Does one have to be a genius to do maths?

This is a fantastic blog post and it needed to come from someone as accomplished as Tao.<p>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.<p>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.

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 &#38; replaced so the text refers to "entrepreneurs" instead of "mathematicians:"<p><i>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.<p>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.</i>

The biggest problem with math is the language. If it were a programming language, we would call it crufty and obscurantist.<p>The language and notation really needs to be rebooted and cleaned up. Math with a sane notation would be significantly easier to learn.<p>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).

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".<p>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.<p>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.

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."<p>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.

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.<p>Sure, hard work may be the way that he experiences himself. But read <a href="http://www.davidsongifted.org/db/Articles_id_10116.aspx" rel="nofollow">http://www.davidsongifted.org/db/Articles_id_10116.aspx</a> 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.)<p>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.

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."

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.<p>In an interview in BBC Music Magazine British violinist Nicola Benedetti says quite bluntly "If you sound like rubbish at age 13, you quit."<p>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]?<p>[0]: <a href="http://en.wikipedia.org/wiki/El_Sistema" rel="nofollow">http://en.wikipedia.org/wiki/El_Sistema</a>

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:<p>"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."<p>To anyone who doesn't know, Martha Graham was to dance what Picasso was to visual art.<p><a href="http://en.wikipedia.org/wiki/Martha_Graham" rel="nofollow">http://en.wikipedia.org/wiki/Martha_Graham</a> <a href="http://joshuaspodek.com/master-speaks-creative-expression" rel="nofollow">http://joshuaspodek.com/master-speaks-creative-expression</a>

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).

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,<p><a href="http://epsiloncamp.org/" rel="nofollow">http://epsiloncamp.org/</a><p>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.<p>Based on this published writing by Tao and various writings by other mathematicians, the Epsilon Camp program provides FAQ pages for parents,<p><a href="http://epsiloncamp.org/FAQ.php" rel="nofollow">http://epsiloncamp.org/FAQ.php</a><p>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.<p>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,<p><a href="http://rjlipton.wordpress.com/2012/08/09/a-new-way-to-solve-linear-equations/" rel="nofollow">http://rjlipton.wordpress.com/2012/08/09/a-new-way-to-solve-...</a><p>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.

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.

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