226

Karl Weierstrass was in his 40's when he got his PHD. There are a dozen other counterexamples, a number fairly recent. A good set of examples can be found in the thread on MO here: https://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners This myth of "science is a game for the young" is one of the falsest and most destructive canards ...


169

21 is not old at all. I personally know heaps of people my age (32) who started out at 18 as salesclarks/BA or BCom majors/lawyers/bookeepers etc and ended up having a PhD degree in some advanced math areas and landed a job in academia or industry. My personal case: I got a lousy BCom degree with little math at 22 and then worked in a primitive banking job. ...


99

There is a continuum in the way one understands a theorem. At one end of the spectrum mathematicians just try to understand the statement and use it as a black box . At the other end they understand the theorem so well that they improve on it: this is called research. An important thing to keep in mind is that your attitude toward a result is not fixed ...


65

In Israel kids are expected to serve in the army when they are 18, and they serve for three years (men do, women serve two years). After this period it is common to find yourself questioning what you should do with yourself and not many people have answers. Therefore it is common to take another two years to work and travel the world before settling down and ...


62

When I was learning introductory real analysis, the text that I found the most helpful was Stephen Abbott's Understanding Analysis. It's written both very cleanly and concisely, giving it the advantage of being extremely readable, all without missing the formalities of analysis that are the focus at this level. While it's not as thorough as Rudin's ...


61

Don't try to memorise the proofs: try to memorise the methods that are used in most analysis proofs. That way you only have to memorise a handful of methods instead of 30-50 proofs, and you can adapt them to prove things you have never seen before as well.


57

In no particular order: Algebraic number theory notes by Sharifi: http://math.arizona.edu/~sharifi/algnum.pdf Dalawat's first course in local arithmetic: http://arxiv.org/abs/0903.2615 Intro to top grps: http://www.mat.ucm.es/imi/documents/20062007_Dikran.pdf Representation theory resources: http://www.math.columbia.edu/~khovanov/resources/ Classical ...


51

I recently read an article on the 40 hour work week and I think it is somewhat related. The basic idea of it was that in the mid 20th century, they had a 40 hour work week and they had lots of research on it showing that it was optimal in many ways. That is, if you increased your work week from 40 hours to 60 hours, you wouldn't gain 50% extra productivity....


51

This could be explained using algebraic transformation but i would rather show a very simple geometric proof for sum: 1 + 1/2 + 1/4 + ... = 2


50

If your goal is to become a research mathematician, then doing exercises is important. Of course, there will be the rare person who can skip exercises with no detriment to their development, but (and I speak from the experience of roughly twenty years of involvement in training for research mathematics) such people are genuinely rare. The other kinds of ...


46

I think it's important to remember the key ideas in long proofs, as these may serve you later. Trying to remember every single equation in a long proof is a loss of time and energy. However, it is a good idea to memorize the statements of theorems in order to be able to recall them without having to reread them every single time. The thing is, the more you ...


44

I think all of us at some point will invoke theorems whose proofs we have forgotten. I would argue that memory is important for mathematics in the sense that it is important for practically every other field. Certainly having good memory will not hurt you and several mathematical giants were undoubted aided by their prodigous memories (notable examples ...


43

Of course you can make a career out of it! When I started reading your question I though you were around 50, but 22 is not old at all to go after a career in anything but sports. This kind of time don't affect your brains ability to think at all. The only thing is, if you have great ambitions, you are probably not gonna be able to win the fields medal ...


41

I definitely think the Putnam exam tests mathematical "maturity" just as much as talent or raw knowledge. By maturity, I mean the somewhat less tangible things we pick up like gauging a problem's difficulty, anticipating the necessary techniques, knowing how to construct an intelligible argument, and knowing when to move on. Ego can be a huge issue as well ...


40

Memorizing proofs doesn’t really do much for you, at least in the long run; instead, you should try to see what makes them tick. First, what is the structure of the argument? What are the main steps, and what are merely details of carrying out those steps? Many proofs at this stage of your studies have just a single main idea, and everything else is details. ...


40

This is a recapitulation and extension of what we talked about in chat. Whatever you do, I recommend that you try a variety of areas in order to find out what you like best. Don’t feel obliged to stick to the most common ones, either; for instance, if you find that you’ve a taste for set theory, give it a try. My own interests are outside the ...


39

I was recently in a similar situation. After finishing precalculus at my high school, when I was 15 I started taking calculus at my local university and studying higher mathematics on my own (out of the book "Modern Algebra: An Introduction" by John Durbin, which in retrospect seems laughably basic but at the time blew my mind). Three years later, I can say ...


39

I'm not sure my personal experiences will be very helpful to a mere kid of 22, but here goes ... I made a complete mess of being an undergraduate when I was 18 (until 21), and followed a career for some decades before I finally got round to seeing if I was actually capable of doing maths at a more advanced level. I was almost 50 by the time I had published ...


38

What you're talking about seems much less like a mathematical or academic complaint than a psychological one. Here's what I read in your post: You seem to be insecure about your understanding of higher-level topics, so you continuously and obsessively revisit lower-level topics, despite that this is probably not necessary: really, if you got into a program ...


38

To me, asking if you need to understand analysis is roughly* like asking "Is it necessary for one to understand how to operate a computer to pursue a career in mathematics?", in that the answer is technically no, but Everyone else does They'll assume that you do too There's no good reason not to know By not knowing, you are making things incredibly ...


37

Could it be that you're simply expecting too much? Mathematical formulas are designed to cram a lot of information into little space, but you still need to process all of that information, so you should expect your reading speed to drop dramatically each time you encounter a formula -- at least as measured centimeter by centimeter. Furthermore (and I can ...


35

I've answered this question several times-once on Math Overflow and once here. I don't really have much to add to that answer because I think it's great advice (not to break my arm patting myself on the back about it) because it was arrived at through a lot of trial and error, pain and suffering. Firstly,although I firmly believe mathematics has to be ...


35

This question partially belongs to the sister SE site: productivity.SE To fight the mental fatigue the following things will help: doing physical exercises, as they improve oxygen supply to the brain (e.g. walking, working out, etc) getting enough sleep keeping a healthy diet Essentially of all the above is to condition the brain to be in the best working ...


33

Eric's answer is quite thorough, but it is missing an important point: in mathematics, everything tends to be related and intertwined. You might think that doing abstract algebra will get you far from analysis, but you would need to restrict your scope a lot more to do that, as there are many things relating the two, just to name a few: Group theory and ...


32

As you say, mathematics is a language. In any language, memorizing vocabulary very quickly becomes useless, if it is not paired with active use. Certainly, memorizing an entire Spanish dictionary before attempting to formulate one's first Spanish sentence is not the right way to learn Spanish. You should learn a few basic words, and practice using them in ...


31

Depends on the textbook, I suppose. Some textbooks introduce a lot of material in the exercises that isn't developed in the main text.


28

One thing that works for me when learning a theorem is to go through all the conditions and find corresponding counter-examples, as well as seeing exactly where the proof fails. Take for instance Rolle's theorem: If a real-valued function $f$ is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then ...


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