study Math. Why? Because Math teaches you how to think." The skills I learned studying Mathematics have been invaluable, the Math that I currently am able to recall is abysmal. The author did a great job calling this out succinctly: Mathematics is an excellent proxy for problem-solving
KKKKkkkk1 11y ago
If you find yourself saying that you gained nothing from your education other than soft skills, maybe you should have passed over the functional analysis part and put the effort directly into learning said soft skills. I'm in the same boat, and I can see how it can be hard to admit this.
late2part 11y ago
That's okay. I never understood it, at least you did for a while.
musesum 11y ago
I had a similar problem with a crypto presentation. Basically, I was angling for a free ticket to an expensive conference. The trick was to propose something that is plausible, but too arcane for practical use. The consolation was a free ticket. Problem was that they accepted the talk. Damn! So, I started to read crypto journals. Basically anything co-authored by Chaum, Micali, Goldreich, Wigderson. After a few weeks, I starting to get the hang of it. Sort of like learning a new language. So, I gave the presentation and then forgot about it. A few years later, I decided to show my powerpoint to someone and describe the process. WTF? How did this lead to that? Didn't understand half of it. Was really embarrassing.
lkrubner 11y ago
He writes: "Mathematics is an excellent proxy for problem-solving... Mathematics, by its concise and logical nature, lends itself to problem-solving (it is not unique in this regard)." But how can we be sure this is true if is unable to read what he wrote? Maybe I'm thinking about the way Clojure programmers tend to use the word "concise" -- concise is meaningful only if it contributes to readability. Otherwise the more accurate description is "terse". And terse does not lend itself to problem solving.
brudgers 11y ago
The Sheetrock was the last step. I myself would do the exterior and interior painting. I told Ted I wanted to do at least that much, or he would have done that, too. When he himself had finished, and he had taken all the scraps I didn't want for kindling to the dump, he had me stand next to him outside and look at my new ell from thirty feet away. And then he asked it: "How the hell did I do that?" --Kurt Vonnegut, Timequake I find the experience common when I look back on things I write or design or build. As Bill Gates said, “Most people overestimate what they can do in one year and underestimate what they can do in ten years.”
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nowprovision 11y ago
This reminds me of revisiting Bash after a period of absence
ISL 11y ago
It takes about three months before a paper we ship becomes the best reference we have on the subject, exceeding our own recollection. Our brains can only hold so much, especially tiny details.
pervycreeper 11y ago
The dissemination of knowledge is at least as important as its discovery. Accessibility (i.e. clarity of exposition, availability to the public, etc.) needs to become a cardinal virtue in research.
jimbokun 11y ago
When I was taking machine learning courses and reading machine learning textbooks a few years ago, I have fond recollections of the derivations from Tom Mitchell's textbook. http://www.cs.cmu.edu/~tom/mlbook.html http://www.cs.cmu.edu/~tom/mlbook.html Where other textbooks tended to jump two or three steps ahead with a comment about the steps being "obvious" or "trivial", Mitchell would just include each little step. Yes, you could argue it was my responsibility to remember all of my calculus and linear algebra. But it is kind to the reader to spell out the little steps, for those of us who maybe forgot some of our calculus tricks, or maybe don't even have all of the expected pre-requisites but are trying to press on anyway. Or actually know how to perform the steps but have to stop and puzzle through which particular combination of steps you are describing as "obvious" in this particular instance. I just remember how nice it was to have those extra steps spelled out, and how much more pleasant it made reading Tom's book. So thanks, Dr. Mitchell!
aldanor 11y ago
As someone who left academia after a PhD in math (been working as a quant in HFT for the last few years, which mostly involves coding in one form or another), I can totally relate! Back then, all those stochastic integrals and measures made much more sense. However, it doesn't seem totally alien -- I'm pretty sure I could go to hacking math if required, but it would require at least several months to get into the flow.
- thearn4 11y ago
  I also left academia after a math PhD. Though in my case, my area was numerical linear algebra, and I entered the aerospace engineering field. A lot of what I was working on was immediately relevant, but I would say only a fraction of what became my dissertation would be counted among it.
egonschiele 11y ago
This happens to me all the time. I have a very popular illustrated post on Monads titled "Functors, Applicatives, and Monads in Pictures"[1]. When I wrote it I thought it was the best monad guide ever. Now, reading back, I can see that some parts are confusing. I still see a lot of people liking it, but three years later I wish some parts of it were better. [1] http://adit.io/posts/2013-04-17-functors,_applicatives,_and_monads_in_pictures.html http://adit.io/posts/2013-04-17-functors,_applicatives,_and_...
- throweway 11y ago
  Its quite good actually. Problem with monads is that no tutorial will make them less confusing. But writing code that does useful stuff with them builds the intuition. I now question the need to fully understand sometimes why not use libraries as per example and cargo cult a little to build up intuition. Then later get a more formal understanding.
JulianMorrison 11y ago
Math is twiddling with formal systems, and discovering how they behave, Some of it has uses, some doesn't, and some of what presently doesn't will in the fullness of time result in further islands of usefulness, as yet not even imagined. But ultimately, it needs no more justification than orchestral music.
fiatjaf 11y ago
The author almost realized the much more important conclusion of the fact he lived. He shouldn't conclude the article by asking "what is the purpose of studying maths?" and then giving an three stupid answers. He should have asked: is this actually "knowledge" as they say academia brings to society? Is the money researchers earn being well spent? Did I actually deserve to be remunerated by this piece of work no one understands -- and, in fact, no one has read except for maybe three people?
- argonaut 11y ago
  Considering that PhDs get paid next to nothing, I actually think they're getting paid pretty adequately.
  - fiatjaf 11y ago
    Nothing would be better, considering this article.
    - argonaut 11y ago
      Except that some % of PhDs go on to be professors (who do a real service), and every once in a while you get a PhD student whose work probably contributes hundreds of millions of value to society. And every couple years or so, you have a PhD student who contributes billions to society in value. Not to mention PhDs usually have to TA (teach students), which accounts for some of their pay.
      - fiatjaf 11y ago
        There's no example of a PhD work that contributed hundreds of millions of value to society. (I could say there's no way to measure "value to society", in fact this concept means nothing, but I agree on settling on "an enourmous amount of productive capital to someone, not exactly the society").
        argonaut 11y ago
fiatjaf 11y ago
How many people have read this and understand it? How much money was spent on the production of this disseration?
option_greek 11y ago
Math is the shadow universe of physics. Most theorems may not look like they are useful for anything real world till someone is able to peg all the variables to real world. And then as if by magic we realize we already know how how the real world behaves. Till someone does this pegging, the theorems sit idle waiting for problems to solve. I believe this is actually a good thing. We are letting people find solutions before someone finds problems to use them for.
- stuxnet79 11y ago
  > I believe this is actually a good thing. We are letting people find solutions before someone finds problems to use them for. This may lead to a few problems too: (1) Prior knowledge can serve as a blinder. Our conceptual toolkits may end up consisting of unwieldy theorems that while useful are cumbersome to use. (2) The number of theorems with no immediately evident practical use is so large that they may end up being "forgotten" anyways, and this will result in us having to do the "mental heavy lifting" all over again.
- daniel-levin 11y ago
  I disagree. A mathematical result doesn't have to have direct application to the real world for it to be useful. Drawing analogies between the real world and mathematical concepts is very powerful. I have a good example: Number theorists (amongst others) are seemingly obsessed with bounding things. There are entire books written about obtaining and then refining bounds - which appear to be nothing more than inequalities. There is great real-world value to be derived from seeing 'inequalities' as tools to leverage. Brian Kernighan once commented that controlling software complexity is the essence of programming [0]. I believe similar thinking applies to other aspects of software engineering, and product and business development. If you can take a hard problem, and bound its complexity, then you can say "the problem is no more complex than this". This is very useful. The chief value proposition of many SaaS businesses is the trivialisation of the upper bounds of complexity of hard problems. For instance, for many developers, Heroku makes the complexity of deployment very low. [0] http://quotes.cat-v.org/programming/ http://quotes.cat-v.org/programming/
deleted 11y ago
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mirimir 11y ago
> Mathematics is an excellent proxy for problem-solving In my experience, earning a PhD in [redacted] was excellent training in problem solving. And in developing working expertise in new areas. I suspect that the choice of field is indeed irrelevant. > Mathematics embeds character in students I'd say that actually finishing a PhD does that. > Mathematics is fun Whatever you pick for your dissertation topic had better be fun ;)
jeena 11y ago
It would have been cool if the original was linked here http://fjmubeen.com/2016/02/14/202/ http://fjmubeen.com/2016/02/14/202/ and not the medium repost. But still interesting.
thaw13579 11y ago
To make an admittedly bad metaphor, it's likely a lot of that knowledge has been moved from main memory to cold storage, and it would take some time to bring it back. It certainly makes the case for why we write things down! Although the part about having to dig for the main result makes me think the abstract could be improved...
rackforms 11y ago
One may be tempted to call this Hodge Theaters syndrome.
omginternets 11y ago
Me neither. My understanding is that this is the rule, not the exception.
danieltillett 11y ago
This post inspired me to re-read my thesis (well browse through it). Although it has been 16 years since I last looked at it, I didn’t have any problem understanding it and I didn’t even really cringe reading it. I guess it depends on your field how bad this effect is.
- jackmaney 11y ago
  I was about to write much the same thing... Last year, I found a hard copy of my dissertation (which I defended in 2004). I skimmed through it and had absolutely no problem whatsoever understanding what I had written 11 years prior. And I've been out of academia since 2008.
peterbraden 11y ago
Could part of it just be because Mathematical notation is just so bad? It's more of a shorthand than an actual tool of conveying meaning. So much context goes into establishing what a notated equation means - and that context is now gone.
- stuxnet79 11y ago
  I always complain about this. While I'd say I have an affinity for math, it was much more difficult to get caught up with my peers in university than let's say programming because there was just so much context / assumed knowledge required to make sense out of what the professor was stating that it took me forever to understand even the most trivial things. To be mathematically mature constant exposure is imperative. As to how to improve mathematical notation so that it is not so subjective or context dependent - I haven't a clue - seems to me like almost everyone is alright with the status quo. Not a good thing in my opinion.
bambax 11y ago
> what is the purpose of studying maths? Mathematics is an excellent proxy for problem-solving / Mathematics embeds character in students / Mathematics is fun Those may be reasons to study maths (although, studying anything seriously probably yields comparable benefits) but doing a PhD and writing a thesis is not only about yourself: it's supposed to advance the field. It's something you do for the general community.
z3t4 11y ago
I've also forgotten basically all high level math from school. And have to re-learn when the occasion comes to use some of it. But one thing that occurred to me is that in school I just learned how to make the calculations, so I never got a deep understanding on how things worked anyway. And that's fine.
- qb45 11y ago
  That's fine if you don't mind the time that could have been spent on something you actually cared about. Sometimes it happened that school attempted to teach me something I was interested in and I ended up understanding it. At other times, however, it all went to /dev/null.
  - z3t4 11y ago
    It's good that we forget stuff, or everyday tasks would be like querying from a fully saturated disk. But we might have a tiny fraction of it in cpu cache, that will let use make heuristic decisions.
osoti 11y ago
Top Places to Eat in U.S. According to Yelp http://goo.gl/18pkmx http://goo.gl/18pkmx
znpy 11y ago
It seems to me that most commenters are ignoring the fact that the author is a guy that basically left high level mathematics after completing his phd. So basically he went doing other stuff not functional-analysis-related and his functional-analysis got rusty. It seems quite reasonable to me. Call it old code syndrome, call it "my math got rusty", it seems quite normal to me. Also: according to http://fjmubeen.com/about/ http://fjmubeen.com/about/, the author got his phd in 2007. It's 2016. Almost ten years. What... What are we talking about?
- refrigerator 11y ago
  He actually finished the PhD in 2011, but started in 2007. Not sure whether that's a significant enough difference to change the point that you're making though.
  - znpy 11y ago
    You're right. I still think my point is valid though. I guess he still has a strong mathematical background, a solid problem solving attitude, but details of the topic of his doctoral thesis are missing. It's been many years, it isn't strange at all.
calibraxis 11y ago
Reminds me to get around to Piper Harron's thesis. Which was made to be seriously readable. Math seems to have a culture of systematically bad writing (as Bill Thurston discussed).
GreaterFool 11y ago
I've been working with Haskell* for a couple of years and it is quite often that I work with code that I don't fully understand. I'll come across a terse bit of code, then carefully take it apart to see what it does (by taking bits and pieces out and giving them names instead of passing in using point-free notation and also adding type annotations). Once I see the whole picture, I make my own change and then carefully re-assemble the original terse bit of code. One could ask the question: wasn't the verbose version better? I'm going to lean on the side of no. If I left this verbose and other bits verbose then it would be hard to see the whole picture. I think doing maths would be better if it was done interactively with software. If equations were code then you could blow it up and look into fine details and then shrink it to a terse form while software keeps track of the transformations to make sure what you write is equivalent. Maybe it's time to add a laptop to that paper pad? * not arguing anything language specific here except that Haskell makes use of variety of notations that makes the code shorter and more like mahts. More so than most languages.
- mikko-apo 11y ago
  I think it's about balancing various things. Having less code is good unless there's a hidden catch you need to be aware of or it takes a few weeks to unravel what the code does. I prefer code that is understandable right away, is consistent and doesn't have any surprises.
  - GreaterFool 11y ago
    A good example is Go. Many have written blogposts describing how great it is that it's such a simple language because the code is easy to read. And I can't deny that. The code is simple to read. But then I read through pages and pages of such code and all with little meaning. Here's a loop, here we check for an error condition, here's another loop, here we check check for another error condition. It makes it harder to see through all that and answer the question "what does this code try to accomplish?". At least for me, the more code there is, the harder it is to see.
    - kyllo 11y ago
wodenokoto 11y ago
Let us give some credit to the professors who read the dissertation and understood what was going on.
Houshalter 11y ago
I took a course on formal logic on courses. I put all of the questions and exercises into anki, a spaced repitition program. This ensures I will always remember it and get it in my head at an intuitive level. Basically it's like flash cards that decay exponentially. The first review is in one day, the second in two days, then 4 days, and so on.
riprowan 11y ago
We are clearly approaching the point where unassisted human intelligence is becoming insufficient to continue to master even specific domain expertise.
lin0 11y ago
This wrapper[1] helps me edit LaTeX fragment in Vim/Emacs... [1]: https://github.com/linktohack/lyxit https://github.com/linktohack/lyxit
CurtMonash 11y ago
Much the same is true of me, and of my best friend in grad school.
kazinator 11y ago
Someone doesn't understand his own work five years later to this extent, this is a strong indication that the work is actually garbage, and the prior understanding five years ago was only a delusion brought on by the circumstances: the late nights, the pressure, and so on. Perhaps it doesn't make sense today because it never did, and the self deception has long worn off, not because the author has gone daft. Several weeks ago, on the last work day before going on vacation, I submitted fixes for nine issues I found in one USB host controller driver. The last time I looked at the code was more than a year ago. I had refactored it and really improved its quality. Looking at some of the code now, I couldn't understand it that well. But that's because it wasn't as good as I thought it was. I was still relying on the fictitious story of how I thought certain aspects of the code worked really well thanks to me, and it wasn't meshing with the reality emanating from freshly reading it with more critical eyes. And, of course, I was also confronted by a reproducible crash. As I'm reading the code, I'm forced to throw away the false delusions and replace them with reality. This is because I'm smarter and fresher today, not because I've forgotten things and gotten dumber! It's taking effort because something is actually being done. Perhaps a similar problem is here: he's reading the paper with more critical eyes and seeing aspects that don't match the fake memory of how great that paper was, which was formed by clouded judgment at the time of writing. Maybe that obscure notation that he can't understand is actually incorrect garbage. His brain is reeling because it's actually digging into the material and trying to do proper work, perhaps for the first time. If you can show that your five year old work is incorrect garbage, that suggests you're actually superior today to your former self from five years ago. So that could be the thing to do. Don't read the paper assuming that it's right, and you've gone daft. Catch where you went wrong. By the way, I never have this problem with good code. I can go back a decade and everything is wonderful. Let's just say there is a suspicious smell if you can't decipher your old work. Good work is clear, and based on a correct understanding which matches that work. There is a durable, robust relationship between the latent memory of that work and the actual work, making it easy to jog your memory.
juped 11y ago
Reading mathematical papers is an acquired skill that needs practice. Writing them also is, but the skills exist somewhat independently.

682 points by refrigerator 11y ago

As a PhD in spplied math, I must say I concur wholeheartedly with the author. The true value of a PhD in a quantitative field is less about specific domain knowledge, and more in the set of general problem solving skills you pick up.

forrestthewoods 11y ago
This raises an interesting question. Is the PhD process the best way to acquire those general problem solving skills? Or is there a better way to learn them?
- Silhouette 11y ago
  From my experience of both academia and life in general, I'd say traditional tertiary education -- meaning lectures and research work from undergraduate upwards in a university setting -- is actually quite a poor way to learn anything. I'd say the ideal way to develop knowledge, understanding and skill in almost any field is a combination of systematic practice and receiving personalised guidance and training from someone who thoroughly understands the field itself to at least the level you are trying to reach and is able to share that understanding effectively based on your current level of understanding. Sadly, this is usually hopelessly unrealistic, because there are nowhere near enough suitable trainers around to give everyone close to 1:1 training in that format. But the further we drift from it, the more impersonal and generic training becomes, the more isolated individual practice becomes, the less immediate and detailed feedback becomes, the less effective the training regime as a whole will be. Given that neither undergraduate-style mass lectures nor postgraduate-style research are particularly efficient at conveying useful information, guiding practice, or promoting rapid and actionable feedback, I personally don't rate either particularly highly on an idealised scale. Perhaps a more practically useful question would be whether there are ways to improve university-level training that are realistic given the time and money constraints and, beyond a certain point, the lack of many if any people who actually are more knowledgeable or skillful in increasingly specialised fields than the research student who is dedicated to exploring them.
  - vidarh 11y ago
    Both the CS and maths undergrad courses at my uni were structured with voluntary lectures coupled with compulsory group study with 10-15 students led by a post-grad TA for many of the larger courses. I skipped most of the lectures, and focused on the group study, and it was far more rewarding.
    - Silhouette 11y ago
      That sounds like an all-too-familiar story. On the subject of lectures, it does slightly surprise me that in 2016 we still have researchers with neither much interest in teaching nor the presentation skills to do it well being asked/compelled to deliver undergraduate lecture courses at individual universities. You'd think with the easy access to video presentations and supporting materials now offered by the Internet, universities might have collaborated by now to build the personal elements of tuition around video lectures given by academics who do have the interest and are gifted presenters. On the other hand, I suppose that would expose how little personal attention many students actually receive in return for the fees and debts they take on, and universities don't want to encourage potential students to question how much real value they provide. Surely it would be more reliable and efficient as an education method if they focused their efforts on small group tuition and individual guidance, though.
- zzleeper 11y ago
  I was thinking the same thing.. 5-7 years is an insane amount of time (and money) to spend just to get better general problem solving skills.
- brohee 11y ago
  Well the PhD process lets you experience failure without putting much more than your self esteem in danger...
- lmm 11y ago
  An interesting question indeed, but too broad to make much progress on. If you have ideas for a) an incremental change to the PhD process that would improve this or b) an alternative that might work better then please do pursue them. (Remember that the credential aspect is important as well as the actual learning).
- hnarayanan 11y ago
  In general (and in my opinion), this is not the best way to acquire such skills. But it is a wonderful way to spend 5–7 years in a relatively stress-free environment while you can exercise your brain, explore different avenues, and pick up some had skills and stick–to–itiveness along the way.

jedberg 11y ago

> Beyond scarcely stretching the boundaries of obscure mathematical knowledge, what tangible difference has a PhD made to my life? The same thing a bachelors degree does for everyone else. You've proven that you can start, stick with, and complete a task that takes multiple years and a complicated set of steps.

tluyben2 11y ago
Which feels like a 'life is suffering' weirdness. I did it and never needed my diplomas for anything. Maybe I benefited somehow but I was a coder and entrepreneur before that; I had a software company before uni. It was over 20 years ago and in hindsight I find it pretty pointless and a waste of time. Maybe I became a better problem solver on some level but unless you are going into research or are not a self starter I would not recommend it.
- eru 11y ago
  At least it keeps you from having regrets about `should have done a degree'.
- vidarh 11y ago
  I went into taking my MSc explicitly to be able to add the letters to my resume - I started not long after the dot-com bubble burst, as a precaution. I don't regret it; I learned some interesting things during my thesis (the rest of it was regurgitating stuff I already knew; but it was distance learning so I didn't have to put in much effort), but similar experience - I don't really use it much. If I'd done it full time, it would have been a tremendous waste of time, though. But a lot of the reason for this experience for me at least was that I started uni after having spent about 15 years learning to program already, and by the time I picked up again and did my masters, I'd had another 10 years of commercial software development experience. These things are not really geared for people like us that came to them with a lot of pre-existing knowledge, but at people like my class-mates first time around that had hardly touched a computer before, and that did need a lot of hard work to come out with a good understanding of the problems. As a hiring manager this is why I rarely care whether someone has a degree or not if they can demonstrate experience. And on the other side of the table, I only took that degree because in the UK there are still sufficiently many employers that have an obsession with degrees regardless of experience...

bitL 11y ago

So, you basically ended up being better coming up with new algorithms?

amelius 11y ago

This is what happens to most programmers as well, when they try to read code that they wrote a while ago.

pmarreck 11y ago

Math seems to have a very ephemeral lifetime in the brain. I skipped a year of college once, and when I returned I realized I had to basically abandon any major with a math requirement, because I had seemingly forgotten everything. I'm currently struggling with an online Machine Learning class (the Coursera one... at the tender age of 43), and I can only take it (so far, at least... just failed my first quiz... fortunately I can review and re-take) because I was rather obsessed with matrices, oh, about 28 years ago. "You mean I can rotate things in an n-dimensional space without using trig?"

baby 11y ago
> because I had seemingly forgotten everything graduated in math, let me tell you this is how all my fellow students felt like every semester in every class.

sbardle 11y ago

A PhD isn't so much a test of intelligence as it is of perseverance.

analog31 11y ago
I think it's hard to generalize about PhDs because of the huge diversity of experiences. A PhD student should have a lot of freedom to define for themselves what they get out of their education. They are responsible adults and if they wanted a "marketable skill," they would have finished with a BS or MS. Predictably, the flexibility of PhD education doesn't always happen, and even when it does, it's both a blessing and a curse.

nanis 11y ago

First, I am not sure Functional Analysis is as obscure as some other areas. But, second, this just shows, once again, that one ought never to use "clearly," "obviously" etc in proofs. It is the same principle as writing programs so they are easier for the next programmer to read. That person may be you.

rafekett 11y ago
it also leads to proof by intimidation which often masks errors.
vidarh 11y ago
I don't read maths papers, but in CS papers too, words like "clearly" and "obviously" etc. serve as red flags to me that screams "hand-waving or big gaps coming up". Often it's understandable short-cuts, but often it also turns out that the author has left out very substantial chunks of knowledge, or sometimes clearly don't understand why they got the results they did. In CS papers there's an additional red flag: Maths. Outside of a few maths heavy areas of CS where it is justified, if a CS paper is full of equations, it's a good sign they'll have glossed over a lot of essential information, such as parameters that often turns out to be essential to be able to replicate their results. Not always, but often enough for me to be vary. I'm guessing it is because in the instances that include pseudo-code or working code, it is instantly obvious that something is missing, both to the author and to reviewers, but when it's obscured in equations it takes more effort to identify the same flaw because so many steps are often legitimately left out because of conventions that it's non-trivial for someone not steeped in the same notation to determine which bits should be defined and which bits are not necessary. I'm sure most of the time it's not intentional. But taking that shortcut seems to make it a lot easier to forget which additional information is actually necessary. And the irony is that I've seen plenty of example where the equations have taken up just as much space as pseudo-code or even working-but-naive implementations would have taken.
- reikonomusha 11y ago
  Wait, let me get this straight. It's a red flag when a computer science paper has math in it? Computer science (in the asymptotic limit) is math. And the papers ideally should read like math papers. Otherwise it's not CS.

chx 11y ago

> Mathematics is an excellent proxy for problem-solving I went to a special, maths focused high school class and this rings true on that lower level too. I am a reasonably successful programmer/architect today and I have -- repeatedly -- attributed my successful attitude toward solving my problems to the 1200 or so problems we solved during those four years. Our maths education was literally nothing else but solving one problem after the other.

closure 11y ago

This does not surprise me in the least. Math was always extremely easy for me growing up. Up through my first differential equations class I found almost everything trivial to learn (the one exception is that I always found proving things difficult). I made the mistake of minoring in math and that slowly killed my enjoyment of it. Once I got to differential geometry and advanced matrix theory it all just became too abstract and I just wanted to get away from it. For several years after college I would routinely pull my advanced calculus text out and do problems "for fun". After a while I stopped doing that. Within a few years of no longer being exposed to math, I found it all incredibly foreign and challenging, to the point where I would say I have a bit of an aversion/phobia to it. I'm trying to reverse that now by tackling a topic I'm interested in but have previously avoided due to the math-heavy nature of it - type theory. Hopefully I can find the joy in math again through this. I think my point is that you can lose competence in math very very quickly through lack of constant exposure. The same is probably true of programming but I hope to never end up in that position.

pmarreck 11y ago
> type theory Not sure if related to your username, but I have a book on it (that was recommended to me here, in fact) on this topic, which I was curious about due to its implications in the programming field
- closure 11y ago
  TAPL by Pierce is as far as I can tell the book everyone recommends, but if you have another you think is good I'm interested.
zarkov99 11y ago
The same is indeed of true of programming, at least in my experience. I took a management/consulting detour that lasted about 18 months and getting back into programming was quite though. I could not focus, could not hold the problem in my head, could not go from design concepts to implementation details. It took me a solid 6 months to get back to my former level.
11y ago

jholman 11y ago

All of these arguments are arguments for replacing the mathematics curriculum with video gaming. Games require generalized problem solving (arguably better-generalized than math, and arguably better-transferrable to other domains). Games build character: grit and tenacity, cautious optimism, etc blah blah etc. And games are fun (for many more people than find math fun). Guess math teachers should start learning to play League of Legends and Pokemon. Alternatively, I guess we need better reasons than those to teach a subject.

deleted 11y ago
[deleted]
YeGoblynQueenne 11y ago
Don't laugh it off. Personally, I learned to code by building Magic: the Gathering decks and playing at tournaments at high school. I learned things about resource management, reducing solutions, integrating disparate components into a functional system and so on, and even a bit of probabilities along the way. Not to mention what it did for my ability to concentrate and analyse an adversarial situation. If you think about it, a lot of education is really a kind of game and games themselves are often educational, usually by accident. Frex, I think a lot of people would recognise the value of teaching kids to play chess in order to improve their concentration and problem-solving skills. Well, why not more modern board games?
j2kun 11y ago
Video games are designed for grinding or following a fixed story, and your creativity is completely limited by the possibilities programmed for you. The exceptions to this (minecrafters designing Turing machines, hacking pokemon) are an insignificant fraction that are entirely discovered and engineered by those who understand mathematics and computer science and apply it to the video game. Why do you think the kinds of problem solving in video games is "better generalized" than math?

csense 11y ago

TLDR: The author independently re-discovered what you may know as Old Code Syndrome. I think that's because mathematical papers place too much value on terseness and abstraction over exposition and intuition. This guy's basically in the position of a fairly new developer who's just been asked to do non-trivial update of his own code for the first time. All those clever one-liners he put into his code made him feel smart and got the job done at the time. But he's now beginning to realize that if he keeps doing that, he's going to be cursed by his future self when he pulls up the code a few months later (never mind five years!) and has zero memory of how it actually works. I'm not intending to disparage the author; I've been there, and if you've been a software developer for a while you've likely been there too. Any decent programmer with enough experience will tell you the fix is to add some comments (more expository text than "it is obvious that..." or "the reader will quickly see..."), unit tests (concrete examples of abstract concepts), give variables and procedures descriptive names (The Wave Decomposition Lemma instead of Lemma 4.16), etc.

bbcbasic 11y ago
Kind of old code syndrome. I would say it is more like changing from Java to Ruby on Rails full time, and looking at your old Java code which is now a rusty language, with rusty memories of the class libraries and frameworks you used. Even with beautiful code it will be a struggle.
gjulianm 11y ago
> I think that's because mathematical papers place too much value on terseness and abstraction over exposition and intuition. If it only happened on mathematical papers... It is all over the place, in books, classes, etc. Few professors tell you what are they doing and why; they just throw formulas, theorems and symbols at you. Intuition is not given its necessary importance, first because it's hard to grasp some concepts, and second because sometimes there's no easy intution behind them unless you have deeper knowledge in the subject. Regarding "unit tests", or descriptive names... That's very difficult once you get into advanced math. Examples are probably going to be trivial or too contrived to be useful. And if you have to give a name to a theorem or lemma, you'll end up describing its full contents.

MicroBerto 11y ago

My response is that for every 100 of these types of papers, one of them may prove to be pivotal or inspirational in something truly groundbreaking and functionally useful. For this reason, I am all for 100 different people spending their time doing things like this, because eventually one of them will make an impact that is greater than 100x the efforts of 100 normal men. It's just a different kind of "brick in the wall" - only the diamonds in the rough can turn out to be hugely important for something else in the future.

mfn 11y ago
Great point. I think this applies to scientific research in general, which is why the constant emphasis on only funding research with clear and immediate economic payoff seems a bit shortsighted. In reality, chances are that most research won't lead to anything significant - but the 1% that does will have outsized impact that will more than pay for the rest. And we don't know which 1% this will be in advance.
- dnautics 11y ago
  Unfortunately, you can very easily tell that what is excused as basic science is often just 'solutions looking for a problem'. Most "basic science" these days is just professors looking to get paid with no accountability. Although there have been a few Fouriers across history, the most compelling brilliant scientists had one foot in applied science working on theory in the spare time. Euler, Gauss, Faraday, Langmuir. Most of the best basic science had goals "to explain something pressing" (which is not the same as "do something because there will be no payoff") anyways, like Planck, Einstein, Peter Mitchell. There's something to be gained about cutting your teeth on problems with results, instead of just lollygagging about in theoryland. Feynman talked about this in an exercise where he described the motion of spinning discs - a very "applied" problem, and remarked that later those insights proved useful in a separate, unrelated problem.
pluteoid 11y ago
Exactly. I have the broadly same issue with my doctorate: I know essentially what I showed, but I only ever really completely understood my thesis's more abstruse passages around the time I was writing them (and being examined on them). But I don't find this surprising or concerning. The overriding motivation for doing a PhD was to try to contribute some original knowledge and insights to a field I find interesting. All this article talks about is, what did my PhD do for me personally, and tangentially how can the PhD experience help to train the mathematical brain. To me those are secondary issues, my reward is that my thesis, although hardly earth-shattering, gets checked out of the library every so often, and I get emails from interested and interesting people asking me about it.

chipsy 11y ago

It speaks to how finite we are around "knowledge." At the moment we reach understanding, we experience a sophomoric feeling of confidence. But as it fades farther and farther from our working memory, we become less fluent and more hesitant. The emerging pattern becomes one of "I can understand these interesting concepts, but it takes a lot of work and they don't last, so I have to choose to understand practically and situationally." And then in the end our bodies and personalities turn out to control our minds more than we might want to believe, as we turn away from one problem and towards a different one on some whim, never able to view the whole. As I recognize this more in myself, I am more inclined to become a bit of a librarian and develop better methods of personal note-taking and information retrieval, so that I lose less each time my mind flutters. At the moment that's turned into a fascination with mind maps - every time I need to critically-think through a problem I start mapping it. In the future I might look into ways of searching through those maps.

timroy 11y ago
You might also check out spaced repetition software. I've started putting anything I want to remember for the long haul into org-drill, an SRS package for Emacs, though Anki and Mnemosyne are more well known. I even schedule articles that I want to return to.

smonff 11y ago

I experiment the same problem with "trivial regular expressions" in two yeats old Perl programs.

eccstartup 11y ago

Functional Analysis or Nonlinear Functional Analysis?

analog31 11y ago

My PhD is in physics, from 20+ years ago, and I would not be able to explain or defend it today without studying it for a while. I've even forgotten the language (Pascal) that I wrote my experimental control and analysis code in. My experiment formed the basis of a fairly productive new research program for my thesis advisor, so at least it lived on in somebody's brain, but not in mine. ;-)

ck2 11y ago

Reminds me of this (well without the forgetting part, but I do that with old code all the time) http://matt.might.net/articles/phd-school-in-pictures/ http://matt.might.net/articles/phd-school-in-pictures/

tokenadult 11y ago

An interesting read. But I think the author should have explicitly written out the point he is really making: you can't be too careful about making your writing clear, even to yourself. I recall reading (I'd point to the book with a link if I could remember in what book I read this) that mathematicians who occasionally write expository articles on mathematics for the general public are often told by their professional colleagues, fellow research mathematicians, "Hey, I really liked your article [name of popular article] and I got a lot out of reading it." The book claimed that if mathematicians made a conscious effort to write understandably to members of the general public, their mathematics research would have more influence on other research mathematicians. That sounds like an important experiment to try for an early-career mathematician. More generally, in the very excellent book The Sense of Style: The Thinking Person's Guide to Writing in the 21st Century,[1] author and researcher Steven Pinker makes the point that the hardest thing for any writer to do is to avoid the "curse of knowledge," assuming that readers know what you know as they read your writing. It's HARD to write about something you know well without skipping lots of steps in reasoning and details of the topic that are unknown to most of your readers. This is one of the best reasons for any writer to submit manuscripts to an editor (or a set of friends, as Paul Graham does) before publishing. And, yes, if you think what I wrote above is unclear, as I fear it is, please let me know what's confusing about what I wrote. I'd be glad to hear your suggestions of how to make my main point more clear. I'm trying to say that anyone who writes anything has to put extra effort into making his point clear. [1] http://www.amazon.com/gp/product/B00INIYG74/ http://www.amazon.com/gp/product/B00INIYG74/

tonyedgecombe 11y ago
It's almost like you want the source code as well as the compiled program.
Houshalter 11y ago
This is a really common experience. See Explainers Shoot High, Aim Low!: http://lesswrong.com/lw/kh/explainers_shoot_high_aim_low/ http://lesswrong.com/lw/kh/explainers_shoot_high_aim_low/ >We miss the mark by several major grades of expertise. Aiming for outside academics gets you an article that will be popular among specialists in your field. Aiming at grade school (admittedly, naively so) will hit undergraduates. This is not because your audience is more stupid than you think, but because your words are far less helpful than you think. You're way way overshooting the target. Aim several major gradations lower, and you may hit your mark. This phenomenon has a name, the Illusion of Transparency: http://lesswrong.com/lw/ke/illusion_of_transparency_why_no_one_understands/ http://lesswrong.com/lw/ke/illusion_of_transparency_why_no_o... Also see some of the other posts there on the issue, it's quite interesting.

shalmanese 11y ago

I'm truly shocked by the multiple people in the thread who claim that Math knowledge can be completely erased through as little as a year of non-practice. For me, Math has always resembled riding a bike more than anything else. Sure, the first few moments, the path is a bit overgrown and all the weeds need to be cleared off but it was always significantly easier revisiting a topic than understanding it for the first time. For those who forget so quickly, I wonder if you felt like you truly understood it in the first place?

marcosdumay 11y ago
I've recently taken a calculus problem (multidimensional real optimization) I guess for the first time since I was an student. While your description has some merit, there's a huge amount of trivia in the format of "I can do X, I just have to do Y first", "X has no known solution, try something else", and "operation X is very useful, try it". That goes away, and everything gets way harder.
- markus2012 11y ago
  I've always felt that if I had done all of my calculus with Mathematica I would have left college with an excellent grasp on how to use higher level functions provided by Mathematica that would have largely abstracted away all of this. Of course, the higher level functions might get covered in cobwebs - but I suspect not the same way; I would have kept these higher level skills up to date because: - I recently went though a couple of books on Bayes and computer vision. I would have used Mathematica - refreshing my memory. - I sometimes need to do some stats / analysis - Refresh... - I recently picked up a Student's guide to Maxwell's equations - Refresh... - I need to help my children with Calculus... If I had been using a high level tool my whole life I think I actually would make use of calculus and other mathematics. I'm curious if anyone else feels the same.
  - pjscott 11y ago
    That's roughly what I did, and it worked out about as well you predict. As soon as I understood what was going on with some kind of math, I'd use whatever tools I had to automate it: calculators at first, then computer algebra systems, numpy, whatever. No regrets; it was a great time-saver with very few drawbacks, and I made good educational use of the time it freed up.

bbcbasic 11y ago

It is not like riding a bike! I am reading up on stats after 15 years away from the subject and even the very basic stuff I have forgotten. Although the 'muscle memory' is there so that perhaps it is a bit easier then when totally new. What I also find is I am more interested in the application/intuition behind something now rather than the mechanics of the formulas. Maybe that has to do with a different aim i.e. usefulness vs. passing an exam..

ikeboy 11y ago

> I have attempted to deliver [these lectures] in a spirit that should be recommended to all students embarking on the writing of their PhD theses: imagine that you are explaining your ideas to your former smart, but ignorant, self, at the beginning of your studies! -Richard Feynman

j2kun 11y ago
After spending the last five years trying to explain math to a general technical audience, I'd agree with this, but with one minor change: s/smart/dumb :)
- repsilat 11y ago
  I find it gets harder and harder to do as you progressively become more affected by the topic. I like to think I'm pretty good at explaining concepts in physics to lay audiences, but the more real physics I do the more I think like a physicist, and the less I can see what an explanation looks like to the layman. Quantum mechanics is the worst. I dislike a lot of the "popular science" language and analogues used to describe it, but the real (academic) pedagogical material is completely inappropriate for regular people. I'm worried that I'll be an inscrutable physicist before I grok it well enough to explain to a highschooler, though.

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admirethemeyer 11y ago

I had several exceptional Math teachers throughout my education, but the piece of advice that stuck with me the most is: "If you're not sure what you want to do with your life, study Math. Why? Because Math teaches you how to think." The skills I learned studying Mathematics have been invaluable, the Math that I currently am able to recall is abysmal. The author did a great job calling this out succinctly: Mathematics is an excellent proxy for problem-solving

KKKKkkkk1 11y ago

If you find yourself saying that you gained nothing from your education other than soft skills, maybe you should have passed over the functional analysis part and put the effort directly into learning said soft skills. I'm in the same boat, and I can see how it can be hard to admit this.

late2part 11y ago

That's okay. I never understood it, at least you did for a while.

musesum 11y ago

I had a similar problem with a crypto presentation. Basically, I was angling for a free ticket to an expensive conference. The trick was to propose something that is plausible, but too arcane for practical use. The consolation was a free ticket. Problem was that they accepted the talk. Damn! So, I started to read crypto journals. Basically anything co-authored by Chaum, Micali, Goldreich, Wigderson. After a few weeks, I starting to get the hang of it. Sort of like learning a new language. So, I gave the presentation and then forgot about it. A few years later, I decided to show my powerpoint to someone and describe the process. WTF? How did this lead to that? Didn't understand half of it. Was really embarrassing.

lkrubner 11y ago

He writes: "Mathematics is an excellent proxy for problem-solving... Mathematics, by its concise and logical nature, lends itself to problem-solving (it is not unique in this regard)." But how can we be sure this is true if is unable to read what he wrote? Maybe I'm thinking about the way Clojure programmers tend to use the word "concise" -- concise is meaningful only if it contributes to readability. Otherwise the more accurate description is "terse". And terse does not lend itself to problem solving.

brudgers 11y ago

The Sheetrock was the last step. I myself would do the exterior and interior painting. I told Ted I wanted to do at least that much, or he would have done that, too. When he himself had finished, and he had taken all the scraps I didn't want for kindling to the dump, he had me stand next to him outside and look at my new ell from thirty feet away. And then he asked it: "How the hell did I do that?" --Kurt Vonnegut, Timequake I find the experience common when I look back on things I write or design or build. As Bill Gates said, “Most people overestimate what they can do in one year and underestimate what they can do in ten years.”

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nowprovision 11y ago

This reminds me of revisiting Bash after a period of absence

ISL 11y ago

It takes about three months before a paper we ship becomes the best reference we have on the subject, exceeding our own recollection. Our brains can only hold so much, especially tiny details.

pervycreeper 11y ago

The dissemination of knowledge is at least as important as its discovery. Accessibility (i.e. clarity of exposition, availability to the public, etc.) needs to become a cardinal virtue in research.

jimbokun 11y ago

When I was taking machine learning courses and reading machine learning textbooks a few years ago, I have fond recollections of the derivations from Tom Mitchell's textbook. http://www.cs.cmu.edu/~tom/mlbook.html http://www.cs.cmu.edu/~tom/mlbook.html Where other textbooks tended to jump two or three steps ahead with a comment about the steps being "obvious" or "trivial", Mitchell would just include each little step. Yes, you could argue it was my responsibility to remember all of my calculus and linear algebra. But it is kind to the reader to spell out the little steps, for those of us who maybe forgot some of our calculus tricks, or maybe don't even have all of the expected pre-requisites but are trying to press on anyway. Or actually know how to perform the steps but have to stop and puzzle through which particular combination of steps you are describing as "obvious" in this particular instance. I just remember how nice it was to have those extra steps spelled out, and how much more pleasant it made reading Tom's book. So thanks, Dr. Mitchell!

aldanor 11y ago

As someone who left academia after a PhD in math (been working as a quant in HFT for the last few years, which mostly involves coding in one form or another), I can totally relate! Back then, all those stochastic integrals and measures made much more sense. However, it doesn't seem totally alien -- I'm pretty sure I could go to hacking math if required, but it would require at least several months to get into the flow.

thearn4 11y ago
I also left academia after a math PhD. Though in my case, my area was numerical linear algebra, and I entered the aerospace engineering field. A lot of what I was working on was immediately relevant, but I would say only a fraction of what became my dissertation would be counted among it.

egonschiele 11y ago

This happens to me all the time. I have a very popular illustrated post on Monads titled "Functors, Applicatives, and Monads in Pictures"[1]. When I wrote it I thought it was the best monad guide ever. Now, reading back, I can see that some parts are confusing. I still see a lot of people liking it, but three years later I wish some parts of it were better. [1] http://adit.io/posts/2013-04-17-functors,_applicatives,_and_monads_in_pictures.html http://adit.io/posts/2013-04-17-functors,_applicatives,_and_...

throweway 11y ago
Its quite good actually. Problem with monads is that no tutorial will make them less confusing. But writing code that does useful stuff with them builds the intuition. I now question the need to fully understand sometimes why not use libraries as per example and cargo cult a little to build up intuition. Then later get a more formal understanding.

JulianMorrison 11y ago

Math is twiddling with formal systems, and discovering how they behave, Some of it has uses, some doesn't, and some of what presently doesn't will in the fullness of time result in further islands of usefulness, as yet not even imagined. But ultimately, it needs no more justification than orchestral music.

fiatjaf 11y ago

The author almost realized the much more important conclusion of the fact he lived. He shouldn't conclude the article by asking "what is the purpose of studying maths?" and then giving an three stupid answers. He should have asked: is this actually "knowledge" as they say academia brings to society? Is the money researchers earn being well spent? Did I actually deserve to be remunerated by this piece of work no one understands -- and, in fact, no one has read except for maybe three people?

argonaut 11y ago
Considering that PhDs get paid next to nothing, I actually think they're getting paid pretty adequately.
- fiatjaf 11y ago
  Nothing would be better, considering this article.
  - argonaut 11y ago
    Except that some % of PhDs go on to be professors (who do a real service), and every once in a while you get a PhD student whose work probably contributes hundreds of millions of value to society. And every couple years or so, you have a PhD student who contributes billions to society in value. Not to mention PhDs usually have to TA (teach students), which accounts for some of their pay.
    - fiatjaf 11y ago
      There's no example of a PhD work that contributed hundreds of millions of value to society. (I could say there's no way to measure "value to society", in fact this concept means nothing, but I agree on settling on "an enourmous amount of productive capital to someone, not exactly the society").
      - argonaut 11y ago

fiatjaf 11y ago

How many people have read this and understand it? How much money was spent on the production of this disseration?

option_greek 11y ago

Math is the shadow universe of physics. Most theorems may not look like they are useful for anything real world till someone is able to peg all the variables to real world. And then as if by magic we realize we already know how how the real world behaves. Till someone does this pegging, the theorems sit idle waiting for problems to solve. I believe this is actually a good thing. We are letting people find solutions before someone finds problems to use them for.

stuxnet79 11y ago
> I believe this is actually a good thing. We are letting people find solutions before someone finds problems to use them for. This may lead to a few problems too: (1) Prior knowledge can serve as a blinder. Our conceptual toolkits may end up consisting of unwieldy theorems that while useful are cumbersome to use. (2) The number of theorems with no immediately evident practical use is so large that they may end up being "forgotten" anyways, and this will result in us having to do the "mental heavy lifting" all over again.
daniel-levin 11y ago
I disagree. A mathematical result doesn't have to have direct application to the real world for it to be useful. Drawing analogies between the real world and mathematical concepts is very powerful. I have a good example: Number theorists (amongst others) are seemingly obsessed with bounding things. There are entire books written about obtaining and then refining bounds - which appear to be nothing more than inequalities. There is great real-world value to be derived from seeing 'inequalities' as tools to leverage. Brian Kernighan once commented that controlling software complexity is the essence of programming [0]. I believe similar thinking applies to other aspects of software engineering, and product and business development. If you can take a hard problem, and bound its complexity, then you can say "the problem is no more complex than this". This is very useful. The chief value proposition of many SaaS businesses is the trivialisation of the upper bounds of complexity of hard problems. For instance, for many developers, Heroku makes the complexity of deployment very low. [0] http://quotes.cat-v.org/programming/ http://quotes.cat-v.org/programming/

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mirimir 11y ago

> Mathematics is an excellent proxy for problem-solving In my experience, earning a PhD in [redacted] was excellent training in problem solving. And in developing working expertise in new areas. I suspect that the choice of field is indeed irrelevant. > Mathematics embeds character in students I'd say that actually finishing a PhD does that. > Mathematics is fun Whatever you pick for your dissertation topic had better be fun ;)

jeena 11y ago

It would have been cool if the original was linked here http://fjmubeen.com/2016/02/14/202/ http://fjmubeen.com/2016/02/14/202/ and not the medium repost. But still interesting.

thaw13579 11y ago

To make an admittedly bad metaphor, it's likely a lot of that knowledge has been moved from main memory to cold storage, and it would take some time to bring it back. It certainly makes the case for why we write things down! Although the part about having to dig for the main result makes me think the abstract could be improved...

rackforms 11y ago

One may be tempted to call this Hodge Theaters syndrome.

omginternets 11y ago

Me neither. My understanding is that this is the rule, not the exception.

danieltillett 11y ago

This post inspired me to re-read my thesis (well browse through it). Although it has been 16 years since I last looked at it, I didn’t have any problem understanding it and I didn’t even really cringe reading it. I guess it depends on your field how bad this effect is.

jackmaney 11y ago
I was about to write much the same thing... Last year, I found a hard copy of my dissertation (which I defended in 2004). I skimmed through it and had absolutely no problem whatsoever understanding what I had written 11 years prior. And I've been out of academia since 2008.

peterbraden 11y ago

Could part of it just be because Mathematical notation is just so bad? It's more of a shorthand than an actual tool of conveying meaning. So much context goes into establishing what a notated equation means - and that context is now gone.

stuxnet79 11y ago
I always complain about this. While I'd say I have an affinity for math, it was much more difficult to get caught up with my peers in university than let's say programming because there was just so much context / assumed knowledge required to make sense out of what the professor was stating that it took me forever to understand even the most trivial things. To be mathematically mature constant exposure is imperative. As to how to improve mathematical notation so that it is not so subjective or context dependent - I haven't a clue - seems to me like almost everyone is alright with the status quo. Not a good thing in my opinion.

bambax 11y ago

> what is the purpose of studying maths? Mathematics is an excellent proxy for problem-solving / Mathematics embeds character in students / Mathematics is fun Those may be reasons to study maths (although, studying anything seriously probably yields comparable benefits) but doing a PhD and writing a thesis is not only about yourself: it's supposed to advance the field. It's something you do for the general community.

z3t4 11y ago

I've also forgotten basically all high level math from school. And have to re-learn when the occasion comes to use some of it. But one thing that occurred to me is that in school I just learned how to make the calculations, so I never got a deep understanding on how things worked anyway. And that's fine.

qb45 11y ago
That's fine if you don't mind the time that could have been spent on something you actually cared about. Sometimes it happened that school attempted to teach me something I was interested in and I ended up understanding it. At other times, however, it all went to /dev/null.
- z3t4 11y ago
  It's good that we forget stuff, or everyday tasks would be like querying from a fully saturated disk. But we might have a tiny fraction of it in cpu cache, that will let use make heuristic decisions.

osoti 11y ago

Top Places to Eat in U.S. According to Yelp http://goo.gl/18pkmx http://goo.gl/18pkmx

znpy 11y ago

It seems to me that most commenters are ignoring the fact that the author is a guy that basically left high level mathematics after completing his phd. So basically he went doing other stuff not functional-analysis-related and his functional-analysis got rusty. It seems quite reasonable to me. Call it old code syndrome, call it "my math got rusty", it seems quite normal to me. Also: according to http://fjmubeen.com/about/ http://fjmubeen.com/about/, the author got his phd in 2007. It's 2016. Almost ten years. What... What are we talking about?

refrigerator 11y ago
He actually finished the PhD in 2011, but started in 2007. Not sure whether that's a significant enough difference to change the point that you're making though.
- znpy 11y ago
  You're right. I still think my point is valid though. I guess he still has a strong mathematical background, a solid problem solving attitude, but details of the topic of his doctoral thesis are missing. It's been many years, it isn't strange at all.

calibraxis 11y ago

Reminds me to get around to Piper Harron's thesis. Which was made to be seriously readable. Math seems to have a culture of systematically bad writing (as Bill Thurston discussed).

GreaterFool 11y ago

I've been working with Haskell* for a couple of years and it is quite often that I work with code that I don't fully understand. I'll come across a terse bit of code, then carefully take it apart to see what it does (by taking bits and pieces out and giving them names instead of passing in using point-free notation and also adding type annotations). Once I see the whole picture, I make my own change and then carefully re-assemble the original terse bit of code. One could ask the question: wasn't the verbose version better? I'm going to lean on the side of no. If I left this verbose and other bits verbose then it would be hard to see the whole picture. I think doing maths would be better if it was done interactively with software. If equations were code then you could blow it up and look into fine details and then shrink it to a terse form while software keeps track of the transformations to make sure what you write is equivalent. Maybe it's time to add a laptop to that paper pad? * not arguing anything language specific here except that Haskell makes use of variety of notations that makes the code shorter and more like mahts. More so than most languages.

mikko-apo 11y ago
I think it's about balancing various things. Having less code is good unless there's a hidden catch you need to be aware of or it takes a few weeks to unravel what the code does. I prefer code that is understandable right away, is consistent and doesn't have any surprises.
- GreaterFool 11y ago
  A good example is Go. Many have written blogposts describing how great it is that it's such a simple language because the code is easy to read. And I can't deny that. The code is simple to read. But then I read through pages and pages of such code and all with little meaning. Here's a loop, here we check for an error condition, here's another loop, here we check check for another error condition. It makes it harder to see through all that and answer the question "what does this code try to accomplish?". At least for me, the more code there is, the harder it is to see.
  - kyllo 11y ago

wodenokoto 11y ago

Let us give some credit to the professors who read the dissertation and understood what was going on.

Houshalter 11y ago

I took a course on formal logic on courses. I put all of the questions and exercises into anki, a spaced repitition program. This ensures I will always remember it and get it in my head at an intuitive level. Basically it's like flash cards that decay exponentially. The first review is in one day, the second in two days, then 4 days, and so on.

riprowan 11y ago

We are clearly approaching the point where unassisted human intelligence is becoming insufficient to continue to master even specific domain expertise.

lin0 11y ago

This wrapper[1] helps me edit LaTeX fragment in Vim/Emacs... [1]: https://github.com/linktohack/lyxit https://github.com/linktohack/lyxit

CurtMonash 11y ago

Much the same is true of me, and of my best friend in grad school.

kazinator 11y ago

Someone doesn't understand his own work five years later to this extent, this is a strong indication that the work is actually garbage, and the prior understanding five years ago was only a delusion brought on by the circumstances: the late nights, the pressure, and so on. Perhaps it doesn't make sense today because it never did, and the self deception has long worn off, not because the author has gone daft. Several weeks ago, on the last work day before going on vacation, I submitted fixes for nine issues I found in one USB host controller driver. The last time I looked at the code was more than a year ago. I had refactored it and really improved its quality. Looking at some of the code now, I couldn't understand it that well. But that's because it wasn't as good as I thought it was. I was still relying on the fictitious story of how I thought certain aspects of the code worked really well thanks to me, and it wasn't meshing with the reality emanating from freshly reading it with more critical eyes. And, of course, I was also confronted by a reproducible crash. As I'm reading the code, I'm forced to throw away the false delusions and replace them with reality. This is because I'm smarter and fresher today, not because I've forgotten things and gotten dumber! It's taking effort because something is actually being done. Perhaps a similar problem is here: he's reading the paper with more critical eyes and seeing aspects that don't match the fake memory of how great that paper was, which was formed by clouded judgment at the time of writing. Maybe that obscure notation that he can't understand is actually incorrect garbage. His brain is reeling because it's actually digging into the material and trying to do proper work, perhaps for the first time. If you can show that your five year old work is incorrect garbage, that suggests you're actually superior today to your former self from five years ago. So that could be the thing to do. Don't read the paper assuming that it's right, and you've gone daft. Catch where you went wrong. By the way, I never have this problem with good code. I can go back a decade and everything is wonderful. Let's just say there is a suspicious smell if you can't decipher your old work. Good work is clear, and based on a correct understanding which matches that work. There is a durable, robust relationship between the latent memory of that work and the actual work, making it easy to jog your memory.

juped 11y ago

Reading mathematical papers is an acquired skill that needs practice. Writing them also is, but the skills exist somewhat independently.