8 ms·

Does one have to be a genius to do maths?

113 points by caustic 14y ago

zakshay 14y ago
Mathematics when studied alone can get a bit difficult. In my opinion mathematical concepts are better understood through applications of it such as Physics, specially Mechanics.
- masterzora 14y ago
  This is definitely a per-person sort of thing. I was a maths major who took several physics courses and I tended to find that they muddied my mathematical understanding as often as they helped whereas I found algebra and number theory (my foci, admittedly) to be easy enough to understand on their own that the classes were just on this side of trivial.
  - csense 14y ago
    Physicists often take shortcuts based on physical intuition, e.g. symmetry arguments. I remember having trouble understanding equilibrium charge distributions when studying E&M -- laws such as "at electrostatic equilibrium, all charge is on the surface of the conductor" -- the questions of whether such a static equilibrium exists, is unique, or is a place you'll always end up from an arbitrary initial configuration weren't really addressed. (The best I could come up with is that any movement of charge will eventually die out due to friction, but this was more of a vague intuition than a satisfying explanation.) Anyway, I guess if you have the physics gene, you just have a strong intuition that tells you the answers to questions like these. I didn't have it. Physics is better for people who like to trust their intuition. Math is more programming-like in that the people who do well tend to be hard-nosed about details and corner cases.
smalter 14y ago
My dad is a math professor at a big state school with a strong engineering program. I was hanging out with him and some of his college classmates, many of whom are also math/science/engineering professors. One of them told me, "If everyone worked as hard as your dad, anyone could be a math professor." That stuck with me, not as a statement of fact, but a testament to the power of hard work to shape outcomes, even in a field considered to be dominated by genius.
- mej10 14y ago
  Reminds me of this anecdote from Richard Hamming's "You and Your Research": Now for the matter of drive. You observe that most great scientists have tremendous drive. I worked for ten years with John Tukey at Bell Labs. He had tremendous drive. One day about three or four years after I joined, I discovered that John Tukey was slightly younger than I was. John was a genius and I clearly was not. Well I went storming into Bode's office and said, ``How can anybody my age know as much as John Tukey does?'' He leaned back in his chair, put his hands behind his head, grinned slightly, and said, ``You would be surprised Hamming, how much you would know if you worked as hard as he did that many years.'' I simply slunk out of the office!
benmccann 14y ago
Written by a guy who was attending university level mathematics courses at the age of nine. Perhaps the sentiment is better expressed as being a genius is not sufficient. Hard work is still required (but so is being a genius).
- cschmidt 14y ago
  I imagine he would say being a genius (which he clearly is) is neither necessary nor sufficient for working in math.
- ColinWright 14y ago
  Certainly hard work is required, but equally certainly, being a genius is not necessary. I'm a mathematician, and I'm no genius.
  - BasDirks 14y ago
    You are not? How does one tell?
    - ColinWright 14y ago
      I've spent time with Tim Gowers, John Conway, Ron Graham, and others who are, without question, geniuses. I'm not in their league.
      - Evbn 14y ago
        Right. So you don't need to be a one in a million mind like Growers and Tao. You just need to be one in a thousand. Still not "anyone can bang it out".
        dkarl 14y ago
        The minute or so starting at 5m40s in this video is apropos: http://video.google.com/videoplay?docid=5935911405946587342 http://video.google.com/videoplay?docid=5935911405946587342 (You may be interested in the rest of the video as well. It's a fascinating show.)
cs702 14y ago
Every word of advice in this blog post by Terry Tao applies verbatim to many other fields -- including entrepreneurship. Here are two key paragraphs from his post, with just a few words searched & replaced so the text refers to "entrepreneurs" instead of "mathematicians:" Even if one dismisses the notion of genius, it is still the case that at any given point in time, some entrepreneurs are faster, more experienced, more knowledgeable, more efficient, more careful, or more creative than others. This does not imply, though, that only the “best” entrepreneurs should start companies; this is the common error of mistaking absolute advantage for comparative advantage. The number of interesting business opportunities and problems to work on is vast – far more than can be covered in detail just by the “best” entrepreneurs, and sometimes the set of tools or ideas that you have will find something that other good entrepreneurs have overlooked, especially given that even the greatest entrepreneurs still have weaknesses in some aspects of business. As long as you have education, interest, and a reasonable amount of talent, there will be some market opportunity where you can make a solid and useful contribution. It might not be the most glamorous idea, but actually this tends to be a healthy thing; in many cases the mundane nuts-and-bolts ideas turn out to actually be more important than fancy ones. Also, it is necessary to “cut one’s teeth” on the non-glamorous parts of a field before one really has any chance at all to tackle hard problems; take a look at the early efforts of any of today’s great entrepreneurs to see what I mean by this. In some cases, an abundance of raw talent may end up (somewhat perversely) to actually be harmful for one’s long-term professional development; if success comes too easily, for instance, one may not put as much energy into working hard, asking dumb questions, or increasing one’s range, and thus may eventually cause one’s skills to stagnate. Also, if one is accustomed to easy success, one may not develop the patience necessary to deal with truly difficult challenges. Talent is important, of course; but how one develops and nurtures it is even more so.
BasDirks 14y ago
Most mathematical genius in my life (in others and me) has been the direct result of DOING. Doing maths at that stage was the result of joy. Joy was often the result of a feeling of newness. A feeling of newness can be the result of a great number of things. Another reason for joy in mathematics can be emulation of ones parents. In an interview in BBC Music Magazine British violinist Nicola Benedetti says quite bluntly "If you sound like rubbish at age 13, you quit." It's lazy to say that this initial "sounding good" is "genius", and leave it at that. To come back to classical music: where is the Mathematics equivalent of El Sistema[0]? [0]: http://en.wikipedia.org/wiki/El_Sistema http://en.wikipedia.org/wiki/El_Sistema
spodek 14y ago
At the risk of going backwards, quoting an artist to a mathematical community about math, I found what Martha Graham said about dance applies to any creative endeavor: "Nobody cares if you can’t dance well. Just get up and dance. Great dancers are not great because of their technique, they are great because of their passion." To anyone who doesn't know, Martha Graham was to dance what Picasso was to visual art. http://en.wikipedia.org/wiki/Martha_Graham http://en.wikipedia.org/wiki/Martha_Graham http://joshuaspodek.com/master-speaks-creative-expression http://joshuaspodek.com/master-speaks-creative-expression
T_S_ 14y ago
Math is particularly tough on the ego. I remember one of my advisors said: "I try to decide if something is true. I work extremely hard to do so. Then when I am done, it seems clear that it was true all along, and the only problem was that I didn't know it."
jboggan 14y ago
This is a fantastic blog post and it needed to come from someone as accomplished as Tao. The cult of genius can be very caustic to young minds, especially in mathematics. I used to do some rather stupid things out of a combined sense of pressure from family, teachers, and peers. I would compare myself unfairly to historical luminaries as a yardstick of what I should be accomplishing at what age. I worked incredibly hard, but on intractable problems and not on reasonable pieces of research for even a precocious mathematician. My grades suffered because I thought I was going to solve some open conjecture instead of learn the tools bit by bit like virtually every other successful mathematician had done before me. Depression can set in when you discover that your 20th birthday has passed and you are not Evariste Galois. I know it sounds stupid when it is phrased like that but human psychology is full of improbable behavior designed around avoiding cognitive dissonance. We're funny meatbots.
- lrs 14y ago
  This rings painfully true.
- _delirium 14y ago
  I think to me it might've been more reassuring if it came from someone other than Tao. I tend to put Tao more in the Galois-like depressing category, given how much he had already accomplished by the time he was 20 (he was promoted to full professor at age 24). There's no amount of hard work that can replicate his trajectory unless you go back and start it at age 8, and even then it's unlikely.
  - cs702 14y ago
    _delirium: I highly recommend you read this short biography of Scottish scientist James Croll, who developed the modern theory of Ice Ages -- with little formal education, he decided to become a scientist while working as a janitor at the age of 38: http://www.guildtownandwolfhill.org.uk/assets/files/pdf/James%20Croll%20longer%20article.pdf http://www.guildtownandwolfhill.org.uk/assets/files/pdf/Jame... Edit: submitted the link as http://news.ycombinator.com/item?id=4370924 http://news.ycombinator.com/item?id=4370924 because Croll's story can be very inspirational for the many entrepreneurs on HN who're coping with the challenges of building a business from scratch.
tokenadult 14y ago
This interesting article submitted here on HN is one I have often recommended to other readers, so I'm glad to see it on HN's main page. It's particularly interesting to read the comments here, mostly largely agreeing with Tao, as I am currently at Epsilon Camp, http://epsiloncamp.org/ http://epsiloncamp.org/ the most advanced mathematics summer program for YOUNG learners in North America, and the parents of the campers here are all pondering the issue of their children's mathematical development. Plainly, at any given age, some young people are more advanced in their mathematical development than many of their age mates, but it is still to be seen how steadily and consistently the mathematical development of the most advanced young learners can be developed if optimal practices are applied to their education. Based on this published writing by Tao and various writings by other mathematicians, the Epsilon Camp program provides FAQ pages for parents, http://epsiloncamp.org/FAQ.php http://epsiloncamp.org/FAQ.php who in many cases are not themselves mathematicians, to serve as food for thought as precocious young mathematics learners are growing up. One of the issues for many of the parents from various parts of the United States is simply finding a flexible local school. Another issue, which the meetings of the parents at the camp has helped to handle, is sustaining friendship relationships among those most advanced young mathematics-learners as they disperse around the country at the end of the summer program. On the whole, the parents participating in the program have great buy-in to Tao's idea that whatever initial dose of "talent" or "native ability" a child starts with, careful and intentional guidance of the child's whole-child development is still very important for the child to have the best enjoyment of advanced study of mathematics and the best success in making a new contribution to human knowledge as an adult, in whatever domain the child chooses. By the way, everyone on Hacker News might enjoy looking at Tao's comments on a blog post that reached the Hacker News page yesterday, http://rjlipton.wordpress.com/2012/08/09/a-new-way-to-solve-linear-equations/ http://rjlipton.wordpress.com/2012/08/09/a-new-way-to-solve-... in which we can see Tao thinking out loud in blog comments about what the blog post really means. This kind of careful, step-by-step thinking is something that every mathematician needs to develop sooner or later.
api 14y ago
The biggest problem with math is the language. If it were a programming language, we would call it crufty and obscurantist. The language and notation really needs to be rebooted and cleaned up. Math with a sane notation would be significantly easier to learn. The other -- and closely related -- problem is with how math is taught. It is taught procedure-first, not language-first and concept-first. It is impossible to understand math without being able to read the notation and translate it into relevant concepts. Doing the mechanics is secondary (and often done by computers these days).
- guscost 14y ago
  I agree 100% with this. Math will be a lot easier to learn with better tools.
  - ChuckMcM 14y ago
    I think the dragonbox app [1] is a pretty solid empirical test of that thesis. [1] http://dragonboxapp.com/ http://dragonboxapp.com/
- MaysonL 14y ago
  Go back and look at Newton's and Leibniz's notation. Math notation, proof structure, and simplicity are continually getting cleaned up. The thing is, math is HARD.
  - johnwatson11218 14y ago
    I read that Leibniz had studied Chinese and that it influenced some of the math notation that he introduced. Like using df/dx to express the derivative of f with respect to x as a kind of ratio. Whereas Newton just put f' to denote the same thing. I think Leibniz also came up with the elongated sigma for representing integrals as a type of summation.
    - btilly 14y ago
      No, Newton put a dot over y to represent a derivative. The f' notation came from Lagrange.
dave_sullivan 14y ago
I started taking an interest in machine learning and AI about a year and a half ago. I don't consider myself any kind of genius (although I'm reasonably intelligent), and I was terrible at math in school--to the point where I'd come to the conclusion that I simply "wasn't good at math". After a good deal of reading, trial and error, and banging my head against the wall, I've managed to get myself to pretty much the cutting edge of ML research as it applies to neural networks. There's quite a bit of math involved, and it would have been easy for me to write it off as "too hard" in the beginning. However, I'm glad I stuck with it because I'm actually using it for some pretty neat applications. My point being, if you have an interest in something that seems like you have to be a genius to be good at it, don't let that stop you because it probably isn't true.
- akshaykarthik 14y ago
  I myself am quite interested in ML research. Do you have any resources that you found useful in learning not only ML, but also the associated math?
  - sidman 14y ago
    Im not sure if you can find many places where they have both in great detal as most machine learning material assume some knowledge in the specific math but you could always write down the terms that you dont understand from any ML material and look it up on khan academy. For all the material you need for Andrew Ng's classes (on coursera) for example you can pretty much find all the required terms (matricies, probability etc) on khan academy
  - dave_sullivan 14y ago
    Yeah, definitely. If you're just getting started, I highly recommend Andrew Ng's online ML class. I had started reading up before this was available, but it really tied a lot of basics together that I was confused about. From there, read papers. For me, my primary interest is in neural networks. Geoffrey Hinton and Yoshua Bengio have two very good groups that both have contributed a great deal to research in this area, and their websites provide lots of good stuff. After spending a bit of time on a survey of the field, try to come up with a practical goal as quickly as you can: I want to use ML to do X. Then, try to do that. When you get stuck, get back to reading until you find the answer. Rinse and repeat. The math naturally falls into this--you'll get stuck on things that you can't fix without a decent understanding of the math. So figure out how to formulate the question you're really asking, hit google, and read. Then try again. Rinse and repeat. If you're interested in neural networks, I can also recommend the deep learning tutorials associated with Theano, a library for compiling python code down to CUDA code to get speed increases on certain operations that will let you train your models about 10-40x faster on GPU than if you tried it on CPU. For me, putting things into practice has helped me make the biggest leaps in understanding, but of course I wouldn't have been able to do that at all without getting a basic grasp of the mechanisms involved. So it's a bit of push and pull between practice and learning, like anything worth doing.
deleted 14y ago
[deleted]
btilly 14y ago
I love this essay but would find it more believable if it did not come from one of the people who best exemplifies genius in mathematics. Sure, hard work may be the way that he experiences himself. But read http://www.davidsongifted.org/db/Articles_id_10116.aspx http://www.davidsongifted.org/db/Articles_id_10116.aspx for an account of his childhood, written when he was 10. Teaching yourself to read and do math before most children can use complete sentences requires something more than pure effort. My son is well above average, but there is absolutely no way that it would be possible to get him to work hard enough to compare with average high school seniors on the SATs before he was 10, let alone scoring near the top. (For those who took the SATs in the last 20 years, the scale used in the 1980s was much tougher than it is now. 700+ would have been easily in the top 1% on the test.) All of that said, he would not have his current success without constantly working hard. And it is possible to succeed without being at Terry's level of genius. But he's the worst possible example to use for saying that what appears to be genius is just hard work. Because sometimes what appears to be genius really is genius if you dig in.
the1 14y ago
there's nothing wrong with being a math genius. there's nothing wrong with not being a math genius.

113 points by caustic 14y ago

Mathematics when studied alone can get a bit difficult. In my opinion mathematical concepts are better understood through applications of it such as Physics, specially Mechanics.

masterzora 14y ago
This is definitely a per-person sort of thing. I was a maths major who took several physics courses and I tended to find that they muddied my mathematical understanding as often as they helped whereas I found algebra and number theory (my foci, admittedly) to be easy enough to understand on their own that the classes were just on this side of trivial.
- csense 14y ago
  Physicists often take shortcuts based on physical intuition, e.g. symmetry arguments. I remember having trouble understanding equilibrium charge distributions when studying E&M -- laws such as "at electrostatic equilibrium, all charge is on the surface of the conductor" -- the questions of whether such a static equilibrium exists, is unique, or is a place you'll always end up from an arbitrary initial configuration weren't really addressed. (The best I could come up with is that any movement of charge will eventually die out due to friction, but this was more of a vague intuition than a satisfying explanation.) Anyway, I guess if you have the physics gene, you just have a strong intuition that tells you the answers to questions like these. I didn't have it. Physics is better for people who like to trust their intuition. Math is more programming-like in that the people who do well tend to be hard-nosed about details and corner cases.

smalter 14y ago

My dad is a math professor at a big state school with a strong engineering program. I was hanging out with him and some of his college classmates, many of whom are also math/science/engineering professors. One of them told me, "If everyone worked as hard as your dad, anyone could be a math professor." That stuck with me, not as a statement of fact, but a testament to the power of hard work to shape outcomes, even in a field considered to be dominated by genius.

mej10 14y ago
Reminds me of this anecdote from Richard Hamming's "You and Your Research": Now for the matter of drive. You observe that most great scientists have tremendous drive. I worked for ten years with John Tukey at Bell Labs. He had tremendous drive. One day about three or four years after I joined, I discovered that John Tukey was slightly younger than I was. John was a genius and I clearly was not. Well I went storming into Bode's office and said, ``How can anybody my age know as much as John Tukey does?'' He leaned back in his chair, put his hands behind his head, grinned slightly, and said, ``You would be surprised Hamming, how much you would know if you worked as hard as he did that many years.'' I simply slunk out of the office!

benmccann 14y ago

Written by a guy who was attending university level mathematics courses at the age of nine. Perhaps the sentiment is better expressed as being a genius is not sufficient. Hard work is still required (but so is being a genius).

cschmidt 14y ago
I imagine he would say being a genius (which he clearly is) is neither necessary nor sufficient for working in math.
ColinWright 14y ago
Certainly hard work is required, but equally certainly, being a genius is not necessary. I'm a mathematician, and I'm no genius.
- BasDirks 14y ago
  You are not? How does one tell?
  - ColinWright 14y ago
    I've spent time with Tim Gowers, John Conway, Ron Graham, and others who are, without question, geniuses. I'm not in their league.
    - Evbn 14y ago
      Right. So you don't need to be a one in a million mind like Growers and Tao. You just need to be one in a thousand. Still not "anyone can bang it out".
      - dkarl 14y ago
        The minute or so starting at 5m40s in this video is apropos: http://video.google.com/videoplay?docid=5935911405946587342 http://video.google.com/videoplay?docid=5935911405946587342 (You may be interested in the rest of the video as well. It's a fascinating show.)

cs702 14y ago

Every word of advice in this blog post by Terry Tao applies verbatim to many other fields -- including entrepreneurship. Here are two key paragraphs from his post, with just a few words searched & replaced so the text refers to "entrepreneurs" instead of "mathematicians:" Even if one dismisses the notion of genius, it is still the case that at any given point in time, some entrepreneurs are faster, more experienced, more knowledgeable, more efficient, more careful, or more creative than others. This does not imply, though, that only the “best” entrepreneurs should start companies; this is the common error of mistaking absolute advantage for comparative advantage. The number of interesting business opportunities and problems to work on is vast – far more than can be covered in detail just by the “best” entrepreneurs, and sometimes the set of tools or ideas that you have will find something that other good entrepreneurs have overlooked, especially given that even the greatest entrepreneurs still have weaknesses in some aspects of business. As long as you have education, interest, and a reasonable amount of talent, there will be some market opportunity where you can make a solid and useful contribution. It might not be the most glamorous idea, but actually this tends to be a healthy thing; in many cases the mundane nuts-and-bolts ideas turn out to actually be more important than fancy ones. Also, it is necessary to “cut one’s teeth” on the non-glamorous parts of a field before one really has any chance at all to tackle hard problems; take a look at the early efforts of any of today’s great entrepreneurs to see what I mean by this. In some cases, an abundance of raw talent may end up (somewhat perversely) to actually be harmful for one’s long-term professional development; if success comes too easily, for instance, one may not put as much energy into working hard, asking dumb questions, or increasing one’s range, and thus may eventually cause one’s skills to stagnate. Also, if one is accustomed to easy success, one may not develop the patience necessary to deal with truly difficult challenges. Talent is important, of course; but how one develops and nurtures it is even more so.

BasDirks 14y ago

Most mathematical genius in my life (in others and me) has been the direct result of DOING. Doing maths at that stage was the result of joy. Joy was often the result of a feeling of newness. A feeling of newness can be the result of a great number of things. Another reason for joy in mathematics can be emulation of ones parents. In an interview in BBC Music Magazine British violinist Nicola Benedetti says quite bluntly "If you sound like rubbish at age 13, you quit." It's lazy to say that this initial "sounding good" is "genius", and leave it at that. To come back to classical music: where is the Mathematics equivalent of El Sistema[0]? [0]: http://en.wikipedia.org/wiki/El_Sistema http://en.wikipedia.org/wiki/El_Sistema

spodek 14y ago

At the risk of going backwards, quoting an artist to a mathematical community about math, I found what Martha Graham said about dance applies to any creative endeavor: "Nobody cares if you can’t dance well. Just get up and dance. Great dancers are not great because of their technique, they are great because of their passion." To anyone who doesn't know, Martha Graham was to dance what Picasso was to visual art. http://en.wikipedia.org/wiki/Martha_Graham http://en.wikipedia.org/wiki/Martha_Graham http://joshuaspodek.com/master-speaks-creative-expression http://joshuaspodek.com/master-speaks-creative-expression

T_S_ 14y ago

Math is particularly tough on the ego. I remember one of my advisors said: "I try to decide if something is true. I work extremely hard to do so. Then when I am done, it seems clear that it was true all along, and the only problem was that I didn't know it."

jboggan 14y ago

This is a fantastic blog post and it needed to come from someone as accomplished as Tao. The cult of genius can be very caustic to young minds, especially in mathematics. I used to do some rather stupid things out of a combined sense of pressure from family, teachers, and peers. I would compare myself unfairly to historical luminaries as a yardstick of what I should be accomplishing at what age. I worked incredibly hard, but on intractable problems and not on reasonable pieces of research for even a precocious mathematician. My grades suffered because I thought I was going to solve some open conjecture instead of learn the tools bit by bit like virtually every other successful mathematician had done before me. Depression can set in when you discover that your 20th birthday has passed and you are not Evariste Galois. I know it sounds stupid when it is phrased like that but human psychology is full of improbable behavior designed around avoiding cognitive dissonance. We're funny meatbots.

lrs 14y ago
This rings painfully true.
_delirium 14y ago
I think to me it might've been more reassuring if it came from someone other than Tao. I tend to put Tao more in the Galois-like depressing category, given how much he had already accomplished by the time he was 20 (he was promoted to full professor at age 24). There's no amount of hard work that can replicate his trajectory unless you go back and start it at age 8, and even then it's unlikely.
- cs702 14y ago
  _delirium: I highly recommend you read this short biography of Scottish scientist James Croll, who developed the modern theory of Ice Ages -- with little formal education, he decided to become a scientist while working as a janitor at the age of 38: http://www.guildtownandwolfhill.org.uk/assets/files/pdf/James%20Croll%20longer%20article.pdf http://www.guildtownandwolfhill.org.uk/assets/files/pdf/Jame... Edit: submitted the link as http://news.ycombinator.com/item?id=4370924 http://news.ycombinator.com/item?id=4370924 because Croll's story can be very inspirational for the many entrepreneurs on HN who're coping with the challenges of building a business from scratch.

tokenadult 14y ago

This interesting article submitted here on HN is one I have often recommended to other readers, so I'm glad to see it on HN's main page. It's particularly interesting to read the comments here, mostly largely agreeing with Tao, as I am currently at Epsilon Camp, http://epsiloncamp.org/ http://epsiloncamp.org/ the most advanced mathematics summer program for YOUNG learners in North America, and the parents of the campers here are all pondering the issue of their children's mathematical development. Plainly, at any given age, some young people are more advanced in their mathematical development than many of their age mates, but it is still to be seen how steadily and consistently the mathematical development of the most advanced young learners can be developed if optimal practices are applied to their education. Based on this published writing by Tao and various writings by other mathematicians, the Epsilon Camp program provides FAQ pages for parents, http://epsiloncamp.org/FAQ.php http://epsiloncamp.org/FAQ.php who in many cases are not themselves mathematicians, to serve as food for thought as precocious young mathematics learners are growing up. One of the issues for many of the parents from various parts of the United States is simply finding a flexible local school. Another issue, which the meetings of the parents at the camp has helped to handle, is sustaining friendship relationships among those most advanced young mathematics-learners as they disperse around the country at the end of the summer program. On the whole, the parents participating in the program have great buy-in to Tao's idea that whatever initial dose of "talent" or "native ability" a child starts with, careful and intentional guidance of the child's whole-child development is still very important for the child to have the best enjoyment of advanced study of mathematics and the best success in making a new contribution to human knowledge as an adult, in whatever domain the child chooses. By the way, everyone on Hacker News might enjoy looking at Tao's comments on a blog post that reached the Hacker News page yesterday, http://rjlipton.wordpress.com/2012/08/09/a-new-way-to-solve-linear-equations/ http://rjlipton.wordpress.com/2012/08/09/a-new-way-to-solve-... in which we can see Tao thinking out loud in blog comments about what the blog post really means. This kind of careful, step-by-step thinking is something that every mathematician needs to develop sooner or later.

api 14y ago

The biggest problem with math is the language. If it were a programming language, we would call it crufty and obscurantist. The language and notation really needs to be rebooted and cleaned up. Math with a sane notation would be significantly easier to learn. The other -- and closely related -- problem is with how math is taught. It is taught procedure-first, not language-first and concept-first. It is impossible to understand math without being able to read the notation and translate it into relevant concepts. Doing the mechanics is secondary (and often done by computers these days).

guscost 14y ago
I agree 100% with this. Math will be a lot easier to learn with better tools.
- ChuckMcM 14y ago
  I think the dragonbox app [1] is a pretty solid empirical test of that thesis. [1] http://dragonboxapp.com/ http://dragonboxapp.com/
MaysonL 14y ago
Go back and look at Newton's and Leibniz's notation. Math notation, proof structure, and simplicity are continually getting cleaned up. The thing is, math is HARD.
- johnwatson11218 14y ago
  I read that Leibniz had studied Chinese and that it influenced some of the math notation that he introduced. Like using df/dx to express the derivative of f with respect to x as a kind of ratio. Whereas Newton just put f' to denote the same thing. I think Leibniz also came up with the elongated sigma for representing integrals as a type of summation.
  - btilly 14y ago
    No, Newton put a dot over y to represent a derivative. The f' notation came from Lagrange.

dave_sullivan 14y ago

I started taking an interest in machine learning and AI about a year and a half ago. I don't consider myself any kind of genius (although I'm reasonably intelligent), and I was terrible at math in school--to the point where I'd come to the conclusion that I simply "wasn't good at math". After a good deal of reading, trial and error, and banging my head against the wall, I've managed to get myself to pretty much the cutting edge of ML research as it applies to neural networks. There's quite a bit of math involved, and it would have been easy for me to write it off as "too hard" in the beginning. However, I'm glad I stuck with it because I'm actually using it for some pretty neat applications. My point being, if you have an interest in something that seems like you have to be a genius to be good at it, don't let that stop you because it probably isn't true.

akshaykarthik 14y ago
I myself am quite interested in ML research. Do you have any resources that you found useful in learning not only ML, but also the associated math?
- sidman 14y ago
  Im not sure if you can find many places where they have both in great detal as most machine learning material assume some knowledge in the specific math but you could always write down the terms that you dont understand from any ML material and look it up on khan academy. For all the material you need for Andrew Ng's classes (on coursera) for example you can pretty much find all the required terms (matricies, probability etc) on khan academy
- dave_sullivan 14y ago
  Yeah, definitely. If you're just getting started, I highly recommend Andrew Ng's online ML class. I had started reading up before this was available, but it really tied a lot of basics together that I was confused about. From there, read papers. For me, my primary interest is in neural networks. Geoffrey Hinton and Yoshua Bengio have two very good groups that both have contributed a great deal to research in this area, and their websites provide lots of good stuff. After spending a bit of time on a survey of the field, try to come up with a practical goal as quickly as you can: I want to use ML to do X. Then, try to do that. When you get stuck, get back to reading until you find the answer. Rinse and repeat. The math naturally falls into this--you'll get stuck on things that you can't fix without a decent understanding of the math. So figure out how to formulate the question you're really asking, hit google, and read. Then try again. Rinse and repeat. If you're interested in neural networks, I can also recommend the deep learning tutorials associated with Theano, a library for compiling python code down to CUDA code to get speed increases on certain operations that will let you train your models about 10-40x faster on GPU than if you tried it on CPU. For me, putting things into practice has helped me make the biggest leaps in understanding, but of course I wouldn't have been able to do that at all without getting a basic grasp of the mechanisms involved. So it's a bit of push and pull between practice and learning, like anything worth doing.

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btilly 14y ago

I love this essay but would find it more believable if it did not come from one of the people who best exemplifies genius in mathematics. Sure, hard work may be the way that he experiences himself. But read http://www.davidsongifted.org/db/Articles_id_10116.aspx http://www.davidsongifted.org/db/Articles_id_10116.aspx for an account of his childhood, written when he was 10. Teaching yourself to read and do math before most children can use complete sentences requires something more than pure effort. My son is well above average, but there is absolutely no way that it would be possible to get him to work hard enough to compare with average high school seniors on the SATs before he was 10, let alone scoring near the top. (For those who took the SATs in the last 20 years, the scale used in the 1980s was much tougher than it is now. 700+ would have been easily in the top 1% on the test.) All of that said, he would not have his current success without constantly working hard. And it is possible to succeed without being at Terry's level of genius. But he's the worst possible example to use for saying that what appears to be genius is just hard work. Because sometimes what appears to be genius really is genius if you dig in.

the1 14y ago

there's nothing wrong with being a math genius. there's nothing wrong with not being a math genius.