How does one truly and fully learn Mathematics?

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What i mean by my question is. How do you learn Maths to a very deep and advanced level that you can utilize it effortlessly while practicing it throughout your life? Currently my maths is not as good as i want it to be and i want to learn Maths to a much more deeper level so that i can understand it and use it in my projects or just use it in my life to quickly calculate things when needed.

Maths is a language that is very very useful for calculating many things in life, the technological advances that are available today would not have been possible without Maths. I also am interested in physics and observing/testing the environment that i am surrounded with to see what i can discover.

So if there any skilled Mathematicians, how would you say the best way to fully understand Maths is? Where would one start?

I am about to start a PhD in mathematics, so I could maybe offer some advice.

It really depends what level you're at - but I'd say the most important thing to do as a mathematician is to make sure you understand everything that is going on. At schools in the UK, there's a very large emphasis on just learning the methods (such as how to integrate and differentiate functions) and using them in exams. There is not enough emphasis on understanding why these methods of valid in British curricula, nor is there much about proof - although these are essential in order to succeed at university level mathematics.

If you want to be a great applied mathematician, it's worth learning about different methods, different situations in which they apply, and their limitations. In particular, a good thing to do if you want to boost your mathematical toolbox is to learn to solve differential equations, and also learn a basic mathematical programming language like MATLAB so that you can run numerical algorithms to solve equations and so forth.

If you want to be a great pure mathematician, it's worth looking through proofs of theorems and propositions and making sure you understand every step. It's also a great idea to expose yourself to as many mathematical proofs as you can, as you can then acquaint yourself with different ideas and approaches towards attacking a mathematical theorem. This is where learning methods can be essential - but at the same time you need to be able to be comfortable with abstract algebraic objects and relations between them.

What sort of mathematics is it you want to practice?

Some philosophical questions ...

1. Is math real ?
2. How is math complete with regards to Gödel's incompleteness theorem?
3. Is math objective or subjective (i.e., does it exist outside the human mind) math realism vs math fictionalism?
4. What assumptions about reality are the foundations of math (i.e., the postulatations about reality)?
5. What is up with irrational numbers, how can something that never ends actually exist in the world?
6. How can math "proofs" use the concept of "infinity" that does not exist in reality?

The best part is that YOU have to figure this stuff out think. You are a detective, and have clues. I won't tell you what I think. You can find online some philosophers and mathematicians who have debated these questions.

You may want to start here:
http://en.wikipedia.org/wiki/Philosophy_of_mathematics
https://www.youtube.com/watch?v=TbNymweHW4E

This more philosophy than math, corsera for example offers some courses on university level mathematics, i would start with something like this. The phillosophical basics of math don't offer much mathematical understanding.

@deep-techno thank your advice, it is deeply appreciated and i just want to learn pure Maths, I want to know most methods and theories about Maths. Maths is very interesting to me.

@LoveNotHate Thank you, I know that it is me that has to find out more about Maths so that is why i asked this question on this forum because there are no doubt very experienced and skilled Mathematicians on this forum who know a lot about Maths so it would make sense if i hear from them, but i am quite a philosophical person and i those philoshipical questions are interesting so thank you .

Good question! I have been struggling with this too, in part because it is time to take my GRE but also because as I get older I find my brain is "shifting" toward an analytical mode of thought which wasn't present when I was younger, perhaps because many mathematical concepts were too abstract for me at the time, and I was better with things I could observe. So thanks for asking! The replies are useful to me, too.

One suggestion regarding creative ways to observe, understand and apply maths: find a copy of the BBC documentary, "The Code." It is outstanding! I saw it a few years ago and just ordered a copy for my son. If someone had explained to me where all these abstract concepts come from, in the way that is described in the video, I know that I would have been much more keen to learn maths at an earlier age. Unfortunately in school the presentation seems very arbitrary, so if you aren't naturally a numbers addict, the lessons seem overwhelming.

If you go to www.thegreatcourses.com there is a company by this name which offers some maths lessons. All university level, by accredited professors. I have not used them yet but have heard positive recommendations. They are not inexpensive but they seem to run regular sales, so it's possible to get a good quality lecture course for under $50. I mean to try one this fall, just haven't decided which

@Naturalist Thank you ! I will definitely check those out! I give you the best graves on your journey to learn Maths as well !

It may help to polish up on you're basic problem solving skills. Some books I recommend to this end include:

How to Solve it by G. Polya

Of Course! The Greatest Collection of Riddles and Brainteasers for Expanding You Mind! by

Zack Guido (Available as an ebook).

Any book with SAT style math problems will also help develop you're problem solving and basic math skills.

Also, for an easy to read introduction to all aspects of math, including higher math, I highly recommend The Math Book by Clifford A. Pickover.

For a free study of math, go to https://www.khanacademy.org/. They have all kinds of courses for free. I've looked at it and it seems like a very good resource...

Maths is a very large field and isn't one subject per se - the term is more of an umbrella. You'll find someone whose speciality is Combinatorics, someone whose speciality is Eisenstein primes etc - in much the same sense that a Classical violinist may know nothing of Jazz - despite the similarities in language.

I don't think any mathematician would describe their efforts as effortless.

This sounds more like arithmetic than mathematics.


Or you could save yourself thousands of years research and simply read/try your best to understand the papers that already exist on the subjects you have an interest in? It would save you a lot of time: no point creating a map for somewhere that is already mapped (unless the process of self-discovery is what you're looking for).

Any mathematician will tell you it's impossible to 'fully understand' maths.

Where to start RE your own journey:

1) Establish what your specific goals are
2) What is your present reality

create a manageable approach for getting from a-b.

If you try to 'learn maths', you'll probably give up after a few weeks. However, if you try to learn specifics i.e. trig, pythagoras theorem, calculus etc, you'll far more likely achieve your goals, as the goal in itself consists of nothing more than the individual components/constituent parts that act as the 'elements of the field'.


You should have a read through Terry Tao's blog - very humble, and he has articles for the layman.

But yes - first establish what you actually would like to be able to do. The more specific you are, the more realizable the outcome will be.

If your goal is to be able to do calculations faster in your head, you don't need to bother wasting your time studying calculus.
If you wish to understand Lattice Theory, studying the Arf invariant of a knot would be something of a diversion.

I am not qualify to answer your question.

The learning of Maths is learning 'to think', not just the learning of formulas.
Learning to think means everything, learning formulas is far less important.
The art of Maths is the art of thinking and solving problems....the formulas are merely the way one communicates what one is thinking.
To fully understand Maths it is essential to understand how Maths philosophically and historically evolved.

I hope these comment find purchase within you.
Be well.

If you want to learn maths for real world /physics purposes then I suggest learning calculus 1-3 (of course) , linear algebra, Fourier analysis, and ordinary and partial differential equations. When doing this, just ask yourself "why is this theorem true " and try to prove it yourself before reading the proof in the book- that way it won't seem like a bunch of steps and manipulations that seem to magically lead to the result. That way you can have an intuitive grasp of why something is true and what the theorem is saying.

Now that is for physical/real world/applied applications. Now if you want to understand pure mathematics "deeply" that is an entirely different question.I would suggest you read books on abstract algebra and real analysis. This would get you into topics such as groups/rings/fields, continuity, limits,sequences, compactness, countability/denumerability, etc. that are important to higher mathematics. From there, what you read would be up to you and would depend on what you want to learn about in mathematics. It branches into so many very different fields. Interestingly, a concept used in almost every mathematical text is the concept called a "set" - but most texts make no attempt to even describe what a set is and what is allowable in set theory- they just say "provided there is a set S". Actually the idea of a "set" troubled mathematicians for some time in the early 20th century because some formulations of set theory led to paradoxes because they allowed for sets like this : "let S be a set of all sets that do not contain itself" (find the paradox-does S contain itself?). If you want to study this , as it is an interesting topic-espescially if you want to study the very foundations of mathematics- read a book on Axiomatic set theory.

The 'Constructivism (philosophy of education)' may be of interest. Focus on the Math-Wars: Discovery-Based Teaching Techniques."

It's helpful to be brilliant (something I'm not).

It depends what you're interested in. I took tons of math courses in college. Most of the time I was just cramming for tests. There is so much math to be known, unless you are really brilliant, it is never going to be effortless. Just focus on on an area that interests you and try to learn the math behind it (if you have a lot of free time or something).

Honestly math gets very very very tough at the upper levels. It's never going to be easy.

As much as I (sort of, sometimes) enjoy math, most people can get by in their day to day life with basic arithmetic. There are a lot of very brilliant people working on very difficult problems and I will leave the difficult problems to them.

Basic algebra is pretty much all you need to know for a foundation For this all you really need is a good textbook.

I would guess that you aren't a mathematician.

I was going to recommend this myself... From where I am currently (pre-algebra ) through to the higher and more arcane forms of calculus, it's all there and it's all free
I'm finding it immensely enjoyable and it's quite confronting come up against things such as my old nemesis, long division -- as soon as I started having to do those kinds of problems I had to address the reasons that I gave it up in the first place... ie, it's difficult and makes my head hurt.

I think you've got some great reasons to want to learn more about it; personally I want to use the skills to make more and more elaborate generative art in programs such as vvvv.

They also have a programming component; I watched a youtube example of somebody coding a Mandelbrot fractal in Khanacademy's in-browser programming environment; pretty much exactly what I want to do. The logic is still beyond me but I am confident that I'll get there. I liken my own motivations to the reason people do Sudoku puzzles and such, just on a broader scale.