What is it like to be a mathematician?(slate.com)
slate.com
What is it like to be a mathematician?
http://www.slate.com/articles/health_and_science/new_scientist/2013/10/edward_frenkel_on_love_and_math_what_is_it_like_to_be_a_mathematician.html
13 comments
That is definitely good and in-depth.
(I had to use the web dev inspector to remove the overlay trying to force me to enable scripts and sign in, but it was worth it.)
(I had to use the web dev inspector to remove the overlay trying to force me to enable scripts and sign in, but it was worth it.)
You can also add ?share=1 to the end of any Quora URL to remove the overlay.
There's a link on that overlay to "close and continue reading" if you get out your microscope. ;)
For me that only shows the top answer.
I'm an applied mathematician (PhD) working partially in software development, partially in research mathematics and statistics proper.
A mathematician is someone who works on things that honestly very few folks can understand completely (not due to complexity per se, just depth and lack of general familiarity). Publication peer-review aside, it's also unlikely that the few folks who can judge your work are in any position to fire you. It's a rewarding and pretty secure gig.
But I still think that the most interesting work is very multidisciplinary in nature, so a CS, engineering or physical sciences background really does enhance the work you can do. Like in any other career, don't let yourself become a one-trick pony.
A mathematician is someone who works on things that honestly very few folks can understand completely (not due to complexity per se, just depth and lack of general familiarity). Publication peer-review aside, it's also unlikely that the few folks who can judge your work are in any position to fire you. It's a rewarding and pretty secure gig.
But I still think that the most interesting work is very multidisciplinary in nature, so a CS, engineering or physical sciences background really does enhance the work you can do. Like in any other career, don't let yourself become a one-trick pony.
My advisor in grad school put it this way: being a mathematician basically amounts to being the world's leading expert on something only about 5 people give a shit about.
> it's also unlikely that the few folks who can judge your work are in any position to fire you
But in the other hand, it means those who can judge your work are not in the same position to hire you, and it is not ruled out that people in position to fire you has to understand your work in order to do so.
But in the other hand, it means those who can judge your work are not in the same position to hire you, and it is not ruled out that people in position to fire you has to understand your work in order to do so.
I did three years of Maths in University. Then, I swap to Computer Science because it was always 'my thing'. When I read this kind of posts I really miss Maths. It is beautiful.
If you never tried, if your only experience is high school, go for it. Learn some Math. You will be happier.
If you never tried, if your only experience is high school, go for it. Learn some Math. You will be happier.
Math is satisfying in that you can study it either for its own sake, or in order to better arm yourself to understand the world.
To each his/her preference. I like both.
To each his/her preference. I like both.
[deleted]
Math is somehow what intrigued me most during university. After my startup endeavours, I want to make my mark in Physics and Maths, because I think once you have mastered earth and humans(with a startup), it is time to take on the universe (Maths & Physics)
Where can I start learning math on my own? I always thought math books are arcane in the sense that perquisites math knowledge for one book are so vast that I couldn't wrap my head around one.
I would recommend 'Concepts of Modern Mathematics' [1], by Ian Stewart. It has some very nice illustrations and humor, and reminds me of 'Learn You A Haskell For Great Good' (though I believe CoMM came out before LYaH). It's a wonderful preview of a variety of topics, and is intended to introduce someone with a poor math background to some of the different fields of math.
From the Amazon description:
In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts underlying "new math": groups, sets, subsets, topology, Boolean algebra, and other subjects. No advanced mathematical background is needed to follow thought-provoking discussions of such topics as functions, symmetry, axiomatics, counting, topology, hyperspace, linear algebra, and more. 200 illustrations.
---
[1]: http://www.amazon.com/Concepts-Modern-Mathematics-Dover-Book...
From the Amazon description:
In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts underlying "new math": groups, sets, subsets, topology, Boolean algebra, and other subjects. No advanced mathematical background is needed to follow thought-provoking discussions of such topics as functions, symmetry, axiomatics, counting, topology, hyperspace, linear algebra, and more. 200 illustrations.
---
[1]: http://www.amazon.com/Concepts-Modern-Mathematics-Dover-Book...
What math do you know, what math do you want to know?
When I did Project Euler, I couldn't come up with elegant solution that was derived from high enough abstraction using math, leaving me using brute force method that was not really efficient.
I was fascinated by how people come up with elegant math solution for the challenges in Project Euler. I am not sure how I can achieve such elegance, proving my math is insufficient for what can be done.
I was fascinated by how people come up with elegant math solution for the challenges in Project Euler. I am not sure how I can achieve such elegance, proving my math is insufficient for what can be done.
> When, later on in life, the subject of mathematics comes up, most people wave their hands and say, "Oh no, I don't want to hear about this, I was so bad at math."
I've noticed this happening a lot. It is interesting that nobody says that about the other two Rs: reading and writing. Have you ever heard anyone say "I don't like books and reading in general"?
As far as I can tell this is the reasoning used by most "math haters"
The first fault with this reasoning is the premise---there are plenty of smart kids who are simply not drawn to math, especially since math is often presented as memorization and not understanding. So whether you liked math in high school or not has almost no bearing on how good you are at math. Chill.
The other fault with the reasoning is that of time-invariance. Math was "tough" for me when I was 8 years old, so it will still be tough for me even though I am an adult now. It takes some level of humbleness to go back to learning something that "kids should know," but it is totally worth doing. Whether you are a coder, an english major, or a designer, learning a bit of math will give you a lot of extra power to do whatever you do already, and other stuff. For example, learning math will suddenly make half a million more wikipedia pages accessible to you (all the ones with lots of equation blocks). That is a lot of knowledge, and knowledge is power...
<plug>What if there was a high school math textbook written for adults? A textbook with no BS, which gets directly to the power part right away. What if learning high school math opened the door for you to learn differential calculus (lim,ƒ'(x),max), and integral calculus (∫ƒdx,∑a_i) in the same sitting. Then knowing calculus, you would be able to pick up mechanics (F=ma, a(t)=x''(t), p=mv, ∑Ei=∑Ef, Acos(ωt+ϕ)) quite easily too. All of this in just 383 pages that you can read four weeks if properly caffeinated!</plug>
I've noticed this happening a lot. It is interesting that nobody says that about the other two Rs: reading and writing. Have you ever heard anyone say "I don't like books and reading in general"?
As far as I can tell this is the reasoning used by most "math haters"
Kids who are smart are good at math
I was not good at math
Therefore I am not smart
The first fault with this reasoning is the premise---there are plenty of smart kids who are simply not drawn to math, especially since math is often presented as memorization and not understanding. So whether you liked math in high school or not has almost no bearing on how good you are at math. Chill.
The other fault with the reasoning is that of time-invariance. Math was "tough" for me when I was 8 years old, so it will still be tough for me even though I am an adult now. It takes some level of humbleness to go back to learning something that "kids should know," but it is totally worth doing. Whether you are a coder, an english major, or a designer, learning a bit of math will give you a lot of extra power to do whatever you do already, and other stuff. For example, learning math will suddenly make half a million more wikipedia pages accessible to you (all the ones with lots of equation blocks). That is a lot of knowledge, and knowledge is power...
<plug>What if there was a high school math textbook written for adults? A textbook with no BS, which gets directly to the power part right away. What if learning high school math opened the door for you to learn differential calculus (lim,ƒ'(x),max), and integral calculus (∫ƒdx,∑a_i) in the same sitting. Then knowing calculus, you would be able to pick up mechanics (F=ma, a(t)=x''(t), p=mv, ∑Ei=∑Ef, Acos(ωt+ϕ)) quite easily too. All of this in just 383 pages that you can read four weeks if properly caffeinated!</plug>
> Have you ever heard anyone say "I don't like books and reading in general"?
Well, I have heard engineery types saying they don't see the point of reading fiction ("it's just made up stories"), and more frequently have heard them disparaging subjects like philosophy and history. Maybe that's the equivalent response?
Well, I have heard engineery types saying they don't see the point of reading fiction ("it's just made up stories"), and more frequently have heard them disparaging subjects like philosophy and history. Maybe that's the equivalent response?
From Edward Frankel also comes this wonderful excerpt on antisemitism in Russian higher education (that doesn't sound exciting now, but you'll like the article):
http://www.newcriterion.com/articles.cfm/The-Fifth-problem--...
http://www.newcriterion.com/articles.cfm/The-Fifth-problem--...
When you're a mathematician, everything is black or white, but you get to define what those two words mean.
I think that's another misunderstood thing about math.
By the time you get into higher mathematics, you get to a place where nearly everything is unknown. I remember reaching the level of ordinary differential equations and realizing that "most" non-linear equations had no exact solution at all.
In ways, math and programming have this in common - as the size of the systems being studied increases, a randomly chosen example becomes harder and harder to deal and thus the "art" consists in choosing important and tractable examples out of a world of huge but intractable systems.
By the time you get into higher mathematics, you get to a place where nearly everything is unknown. I remember reaching the level of ordinary differential equations and realizing that "most" non-linear equations had no exact solution at all.
In ways, math and programming have this in common - as the size of the systems being studied increases, a randomly chosen example becomes harder and harder to deal and thus the "art" consists in choosing important and tractable examples out of a world of huge but intractable systems.
When I was 17, high school, and really enjoying maths a maths professor came to our school to speak to a few of us about continuing maths at University. She really annoyed me. She asked what about maths we liked and a friend and I answered that it was the certainty - that things were right or wrong and you could most often tell plainly which they were.
She of course responded that we couldn't be more wrong.
Annoyed the hell out of me until I came across Godel and began to understand maths from a different perspective.
Wish I'd had chance to study under that professor.
She of course responded that we couldn't be more wrong.
Annoyed the hell out of me until I came across Godel and began to understand maths from a different perspective.
Wish I'd had chance to study under that professor.
> I remember reaching the level of ordinary differential equations and realizing that "most" non-linear equations had no exact solution at all.
This may be a bit theoretical for most readers, but every ODE that has one and only one solution has an exact solution. That the best definition for that function is "the solution to that ODE" does not mean that the function is any less exact than something you can construct out of exp, sin, cos, etc. In fact it would be completely reasonable to define exp as a solution to a differential equation and then to prove afterwards, that exp is equivalent to the usual power series and accidentally also has some other nice properties :).
This may be a bit theoretical for most readers, but every ODE that has one and only one solution has an exact solution. That the best definition for that function is "the solution to that ODE" does not mean that the function is any less exact than something you can construct out of exp, sin, cos, etc. In fact it would be completely reasonable to define exp as a solution to a differential equation and then to prove afterwards, that exp is equivalent to the usual power series and accidentally also has some other nice properties :).
Ah,
I spoke too loosely. I should have written "closed-form symbolic solution" rather than "exact solution" (though "exact" is used colloquially to mean this, it's not clear). Yes, all solutions to an ode would be "exact solutions" in the literal sense.
It is worth noting also that an ode having a solution around a point doesn't imply that it can be extended indefinitely - you can hit "singularities" that make extension impossible.
I spoke too loosely. I should have written "closed-form symbolic solution" rather than "exact solution" (though "exact" is used colloquially to mean this, it's not clear). Yes, all solutions to an ode would be "exact solutions" in the literal sense.
It is worth noting also that an ode having a solution around a point doesn't imply that it can be extended indefinitely - you can hit "singularities" that make extension impossible.
every ODE that has one and only one solution has an exact solution
Upvoted, but - is that a particular theorem, or are you just basically stating the definition of "having a solution"? Wouldn't your statement be true of any system of differential equations, not just ODEs?
Upvoted, but - is that a particular theorem, or are you just basically stating the definition of "having a solution"? Wouldn't your statement be true of any system of differential equations, not just ODEs?
I just stated the definition of having one and only one solution, yes :). That was kind of the point I was making.
I originally intended to write a statement specific to ODE, but was then to lazy to look the specifics up ^^.
I originally intended to write a statement specific to ODE, but was then to lazy to look the specifics up ^^.
I read logicalee's comment as a reference to the way most (all?) branches of mathematics may be built using sets and their axioms. Set theory boils down to "in the set" and "not in the set", ie. black and white.
Quite interestingly, the axioms mathematics is build upon do not let you decide every question that you might come up with. As such in mathematics not everything is black or white. For example, the continuum hypothesis is independent of the most commonly used axiom system ZFC. To explain what that means imagine ZFC would be describing an apple. Out of the definition of "apple" you can get theorems that tell you "if you start on any point of the apple and dig down in a straight line, at some future point in time you will reach the 'other side' of the apple" or some statements you can disprove "if you walk on the surface of the apple in a straight you will never come back to point where you started". What you will not get is the colour of the apple. Everything you know about the apple is consistent with it being green. But all that is also consistent with it being red. The colour of the apple is independent of everything that you are interested in in an apple. Therefore there is no reasonable answer to the question "which colour does the apple have", except maybe: It should not matter. The same is true of the continuum hypothesis. I remember reading that if you need the CH to prove something, then you should definitely reconsider the statement you are trying to prove :).
> I remember reading that if you need the CH to prove something, then you should definitely reconsider the statement you are trying to prove
This is not entirely true. There are many proofs that begin by assuming CH. There are also two ways of interpreting "need CH to prove". One way is that the thing you are trying to prove is equivalent to CH, which is an interesting/useful result. Another way to interpret it is that you can not come up with another way of proving it. In the latter case, the CH based proof justifies that your proposition is not inconsistent, and may even lead you (or others) to proof that it is true regardless of CH. If I recall correctly some statements have been proved by proving the statement when CH is true, and also when CH is false.
This is not entirely true. There are many proofs that begin by assuming CH. There are also two ways of interpreting "need CH to prove". One way is that the thing you are trying to prove is equivalent to CH, which is an interesting/useful result. Another way to interpret it is that you can not come up with another way of proving it. In the latter case, the CH based proof justifies that your proposition is not inconsistent, and may even lead you (or others) to proof that it is true regardless of CH. If I recall correctly some statements have been proved by proving the statement when CH is true, and also when CH is false.
Ah but the real story is that like sophisticated math systems, in relation to any fixed axiomization, set theory involves provably, provably false and unprovable/independent statements. And in relation to any fixed model, set theory involves true and provable, true but unprovable, false but un-disprovable and false and disprovable.
And when you're math as a human endeavor, you can also add "provable but not yet proven" and "proven independent"
So, learn some stuff, see how far from black and white higher math can be.
And when you're math as a human endeavor, you can also add "provable but not yet proven" and "proven independent"
So, learn some stuff, see how far from black and white higher math can be.
Great article. I think we lose the forrest and trees in math. Similar in computer science. Many times we get hung up on language syntax to miss the beauty underneath. (For example: Godel's incompleteness theorum or the implications of Turing machines)
On one tangents....
I think we mathematicians are a little bit behind the curve. We are not fully aware of the Frankenstein that we may have already created or could create. I think that's another aspect of this responsibility of mathematicians to take a more public role—to educate the public by giving them access to the beauty and power of mathematics.
Interesting tie in to the importance of ethics for mathematics. It's not just a bunch of folks writing proofs about numbers in journals.
On one tangents....
I think we mathematicians are a little bit behind the curve. We are not fully aware of the Frankenstein that we may have already created or could create. I think that's another aspect of this responsibility of mathematicians to take a more public role—to educate the public by giving them access to the beauty and power of mathematics.
Interesting tie in to the importance of ethics for mathematics. It's not just a bunch of folks writing proofs about numbers in journals.
Ed Frenkel made a film to convey the beauty of math - some nudity.
http://www.imdb.com/title/tt1530994/
I am big fan of Ed Frenkel's lectures on youtube but this movie was one of the worst movies I've ever watched.
[deleted]
I'd like to add a little to the puzzle analogy. Not only is doing research like putting together a puzzle without an idea what the picture is, but the puzzle is irregularly shaped.
Same as to be a biologist, or a programmer or a doctor.
I've done some research in theoretical physics, and now I work as a programmer. And the two are very, very different in their reward structure.
In research it was usual to go for a few months without any relevant feelings of success (and not just me, many of my colleagues felt the same). In programming I usually get some positive feedback a few times a day, at the very least every few days.
That's a huge difference.
In research it was usual to go for a few months without any relevant feelings of success (and not just me, many of my colleagues felt the same). In programming I usually get some positive feedback a few times a day, at the very least every few days.
That's a huge difference.
As far as I can see, being a programmer has almost nothing in common with being a doctor...
Debugging.
Funny... that reminds me that in 'Programming Pearls' (http://netlib.bell-labs.com/cm/cs/pearls/) Jon Bentley recommends the book 'The Medical Detectives' by Berton Roueche, as both entertaining and instructive for the kind of lateral thinking needed to debug.
Not in my experience.
I've worked both as a research mathematician and as a programmer working on production software.
Even applied to the "same" types of problems, I've found them to be very different in day to day activities, goals, incentives ... really they are quite far apart.
I've worked both as a research mathematician and as a programmer working on production software.
Even applied to the "same" types of problems, I've found them to be very different in day to day activities, goals, incentives ... really they are quite far apart.
Certainly not the same, but I do believe that when a computer model is having some kind of math root, it gets its own coherency and can be developed further. By math root it does not imply a math model. but the idea of creating a set of object with given rules and behaviors like for example a vectorial space. Very often it starts with real "computer tries" but the best part are always when there is an underlying model. I tend even to believe that the level of math knowledge in modern programming has grown up in regards to 15 years ago.
I'd have to concur. I highly value both kinds of experience, for this diversity actually.
i'm pretty sure that title should be "mathemagician", cuz that shit is freakin mysterious
The article mentions how most of us are introducted to math and the reaction most of us have to math in later life: "Oh no, I don't want to hear about this, I was so bad at math." The article continues about how wonderful math is and what not.
The article is wasting our time.
Those of us who were force-fed useless mathematics since 1st grade have already closed our minds completely to any attempts at beautifying something that we normal people, after +-/, see as completely fucking useless, boring and hate-inducing. At least I feel that way, but I assume I'm not alone.
I was force fed not only simple +-/ but several university level courses. Why? Because I wanted to get myself a programming degree and for that one, somehow, needed a whole bunch of weird-ass algebra and shit. I could see no reason for learning that at 15, when I was writing Pascal and PCBoard PPEs, no reason for learning that at 20 when I was writing C/C++ shit and no reason now at 30+ when I'm writing webapps in PHP/js/html5/etc. But time and time again I was told that without that complicated mathematics I wouldn't become shit.
Well, now I'm making a living programming, without a degree, and without being able to derive x2 from god-knows-what. My mind has PROVEN to me that I don't need complicated math and therefore cemented my previously belief that advanced math is necessary to program.
Not once during all my years of school and university was I ever shown a use for that advanced math, even though I _begged_ for at least some connection to reality, some proof that the math is going to be useful for anything else other than passing an exam. Do my SQL queries speed up if I use cosine? Will my code autoindent properly if the square root of x^y is used? No. Instead I was told to just learn it because I'd use it "some day".
My brain is now completely, 100% closed to advanced math. So what this Edward guy in the article should do is not try to change our minds, or even change the minds of the kids of today that are being force-fed math. Instead he should concentrate his efforts on bringing the teachers (and curriculum) down to earth. Change the way math is taught. Bring in real-world examples of why derivations are somehow important. After that is done then the NEXT generation won't fucking hate math, and mathematicians, as much as use older folk do.
Fuck I hate math.
The article is wasting our time.
Those of us who were force-fed useless mathematics since 1st grade have already closed our minds completely to any attempts at beautifying something that we normal people, after +-/, see as completely fucking useless, boring and hate-inducing. At least I feel that way, but I assume I'm not alone.
I was force fed not only simple +-/ but several university level courses. Why? Because I wanted to get myself a programming degree and for that one, somehow, needed a whole bunch of weird-ass algebra and shit. I could see no reason for learning that at 15, when I was writing Pascal and PCBoard PPEs, no reason for learning that at 20 when I was writing C/C++ shit and no reason now at 30+ when I'm writing webapps in PHP/js/html5/etc. But time and time again I was told that without that complicated mathematics I wouldn't become shit.
Well, now I'm making a living programming, without a degree, and without being able to derive x2 from god-knows-what. My mind has PROVEN to me that I don't need complicated math and therefore cemented my previously belief that advanced math is necessary to program.
Not once during all my years of school and university was I ever shown a use for that advanced math, even though I _begged_ for at least some connection to reality, some proof that the math is going to be useful for anything else other than passing an exam. Do my SQL queries speed up if I use cosine? Will my code autoindent properly if the square root of x^y is used? No. Instead I was told to just learn it because I'd use it "some day".
My brain is now completely, 100% closed to advanced math. So what this Edward guy in the article should do is not try to change our minds, or even change the minds of the kids of today that are being force-fed math. Instead he should concentrate his efforts on bringing the teachers (and curriculum) down to earth. Change the way math is taught. Bring in real-world examples of why derivations are somehow important. After that is done then the NEXT generation won't fucking hate math, and mathematicians, as much as use older folk do.
Fuck I hate math.
That's pretty sad. Programming and mathematics are normally things that go together, like burgers and beer. I can't imagine loving one and not the other. I do sympathise with the wish to see a practical application. I have a distinct recollection of an ah-ha moment for me, at about 15 years of age. We were being taught basic calculus. The idea was introduced that a function f(x) could be used to represent the yield of some industrial process. You put in x, you get out y = f(x). So far, so obvious, so boring right? But then came the cute part; If you can calculate the derivative of f(x), you get another function f'(x) which is the slope of f(x) for any value of x. Then to find the maximum yield you seek the value of x where f'(x) = 0, because the slope of f(x) will be zero 'at the top of the hill'.
I guess it's a personal thing, but for me this fired my enthusiasm for both learning the various techniques for differentiation, and for equation solving. Then integration followed as simply reverse differentiation [amazing - one is the area under the graph - one is the slope of the graph - why should they be opposite ? Because maths is beautiful is the short answer - a universe of discovery is the long answer]. Around this point maths became fun, the work part evaporated. Later I got interested in computation because you can't always differentiate or solve equations analytically (with pure math), but you can do it numerically, to any desired degree of accuracy with a computer.
I guess it's a personal thing, but for me this fired my enthusiasm for both learning the various techniques for differentiation, and for equation solving. Then integration followed as simply reverse differentiation [amazing - one is the area under the graph - one is the slope of the graph - why should they be opposite ? Because maths is beautiful is the short answer - a universe of discovery is the long answer]. Around this point maths became fun, the work part evaporated. Later I got interested in computation because you can't always differentiate or solve equations analytically (with pure math), but you can do it numerically, to any desired degree of accuracy with a computer.
> The article is wasting our time.
indeed. here's a better link:
http://en.wikipedia.org/wiki/Langlands_program
> Fuck I hate math.
this is the only other sentence of your post i had time for, and all i have time to say in response is that i don't.
indeed. here's a better link:
http://en.wikipedia.org/wiki/Langlands_program
> Fuck I hate math.
this is the only other sentence of your post i had time for, and all i have time to say in response is that i don't.
As a more advanced examples take a look at perfect hashing. Without some randomization an attacker can produce collisions for any fast hash functions and thus degrade your hash table to the worst case possible. This was a real DoS attack vector in real web applications (see for example https://mail.python.org/pipermail/python-dev/2011-December/1... ). With a little advanced math this matter is trivial to solve.
Something else I remember was a computer graphics professor saying that the fast fourier transform was the greatest invention after the wheel.
Generally speaking you can find programming jobs where you need basically no understanding of mathematics. But computational engineering science is so much more than programming. Conversely often enough the abstract thinking that is the core of mathematics can greatly help you to tackle the more complex problems you can encounter in programming.
Something else I remember was a computer graphics professor saying that the fast fourier transform was the greatest invention after the wheel.
Generally speaking you can find programming jobs where you need basically no understanding of mathematics. But computational engineering science is so much more than programming. Conversely often enough the abstract thinking that is the core of mathematics can greatly help you to tackle the more complex problems you can encounter in programming.
Software is a very broad field and you are painting with a very broad brush. It is true that for many branches of software you do not need much in the way of advanced mathematics. But EVERY graphics programmer I have ever met has told me he wishes he had paid more attention in higher level math courses. The same holds true for folks doing signal and image processing, computer vision, artificial intelligence, and on and on...
You've got a list of stores, and a person's location. Which store is closest?
You are buying ads online, and want to maximize your ROI.
You have 6 types of EC2 nodes you can use, and you want to pick the most efficient set for your mildly parallel work.
You are buying ads online, and want to maximize your ROI.
You have 6 types of EC2 nodes you can use, and you want to pick the most efficient set for your mildly parallel work.
See that https in your browser bar right now? You can thank math for that.
From the examples in your post, it seems like you only got to calculus. Which is not really advanced math.
http://www.quora.com/Mathematics/What-is-it-like-to-understa...
I'm not a mathematician, but was a theoretical physicist for more than a decade, and the Quora article accords well with my experience in understanding mathematics.
(Fairly) readable examples of some of the points made in the above article may be found in Terry Tao and Scott Aaronson's comments about how to think intuitively in high dimensions:
http://mathoverflow.net/questions/25983/intuitive-crutches-f...