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.
Mathematicians are chronically lost and confused (2014)
11–20 of 147 posts
Re: Mathematicians are chronically lost and confused (2014)
#12Does 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.
Step 1: Read Lockhart's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament....
Step 2: Download the Book of Proof: 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.
Re: Mathematicians are chronically lost and confused (2014)
#13To 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.
Re: Mathematicians are chronically lost and confused (2014)
#14Does 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.
Since philosophy was your favourite, you might enjoy the novels of Greg Egan. As far as I can tell, he puts good mathematics in his stories.
Re: Mathematicians are chronically lost and confused (2014)
#15I 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.
I always thought that in the end, one has got to like facing these hard problems every day on the job in order to thrive in this field (I am in tech too). This is something I try to detect in candidates in job interviews I conduct.
I noticed people can be roughly split in two categories: -- ones who spend a couple of years dealing with this sort of job activities, then want out by any means, on to something different; -- others enjoy it more and more as they gain expertise, start having more and more fun -- get more expertise, and with results (which necessarily come up), the right to pick and choose the subjects and so the hard problems that come with them.
I happen to belong to the second category (and it has been a while I stay in the same domain), and I don't believe someone in that category would feel stupid and confused. it rather feels like investigation every time.
Though over the years, I've become wary of the stress always associated with being put on such issues -- a last-minute demo for a trade show that doesn't work, a customer who has escalated to upper management, a delivery which is hopelessly late... -- with all attention of the management attracted. I keep telling myself that a soldier has got to participate in war campaigns, not be doing paperwork at the headquarters, and literally force myself in last years.
But I don't recall I have ever felt stupid or confused, even in the beginning of the career.
Re: Mathematicians are chronically lost and confused (2014)
#16I 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.
Re: Mathematicians are chronically lost and confused (2014)
#17I 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 m…
Wow, that's oddly similar to my story.
LinAlg in college, nothing but a blur of matrices.
Algo trading for work. Now there's a reason to do it, it makes sense.
About SVD, btw, it is another path to PCA, one that solves certain problems that PCA does not.
https://jeremykun.com/2016/04/18/singular-value-decompositio...
Re: Mathematicians are chronically lost and confused (2014)
#18Earlier quoted context omitted.
a) get started on something - if you can't decide, roll the dice and pick something at random b) ignore everything else until you are done c) repeat Don't worry that at your pace it will take ages - very soon you will develop an idea how to rank what you should look at next. If you don't, go back to a). Obviously it's useless to think about it too much when your knowledge about a subject consists mostly of holes and…
But some areas require knowledge base of others
Which you won't find out unless you start - somewhere, anywhere. How are you going to find out when you just sit there and think about something you know nothing about? Wait for sudden enlightenment? Math is one off the easiest subjects to get started, soooooo many starter books and by now even great online courses (again: Khan Academy, for the basics, probably many more). So much guidance. There is so much out there - free and easily accessible thanks to the Internet, the problem isn't "how do I get started" but "what resource do I use to get started".
Re: Mathematicians are chronically lost and confused (2014)
#19tao's stages of math is relevant
https://terrytao.wordpress.com/career-advice/there%E2%80%99s...
Re: Mathematicians are chronically lost and confused (2014)
#20Does 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.
Step 1: Read Lockhart's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament.... Step 2: Download the Book of Proof: 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, w…
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.