9 ms·

Any idea where to start? I've always found it extremely challenging, in regards to both motivation and direction.

moralestapia 7y ago
I don't like giving you a vague answer to this but math is truly vast so it depends a lot on what you're doing. Tell me more and I could give you more suggestions. In my particular case, I started with Linear Algebra and that led me to some Applied Geometry and now I'm reading some (basic) things about Topology as well. I've also read a lot about Category Theory, in part thanks to HN as it keeps surfacing here often.
hamiltonc 7y ago
Statistics and probability. I've lost count how many times I inferred stuff by common sense and hunch, just to be proved wrong. Our intuition is mostly off for statistical phenomena like confidence levels, odds of something happening again given some sample, some event being (statistically) relevant or not, etc.
- ahpearce 7y ago
  I have a Bachelor's and Master's in stats, more or less, but my confidence is still extremely low.
sn9 7y ago
It depends both on your current background and your goals. What's the highest level of math that you know so well you could walk into any classroom and teach it? For many, it's been so many years since they've had to use anything from even middle school algebra that they might have to start there. Make sure you're building on a strong foundation if you wish to build something grand that persists. You want to find the highest level that feels too easy, and then gradually increase difficulty from there. You can use sites like Khan Academy to find that point. Alternatively, you can just embrace the fact that you may be overconfident and work through books that seem "below your level" just to verify that you really have mastery over the subject matter. You could spend a year or two going through the Art of Problem Solving books which cover pre-collegiate math at a deeper level and use SRS like Anki to schedule review of concepts and problems you've understood and solved. At the college level, assuming you're a programmer, the first subjects to master are probably how to write proofs well, calculus in single and many variables (e.g., Spivak followed by Hubbard and Hubbard), discrete mathematics (e.g., Rosen or Epps), linear algebra (e.g., Strang, Axler, Shilov, etc.), and statistics and probability. Hang out on math forums like the math subreddit and you'll see many books recommended over and over again. Be sure to examine the preface and introduction of any book you're thinking about buying. They usually include a section on the intended audience and what their background should be to productively engage with the text. The most important thing is that you solve or attempt to solve many problems, and avoid looking at the solution until you've really spent a good deal of time banging your head against the wall trying to figure it out. How much time you should wait scales with the difficulty of the subject (e.g., less than an hour for high school level material, a week or more for advanced undergraduate level material, etc.).
- ahpearce 7y ago
  Appreciate your response to this. I guess it's tough to go back and try to pick things up and move forward from a particular point. I feel like getting a much more solid grasp on Probability and Linear Algebra would be the most beneficial to my profession (data science), but again, not sure that I have the motivation.
  - sn9 7y ago
    It absolutely wasn't my intent to discourage you! The way I think of it, we all likely have decades left in our career. Consistently spending a few hours per week studying mathematics or some other technical subject related to our goals will only take a few years to reach a non-trivial level and then you still get to enjoy decades of returns off that initial investment of time. How much more effectful will you be with a deep understanding of the fundamentals of your craft? How much more fulfilling will your work be when you have a top X percentile set of skills within your field or a unique combination of skills that give you unique insights or career leverage? And that doesn't even touch on the aesthetic beauty of the subject matter itself! How will it affect the way you view and understand the world around you? Remember the lessons of compound interest: a small increase of investment consistently made early on will likely have outsized effects down the line.

I don't like giving you a vague answer to this but math is truly vast so it depends a lot on what you're doing. Tell me more and I could give you more suggestions. In my particular case, I started with Linear Algebra and that led me to some Applied Geometry and now I'm reading some (basic) things about Topology as well. I've also read a lot about Category Theory, in part thanks to HN as it keeps surfacing here often.

hamiltonc 7y ago

Statistics and probability. I've lost count how many times I inferred stuff by common sense and hunch, just to be proved wrong. Our intuition is mostly off for statistical phenomena like confidence levels, odds of something happening again given some sample, some event being (statistically) relevant or not, etc.

ahpearce 7y ago
I have a Bachelor's and Master's in stats, more or less, but my confidence is still extremely low.

sn9 7y ago

It depends both on your current background and your goals. What's the highest level of math that you know so well you could walk into any classroom and teach it? For many, it's been so many years since they've had to use anything from even middle school algebra that they might have to start there. Make sure you're building on a strong foundation if you wish to build something grand that persists. You want to find the highest level that feels too easy, and then gradually increase difficulty from there. You can use sites like Khan Academy to find that point. Alternatively, you can just embrace the fact that you may be overconfident and work through books that seem "below your level" just to verify that you really have mastery over the subject matter. You could spend a year or two going through the Art of Problem Solving books which cover pre-collegiate math at a deeper level and use SRS like Anki to schedule review of concepts and problems you've understood and solved. At the college level, assuming you're a programmer, the first subjects to master are probably how to write proofs well, calculus in single and many variables (e.g., Spivak followed by Hubbard and Hubbard), discrete mathematics (e.g., Rosen or Epps), linear algebra (e.g., Strang, Axler, Shilov, etc.), and statistics and probability. Hang out on math forums like the math subreddit and you'll see many books recommended over and over again. Be sure to examine the preface and introduction of any book you're thinking about buying. They usually include a section on the intended audience and what their background should be to productively engage with the text. The most important thing is that you solve or attempt to solve many problems, and avoid looking at the solution until you've really spent a good deal of time banging your head against the wall trying to figure it out. How much time you should wait scales with the difficulty of the subject (e.g., less than an hour for high school level material, a week or more for advanced undergraduate level material, etc.).

ahpearce 7y ago
Appreciate your response to this. I guess it's tough to go back and try to pick things up and move forward from a particular point. I feel like getting a much more solid grasp on Probability and Linear Algebra would be the most beneficial to my profession (data science), but again, not sure that I have the motivation.
- sn9 7y ago
  It absolutely wasn't my intent to discourage you! The way I think of it, we all likely have decades left in our career. Consistently spending a few hours per week studying mathematics or some other technical subject related to our goals will only take a few years to reach a non-trivial level and then you still get to enjoy decades of returns off that initial investment of time. How much more effectful will you be with a deep understanding of the fundamentals of your craft? How much more fulfilling will your work be when you have a top X percentile set of skills within your field or a unique combination of skills that give you unique insights or career leverage? And that doesn't even touch on the aesthetic beauty of the subject matter itself! How will it affect the way you view and understand the world around you? Remember the lessons of compound interest: a small increase of investment consistently made early on will likely have outsized effects down the line.