Understand Pure Mathematics Pdf
Margit Szermer
In the early stages of your education, only after trying your absolute hardest to understand the proof, consulting other references if need be, should you perhaps ask for a hint about a proof or a solution. Be honest to yourself - only ask for help when you have actually exhausted all other options. You only hurt yourself by asking for help prematurely.
Yes. It's hard, and that makes it worth doing. Individual results in mathematics (pure or otherwise) are culminations of weeks to months to years to decades of thoughts and ideas. Centuries if you include your predecessors that added to the foundations of these ideas. If you are studying mathematics full-time and your problems do not take you very much time to solve, chances are you should be looking at harder problems.
If you are interested in pure mathematics, and would like to give a try, I recommend Linear Algebra (it is on its 4th edition now, i guess) written by Friedberg and other two authors. In fact, this was my first book studying in this field, and I found it profitable to me.Most of students encounter the two problems you mentioned, so don't be that depressed. One of the reason (and I believe, the reason of most time) of being unable to construct a proof is not understanding all the definitions, which let one be confuse with what should be proved. The book starts from the definition of vector space and also offers enough explanations with intuition. Moreover, I believe the problem sets it contain helps a lot too, especially the true-or-false problems. Try to give every details in the beginning to support your proofs, and you will improve.
It is practice sir. Math is one of those things where, if you have a calm, relaxed mind, things will just come if you give it time. So take your hours to understand your proofs, you will find which way you learn best.
While a basic understanding of calculus can be helpful, it is not a requirement for learning pure mathematics. Pure mathematics involves abstract concepts and logical reasoning, rather than calculations and applications of calculus. However, some areas of pure mathematics, such as differential geometry, do require knowledge of calculus.
It is not recommended to skip learning calculus altogether, as it provides a strong foundation for understanding many concepts in pure mathematics. However, if you have a good grasp of algebra and geometry, you may be able to start learning some areas of pure mathematics without prior knowledge of calculus.
Calculus is a branch of mathematics that deals with rates of change and accumulation. It is often used as a tool in pure mathematics to solve problems and prove theorems. Many concepts in calculus, such as limits and derivatives, are also important in pure mathematics.
Yes, it is possible to learn pure mathematics without learning calculus. However, it may be more challenging as calculus provides a useful framework for understanding many concepts in pure mathematics. It is important to have a strong foundation in algebra and geometry before attempting to learn pure mathematics without prior knowledge of calculus.
Yes, calculus can be used to solve problems in pure mathematics. Many concepts in calculus, such as optimization and integration, have applications in pure mathematics. However, pure mathematics also involves abstract reasoning and logical thinking, which cannot always be solved using calculus.
If you never learn anything besides what other physicists do, the only advantage you will have over them is being smarter or luckier, which means that you will have to be really smart or lucky to get a job/get tenure at a good university/win a Nobel prize.
However, if you learn some pure math that most physicists don't know, you might be able to apply it to physics somehow. This could help you get good results, which could help your career. If you enjoy learning pure mathematics, then by all means learn some. If you don't, then you probably don't need to, but you might want to consider studying a broader range of topics in physics.
I will also give my own story: in 1959 I took a math course on sets and group theory, against the advice of my physics major adviser, because I found it interesting. Even though an experimental physicist, it sure came handy when the eightfold way came my way :).
I will tell a story about the Nobel prize winning physicist, Murray Gell-Mann. He tells it himself, I forget where. As a grad student he took, purely out of intellectual curiosity, a course in the maths dept. where he was, a course in pure maths, representations of Lie groups, and in particular he learned pretty well SU(3) (since, after all, it is one of the ones easy to visualise, the Cartan weight diagram is two-dimensional). Later in life when graphing the properties of some of the elementary particles known at that time he saw these weight diagrams in them, except for one, where one weight was missing. So he hypothesised the existence of a new particle to fill in the missing weight...the omega minus particle, and called this arrangement "the eightfold way". If you never want to make a discovery like that, then go ahead, never learn any pure maths simply out of intellectural curiosity...
Yes. If you attempt to learn quantum mechanics, understanding of the Heisenberg picture requires a reasonable grasp of linear algebra, which will at least require the correct definition of a vector space and some facts about matrix diagonalisation. This is usually rated "pure math". Every $21^st$ century course on quantum mechanics I have ever heard of has a huge fail rate because the students are not taught a sufficient amount of linear algebra. You have been warned.
Abstract algebra (Group and representation theory),Topology and functional analysis are very interesting and useful. More than learning a bunch of stuff, just as there is a "physics way" of thinking, there is also a 'math way" of thinking which is good to acquire. Believe me, the two are quite different!
Although you may find it difficult and superfluous at first, you develop an appreciation for the need for rigor. It is hard to put in words, but just stick with it and hopefully you will understand what I am trying to convey.
There is definitely plenty of math you need to learn that will not be in your required math coursework and will only be given a shaky basis if it is covered at all in the physics curriculum. But I wouldn't say its necessary to sit through lots of math classes or wade through math textbooks (especially since a lot of math is taught in a terrible fashion).
For example, I was never actually taught topology in a class, nor has any physics student I know. But the idea of a theoretical physics who doesn't know any topology is absurd. So you'll probably have patch these gaps yourself when the time comes.
For some things such as category theory, most theoretical physicists can do without (sorry John Baez). However, the less mathematics you understand, the harder it is to advance into many realms of physics. For example, it would be near impossible to study, say, canonical quantum gravity without knot theory. Even classical theoretical physicists need to be well versed in symplectic topology and jet manifolds.
However, this applies only to theoretical physics. I would also like to state that your education in mathematics need not be a technical. A lot of higher mathematics (like algebraic topology) is very intuitive and feels more like an art than a science so don't stress out. :)
I'm a Math PhD student in pure math. Many times I have encountered people who just doesn't value pure math or does not see the beauty of pure math. Most of the time these are those people who haven't done any "serious mathematics" or any advanced level mathematics. I forgive these people because I understand that their ignorance is coming from their lack of knowledge and it's going to be hard for me explain or show them the beauty of mathematics simply because they don't know advanced mathematics.
I've also had these kind of experiences with many undergrad students who take some math courses but are not math majors. I have tried to explain it to them by saying that math is not just about using formulas and applying them in the real world, but it's also about the quantitative skills, critical thinking and problem solving skills that you develop along the way, which will be useful throughout your life and you'll need it everywhere.
But the real problem is dealing with those people who insult pure mathematics and are (have) pursuing (pursued) advanced degree in applied math or who come into mathematics from engineering background or some other more applied discipline. I find it surprising that even though these people have been exposed to quite a lot of mathematics, they still say such things. How do they not understand that just because people haven't found a real world application of some of the abstract concepts in pure math at the present that doesn't make it useless, neither does that mean real world applications can't be found? There are so many instances where people have been able to use some of these abstract concepts into the real world many years after their discovery.
(I do not intend to insult or humiliate entire group of people doing applied math or are into other more applied disciplines. I'm just surprised to hear such things from some of the people in these groups)
I think an objective perspective on pure mathematics would probably have to include that the majority of what pure mathematicians (however you want to define that) work on is not driven by any current applications and is unlikely to ever find applications. Of course there are papers that 50 years later have found uses, but pointing to these one-in-perhaps-millions is not a good argument. Nor is that applied mathematicians have "frequently" drawn onto material created decades before in addressing current problems -- perhaps they have drawn onto some basic concepts, and use the language previously created, but more often than not, a lot of work still needed to be done to make this material useful to applications, and that's work applied mathematicians then did. (They were "applied" mathematicians by definition here: They wanted to solve an actual problem. Of course, they may previously have been pure mathematicians.)
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