
Mathematicians are chronically lost and confused (2014) - Halienja
https://j2kun.svbtle.com/mathematicians-are-chronically-lost-and-confused
======
laingc
To me, this is about "mathematical maturity".

My observation is that many programmers, especially those who have come of age
by working in startups, tend to value ability and sometimes experience over
formal education. This is a result, I believe, of noticing that they can
outperform many people who have a classical education, and also seeing that
many of the people to whom they look up also do not have much in the way of
formal education.

However, I truly believe that Mathematics is a discipline that is very hard to
engage with outside of formal education - or at least nobody has really found
a great model for doing so yet.

Learning Mathematics in a classical, structured way really does change the way
you think. I notice a substantial difference even between those of my
colleagues who entered industry straight after their Masters or even
Bachelors, and those who completed Doctorates or even held postdoctoral
positions.

In my opinion, it is this lack of mathematical maturity that makes the switch
from general programming to scientific programming more challenging than the
converse.

~~~
agentultra
I disagree that mathematics itself is difficult to engage with outside of
academia.

There are some academic mathematicians who think mathematics has to be hard
because it was difficult for them to understand. There are others for whom
sharing their insights is more important than searching for new ones. I think
that the set of mathematicians together have done a rather good job in the
20th century of producing mathematical literature and pedagogy that laypersons
and the polymaths can understand.

I skipped university when I was young due to circumstance. I still had a love
for maths but couldn't pursue formal training. When I came around to
programming as a career instead of odd-loner-hobby it was as you say. I was
quite impressed with my ability to code circles around CS graduates who, for
all their theory and training, weren't prepared for "coding in the real
world."

I never stopped studying mathematics. I've recently been going through
temporal logics. If only more programmers were aware of the beauty of the
proof of weak fairness. And its implications in system specifications.

Where I can see formal training being useful is exposing you to subjects you
were not aware of in short order. My experience has been more akin to the
mansion metaphor. Where I encounter limitations or difficulties in my
programming I look to maths for the right tool to simplify the task and ensure
it is correct.

So I agree about "mathematical maturity." I just don't believe it is
exclusively acquired in an academic setting. I think some people have an
inclination or awareness that pushes them to think this way.

~~~
yequalsx
Not knowing your level of mathematical insight and knowledge makes it hard to
know if your belief about your mathematical talent is a self deception. I've
never encountered anyone who understood typical second year graduate level
mathematics without formal training. I know such people could exist. I've just
never met any. I have met people who claimed to be self taught in mathematics
and it was obvious that they didn't really understand the concepts though they
were absolutely convinced they knew what they were talking about.

Outliers exist. But have any good mathematicians been self taught in the last
50 years?

~~~
vinceguidry
> I have met people who claimed to be self taught in mathematics and it was
> obvious that they didn't really understand the concepts though they were
> absolutely convinced they knew what they were talking about.

This is an interesting contrast with my experience with learning programming,
where, it doesn't matter how much formal education / training you get, you're
still going to have to do a massive amount of self-learning before you can
hope to achieve excellence.

All good programmers are self-taught. No good mathematicians (currently) are.
I wonder why this is. Probably boils down to what 'math' is trying to
accomplish vs. what 'programming' does.

~~~
thevardanian
Simple profit incentives.

You can't really make a living off of being a self-taught mathematician,
whereas you can by knowing how to program, and run your own business.

The only way to make a living as a mathematician is to into academia, so that
why you see all mathematician in academics. Basically it isn't that academia
produces mathematician, although it helps, but that it attracts
mathematicians. Just as, say, Y-Combinator attracts good startups.

~~~
NobleSir
As someone who just finished a ph.d. in (non-applied) math I would agree
profit incentives are a large percentage of the reason - in programming, you
can learn enough basic skills to make money, and then smoothly transition that
to higher skill / profit levels just through the experience you gain doing
more actual programming. In math, you can't really make money with a lower
level of math skill (your options of making money at a lower math level are
maybe accounting or teaching, and neither of these provides a transition to
deeper math skill levels). Thus your option to increase your math skill is to
tough it out through 5 years of grad school making about minimum wage, and
then another three years of a postdoc, making a 1/3-1/2 of the average
software developer salary, and then you can start making more money if you are
lucky enough in the problems you choose, and work hard enough.

I would guess the more applied math you care about / the closer to software
your math is, the better you can do as a self-taught. I actually know of one
guy who transitioned from an engineering ph.d. to teaching applied math doing
self-taught / working on fluid dynamics in aerospace, but I expect the
examples of someone doing this in more abstract / pure areas are extremely
few.

------
vecter
I have a great personal story that highlights how long the journey of
understanding mathematics is.

I took linear algebra my freshman year of college. It was the non-math major
course, so it didn't require proofs. I got an A+ in the class. Not just an A,
an A+. I was able to obtain such a high grade by taking tons of practice
tests, and since the actual tests were basically mildly veiled calculations, I
just had to map the question to the right calculation. So for instance, if
after a little interpretation, I figured out that the question was asking for
me to calculate the singular value decomposition of a given matrix, I would
mindlessly compute, check my algebra, and move on.

However, it was very clear to me by the end of the course that I didn't really
understand what the heck linear algebra was about.

Five years later, I started a job as an algorithmic trader. One of the first
things my boss wanted to do was to do a Principal Component Analysis (PCA) of
bond price movements. This is a very common thing to do. I didn't know what
PCA was, but I read a short paper he gave me and I was able to grok it. After
reading that paper and actually performing the PCA (which by the way was
basically one line of R code), I finally came to understand the core essence
of linear algebra, which is the idea of linear transformations. I was able to
connect the equation Ax=lambdax to the geometry of what an eigenvector meant.
Through a little more reasoning, I realized that every real matrix
corresponded to a linear transformation of that space via a rotation, a
reflection, a stretching, a shearing, etc. At that point, all of the mindless
calculations I had been doing half a decade earlier instantly clicked, and I
was enlightened.

This was literally half a decade later after I "aced" my linear algebra class.
I know that it seems absolutely ridiculous that I could "score so well" in a
math class yet so clearly miss the core idea behind the entire class, but
that's been my experience with math for as long as I can remember. You start
by doing the calculations and just getting comfortable with them. Some
arbitrary time later, you have an insight and suddenly everything is so
crystal clear and trivial that you wonder how you could even _not_ have
understood it before.

Oh, and even to this day, I don't understand what singular values actually
are. Something to do with a mapping from the row space to the column space,
blah blah. I'm sure if I spent an hour to read about them and picture the
geometry, I could figure it out, but I just haven't gotten around to doing it.

~~~
wmsiler
Linear algebra is the sort of topic where it seems like you can always go
deeper. I also got an A in my linear algebra class in college. When I got to
grad school for math, I took the first year graduate course on algebra and saw
linear algebra in terms of algebraic things like modules and representations.
I decided that I hadn't really known linear algebra before, but that I did
then. Then I took differential geometry, which involves studying infinite
collections of vector spaces parameterizes by points on a manifold. I realized
I still hadn't know linear algebra, but after that I certainly did. Then I
took a functional analysis class, where we did infinite dimensional linear
algebra, where a lot of the finite dimensional theory has analogues but
everything is more complicated (e.g. instead of finite bases and dot products,
you consider things like Fourier analysis and Hilbert spaces).

The same realization that I don't know linear algebra came when taking a Lie
algebras class, and again when learning homological algebra, and probably a
few other times as well. There are certainly lots of areas of math that I've
never explored that take linear algebra in some other direction (for example,
I don't have any idea what the applied math guys do with....).

It's really an amazingly vast subject, especially considering that it's
usually just thought of as a tool used to study more advanced topics.

~~~
bglazer
I've been learning (trying to learn) linear algebra by studying Gilbert
Strang's "Introduction to Linear Algebra" book. There are some really mind
bending sections in which he explicitly treats a concept like linear
regression in terms of geometry and then algebra and then calculus and then
probability and then ties them all back together.

What's really struck me is when I dive into a Wikipedia rabbit hole of linear
algebra, following links for terms I don't know. I start on the Topology page,
read the introduction section of 10 articles that start with "x is a
generalization of y" and somehow end up back on the Topology page. Shallow
exposure to the sheer volume and conceptual depth of stuff like topology and
algebra has really made me respect modern math.

------
j2kun
Author here.

I can't help but plug my mailing list for a book I'm writing, called "A
Programmer's Introduction to Mathematics." Cheers, and thanks for reading!

[https://jeremykun.com/2016/04/25/book-mailing-
list/](https://jeremykun.com/2016/04/25/book-mailing-list/)

------
te_platt
This reminds me of the book "The Perfect Wrong Note". The book is focused on
learning to play music but the principles it teaches apply to learning just
about anything. The core message to not be afraid of mistakes during practice.
Little kids fall over when they learn to walk, you'll have moments of
confusion learning new things. There's a time to get things done well, like
playing at a recital or releasing production code. There also needs to be time
to practice and part of practicing is the expectation that there will be
mistakes.

------
friendly_chap
I feel the same way when solving tasks in my day job. The thing I tell to
young people learning programming/tech that I hope they don't get frustrated
easily, because they will spend every day of their life feeling rather stupid
and confused, never knowing when will they discover a solution for a
particular problem.

This is something that was a great source of stress early in my career.

~~~
Swizec
It's a game of educated guessing.

Experience helps.

~~~
friendly_chap
Indeed, but the more experienced I get the harder/bigger problems I work on,
so that balances it out. I still feel rather stuck on a daily basis, but I
accept the situation and try to deal with it with a calm focus.

------
jonstokes
A mathematician was walking home from campus one day, and as he walked he was
pondering a particularly thorny problem. At one point, he snapped out of his
reverie and looked around and realized that he had no idea where he was. He
saw a young boy playing with a ball in a yard, and figured maybe the boy could
tell him the way home. So he says to the boy, "young man, do you know where
Prof. So-and-so lives?"

The boy looked at him and said, "Dad, what's wrong with you?"

~~~
selimthegrim
This story is usually told about Norbert Wiener and his daughter.

~~~
jonstokes
Good to know.

------
jondubois
My favourite quote about Mathematics is from John von Neumann: "In mathematics
you don't understand things. You just get used to them." \- This quote
highlights precisely why I ended up choosing software engineering over maths.

I'm just not very good at applying processes/methodologies which I don't fully
understand.

For example, I wasn't very good with linear algebra until I was able to
visualize the equations in my head. For example, now, when I think about the
equation 'f(x) = ax^2 + bx + c' \- I can see that this represents the set of
all possible quadratic equations and I can roughly visualize what that looks
like on a cartesian plane (well it would turn the whole plane black because
there would be an infinite number of graphs). Then if I choose any three
points on that crowded cartesian plane, I can visualize that among this
infinite set of curves, one of them passes through all three points. Thinking
about it in that way allows me to make sense of Gauss-Jordan Reduction and
other mathematical processes related to linear algebra.

Programming is much easier for me because I can visualize the results
instantly on a computer - I don't need someone else to explain it to me. Any
uncertainty can be quickly resolved by simply running some code.

~~~
pixl97
>Any uncertainty can be quickly resolved by simply running some code.

Until you have to interact with a black box of someone else's code. You can
only be certain that the data you sent that particular time works, not that
all possibilities of valid data work.

~~~
jondubois
I guess with programming, you can never get 100% certainty when it comes to
correctness of your code - Unless you perform formal analysis which is
impractical for most use cases.

~~~
chipsy
You might not succeed at 100%, but you can remove "classes of error" by
adopting particular styles or techniques backed by formal analysis. That's one
of the biggest appeals of compiler technology - it can encode an understanding
of patterns proven to detect failure, and in so doing lower your resulting bug
count.

------
kinai
Does anybody know a good guide on where to begin? Resources are not the issue
here, but usually the overwhelming amount of information regarding all those
topics and areas of mathematics. I was always very interested but got
discouraged rather quickly, even after a semester at university. So far my
favorite access to math was through philosophy.

~~~
laichzeit0
Step 1: Read Lockhart's Lament:
[https://www.maa.org/external_archive/devlin/LockhartsLament....](https://www.maa.org/external_archive/devlin/LockhartsLament.pdf)

Step 2: Download the Book of Proof:
[http://www.people.vcu.edu/~rhammack/BookOfProof/](http://www.people.vcu.edu/~rhammack/BookOfProof/)
You read through it and do all the odd numbered exercises (the solutions are
at the end of the book).

Step 3: Get a book called Real Mathematical Analysis by Charles Pugh and you
work through that and attempt as many problems as you can, with a view not to
rush through it, but to expand your mind through each problem.

Step 4: Pick any of these books that interest you the most and do the same:

\- Calculus by Spivak

\- Algebra: Chapter 0 by Paolo Aluffi

\- Linear Algebra Done Right by Axler

By then you should have enough mathematical maturity to know what to do next.

~~~
nextos
I'd also recommend Hubbard & Hubbard for a beautiful and beginner-friendly mix
of algebra and analysis.

My preferred starter kit is Rudin plus Halmos or Axler, but treating Rudin as
a summary. So a helper would be needed, like Counterexamples in Analysis. This
is what Math 55 used to do.

~~~
pmiller2
I don't really recommend Rudin for a true beginner at all (unless, by "as a
summary," you mean not really digging into the proofs themselves, in which
case any good analysis book will do). Rudin will always try to take the most
elegant route to the theorem, regardless if that route goes anywhere near
where the rest of the text has been. The result, for me, has been that many of
his proofs seem to just meander about for a little while until, at the very
end, you arrive at the theorem. It's a bit like driving to work on auto pilot,
and just as disconcerting to me.

~~~
nextos
I should have said as an outline, instead of a summary.

I think the beauty of Rudin is how compact it is. But of course you need an
alternative book to be able to digest it.

------
reachtarunhere
As an undergrad who recently became serious about math (thanks to its
importance in areas I am interested in) this is very inspiring. I have been
trying to grok mathematics for some time and sometimes being too frustrated
with problems I can't handle. I have experienced the phenomena of giving up on
something and coming back to it and finding it trivial. This is exactly what I
needed.

------
riazrizvi
Beautiful article. Love the advice at the end!:

"What’s much more useful is recording what the deep insights are, and storing
them for recollection later. Because every important mathematical idea has a
deep insight, and these insights are your best friends. They’re your
mathematical “nose,” and they’ll help guide you through the mansion."

~~~
abc_lisper
I can't think of a better argument to goad programmers into writing code
comments, especially for elegant code.

For some reason, programmers think good code should be self explanatory. It
doesn't have to - if your insight is outside the flow (or the argument) of
your code.

I see a lot of Java code written like this; if the programmer's job is to
multiply two numbers, they add number a number b times, and say comments are
not needed!

------
auvrw
tao's stages of math is relevant

[https://terrytao.wordpress.com/career-
advice/there%E2%80%99s...](https://terrytao.wordpress.com/career-
advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/)

------
l3robot
What a great blog post! Totally agree with him. And, personnaly, it is why i'm
having so much fun doing maths. Everytime it's a new exploration, a new
challenge. My best math teacher I had was seeing math with this philosophy in
mind and his class was like discovering new lands every time. I'm sure that if
we explained in a way that failing a math problem is as normal and challenging
that failing a Mario Bros Level, more people would be in peace with it.

------
pjlegato
> _Finally, after six months or so, you find the light switch, you turn it on,
> and suddenly it’s all illuminated. You can see exactly where you were. Then
> you move into the next room and spend another six months in the dark..._

What are you supposed to do if you like math and the idea of grokking it, but
you also have a job and a family and can't afford to spend six months
contemplating each room in the mansion?

~~~
dragonwriter
> What are you supposed to do if you like math and the idea of grokking it,
> but you also have a job and a family and can't afford to spend six months
> contemplating each room in the mansion?

Learn to come to grips with the idea that the universe isn't always going to
support your mutually exclusive preferences?

~~~
pjlegato
Yep, that's probably the only option, huh.

------
Koshkin
One of the ways to acquire a taste for mathematics is to try solving
elementary but challenging problems, such as those included in MathCamp's
qualifying quizzes:
[http://www.mathcamp.org/prospectiveapplicants/quiz/pastquizz...](http://www.mathcamp.org/prospectiveapplicants/quiz/pastquizzes.php).

------
pm24601
All they have to do is read this article:
[https://news.ycombinator.com/item?id=11874395](https://news.ycombinator.com/item?id=11874395)

------
MikeNomad
Shouldn't the year the article was written (2014) be included in the title?

------
justifier
any discipline where you are attempting to answer yet to be answered questions
leaves you in a state of chronic confusion and lacking direction

math the same

------
gauruvbose
"mathematicians don’t work like this"? Sure they do. Reading textbooks is
normal. As a mathematician, this post is foreign.

