Foundations A rigorous approach to learn Mathematics

[tuxscholar]

Hello Intellectuals! So far it seems to be reasonable to learn mathematics in a rigorous way by not solely considering the techniques of problem solving or the applications of a particular subject or concept.

Also to truly appreciate the beauty of mathematical endeavor one need to learn the reasoning behind the origination of concepts in mathematics, so as a beginner it appears to be worthwhile to learn the highly abstract aspects of mathematics like proofs, logic, and topics in pure mathematics (number theory, analysis etc). To be able to acquire such a decent sophistication in mathematics one need to learn the works of the masters like that of Euclid, Euler, Gauss and Hardy, as one says "study the masters not their pupils".

So it's not an academic preaching or pedagogical prescription of any sort but an approach to learn the value of the subject which most of the time seems to be useless (especially pure math). But as the saying goes that beautiful things are useless and useful things are ugly, so it's absolutely worth it to learn pure math.

So as a beginner is it good to start one's mathematical endeavor to learn pure math in a rigorous way from the very start with the help of this works :

• Elements by Euclid
• Elements of Algebra by Euler
• Gelfand's Algebra, Trigonometry and calculus of variations
• Introduction to the Analysis of the Infinite by Euler
• A Course of Pure Mathematics by G. H. Hardy
• Thomas' or Spivak's Calculus
• .....well what can be a decent and rigorous but comprehensive text to learn proof theory or logic as a beginner

Also what would be your approach as a beginner to learn pure mathematics?

 

[.Scott]

I would suggest you start by checking out the work of Procrustes, who created a bed that was the perfect fit for anyone.
* Don't set out a curriculum for yourself that you would hate.
* Recognize that there is no single "Mathematician" target.

The "very start" is no older than age 11 or 12. If you're older than that, you have already started.

From about 2nd grade through college, I was always very good at Math and I was a Math major in college. But I have never considered myself to be a "Mathematician". I latched onto computer programming in High School and that has been what I have followed ever since. Software Engineers do not think about problems in the same way as Mathematicians. And, neither Software Engineers nor Mathematicians think about problems in the same way as their peers.

And it's worth mentioning, especially in these Forums, that Mathematicians and Physicists do not think about problems the same way either.

 

[mathwonk]

Please forgive me for reminding you that it is much easier to make a list than it is to read profitably through that list. In my opinion, the books on your list would require many years of hard study for a very bright focused person to actually master. Moreover, I doubt if anyone I know has read, or even consulted, them all; at least I have not, even in my long lifetime of studying and teaching math.

As a suggestion to you, for actually learning mathematics, choose any book that "speaks to you", i.e. that you can read and enjoy, and feel you are learning from. Occasionally dip into one of the books on your list, one of the Elements, or even Hardy or Spivak, but do not be discouraged if those seem impenetrable.

If you are truly a beginner, you might perhaps start from a book by Harold Jacobs, like his Elementary Algebra, which I bought for my young children and grandchildren.
https://www.amazon.com/Elementary-Algebra-Harold-R-Jacobs/dp/0716710471?tag=pfamazon01-20

Euler's Elements of Algebra is also for beginners; a modern reader may find it lengthy, but his patience will probably feel rewarded.

To get into Euclid's Elements, I recommend Hartshorne's guide, Geometry: Euclid and Beyond, and the Green Lion edition of Euclid.
https://www.greenlion.com/books/EuclidsElements.html

 

[gleem]

A book that I found absorbing and informative is Eric Temple Bell's book "Mathematic: Queen and Servant of Science" 1951 and republished in 1986, covering pure and applied math, their interconnection, with discussion on algebra, number theory, logic, probability, infinite sets, the foundations of mathematics, rings, matrices, transformations, groups, geometry, and topology.

A PDF copy is available online here.

 

[Frabjous]

“What is mathematics” by Courant and Robbins is a good starting point.

 

[MidgetDwarf]

Is op an AI bot?

 

[mathwonk]

wouldn't an AI bot have more perfect grammar?

 

[berkeman]

Is op an AI bot?

The Mentors/Admins have no indication of that yet.

 

[tuxscholar]

Thank you for your concerns and recommendations. Well I'm not a bot at all, neither english is my first language. So far I've a studied mathematics in school up to high school level and now I've took an autodidactic approach in learning mathematics because it seems to be essential prerequisite to explore other field like Physics, CS etc.

So far all I've learned in mathematics in school is mere manipulation of expression or arithmetic calculation but find it quite terse. But still i wonder why those so called mathematician devoted their whole life to learn and do research in so abstract a field like mathematics. What made Ramanujan and many other intellectuals to be so obsessed with this subject. What really is so intriguing and profoundly valuable behind those framework of symbols especially in pure math.

So I've realized that to be able to truly appreciate this subject i need to spend decades in exploring this subject, so I've decided I'll devote my whole life to have a profound grasp in this subject, and there seems to be no better way than the rigorous approach or taking the hard way to not being discouraged later when intricacy increases.

So I'm currently working with Euler Elements of algebra which i find very comprehensive and insightful, like the way i never saw the subject before and every uninitiated aspiring mathematician should start his mathematical journey by working with this book. Also is it necessary to read Euclid's elements to fully appreciate the elegance of Geometry.

 

[Muu9]

To be able to acquire such a decent sophistication in mathematics one need to learn the works of the masters like that of Euclid, Euler, Gauss and Hardy, as one says "study the masters not their pupils".

That quote is generally said in the context of the humanities - math is significantly easier to learn these days thanks to improved notation and us being comfortable with contemporary language. But if you're enjoying Euler, stick with it.

I would guess the majority of modern geometers have never read Euclid and I don't think they are worse off for it.

A modern book that is known for its rigorous approach to pre-college mathematics is "Basic Mathematician" by Lang, who I would say is a master in his own right.

 

[Hornbein]

I would suggest you start by checking out the work of Procrustes, who created a bed that was the perfect fit for anyone.
* Don't set out a curriculum for yourself that you would hate.
* Recognize that there is no single "Mathematician" target.

The "very start" is no older than age 11 or 12. If you're older than that, you have already started.

From about 2nd grade through college, I was always very good at Math and I was a Math major in college. But I have never considered myself to be a "Mathematician". I latched onto computer programming in High School and that has been what I have followed ever since. Software Engineers do not think about problems in the same way as Mathematicians. And, neither Software Engineers nor Mathematicians think about problems in the same way as their peers.

And it's worth mentioning, especially in these Forums, that Mathematicians and Physicists do not think about problems the same way either.

I was a professional programmer and a student of math. If you want to write reliable programs then I recommend writing them much like mathematical proofs. They are quite different though : the language of programming is much simpler and concrete. And it's much easier to get a job.

 

[.Scott]

I was a professional programmer and a student of math. If you want to write reliable programs then I recommend writing them much like mathematical proofs. They are quite different though : the language of programming is much simpler and concrete. And it's much easier to get a job.

Math statements are declaratory. For most programming languages, statement are imperative.
Reliable programs are created when the requirements are well-captured, the code is comprehensively tested against those requirements, and the projects development procedures are well-established and followed.

 

[Mamta25]

Well, the right way to start learning pure mathematics is to first learn proofs, logics and set then build a strong foundation in analysis, algebra, etc. through rigorous problem solving. Here are some of the best book for proofs and logics.
1. Book of Proof by Richard Hammack
2. How to Prove it by Daniel Velleman
3. Mathematical Proof by Chartrand
4. A Mathematical Introduction to Logic by Enderton
5. Computability and Logic by Boolos and Jeffrey.
I hope this will help you.

 

[wrobel]

To be able to acquire such a decent sophistication in mathematics one need to learn the works of the masters like that of Euclid, Euler, Gauss and Hardy, as one says "study the masters not their pupils".

Actually not. Moreover in most cases it is counterproductive.

 

[PeroK]

Also to truly appreciate the beauty of mathematical endeavor one need to learn the reasoning behind the origination of concepts in mathematics, so as a beginner it appears to be worthwhile to learn the highly abstract aspects of mathematics like proofs, logic, and topics in pure mathematics (number theory, analysis etc).

I'm not convinced by this. Introductory books on Abstract Algebra, Number Theory and Real Analysis etc. would cover a lot of general and specific ground. If you study them thoroughly, this will take significant time. Probably a year or two of serious part-time study. Also, such textbooks are written by university professors with lots of experience of teaching undergraduate students and are focused on students trying to learn the subject for the first time.

As a beginner, you will probably be better served by text books written specifically for undergraduate students. This assumes, of course, that you have already mastered high-school mathematics.

 

[MidgetDwarf]

I would suggest getting a copy of Hammock : Book of Proofs

Pommersheim: A Lively Introduction to the theory of numbers

Gallian: Contemporary Algebra

Abbot: Understanding Analysis

Moise/Downs: Geometry

Work through the proofs book together with the number theory and geometry book.

Once you are done, give Gallian a try.

This is the easiest route from my experience to get into proof based mathematics