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Ask HN: How or where to begin learning mathematics from first principles?

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Re: Ask HN: How or where to begin learning mathematics from first principles?

#71

Post by ivansavz »

I recommend you start with a review of all the topics from high school math which are not clear to you, e.g. functions, solving equations, geometry, and algebra. This may take some time, but it's totally worth it. Building your math knowledge is like building a house---you want to start from a solid foundation.

Next, the traditional "pillars" of STEM are calculus and mechanics. Calculus will beef-up your skills for understanding and manipulating function. Mechanics is important because it teaches you about modelling real-world phenomena with mathematical equations.

Perhaps of even greater importance are the subjects of probability and linear algebra. Probabilistic reasoning and linear algebra techniques (e.g. eigendecomposition) are used for many applications.

RE problems, I think you should reconsider your stance about that. It is very easy to fall into the "I learned lots of cool stuff today" trap, where you think you're making progress, but actually you haven't integrating the knowledge fully. Solving problems usually will put you outside of your comfort zone and force you to rethink concepts and to form new "paths" between them. That's what you want---ideally the math concepts in your mind to be a fully connected graph. Speaking of graphs, here's a concept map from my book that shows (a subset of) the links between concepts from high school math, physics, calculus, and linear algebra: http://minireference.com/static/tutorials/conceptmap.pdf

Good luck with your studies!

Re: Ask HN: How or where to begin learning mathematics from first principles?

#72

Post by chris_wot »

Here is what I've read so far (I've been starting from first principles). Basically, I've been reading through the series "Humungous Book of x Problems", having read (and about to read) the following:

1. Humungous Book of Basic Math and Pre-Algebra Problems

2. Humungous Book of Algebra Problems (still actually going through this)

3. Humungous Book of Trigonometry Problems (up to vectors)

4. Will be reading Humungous book of Calculus Problems soon and then the Humungous Book of Geometry Problems.

I know this sounds slanted, but I'm a big fan of these books. However, I also had to do what you did with Trigonometry - one of the extremely irritating things is that nobody teaches you why the names are sine, cosine, tangent, cotangent, secant and cosecant. For me, I had to go to do some elementary research and draw the lines on a circle to understand that sine was "gap" or "bow" (corruption of the original Arabic), tangent was based on tangens (to touch) and secant was based on secans (to cut). Once I worked that out, actually everything started to get rather a lot easier. Sure wish I'd had easy Internet access in high-school and a more adventurous intellect!

Re: Ask HN: How or where to begin learning mathematics from first principles?

#73

Post by eru »

ivansavz wrote: post #71

I recommend you start with a review of all the topics from high school math which are not clear to you, e.g. functions, solving equations, geometry, and algebra. This may take some time, but it's totally worth it. Building your math knowledge is like building a house---you want to start from a solid foundation. Next, the traditional "pillars" of STEM are calculus and mechanics. Calculus will beef-up your skills for u…

All good points. By the way, if you go from first principles, do measure and integration theory before probability theory.

Re: Ask HN: How or where to begin learning mathematics from first principles?

#74

Post by jpfr »

nextos wrote: post #65
jpfr wrote: post #59

Wanting to learn mathematics from "first principles" brought a lot of comments from graduate-level mathematicians. While their advice applies very much for mathematics students, I can't recommend going down that road for engineering types. In mathematics, everything is connected. One can build up a specific topic from first principles only. But with a too narrow focus one looses these lovely connections between diffe…

Is Hubbard & Hubbard an engineering-type text? I was under the impression that it's very rigorous. It was even used as a textbook at Harvard Math 55.

As a mathematics textbook it of course provides rigorous proofs. But it doesn't neglect to develop intuitive insight (that can be less rigorous yet still very effective).

Here's a quote given in chapter two. That's some motivation to develop more mathematical maturity as an engineer!

  In 1985, John Hubbard was asked to testify before the
  Committee on Science and Technology of the U.S. House of
  Representatives. He was preceded by a chemist from DuPont,
  who spoke of modeling molecules, and by an official from the
  geophysics institute of California, who spoke of exploring
  for oil and attempting to predict tsunamis. When it was his
  turn, he explained that when chemists model molecules, they
  are solving Schrödinger’s equation, that exploring for oil
  requires solving the Gelfand-Levitan equation, and that
  predicting tsunamis means solving the Navier-Stokes equation.
  Astounded, the chairman of the committee interrupted him and
  turned to the previous speakers. “Is that true, what
  Professor Hubbard says?” he demanded. “Is it true that what
  you do is _solve equations_?”

Re: Ask HN: How or where to begin learning mathematics from first principles?

#75

Post by sudorank »

I started to learn maths again in university. The library was a great place to learn the history of it (I started to get into the history of Encryption)

As for learning how to do stuff with maths. I'm a huge fan of being taught it - then again i'm the sort of learner who really gains when showed how to do something and then left to practice.

Re: Ask HN: How or where to begin learning mathematics from first principles?

#76

Post by DavidWanjiru »

nicklaf wrote: post #64
begriffs wrote: post #55

If you really want to go back to first principles, try "Foundations of Analysis" by Edmund Landau. It builds the integers, fractions, Dedekind cuts, and the real and complex numbers from scratch. It's totally rigorous and starts from, "the ability to read English and to think logically -- no high-school mathematics, and certainly no advanced mathematics."

I would be very careful before unleashing a beginner on this book. It would be too easy, IMO, for the reader to end up with the wrong idea that mechanical proofs like the ones in this book are all that are needed in mathematics, since it's possible to get as far as the real numbers (or complex numbers) with so little geometric intuition. Furthermore, the real numbers are the most concrete, familiar setting to do anal…

Are you on twitter, nicklaf? You sound like someone I'd want to be following.

Re: Ask HN: How or where to begin learning mathematics from first principles?

#77

Post by learnstats2 »

I've worked with several adult learners on this but almost no adult is able to learn math just for fun - unless they have a project or crucial examination to work towards. But, you do have a project. You should use that.

I think you're mistaken that your existing tactics won't get you further. This is how most people learn, by trying things out and building on them until they understand what works and what doesn't - standing on the shoulders of giants.

For practical purposes, your satisfactory solutions are a great piece of learning and the start of your understanding. You're maybe experiencing some discomfort about it, but that's normal. Keep going!

P.S. You said "from first principles", which has a specific meaning in math. It's a kind of philosophical ideal in math that you start from nothing (forget about high school math) and carefully and precisely build the subject of mathematics on top. Some of the answers here picked up on that phrase, but it probably isn't relevant to the other interests you mentioned - to do programming or electronics, you will want to build on the knowledge that you have learned already.

Re: Ask HN: How or where to begin learning mathematics from first principles?

#80

Post by tjl »

kellros wrote: post #34
haxiomic wrote: post #4

Try http://www.khanacademy.org (free), their math series starts from basic arithmetic and walks all the way through to undergrad-level mathematics. I personally preferred khanacademy to my math teaching at school and it's been handy during my degree. For more advanced stuff i've found Stanford's online courses ( https://www.youtube.com/user/StanfordUniversity/playlists ) and MIT OpenCourseWare ( http://ocw.mit.edu/in…

I second this and would strongly advise this approach. The courses available on Khan Academy help you visualize the math and gain a better understanding on the 'why' (reasoning) while also teaching you the 'how' (application). There's sufficient math courses available to teach you everything from pre/primary school arithmetic to first year university/college level calculus/linear algebra.

I hope that their math section is better than some of their engineering courses. See the critique at,

http://www.leancrew.com/all-this/2012/12/khan/

I've also been involved in teaching similar material as Dr. Drang and agree completely with his critique.

I've come across students who've had similar sloppy teaching and had to re-teach material so they could unlearn what they'd learnt and get a proper foundation for moving forward. Consistently, they would have very poor assignments for the first few weeks until they had that foundation.

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