5 ms·

The Art of Mathematics

163 points by mathgenius 8y ago

j7ake 8y ago
Reminds me of Lockhart's Lament (https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament....). Lockhart has a description of an analogous world where music education from K-12 is pushed as important and all kids must learn it to do well in the world. The kids are educated by music teachers who have never wrote or have any deep connection with music. Kids are rewarded for filling in note correctly on a scoresheet and doing difficult things such as transposing music to one key to another, but they never learn to write or play music.
- deleted 8y ago
  [deleted]
- wyldfire 8y ago
  > “Oh we don’t actually apply paint until high school,” I was told by the students. ... “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. This was worth a chuckle. I recall taking pre-calculus/calculus in high school from a particularly passionate teacher. He really enjoyed his job and I felt like I got a greater appreciation for mathematics from his teaching. He did live graphing of equations with a then-novel overhead-transparency peripheral display for his computer. He included lots of practical engineering applications. After reflecting on this article I wonder if he were unconstrained in the curriculum whether he would've been yet more inspiring still.
- agumonkey 8y ago
  Some French education researchers / teachers had a saying: "if riding a bike was taught at school, no one would know how to do" People naturally learn things at the right dose and context. Music is a huge pleasure and desire. And music school doesn't really teach you music, it barely put you on rails and keep you digging in music for 5+ years (which makes one learn anything). They don't teach you what groove/swing/funk means (it's not even described except with informal tags). I had the same sense about maths. You go through the motion, but the actual thinking process, the reasoning behind even ~obvious notions like variable, equality, finite domains .. You have to read books about history to know that.
HiroshiSan 8y ago
Ha, I'd say 4 in 10 is quite a lot of people that hate math...so in that sense it is a majority. It's more people than I'd like, that hate math.
- nkoren 8y ago
  But that isn't what "majority" means...!
  - 1000units 8y ago
    This is the precision of a man who likes math.
  - unhkbfukgfjb 8y ago
    It is, if half of the rest doesn't care either way.
    - mkl 8y ago
      No, then it would be a plurality.
roenxi 8y ago
One way to divide people on mathematics is to separate those who see it as a technology (have problem -> apply mathematics -> problem is solved) from those who see it as a way of interpreting the world (eg, the stock market /is/ a set of correlated random variables, adding things to a heap /is/ addition, geometry /is/ how the world fits together). Usually there isn't a huge amount of difference between these competing philosophies, but it really rears its head in probability and statistics where a huge amount of damage is done, eg, p-value-hacking by people who think statistics is a technology rather than a perspective.
- llamaz 8y ago
  There is a divide between pure and applied math, one that arguable should be bridged, but the divide you talk about /within/ applied math seems artificial: Either you learn math properly, at least with regard to depth as opposed to breadth, or you shouldn't be working with it in the first place. Within electrical engineering the design process is 1. pen/paper + math 2. simulation 3. building/testing. You learn quickly that jumping straight to 3, or even 2, is a /bad/ idea. This is because in order to isolate where the problem lies, you need to know what is expected. A similar process is used within any engineering discipline. There is a such a thing as software engineering, where in the interest of safety the software system is documented completely. But in the vast majority of case, I hold the outre opinion that programming cannot in good conscience be considered engineering. In conclusion: If you are working in engineering or applying math, you should learn math properly, and shouldn't treat it as a black box.
- globuous 8y ago
  The otherday, I realized that slugging a string is a projection ! It's a surjective map with a reduction of dimensions (aka I give you a slug and in nearly all cases you won't be able to find the original string), and is idempotent, that is, the slug of a slug is itself. As is, this realization is completely useless since I don't really have any useful way to apply it, but it's a fun way of looking at slugs. And while we're at it, we can also see that there is a reduction of dimension in the process of hashing but that it is not a projection since the hash of a hash generally isn't itself. I'm almost sure about what I just said (I'm not a mathematician), but please correct me if I'm wrong !
eftychis 8y ago
"Even mathematicians will tell you that mathematics has a bit of an image problem, particularly in the U.S." That is something that strikes me constantly and I try to understand. I was amazed the first time I heard the phrase "just do the math" to imply do the menial calculations et cetera. This embodies the negative perception and misunderstanding that a lot of people have in general, but perhaps at a different level entirely in the U.S. What is more surprising is this is despite a math degree opening a lot of opportunities in the U.S. In comparison, in a lot of countries one can easily be stuck between teaching or research (if they are lucky and have the passion), or abandoning their degree. Consider yourselves lucky to have been exposed early on to math, is not a good approach or feeling to have. I think the demeanor of "you are good at it, or you are just not" is too damaging to the field in general. People that are passionate, innately or via some environmental factor acting as nurtury and "pylon", are perhaps going to persevere, like in other fields. Nobody told a kid that they should stop drawing, writing, or playing the guitar or piano, because they are bad at it. Math though apparently...
xtiansimon 8y ago
"...you can find Kitchloo at a cafe. He admits he’ll spend hours there, nursing a coffee or tea, and staring off into space, deep in thought...trying to get a feeling for the object he’s studying." That's how people who 'hate' math feel. They stare at the page, run their minds in circles and feel like they're lost in space. I came back to mathematics as I was learning programming as an adult. One thing I cannot stand is the general lack of definitions. My mind is oriented to programming where anything outside of the language must be defined in scope. Or maps, which have a legend and define abstract symbols. But that's just not the case in mathematics. You read the book. Start on page one. Eventually you get to a formula. You can scrape the text all you want, but there are characters without a definition. What's more, if you try to use other texts to find the definition, some special characters have different values. Or the forumla is defined differently. OK. I get it. Just as with a programming language, once you pass a certain threashold, it's possible to reason past these inconsistencies and over the gaps. My question to math-heads, is abstraction more perverse in Mathematics? Does it serve a purpose?
- earthicus 8y ago
  >You read the book. Start on page one. Eventually you get to a formula. You can scrape the text all you want, but there are characters without a definition. If I remember correctly, this is much more common with more elementary math books which are presumably trying to be more informal and 'user friendly'. You might seek out a book pitched at a more advanced level which will be much more formal (and pedantic) if that is the kind of thing that's bothering you. Also many books will have an 'index of notation' inserted at either the beginning or the end, so make sure to look for it - if it's there, the author might assume you know the convention and won't necessarily define everything in the exposition. ----- One note about abstraction in mathematics vs computer programming. When model something in a program, you are often modeling a particular thing, or kind of thing, so it makes sense to give it a descriptive name. Contrast that with, say, a number: I have a small or medium or large pile of apples. I make the adjective more precise and say I have a 3 or 6 or 25 -pile of apples. Then i notice there's nothing special about apples, I can do the same thing for piles of rocks or tea bags. And I can do the same thing for non-piles, like links in a fence, or countries, or more abstract things like counting permutations or combinations or discrete transformations of whatever you can imagine. The point being: in mathematics you are not working with a particular kind of thing, you are noticing patterns satisfied by many very very different kinds of things (hence the lack of descriptive names - there is often not much to describe). You stop asking questions about what it is, and learn to characterize things entirely by the logical properties they satisfy. Its a different thought process than what you might be accustomed to.
knightswhosayni 8y ago
Nice to know that more people are aware of the "pipeline issue" as noted in the article. So many people have experienced implicit bias as mathematicians, especially people who "shouldn't be good at math".

163 points by mathgenius 8y ago

Reminds me of Lockhart's Lament (https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament....). Lockhart has a description of an analogous world where music education from K-12 is pushed as important and all kids must learn it to do well in the world. The kids are educated by music teachers who have never wrote or have any deep connection with music. Kids are rewarded for filling in note correctly on a scoresheet and doing difficult things such as transposing music to one key to another, but they never learn to write or play music.

deleted 8y ago
[deleted]
wyldfire 8y ago
> “Oh we don’t actually apply paint until high school,” I was told by the students. ... “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. This was worth a chuckle. I recall taking pre-calculus/calculus in high school from a particularly passionate teacher. He really enjoyed his job and I felt like I got a greater appreciation for mathematics from his teaching. He did live graphing of equations with a then-novel overhead-transparency peripheral display for his computer. He included lots of practical engineering applications. After reflecting on this article I wonder if he were unconstrained in the curriculum whether he would've been yet more inspiring still.
agumonkey 8y ago
Some French education researchers / teachers had a saying: "if riding a bike was taught at school, no one would know how to do" People naturally learn things at the right dose and context. Music is a huge pleasure and desire. And music school doesn't really teach you music, it barely put you on rails and keep you digging in music for 5+ years (which makes one learn anything). They don't teach you what groove/swing/funk means (it's not even described except with informal tags). I had the same sense about maths. You go through the motion, but the actual thinking process, the reasoning behind even ~obvious notions like variable, equality, finite domains .. You have to read books about history to know that.

HiroshiSan 8y ago

Ha, I'd say 4 in 10 is quite a lot of people that hate math...so in that sense it is a majority. It's more people than I'd like, that hate math.

nkoren 8y ago
But that isn't what "majority" means...!
- 1000units 8y ago
  This is the precision of a man who likes math.
- unhkbfukgfjb 8y ago
  It is, if half of the rest doesn't care either way.
  - mkl 8y ago
    No, then it would be a plurality.

roenxi 8y ago

One way to divide people on mathematics is to separate those who see it as a technology (have problem -> apply mathematics -> problem is solved) from those who see it as a way of interpreting the world (eg, the stock market /is/ a set of correlated random variables, adding things to a heap /is/ addition, geometry /is/ how the world fits together). Usually there isn't a huge amount of difference between these competing philosophies, but it really rears its head in probability and statistics where a huge amount of damage is done, eg, p-value-hacking by people who think statistics is a technology rather than a perspective.

llamaz 8y ago
There is a divide between pure and applied math, one that arguable should be bridged, but the divide you talk about /within/ applied math seems artificial: Either you learn math properly, at least with regard to depth as opposed to breadth, or you shouldn't be working with it in the first place. Within electrical engineering the design process is 1. pen/paper + math 2. simulation 3. building/testing. You learn quickly that jumping straight to 3, or even 2, is a /bad/ idea. This is because in order to isolate where the problem lies, you need to know what is expected. A similar process is used within any engineering discipline. There is a such a thing as software engineering, where in the interest of safety the software system is documented completely. But in the vast majority of case, I hold the outre opinion that programming cannot in good conscience be considered engineering. In conclusion: If you are working in engineering or applying math, you should learn math properly, and shouldn't treat it as a black box.
globuous 8y ago
The otherday, I realized that slugging a string is a projection ! It's a surjective map with a reduction of dimensions (aka I give you a slug and in nearly all cases you won't be able to find the original string), and is idempotent, that is, the slug of a slug is itself. As is, this realization is completely useless since I don't really have any useful way to apply it, but it's a fun way of looking at slugs. And while we're at it, we can also see that there is a reduction of dimension in the process of hashing but that it is not a projection since the hash of a hash generally isn't itself. I'm almost sure about what I just said (I'm not a mathematician), but please correct me if I'm wrong !

eftychis 8y ago

"Even mathematicians will tell you that mathematics has a bit of an image problem, particularly in the U.S." That is something that strikes me constantly and I try to understand. I was amazed the first time I heard the phrase "just do the math" to imply do the menial calculations et cetera. This embodies the negative perception and misunderstanding that a lot of people have in general, but perhaps at a different level entirely in the U.S. What is more surprising is this is despite a math degree opening a lot of opportunities in the U.S. In comparison, in a lot of countries one can easily be stuck between teaching or research (if they are lucky and have the passion), or abandoning their degree. Consider yourselves lucky to have been exposed early on to math, is not a good approach or feeling to have. I think the demeanor of "you are good at it, or you are just not" is too damaging to the field in general. People that are passionate, innately or via some environmental factor acting as nurtury and "pylon", are perhaps going to persevere, like in other fields. Nobody told a kid that they should stop drawing, writing, or playing the guitar or piano, because they are bad at it. Math though apparently...

xtiansimon 8y ago

"...you can find Kitchloo at a cafe. He admits he’ll spend hours there, nursing a coffee or tea, and staring off into space, deep in thought...trying to get a feeling for the object he’s studying." That's how people who 'hate' math feel. They stare at the page, run their minds in circles and feel like they're lost in space. I came back to mathematics as I was learning programming as an adult. One thing I cannot stand is the general lack of definitions. My mind is oriented to programming where anything outside of the language must be defined in scope. Or maps, which have a legend and define abstract symbols. But that's just not the case in mathematics. You read the book. Start on page one. Eventually you get to a formula. You can scrape the text all you want, but there are characters without a definition. What's more, if you try to use other texts to find the definition, some special characters have different values. Or the forumla is defined differently. OK. I get it. Just as with a programming language, once you pass a certain threashold, it's possible to reason past these inconsistencies and over the gaps. My question to math-heads, is abstraction more perverse in Mathematics? Does it serve a purpose?

earthicus 8y ago
>You read the book. Start on page one. Eventually you get to a formula. You can scrape the text all you want, but there are characters without a definition. If I remember correctly, this is much more common with more elementary math books which are presumably trying to be more informal and 'user friendly'. You might seek out a book pitched at a more advanced level which will be much more formal (and pedantic) if that is the kind of thing that's bothering you. Also many books will have an 'index of notation' inserted at either the beginning or the end, so make sure to look for it - if it's there, the author might assume you know the convention and won't necessarily define everything in the exposition. ----- One note about abstraction in mathematics vs computer programming. When model something in a program, you are often modeling a particular thing, or kind of thing, so it makes sense to give it a descriptive name. Contrast that with, say, a number: I have a small or medium or large pile of apples. I make the adjective more precise and say I have a 3 or 6 or 25 -pile of apples. Then i notice there's nothing special about apples, I can do the same thing for piles of rocks or tea bags. And I can do the same thing for non-piles, like links in a fence, or countries, or more abstract things like counting permutations or combinations or discrete transformations of whatever you can imagine. The point being: in mathematics you are not working with a particular kind of thing, you are noticing patterns satisfied by many very very different kinds of things (hence the lack of descriptive names - there is often not much to describe). You stop asking questions about what it is, and learn to characterize things entirely by the logical properties they satisfy. Its a different thought process than what you might be accustomed to.

knightswhosayni 8y ago

Nice to know that more people are aware of the "pipeline issue" as noted in the article. So many people have experienced implicit bias as mathematicians, especially people who "shouldn't be good at math".