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One day I'll retire and go back to school. The idea of learning Math - really learning & understanding Math - as a fun pastime is so appealing. What's stopping

One day I'll retire and go back to school. The idea of learning Math - really learning & understanding Math - as a fun pastime is so appealing. What's stopping me now? That sweet overpaid SDE salary and the endless obligations that come from being an adult. I suspect I am not alone...

paulpauper 4y ago
keep the salary. if the goal it to change the world, to understand reality, to have an impactful life, have a good standard of living, etc. math is one of the hardest ways of achieving that. It's such a saturated field. Almost everything you can imagine has been done to the highest possible degree of abstraction. Every stone overturned except for things which may take a lifetime to even try to understand. Writing a blog post is probably way more fulfilling and also a doable challenge. A top mathematician may spend years working on a result that maybe if he is lucky be worthy of a footnote somewhere. As a field I think math is well past its diminishing returns imho. It's like 'what was the last big philosophical discovery'...yeah...hard to think of one. Maybe the P zombie concept or the simulation hypothesis. But new and important books, fiction and non fiction, are being written all the time.
- bombcar 4y ago
  I think the whole point would be to retire and do math for fun, not desiring anything more than the joy of discovery, no footnotes, no recognition, just math. 99.99% of everyone won't be remembered for their "contributions" so why not do something you enjoy?
  - lupire 4y ago
    Even if no one remembers you, your work is a contribution. Learning for its own sake is a hobby for yourself, like watching TV or reading a book.
    - eyelidlessness 4y ago
      Let me be remembered for watching TV. I would be utterly delighted if that’s my long term legacy. “They finally rested and enjoyed the most banal show imaginable. It was relaxing. Any other lasting contributions will be in other records should you care.” There I’ve written my eulogy.
      - formerkrogemp 4y ago
        My grandfather's last 10 years were spent developing diabetes, watching those three shows, and playing Microsoft solitaire until death. If that's what you want, then, sure, you can be remembered that way.
- andi999 4y ago
  About the simulation hypothesia, how is that different from Descartes evil demon? https://en.m.wikipedia.org/wiki/Evil_demon https://en.m.wikipedia.org/wiki/Evil_demon (apart from saying that the evil demon is a future type computer)
  - postingposts 4y ago
    The simulation stuff is just “Plato’s Cave”:Reloaded for people who never understood the concept of the cave in the first place. At a basic level you can interpret it from Gödel’s incompleteness theorems, which state that systems of logic need some form of observer to function. That observer concept reaches way back across different philosophical domains and authors as well.
    - hither_shores 4y ago
      That is not at all what the incompleteness theorems say. The first incompleteness theorem says that for any consistent formal system T (with a recursively enumerable set of axioms) capable expressing of elementary arithmetic, T can express a statement which it can neither prove nor disprove. The second incompleteness theorem says that T can't prove the statement "T is consistent". (I've still glossed over a number of technical details here; pick up a book on model theory if you want all the messy internals.) First order logic is notably not capable of expressing elementary arithmetic. And observers aren't involved in any way.
      - postingposts 4y ago
        Yes, it is. Sorry if you can’t read deeply into it or something. I’m not posting for grad students. I can sense you just like to correct people. Ahhhh I’m so wrong, you’re right?
        MichaelBurge 4y ago
        I usually think of logics as search algorithms, since that's how the semantics for the meta-language describing them are defined. I guess you could call a Turing Machine implementing the search algorithm for proofs implied by a logic an "observer", since it produces "reachability observations" i.e. proofs.
- xphos 4y ago
  This is what a imagine late stage security after OS mature there stacks and such. The advent/widespread use of robost memory protections like PAC and Cheri are going to be so depressing for those on the offensive. I wish I just had more time to do math like the OP but the saturation of the field especially with people who can deeply understand the abstraction is very very intimidating.
- ratzkewatzke 4y ago
  I'm not going to gainsay your experience, but it doesn't match mine. Much of what you do in graduate school is to discover where the accessible areas of research are--where do we have a foothold, and are making progress, and what are some achievable results? There are big and hairy problems that are bad investments for a young mathematician. I would steer students clear of the Collatz conjecture. But once you get up to speed in your research area, you usually find interesting problems thick on the ground. Tenure-track positions are competitive, but I don't think there are a lack of interesting things to work on.
  - paulpauper 4y ago
    when I worked on math I found that no matter what problem I was working on, someone had already solved it completely or to high level of abstraction than I had. got discouraging after while.
- DecentAI 4y ago
  What was this utter gibberish I just read? I’m absolutely positive you have the least idea of what modern mathematical research entails or is even about. Please refrain from sharing your opinions on subject matters which you clearly lack any understanding of.
- ffhhj 4y ago
  I couldn't wait that much, started researching stochastic processes a few years ago and been developing a theory for supertasks.
- lordnacho 4y ago
  This is why I'm happy to simply read the well established facts of a variety of fields. There's more than enough to learn for a lifetime, and from what I read about research there's a heck of a lot of BS in the way for small incremental gains. Having said that, if something does come along that interests someone, they should try it. A friend of mine is doing his 2nd PhD 40 years after his first, having found his way into it via a love of jazz music.
- nextos 4y ago
  I totally agree with what you say, but there's a lot of low hanging fruit in mathematizing biology. It's not easy, but it's certainly very impactful.
  - 07121941 4y ago
    can you expand on that? sounds really interesting
    - nextos 4y ago
      Drop me an email, contact info in profile :) Otherwise, I will try to come back to this comment in a few days. I'm a bit busy with an urgent deadline, but I'm glad to expand on this later on!
- jamesrcole 4y ago
  just a minor thing to note here: there's the foundations of mathematics... questions like, what are numbers and other mathematical structures? How is it that math, any math at all, can potentially accurately describe parts of reality? Symbolic representations of math are just squiggily lines... why should squiggily lines have any special relationships to the nature of the universe? Details like that are far from being understood.
  - Idiotfucker 4y ago
    maybe I'm misunderstanding you, it sounds like you're just describing abstract algebra. See linear algebra or group theory as topic titles.
foobarian 4y ago
I'm still grumpy that I accepted I knew what real numbers were just because I could recite back the definition given by the teacher. There is so much depth there if you go looking...
- gxs 4y ago
  The way I explain this super simply to people is absolute value. To most people, it just means that when you take the absolute value of a negative number it becomes positive, and the absolute value of a positive number stays positive. Now there is more to it, but how you might think of absolute value instead is as a distance function, particularly how far away from zero you are on a number line. This is way over simplified, but an example of how there can be a little more buried beneath the surface.
- l33t2328 4y ago
  I contend no one understands real numbers if they can’t explain (without resorting to essentially purely symbolic rigor) why you can cover the rationals with intervals of arbitrarily small total length, but you can’t do the same with R.
  - Koshkin 4y ago
    Well, to understand an explanation one must already know something, otherwise first you have to explain those other things, e.g. the simple fact that unlike the reals the set of rational numbers is countable…
    - l33t2328 4y ago
      Cardinality hasn’t much to do with it since there are uncountable sets which you may cover like that.
      - Koshkin 4y ago
        Sure, but for countable subsets (such as the rationals) it is easy to show.
qsort 4y ago
One of my biggest regrets is not having studied (more) math. But would I regret not studying CS had I studied math? You really can't win :/
- qiskit 4y ago
  CS is math...
  - dumpsterlid 4y ago
    In a practical sense, not really.
  - l33t2328 4y ago
    I’d argue math is actually (a proper subset of) CS.
    - Koshkin 4y ago
      But the real numbers cannot be faithfully represented in a computer!
      - l33t2328 4y ago
        So? Neither can the list of possible computer programs, yet CS studies them
        Koshkin 4y ago
        It does not look like CS studies the real numbers.
        l33t2328 4y ago
        Each real number can be represented as a subset of the natural numbers. CS surely studies these.
stocknoob 4y ago
Aim for financial independence on that SDE and you can retire a few decades ahead of schedule. And you’ll have plenty of time for that sweet math learning.
hebrox 4y ago
I'm actually looking into this. Just an hour ago I mailed the local university that I won't be doing any courses there. My initial plan was to do a bachelor at around 50% speed. But working and having to girls (1,5 years and 3 weeks) makes that quite impossible. And looking at photos of myself at 17 makes me feel rather out of place at a university at age 42. The Open University has an AI master that I'm thinking about right now. It has about 25% of the math that I want to learn, so that would be a good start. I did some prep work (an official high school math certificate) last few months and I noticed that I need a schedule to keep me going. One thing that I'm quite certain about, is that doing math is the most important thing. And doing math leads to more doing math.
- ivansavz 4y ago
  RE learning math: you might want to check out some of my (non-free) books on MECH+CALC https://minireference.com/static/excerpts/noBSmathphys_v5_preview.pdf https://minireference.com/static/excerpts/noBSmathphys_v5_pr... and LINEAR ALGEBRA https://minireference.com/static/excerpts/noBSLA_v2_preview.pdf https://minireference.com/static/excerpts/noBSLA_v2_preview.... They are written especially with adult readers in mind. RE schedule to keep you going: I've had some students use the concept maps as a "world map" (like the stage map in Mario World) and check off boxes as you progress through them (each concept corresponds to roughly one section). See https://minireference.com/static/conceptmaps/math_and_physics_concepts.pdf https://minireference.com/static/conceptmaps/math_and_physic... and https://minireference.com/static/conceptmaps/linear_algebra_concepts.pdf https://minireference.com/static/conceptmaps/linear_algebra_... I guess you could time-box these and do N of sections each week to make this into a schedule. Make sure you dedicate lots of time for the exercises/problems, because that's when the real learning happens...
markus_zhang 4y ago
In the same boat. One of my dreams is to go back to school, learn vector analysis, differential equation, differential geometry, classic mechanics, electromagnetic, special relativity and finally general relativity. Actually quite doable as long as one can grit through the Math, some of which do not need a back to back read.
- lordnacho 4y ago
  Why would you need to go back to school to learn those things? At best school is a syllabus telling you what people think you should know, fairly easy to get a hold of, and maybe a bit of accountability to make sure you actually learn it.
  - markus_zhang 4y ago
    Yeah I agree with that. I can also buy a few books and go from there. But essentially O cannot do that now because of X, Y and Z.
stult 4y ago
I’ve had a very similar experience and my solution was to incrementally move into more and more math intensive jobs, starting from regular old SWE working on a web app to working on an aerospace-related web app that involved lots of physics and geospatial calculations and then moving toward MLE/DS jobs. All without a STEM degree, teaching myself the math as I go. It hasn’t been easy but I enjoy what I do more and more over time.
- jtaft 4y ago
  Impressive :)
- darksensei 4y ago
  This is a bit tangential, but was the lack of a STEM degree ever an obstacle in getting those jobs? Just asking because I'm currently in a similar position, regular SWE job, but I'd like to eventually move to something more mathy.
Koshkin 4y ago
Be sure not to retire too late. At an older age, with all the experience you will have under your belt, picking up new concepts and methods will not be a problem, but retaining details in memory will. You have no idea how incredibly hard it will be. So start early.
agumonkey 4y ago
not alone, or more like the vast majority I'm still trying to find a part time dev gig so I can just focus on graphs and advanced combinatorics
lupire 4y ago
Yes, you can't do something if you chose to do something else instead. Clearly you find earning/spending money more appealing than math.
- pontus 4y ago
  What a ridiculous oversimplification of life...
postingposts 4y ago
You can’t really learn Math at school. Not really from the perspective of understanding mathematical beauty deeply. I wish there was an outlet or means to do so. I always found schooling to be woefully inept at assisting in learning the craft.
mbrodersen 4y ago
https://arxiv.org/pdf/1910.09336.pdf https://arxiv.org/pdf/1910.09336.pdf
eyelidlessness 4y ago
My lofty future academic pursuit for fun when I can relax from my professional programming career would probably be compsci and then research if I’m still interested. Gonna get really old and tired working in the semicolon mines, then turn around and learn all the stuff I was supposed to learn when I started. At least that’s my picture of my casual retirement interests.
ifokiedoke 4y ago
Same situation, but I've started! I only spend about 30 minutes every other day, and it's extremely slow-going, but so fulfilling. I had the same goal as you -- _really_ learning & understanding math. I finished pre-Algebra last year and I'm halfway through an Algebra text by Gelfand & Shen now. My friends look at me funny when I tell them I'm re-learning Math from the ground up for fun (esp. with a degree in CS) but it has been so rewarding. I probably won't get to finishing Calculus for another couple years but I'm already having so much fun. Stumbled upon deriving some exponent laws last month by accident and truly understanding the sum and difference of squares has been awesome.
yobbo 4y ago
If your goal of learning is for your own benefit, it's much easier than actually going to school and doing exams, assignments, and getting credits which involves a massive amount of formalities. A good starting point could be MIT OCW 18.01, 18.02, and 18.03. Do all their problem sets and get as much understanding you want. It corresponds to a first year university engineering curriculum.

paulpauper 4y ago

keep the salary. if the goal it to change the world, to understand reality, to have an impactful life, have a good standard of living, etc. math is one of the hardest ways of achieving that. It's such a saturated field. Almost everything you can imagine has been done to the highest possible degree of abstraction. Every stone overturned except for things which may take a lifetime to even try to understand. Writing a blog post is probably way more fulfilling and also a doable challenge. A top mathematician may spend years working on a result that maybe if he is lucky be worthy of a footnote somewhere. As a field I think math is well past its diminishing returns imho. It's like 'what was the last big philosophical discovery'...yeah...hard to think of one. Maybe the P zombie concept or the simulation hypothesis. But new and important books, fiction and non fiction, are being written all the time.

bombcar 4y ago
I think the whole point would be to retire and do math for fun, not desiring anything more than the joy of discovery, no footnotes, no recognition, just math. 99.99% of everyone won't be remembered for their "contributions" so why not do something you enjoy?
- lupire 4y ago
  Even if no one remembers you, your work is a contribution. Learning for its own sake is a hobby for yourself, like watching TV or reading a book.
  - eyelidlessness 4y ago
    Let me be remembered for watching TV. I would be utterly delighted if that’s my long term legacy. “They finally rested and enjoyed the most banal show imaginable. It was relaxing. Any other lasting contributions will be in other records should you care.” There I’ve written my eulogy.
    - formerkrogemp 4y ago
      My grandfather's last 10 years were spent developing diabetes, watching those three shows, and playing Microsoft solitaire until death. If that's what you want, then, sure, you can be remembered that way.
andi999 4y ago
About the simulation hypothesia, how is that different from Descartes evil demon? https://en.m.wikipedia.org/wiki/Evil_demon https://en.m.wikipedia.org/wiki/Evil_demon (apart from saying that the evil demon is a future type computer)
- postingposts 4y ago
  The simulation stuff is just “Plato’s Cave”:Reloaded for people who never understood the concept of the cave in the first place. At a basic level you can interpret it from Gödel’s incompleteness theorems, which state that systems of logic need some form of observer to function. That observer concept reaches way back across different philosophical domains and authors as well.
  - hither_shores 4y ago
    That is not at all what the incompleteness theorems say. The first incompleteness theorem says that for any consistent formal system T (with a recursively enumerable set of axioms) capable expressing of elementary arithmetic, T can express a statement which it can neither prove nor disprove. The second incompleteness theorem says that T can't prove the statement "T is consistent". (I've still glossed over a number of technical details here; pick up a book on model theory if you want all the messy internals.) First order logic is notably not capable of expressing elementary arithmetic. And observers aren't involved in any way.
    - postingposts 4y ago
      Yes, it is. Sorry if you can’t read deeply into it or something. I’m not posting for grad students. I can sense you just like to correct people. Ahhhh I’m so wrong, you’re right?
      - MichaelBurge 4y ago
        I usually think of logics as search algorithms, since that's how the semantics for the meta-language describing them are defined. I guess you could call a Turing Machine implementing the search algorithm for proofs implied by a logic an "observer", since it produces "reachability observations" i.e. proofs.
xphos 4y ago
This is what a imagine late stage security after OS mature there stacks and such. The advent/widespread use of robost memory protections like PAC and Cheri are going to be so depressing for those on the offensive. I wish I just had more time to do math like the OP but the saturation of the field especially with people who can deeply understand the abstraction is very very intimidating.
ratzkewatzke 4y ago
I'm not going to gainsay your experience, but it doesn't match mine. Much of what you do in graduate school is to discover where the accessible areas of research are--where do we have a foothold, and are making progress, and what are some achievable results? There are big and hairy problems that are bad investments for a young mathematician. I would steer students clear of the Collatz conjecture. But once you get up to speed in your research area, you usually find interesting problems thick on the ground. Tenure-track positions are competitive, but I don't think there are a lack of interesting things to work on.
- paulpauper 4y ago
  when I worked on math I found that no matter what problem I was working on, someone had already solved it completely or to high level of abstraction than I had. got discouraging after while.
DecentAI 4y ago
What was this utter gibberish I just read? I’m absolutely positive you have the least idea of what modern mathematical research entails or is even about. Please refrain from sharing your opinions on subject matters which you clearly lack any understanding of.
ffhhj 4y ago
I couldn't wait that much, started researching stochastic processes a few years ago and been developing a theory for supertasks.
lordnacho 4y ago
This is why I'm happy to simply read the well established facts of a variety of fields. There's more than enough to learn for a lifetime, and from what I read about research there's a heck of a lot of BS in the way for small incremental gains. Having said that, if something does come along that interests someone, they should try it. A friend of mine is doing his 2nd PhD 40 years after his first, having found his way into it via a love of jazz music.
nextos 4y ago
I totally agree with what you say, but there's a lot of low hanging fruit in mathematizing biology. It's not easy, but it's certainly very impactful.
- 07121941 4y ago
  can you expand on that? sounds really interesting
  - nextos 4y ago
    Drop me an email, contact info in profile :) Otherwise, I will try to come back to this comment in a few days. I'm a bit busy with an urgent deadline, but I'm glad to expand on this later on!
jamesrcole 4y ago
just a minor thing to note here: there's the foundations of mathematics... questions like, what are numbers and other mathematical structures? How is it that math, any math at all, can potentially accurately describe parts of reality? Symbolic representations of math are just squiggily lines... why should squiggily lines have any special relationships to the nature of the universe? Details like that are far from being understood.
- Idiotfucker 4y ago
  maybe I'm misunderstanding you, it sounds like you're just describing abstract algebra. See linear algebra or group theory as topic titles.

foobarian 4y ago

I'm still grumpy that I accepted I knew what real numbers were just because I could recite back the definition given by the teacher. There is so much depth there if you go looking...

gxs 4y ago
The way I explain this super simply to people is absolute value. To most people, it just means that when you take the absolute value of a negative number it becomes positive, and the absolute value of a positive number stays positive. Now there is more to it, but how you might think of absolute value instead is as a distance function, particularly how far away from zero you are on a number line. This is way over simplified, but an example of how there can be a little more buried beneath the surface.
l33t2328 4y ago
I contend no one understands real numbers if they can’t explain (without resorting to essentially purely symbolic rigor) why you can cover the rationals with intervals of arbitrarily small total length, but you can’t do the same with R.
- Koshkin 4y ago
  Well, to understand an explanation one must already know something, otherwise first you have to explain those other things, e.g. the simple fact that unlike the reals the set of rational numbers is countable…
  - l33t2328 4y ago
    Cardinality hasn’t much to do with it since there are uncountable sets which you may cover like that.
    - Koshkin 4y ago
      Sure, but for countable subsets (such as the rationals) it is easy to show.

qsort 4y ago

One of my biggest regrets is not having studied (more) math. But would I regret not studying CS had I studied math? You really can't win :/

qiskit 4y ago
CS is math...
- dumpsterlid 4y ago
  In a practical sense, not really.
- l33t2328 4y ago
  I’d argue math is actually (a proper subset of) CS.
  - Koshkin 4y ago
    But the real numbers cannot be faithfully represented in a computer!
    - l33t2328 4y ago
      So? Neither can the list of possible computer programs, yet CS studies them
      - Koshkin 4y ago
        It does not look like CS studies the real numbers.
        l33t2328 4y ago
        Each real number can be represented as a subset of the natural numbers. CS surely studies these.

stocknoob 4y ago

Aim for financial independence on that SDE and you can retire a few decades ahead of schedule. And you’ll have plenty of time for that sweet math learning.

hebrox 4y ago

I'm actually looking into this. Just an hour ago I mailed the local university that I won't be doing any courses there. My initial plan was to do a bachelor at around 50% speed. But working and having to girls (1,5 years and 3 weeks) makes that quite impossible. And looking at photos of myself at 17 makes me feel rather out of place at a university at age 42. The Open University has an AI master that I'm thinking about right now. It has about 25% of the math that I want to learn, so that would be a good start. I did some prep work (an official high school math certificate) last few months and I noticed that I need a schedule to keep me going. One thing that I'm quite certain about, is that doing math is the most important thing. And doing math leads to more doing math.

ivansavz 4y ago
RE learning math: you might want to check out some of my (non-free) books on MECH+CALC https://minireference.com/static/excerpts/noBSmathphys_v5_preview.pdf https://minireference.com/static/excerpts/noBSmathphys_v5_pr... and LINEAR ALGEBRA https://minireference.com/static/excerpts/noBSLA_v2_preview.pdf https://minireference.com/static/excerpts/noBSLA_v2_preview.... They are written especially with adult readers in mind. RE schedule to keep you going: I've had some students use the concept maps as a "world map" (like the stage map in Mario World) and check off boxes as you progress through them (each concept corresponds to roughly one section). See https://minireference.com/static/conceptmaps/math_and_physics_concepts.pdf https://minireference.com/static/conceptmaps/math_and_physic... and https://minireference.com/static/conceptmaps/linear_algebra_concepts.pdf https://minireference.com/static/conceptmaps/linear_algebra_... I guess you could time-box these and do N of sections each week to make this into a schedule. Make sure you dedicate lots of time for the exercises/problems, because that's when the real learning happens...

markus_zhang 4y ago

In the same boat. One of my dreams is to go back to school, learn vector analysis, differential equation, differential geometry, classic mechanics, electromagnetic, special relativity and finally general relativity. Actually quite doable as long as one can grit through the Math, some of which do not need a back to back read.

lordnacho 4y ago
Why would you need to go back to school to learn those things? At best school is a syllabus telling you what people think you should know, fairly easy to get a hold of, and maybe a bit of accountability to make sure you actually learn it.
- markus_zhang 4y ago
  Yeah I agree with that. I can also buy a few books and go from there. But essentially O cannot do that now because of X, Y and Z.

stult 4y ago

I’ve had a very similar experience and my solution was to incrementally move into more and more math intensive jobs, starting from regular old SWE working on a web app to working on an aerospace-related web app that involved lots of physics and geospatial calculations and then moving toward MLE/DS jobs. All without a STEM degree, teaching myself the math as I go. It hasn’t been easy but I enjoy what I do more and more over time.

jtaft 4y ago
Impressive :)
darksensei 4y ago
This is a bit tangential, but was the lack of a STEM degree ever an obstacle in getting those jobs? Just asking because I'm currently in a similar position, regular SWE job, but I'd like to eventually move to something more mathy.

Koshkin 4y ago

Be sure not to retire too late. At an older age, with all the experience you will have under your belt, picking up new concepts and methods will not be a problem, but retaining details in memory will. You have no idea how incredibly hard it will be. So start early.

agumonkey 4y ago

not alone, or more like the vast majority I'm still trying to find a part time dev gig so I can just focus on graphs and advanced combinatorics

lupire 4y ago

Yes, you can't do something if you chose to do something else instead. Clearly you find earning/spending money more appealing than math.

pontus 4y ago
What a ridiculous oversimplification of life...

postingposts 4y ago

You can’t really learn Math at school. Not really from the perspective of understanding mathematical beauty deeply. I wish there was an outlet or means to do so. I always found schooling to be woefully inept at assisting in learning the craft.

mbrodersen 4y ago

https://arxiv.org/pdf/1910.09336.pdf https://arxiv.org/pdf/1910.09336.pdf

eyelidlessness 4y ago

My lofty future academic pursuit for fun when I can relax from my professional programming career would probably be compsci and then research if I’m still interested. Gonna get really old and tired working in the semicolon mines, then turn around and learn all the stuff I was supposed to learn when I started. At least that’s my picture of my casual retirement interests.

ifokiedoke 4y ago

Same situation, but I've started! I only spend about 30 minutes every other day, and it's extremely slow-going, but so fulfilling. I had the same goal as you -- _really_ learning & understanding math. I finished pre-Algebra last year and I'm halfway through an Algebra text by Gelfand & Shen now. My friends look at me funny when I tell them I'm re-learning Math from the ground up for fun (esp. with a degree in CS) but it has been so rewarding. I probably won't get to finishing Calculus for another couple years but I'm already having so much fun. Stumbled upon deriving some exponent laws last month by accident and truly understanding the sum and difference of squares has been awesome.

yobbo 4y ago

If your goal of learning is for your own benefit, it's much easier than actually going to school and doing exams, assignments, and getting credits which involves a massive amount of formalities. A good starting point could be MIT OCW 18.01, 18.02, and 18.03. Do all their problem sets and get as much understanding you want. It corresponds to a first year university engineering curriculum.