
Does one have to be a genius to do maths? - caustic
http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/
======
jboggan
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.

~~~
_delirium
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
_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/Jame...](http://www.guildtownandwolfhill.org.uk/assets/files/pdf/James%20Croll%20longer%20article.pdf)

Edit: submitted the link as <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.

~~~
gwern
Why is an exception like Croll, who is literally one out of thousands (how
many scientists make their breakthrough in their late 30s/40s, while working
at menial labor? now, how many _cranks_ do that...) of any interest?

Exceptions are exceptional, hence the name.

~~~
cs702
gwern: I find his story of dogged persistence an inspiration.

------
cs702
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._

------
api
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).

~~~
ColinWright
You know, I don't usually do this, but in this case I'll make an exception.

    
    
      > The biggest problem with math is the language.
    

Citation needed.

    
    
      > Math with a sane notation would be significantly
      > easier to learn.
    

Citation needed.

    
    
      > Doing the mechanics is secondary
    

Citation needed.

OK, so I speak from a different perspective from yours, because I have a PhD
in math. I also do a lot of outreach, enrichment, and enhancement, dealing
with reasonably bright (but very rarely genius level) kids, and in my direct
personal experience I have found that

(a) the notation is not a problem, and that

(b) doing the mechanics opens the door to internalising the structures and
enables the understanding of the ideas.

In my experience, trying to grasp the ideas and concepts without working on
what seem to be tedious, mechanical and apparently unnecessary processes
results in a flailing about, an inability to anchor those concepts in
experience or intuition. It's the mechanics and the processes that let you
internalise the patterns and start to build your own structures, into which
the ideas and concepts can then be fitted, making a coherent structure.

So forgive me if I regard your pronouncements with a degree of suspicion and
scepticism.

~~~
Dn_Ab
Spivak and Sussman have made excellent cases as to the problem of notation in
mathematics. I have long had much the same opinions as the grand parent so was
delighted to see such luminaries in agreement. Now, mathematics is compact but
that is not the problem. The problem is ambiguity. You can with time learn to
pack the and unpack the extremely dense notation and the upfront costs are
worth it but the ambiguity (not to mention differing conventions across
branches) is an inexcusable mess. Calculus for example is replete with the
abuse of variable binding which is just cruel to the beginner. Quoting:
<http://mitpress.mit.edu/sicm/book-Z-H-5.html#footnote_Temp_4>

_" 1 In his book on mathematical pedagogy [17], Hans Freudenthal argues that
the reliance on ambiguous, unstated notational conventions in such expressions
as f(x) and df(x)/dx makes mathematics, and especially introductory calculus,
extremely confusing for beginning students; and he enjoins mathematics
educators to use more formal modern notation.

2 In his beautiful book Calculus on Manifolds [40], Michael Spivak uses
functional notation. On p. 44 he discusses some of the problems with classical
notation. We excerpt a particularly juicy passage:

The mere statement of [the chain rule] in classical notation requires the
introduction of irrelevant letters. The usual evaluation for D1(fo(g,h)) runs
as follows:... This equation is often written simply...

Note that f means something different on the two sides of the equation!

3 This is presented here without explanation, to give the flavor of the
notation. The text gives a full explanation.

4 ``It is necessary to use the apparatus of partial derivatives, in which even
the notation is ambiguous.'' V.I. Arnold, Mathematical Methods of Classical
Mechanics [5], Section 47, p. 258. See also the footnote on that page. _"

~~~
hazov
In my experience I think notation would not make the life of a student much
easier if you're learning things at the level Terry Tao writes in his blog,
he's not writing college level math for engineers (ie. Calculus, LinAlg, Basic
Fourier Analysis and some other topics) but Graduate level math for
mathematicians and people interested in pure and/or applied mathematics.
Notation is not the problem when you're having a hard time studying Functional
Analysis, Algebraic Topology, Advanced Probability (using Lebesgue integral)
or trying to understand the proof of the Prime Number theorem.

The problem actually is that notation preference is a matter of personal
preference, the physicists love the bra and ket notation I think it is very
confusing, maybe because I'm not a physicist. I think any time someone comes
with new notation for settled things they just turns the matter worse.

Partial Differential Equations is a perfect example how these things works,
there's generally different notations being used by mathematicians, physicists
and engineers, at least in my experience, and because every couple of years
someone comes with a new notation to "simplify" everything to his field of
study, but a introductory note of a page or two in any book is enough to
explain the differences between the notations.

The grandparent argument is valid for some math below graduate level and some
confusing bits in advanced mathematics but it's generally not a problem for
anything above. I actually think notation is a problem for students that are
not that much interested in mathematics, high schoolers and some engineering
students that I saw in my life generally are confused why some things are
written the way they are without any justification. This is a problem with
learning methods and if you just give a new arbitrary notation to these
students I believe the problem will persist.

I also think that trying to reboot the entire mathematical notation to fit
areas such as aerospace control theory, algorithm complexity theory, abstract
algebra, biostatistics and thermodynamics, among other fields would probably
result in failure, like creating a universal language like esperanto that
would never be used.

~~~
Dn_Ab
Right, that's why the focus is on _introductory classes_. As I noted, yes a
sufficiently motivated individual will get used to the notation in time but
that doesn't mean things are okay.

These things that seem minor to the expert actually make a big difference
before _chunking_ is achieved and can hinder all but the most motivated. If
you are taxing short term memory by using unhygenicly bounded variables then
no, it is not just a matter of who is interested. If you are not pointing out
the difference between higher order functions and regular functions nor
separating the notion of function from application then you are causing
unnecessary representational couplings that create a lot of friction. These
things have real cognitive and physiological costs. The design should
streamline thought for expert and novice alike, it should not be arbitrary.
And in the absence of anything better we cannot say that the issue of notation
is not a problem at high levels. Sure learning is no longer the problem but
what of adroit mental manipulations? I tend to agree with Alfred Whitehead who
said

" _By relieving the brain of all unnecessary work, a good notation sets it
free to concentrate on more advanced problems, and in effect increases the
mental power of the race._ ".

We can't lament the lack of scientists and engineers on one hand and not try
to do reduce uptake friction on the other. There's a real problem with math
education if the experts are not trying to relate to the ones who are
struggling.

btw braket is a wonderful notation in my book and I'm not a physicist, it is
an elegant way of writing sparse vectors.

------
dave_sullivan
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
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?

~~~
dave_sullivan
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.

------
smalter
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
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!

------
btilly
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> 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.

------
T_S_
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."

------
BasDirks
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>

------
spodek
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://joshuaspodek.com/master-
speaks-creative-expression>

------
benmccann
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).

~~~
ColinWright
Certainly hard work is required, but equally certainly, being a genius is not
necessary. I'm a mathematician, and I'm no genius.

~~~
BasDirks
You are not? How does one tell?

~~~
ColinWright
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
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
The minute or so starting at 5m40s in this video is apropos:
<http://video.google.com/videoplay?docid=5935911405946587342>

(You may be interested in the rest of the video as well. It's a fascinating
show.)

------
tokenadult
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/>

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>

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-...](http://rjlipton.wordpress.com/2012/08/09/a-new-way-to-solve-linear-
equations/)

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.

------
the1
there's nothing wrong with being a math genius. there's nothing wrong with not
being a math genius.

------
zakshay
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
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
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.

