I wish the rhetoric around "screens" was less focused on the delivery mechanism and instead more focused on the problematic thing behind those screens.
The issues raised here seem to not be related to glowing LCDs, but rather particular apps and behaviors in those apps that lead to addictive and problematic behavior.
"Screens" is even broader than saying things like "the internet is bad". The latter statement already sounds unhelpful because everyone has a better understanding of just how diverse of a platform "the internet" is.
The problem, of course, is that it's really difficult for a parent to differentiate between the various types of things kids can do on their screens, especially for someone who is less tech savvy in general. This isn't to say that it shouldn't be done, just that it's hard.
Personally, if I had to select one "easy avenue" to ban or restrict, it would be not screens but internet access. This has the side effect of putting the parent in charge of acquiring new content.
I built one of these! With a shifter on the palm, and a separate control toggle, you'd get pretty much the full ASCII set.
Even though I'm right handed, it worked best on my left hand. This is because of how surprisingly nice it was to use my dominant hand to do something (use a mouse, eat, etc), while still typing with my left.
Alas I assume the reason this never caught on was due to the learning curve. People will never leave qwerty. Also it was pretty slow. When I built this back in 2005, I was thinking for PDAs and early cell phones and was only competing against T9 and early palm keyboards. After a few months of practice, I never topped about 45 wpm.
> Chorderoy is a an attempt at crafting an optimal method of text input for mobile and wearable devices aimed at those who enjoy the rewards that come with free climbing steep learning curves.
These videos are frankly better explanation of college-level math concepts than most college classes.
Also, now you probably care much more about the intuitions of mathematics over the raw mechanics of it. Once again, this channel perfectly exemplifies this concept.
These videos are posted any time linear algebra is mentioned. I find it almost comical at this point. What good is this intuition? I struggle to understand the value that these videos bring, but I'm not saying there is no value. I'm just lost, and kind of jealous because I need a deep understanding of linear algebra for work.
Those are good videos, and I also endorse them. However, you will not truly know what is going on until you solve some real problems with them.
One interesting option that computer programmers have to understand linear algebra that most people do not is that you could go do some stuff with computer graphics. Grab three.js, and then once you've followed some tutorial somewhere to get a triangle on the screen, start doing things, but do them manually, including implementing matrix multiplication yourself. Modern graphics has moved so far up the stack nowadays that you probably shouldn't say that you "know 3D graphics" after that exercise, because you'll know 3D graphics circa 1995. But you will have a much better intuition for linear algebra, and those videos will either make sense, or be trivially obvious to you.
(One of the reasons why the linear algebra videos can be so helpful is that it has historically been very easy to take an entire class on the topic and just grind numbers, without ever getting to that level of intuition. Differential equations, if you took physics that did not use them, can have a very similar problem, where you just grind through problems for a semester with no motivation.)
It's not just linear algebra, 3blue1brown also has an entire series on undergraduate calculus, and series on Statistics, Linear Algebra II and Group Theory are in the works. Plus a large number of excellent videos on miscellaneous math topics.
I think I understand what you're saying -- one needs more than just cool videos and cool intuition. You need to do exercises. This is a point made multiple times in those very videos.
But, the intuition provided in those videos is absolutely excellent. As an example, look at the explanation of change-of-basis in the linear algebra I series.
Ah, the videos come with pointers to exercises? I'm quite excited about https://www.edukera.com/ and possibilities for interactive learning through automated theorem proving. Thanks!
Actually, there are not many pointers to exercises. I believe they may be planning to add some written materials, so maybe in the future, but not currently.
The videos suggest pausing and trying to figure the next bit out yourself and a couple of the videos do end with a suggestion to prove something yourself.
For me personally, understanding why it is done on a deeper level than is commonly taught helps me consolidate the concept more comprehensively and permanently.
I think what stands out about those videos is that people who have no prior higher math education still get a shadow of intuition of what's actually going on, and people who already 'grokked' the concepts still got an alternative, simpler view on those concepts, resulting in at least a view 'a-ha!' moments for almost everyone.
Grant (the 3Blue1Brown guy) has an uncanny ability to explain difficult concepts and the fundamental intuition behind them. In many of his videos, he explains topics from the perspective of a person inventing that topic (such as in his first Calculus video [1]). I can't recommend his videos enough.
It works for a lot of people, I'm just looking for something else, exercise-based computer-checked proofs to teach mathematics, something like that. Want it bad.
I second this. I was refreshing some of the linear algebra I learned in college recently and his videos gave far more insight into what linear algebra is actually about than I was taught in college. If everyone studying linear algebra in school watched his videos before taking the course they would have a much easier time learning it. His calculus series is of similarly high quality and I would imagine his other videos are too.
It is important to really pause the videos at some points and do the "exercises". Doing the exercises is always a very important part of learning (I believe this holds in any field). Do them with pen and paper, not just in your head.
I would love to see a list compiled somewhere of two articles with exactly 6 degrees of separation. This is proving to be extremely difficult. In the entire HN thread so far, I only see one so far by dkuder https://www.sixdegreesofwikipedia.com/?source=Six%20Degrees%...
I'm storing all the search results and will do some analysis on the data. Maybe I'll even get around to adding a new page with some of the interesting stuff I find.
These videos do an incredible job of illustrating how to intuitively arrive at an answer by composing many of the parts you need to build a proof for more complex topics.
In a similar vein, I've been enjoying PBS Infinite series. The original host recently stepped aside to finish her dissertation and the new hosts are doing great but still finding their footing a bit - I suggest looking at some of the older videos to get a good sense of the channel.
It always saddens me how mathematicians seem to look down on “intuition”. Maybe higher math as a full-time job is just hard work and stubborn precision but for me, an intuitive, visual look is probably getting me closer to understanding than any cold-hard-facts book does. Not to mention how much easier it is to appreciate the beauty of it.
I recommend reading the entire post - it's not too long, and links to other great materials - but this is a good relevant quote:
> The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.
Mathematians (such as myself) don’t look down on intuition. It is the only way to construct a map through a complex proof.
However what you may have picked up on is we disstain people who insist they have a brilliant idea for which someone should prove they are right. That’s like saying a car engine consists of cyclinders and piston; someone else just needs to do the work to put the pieces together.
> It always saddens me how mathematicians seem to look down on “intuition”.
They do this for good reason. I had to take, as part of my philosophy concentration, a bunch of mathematical logic classes. First Order and Second Order logic, as you might imagine, are pretty simple. The rules make sense in a very intuitive way. I'd get 100 on exams just because I'm good at programming. But when I had to study metalogic, model theory, Henkin Proofs, and Godel's Theorems, that intuition quite literally flies out the window. I guess my point is that most interesting stuff is rarely intuitive.
That is ludicrous. The intuition is just harder to find, harder to grasp. But it is still there because these theorems are logical consequences of the base axioms. And the base axioms are logical consequences of our desire for sensible assumptions.
This is provably untrue and all I need to do is think about how weird stuff gets when increasing dimensionality of "household" shapes like circles and squares. It literally goes against every intuitive fiber of my body.
If you think that coming up with theorems about, e.g. infinitary algebra, is intuitive, then I guess you're just a lot smarter than I am. Infinities always throw a wrench into our intuitions. Something being a "logical consequence" of something else is an incredibly simplistic way of looking at things, not to mention that I'm not exactly sure what that has to do with intuition anyway.
It /becomes/ intuitive with time, thought, and exposure, as you pick up more tools for understanding the problem space.
That 'non-intuitive' aspect of high-dimensional spheres having very small volume relative to a hypercube becomes a point that builds intuition for how those spheres behave, once you're familiar with it. And then you can start throwing out things that seem intuitively wrong, and following intuition towards new ideas.
Intuition is a part of the mind that makes approximate models based on experience up to that point. The experiences without anything that would help model infinites would make them non-intuitive. Experiences that would help model infinites would make them at least more intuitive. Experience with rational investigation of them going through examples is actually one way to do that. There's been organizations like Marine Corps incorporating intuition into training for a long time where they set out to program it using realistic scenarios in drills that people do over and over until the intuitive processes pick up the patterns/model. It will happen as many people study that topic as well.
The trick is that you need good data to feed it that makes the patterns clear. You might have not gotten a lot of that on the topic you had no intuition of. On some topics, there hasn't been much of an attempt to do that since people expect it to be hard with everyone forced to do the methods that are non-intuitive. Lots of traditions are like that. Then, there's the possibility something can't be intuitively handled at all. I don't know if those exist but they might. That many activities of people doing math seem to get an intuitive boost during their day-to-day work makes me lean towards intuitive possibilities for about everything that can be approximated (esp with heuristics).
At this point a mathematician might step in and say that you all need to define your terms. You're basing your arguments on two subtly different definitions of the word "intuition."
The differences aren't subtle. I'm using the run-of-the-mill definition you might find in a dictionary[1], what other definition could you possibly be using?
It depends how you define intuition. Maybe we could define intuition as your personal "model" for making guesses on how things work.
Let's claim that writing a proof is similar to writing a program. You start from a state, your premise, and have your assumptions, input, which you process through a series of steps, maybe taking cases in between (branching) and return an output (hopefully True).
Then considering this correspondence, it will be fine to say that you, as a programmer, can guess the output/behavior/semantics of any program without running it. Similarly, a mathematician would be able to reach the conclusion that the theorem holds, without going through all the details of the proof.
Sure, in most cases you can, and as you get more and more experience and knowledge as a programmer you can say you improve your guesses. But this ends up backfiring with a good probability, when our model skipped a step, e.g. the interaction of two concurrent data structures.
Thus I don't think it is right to say "as you are able to know all axioms and rules of inference, you should be able to achieve perfect intuition". You can know the fundamentals of each layer on your stack and you would find it insane if one asked you to guess the output of any non-trivial terminating program.
Note also that the way of inference (logical consequences) can and does change (e.g. ZF vs ZFC).
() I may be misusing the word model here -- it is usually defined (at least to things I work with) as the interpretation of a theory that satisfies all theorems (inferences).
() constructive proofs yes. No flame wars please :)
Working mathematician does derive his ideas from axioms, it is ridiculous perception of mathematical process. And axioms are not logical consequences of sensible assumptions, it is try and fail process, long historical process. Intuition in mathematics has nothing to do with common sense intuition. Respectfully.
In abstract mathematics, one can choose any axioms one wants. To derive all kinds of consequences/theorems.
But respectfully, ZFC came about because of logical paradoxes that couldn't be accepted as consistent.
You can create a new number system and derive all kinds of consequences but the truth is, most mathematicians care more entirely about prime number theory on the naturals that are entirely based on counting.
Most of modern math is based on the natural numbers. You can't remove all intuition. The thread that holds our love for math is also the same one that tells us we are exploring consequences that tell us a dearth about our universe.
You'll find that no such thing is true. Intuition is something you develop over time in mathematics, and mathematicians will often use this intuition to have certain "feelings" about problems e.g. a feeling that a certain conjecture must be true.
There's nothing wrong with intuition, but intuition does not write proofs. It can help you write proofs, but you need more than that.
To pick an example:
Intuition is great for understanding the intermediate value theorem. If the temperature was 40 degrees at 8 am, and is 60 degrees at noon, it must have been 50 degrees somewhere in between.
Why, though? How can you guarantee that there will always have to be a time where it was 50 degrees[1]?
[1] provided that temperature is real-valued and temperature change is continuous, the debate of which I'll leave to the science people
Am I correct in understanding that it's not that intuition is invalid, but that it's insufficient for communicating a proof to another mathematician.
What I believe to be true is that a solid intuitive proof will be sound, but not necessarily transcribable until formalized. Formalization of the proof is essentially the matter of making a proof communicable.
Putting it one more way: purely-intuitive proofs would be just fine if one could share one's mental state with another.
> Am I correct in understanding that it's not that intuition is invalid, but that it's insufficient for communicating a proof to another mathematician.
is close to correct. It's a very useful tool, but it can sometimes lead you into incorrect assumptions.
However, the rest of it incorrect. Math, and in particular, probability and statistics, are full of things that seem intuitive and easy at first glance, but are actually a bit more nuanced.
Take for example, the Monty Hall Problem. It's probably familiar to most here, but essentially you have three doors. Behind one, is a car or some other desirable object, and behind the other two are goats or something undesirable. Select a door, and you get whatever is behind that door. What's the probability of getting the car in this scenario?
Easy enough right? It's just 1/3. However, if Monty Hall opens one of the doors after you select a door, he always reveals a goat, and he offers to let you switch your door now, should you switch or keep your door?
What's the probability that you get a car if you switch? Is it 1/2 or 1/3? What's the probability that you get a car if you stay? 1/3?
Well, the somewhat surprising answer is that you have a 2/3 probability of winning the car if you switch, and a 1/3 probability of winning the car if you stay.
Often things will break down in the limits as well, so if you try to apply your finite dimensional intuition to something that corresponds to an empty object or an infinite number of objects, you'll find that you're often wrong.
Not who you're asking, but I think that is incorrect. Intuitive explanations are how mathematicians usually communicate proofs to each other in person, when discussing mathematics. The receiving mathematician is convinced of the proof's validity if their own intuition and experience tells them that the nitty-gritty details can be formalised correctly.
3Blue1Brown makes great videos and offers good explanations. However, particularly with his latest Fourier Transform video, he not only gives the incorrect explanation, he makes it unnecessarily and overwhelming complicated. It is still better than the linked circles approach, but the Fourier transform can be explained more accurately in around two minutes and in three if you were to include the the definition of frequency.
3Blue1Brown is better than most but far from best.
He has a convoluted (no pun intended) center of mass explanation that the equation does not describe, admits that this is incorrect, and then goes onto another incorrect explanation of a strange ellipse where there is another center of mass.
I've never been able to find a good explanation online, only in some older textbooks. I've begun to fear that there are few people who are able to read equations while the others interpret the explanations of those people, where then those same old explanations gradually mutate as they propagate without reference to their source. This Stanford lecture is the best I could find: https://www.youtube.com/watch?v=1rqJl7Rs6ps
What I still don't understand is why there isn't a Wikipedia for explorables.
3b1b's videos are excellent, but if they were interactive, open-source and in a similar model to Wikipedia, people could contribute and improve upon content that is already great.
There are great examples of what is possible with explorables, see Bret Victor, D3.js, distill.pub, observablehq etc. Just no encyclopedic model yet.
I nice companion project for it would be a tool for creating these 'explorable explanations'. Anyone know if any good attempts have been made at that yet?
To add to your list, I recently saw an HN submission where a PhD student made a markdown-derivative with some explorable features, not sure what it was called though.
That book got me back into math. Using functional equations to teach concepts is very powerful. You get to say I want these properties and derive things like the exponential function.
Here's diagrams of all of the tallest buildings by 2023 (including built ones) visually next to each other sorted by official height thanks to SkyscraperPage: http://skyscraperpage.com/diagrams/?searchID=207
The Skyscrapercenter doesn't include the Dubai Creek Tower which is also already under construction.
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The issues raised here seem to not be related to glowing LCDs, but rather particular apps and behaviors in those apps that lead to addictive and problematic behavior.
"Screens" is even broader than saying things like "the internet is bad". The latter statement already sounds unhelpful because everyone has a better understanding of just how diverse of a platform "the internet" is.