What's one thing your learned at college/university that blew your mind?
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More impressively, you can have function that is continuous, but you cannot find a connected path on it (i.e. not path connected). In plain words, if anyone told you "a function is continuous when you can draw it without lifting your pen". They have lied to you.
You are misrepresenting an analogy as a lie. Besides that, in the context where the claim is typically made, the analogy is still pretty reasonable and your example is just plain wrong.
People are talking about continuous maps on subsets of R into R with this analogy basically always (i.e., during a typical calc 1 or precalc class). The only real issue are domain requirements in such a context. You need connectedness in the domain or else you're just always forced into lifting your pen.
There are a couple other requirements you could add as well. You might also want to avoid unbounded domains since you can't physically draw an infinitely long curve. Likewise you might want to avoid open endpoints or else things like 1/x on (0,1] become a similar kind of problem. But this is all trivial to avoid by saying "on a closed and bounded interval" and the analogy is still fairly reasonable without them so long as you keep the connectedness requirement.
For why your example is just wrong in such a context, say we're only dealing with continuous maps on a connected subset of R into R. Recall the connected sets in R are just intervals. Recall the graph of a function f with domain X is the set {(x,f(x)) : x is in X}. Do you see why the graph of such a function is always path connected? Hint: Pick any pair of points on this graph. Do you see what path connects those two points?
Once you want to talk about continuous maps between more general topological spaces, things become more complicated. But that is not within the context in which this analogy is made.
wrote last edited by [email protected]Sure, I have no problem with analogy. I called them lie simply to peak people's interest, but in research and teaching, lies can often be beneficial. One of my favorite quote (I believe from Mikołaj Bojańczyk) is "in order to tell a good story, sometime you have to tell some lies".
At the begining of undergrad, "not lifting pen" is clearly a good enough analogy to convey intuition, and it is close enough approximation that it shouldn't matter until much later in math.
I can say "sin(1/x) is a continuous function on (0,1] but its graph is not path connected", which is more formal, but likely not mean anything to most of the reader.
In that sense, I guess I have also lied
However, I like to push back on the assumption that, in the context of teaching continuous function, the underlying space needs to be bounded: one of the first continuous function student would encounter is the identity function on real, which has both a infinite domain and range.
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Did you believe otherwise growing up?
Yes. I grew up in the Dutch Bible Belt, with very strict evangelical parents. They sent me to a Christian school that taught a literal interpretation of the Bible. So I was taught at home, in church, and in school that Earth was created about 6000 years ago.
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Which part
locality. what was it supposed to be? and why doesn't it actually exist? what model "replaces" it, if any?
maybe the meaning of spacetime isn't obvious either, and I just misunderstand what it means
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If you were to put a big fan on a sailboat and point it at the sail, it would move the sailboat in a similar way as if the wind was pushing the sail.
wrote last edited by [email protected]Which actually makes sense if you understand it's not the wind pushing but the generated updraft at the sail.
(also not point at, but sideways)
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Sure, I have no problem with analogy. I called them lie simply to peak people's interest, but in research and teaching, lies can often be beneficial. One of my favorite quote (I believe from Mikołaj Bojańczyk) is "in order to tell a good story, sometime you have to tell some lies".
At the begining of undergrad, "not lifting pen" is clearly a good enough analogy to convey intuition, and it is close enough approximation that it shouldn't matter until much later in math.
I can say "sin(1/x) is a continuous function on (0,1] but its graph is not path connected", which is more formal, but likely not mean anything to most of the reader.
In that sense, I guess I have also liedHowever, I like to push back on the assumption that, in the context of teaching continuous function, the underlying space needs to be bounded: one of the first continuous function student would encounter is the identity function on real, which has both a infinite domain and range.
wrote last edited by [email protected]I can say "sin(1/x) is a continuous function on (0,1] but its graph is not path connected", which is more formal, but likely not mean anything to most of the reader. In that sense, I guess I have also lied
It's also false. Take any pair of points on the graph of sin(1/x) using the domain (0,1] that you just gave. Then we can write these points in the form (a,sin(1/a)), (b,sin(1/b)) such that 0 < a < b without loss of generality. The map f(t)=(t,sin(1/t)) on [a,b] is a path connecting these two points. This shows the graph of sin(1/x) on (0,1] is path connected.
This same trick will work if you apply it to the graph of ANY continuous map from a connected subset of R into R. This is what my graph example was getting at.
The "topologists sine curve" example you see in pointset topology as an example of connected but not path connected space involves taking the graph you just gave and including points from its closure as well.
Think about the closure of your sin(1/x) graph here. As you travel towards the origin along the topologists sine curve graph you get arbitrarily close to each point along the y-axis between -1 and 1 infinitely often. Why? Take a horizontal line thru any such point and look at the intersections between your horizontal line and your y=sin(1/x) curve. You can make a limit point argument from this fact that the closure of sin(1/x)'s graph is the graph of sin(1/x) unioned with the portion of the y-axis from -1 to 1 (inclusive).
Path connectedness fails because there is no path from any one of the closure points you just added to the rest of the curve (for example between the origin and the far right endpoint of the curve).
A better explanation of the details here would be in the connectedness/compactness chapter in Munkres Topology textbook it is example 7 in ch 3 sec 24 pg 157 in my copy.
However, I like to push back on the assumption that, in the context of teaching continuous function, the underlying space needs to be bounded: one of the first continuous function student would encounter is the identity function on real, which has both a infinite domain and range.
This is fine. I stated boundedness as an additional assumption one might require for pragmatic reasons. It's not mandatory. But it's easy to imagine somebody trying to be clever and pointing out that if we allow the domain or range to be unbounded we still have problems. For example you literally cannot draw the identity function in full. The identity map extends infinitely along y=x in both directions. You don't have the paper, drawing utensils or lifespan required to actually draw this.
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That I spent years developing proficiency in my language and expanding my vocabulary to get accepted, only to be told to write simplified English in journalism school. Then they doubled down in my business classes to write for a 6th grade education and those who don't speak natively.
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Computer science students multiple years into the course think I'm a hacker for using the linux terminal
I once showed that
trickfeature - opening the terminal - to an Apple Genius Bar employee once. His brains almost fell out of his ear he was so surprised. -
The Twin Paradox (special relativity). Every time I wrap my head around the idea I lose it a few weeks later an it's a mystery all over again.
Twin Paradox TL;DR: Identical twins—one stays on Earth, the other rockets off near light speed and returns. Relativity says time slows for the traveler, so they age less (e.g., returns 20 while sib is 50). "Paradox" cuz from traveler's view, Earth seems to move, but acceleration/turnaround breaks the symmetry, so no real contradiction. Mind-bendy Einstein stuff.
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At a certain point, you need to be force which pushes you forward. I saw a lot of intelligent people fail because they no longer had the external stimulus to go to class.
Also, it is easier to manipulate people in positions of power, but you have to understand how they think and are rewarded. There is a reason why a lot of liberal arts education is focused on having people understand others.
Also, the liberal arts education of a century ago was basically a degree which was intended to make managers. Along with it, the extra-curricular activities were an important part of the education, but just what happened in class.
Why is it easier to manipulate people in power? What makes them more vulnerable to manipulation?
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Why is it easier to manipulate people in power? What makes them more vulnerable to manipulation?
A lot of the official liberal arts college education goes into understanding the perspectives of others, with a bias to people in power and their power structures. While not an explicit thing they are teaching you, college is teaching you how to understand power structures and the people within them.
If you have a better understanding of power structures, it becomes easier to push said structures to achieve your own goals since you can speak to power structures in their language instead of your own in order to get what you want.
Also, a lot of the clubs and other extra-curricular activities are designed to create small power bases to practice these techniques on.
It is a lot easier to get what you want when you can speak on other people's terms.
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I can say "sin(1/x) is a continuous function on (0,1] but its graph is not path connected", which is more formal, but likely not mean anything to most of the reader. In that sense, I guess I have also lied
It's also false. Take any pair of points on the graph of sin(1/x) using the domain (0,1] that you just gave. Then we can write these points in the form (a,sin(1/a)), (b,sin(1/b)) such that 0 < a < b without loss of generality. The map f(t)=(t,sin(1/t)) on [a,b] is a path connecting these two points. This shows the graph of sin(1/x) on (0,1] is path connected.
This same trick will work if you apply it to the graph of ANY continuous map from a connected subset of R into R. This is what my graph example was getting at.
The "topologists sine curve" example you see in pointset topology as an example of connected but not path connected space involves taking the graph you just gave and including points from its closure as well.
Think about the closure of your sin(1/x) graph here. As you travel towards the origin along the topologists sine curve graph you get arbitrarily close to each point along the y-axis between -1 and 1 infinitely often. Why? Take a horizontal line thru any such point and look at the intersections between your horizontal line and your y=sin(1/x) curve. You can make a limit point argument from this fact that the closure of sin(1/x)'s graph is the graph of sin(1/x) unioned with the portion of the y-axis from -1 to 1 (inclusive).
Path connectedness fails because there is no path from any one of the closure points you just added to the rest of the curve (for example between the origin and the far right endpoint of the curve).
A better explanation of the details here would be in the connectedness/compactness chapter in Munkres Topology textbook it is example 7 in ch 3 sec 24 pg 157 in my copy.
However, I like to push back on the assumption that, in the context of teaching continuous function, the underlying space needs to be bounded: one of the first continuous function student would encounter is the identity function on real, which has both a infinite domain and range.
This is fine. I stated boundedness as an additional assumption one might require for pragmatic reasons. It's not mandatory. But it's easy to imagine somebody trying to be clever and pointing out that if we allow the domain or range to be unbounded we still have problems. For example you literally cannot draw the identity function in full. The identity map extends infinitely along y=x in both directions. You don't have the paper, drawing utensils or lifespan required to actually draw this.
Yes, you are right, topologists' sine curve includes the origin point, which is connected but not path connected. I guess I didn't do very well in my point-set topology. I will change that in my answer
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Which actually makes sense if you understand it's not the wind pushing but the generated updraft at the sail.
(also not point at, but sideways)
Even if you are sailing directly downwind, it works. That was actually the professor's demonstration. He said that at the time it was accepted as a physical phenomenon, there were many physicists who said it wasn't possible, but it was being actively used by some engineers to make jets go in reverse.
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locality. what was it supposed to be? and why doesn't it actually exist? what model "replaces" it, if any?
maybe the meaning of spacetime isn't obvious either, and I just misunderstand what it means
In short locality states that things are only directly affected by their immediate surroundings. This is provably untrue in quantum mechanics.
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It has been proven that each mathematical reasoning system* either has a statement that cannot be proven true or false, or a statement that can be proven both true or false. In simpler terms, it has been proven that we can't prove everything.
Gödels incompleteness theorem if anyone wants to look it up.
- only holds for reasoning systems that can reason about numbers
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A lot of the official liberal arts college education goes into understanding the perspectives of others, with a bias to people in power and their power structures. While not an explicit thing they are teaching you, college is teaching you how to understand power structures and the people within them.
If you have a better understanding of power structures, it becomes easier to push said structures to achieve your own goals since you can speak to power structures in their language instead of your own in order to get what you want.
Also, a lot of the clubs and other extra-curricular activities are designed to create small power bases to practice these techniques on.
It is a lot easier to get what you want when you can speak on other people's terms.
Where can I learn more about this? Recommend any books or any techniques? I'd love to learn more about power structures, and people in power.
In workplaces, I've seen people put themselves into positions of power, get roles their not qualified for, and influence managers to dislike people. Office politics.
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Wait, what's the other option for past tense 'to prove'?
OP has just proven their point.
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It’s also interesting how the past-tense of “to dive” has changed over recent generations. “Dived” is supposed to be standard, yet people turn it into “dove” so frequently, it’s becoming the new normal.
I resist because I'm literate enough to know that "dove" is a bird that rhymes with "love".
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Higher education is a waste of money for the vast majority of degrees, even STEM ones.
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For real? A lot of high school subjects were pre requisites for enrolling in my degree here and it'd be quite tough to get through the degree without the foundation laid in those subjects. At the very least they'd have to extend the university course by probably a year or so.
Probably varies largely on where you're talking about, and even then, which university program you're looking at enrolling in. If you go and look at universities in the UK, for example, a BA studying a foreign language generally seems to assume that this is a language you've already been studying for several years in secondary education. You're meant to be entering the program with roughly a B1 level in the language, and allegedly develop up to C1 over the course of 3 years of study. Meanwhile, in the US, you can rock up to a university and be a Japanese language major with nothing more than "Well, he says he likes anime and his grades are okay." and the degree program will start you off in a 100-level class that expects negligible prior knowledge, if any.
Then again, having attempted university in the US, and now doing it at a UK school, university education is pretty drastically different. The US schools take 4 years to grant the same degree, and you spend almost the whole of the first year and a good chunk of the second just doing general education requirements that are, at best, only tangentially relevant to your chosen field of study. If I were doing my current degree program for a BA in French and Spanish as a first time student in the US, unless I did a bunch of AP courses or took night classes at a community college on the side, I'd need to do a general English composition class, a few math classes, probably get to pick between a biology or chemistry course, something to do with world cultures or music and the arts, and a handful of other electives I'm forgetting about. For that degree in the UK, from start to finish over the course of 3 years, I exactly 2 modules that aren't either French or Spanish, with one being the "Hey, we need to make sure you can actually write in English competently, too" module, and the other being a free choice of an introductory language module for something else.
I'd also assume the US' lack of a national curriculum also plays into how things work out with universities here, as well. Since things can be so variable at a regional and local level, not only in terms of the established curriculum, but what courses your particular secondary school has the funding to offer, universities can't really assume much of incoming students' education. You can have a kid from one state whose school was a Spanish language immersion school offering bilingual education from day 1 of Kindergarten, and later offering French, German, Japanese and Arabic as a third language for the final 4 years of compulsory education sat side-by-side with another from a different part of the country who only had the chance to take 2 years of Spanish classes. Even for subjects with a better baseline, someone whose studies covered all the available math classes up to geometry and algebra is going to have a totally different starting point from another whose school partnered with a local college to offer college level courses in calculus and statistics in high school.
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What does that mean?
You know how you get COVID from being in the same room as someone who sneezes on you or something? It's that.
Non locality is when you get COVID from someone the next state over.
