u/Ancient_Yoghurt2481

I recently read a mind blowing story about "Alexander Grothendieck" . When he first enrolled in university he knew very little advanced mathematics and possessed only a basic, standard high school education. He felt his textbooks and lectures were insufficient so he tossed them aside and without using any advanced books or references , he spent three years in absolute isolation rediscovering mathematics from scratch. He had no idea that his solitary notes perfectly duplicated the famous Lebesgue integral and measure theory!

A remarkably similar case is Srinivasa Ramanujan. Lacking any formal university education, Ramanujan’s only window into advanced mathematics was a single, proofless reference book. Working alone with just a piece of chalk and a slate, he independently derived thousands of complex theorems, completely unaware that European mathematicians had discovered them generations prior.

These cases make me question how we are doing math and it feels very strange that how can someone do all this without any guidance. I know these two were once in a century geniuses but they succeed due to lack of resources that forced them to rediscover on their own. Nowadays there is an abundance of resources and nobody is focused on rediscovering (It might feel like a waste of time and extremely slow) , there is more focus on consumption of knowledge. While it is true that they were geniuses it is also true that nobody wants to follow the path of these people since it's slow and risky.

I want to ask is it really possible to do math with a minimum use of textbooks ? Am I deriving a wrong conclusion from these stories? I am interested in this because we have been conditioned to follow textbooks line by line and follow lectures from a tutor or teacher. What are your opinions on this and has any one of you have tried doing mathematics this way? Or any different point of views you have discovered on learning math. Feel free to share whatever opinions you have.

Hi I am in 12th grade and I find people around me are not really interested in mathematics and those who claim to be interested in mathematics are the ones who only like solving problems and are focused on using mathematics as a tool for other subjects of their interests. They are busy solving the problems proposed by others and are not interested in finding answers to their own problems. I don't think mathematics is all about solving problems, I am more interested in asking my own questions, try to rediscover something on my own and rebuild a theory or concept. If the theory is totally understood and if I build a strong theory the problems become trivial. I don't think anyone around me is interested in seeing the essence of mathematical structures and the faces of these structures. No one is interested in finding new point of views that lead to a beautiful vision.

Most of the students are focused on competitive exam like Olympiads, JEE, ISI, CMI entrance etc. I think these competitive exams are the main reason behind lack of interest in mathematics. In class 11th I was preparing for JEE and joined a coaching institute where the focus was only solving problems in limited time , teachers were teaching to solve Logarithm problems but they never really taught "What is Logarithm?" , set thoery was considered trivial and problems were solved using tricks and nobody would ever question these tricks. We were told to solve 30-40 problems in mathematics everyday, this is just memorisation of patterns and there is no creativity in it, this makes the mind dull and weary. Competition is just murder of curiosity whether it is olympiads or JEE.

The indian books are very poorly written they just state Theorems and examples , students use them just for problems because they are only told that mathematics = solving problems . Students are not interested in reading books and not only mathematics the same applies for physics or chemistry since coaching institutes are their cradles. Students are not even interested in reading the theory from the book they just skip to examples or exercises, this is the definition of their "Self study" . They aren't even solving problems properly they are just using the patterns they learnt from a problem they memorised earlier. The problems proposed in these books are also not thought provoking and they don't require creativity to solve them. These books are not even written for teaching students they are written for "JEE mains" , "JEE advanced" , "Olympiads" the tougher the problems and relevant to the exam the better the books is considered. Many Foreign book authors like Apostol, Spivak and many more actually write to teach and the problems are really good, there is an importance of proof writing unlike Indian books. Indian books don't even teach completeness axiom for calculus. These books are of no use to understand the subject and they don't even have any historical notes , I find it interesting to read about mathematicians and the minds who discovered these mathematical ideas but unfortunately these books are not interested in making the students aware about these minds.

Mathematics is not a hurdle to be cleared; it is a language of absolute truth. By stripping away the proofs and the axioms, students are taught the grammar of a language they will never be able to speak. We may win the race to the IITs or top colleges, but if we lose the ability to think critically and creatively, we have lost the subject entirely. This is not education, it is a betrayal. By reducing the infinite architecture of the mind to a series of tricks and timed drills, we are murdering our curiosity. We must burn the manuals of pattern recognition and return to the solitude of the blank page. We must refuse to be 'problem solvers' for an empire of exams and dare to be dreamers of structures. It is better to fail in the pursuit of a single beautiful axiom than to succeed in a system that demands we stop thinking altogether.

In my previous post I mentioned about learning calculus from Tom Apostol calculus and it is a rigorous introduction to the subject. It is my first encounter with proof based mathematics , in fact I didn't even knew that you need to learn something called "Proof writing" before touching a rigorous text ( many people recommend this) . I just searched about a good calculus book on the net and Tom Apostol's book was there , I read the introduction of this book and it was very interesting so I continued . I didn't knew about Proof writing because I was never educated about it and there are gaps in my learning because of bad schooling ( I am still in high school ) and no mentorship or guidance therefore i am largely self taught. I also didn't liked the teaching in school and was more interested to find answers to my own questions.

Some people suggested me that i should first learn to read and write proofs before reading a rigorous text and suggested books like "How to prove it" by Daniel J velleman. This idea does not interests me and to be honest I am very lazy to read another 200-300 pages book on this proof writing instead of learning real mathematical subjects , I prefer to learn these methods of proof writing as I go through my mathematical adventures like building a bridge while crossing it instead of reading a seperate book on this subject. In fact I learnt a "method of Sherlock Holmes" from Tom Apostol ! Where we eliminate everything which is impossible or incorrect until we are left with truth, I want to continue with the calculus book (since I am enjoying it) instead of reading a seperate proof writing book.

I would like to hear from you all and get an advice on this topic. Please feel free to express anything you feel relevant to this topic and at last answer my question "is a dedicated proof writing book necessary?" Thank you for reading.

Hi i am a high school student I recently started learning calculus from Tom Apostol vol 1 there he mentions that we don't use the concepts of infinitesimal anymore. But why? When Leibniz was able to solve all the problems and it was logically sound then what's the point of introducing limits? Why don't we use the original concept created by the father of calculus himself ? Why do we now rely on limits?