> the derivative (tangent) of the minimum (lowest point) of any curve is zero (horizontal).
> holding their pencil up to it at the lowest point and laying it along, and discovering that, sure enough, the tangent is horizontal.
Any gifs showing that? just really curious and can't picture that in my head.
Try holding your mouse* upside-down on the surface of your desk to get the same effect. The point at which the mouse meets the desk will always be the point where the tangent of the curve is horizontal.
It might help to think of a specific curve. An example would be a simple parabola: f(x) = x^2. At the bottom of the curve (x = 0), the slope (aka tangent, aka derivative) is 0, or a horizontal line.
One way to imagine the the horizontal tangent is that it's where the curve stops going down (negative slope) and starts going up (positive slope). Well, that is also basically the definition of a local minima, and one of those minima is also the global minima.
This was my "ah-ha" moment as well. I always knew that derivative of the lowest point is 0 but I didn't really understand what it meant. Take any arbitrary curve. Tangent is a line that touches a single point on the curve. Draw a tangent aka pencil touching the lowest point of the curve. It's derivative ie dy/dx apply the formulae to the curve's formulae. It will always be 0 which means it will have no "slope" ie parallel to the x axis
The "pencil" (the short line representing the tangent) is green when positive, black when zero, and red when negative. Note that when it is black/zero, the "pencil" line is flat. This occurs at both local maxima and minima.
This is why it's important for STEM and tech people to have some understanding of liberal arts. "Understanding" means having perspective, it's easy to get caught up in our world, thought patterns, echo chambers, and biases. I think that's why curiosity is such an important trait, it promotes understanding, not the accumulation of facts. Next time you're at a book store, pick something up on a topic that sounds interesting that's outside of your traditional scope.
So I'm totally on board with everyone knowing something about the liberal arts and trying to expand your world outside your traditional scope.
But I think "understanding" can mean just making connections within a single tech field, even without involving liberal arts. For example, a grade school math problem: "Assume the earth is a perfect sphere with radius 6378 km, and you have a piece of string just long enough to reach all the way around the earth's equator at the earth's surface. How much longer would your string have to be to make a perfect circle exactly one meter above the earth's equator at every point?"
The answer is 2(pi) meters. That's true for any spherical planet of any size -- that's what it means for the derivative of 2(pi)(r) with respect to r to be 2(pi). That is sometimes not the first thing people think of though, because of the grade-school context they associate with this problem....
I really like the graphics in this. I like how "information" and "knowledge" are contrasted in the six squares. This is really what it feels like. Knowledge is structured, tightly connected in predictable ways, whereas information is just a kind of soup, a mess of facts. What the other distinctions are supposed to mean is less clear, but at least they're entertaining. I also really hope that the "knowledge" vs "understanding" plot is accurate. That would be quite encouraging if true. Seems plausible.
Not a psychology researcher and this wasn’t a study so I can’t vouch for “accurate”. Think of that graph as a distillation off my observations and experiences into 4 lines.
The idea I’m trying to share is that understanding compounds. The more you understand, the faster you can gain new understanding. And the more frameworks you have for the world, the faster you can gain knowledge because you need fewer examples to grok something.
For example, once you understand how browser-server communication works, picking up a new library is just a matter of syntax and names. You already know the concepts and what to expect. You might even be able to predict/guess what the functions are called based on knowing what the necessary operations are.
>> Ask you that same question in a different way and you have no idea. Never heard of it. This is why whiteboard interviews are hard – questions that you know, but out of context
In my experience this is true. I "test" my friends on these and most of them always screw up an answer to the questions I asked them just a couple of days ago when the question is slightly different but the underlying principle is the same. This is also why I spend hours on a question I get an answer to easily. Because it's important to understand the principle rather than getting the right answer. You can also conduct this test. The simplest and the most basic test is to ask someone what "binary search" is. Obviously, every developer knows what it is but when you ask them to apply binary search in a slightly different context(pick up any medium binary search question on Leetcode, Hackerrank etc) they will fudge it up. I get the hate "trivia" algo interview questions get on HN. However, I just love them because I have always loved "puzzles"
Interviewed at a company that wanted to hire me a PHP web dev. I mentioned right out of the gate, "I have zero PHP experience and most of my roles have been back-end/infrastructure related, but since we are chatting, here are examples of similar projects in other languages. And I'll also discuss a bit about my understanding of web development in general".
Hired and started within the week and had a very productive working relationship with them for quite some time.
We often mislabel 'information' as 'knowledge'. But information is just information. A 'knowledge base' is no such thing; it's an information base. Knowledge is not 'memory of facts'.
Knowledge == information + understanding.
Wisdom == information + understanding + experience.
Example: Python syntax is information. Understanding and using Python syntax is knowledge. Knowing when Python isn't what you need is wisdom.
it seems like you're saying the same thing, but with different definitions.
The key to both you and the article is that the base memorization of discrete rules and facts is not sufficient -- the real value comes from being able to dynamically connect the rules yourself in new contexts (rather than having them simply handed to you, to be memorized)
Having information does not imply having knowledge of it. For example, an engineer may not know which material can withstand a given environment (temperature, corrosion, pressure, etc) but they may be able to look it up in a book and select the appropriate material.
Knowledge is similar, except it's within the mind. For example, an engineer may know that the thermal coefficients for steel and concrete are similar, so the expected deformation under temperature change should be similar. But this doesn't mean that they necessarily understand why these two materials have similar coefficients, only that they do.
Understanding is a matter of cognitive appreciation of the underlying situation and is built around the parameters of a situation. For example, someone drowning understands that they need to grab something buoyant in order to breath. It's built on a base of knowledge, but it doesn't necessarily imply that they have knowledge of the complete situation. Even in science or software, there are things that one can know without understanding. Personal example: There was once a bug in ruby when built from source. I was tracking down a bug in production and I was unable to replicate it locally. So I finally got someone to shell into prod and run three lines of code for me. That code crashed even though it worked fine on literally every other computer I tried it on, including staging. I knew that the issue was with the production server, but I didn't understand why. It took further work to go from knowledge to understanding.
Wisdom is squishy. It correlates with experience, yes, but there are other factors that play into it as well. A child can be wise and choose not to associate with, say, a gang not because they've experienced it before, but because of something else. An inner reflection of the dangerousness of the situation and the courage or inner resolve to avoid it.
Or we could take the piece charitably and assume the best instead of asserting incorrect definitions. Especially given that common terminology is loose.
Alternately, we need formal language to talk about these concepts or we risk another "evolution is just a theory."
Yes, at some level both are just memorising something. But in the way I like to think about it, they are different things.
For example, you could figure out if someone is overweight by memorising a whole pile of weight & height stats. If someone height 170cm and they are over 75kg, they're overweight. If they are 180cm and over 81kg they're overweight, and so on. That's knowledge.
But if you understand being overweight is a statement about height versus weight, and memorise the formula for BMI you've compressed all that knowledge into a few bytes worth. That's understanding. Both are just memorising something.
Taken a bit further, what AI's turn the knowledge encoded in whole pile of specific examples (the training data) into a understanding (ie, a formula encapsulating the knowledge).
A very formulaic article, start with an anecdote that could fit many other articles (it seems Feynmann always shows up when we are talking about learning), refer to some Twitter conversation, add a formula or graph which provides no insight, sprinkle memes everywhere and that's all.
In the end, it seems the entire article can be summarised to, "it is all good to google for some recipes to solve your problems, but you also should care about some deeper understanding". Is there anything else to it?
Having skimmed through the article, I agree that the insight could have been summarized as a one-liner.
One could raise the same criticism for majority of content on the Internet, or even the field of literature.
On the other hand, starting with a Feynmann anecdote got my attention; the article takes the reader on an exploration of that single insight - at least it can be credited with staying on point - with some amusing illustrations/memes, and concludes by explaining the contrast between top-down/bottom-up approaches. OK, this last point could have been elaborated, what it means to "work from first principles".
Over all, I enjoyed following the author's thinking process, and felt that the article had enough substance to keep the reader's interest and stimulate some reflection and discussion, to re/consider one's approach.
> Is there anything else to it?
Like you, I seek more philosophical and intellectual substance. I think articles like this one are aimed at a general audience, like "popular science" - it's second-hand digested information, with some opinions based on experience. To get to the real meat, one must get closer to the source of the insights, like Feymann's own work.
On the other hand, this article took me 6 months of having a thought I couldn’t put into words until it clicked earlier this week and the Feynman anecdote is just a stylistic choice for structuring the article.
And you can summarize anything in 1 sentence if you don’t care about readers getting it, are trying to be funny, or are cynical enough. The world is not so complex after all. Everything’s just about increasing or decreasing entropy.
31 comments
[ 2.4 ms ] story [ 72.1 ms ] threadAny gifs showing that? just really curious and can't picture that in my head.
*Assuming your mouse is curved on top.
One way to imagine the the horizontal tangent is that it's where the curve stops going down (negative slope) and starts going up (positive slope). Well, that is also basically the definition of a local minima, and one of those minima is also the global minima.
Just draw any curve in a piece of paper, you'll see it happening.
The "pencil" (the short line representing the tangent) is green when positive, black when zero, and red when negative. Note that when it is black/zero, the "pencil" line is flat. This occurs at both local maxima and minima.
But I think "understanding" can mean just making connections within a single tech field, even without involving liberal arts. For example, a grade school math problem: "Assume the earth is a perfect sphere with radius 6378 km, and you have a piece of string just long enough to reach all the way around the earth's equator at the earth's surface. How much longer would your string have to be to make a perfect circle exactly one meter above the earth's equator at every point?"
The answer is 2(pi) meters. That's true for any spherical planet of any size -- that's what it means for the derivative of 2(pi)(r) with respect to r to be 2(pi). That is sometimes not the first thing people think of though, because of the grade-school context they associate with this problem....
What has "an understanding of liberal arts" got to do with this? Sounds like an unnecessary reach.
The idea I’m trying to share is that understanding compounds. The more you understand, the faster you can gain new understanding. And the more frameworks you have for the world, the faster you can gain knowledge because you need fewer examples to grok something.
For example, once you understand how browser-server communication works, picking up a new library is just a matter of syntax and names. You already know the concepts and what to expect. You might even be able to predict/guess what the functions are called based on knowing what the necessary operations are.
Oh, of course! This is a really good argument and now that you say this, I realize I've also noticed this happening.
In my experience this is true. I "test" my friends on these and most of them always screw up an answer to the questions I asked them just a couple of days ago when the question is slightly different but the underlying principle is the same. This is also why I spend hours on a question I get an answer to easily. Because it's important to understand the principle rather than getting the right answer. You can also conduct this test. The simplest and the most basic test is to ask someone what "binary search" is. Obviously, every developer knows what it is but when you ask them to apply binary search in a slightly different context(pick up any medium binary search question on Leetcode, Hackerrank etc) they will fudge it up. I get the hate "trivia" algo interview questions get on HN. However, I just love them because I have always loved "puzzles"
Interviewed at a company that wanted to hire me a PHP web dev. I mentioned right out of the gate, "I have zero PHP experience and most of my roles have been back-end/infrastructure related, but since we are chatting, here are examples of similar projects in other languages. And I'll also discuss a bit about my understanding of web development in general".
Hired and started within the week and had a very productive working relationship with them for quite some time.
We often mislabel 'information' as 'knowledge'. But information is just information. A 'knowledge base' is no such thing; it's an information base. Knowledge is not 'memory of facts'.
Knowledge == information + understanding.
Wisdom == information + understanding + experience.
Example: Python syntax is information. Understanding and using Python syntax is knowledge. Knowing when Python isn't what you need is wisdom.
The key to both you and the article is that the base memorization of discrete rules and facts is not sufficient -- the real value comes from being able to dynamically connect the rules yourself in new contexts (rather than having them simply handed to you, to be memorized)
Having information does not imply having knowledge of it. For example, an engineer may not know which material can withstand a given environment (temperature, corrosion, pressure, etc) but they may be able to look it up in a book and select the appropriate material.
Knowledge is similar, except it's within the mind. For example, an engineer may know that the thermal coefficients for steel and concrete are similar, so the expected deformation under temperature change should be similar. But this doesn't mean that they necessarily understand why these two materials have similar coefficients, only that they do.
Understanding is a matter of cognitive appreciation of the underlying situation and is built around the parameters of a situation. For example, someone drowning understands that they need to grab something buoyant in order to breath. It's built on a base of knowledge, but it doesn't necessarily imply that they have knowledge of the complete situation. Even in science or software, there are things that one can know without understanding. Personal example: There was once a bug in ruby when built from source. I was tracking down a bug in production and I was unable to replicate it locally. So I finally got someone to shell into prod and run three lines of code for me. That code crashed even though it worked fine on literally every other computer I tried it on, including staging. I knew that the issue was with the production server, but I didn't understand why. It took further work to go from knowledge to understanding.
Wisdom is squishy. It correlates with experience, yes, but there are other factors that play into it as well. A child can be wise and choose not to associate with, say, a gang not because they've experienced it before, but because of something else. An inner reflection of the dangerousness of the situation and the courage or inner resolve to avoid it.
Alternately, we need formal language to talk about these concepts or we risk another "evolution is just a theory."
Wish I could figure out the original source. Do you know?
Basically, these are the same as the building blocks of library and information science.
Yes, at some level both are just memorising something. But in the way I like to think about it, they are different things.
For example, you could figure out if someone is overweight by memorising a whole pile of weight & height stats. If someone height 170cm and they are over 75kg, they're overweight. If they are 180cm and over 81kg they're overweight, and so on. That's knowledge.
But if you understand being overweight is a statement about height versus weight, and memorise the formula for BMI you've compressed all that knowledge into a few bytes worth. That's understanding. Both are just memorising something.
Taken a bit further, what AI's turn the knowledge encoded in whole pile of specific examples (the training data) into a understanding (ie, a formula encapsulating the knowledge).
In the end, it seems the entire article can be summarised to, "it is all good to google for some recipes to solve your problems, but you also should care about some deeper understanding". Is there anything else to it?
One could raise the same criticism for majority of content on the Internet, or even the field of literature.
On the other hand, starting with a Feynmann anecdote got my attention; the article takes the reader on an exploration of that single insight - at least it can be credited with staying on point - with some amusing illustrations/memes, and concludes by explaining the contrast between top-down/bottom-up approaches. OK, this last point could have been elaborated, what it means to "work from first principles".
Over all, I enjoyed following the author's thinking process, and felt that the article had enough substance to keep the reader's interest and stimulate some reflection and discussion, to re/consider one's approach.
> Is there anything else to it?
Like you, I seek more philosophical and intellectual substance. I think articles like this one are aimed at a general audience, like "popular science" - it's second-hand digested information, with some opinions based on experience. To get to the real meat, one must get closer to the source of the insights, like Feymann's own work.
And you can summarize anything in 1 sentence if you don’t care about readers getting it, are trying to be funny, or are cynical enough. The world is not so complex after all. Everything’s just about increasing or decreasing entropy.