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Ask HN: Concepts that clicked only years after you first encountered them?

579 点作者 luuuzeta超过 2 年前
I&#x27;m reading Petzold&#x27;s <i>Code</i> [1], and it dawned on me that I didn&#x27;t understand logic gates intuitively until now. I took a Computer Architecture course back in college, and I understood what logic gates meant in boolean algebra but not empirically. Petzold clarified this for me by going from the empirical to the theoretical using a lightbulb, a battery, wires, and relays (which he introduces when he talks about the telegraph as a way to amplify a signal).<p>Another concept is the relationship between current, voltage, and resistance. For example, I always failed to understand why longer wires mean more resistance while thicker wires mean less resistance.<p>[1]: <a href="https:&#x2F;&#x2F;www.codehiddenlanguage.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.codehiddenlanguage.com&#x2F;</a>

198 条评论

fortituded0002超过 2 年前
1. Everyone is the main character in their own story.<p>This manifest in all sorts of ways - from people not being there when you need them the most, from friends dying off as soon as proximity changes, to how and why get people get promoted in jobs. This isn&#x27;t necessarily bad, but if you don&#x27;t know how to navigate this it can be quite painful and confusing.<p>2. Representation matters.<p>I knew this for a long time, but it didn&#x27;t fully click until years had gone by and I realized I had unconsciously held myself back from pursuing a wide range of things because I just didn&#x27;t see anyone like me there.<p>3. Rules in life are just constructs that we as humans have created.<p>Starting a business helped the most on this one. That&#x27;s when I started to see that &quot;rules&quot; or &quot;procedure&quot; are all made up and exceptions can always be made.<p>(Edit: typos)
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machina_ex_deus超过 2 年前
Learning history &#x2F; literature in school is important.<p>I was a total STEM math nerd in school. I used to frequently complain how I don&#x27;t get what&#x27;s the point of it, or how it&#x27;s a waste of time and I&#x27;m learning nothing. I still think the emphasis of school was off, but I get the point of it now.<p>Stories are like code for humans. You can&#x27;t tell someone what it means to be good or bad, or to give them a course in philosophy and they will become good people. But you can tell them a good story, that engages with them emotionally, and it will change their perception. And history shows that in fact, those stories being told and repeated aren&#x27;t just interesting minor curiosity, but they have shaped the direction of humanity and they are driving it. A single person with a single story can change history in such a way that it would be completely different without it. And some stories about stories need to be told as a warning so that people will not fall for those kinds of stories again.
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thyrox超过 2 年前
The power of follow ups (especially in sales)<p>One thing which held me back for a very long time was not following up with people who didn&#x27;t show much interest initially.<p>I wasted so many good leads thinking it is impolite to follow up with people after contacting them once. My whole life changed once I understood the power of follow ups and understanding that most people are so busy that it takes at least 6 reminders before most people will take any substantial action.<p>The reverse is also true. People say a lot of things and most of the times you never cross the bridge or reach it. Nowadays, I rarely argue about anything and don&#x27;t act on stuff until a person reminds me once or twice. This small filter can be like a miracle for saving your time and energy.
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geocrasher超过 2 年前
Narcissism.<p>My grandmother was a terrible narcissist. I loved her dearly and she had a lot of wonderful qualities, but The quality that stood out the most, sadly, was narcissism.<p>My mother was also a narcissist to a somewhat lesser degree. It didn&#x27;t occur to me that I too was a narcissist until I was about 35 years old. It took waking up in the corner of the living room in my friends one bedroom apartment Early one morning to see it.<p>I had pushed away my wife and kids because in my mind all of my problems were their fault. I had blamed others for every thing that had ever happened to me or every feeling that I had felt. And in that moment I realized:<p>It&#x27;s ME.<p>Everything changed in that instant. It was no longer just about me anymore. I stopped seeing the people closest to me as opponents and started seeing them as what they were, family. My support system. The love of my life.<p>As the years have gone by since then I have seen more of my past through that light and things have become so much more clear.<p>Understanding that my grandmother was a very damaged person who turned a narcissist to deal with it, then raise my mother similarly, help me understand two things. The first was that the things I blamed myself for in the past weren&#x27;t my fault. Secondly, it helped me forgive them for some of the awful things that happened. I&#x27;m not saying it&#x27;s okay to be a narcissist. But recognizing that their narcissism affected my life, and it was something that I could shed in my own personality was a serious life changer. And the funny part is that after I realized all of this, my debilitating depression essentially went away. And that was a big deal.<p>I also learned not even 2 years ago that I have ADHD which was like a light bulb moment for me as well because it explained so much of my life.
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theptip超过 2 年前
Domain Driven Design. The book by Eric Evans lays out a bunch of concepts and as a developer that had not owned the architecture of a big domain, it was hard for me to see exactly where they fit. But after reading the book a couple times, and then encountering a few tricky domain modeling challenges, I started to see where these patterns add value. Also, as I started trying to describe the cohesive domain architecture of the system to a growing engineering organization, I also clicked on the advantage of having a standardized set of terminology for the problem, rather than inventing your own. It&#x27;s nice to be able to link to an existing explanation of what a Repository is for, instead of having to name and document your own ad-hoc architectural patterns (or more likely, end up with you ad-hoc architecture being under-documented).<p>Things like Repositories, Aggregates, Bounded Contexts, and so on are going to be a net drag on your system if you only have a few 100 kloc of code in a monolith. But they really start to shine as you grow beyond that. Bounded Contexts in particular are a gem of an idea, so good that Uber re-discovered them in their microservices design: <a href="https:&#x2F;&#x2F;www.uber.com&#x2F;blog&#x2F;microservice-architecture&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.uber.com&#x2F;blog&#x2F;microservice-architecture&#x2F;</a>.<p>(Edited to clarify the book author)
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rolenthedeep超过 2 年前
It took me an embarrassing amount of time to fully grok what object oriented programming really means. It&#x27;s one thing to have someone tell you what an object is, but it&#x27;s another thing entirely to build a fully object oriented system.<p>I remember what made it click: I was designing an animation system, which had a bunch of different interdependent moving parts. Once I started treating each part like an object and letting it manage its own state it all just clicked. I started with this massively complex functional-like system that managed four or five different motions, but once it was broken into objects most of the code just fell off and it became a nice clean system.<p>I was super proud of it at the time, but it&#x27;s pretty bad by my current standards.
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bsuvc超过 2 年前
Unit testing and using dependency injection to write test-able code.<p>I&#x27;m not sure if it was years, but it wasn&#x27;t immediate. I just didn&#x27;t understand why dependency injection was good at first, and not just someone&#x27;s weird personal code style choice.<p>I thought it was just people being &quot;Enterprisey&quot; which I&#x27;d encountered many times over the years.<p>Once I committed to unit testing, I realized how necessary it is.<p>Unfortunately I still encounter customers who haven&#x27;t bought into it and as a result have untestable code. It&#x27;s so hard to go back and retrofit testing.
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a_c超过 2 年前
People never change because of YOU. This applies to work, relationship and family.<p>It made me click about the saying that science advances one funeral at a time. It is easier to rally people of similar thought than to change people of opposite opinion. Not impossible, just more difficult. It explains a lot of thing in my opinion.<p>1. It is easier to start a start up than to convince your boss to take a certain product direction. E.g to not pursue certain pursuit, as outlined by John carmack&#x27;s departure from meta. The ultimate judgement will be whether YOUR idea survive rather than whether your boss buy your idea. And I prefer bootstrap, at least for now, for that reason.<p>2. Never attempt to change your spouse. Find the common ground instead.<p>3. Empathy is mostly about experience sharing. You can&#x27;t have people feel something they never experience before. If you can empathize, it means you have experience to draw similarity between. Imagine teaching a 18yo to be a father, that&#x27;s how preaching people to be empathetic felt like.
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itsmemattchung超过 2 年前
The power of an outline when writing.<p>Over the past few years, I&#x27;ve been teaching myself how to write better. I&#x27;m not talking about elementary syntax or grammar. I&#x27;m not talking about writing the traditional, American English five paragraph essay. I&#x27;m talking about writing longer pieces of prose, articles or blog posts or short chapters with word counts ranging anywhere between 1500-3000 words. On this journey of improving the craft, I realized that one of my biggest struggles was writing cohesively. Although I&#x27;ve been able to get lots of words on (digital) paper, eventually I&#x27;d get lost in my own web of thoughts, the article itself totally incoherent, no structure, no organization.<p>Constructing outlines and <i>reverse</i> outlines[0] has helped me tremendously. It&#x27;s not easy ... but the concept itself is finally — years later — starting to click.<p>[0] - <a href="https:&#x2F;&#x2F;explorationsofstyle.com&#x2F;2011&#x2F;02&#x2F;09&#x2F;reverse-outlines&#x2F;" rel="nofollow">https:&#x2F;&#x2F;explorationsofstyle.com&#x2F;2011&#x2F;02&#x2F;09&#x2F;reverse-outlines&#x2F;</a>
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sowbug超过 2 年前
Taking the point of view of others, and learning to view a situation from a &quot;perspective-less perspective.&quot;<p>It&#x27;s easy to think you have people skills because you listen to others and repeat their point of view back to them before telling them they&#x27;re wrong. And unfortunately you can get quite far in the business world simply by being good at demolishing other people&#x27;s positions.<p>As a mental exercise, a few years ago in meetings I started deleting the names from the running transcript I keep in my head. &quot;Joe said X and Jane said Y and then I said Z&quot; was replaced with &quot;we said X, then Y, then Z.&quot; It was a remarkably effective device to rise above the &quot;who&#x27;s going to win?&quot; attitude and instead think about the best way for everyone to proceed as a group. I suddenly started to get what people say about meditation and removing the &quot;I&quot; perspective from your life. If instead of being you, you&#x27;re a quadcopter hovering near the person you call yourself, it&#x27;s so much easier to get your ego to shut up and start listening for once.
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DylanSp超过 2 年前
Matrix multiplication. First encountered it in high school, where the textbooks presented matrices without any real motivation, and matrix multiplication just seemed like a weirdly-defined operation. Once I got to linear algebra in college and matrix multiplication was presented as the way to compose linear transformations, it made a lot more sense.
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tezza超过 2 年前
1&#x2F;2) Trusting Institutions<p>Institutions such as Police, Universities, NHS, Scouts, MsF, Religions, Churches, YCombinator et alia have a hierarchy of internal loyalties in strict precedence:<p>* The Staff Member<p>* The Staff Member’s Family<p>* The Friends of the Staff Member<p>* The Colleagues of the Staff Member<p>* The Group within the Institution the Staff Member belongs to<p>* Wider Groups in the Institution<p>* The actual powerbrokers within the Institution<p>* The acknowledged Leadership of the Institution (may be different to actual powerbrokers)<p>* The actual goals of the Institution<p>* The acknowledged goals of the Institution (may be different to actual goals)<p>* Helping YOU in accordance with what the acknowledged goals of the Institution are…<p>Only when all the loyalties in that list are satisfied is there the slightest chance you may get anything positive from the Institution.<p>Despite the long list of higher precedence loyalties it is still frequently possible to have positive outcomes…<p>But because it is a long list of loyalties far more important than helping YOU, there are often breaks.<p>And because people and families and relationships are involved they can change at any moment.<p>So trusting Institutions to do the best for you or act honourably needs to be carefully weighed against the likelihood that will happen<p>2&#x2F;2) Mortgages<p>How mortgage repayments change over time as you pay off some of it (YMMV)
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robg超过 2 年前
The autonomic nervous system and the adrenal cortex. Homeostasis is taught as a textbook fact, the body reverting to a baseline over time. What’s not taught is how much the impacts of daily life events drive a continuous stress response. Fight or flight is not just a reaction to deadly threats. It’s active every moment of every day to ensure survival. The adrenal cortex is always active, to traffic, your boss and colleagues, relationship struggles, and overall health and wellness like sleep and nutrition. Yes, the system reverts to a baseline over time but how much that baseline varies is obvious in tracking resting heart rate.
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khaledh超过 2 年前
YAGNI and KISS.<p>When I was a junior developer I used to overthink and overdesign solutions, most of which was never needed. It took many years and a lot of battle scars to realize that less abstraction is more. Today I see a lot of juniors do the same mistake and I ask them to revise their designs to keep it simple.
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_huayra_超过 2 年前
Pretty much anything related to <i>personal growth</i> (however you define it) takes time to click, but it took a while to understand two things at a deep enough level to begin to make change:<p>* everyone has a set of habits that served them well during childhood, but may be maladaptive as an adult (e.g. getting angry is almost universal a sign of this maladaptive childhood habit rearing up). Book recommendation [0]<p>* the <i>fear of hope</i> is a key thing to understand when it comes to why people (myself included) hold themselves back. Taking the first step can feel terrifying because it demonstrates that one is responsible for creating one&#x27;s own life. Book recommendation [1]<p>The only way out of this is to deliberately design your days so that you get the most out of them, even if it is fumbling at the start. Add in some times to relax here and there, but if you have a plan, it&#x27;s much easier than staring at a blank Saturday with the vague goal of &quot;I MUST learn JS&#x2F;Rust&#x2F;Go&#x2F;Scala or my career will be over!!!&quot; and then getting nothing done.<p>[0] <a href="https:&#x2F;&#x2F;www.goodreads.com&#x2F;book&#x2F;show&#x2F;23129659-adult-children-of-emotionally-immature-parents" rel="nofollow">https:&#x2F;&#x2F;www.goodreads.com&#x2F;book&#x2F;show&#x2F;23129659-adult-children-...</a><p>[1] <a href="https:&#x2F;&#x2F;www.harpercollins.com&#x2F;products&#x2F;how-we-change-ross-ellenhorn" rel="nofollow">https:&#x2F;&#x2F;www.harpercollins.com&#x2F;products&#x2F;how-we-change-ross-el...</a>
MichaelDickens超过 2 年前
What an algebra equation is. I could solve linear equations, quadratic equations, and systems of equations by adding&#x2F;subtracting&#x2F;multiplying&#x2F;dividing both sides of the equation and by adding two equations together. But I didn&#x27;t know why you were allowed to do any of those things, or what else might be allowed. Are you allowed to square both sides? Raise both sides to a power?<p>It wasn&#x27;t until maybe my third year of algebra that I realized an equation means <i>both sides of the equation are equal to each other</i>, which means you can perform any operation at all to both sides, and the result will still be equal.
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alexmolas超过 2 年前
Social constructs aren&#x27;t less real than other tangible concepts. For instance, I used to complain a lot about the non-sense that&#x27;s to wear a tuxedo in weddings. I was the typical kid saying that I would like to go wearing my best sweatpants to weddings, and that probably my sweatpants are more expensive than some of the tuxedos there. Then, I matured and realized that what we wear have a meaning, wether we like it or not. It&#x27;s like words, maybe you don&#x27;t like the definition of the word &quot;car&quot;, but using an alternative word would be useles since nobody else would understand you. So if you care about the people getting married and you want to communicate it you need to wear the damn tuxedo.
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parasti超过 2 年前
One of the biggest realizations recently for me was realizing that a nearly all of software development is basically about turning a slow, manual process into a faster, automated process. Modern CI&#x2F;CD stems from a bunch of shell commands that somebody wrote and manually executed to test an app and upload it to a server. Modern automated software testing stems from humans writing small test apps and running them to confirm correct behavior. Many modern development practices stem from allowing small test apps to be written easier and faster. It&#x27;s all just a giant manual process-to-automated process time-saving machine.
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nkrisc超过 2 年前
Two concepts that I never really understood until I encountered them together: functional programming and recursion.<p>When I first encountered recursive functions when I began learning programming I had a really hard time understanding what the function would do and how it would play out as it called itself. I couldn&#x27;t think through the recursion and imagine what would happen. Nor did I really understand how to usefully apply one to a situation. I would either use a loop or a recursive function that utilized lots of external state to work.<p>When I later encountered functional programming, having learned programming with OOP languages, it was a real mind-bender. I finally started to understand it but when I encountered the need for recursion in FP, it really threw me for a loop. How the hell was I supposed to do this without external state? So it was that restriction that really let me understand how to create a recursive function that could return something useful with nothing more than the initial input. This new understanding also gave me a better appreciation of functional programming and the idea of pure functions.
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olalonde超过 2 年前
It took me a long time to grasp market economics. I knew it worked somehow but I didn&#x27;t quite understand why. What really made it click was Milton Friedman&#x27;s &quot;Free to Choose&quot; TV series[0].<p>[0] <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=dngqR9gcDDw&amp;list=PLXD32Z5YYifX8J3Kp-WBA9Ongy0p1C1u9">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=dngqR9gcDDw&amp;list=PLXD32Z5YYi...</a>
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JanisErdmanis超过 2 年前
Superconductivity. I did four PhD projects regarding superconductivity properties at nanoscale without really getting hold on the concept. For instance, how zero resistance is related to cooper pair tunnelling and BdG Hamiltonian. Only when I was forced to write introduction a month before my thesis submission I gave my best last try for the farewell and it clicked. Note the title of my thesis “Quantum effects of superconducting phase” ;)<p>Quantum computers. I was so deceived by all the hype that it took me long to see it as SU(N) matrix product accelerator.<p>Zero knowledge proofs and Shamir transform to non-interactive protocol. The tunnel analogy did quite a harm to me, also the math was just unbearable (defining set of languages and etc). Only a year ago when I got an old paper on “Observational wallets” describing how to prove ‘log_g A = a’ and ‘log_g A = log_g B’ the whole ‘proving statement without revealing secret’ after long frustration and angst made sense to me. If I had encountered finding Vodo analogy first, It would have clicked much quicker to me.
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slotrans超过 2 年前
For me the big one was relational modeling &amp; normalization. I took a database class in college and I thought normalization was idiotic. It seemed obviously foolish and wasteful and I couldn&#x27;t imagine why anyone would do this to themselves by choice. Then ~2 years later I got a job working with data, at a company with quite a big database that was designed largely by experts. Within a few months of starting to work with real data I had internalized the Why of normalization and become a believer. Ironically I&#x27;ve been doing database work ever since.<p>Lately I&#x27;ve worked for a number of companies where no one -- like seriously, absolutely zero developers on the team -- understand normalization. They may perform a shadow of it, because someone somewhere told them to, or some article said this is how you model data that looks like X, but they don&#x27;t <i>get</i> it. Predictably, their data models are absolute garbage. Teaching them is an uphill battle because they don&#x27;t care, they&#x27;re not curious, and their web frameworks (<i>cough</i> Rails <i>cough</i>) have taught them to distance themselves from the database and treat it as dumb.<p>The other one was Rich Hickey&#x27;s &quot;Simple Made Easy&quot;, which I first watched in 2013. I enjoyed it even then but I didn&#x27;t really understand what he was getting it. After working with, and building, some systems done in a functional style, I feel like I figured most of the ideas. That would have been some time in 2017 so, definitely took a while to soak in.
whacked_new超过 2 年前
I love this as a research topic. Here are a few examples close to me.<p>1. The nature of variance in regression methods. The first time I heard about &quot;soaking up the variance&quot; in modifying stat models in brain imaging I had no idea what was going on. Then I spent a few years doing brain imaging and modified models to &quot;soak up the variance&quot; differently. Over the few years I came to grok the concept. I had a electrical engineering PhD colleague with an impressive resume who would argue with me about the effects of models. I realized that he knew the textbook stuff (which I didn&#x27;t), but he didn&#x27;t actually grok the concepts (which I did).<p>2. Once in an office was I mystified by hot spots and cold spots of wifi signal. One of my colleagues, a brilliant engineer, then explained to me what might affect the shape of the hot and cold spots, which is why he sits in certain places. I asked him if he had an intuition of how RF fields are distributed and he said yes, so I asked him how. He said, &quot;I used to do a bit of tensor calculus&quot;.
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shockeychap超过 2 年前
The quote below is from a 1995 interview with Steve Jobs. Forgive the length, but the whole thing is quite poignant and there wasn’t much to be trimmed. This particular interview confirmed my own experiences with esoteric processes that people do just because that’s how it’s always been done, and the opportunities that exist because of it.<p>————————<p>You know, throughout my years in business I discovered something. I would always ask why you do things. The answers that I would invariably get are: “Oh, that’s just the way things are done around here.” Nobody knows why they do what they do. Nobody thinks very deeply about things in business. That’s what I found.<p>I’ll give you an example. When we were building our Apple computers in a garage, we knew exactly what they cost. When we got into a factory in the Apple II days, the accountants had this notion of a standard cost, where you kind of set a standard cost and at the end of the quarter, you would adjust it with a variance. I kept asking: why do we do this? The answer was, “That’s just the way it’s done.”<p>After about six months of digging into this, I realized that the reason they did this is that they didn’t have good enough controls to know how much it’s going to cost. So you guess. And then you fix your guess at the end of the quarter. And the reason you don’t know how much it costs is because your information systems aren’t good enough. But nobody said it that way.<p>So later on, when we designed this automated factory for the Macintosh, we were able to get rid of a lot of these antiquated concepts and know exactly what something cost.<p>So in business a lot of things are folklore. They are done because they were done yesterday. And the day before. What it means is, if you are willing to ask a lot of questions and think about things and work really hard, you can learn business pretty fast. It’s not the hardest thing in the world. It’s not rocket science.
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samhuk超过 2 年前
Computer networks. For years I was dumbfounded how IP addresses, VPN, ports, etc, all worked and tied together. Then, when I was interning as a software developer a fair while ago, a colleague drew the analogy of &quot;IP address&quot; = house on street, &quot;port&quot; = something you ask for when you knock on the door of a house.<p>Then it all just clicked. I still remember that day all these years later.<p>Other notable mentions, in no particular order:<p>* Perfect is the enemy of good enough. Did n&#x27;t really appreciate this idea until around 2 years into my career.<p>* How to be professionally displeased at something. Early on my career I would get way too angry at incompetent colleagues peeing in the pool, e.g. bad code, design, management, etc. I would complain quite a bit! It only clicked ~2-3 years later into my career when I figured that one&#x27;s displeasure at a situation should be a function of <i>both</i> how bad the situation is <i>and</i> how able you are to improve it. When you offer constructive solutions to incompetence (suggest alternative algo, management style, library, tool, etc) whilst not actually mentioning what is wrong, instead of just fruitlessly reminding people of what they did wrong, people become far more cooperative and receptive, etc.<p>My tin-foil-hat pet-theory is that the relatively recent tyranny of low expectations and &quot;participation award&quot; society has on average made younger people much more sensitive to negative comments about their work. The extension of that is that people end up robbed of more detailed reasoning about what they did that was wrong.<p>People are interesting!
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ergonaught超过 2 年前
Basic music theory.<p>Almost no one I encountered bothered to actually explain anything. They simply regurgitated things and I guess expected me to somehow intuitively understand something or other.
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umarniz超过 2 年前
Self fulfilling prophecy.<p>It took several near burn-outs and tons of hours of my life lost due to frustrations before I truly understood this. The world is what you want it to be.<p>Hopeless because everyone in the world is so selfish? You create the world you live in. Live selfless and see the world transform around you.<p>Frustrated you have to constantly interfere to make someone good at their role? You have to yourself believe they are the best in that role and then see them flourish.<p>Afraid your relationship might not workout? The fear itself will make your worst thought a reality.<p>Your life becomes extremely easy once you truly understand the world is what you believe it to be.<p>Note: There are of-course strings attached to this concept, but understanding the power of this is life changing.
evanpw超过 2 年前
This is true of almost everything in math. You learn some definitions and techniques in one class, and it doesn&#x27;t all become clear what&#x27;s going on until you&#x27;ve used those to as the base layer for solving some other problems in the next class. Part of it is just that it&#x27;s hard to teach you need the first concept in order to understand the second, but you need the second to understand why you should care about the first, so it&#x27;s all a bit circular.
ISL超过 2 年前
That professors were thinking real hard about which problem sets to give us in hopes that we would actually learn something.<p>I don&#x27;t think I really understood that completely until I started TA-ing.
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lamontcg超过 2 年前
Big bang didn&#x27;t happen in a single point in our space-time like a firecracker, it happened everywhere and was just a uniformly much hotter and denser universe that didn&#x27;t really explode into anything, but space-time just expanded to make the universe less dense.
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djmips超过 2 年前
One of the big aha moments which clicked for me only fairly recently was staring at some physics equations my own internalized realization that light doesn&#x27;t experience time and then I had to tell everyone haha - but that one eureka moment unlocked a whole lot of understanding and certainly a lot more questions. This of course was after all the schooling and physics where it somehow sailed over my head the whole time.
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darawk超过 2 年前
Tensors. Years ago, I would read the Wikipedia page about them every so often, and completely fail to understand what they were.<p>Then one day I was modeling something in a spreadsheet, and I thought to myself &quot;you know, what I really need here, is 3rd axis to this spreadsheet&quot;. And for a few minutes, I thought I had invented a new form of spreadsheet&#x2F;structure, and was considering trying to build a 3d spreadsheet program.<p>But, of course, suddenly everything I&#x27;d read about tensors and not understood before snapped into place, and I realized that this is exactly what they are (to be precise, this is a special case of a tensor, the contemplation of which caused me to understand what the general case is, and why e.g. a scalar is a rank 0 tensor, a vector is rank 1, a matrix is rank 2 my 3d spreadsheet idea is rank 3, etc).
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thesnide超过 2 年前
Working with others.<p>Really.<p>I was super annoyed and insanely annoying to work with for years.<p>Then I understood that difference is hard to cope with, but more often than not, good. As it trades some short term efficiency to long term one.<p>Same with your output. If it is dumbed down to a level everyone can understand it, you <i>really</i> learned it.
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whalesalad超过 2 年前
We are all just children. As kids we always want to be adults but then we become adults and ultimately everyone around us is just a child who’s been on the earth longer.
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gpsx超过 2 年前
This is something that clicked after a few hours rather than a few years, but it is similar in a way. A long time ago when I was a graduate student I was in the library one afternoon studying some idea in field theory (physics). It covered maybe three pages in a small book, but it wasn&#x27;t making any sense. Finally, hours later it suddenly became clear, and it was so simple. I remember saying to myself, &quot;Why didn&#x27;t they just say that?&quot; I looked back at the book and they said exactly what I would have said. I guess sometimes words are not enough.
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I_complete_me超过 2 年前
Two are very pertinent for me at the moment:<p>1. The Monty Hall problem<p>2. The Wason Selection Task.<p>I read about the first one eons ago and was impressed that Marilyn Von Savant was vindicated. Shoutout to her!<p>The name Wason was unknown to me but I bought a book called The Oxford Companion to The Mind (new in 1987!) and his &#x27;task&#x27; was featured in it (pg. 639 in my edition). I spent a lot of time satisfying myself as to the answer and kind of got there. This Christmas I am reading Stephen Pinker&#x27;s &quot;Rationality&quot; and the description of both have allowed the penny to fully drop. I hadn&#x27;t been thinking about these puzzles for ages and then when they cropped up &quot;unexpectedly&quot; I grokked them easily. They &quot;clicked&quot;. There is a certain type of wisdom that comes with passed time.
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buzzert超过 2 年前
For me, it was the difference between “is” and “ought”.<p>It sounds obvious when you first think about it, but I spent a vast majority of my 20’s with a view of the world filled with “oughts” instead of “is’s”, especially when it comes to things I can’t control. I.e., people “ought” to behave a certain way, versus the way people actually behave. The way companies&#x2F;governments “ought” to operate, versus the way they actually do.<p>Thinking in terms of “oughts” can really cloud your judgement in ways that you might not be totally aware of. I guess this is kind of like the “realpolitik” philosophy, but coming to this realization in my early 30’s actually made me a lot happier and less confused about why the world is the way it is.
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btrettel超过 2 年前
It took a couple years in college for me to understand entropy.<p>Entropy in classical thermodynamics is presented in a mysterious way that leads to confusion.<p>Entropy in statistical thermodynamics, however, is logical. Once one understands basic statistical thermodynamics, entropy isn&#x27;t mysterious.<p>The book in my statistical thermodynamics class was <i>An Introduction to Thermal Physics</i> by Daniel Schroeder, which is an excellent book that I&#x27;ve referred to many times since.
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Tade0超过 2 年前
Singing.<p>I was part of a choir in high school and early college, but never managed to get good at this, even though I knew well how skilled singing should sound.<p>More than a decade later, when I was really exhausted after tending to my infant child I let out a sigh of the same type that we did as exercise back in the choir(&quot;Aaaaaaaah&quot;). To my surprise it was almost effortless despite making quite the sound due to it being more or less at the resonance frequency of the bones in my skull.<p>And then it dawned on me. It&#x27;s essentially turning yourself into an elaborate fart pillow.<p>Of course there&#x27;s much more to that in things like posture and the mentioned resonance, but I had that covered from years of training. The only piece missing was using the diaphragm correctly.<p>Now I sing daily and while my range is pretty average, I don&#x27;t strain myself any more and can do it for hours. Also the tone is just so much better now.
sdenton4超过 2 年前
Lie Groups... I signed up for a grad level course in my second year, and had no idea what was going on. Eventually I did my PhD on algebraic combinatorics, which works with Lie Groups quite a lot, but it took years to internalize all the ideas needed to have any intuition at all.<p><a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Lie_group" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Lie_group</a>
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college_physics超过 2 年前
Graph theory. For the longest time I thought it is an interesting and even beautiful branch of mathematics but somewhat &quot;recreational&quot; and disconnected from my various bread-and butter workhorses for applied work (from linear algebra and PDE&#x27;s to differential geometry etc). So I never bothered to dig and connect the dots.<p>At some point I realised the connections between the two worlds e.g., graph operations as linear algebra [0] or the transition from continuous Laplacian to graph Laplacian via a discrete Laplace operator [1]<p>[0] <a href="https:&#x2F;&#x2F;graphblas.org&#x2F;" rel="nofollow">https:&#x2F;&#x2F;graphblas.org&#x2F;</a><p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Discrete_Laplace_operator" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Discrete_Laplace_operator</a>
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deafpolygon超过 2 年前
Life is actually short. As a kid, you don&#x27;t really put much stock into that being bandied about by your elders. You feel like things will go on forever, and that you&#x27;re basically immortal. As an adult, I now have less time than the time I&#x27;ve been alive to make of what time I have left. I&#x27;m not left with cripping regret, but I have wondered what I could have done to make better use of my time.. and that time is just simply gone.
ofcrpls超过 2 年前
Fourier Transform.<p>I&#x27;d been writing DSP code in C and MATLAB during undergrad while thinking I knew every possible aspect thanks to the deep study we had as part of coursework across all the transforms methods in our ECA &amp; Communications coursework in India. Also, dad was a self-taught hands-on Analog electronics whiz whose day job was in Telecommunications Training, and I had uncles and aunts in the Telecom&#x2F;Electronics&#x2F;ATC industry, so I had all these resources from a very young age to revisit the concepts that were being taught to me in school, against practical applications over and over.<p>In week 2 or 3 of ECE490 at UofR, Prof. Heinzelman[1] broke a barrier that I did not know existed in my understanding of DSP. It was pinned to me realizing that the intuition between Fourier and Laplace transforms being the same. It was a moment that I haven&#x27;t experienced since, in that I felt my brain got re-wired within that hour. It must help that her father[2] basically wrote the textbook on speech processing.<p>[1] <a href="http:&#x2F;&#x2F;www.hajim.rochester.edu&#x2F;ece&#x2F;heinzelman&#x2F;" rel="nofollow">http:&#x2F;&#x2F;www.hajim.rochester.edu&#x2F;ece&#x2F;heinzelman&#x2F;</a><p>[2] <a href="https:&#x2F;&#x2F;www.ece.rutgers.edu&#x2F;lawrence-rabiner" rel="nofollow">https:&#x2F;&#x2F;www.ece.rutgers.edu&#x2F;lawrence-rabiner</a>
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ranting-moth超过 2 年前
Music theory. I&#x27;m not claiming I understand it all.<p>I&#x27;ve had many eureka moments when I suddenly realise what or why this thing from years ago is important.<p>I think teaching music theory with both <i>what</i> and <i>why</i> (with examples) is essential.<p>I still have haven&#x27;t found a book that does a good job of it and keeps it interesting. Any recommendations?
jethkl超过 2 年前
The axiom of choice, Tychonoff&#x27;s theorem, and the open mapping theorem [0, 1, 2]. Each one took me, in my view, much too long to grasp.<p>[0] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Axiom_of_choice" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Axiom_of_choice</a><p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Tychonoff%27s_theorem" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Tychonoff%27s_theorem</a><p>[2] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Open_mapping_theorem_(functional_analysis)" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Open_mapping_theorem_(function...</a>
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recursivedoubts超过 2 年前
i didn&#x27;t really understand hypermedia, and, in particular, the uniform interface&#x2F;HATEOAS until a few years after i started building intercooler.js (htmx predecessor)<p><a href="https:&#x2F;&#x2F;intercoolerjs.org&#x2F;2016&#x2F;01&#x2F;18&#x2F;rescuing-rest.html" rel="nofollow">https:&#x2F;&#x2F;intercoolerjs.org&#x2F;2016&#x2F;01&#x2F;18&#x2F;rescuing-rest.html</a><p>much later:<p><a href="https:&#x2F;&#x2F;htmx.org&#x2F;essays&#x2F;hateoas&#x2F;" rel="nofollow">https:&#x2F;&#x2F;htmx.org&#x2F;essays&#x2F;hateoas&#x2F;</a>
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rramadass超过 2 年前
The use of Predicate Calculus in coming up with the Proof along with the Program.<p>Predicate Calculus is used to show that the path followed by a Process through a Cartesian Product space (created from all the memory variables in a Program) is the one you had in mind w.r.t. its Specifications. Suddenly you start to understand basic Set Theory, Types, Relations (Functions) and Logic.
brunooliv超过 2 年前
The biggest one has to be design patterns in OO design!<p>I was fortunate to have a great teacher which taught OOP by focusing on design patterns, dynamic dispatch, composition over inheritance, etc, etc, the really foundational concepts.<p>I was too young and naive to fully understand all the concepts, I just thought writing Java was cool! Ah.<p>Many years later, I really realized how to apply design patterns to solve real problems in a large production codebase and it all clicked then and in retrospect that teacher was really amazing, I just was too naive to realize it back then...
jeffreyrogers超过 2 年前
Dynamic programming made very little sense to me when I first encountered it, then a few years later I read the DP section of Algorithms by Dasgupta, Papadimitriou, and Vazirani and it somehow clicked for me. Now I enjoy dynamic programming problems when I get them in interviews because they are usually pretty easy once you understand the trick.
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hawski超过 2 年前
Parenthood. Now I see how much my mom was right, but couldn&#x27;t put things in proper words.
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ryanschneider超过 2 年前
Calculus, even the basics like derivatives, 2nd derivatives and integration didn’t really click with me until I did some scientific computing&#x2F;signal processing with SciPy and Pandas a couple years ago. Graphing the discrete versions on time series data, e.g. graphing the difference between consecutive values and seeing the derivative pop out, or a rolling sum for integration, etc finally made it all click.
leftcenterright超过 2 年前
Visual thinking! [1]<p>I had read a bit about different types of thinking but I didn&#x27;t really understand it so well until I saw a kid who had amazing visual thinking capability and could visualize years&#x27; old memories in a very detailed manner. It has also been related to photographic memory according to some research.<p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Visual_thinking" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Visual_thinking</a><p>Nikola Tesla&#x27;s Creative Thinking Secrets: <a href="https:&#x2F;&#x2F;www.sers.si&#x2F;wp-content&#x2F;uploads&#x2F;2014&#x2F;10&#x2F;angle%C5%A1%C4%8Dina.pdf" rel="nofollow">https:&#x2F;&#x2F;www.sers.si&#x2F;wp-content&#x2F;uploads&#x2F;2014&#x2F;10&#x2F;angle%C5%A1%C...</a>
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Agingcoder超过 2 年前
I learned classical piano, and a bit of music theory, which was mostly about analyzing what I had in front of me.<p>I always found classical music to be a bit dead - beautiful, but dead : I guess I see it as a dead language, like latin or greek. The jazz I was given was fully written, so it was dead too, except in a few cases where I was supposed to magically improvise.<p>Then I &#x27;got&#x27; Jazz, 25 years later : it&#x27;s entirely about learning the language and grammar and vocab of music, and speaking it, creating new sentences and expressing yourself. The process probably matters more than the style itself.
bob1029超过 2 年前
I&#x27;d have to say... Linear Algebra &amp; SQL.<p>SQL is something that is a permanent journey for me. It is a domain-specific language, so the nature of its use depends almost entirely on how the domain was modeled in the first place. Much of my interest has been drawn towards the modeling aspects and how we can arrive at schemas that business experts can tolerate.<p>Linear algebra properly clicked for me when I started getting into writing my own 3d rendering software. Systems of linear equations are absolutely everywhere. I didn&#x27;t really tie together their use in control theory, DSP and graphics until I was forced to learn them &quot;for real&quot; to achieve my project goals.
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jb3689超过 2 年前
Mathematical proofs. I studied math in college and didn&#x27;t really understand anything I was doing until well after school. When I read Book of Proof a few years later it all suddenly clicked for me and made perfect sense. Half of studying math is just surviving. Same thing for my Fourier analysis and signal processing classes. I didn&#x27;t grok it until a few years after my classes. Don&#x27;t even get me started on statistics which I still find difficult to really understand &quot;why do I trust this magic formula?&quot;. The hard science behind statistics is actually incredibly complicated when you start to dig into it
tails4e超过 2 年前
The fourier transform. Encountered it first in my undergrad engineering degree, it was presented as dry mathematics, with no real explanation, just threw complex exponentials at us, and pages of derivations. Years later I actually use it in my job, and through that and other material can see its beauty, and how its actually not that complicated. Some great resources like this helped a lot:<p>[0] <a href="https:&#x2F;&#x2F;betterexplained.com&#x2F;articles&#x2F;colorized-math-equations&#x2F;" rel="nofollow">https:&#x2F;&#x2F;betterexplained.com&#x2F;articles&#x2F;colorized-math-equation...</a>
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hospadar超过 2 年前
Voltage and amperage for me, years after a college electronics course and a childhood of soldering kits I finally started to get it. I was always tripped up with analogies about how it&#x27;s like water in a pipe or something, which can be useful, but aren&#x27;t quite right.
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shrimpx超过 2 年前
I studied functional programming and I took terms like &quot;expression&quot;, &quot;evaluation&quot;, and &quot;value&quot; for granted until it clicked that these terms have intuitive meaning when interpreted in colloquial&#x2F;human terms.<p>- Expression as in someone expressing themselves, a tentative gesture.<p>- Evaluation as in scrutinizing something to learn what it&#x27;s really about.<p>- Value as in &quot;valuable&quot;.<p>We don&#x27;t know if an expression is valuable until we subject it to evaluation, or we get an interpreter to interpret it for us.
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W0lf超过 2 年前
Discrete mathematics and all sorts of its application in real-world (software development) related problems. Also how any given solution to a problem in one problem domain can be transferred to a problem in another unrelated domain. Think Galois theory but waaaay less fancy :-)
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TheAceOfHearts超过 2 年前
It felt like my understanding of pointers was a bit vague until I learned assembly. I think programmers would benefit from starting off with a simple virtual machine that runs a tiny set of instructions with everything presented cleanly in a visual format which can be inspected and stepped through.<p>The biggest a-ha! moment I&#x27;ve experienced was in my Comptuer Architecture course when it finally clicked how you could build up a bunch of logic gates into a real modern computer.
richardw超过 2 年前
In OO, the interface. It seemed like the most useless construct. Zero implementation. Why make something with no implementation?<p>Later I realised that the benefit isn’t the code, it’s the freedom you gain later in choosing the implementation. You can create an interface and add a simple implementation, then later swap it out for something more robust. All you’re agreeing to now is the contract of what needs to be done, without any restrictions on how.
user3939382超过 2 年前
f(x) = y etc in linear algebra in school. By the time I was learning this in high school I’d already been programming for several years and perfectly understood the concept of function inputs and outputs. It wasn’t until my early twenties that I realized this was just an alternate notation for functions and was so simple.<p>Sad that my math education was just focused on “memorize steps” for concepts that weren’t clearly explained.
mmcconnell1618超过 2 年前
Compound Interest. The idea and the numbers made sense but it is only with many more decades of life that I understand how much impact time has on compounding. If I had understood this when I was younger, I would have made different choices in regards to saving more. Small sacrifices when you&#x27;re younger and can easily tolerate them, have massive dividends later in life.
ChrisMarshallNY超过 2 年前
In my case, almost everything has been like that.<p>1) I learn rote.<p>1.1) Almost no improvement.<p>2) I &quot;get it.&quot;<p>2.1) My development in that area suddenly <i>explodes</i>.<p>This has happened with almost every software concept, from calculus, to Structured Programming (&quot;Whatever will I do without my precious GOTOs?&quot;), to Object-Oriented Programming, to Design Patterns, to Protocols, etc.<p>It usually only takes weeks or months, but I suspect some have taken years.
hakanderyal超过 2 年前
Power of static typing, that it allows one to develop complex programs faster and better, not slower.
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23B1超过 2 年前
Simple one: understanding probabilistic vs. deterministic thinking = the serenity prayer*<p>The former is for dealing with any situation where we can&#x27;t control the inputs and outputs. The latter is for when we can.<p>*&quot;Grant to us the serenity of mind to accept that which cannot be changed; courage to change that which can be changed, and wisdom to know the one from the other.&quot;
mejutoco超过 2 年前
Complex numbers, as a convenient representation for certain operations, in the same way as we use negative numbers to represent debts.
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criddell超过 2 年前
For me it was Microsoft’s COM and how it’s all powered by interfaces. It took a long time before it clicked for me. Fortunately, it didn’t really matter because 20+ years later, it’s still relevant. I’ve had plenty of time to work it out.<p>The other Windows-related idea that took an embarrassingly long time to work out is device &#x2F; printer &#x2F; memory contexts.
mablopoule超过 2 年前
&quot;The road to hell is paved with good intentions.&quot;, &quot;Perfect is the enemy of good&quot;, and more generally idioms that sound like either paradoxical assertions or obvious platitudes.<p>It wasn&#x27;t until late teens&#x2F;early twenties that lots of those piece of wisdom went from &quot;cliché phrases you&#x27;ve heard all your life&quot;, to (often) deeply impactful aspect of human conditions.<p>The &quot;road to hell&quot; idiom explicitly didn&#x27;t click until I had enough experience with counter-productive efforts who looks good on the outside, but actually fail to take into account a more nuanced reality.<p>It also kinda works with the distinction between &quot;values I believe in&quot; and &quot;arguments articulated around shared values&quot;. Being able to disagree on something even when (especially when) it&#x27;s invoking values I believe in is one of the most important thing I&#x27;ve gained as I matured, and yet it needs experience to &#x27;click&#x27;.
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em-bee超过 2 年前
object oriented programming.<p>i must have encountered it during my computer science classes and i certainly did some form of oo programming with LPC in MUDs, but only when i was programming modules in Pike for the Roxen webserver, it really clicked how oo programming worked.<p>in Roxen each http request causes a request object to be instantiated which lives for the lifetime of the request. further, each Roxen module gets instantiated as an object for the lifetime of the server process. the request object would call module objects to process the request. the modules would make changes to the request object which would produce the response to the http request, whereas storing data in a module object caused it to be persistent.<p>i have been working on this for a while when one morning i woke up with a literal eureka moment as i realized how objects and encapsulation worked.<p>this happened a few decades ago and it was the most clear moment of this kind that i can still remember.
skatanski超过 2 年前
I think I&#x27;ve read about git multiple times before I started using it. And even then at first it was with a dash of scepticism. I can&#x27;t put my finger on that moment, the moment of clarity but it had to be quite glorious. Before, I was &quot;what else do you need we have svn&quot;. Sounds horrible now that I think about it.
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jefftime超过 2 年前
For me it was pointers. I used C throughout college and I never really grokked the &quot;address to a value&quot; description, but one day it finally just clicked and now I love pointers
dudul超过 2 年前
A lot of the math I learned in high school didn&#x27;t make any sense until I revisited it in college. The difference between stupidly copy&#x2F;pasting rules and theorems without understanding them and having to demonstrate them from the ground up before being able to apply them made a humongous difference for me.
andyjohnson0超过 2 年前
The attachment theory of adult relationships [1].<p>Sometimes you just have to go on the journey to understand what stopped you getting to your intended destination.<p>[1] <a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Attachment_in_adults" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Attachment_in_adults</a>
csomar超过 2 年前
1. Functional programing<p>It took me a while, but once it clicked, everything changed. Programming became a process of data consumption, transformation and then presentation. It was if&#x2F;then&#x2F;loops&#x2F;classes&#x2F;libraries&#x2F;languages before that.<p>2. Entropy<p>The example of the messy room that gets messier with time is probably the worst. Entropy is everywhere (in real life or in IT); and it’s actually the reverse of that example. For me now, entropy is life.<p>3. Bitcoin<p>I was quite a crypto&#x2F;bitcoin skeptic back until 2014-2015. The whole thing seemed like a ponzi&#x2F;pump&amp;dumb scheme. Until I read the book “Mastering Bitcoin”. It goes into the details of how Bitcoin works. That’s when Bitcoin (as a cool technology) clicked to me. I still see the trading activity as suspicious but I made my peace with it as humans doing what humans always do: speculation.
Overtonwindow超过 2 年前
Staying out of debt. A simple concept that took me until I was 37 to appreciate and understand.
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ngalaiko超过 2 年前
for me it was functional programming. and not just fp as is, but it&#x27;s relation to another popular concept - oop.<p>i was properly introduced only to oop, and grasped a little of fp here and there, especially learning go and javascript.<p>what i consider a &#x27;click&#x27; for me is when i realised that all of these paradigms are interchangeable. like, an abstract method is just a function, or a function signature is the same as in interface with a single method.<p>after that i write code however it feels more appropriate for the situation i am in and don&#x27;t think too much about fancy words and patterns. it really feels like programming languages are becoming &#x27;native&#x27; for me.
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dionidium超过 2 年前
I was at a Denny&#x27;s around 2AM once and in an instant became totally convinced that I understood the Ontological Argument perfectly and that it was 100% correct and undeniable proof of the existence of God. But then I lost it.
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screye超过 2 年前
Related : I have a request for the 1 major concept that has never clicked for me --&gt; Can someone help me understand &#x27;productive value&#x27; in an economy ?<p>I have tried hard, and can&#x27;t for the love of me understand what lies at the bottom of trade, what is the base value of assets &#x2F; activity. It just doesn&#x27;t click.<p>Questions like:<p>- Can a services based economy work if the consumers of the services weren&#x27;t producing some non-services based value ?<p>- How can there be any economic value in middle-men ?<p>- Is speculation on artificially limited assets (housing) just a pyramid scheme ?<p>- How is it NOT a zero sum game ?<p>- Is a Fiat currency just &#x27;vibes based economics&#x27; ?
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berniedurfee超过 2 年前
Leg drive when bench pressing. The idea that anything you could do with your legs could possibly help drive the bar upward was always ridiculous.<p>Many, many years later, I finally found the right combination of articles and videos where it finally made total sense.<p>Instant improvement and I wished I would have understood it sooner.<p>The problem is really the name, which leads to a particular mental image, which is incorrect, but difficult to break.<p>I think naming is important and for me, can cause enough cognitive dissonance that I can’t get past what something is called to understanding very easily.
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catears超过 2 年前
I have one example and an anti-example. Both are related to algorithms and computer science.<p>A) Packing a binary tree into an array. Anyone that has attended an algorithms course has likely created a binary tree with nodes, leaf nodes, left and right child etc. Seen the pine-tree like sketch with a larger example where each node except the leaf nodes have a left and right child. So how do you pack this tree into an array and traverse it efficiently?<p>Well you turn it 90 degrees side-ways, slightly shift all nodes on the same level so that none align and put them into an array by going from leftmost to rightmost. (or other way depending on if you shifted 90 degrees or -90 degrees). Congrats, you&#x27;ve packed nodes into an array. How do you traverse it? Our root node is at index 1 and if you packed the array correctly then `idx = (idx * 2) + 1` will move down one side and `idx = (idx * 2) + 0` moves down the other. I don&#x27;t have a good visual explanation of this but you can think of the integer&#x2F;index as a bit-sequence describing when in the tree a left vs. right path was taken (with the exception of the root node).<p>B) Anti-example: Ford Fulkerson algorithm for finding shortest paths between all nodes in a graph. The algorithm is basically just three for-loops stacked on top of each other, but I still can&#x27;t grasp why it works. Something with dynamic programming and incrementally building on prior established intermediate paths. The algorithm is truly the product of a beautiful mind.
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ezedv超过 2 年前
One concept that took me a while to fully understand was the concept of decentralization in blockchain technology. When I first learned about blockchain, I understood the basic principles of how it worked, but it wasn&#x27;t until I started working for a blockchain company that the concept of decentralization really clicked for me.<p>Decentralization is a key feature of blockchain technology, and it refers to the fact that the blockchain is not controlled by a single entity or organization. Instead, it is maintained by a network of computers working together, and every transaction is recorded on a distributed ledger that is available to everyone in the network.<p>Realizing the potential implications of this decentralization - such as greater transparency, security, and accessibility - was a big moment for me, and it solidified my belief in the power of blockchain to transform industries and the way we do business.<p>If you&#x27;re interested in learning more about blockchain and decentralization, I highly recommend checking out Rather Labs (<a href="https:&#x2F;&#x2F;www.ratherlabs.com" rel="nofollow">https:&#x2F;&#x2F;www.ratherlabs.com</a>). We&#x27;re always looking for passionate people to join our team and help drive the adoption of this exciting technology.
jsonne超过 2 年前
You should look up a concept called &quot;Gestalt Learning&quot;. I liken it to the &quot;Eureka&quot; effect that Thomas Edison talked about.<p>For any professional educators or other folks more familiar with this I am about to butcher this concept I am sure but my layman&#x27;s understanding is the following. Basically rather than a bottom up approach to learning it&#x27;s a top down approach where you see the whole picture first rather than each piece and then dance around it until you just &quot;get it&quot;. Think of imagining a house and then learning about it until the entire house system makes sense. You might imagine the house, then learn a large &quot;chunk&quot; which is the foundation, then another chunk which is the plumbing, etc. etc. A more typical learning method might be you learn what a brick is and then you learn how they&#x27;re put together, then you understand a wall, etc. etc. This style of learning is more common in folks on the ASD spectrum but isn&#x27;t exclusive to ASD folks. I find for me I&#x27;ll encounter a concept, then think on it periodically over some period of time until one day I just sort of &quot;get it&quot;. It&#x27;s not better or worse than other ways of learning but it just means you need to approach things differently to learn more efficiently.
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dmak超过 2 年前
This might be a good thread to ask this. I have done all the math in college, but I don&#x27;t fundamentally understand sine, cosine, tangents, and logs. For sine, cosine, and tangents, I understand how they may correlate on a graph, but that doesn&#x27;t really give me additional insight into how to apply or use any of this. Similar thoughts on logs, especially when it comes down to O(logn).<p>Maybe someone can direct me to a video to help me develop a deeper insight?
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iKlsR超过 2 年前
I self taught myself c++ many years ago as my first lang (no idea why, I must have googled how to make games and it came up). No internet, just some downloaded word docs and html pages I saved on flash drive and went to a friends house. I was so bad when I think back that it&#x27;s actually funny now that I&#x27;ve become so proficient in it. I used to think the runtime was the compiler and my very first hello world was done in Word so you can imagine the torture I went through trying to get that to compile.<p>So the topic that took me about 2 years to get at this inefficient setup, (had finally gotten a sam&#x27;s teach yourself in 24 hours at some point iirc) was pointers. I could never grasp them. It suddenly came to me one night when reading something about letters that it clicked, the concept had something to do with delivering letters to houses and each house has an address. Can&#x27;t recall the exact details but man was it an enlightening moment.<p>I googled pointer mailbox analogy and this is very close if not what made it click for me - <a href="https:&#x2F;&#x2F;www.eskimo.com&#x2F;~scs&#x2F;cclass&#x2F;notes&#x2F;sx10.html" rel="nofollow">https:&#x2F;&#x2F;www.eskimo.com&#x2F;~scs&#x2F;cclass&#x2F;notes&#x2F;sx10.html</a>
klysm超过 2 年前
The expression problem. I had vague notions of “horizontal” and “vertical” abstraction but they weren’t concrete enough to discuss or make informed decisions about.
linuxlizard超过 2 年前
Blue Noise dithering based on a HN post from @todsacerdoti. <a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=25633483" rel="nofollow">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=25633483</a> At my previous job we had an ASIC hardware block to implement blue noise dithering. No one, even the people who created it, could explain to me how I needed to use it. Years latter, I read their blog post and a light bulb went on.
ROTMetro超过 2 年前
Dancing. I was both horrible at it and or disaster happened when doing it* so I became &#x27;too cool&#x27; to dance.<p>One New Year&#x27;s I finally had a girlfriend I could be authentic with, we were at the city&#x27;s &#x27;First Night&#x27; party, so different activities all over town, and I told her &#x27;I really want to go dance to the big band in the grand ballroom. But I am a horrible dancer and every woman I have danced with has made fun of me&#x27;. She was down, was patient, and it was one of the most fun nights of my life. After that we started taking salsa lessons. So just dance. And if the people you are with make fun of you, don&#x27;t stop dancing, stop being with those people.<p>* Example: Junior prom, slow dancing with my date in her fancy dress with her $200 hairdo. The gum I was chewing (because I took her out to sushi, which she informed me was gross, and therefore would not kiss me until I got the taste out of my mouth) somehow attached to a hair of hers in my mouth. Before I could unstick it she flicked her hair, yanking the gum from my mouth and into the rest of her long hair. Fun times were not had by all.
wtcactus超过 2 年前
I wouldn&#x27;t say years, but it took quite a few weeks, maybe even months:<p>The idea that the expansion universe is akin to the surface of a balloon expanding (albeit in 4D).<p>I don&#x27;t know why. It is so absurdly simple in my mind now. But when I was first told this (after naively asking where would the center of the expanding universe be), I just couldn&#x27;t put my head around it, it seemed absolutely gibberish to me at that moment.
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gloosx超过 2 年前
Fractality of things is all around us. The whole world is inside you, but at the same time the same world contains all of us. Most simple things are fractal, and most complex things are made using this concept. It&#x27;s very applicable to code as well. I kinda encountered this my whole life, but it started clicking only few years ago on the 5th year of spitting code out of my head
gregfjohnson超过 2 年前
1. The Fast Fourier Transform<p>2. Quantum computing<p>3. Godel’s Incompleteness Theorem<p>4. Denotational semantics<p>In all four cases, I make no claims to real expertise. However, in each of these cases there was a powerful moment of epiphany, when after much groping in a conceptual fog a light seemed to turn on, and an essential clarity, simplicity, and intuition clicked.
codethief超过 2 年前
Special Relativity. E.g. how could it be that observers A and B (moving relatively to each other) <i>both</i> think that time passes more slowly for the other one? How is that not a contradiction?<p>Man, that took me a while. The solution: A and B have different notions of simultaneity and won&#x27;t agree on which events occur at the same time. In particular, they won&#x27;t agree on time measurements.
nmca超过 2 年前
The meaning of &quot;adaptive&quot; in the sense of evolutionary theory, and the capacity of species to evolve to extinction. Reading the selfish gene as a youth I missed the point; &quot;an alien god&quot; [0] and some Hanson articles got the point across finally.<p>[0] <a href="https:&#x2F;&#x2F;www.readthesequences.com&#x2F;An-Alien-God" rel="nofollow">https:&#x2F;&#x2F;www.readthesequences.com&#x2F;An-Alien-God</a>
Berniek超过 2 年前
Accepting the fact that some people are just arseholes and it is no reflection on me. I used to worry that it was something I had or had not done that made them interact in the way they did. Now I will initially give them the benefit of the doubt, (bad day, tired, hangover) but if it continues I no longer interact with them and move on. I no longer even consider or think about them.
ruune超过 2 年前
Long time ago, but mathematical functions.<p>In hindsight stupid, but it took me embarrassingly long to understand that a function just takes a value for x, calculates it with the rest of the numbers in the function and that way basically assigns a y-value to every x-value. No idea why it took me so long, I didn&#x27;t have any similar problems with high school math
krosaen超过 2 年前
The Kalman filter. I felt like I understood it watching some videos but it always turned out I didn&#x27;t really understand it well enough to put it to use or explain it to others.<p>Gaining a better understanding ultimately just took a lot of time playing with the equations and understanding how measurement and process noise&#x2F;uncertainty get incorporated into the Kalman gain, and how that in turn affects the updated state estimate - e.g if you have zero noise in the measurement, you end up fully trusting your measurement. This tutorial [1] is the one I ended up studying. This is a case where memorizing the equations (with the help of Anki) helped me reflect on them and keep everything in my head long enough to improve my understanding.<p><a href="http:&#x2F;&#x2F;www.cs.unc.edu&#x2F;~welch&#x2F;media&#x2F;pdf&#x2F;kalman_intro.pdf" rel="nofollow">http:&#x2F;&#x2F;www.cs.unc.edu&#x2F;~welch&#x2F;media&#x2F;pdf&#x2F;kalman_intro.pdf</a>
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pizlonator超过 2 年前
My top ones:<p>- abstract interpretation. It was at least a year after I first heard of the concept that I grokked it enough to try to write my own, and it wasn’t until many goofy failed attempts and something like 6–7 years that I actually understood it enough to write a good one. And then it was another 5-6 years before I understood the theory well enough to understand abstract interpreters designed by others. So, like, that took me more than a decade to understand. Maybe I still don’t fully understand it.<p>- SSA IR design. It’s so easy to understand the facts of what SSA is, and it’s not that hard to use an SSA IR designed by others. But it took me over ten years from when I first read the papers and did my first attempt until when I actually <i>got</i> it. And I still feel like there are aspects of SSA that I don’t fully understand.
adamnemecek超过 2 年前
The fact that essentially all concepts in math, programming, physics etc. are just applications of fixed points which is predicated on the idea of nilpotence. Fixed points go by many names like invariance, spectra, diagonalization, embedding, braids etc.<p>By fixed point I mean something like the &quot;Lawvere&#x27;s fixed point theorem&quot;. <a href="https:&#x2F;&#x2F;ncatlab.org&#x2F;nlab&#x2F;show&#x2F;Lawvere%27s+fixed+point+theorem" rel="nofollow">https:&#x2F;&#x2F;ncatlab.org&#x2F;nlab&#x2F;show&#x2F;Lawvere%27s+fixed+point+theore...</a><p>I have a braindump on this <a href="https:&#x2F;&#x2F;github.com&#x2F;adamnemecek&#x2F;adjoint">https:&#x2F;&#x2F;github.com&#x2F;adamnemecek&#x2F;adjoint</a><p>I also have a discord <a href="https:&#x2F;&#x2F;discord.gg&#x2F;mr9TAhpyBW" rel="nofollow">https:&#x2F;&#x2F;discord.gg&#x2F;mr9TAhpyBW</a>
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iddan超过 2 年前
Someone once told me he derives his self-confidence from the dialogue with himself. It took me a few years to understand it:<p>1. A clear dialogue with oneself is establishing certainty about the inner self<p>2. Certainty with the inner self enables one to see the outside in a clear way<p>3. That clarity contributes to confidence in one&#x27;s actions.
mumblemumble超过 2 年前
The monitor model in second language acquisition. Or, more accurately, the more contemporary synthesis that it&#x27;s developed into in the decades since it was first proposed.<p>The model itself is easy enough to grasp. But concretely understanding what it implies about how I should be studying took much, much longer.
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semireg超过 2 年前
I read “Deep Survival” a few years ago pre-pandemic and was pleasantly surprised it wasn’t a book of survival stories. The main takeaway for me was that sure, nature doesn’t care if you live or die, yet surviving in modern life is an illusion - a papering over of the brutal, unfeeling and inescapable “nature.”<p>The author wrote of survivors coming home after a harrowing near-death experience and realizing that survival is one day at a time, even in the comfort of your own home. Once you taste true survival it may haunt you. Survival is an easy concept with subtle and deep physical and mental consequences.<p>Kind of reminds me of David Foster Wallace’s “This is water” - <a href="https:&#x2F;&#x2F;youtu.be&#x2F;eC7xzavzEKY" rel="nofollow">https:&#x2F;&#x2F;youtu.be&#x2F;eC7xzavzEKY</a>
zweifuss超过 2 年前
1. Object-oriented programming. At that time, no one gave a clear explanation of what was actually meant by it and how to implement it. The various schools of OOP often did not acknowledge the existence of other (heretical) schools, which caused a lot of confusion in my brain. Critical voices were hounded and silenced. Also, the gap between implementation and theory was always quite wide. I did not know Smalltalk at that time and have not seen it in use to this day. 2. Electromagnetic fields. I&#x27;m still not sure I understand them. I found this video helpful: <a href="https:&#x2F;&#x2F;youtu.be&#x2F;XoVW7CRR5JY" rel="nofollow">https:&#x2F;&#x2F;youtu.be&#x2F;XoVW7CRR5JY</a>
progJustMath超过 2 年前
Programming as “just math”.<p>I got wrapped up sitting there memorizing ins and outs of each language and the ecosystem, others compositions in the form of Apache and the like.<p>Now it seems bizarre we’d think of it as anything but add, div, compare of electrical state in a memory address. It’s not a 1:1 machine translation but it’s the abstraction that’s made me most productive.<p>A whole lot of baggage comes with software that I’m hopeful ML libraries will allow us to retire. Currently wrapping UI around OpenCV to make my own “Photoshop”, for example.<p>I really can’t see any other way of approaching programming as anything but a waste of human agency. There are social problems we could be focused on if we were less focused on butts in chairs cranking out code.
harimau777超过 2 年前
Socialism<p>When I got out of my middle class bubble, made friends with people working multiple jobs and struggling to make ends meet, and experienced a period of financial instability; I began to realize that something wasn&#x27;t working in our current system.<p>Then I started struggling with burnout and other issues and found that corporations were happy to just replace me. I also found that management wanted programmers to be replaceable cogs instead of professionals. At that point I started to suspect that the idea of a dignified professional lifestyle may not be true.<p>I observed that technology and products were getting worse over time. For example, Google search has become mostly useless and it&#x27;s hard to find products that are made to be repaired. I concluded that the invisible hand and&#x2F;or the price theory of value were not true.<p>Then I saw Republicans gain power and run up the national debt. I also observed that when wages actually started increasing the economy fell apart and the Fed started taking steps to prevent wage increases. I concluded that &quot;free market&quot; rhetoric was a lie.<p>At that point I looked for alternatives and found socialism. In particular the &quot;social democracy&quot; strains of socialism as opposed to those advocating central planning or anarchist organization.
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Eddy_Viscosity2超过 2 年前
Here&#x27;s a concept I still don&#x27;t get, even after multiple attempts: quaternions!<p>I mean I get what they are and how they are used and could do operations with them by following the rules. But... I still don&#x27;t really understand them.
more_corn超过 2 年前
When I was 12 my wrestling coach was getting his PHD in fluid dynamics. He gave a tour of the lab where he worked. On screen was a simulation. He explained that they fly a plane back and forth through a certain column of air in the stratosphere to get starting data. Then, using his model he predicts what the flow will look like over a period of time. There were specks flowing through the space onscreen. I asked what they were. “Bugs” he said. Six years later I was sitting in the library studying and I slapped my forehead. “Bugs! Of course! Ha!”
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danielschonfeld超过 2 年前
One thing I feel is clicking for me this year is that of inelastic systems. Be it monetary systems or one’s life occupation. The name of the game is being elastic and rigidity is the end all. Unfortunately as we age ?and as systems age) rigidity seems to be the default path and it takes greater and greater power and aptitude to fight that.<p>One I’m working on still is the unfairness of life. I understand it is unfair down to the level of the cell but how to come to terms with that I still grapple with.
blaser-waffle超过 2 年前
&gt; no one is coming to save you<p>Thought that was about working extra hard to make money and CYA, but really it&#x27;s about finding your own happiness, regardless of what it may be in a money &#x2F; status &#x2F; property sense.<p>&gt; everything related to RSA &amp; trapdoor algos<p>Like I go the rough idea and implications, but didn&#x27;t &quot;click&quot; until I was trying to explain what the square root of 11 was to my nephew. Was helping him with homework and was doing everything in my head just fine until we hit that and I had to stop and think and it clicked.
apienx超过 2 年前
Bayes theorem really clicked when I saw 3Blue1Brown&#x27;s visual representation. <a href="https:&#x2F;&#x2F;youtu.be&#x2F;HZGCoVF3YvM" rel="nofollow">https:&#x2F;&#x2F;youtu.be&#x2F;HZGCoVF3YvM</a>
hammock超过 2 年前
&gt;longer wires mean more resistance while thicker wires mean less resistance.<p>Intuitively (perhaps incorrectly) I would assume that it&#x27;s like trying to force a fluid through a skinny long pipe vs a wide pipe - the skinny long pipe will have higher pressure inside.<p>Or, the long thin wire has more distance for the current to travel (= more resistance), while the thicker wire has more &quot;options&quot; for the current to choose a path of least resistance (literally), tending towards a lower overall resistance compared to a thinner wire of the same length
thewebcount超过 2 年前
For me it was design patterns. I read the book in the late 90s. I was working mostly on Mac development using PowerPlant. But despite the fact that it used several of the design patterns described in the Gang of Four book, many of them still didn’t seem very clear to me. Then when MacOS became OS X, and I started using Cocoa, it just clicked. Having delegates finally made sense, for example. Previously, it was so abstract, but once I saw it in action within Cocoa, it made a lot more sense for some reason.
99failures超过 2 年前
&quot;The Innovator&#x27;s Dilemma&quot;. I read the book and I took the class in b-school.<p>At the moment I thought it was OK, fast forward a decade and it&#x27;s one of the pillars of my business thinking.
rolisz超过 2 年前
Differential equations took a couple of years to grok. I first encountered them in high school, while preparing for physics Olympiad. I could solve basic differential equations by following the &quot;rules&quot; (such as for dampened oscilator), but I didn&#x27;t understand what was going on under the hood. When in university I did some more math courses, suddenly differential equations clicked and made sense (and I could even derive some of the rules)
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sokoloff超过 2 年前
Systems in equilibrium. A lot of my college engineering courses had these (what seemed to me to be) hand-wavy assertions of equality and what seemed like just an assumption that the system would converge to that point.<p>I was probably in my late 30s or early 40s before I really grokked <i>why</i> that tended to be true. (I could blindly accept and grind through the equations to get the answers in college, but it was decades later that I developed a feel for <i>why</i>.)
fouric超过 2 年前
Monads in a software engineering context (&quot;a particular set of rules for composition of two pieces of code&quot;); defunctionalization; Lisp structural macros; fexprs; the Rust lifetime system; how to structure functional programs. All of these things had a delta in years between when I first encountered them and when I finally understood them, with repeated (4-8) spaced exposures over that time period contributing.
spritefs超过 2 年前
1. Being able to derive bottom up dynamic programming leetcode solutions. The only way I really &quot;figured out&quot; how to do this was by reading the dp chapter out of the Cormen algorithms book. It&#x27;s crazy how illuminating (and rare) clear explanations are<p>2. The chain rule. I knew the actual rule from a calc class, but the intuition behind it didn&#x27;t make sense to me until I read a couple pdfs on backpropagation<p>3. Money brings out the worst in people
high_priest超过 2 年前
Homologation implies not only &quot;the granting of approval by an official authority&quot;, but compliance with social agreement about unification of solutions &#x2F; standarisation of thought.<p>I&#x27;ve always thought that homologated solutions are parts of a bigger whole, bigger &quot;homo&quot;. Stating that the producing party has all the right knowledge to make the part fit and adhere to safety guarantees.
revskill超过 2 年前
Everything is a monoid to be useful.<p>Now i understand more about why 1+1=2.
XCSme超过 2 年前
It took me like 4 years of playing&#x2F;training table-tennis to understand what being relaxed when hitting means and how it feels like. It&#x27;s very easy to tense up and not have all your muscles work harmoniously, but it takes years of practice to simply &quot;be relaxed&quot; when playing.
ww520超过 2 年前
Shader and OpenGL programming. It was many foreign jagons, concepts, and computation models. Took me a while and couple tries to get a hang of it.
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ved_a超过 2 年前
For me, it’s the determinism of nature. The clear hard fact that nothing I can do will impact the pre-determined outcomes. This is cold physics as we know of today. Also, the ‘present’ feeling different from past is just a trick our mind plays on us. This realisation has had profound impact on me - but this dawned on me quite late despite reading a lot on this topic for years.
pfoof超过 2 年前
The importance on clean code, architecture and testing. That was during my university when we had a serious diploma group project on the last semester.<p>Also importance of reading books even later. My mother told me this all the time but school books were not the best choice. In high school I started reading fanfics, then occult books and finally I landed on self-helps and ones related to my SWE career.
nathanmcrae超过 2 年前
For me it was lambda calculus. I remember in high-school going through the motions of beta-reduction but having no idea what it meant. Much later I think I finally saw an implementation of natural numbers and addition using bare lambda calculus and it clicked that you really can represent any computation with just variables, abstraction, and application.
supernova87a超过 2 年前
Quantum mechanics as continuous probability distributions.<p>It took me about 3 tries in taking QM classes to learn and understand that the whole thing was about continuous probability distributions (at least, the beginner, static stuff). And finally to understand <i>how</i> to understand what continuous probability distributions are, and how to use them.
jackcosgrove超过 2 年前
It&#x27;s not really a discrete concept, but the course western philosophy took over the past 500 years, and its dialogue with science, was something that took me decades to appreciate. In college I sort of understood the outlines of this development, but years of learning, reflection, and life experience have really solidified it for me.
narag超过 2 年前
That Pareto is everywhere and, when applied to knowledge, a comprehensive study plan, like courses or a book, for most subjects, tend to obscure what&#x27;s the meat.<p>Waking call:<p><a href="https:&#x2F;&#x2F;sive.rs&#x2F;kimo" rel="nofollow">https:&#x2F;&#x2F;sive.rs&#x2F;kimo</a><p>Corollary: a place where there&#x27;s a fat-ass book of &quot;best practices&quot; has probably lost its focus.
mo_42超过 2 年前
Bayesian probability or Bayesian thinking in general<p>Like many, I learned Bayes&#x27; theorem for an exam. I even did well. But it clicked only when I was reading The Scout Mindset (by Julia Galef). I cannot really tell why. I think it helped me connect math formalism with more real-world examples outside of statistics.
CSDude超过 2 年前
Sadly, React. Was using Angular 1, and Vue made much more sense. I&#x27;m not a Frontend Developer though, but needed to learn React but it happened to late. At least, when I needed to learn, components were not classes anymore but functions which made learning curve somehow simpler for me although its not much difference.
foobarbecue超过 2 年前
The Nyquist frequency. Looking back, it&#x27;s hard for me to understand why I had trouble with it. I remember a friend telling me it was really simple but at the time I just thought it was because he was smarter than me or something. I guess I didn&#x27;t see a good demonstration of aliasing until years later?
foota超过 2 年前
Basically all of calculus, but especially multivariable and differential equations. I was first introduced to them in high school and again in college, I struggled with them in high school, but breezed through them in college. Not sure if I just got smarter, worked harder, or some combination of the two :)
nobodyandproud超过 2 年前
A bit more light hearted, but I would’ve had far more dates (and one nighters) if I’d smiled more and worked on my small talk.<p>It only dawned on me many years later that I was:<p>a. physically attractive to the opposite sex, especially during my teens and twenties, regardless of my ethnicity.<p>b. my blank expression was intimidating.
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KolenCh超过 2 年前
MPI.<p>A prof. teaching a grad level course on parallel computing mentioned in class that even he himself took a few years to get used to program in message passing protocol like MPI.<p>Funny enough, even when I don&#x27;t need to program in MPI extensively, after a few years my fear of writing in MPI just went away.
wiseowise超过 2 年前
Programming languages don&#x27;t matter.
ttul超过 2 年前
When we think about history over any time scale, we tend to think about one region, person, group, etc within that period of time. In reality, “everything” was going on during that time.<p>History is far richer than we tend to realize. So much is going on all the time; it’s impossible to grasp it all.
itsgrimetime超过 2 年前
programming in general, i tried probably half a dozen different times throughout my early teens to understand how to write programs in pascal, C, python, java, and it wasn&#x27;t until I was 18 when I found Zed Shaw&#x27;s &quot;Learn Python the Hard Way&quot; that it clicked.
dplgk超过 2 年前
Steely Dan
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vegetablepotpie超过 2 年前
UML<p>When I first saw it, I thought it was a <i>process</i> step that was unnecessary.<p>Then I thought it was a way to program visually, but generated code only works in narrow domains.<p>I realize that UML is tool to define your system, and it lays bare your assumptions, which is one of the hardest problems in computing.
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tevon超过 2 年前
Anyone who is excels is out of their depth.<p>By definition if you continue to progress (in career or life), you will be doing something you have never done before. This means everyone that we look up to who appears to be on a upward trajectory is making it up as they go.
nycdotnet超过 2 年前
I didn&#x27;t really understand HTTP until I started using Fiddler which was probably 10 years after I had written my first web server application. &quot;What do you mean I can&#x27;t change the headers because they&#x27;ve already been sent?&quot;
cm2187超过 2 年前
Coriolis force. It&#x27;s pretty basic but I never really got it until I watched the Tom Scott video <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=bJ_seXo-Enc">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=bJ_seXo-Enc</a>
dgudkov超过 2 年前
Data visualization. I had many years of experience with data analytics, but only when I read &quot;Now you see it&quot; by Stephen Few I started understanding how data visualization is linked to the way the human eye and brain work.
friend_and_foe超过 2 年前
Understanding your data representation is more important than understanding how to write code. It makes sense once you&#x27;ve used code to solve problems a few times, but when you&#x27;re just starting out you don&#x27;t get it.
zenmacro超过 2 年前
Central Banking 101 by Joseph Wang does a great of explaining the repo and reverse repo markets.<p><a href="https:&#x2F;&#x2F;books.google.com&#x2F;books?id=wPs_EAAAQBAJ" rel="nofollow">https:&#x2F;&#x2F;books.google.com&#x2F;books?id=wPs_EAAAQBAJ</a>
andrewflnr超过 2 年前
call-with-current-continuation. First ran into it as a teen and the concept pretty much completely bounced off my brain. I think it took a couple more tries before I started getting it, and arguably I&#x27;m still working on it.
Simon_O_Rourke超过 2 年前
The concept of decorators in Java and Python - I know what they&#x27;re supposed to do, but each time I find myself grinding my teeth when using them, thinking there&#x27;s a better way of doing the same things more simply.
mixmastamyk超过 2 年前
Not my click exactly, but hearing the “Unicode sandwich” described ~15 years after studying Unicode and using it in production was the last piece of the puzzle no one ever mentioned before. Or perhaps in not so clearly.
mattsears18超过 2 年前
Functions in calculus... I never understood them until I learned about functions in programming several years later. They&#x27;re the same damn thing! I really wish I had learned to program before learning calculus.
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ahmadnoid超过 2 年前
That sometimes the extreme needs to manifest in order to pull toward the middle
abyesilyurt超过 2 年前
Specification vs Implementation: There could be only one specification, but many implementations. There is only one Python specification, but it has many implementations, cPython being the most popular.
atemerev超过 2 年前
The relationship between entropy and states of matter (including pressure and temperature). How “degrees of freedom” are calculated and compared for gas molecules, if they are infinite. All that stuff.
deepersprout超过 2 年前
Haskel monads.
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justinhj超过 2 年前
A bon mot I saw on a sign outside a liquor store: When I was young I thought a $1000 was a lot of money. Now I’m older I realize that $1000 is a great deal of money indeed.
2devnull超过 2 年前
The importance of play.
kozak超过 2 年前
Entropy. Both in information and in thermodynamics, and how brilliantly they are connected. The audiobook &quot;The Big Picture&quot; by Sean Carroll has helped a lot.
ojr超过 2 年前
Objective-C message passing&#x2F;variable declaration&#x27;s the syntax is so different from C, C++, Java, Javascript, Ruby, Python and other popular OOP languages
gawin超过 2 年前
Ram Dass - Be Here Now<p>Amazing book with mind blowing illustrations about life, purpose and spirituality. Changed my life. Every time I re-read it I discover something new.
jesprenj超过 2 年前
• Basic concepts of dynamic programming and precomputation.<p>During lockdown I attended a course that prepared students for IOI. I couldn&#x27;t focus&#x2F;follow at home via web conferencing. One year after I attended some lectures from the same course, but physically. I understood everything when it was explained in person.<p>• Basic concept of buckets in distributed hash tables. After reading BEP-0005 I was left puzzled and left the topic for some months. Reading a Wikipedia article about Kademila, first understanding static buckets and transferring this to dynamic buckets with splitting was way easier to grasp.
wunderlust超过 2 年前
Socrates’s “all I know is that I know nothing” and “know thyself” more and more reveal to me something new about life, philosophy, and the nature of being.
amorroxic超过 2 年前
the null concept being referred to as “the billion dollars mistake” - until discovering optionals and gradually opening up to FP, category theory, etc
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mustafabisic1超过 2 年前
For me it was basic statistics. Statistical significance and variability.<p>Had Statistics in college, but only understood it now that I use it to understand data.
LAC-Tech超过 2 年前
I&#x27;m currently doing a dive into classic distributed systems papers, mainly from the 70s (actor model, logical clocks, that kind of stuff).<p>I&#x27;d &quot;understood&quot; the concepts before, but now because I am:<p>- brushing up on my math to understand every equation or proof they drop in there<p>- reading them in combination with applied stuff that uses the same concepts, ie the &quot;designing data intensive applications book&quot;<p>- reading over them slowly, I want my fundamentals to be strong and etched into my head<p>Things are clicking in a way they never did before.<p>TL;DR - studying compsci concepts, slowly, from multiple angles (completely mathematical to practical engineering) is just a different level of understanding from doing one or assuming your mind will bridge the gap.<p>YMMV.
jcelerier超过 2 年前
it was very hard for me to understand what was the point of classes when I opened my first C++ book when I was.. 11&#x2F;12?<p>thankfully it became very clear as soon as I started having actual projects and noticing all the C apis I was writing would all look like<p><pre><code> foo* foo_create(); void foo_destroy(foo*); void foo_set_stuff(foo*, int); void foo_bloberize(foo*);</code></pre>
lqdc13超过 2 年前
Forward-Backward algorithm before there were all sorts of resources and explanations on how it works online.<p>The wikipedia page for it explains it well.
aj7超过 2 年前
That the integral I-V relations for an inductor and capacitor are the fundamental forms, not the differential relations.
jonsen超过 2 年前
Recursion. I tried and tried and tried. Only when I untried the exact same number of times did it click.
jmorenoamor超过 2 年前
Orbital mechanics. It was after playing KSP that I had that &quot;woah! this makes sense!&quot; moment.
rubicon33超过 2 年前
Eating fiber.
throwaway81523超过 2 年前
The purpose of C++ template generics only made sense to me after I started using Haskell.
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gerbilly超过 2 年前
That there will <i>always</i> be work to be done.<p>Whether you define it as working on yourself, on your relationships, fixing up the house, the car, washing the dishes.<p>I guess I sort of lived as if there would be some kind of &#x27;over the rainbow&#x27; someday when I could retire and just chill, but this year when watching &#x27;Stutz&#x27; on Netflix, where he states that there are three inescapable things in life: &quot;pain, uncertainty, and constant work,&quot; the work part really sunk in for me.<p>I&#x27;ve been watching my mother try to create a kind of peaceful cocoon with her new house and her manicured lawns and list of friends curated down to only the ones which don&#x27;t ever challenge her etc, only to find that even nearing her eighties she constantly has surprises and challenges intruding on the peace she is trying to cultivate.<p>The tldr; of it is that I&#x27;ve been looking at work the wrong way, i.e. I&#x27;ve been trying to avoid it or push it away, but I feel I should embrace it. Some recurring tasks can even be sort of comforting in that it creates a routine.
tobyhinloopen超过 2 年前
Shaders.<p>I tried shader tutorials multiple times for years and they only started to click recently.
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lebaux超过 2 年前
theory of reliativity, took me a while, but now I think I got the main gist
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keepquestioning超过 2 年前
People like to pretend they didn&#x27;t work hard for what they have.
SNosTrAnDbLe超过 2 年前
Internet Protocols. Specifically the beauty and simplicity of IPv4
grogenaut超过 2 年前
It took me 4 tries over 10 years to get antlr to work&#x2F;grok.
pcurve超过 2 年前
object oriented programming, with public, private, classes. inheritance, parameters... ugh. It took much longer to internalize these than I&#x27;d care to admit.
chx超过 2 年前
git.<p>Only <a href="https:&#x2F;&#x2F;www.cduan.com&#x2F;technical&#x2F;git&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.cduan.com&#x2F;technical&#x2F;git&#x2F;</a> worked for me.
furyofantares超过 2 年前
Gender as a social construct, separate from biological sex.
sciencesama超过 2 年前
Basics of software planning by reading phoenix project.
dqpb超过 2 年前
Data structures are more interesting than algorithms.
iancmceachern超过 2 年前
The simple beauty of the calculus integral.
topologie超过 2 年前
Refactoring.
colinwilyb超过 2 年前
Formal logic.
alexmolas超过 2 年前
Monads are just a design pattern.
anacleto超过 2 年前
Krebs cycle.
throwaway98797超过 2 年前
blockchains as a single source of truth<p>always seemed like a shitty expensive database for 7+ years
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erdos4d超过 2 年前
Frechet differentiation.
derangedHorse超过 2 年前
Bitcoin and the need for an alternative to the US dollar and gold as reserve assets
zagrebian超过 2 年前
The cascade in CSS.
IceHegel超过 2 年前
There are no rules.
Nimitz14超过 2 年前
Fourier transform
sampo超过 2 年前
Eigenvectors
hydroid7超过 2 年前
OOP on Java.
antiquark超过 2 年前
The idea that atheism is a form of religion.
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rajnathani超过 2 年前
Privacy.
snappy173超过 2 年前
The bowline
ryanlitalien超过 2 年前
Docker.
brailsafe超过 2 年前
Trig
lakeshastina超过 2 年前
(1) Pointers<p>(2) Recursion