The Art of Mathematics(krieger.jhu.edu)
krieger.jhu.edu
The Art of Mathematics
https://krieger.jhu.edu/magazine/fall-2018-v16n1/the-art-of-mathematics/
23 comments
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.
"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.
[deleted]
> “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.
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.
"...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?
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?
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.
My first year linear algebra textbook is nothing like this. On page 1 (the very first page after the Roman numeral pages including the title page, table of contents, and forward) is a bright yellow box defining the set R^n.
The remainder of the book is chock full of these definition boxes. Additionally, there are blue boxes for remarks, green boxes for examples, mauve boxes for theorems, and red boxes for exercises.
It's really that straightforward. You could probably skip all of the prose text and just use these boxes to learn what you need to get through the exercises and pass the course. Other math texts I've read may not have this nice colour scheme, but they're still pretty easy to follow.
The exercises, on the other hand, will challenge you to think creatively and demand that you learn how to apply the definitions and theorems properly in your proofs.
My first year linear algebra textbook is nothing like this. On page 1 (the very first page after the Roman numeral pages including the title page, table of contents, and forward) is a bright yellow box defining the set R^n.
The remainder of the book is chock full of these definition boxes. Additionally, there are blue boxes for remarks, green boxes for examples, mauve boxes for theorems, and red boxes for exercises.
It's really that straightforward. You could probably skip all of the prose text and just use these boxes to learn what you need to get through the exercises and pass the course. Other math texts I've read may not have this nice colour scheme, but they're still pretty easy to follow.
The exercises, on the other hand, will challenge you to think creatively and demand that you learn how to apply the definitions and theorems properly in your proofs.
For me I need to see examples. The abstraction almost always just confuses me. Unfortunately most (advanced) mathematics texts are very thin on examples. It wasn't always like this. Mathematics used to (hundreds of years ago) be done with all examples and no theory! That would suit me just fine. It seems that some people are better at one way or the other. The present trend is towards the abstract side.
I've found I have to turn mathematics algorithmic to understand it, thinking through what each symbol does to the things passing "through" it. Attempting to understand equations per se does nothing for me. They're important for proofs, I get it, but aid me not at all figuring out what it all actually means. Yet equations are the primary tool and presentation method used in most math books.
I'm pretty sure this is why I've never had trouble with programming, despite feeling the way I imagine a dyslexic must when I try to read very "mathy" material even about things that I already understand. Programming's algorithm-first. Except the few languages that aren't, and I bounce off those, too.
I'm pretty sure this is why I've never had trouble with programming, despite feeling the way I imagine a dyslexic must when I try to read very "mathy" material even about things that I already understand. Programming's algorithm-first. Except the few languages that aren't, and I bounce off those, too.
>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.
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.
I call this effect cultural mathematics. After college I avoided maths, went into programming, especially Functional Programming, which has a big math envy. With years I finally clicked on how mathematicians think and write. It's hyper cryptic when not explained or if you're just below the math level brainiac who does grok high level abstraction rapidly and thus will play that formula game with ease.
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.
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.
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.
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.
> the stock market /is/ a set of correlated random variables
No. this is neo-platonic bullshit. In this context is here must be a type-error. What could be imagined in not what really is - reality is one, interpretations are infinite. Stock Market is the result of human (and computer) actions - the result of actions of all institutional participants. This is what it is.
A model could be created to be superimposed on these partially-observable stochastic processes (and, boy, there is no shortage of these, and no one is right, of course) but it will be only a model, usefulness and applicability of which depends upon who is using it and for what means (usually, bullshitting).
Geometry is closer to "what is" because it abstracts out "concrete" patterns this universe is able to produce (and lots of imaginary bullshit, of course).
Let me stress is a gain - maths are generalized and abstracted out patterns of what is, plus abstract sectarian bullshit. Reality comes first, (it is a closure) interpretations and abstractions are bound by it.
No. this is neo-platonic bullshit. In this context is here must be a type-error. What could be imagined in not what really is - reality is one, interpretations are infinite. Stock Market is the result of human (and computer) actions - the result of actions of all institutional participants. This is what it is.
A model could be created to be superimposed on these partially-observable stochastic processes (and, boy, there is no shortage of these, and no one is right, of course) but it will be only a model, usefulness and applicability of which depends upon who is using it and for what means (usually, bullshitting).
Geometry is closer to "what is" because it abstracts out "concrete" patterns this universe is able to produce (and lots of imaginary bullshit, of course).
Let me stress is a gain - maths are generalized and abstracted out patterns of what is, plus abstract sectarian bullshit. Reality comes first, (it is a closure) interpretations and abstractions are bound by it.
I'm having difficulty following what you're saying here. I think you're saying mathematical models are only an (imperfect) interpretetion of reality, which sounds fine.
But then you're talking about "sectarian bullshit" and "neo-platonic bullshit". And at the same time, you called the stock market a "partially observable stochastic process" - why do you have an issue with calling something in reality a set of random variables, but you don't have an issue with calling it a stochastic process? Those are the same thing.
Honestly it feels like you launched into a tirade over a perfectly innocent and fine choice of words. I don't think the commenter you responded to meant to push some kind of weird philosophical agenda.
But then you're talking about "sectarian bullshit" and "neo-platonic bullshit". And at the same time, you called the stock market a "partially observable stochastic process" - why do you have an issue with calling something in reality a set of random variables, but you don't have an issue with calling it a stochastic process? Those are the same thing.
Honestly it feels like you launched into a tirade over a perfectly innocent and fine choice of words. I don't think the commenter you responded to meant to push some kind of weird philosophical agenda.
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 !
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 !
It’s a good observation, and the slug(slug(x)) = slug(x) property means that if you have an x and you’re not sure it’s a slug, it’s always ok to slug it to be sure it is. This property can be really handy in building robust systems, where you might have to attempt a couple of times to complete an operation for sure.
"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...
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...
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".
So many people have experienced implicit bias as mathematicians, especially people who "shouldn't be good at math".
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.
But that isn't what "majority" means...!
This is the precision of a man who likes math.
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.