Should I Become a Mathematician?

[gunslinger]

I started the Coursera course "Introduction to Mathematical Thinking" which is a course that helps students shift from high school level mathematical thinking to university level mathematical thinking. I couldn't continue with Apostol I was way too slow. I might continue tho, after I get a solid background on logic and proofs.

 

[IGU]

I started the Coursera course "Introduction to Mathematical Thinking" ...

This panel discussion includes Keith Devlin, and he has some comments on the MOOC you are taking. I like what he says about using peer grading as a pedagogical device.

I'd suggest you get a bit more serious than just that class, perhaps Courant's What is Mathematics? would suit you well.

-IGU-

 

[gunslinger]

ok then, the MOOC + the book it is. thanks for the advice.
https://fbcdn-sphotos-a-a.akamaihd.net/hphotos-ak-ash3/575518_3878275874048_1306831904_n.jpg

 

[Dens]

that should be fine. just tell the truth. it helps you find your right place. good luck.

Wouldn't they find it stupid that you are listing a first year calculus book in a graduate personal statement?

 

[dkotschessaa]

never be shy about asking for credit for your own work. no matter how little it is, the credit should be there.

This is timely advice for me. I'm working on a paper with another student who is way ahead of me mathwise, but has very poor english. By our collaboration I'll essentially be writing the paper though she will have done most of the mathematical work (the proposed topic is also based on something suggested by me.) I feel a bit redundant to the process right now, but I think it will still be a good experience.

 

[mathwonk]

@dens: au contraire. going successfully through Spivak is exactly what many grad math programs would like to know about you. It's not going to impress Harvard, but at the University of Georgia (my university), it should count for something.

We recently began a remedial program for grad students because today many come to us not knowing advanced calculus, or even how to really make proofs. Spivak is one variable calculus sure, but it is that topic done well, and thoroughly, and deeply. It is the sort of thing many programs hope their seniors can master, not their freshmen. Calling it a first year calc book, is not descriptive. This is only a first year calc book at places like the University of Chicago, and even there it is only for their best students.

But again the point of describing yourself includes telling the truth so you can find the right place for you. Sure a lot of things i say might sound stupid, but as one of my friends said about me, I became a mathematician when others around me did not, because I was not afraid to ask stupid questions, even at Harvard.

 

[deluks917]

I actually mentioned Spivak's Calculus prominently in my personal statement. Since reading it the summer after my freshmen year was what showed me math was interesting and beautiful (the difference between Spivak and the math I had in HS should be apparent to readers of ths thread). I got in a decent grad school (despite only taking math serious after my freshmen year).

 

[Alpharup]

I have taken electrical engineering in a top notch college in India...I wanted to be mathematician in 10th grade but my family background, lack of awareness, the social pressure all prevented me. Neverthless, I liked physics and mathematics and so I took engineering...Here engineers are respected more than physics and math students...This is because, they see that engineers can earn more than physicists and mathematicians...There is a general lack of awareness in the society...Many students don't know what is engineering but want to take it in college!(some want to get placed in top companies)...

Now, I will come to the discussion... It is a well known fact than engineering mathematics is less rigorous than actual mathematics...But I want to learn almost all the concepts of mathematics atleast to the point of understanding general relativity in physics... I know that it is a painstaking job and I should spend a lot of time on it...Here there is a general saying "Dont learn what is outside the syllabus as you will waste 'time'...Do what you can to get good marks or grades"...This is the attitude of general population.If I fail in my college due to reading mathematics, they would blame me for wasting time...So, Iam in a position to learn what is only needed...I have decided that I would learn mathematics after I finish engineering...Would learning mathematics in depth and rigour make me a successful engineer(which the society expects)? Would it help me applying engineering concepts to real life problems? Will I have edge over other engineers of my time?

 

[IGU]

I have decided that I would learn mathematics after I finish engineering...Would learning mathematics in depth and rigour make me a successful engineer(which the society expects)? Would it help me applying engineering concepts to real life problems? Will I have edge over other engineers of my time?

It is a good thing to understand the tools you use, especially their limitations. It will make you a better engineer and certainly help to distinguish you from the common herd. You are, I think, showing maturity in your decision to put it off. There are several reasons this is wise:

• for an engineer, theoretical understanding is secondary to being able to use the tools proficiently
• rigorous understanding will come more quickly and efficiently after you are proficient at techniques
• you will understand how proofs are truly important better later on (you'll quickly see how the assumptions limit where the results can be used)
• you'll be in a better position to know what subset of pure math is important to you

All that said, it wouldn't hurt to take one class meant for pure mathematicians now, so you can see whether you like it. Maybe a semester of group theory if you want it to be hard, or number theory if you want it to be totally irrelevant to engineering, or an introduction to analysis if you want it to be somewhat useful. Easy to drop the class if you find it not worth the effort.

Be aware that your tastes will change as you learn more, and you might get much more busy as opportunities come along, so if you put off learning any pure math you may never find the time. C'est la vie. It's one of the down sides of being mature rather than impulsive.

Also there are many things you can learn to help you become a better engineer: philosophy, writing, drawing, architecture, astronomy, biology (especially bio-electricity). Pure math may not be the best use of your time, even if it is a good idea.

-IGU-

 

[mathwonk]

These little pieces of wisdom seem highly culturally motivated. there is a famous western quote: "A man should read exactly as his interests lead him, for what he reads as a task will do him little good." attributed to Samuel Johnson.

Also in India, the greatest gurus and scholars despise learning primarily for gain, according to my limited understanding. I myself admire Ramana Maharshi and Sri Ramakrishna. These saddhus teach that the primary obstacle to realization is "woman and gold", and that "desirelessness is wisdom".

there is little room in these philosophies for grade grubbing and money seeking. But I must add that life is difficult without prudent concern for ones well being in some form. Thus the hard task is to survive, to pursue ones true and pure passion, without becoming tainted by shallowness and greed. One must also learn to preserve respect for our forebears, even as they urge us to abandon our intellectual dreams for material stability.

 

[Alpharup]

@ mathwonk..Yes many Indian philosophers dislike the work being done to get material benefits...They feel that such materialism brings bondage and results in "Fittest will survive" nature...I personally appreciate you for seeing the good in other traditions(that too being in USA)...Afterall, truth is truth...Indian philosophers have developed a method for doing work only for attaining god and not for the fruits which the work gives...This method is called "Karma yoga"...In this method, you should do the work which you are interested in( which may depend upon the inbuilt traits) and chanting the holy name of any god...You should not think about success or failure...

But whatever I mentioned here is not properly followed by Indians themselves...Almost all the people in different strata and culture of country respect only certain kinds of people..These include politicians, engineers and people who have some technical knowledge and some power...Some occupations are seen as superior and others as inferior like that of mechanic, etc..This attitude is highly prevalent among the middle class.. Here many people work day and night for just a few rupees...They lose their sleep in these processes...Almost every person has social insecurities.. India is in a transition state and this has affected education a lot...

Education is in the same way as it was in Europe in early 1900's..Rote learning is prevalent here...In my mathematics board exam, they never ask questions which is outside the textbook..If you show some creativity in answering some math questions, you have to leave the fate to the teacher who corrects it!Financial insecurity is also a problem..Unlike USA where bright students take teaching jobs, here only the students who got low grades take them...So, professors and teachers are looked down upon...
The basic principle is; memorise->marks->good course->good job...Thus almost all the activities pertaining to education is against this Karma Yoga...Private schools are run mostly for money...Creativity which is a part of Karma Yoga is lost in this process...But USA has a very different culture than that of India...Students atleast have the freedom to raise questions in class,I suppose...But here it is not the case...If you ask, you are a blasphemer...

All these factors are present in India...I may take mathematics(or physics) and I may like to do it..But I don't know whether I will be able to get material success which I don't prefer much...I may not enjoy what engineers enjoy in society..Leave all these..Inspite of all these,Even if I take pure mathematics(or physics) and I don't succeed, my parents will be disappointed..Atleast for their sake, I took engineering(Engineering syllabus is dependent on mathematics and physics)...

 

[mathwonk]

It is perhaps only when one finds disappointment in pursuing material goals that one begins to turn to philosophical ones instead, in search of peace, or understanding. Paradoxically, at this time one may find that practical success is also more within reach.

Those with understanding of themselves and others, may find it easier to obtain jobs, grants, promotions, and to manage others, than those who are consumed with self interest. Even if we lack recognition for our work, or material success, it matters less if the fruits of that work are "dedicated to God". Indeed that is one coping mechanism, in a situation where one is unappreciated by superiors or peers. One cannot control the response to ones efforts, but one can try to control the spirit in which those efforts are given.

There is a beautiful line in Nan Yar?, something like: "when one enters the train, one does not any longer carry ones little bag on ones head, but puts it down, for the train carries all loads. In the same way the great God supports us by His grace."

With some little understanding, and the peace it brings, one may find more time to study. At least the scriptures seem to tell us this. Of course sometimes divine wisdom speaks to us also through our parents.

 

[Alpharup]

It is perhaps only when one finds disappointment in pursuing material goals that one begins to turn to philosophical ones instead, in search of peace, or understanding. Paradoxically, at this time one may find that practical success is also more within reach.

Those with understanding of themselves and others, may find it easier to obtain jobs, grants, promotions, and to manage others, than those who are consumed with self interest. Even if we lack recognition for our work, or material success, it matters less if the fruits of that work are dedicated to God. Indeed that is one coping mechanism, in a situation where one is unappreciated by superiors or peers. One cannot control the response to ones efforts, but one can try to control the spirit in which those efforts are given.

.

These are all advantages of Karma Yoga..When we attain materialistic success through Karma Yoga, we don't rejoice instead we take it as a blessing of god...This reduces our ego..When we realize that success or failure is due to god, we can never have ,"I'am the greatest. I can do whatever I want with hardwork(without god).I'am an expert. Iam a success. Everyone is below me".This attitude will surely bring depression..There are many self help books which stress the importance of success...One book says "Why do you need success? Without success, there will be few friends and there will be less enjoyment in life" This is what it gives on why we need success! In fact we are born to die but people are mad for success by compromising their health and family...Iam sorry to say but these books and motivational speakers are breeding a whole class of egotistic people who recognise themselves as successes.

Many successful people lack humility and are indulgent in pleasure giving work without heeding to moral values. Almost all the so called successful people(including people in academia, business, sports, politicians, serviceperson and so on) have atleast a little attitude like these. They officially or unofficialy become atheistic(I too became one)...When we believe that fruits, intelligence, well-abled body, strength are given by god, we will have a peaceful life(our mind is happy in the presence of god) without endangering other species in nature...

 

[mathwonk]

I confess that I struggled hard for years to make my way in my career, and was helped greatly by a study of various yogic disciplines, karma, swar, and others (yantra, tantra, mantra,...). I have great respect for the wise seers and gurus who made their teachings available to us. But there is not only one path to enlightenment. One should think hard about his definition of success before pursuing it with all his self. A man who knows who he is does not need to tremble when his boss or professor calls him. If he does so, it may be a sign he should reconsider his priorities and recapture his self.

 

[Tim92G]

I have a question for the professionals in mathematics here: Do you think competing in the IMO in high school is necessary in order to become a successful mathematician? Currently I am an undergraduate in math, and have considered taking the Putnam exam to help build my resume for grad school, but I am ashamed to say I have never participated in a formal mathematics competition before. Growing up I had always done GOOD in maths, but it was not until later (around 16) that I found an interest in advanced maths. I chose not to go to a STEM high school ( a decision I bitterly regret), therefore I never went to any summer camps like some of my friends did, or ever had any formal competition training. Many of my friends in my undergrad program even qualified on their AIME tests. I feel as if there is an exclusive industry of training students with an interest in mathematics early on to do well on competitions such as the IMO, and later the Putnam (in college), grooming them to become the mathematical prestige, and that I have missed out on this. Although my creative thinking abilities (high understanding of proof writing, and developing my own intuition behind theorems) make me believe I have what it takes to become a mathematician, I fear my lack of competition experience will limit me. Lately, this has discouraged me to the point that I have considered abandoning the field of math entirely, and changing my major to engineering or economics. What are your opinions?

 

[mathwonk]

No your situation is not hopeless, even without the advantages you lack. The strengths you mention are more than enough to succeed. But you are advised to proceed based on how much satisfaction and pleasure you gain from doing your subject. The rewards for a mathematician are not great monetarily, so one needs to enjoy the work. My friends with degrees in economics earn far more.

 

[IGU]

I have a question for the professionals in mathematics here: Do you think competing in the IMO in high school is necessary in order to become a successful mathematician?

I'm not a professional mathematician, but I am involved with math kids and competitions. From talking to many professional mathematicians, it's pretty clear that they are divided on the value of competitions. Certainly nobody thinks that competitions are a prerequisite for becoming a real mathematician. Many think they are a bad idea, pushing promising kids into wasting their time on irrelevant nonsense. I haven't found anybody who thinks that ignoring competitions entirely is a problem for kids who love math. So I'd say your worries come from paying attention to the wrong people.

What I see as the main good thing about competitions is the social side -- they are a framework for like-minded kids to meet each other, work together, and play together. But doing well at competitions takes time and energy, so if you spent your time and energy on other things you didn't miss out on anything important. Here's something on the pros and cons of competitions by Richard Rusczyk, who's always worth reading.

From what you describe of your situation, I see no value in taking the Putnam. You might find going to the club or class or training sessions interesting -- you might meet people worth meeting and learn some things worth learning and have some fun. But you are unlikely to do well on the test; almost nobody does. So don't sweat it.

I'll tell you what I tell the kids who do competitions with me: if you're not having fun then you're doing the wrong thing. Do something else that is fun. Here's an idea. Start a study group to work through some interesting math book or paper or MOOC or whatever. Finding like-minded people who want to grapple with some difficult math during their recreational time is more likely to be fun than trying to compete on somebody else's agenda. Most important is that if you don't find math fun then you ought to be pursuing something else. But competitions are not math, and aren't even a little like what real mathematicians do.

-IGU-

 

[mathwonk]

I agree with most of what the previous poster said except the somewhat cynical tone. Also i would suggest trying the Putnam just for fun and education. And I am a professional mathematician.

 

[Alpharup]

I recently bought Apostle calculus for self study...It is much cheaper than Spivak in India...I love Apostle's calculus and it is very thought provoking...I have read a few pages and the way the subject is presented is great...The use of inequalities and method of contradiction, induction for proofs is much logical...I have never known how simple axioms can be used to prove many results...But it is a little bit time consuming...For undertstanding a single result, it takes many strategies like linking many axioms, using comparisons, etc...All these are little difficult for beginner like me...So please suggest some simple strategies for undertstanding the mathematics of Apostol in a much easier way by and in much lesser time...
(Note; Iam in vacation and after my college is opened, I won't have time...So, I want to cover as much material as possible within short duration)

 

[IGU]

So please suggest some simple strategies for undertstanding the mathematics of Apostol in a much easier way by and in much lesser time...

This stuff is hard. What you're doing is learning a new way of thinking. Apostol will give you the best kind of start, but I don't think you'll find a way to make it easy and quick. Even just doing a couple of chapters, working the hard problems (not just the ones that are for practice), will give you a big advantage going into an engineering calculus class. You'll notice when they're not being rigorous (this proof is beyond the scope of this book, or we'll assume this lemma), and you'll feel more in control.

It's somewhat of a cliche, but the more you put into it (the harder you work), the more you'll get out of it. And once you work through some Apostol, the class you take will probably seem easy in comparison.

-IGU-

 

[mathwonk]

the time cannot be decreased. the point is to try to realize how much you are learning in a few pages of apostol. i.e. time spent on apostol expands. a few pages will last you a long time and take you a long way.

 

[Alpharup]

I don't know why Apostol like books can't be used for engineering mathematics..
I compared the topics covered in Engineering Mathematics Textbook(Erwin Kreyszig) and Apostol and found that they almost match in topics. Moreover, The engineering mathematics is not so rigorous in the approach. What I feel is that lack of rigour discourages mathematical learning. There should be continuity in ideas. I feel that Apostol gives the continuity of ideas. After reading a few pages, I got immersed in it and I didn't refer any other textbook. I think it is more self contained in concepts. On the contrary, when I read engineering mathematics, there is a need to refer some other book for results, proofs, etc..Many tough proofs are omitted and it irritates a lot. Please do comment on the idea whether Apostol like textbooks can be used for Engineering mathematics.

 

[IGU]

Please do comment on the idea whether Apostol like textbooks can be used for Engineering mathematics.

Obviously you can use Apostol, but for most engineering students the proofs are uninteresting and irrelevant, taking time away from practicing usage of the new mathematical tool. Apostol teaches math, not engineering. And he created and refined his books while teaching the material to Caltech freshmen and sophomores with no calculus background, who were much more about science than engineering (still applied math, but a little less so).

As I wrote earlier, you will benefit from learning calculus from Apostol. It will give you an advantage over your peers, engineering students who don't understand the math as well. So go for it! Just make sure you don't neglect practicing the application of what you learn to real-world problems.

-IGU-

 

[QuantumP7]

I became fully deaf about a year and a half ago. I've always had problems with my hearing and severe depression, so no degree yet. I've been studying finance so that I can try to make some money and get some cochlear implants (Medicaid in my state doesn't pay for it), and get off of SSI. I REALLY miss studying pure math, though. *sighs*

 

[mathwonk]

I have learned what I know of calculus by teaching it from several different books, learning something different from each one.

They include Spivak, Courant, Kitchen, Apostol, Thomas (an older edition), Cruse and Granberg, Edwards and Penney (several editions), Fleming, Loomis-Sternberg, Bers, Sylvanus P. Thompson, Stewart, Lang, ...

 

[nitro_gif]

Ok, got a few books on the go right now, in particular Lang's Basic Mathematics.

I like the content, but how can I retain and absorb more information? I feel like I read stuff but don't retain what I should, so I reread it again and still don't retain enough.

When reading a math text, how does one approach it from an active standpoint rather than a passive standpoint?

Is it worth writing notes from the text as you are reading?

 

[epenguin]

Is it worth writing notes from the text as you are reading?

Maybe it is worth making notes after reading and then find out if you know what you have read.

(Do what I say not what I do. )

 

[dkotschessaa]

When reading a math text, how does one approach it from an active standpoint rather than a passive standpoint?

Is it worth writing notes from the text as you are reading?

Yes, you pretty much have to. Except perhaps for some exceptional people, if you're not at least doing some pencil and paper work while reading, you're not really going to learn much.

Math textbooks are dense and leave a lot of stuff out, intentionally. Proofs in particular, with good reason, do not show all the "background" steps involved in getting from point A to point B. So you need to fill in those blanks, and you need to "convince yourself" that the things the books is saying are true.

If something is abstract, you may need to scratch out some concrete examples. For example, if you were reading an algebra text that tells you that a^xa^y=a^(x+y) then you'd want to plug some numbers in there to see that it "works."

-Dave K

 

[mathwonk]

one of my best math teachers, the great maurice auslander, said if you are not writing 5 pages for every page you read you are not learning anything.

 

[dkotschessaa]

my best math teacher, the great maurice auslander, said if you are not writing 5 pages for every page you read you are not learning anything.

Fantastic!

I've discovered the joy of the whiteboard now. I have a standard one, plus sticky-whiteboard sheets plastered all over my office wall. I am enjoying the hours of lively activity, working out examples, proving theorems, writing definitions until I know them from memory, ironing out all the details and just generally mathematically playing around. I've found it is better for someone as hyper as me, rather than trying to sit still, hunched over a desk. I'm learning quite a bit this way.

-Dave K