Other Should I Become a Mathematician?

[Dougggggg]

I do not know what to suggest as the best way to learn proofs. This happens gradually as you practice them. I got my start in high school in a geometry course back when they were proof based, but I still had a lot of trouble with the language. Than as a senior we had a special course out of Principles of mathematics, which began with a chapter on sets and logic. It had a little intro to propositional calculus and truth tables, and I finally found out what a converse was, and a contrapositive, and how to negate things. Basically, in order to prove something you need to know what it would mean for it to be false. And a basic technique is proof by negation, so you need to know that A implies B if and only if notB implies notA. Then I still had trouble in a first year Spivak type proof based calculus course, but it helped more. Then later I had an abstract analysis course, where we proved set theoretic statements about measure theory, and I internalized how to negate lengthy quantified statements.


And if you are getting a C from the good physicist, try for a B. It is often more instructive to get a B from a hard prof than an A from an easy one. (But a D means little from anyone.)
There are some elementary books that claim to teach you how to do proofs but I don't think any of them are much good. One of the best, and maybe hardest, general books to improve your math knowledge is "What is mathematics" by Courant and Robbins. Otherwise it probably helps just to study a proof based book on some specific topic like linear algebra, such as Halmos' Finite dimensional vector spaces.

Thank's I will check out that first book you mentioned during Christmas break.

 

[dkotschessaa]

WRT the original post, and oldie but goodie, at what point do you recommend starting to look at the works of the early mathematicians you mentioned (Gauss, Euclid). And which works? I came across Euclid's Elements in the bookstore the other day and put the thick volume down deciding I would be getting a bit ahead of myself. What I found of Gauss online was all in German.

I have one semester of calculus under my belt but it is from some time ago, so I'll be starting over this spring.

-DaveKA

 

[mathwonk]

get the green lion edition of euclid without all the commentaries. gauss is available in english but may be a little pricey. you are certainly ready to read euclid and probably gauss.

 

[dkotschessaa]

Found that, thanks. I am always interested in reading fundamental texts when exploring a subject. There is nothing quite like it.

Along the same lines, do you recommend Newton's Principia?

 

[mathwonk]

yes. one of my big regrets was selling my copy in 1977 when lightning my load to move, for a dime! I would not have let it go, but the book buyer obviously loved books and smoothed them lovingly with his hand as he stacked them up, so i thought it was going to a good home. I also highly recommend reading euler. also archimedes.

 

[dkotschessaa]

Ok, all will be added to my light summer reading list. :) You've been tremendously helpful.

-DaveKA

 

[Shackleford]

My calculus textbook says the three greatest mathematicians are Newton, Gauss, and Archimedes.

 

[andyroo]

My calculus textbook says the three greatest mathematicians are Newton, Gauss, and Archimedes.

Archimedes is probably my favorite actually!

 

[Dougggggg]

I found this https://www.amazon.com/dp/0521045959/?tag=pfamazon01-20 in my library, I was enjoying reading it so much. Only problem was that I had to leave to do homework. They had all 7 volumes. The notation is way over my head but I found where he originally showed the definition of a derivative. Granted I only realized that because of the notes at the bottom.

 

[mathwonk]

Don't expect to read it all, or even a large amount of these books, in one summer. Just read some of it, any really. Even one page of one of those greats will bless you with a special dispensation of knowledge for the rest of your life. Honest.

Once, in grad school, while studying algebraic geometry, I went to the library and struggled through a single page of Zariski. I was very frustrated because it took me all afternoon to read that page, and I felt discouraged. The next day in class however, I answered every question the professor asked, until he asked me to stop answering since I seemed to know everything.

Similarly in Archimedes there is a single sentence that says something like: a sphere is essentially a cone whose base is the surface of the sphere, and whose apex is the center of the sphere.

That one sentence, if you understand it, tells you why the volume of a sphere equals 1/3 the surface area times the radius of the sphere. Not even all mathematicians have this insight.

Just open these books and you will begin to tread on the higher ground that few people tread.

 

[QuantumP7]

Once, in grad school, while studying algebraic geometry, I went to the library and struggled through a single page of Zariski. I was very frustrated because it took me all afternoon to read that page, and I felt discouraged. The next day in class however, I answered every question the professor asked, until he asked me to stop answering since I seemed to know everything.

Which book by Zarinski do you mean?

 

[Shackleford]

Don't expect to read it all, or even a large amount of these books, in one summer. Just read some of it, any really. Even one page of one of those greats will bless you with a special dispensation of knowledge for the rest of your life. Honest.

Once, in grad school, while studying algebraic geometry, I went to the library and struggled through a single page of Zariski. I was very frustrated because it took me all afternoon to read that page, and I felt discouraged. The next day in class however, I answered every question the professor asked, until he asked me to stop answering since I seemed to know everything.

Similarly in Archimedes there is a single sentence that says something like: a sphere is essentially a cone whose base is the surface of the sphere, and whose apex is the center of the sphere.

That one sentence, if you understand it, tells you why the volume of a sphere equals 1/3 the surface area times the radius of the sphere. Not even all mathematicians have this insight.

Just open these books and you will begin to tread on the higher ground that few people tread.

Interesting. So, something like this?

https://www.amazon.com/dp/0486420841/?tag=pfamazon01-20

 

[mathwonk]

yes.

 

[Shackleford]

yes.

okay.

 

[ephedyn]

... most of which are just lying on my computer in an outdated font and a word processor that isn't even readable by current versions of the same program (guess what famous software company produced this marvel of usefulness), and publish them for profit instead of giving them away. I am told however that publishers pay authors so little that it is hardly worth it. This may be why Mike Spivak publishes his own works.

...Microsoft? I don't know if there's anything I can do to help, but if you're interested, we could work something out and I'm more than willing to help lessen the workload of typesetting it in LaTeX.

For zero credit or profit, of course; I just love the feeling of mathematics flowing off the tips of my fingers in LaTeX, and the learning experience is more than profit. (I remember you had a set of notes on linear algebra which was sweet and even now I try to style my notes in a similar format).

 

[qspeechc]

Aren't there programs available for free from Microsoft which you download and it let's you open your old files on the new program on your computer? My sister downloaded such a program for Microsoft Word. Maybe you should check the website, or e-mail them to find out, or go to one of their shop with the technicians.

 

[mathwonk]

As far as I know there is no fix for this. My technical support professional surfed the web for some time and I also tried all the chat sites and so on that I could find. Apparently microsoft just cut loose all their own old customers.

A second problem I cause myself was using an old special font that is not supported on newer computers.

ephedyn, if you want to try texing my brief linear algebra notes from my webpage, you are welcome to do so. That's less than 15 pages, and might be feasible. I think you would get tired before long though on all that other lengthier stuff. If you do, please add a credit to yourself.

 

[bhuavnxp]

as to those planning a career in math, here is a relevant joke i got from a site provided by astronuc:

Q: What is the difference between a Ph.D. in mathematics and a large pizza?
A: A large pizza can feed a family of four...

ha haha good one :)

 

[bhuavnxp]

i am really interested in lwearning maths..alll i am doing in now studying ENGINEERING MATHEMATICS sruggling with partial differential equations...i am a maths tutor now...teaching students... :)
i love mathematics although i am not to good i it but i love it anyhow :)

 

[QuantumP7]

How do you know if you have a talent for mathematics?

 

[hunt_mat]

Because all your maths grades are higher than you other grades and it's something wihich you look forward to doing. That was my route anyway.

 

[RoughRoad]

I don't know if this is the right thread to ask my query, but anyways...


I am 17, currently in my 12th grade, and I live in Mumbai, India.

Now, my problem is that I really like how mathematics can be used as a tool for explaining the phenomenas nature. I love reading biographies of various physicists and mathematicians, and really get motivated when I ready some quotes by other eminent mathematicians. By doing that, I really get motivated to study maths.
But when I ACTUALLY sit down to solve problems, withing an hour, I get bored. It's not that I don't like the subject, but it's just that I don't have the concentration power to let myself study the subject.

What should I do? Is it that I don't have an amplitude for Maths?
Pls help, and if possible, pls share your childhood experienes while studying this subject, and you gained motivation to solve more and more problems without getting frustated...

 

[muppet]

As regards people minding their own business, you've posted this in a discussion thread that aims to provide people help on how to become mathematicians. If those of us who've actually attained degrees from those universities and their silly courses think that e-mailing some guy to get answers is a lousy way to make progress, that's our contribution to the discussion. If the derision of university courses as "silly" arouses our suspicion as to the author's credentials or common sense, that's also our fair comment to make.

EDIT: This reply appears to be to a post that no longer exists...

 

[muppet]

Mathwonk: I'm intrigued by your post about the cone and the sphere! My initial way of rationalising it was to think about somehow wrapping the base around the apex, but as they have different curvatures I decided that that was a lousy way to think about it. It seems to me that (from a modern perspective, which is the only one I really have) a better way to think about it might be in terms of the symmetry groups- you can rotate a cone into itself around the axis from the apex to the centre of the base, although this doesn't hold for every point on the base the way it does for the sphere...

In any event I'm baffled by how Archimedes could have arrived at such a conclusion (something about sweeping out circles of increasing radii?), but then I'm no Archimedes. I'm really a theoretical physicist anyway, as probably shows.

It was also nice to read that you could spend an afternoon reading a single page of a textbook and struggle with it- I don't know anything about algebraic geometry, but my study of quantum field theory yields similar experiences on a regular basis.

 

[mathwonk]

A cone is a union of straight lines emanating from a point. They end at the base. Since a sphere is a union of lines emanating from the center and ending at the surface of the sphere, it follows that the solid sphere is a cone with vertex at the cnter and base is the surface. grok?

 

[Shackleford]

A cone is a union of straight lines emanating from a point. They end at the base. Since a sphere is a union of lines emanating from the center and ending at the surface of the sphere, it follows that the solid sphere is a cone with vertex at the cnter and base is the surface. grok?

Is the base curved or flat? So, a solid sphere is a rotated cone?

 

[mathwonk]

excellent question. normally of course the base is flat. but recall that differential calculus is the science of approximating curved things by flat ones. so if you approximate the surface of the sphere by small pieces of tangent planes, you will also approximate the volume of the sphere by many volumes of pyramids with flat bases. since the ratio of volume to area of base times radius is 1/3 in all these flat cases, it remains true in the limit. have you had calculus and limits? if so, now you can begin to see how to think in those terms as archimedes did.

 

[Shackleford]

excellent question. normally of course the base is flat. but recall that differential calculus is the science of approximating curved things by flat ones. so if you approximate the surface of the sphere by small pieces of tangent planes, you will also approximate the volume of the sphere by many volumes of pyramids with flat bases. since the ratio of volume to area of base times radius is 1/3 in all these flat cases, it remains true in the limit. have you hD CALCULUS AND LIMITS? if so, now you can begin to see how to think in those terms as archimedes did.

Yes. I've had taken Differential Equations and Vector Analysis. I think this line of thought is pretty interesting. These seem to be the very basic notions that underlie mathematics. I'm beginning to think that I enjoy - as I've seen it called before - the language of and logic behind mathematics. Is that the correct way to look at it?

 

[-IamLittle-]

do i need to get first hornor to be a mathematician?

 

[mathwonk]

This is a long thread about becoming a mathematician, but i recommend going back and reading at last page one of it. There is nothing mentioned anywhere here to my knowledge about getting first honors. Indeed I do not know what they are. Essentially, if you think you are a mathematician, you are making a good start.

 

[muppet]

Grok. Thanks for your reply Mathwonk.

 

[-IamLittle-]

This is a long thread about becoming a mathematician, but i recommend going back and reading at last page one of it. There is nothing mentioned anywhere here to my knowledge about getting first honors. Indeed I do not know what they are. Essentially, if you think you are a mathematician, you are making a good start.

You can get first honor if your gpa in university is A,in my university, if you don't get A, you cannot be a postgraduate. Can i be a mathematician after i leave the university? I am worry about it.

 

[mathwonk]

I see you have read sentence 3 of my answer. Now please read sentences 1,2, and 4. And good luck to you.

 

[mathwonk]

Let me be more specific. I myself made a (D-) in 2nd semester freshman honors calculus (largely from not having adequate study habits, not from having no talent). I was later asked (i.e. required) to leave school for one year to do some maturing, and then re - apply. After doing so, I learned to get reasonable grades, i.e. go to class, do the work, do extra work if need be. But I did not graduate with any kind of honors, neither 1st , 2nd, nor 3rd... But my good performance senior year enabled me to enter a grad program.

But again in graduate school, I at first try only managed to earn a masters degree, again from losing focus. Eventually I found another chance at another school and, after further seasoning in life skills, graduated with a PhD. That was over 30 years ago. So no, life does not end at age 21, nor at the end of undergraduate school, regardless of the current situation. At some point however you must perform.

So it seems that grad school at your present university may be out of the mix, but there may well be other choices, if you can convince someone you can do significant work. But be flexible. Maybe some other work also interests you.

 

[-IamLittle-]

nice sharing! thank you very much

 

[dkotschessaa]

New question:

At my school there are quite a few seminars and colloquia and such that go on during the semester in the math department. (Physics as well). Titles like "Integrable discretizations and soliton solutions of KdV and mKdV equations" and "Making Sense of Non-Hermitian Hamiltonians."

I have to admit I haven't a clue what these are even about, but the question is - should I attend? I'm kind of reflecting on how one can learn a language through a process of immersion and wondering if there is a similar effect in mathematics, so long as I continue to work on the fundamentals in the meantime.

Edited to add: These talks tend to be grad students, professors, etc. So whatever they are talking about I probably won't be doing for another 3 years at least.

-DaveKA

 

[uman]

It might hurt your self-esteem...

 

[dkotschessaa]

It might hurt your self-esteem...

Meaningless concept.

 

[dkotschessaa]

I believe the word is "humbling." That's ok.

 

[uman]

Not at all. If it humbles you so much that you think "I will never be at this level... I should quit math", then harm was done.

On the other hand, if you think you're immune to that, go for it.

 

[mathwonk]

as sylvanus p thompson put it, what one fool can do, another can.

 

[dkotschessaa]

Not at all. If it humbles you so much that you think "I will never be at this level... I should quit math", then harm was done.

On the other hand, if you think you're immune to that, go for it.

Insecure,self defeating attitudes are not at all my style. I would think if anything it would make me want to head back to my study and learn more.

I learn a lot of Spanish from my wife and family especially when we travel. You hear what phrases tend to pop up over and over and what frequency certain words and idioms have. Using the oft heard metaphor of math as a language I would think the process might be similar. I'm just wondering if I'm corrext in applying the metaphor this way or if it might not be a good use of time.

 

[mathwonk]

I think most people recommend this as a good way to learn things you cannot learn any other way.

 

[qspeechc]

New question:

At my school there are quite a few seminars and colloquia and such that go on during the semester in the math department. (Physics as well). Titles like "Integrable discretizations and soliton solutions of KdV and mKdV equations" and "Making Sense of Non-Hermitian Hamiltonians."

These seem to be physics or applied math topics, which I do not think will be of much interest to a math student.

 

[symbolipoint]

dkotschessaa said this:

I learn a lot of Spanish from my wife and family especially when we travel. You hear what phrases tend to pop up over and over and what frequency certain words and idioms have. Using the oft heard metaphor of math as a language I would think the process might be similar. I'm just wondering if I'm corrext in applying the metaphor this way or if it might not be a good use of time.

Acquiring Mathematics is a little bit different than acquiring a human language, but your attempt at the metaphor is at least encouraging if not exactly enthusiatic (which for you it may very well be). I found that physical sciences lecturea AND LABORATORY courses, and especially Fundamental Physics couses forced some acquired skill with Algebra and Trigonometry and some Calculus; and such skill would not have developed as effectively from just the Mathematics courses alone.

With human languages, people can learn to use and understand them if some intelligent person shows them what the words and phrases are and how they work and gives them exercises in using the words and phrases. This stuff can be both formal and informal.

Topics in Mathematics are best taught formally first, and then the student should (and often IS) put into situations to use and THINK in those topics.

 

[dkotschessaa]

Thanks for that input. In addition to my math courses basically my plan (actually its more of a plunge) is to get involved with undergraduate research in the physics and/or math departments, in addition to showing up at lectures. Its a kind of fake it till you make it approach.

Ready, fire, aim!

 

[QuantumP7]

So, for a math PhD, do they look favorably upon physics research experience, or should you just put your math research experience on your application?

 

[dkotschessaa]

I would think any research would show favorably for math, especially something as math intensive as physics, unless I'm mistaken.

 

[Dougggggg]

The conversation about other languages has me wondering if when you go to different countries how much does the mathematical language change, in both english and non english speaking countries?

 

[dkotschessaa]

The conversation about other languages has me wondering if when you go to different countries how much does the mathematical language change, in both english and non english speaking countries?

Maybe another thread should be started? This is a topic I find interesting.

What I've found with languages is that technical terminology is less likely to have evolved far from it's latin roots, so many of the words are cognates. Look up "quadratic" in Google translate and you'll find that the term is similar. (cuadrático in spanish and portuguese, quadratisch in German).

I think the non Indo-European languages have adopted the latin or english terms, so they might still have cognates, but I have no evidence of this since google translate renders the translations in whatever script the language uses.

Though I did find that Icelandic translates "quadratic" as "stigs."

I'm not sure what you're asking in reference to English speaking languages though. You mean perhaps British English as opposed to American English or something? I've found that when languages start to diverge, it's usually the more "common" dialog that changes - and that technical terms, again, don't change much, probably because they are more precise. Though in England you might say "formuler." :)

-DaveKA