Should I study math?

For the discussion of math. Duh.

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confused314
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Should I study math?

Postby confused314 » Wed Mar 16, 2011 6:08 am UTC

Hello, I am a high school student and I really love mathematics. I remember myself studying math for my entire life. I have participated in competitions and carried out research it all made me very excited. I have also been accepted to study mathematics at Cambridge. But, I am not really sure that is what I want to do for the rest of my life. I feel that here there are people who love math and have studied math for a long time and I wish to pose a question.
Do theoretical mathematics have real world applications?
I know that cryptography, computational math, complexity theory, dynamical systems have applications, but I would really appreciate a detailed response on these applications. Can a math major help the world through interdisciplinary work? I want to do something worthwhile and not something purely for my self (I realise that adding to knowledge in general is worth it, but please could you please elaborate a bit on the real life applications of math?). Thank you!
I have read Hardy's A mathematician's apology and I was somewhat discouraged (I realise it's quite old but still). :)

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Re: Should I study math?

Postby mcvoid » Wed Mar 16, 2011 12:27 pm UTC

I had a long response all typed out when my browser unexpectedly quit and I'm too lazy to retype, so here's the summary:

Yes.

greengiant
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Re: Should I study math?

Postby greengiant » Wed Mar 16, 2011 1:56 pm UTC

I'd say if your only worry is having real world applications, go for it. Not 100% sure about Cambridge, but chances are you'll start your degree with lots of compulsory pure maths modules but then have a lot of choice after that. Just talk to a tutor and explain what you'd like to be do with your degree and ask what modules you should take. When I was at uni, there seemed to be loads of opportunities to go do work (or more likely some sort of study) in chemistry/biology/comp-sci after a maths degree (assuming your module choice fitted). Which areas are looking for mathematicians probably varies from time to time but as long as other subjects are based on pretty hard maths, there should always be room for a mathematician on their team.

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Re: Should I study math?

Postby Yakk » Wed Mar 16, 2011 4:35 pm UTC

Math is far from useless.

There once was a famous mathematician who, frustrated with the use of calculus (or the era-appropriate equivalent -- I think it was bombardment tables?) in warfare decided to study math without application. In particular, number theory.

Number theory is now the foundation of cryptography, which allows US pilots to "safely" control drones 100s of km away and blow people up with them.

Another example is non-Euclidean geometry, in which an esoteric corner of it was picked up by Einstien and turned into the theory of general relativity, which is used (directly) in making GPS systems work, and (mostly indirectly) in many other areas.
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Re: Should I study math?

Postby confused314 » Wed Mar 16, 2011 5:59 pm UTC

First, thank you for your posts!
Second, is the mostly theoretical non-euclidean geometry and number theory actually applied in GPS systems or cryptography respectively? My question mainly is, is it just a small part of these fields that is actually applicable and allows people in other disciplines to develop several systems, or are they massively applied? If yes, is there room for research in them? (if anyone has more details I would appreciate it)

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Re: Should I study math?

Postby Yakk » Wed Mar 16, 2011 6:10 pm UTC

The mostly theoretical non-euclidean geometry was used to build general relativity.

Once a field of mathematics has an application, it gets a name associated with that application (and is sometimes called applied mathematics).

So there are kinds of abstract geometry that have no connection to general relativity. But the theory of general relativity was based off of what was, at the time, utterly abstract mathematics with no known application.
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Re: Should I study math?

Postby mdyrud » Wed Mar 16, 2011 6:17 pm UTC

It's really hard to discuss how useful and practical different branches of math are. This thread has a pretty good discussion on some of the things that seem useless but aren't. With respect to your original question, I would say study math. Right now, it sounds like that is your favorite subject, and you aren't really leaning toward anything else. Declaring a major that you are interested in but not quite sure about is no problem. If it doesn't work out, look into other fields.

confused314
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Re: Should I study math?

Postby confused314 » Wed Mar 16, 2011 6:22 pm UTC

Thanks, I get the point.:idea:

Would you say that research that is being done know in non-euclidean geometry has further applications?
Anyone having any idea of how different is applied math from pure math?

Changing majors at cambridge is not that simple, which is now I am considering accepting my offer for a us university
But, I can't think of a better place to study maths than cambridge.
The only reason I would chose my us offer would be to be able to change major.
Then again, nothing fascinates me to the extent mathematics do (theoretical physics might be an exception). I just need to know whether I can help other people through my work. :)

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Re: Should I study math?

Postby Quaternia » Wed Mar 16, 2011 6:43 pm UTC

confused314 wrote:I have read Hardy's A mathematician's apology and I was somewhat discouraged (I realise it's quite old but still).

He wrote this in 1940. Hitler was carving up Europe. Mussolini was in power in Italy and Fascists controlled Spain. His extremely close friend, Srinivasa Ramanujan, had died.
This had a great impact on Hardy's mood, as you can imagine.
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Re: Should I study math?

Postby greengiant » Wed Mar 16, 2011 7:38 pm UTC

confused314 wrote:Anyone having any idea of how different is applied math from pure math?


I have to say the distinction's not as clear as is sometimes implied. I've heard areas like combinatorics referred to as both. You can use combinatorics as a tool to prove all sorts of pure maths results, but it also has plenty of applications in computing/optimisation/etc.

I'm assuming you're a UK student who's just finishing up A-levels including further maths. In which case you probably won't have covered much stuff that's really considered pure maths (unless your teacher goes above and beyond). From memory, you'll have done a bit of group theory and matrix theory which are both topics algebra. I don't think you'll have really done anything from other pure disciplines like number theory and analysis. For this reason, I wouldn't worry about the distinction too much at this point - lot's of people I was at uni with were very surprised by which bits of maths they ended up being interested in. I certainly couldn't have predicted during my A-levels which areas I'd come to enjoy.

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Re: Should I study math?

Postby eSOANEM » Wed Mar 16, 2011 7:54 pm UTC

I'm going to address the question in the subject rather than the body of your post as the latter has been answered fairly comprehensively.

Yes. You say you like maths and have always enjoyed it and that nothing fascinates you so much; that all points very heavily towards a big yes.

As for whether you should do it at Cambridge rather than in the US, one important thing to consider is that maths wouldn't be your major, it would be your course. Here in the UK, degree courses tend to be a lot more specialised than in the US which is both a good thing and bad thing depending on your point of view; on the plus side, you will get to study more maths, on the down side, you won't be able to study as much other stuff.

As for the ability to change course (because you don't, in general, have a major in the UK), compared to other universities in the UK, is much greater at Cambridge however it still isn't great. As far as I can tell looking at the prospectus here, the only option they like people taking is transferring to physics after the first year but that's not to say you can't change to another course if you can convince you college that it would be a good idea. It's also worth bearing in mind that the Cambridge maths course specialises as it goes on (as do all the courses) so, over the course of your degree you could have strayed pretty far away from pure maths into applied things such as cryptography, cosmology and quantum mechanics.

Lastly, you say you're not sure whether you want to do maths for the rest of your life; this is no reason to not study it, what you do as a degree has very little bearing on what you can go on to do for a career (assuming your degree is in a purely academic subject rather than say medicine), your degree doesn't just show to employers that you can do maths and so limit you to only doing maths related jobs, but rather shows an ability to think logically, rationally and solve problems, if you don't plan on being a research mathematician (and, from your posts, I get the impression you don't), you're very unlikely to use the vast majority of the stuff covered on your course even if you did get a maths related career.

So basically, yes, you should, don't let the specialism of coming to a university here in the UK scare you, in that regard, Cambridge isn't too bad, you could be forced into a much more rigid path had you applied elsewhere.
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Re: Should I study math?

Postby confused314 » Wed Mar 16, 2011 9:04 pm UTC

Thank you all for your post.
@Quaternia: You are right! Hardy was very affected, but (un)fortunately he made disappointed me a bit.
@greengiant: assumptions can be wrong 8) I am an international student, doing the IB and I have significant experience with number theory through competitions and personal readings and (un)fortunately it's the part of mathematics that interests me the most at the moment. I have had some contact with calculus, graph theory, group theory, statistics and combinatorics. I get your point that I have of course not had an exposure to math similar to that of undergraduates but I have seen very beautiful areas. I guess an infinite number of fields are waiting for me and I don't want to miss the opportunity to come across them.
@eSOANEM: Reading Cambridge's descriptions about math I also figured out that the most probable change that can happen to a math person is going to physics and everything else is rare. Also, I am very interested in a research career (I have a research experience in number theory and I enjoyed every part of it). My teacher at school told me that what he did for his PhD was so specialized that very few people could understand it and talk to him about it and that this is the case for many research-mathematicians (his PhD was on theoretical calculus) and that he could not see any applications to it. What I am afraid is that I will wake up one day and realize that what I have done won't help anyone. :)

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Re: Should I study math?

Postby torgos » Wed Mar 16, 2011 10:10 pm UTC

confused314 wrote:Thank you all for your post.
@Quaternia: You are right! Hardy was very affected, but (un)fortunately he made disappointed me a bit.
@greengiant: assumptions can be wrong 8) I am an international student, doing the IB and I have significant experience with number theory through competitions and personal readings and (un)fortunately it's the part of mathematics that interests me the most at the moment. I have had some contact with calculus, graph theory, group theory, statistics and combinatorics. I get your point that I have of course not had an exposure to math similar to that of undergraduates but I have seen very beautiful areas. I guess an infinite number of fields are waiting for me and I don't want to miss the opportunity to come across them.
@eSOANEM: Reading Cambridge's descriptions about math I also figured out that the most probable change that can happen to a math person is going to physics and everything else is rare. Also, I am very interested in a research career (I have a research experience in number theory and I enjoyed every part of it). My teacher at school told me that what he did for his PhD was so specialized that very few people could understand it and talk to him about it and that this is the case for many research-mathematicians (his PhD was on theoretical calculus) and that he could not see any applications to it. What I am afraid is that I will wake up one day and realize that what I have done won't help anyone. :)


Well, if you someday wake up and feel that way, you can always switch over to applied mathematics.

Even if you end up deciding that pure math isn't your cup of a tea, a solid background in math will be in invaluable if you decide to pursue a graduate degree in engineering, physics, economics, statistics, or computer science; it's even a boon if you want to do a quantitative(or perhaps even non-quantitative; people with experience tackling problems that have right and wrong answers are always respected) MBA, or go into law. I'm sure there are plenty of fields that are under-utilizing mathematical tools, and you can be remembered as a pioneer if you identify them and do some good work. I personally have long wanted to apply 'advanced' statistics to linguistics, my original college subject of interest; I'll have to discuss it with some of the linguists I know.
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Re: Should I study math?

Postby greengiant » Wed Mar 16, 2011 10:15 pm UTC

confused314 wrote:@greengiant: assumptions can be wrong


Yep, sorry about that. I'm afraid I have no personal insight from the point of view of an IB student. The thing about not knowing which areas you'll enjoy 'til you try them probably still holds. Anyway if you do decide to go for a maths degree, hope you enjoy it as much as I did.

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Re: Should I study math?

Postby eSOANEM » Wed Mar 16, 2011 10:48 pm UTC

confused314 wrote:@eSOANEM: Reading Cambridge's descriptions about math I also figured out that the most probable change that can happen to a math person is going to physics and everything else is rare. Also, I am very interested in a research career (I have a research experience in number theory and I enjoyed every part of it). My teacher at school told me that what he did for his PhD was so specialized that very few people could understand it and talk to him about it and that this is the case for many research-mathematicians (his PhD was on theoretical calculus) and that he could not see any applications to it. What I am afraid is that I will wake up one day and realize that what I have done won't help anyone. :)


This is the thing, the purer your area of expertise the longer it will take for a use to be found (because it has to filter through all the less pure subjects before eventually hitting the engineers who design a useful product that relies on it) but that doesn't mean that it won't ever have an application. Besides, if you want to go into research, surely the act of helping increase the sum of human knowledge is sufficient contribution to society without having to find a specific application for it, someone else can do that.
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Re: Should I study math?

Postby Quaternia » Thu Mar 17, 2011 5:05 pm UTC

Others have pretty much said it all already, so I just wanted to give an example.
I have a friend who's in mathematical logic, namely category theory. Now, that didn't look remotely applicable to anything other than pure math problems to me, but then he explained that his research is in computational linguistics, and he needs category theory to design new algorithms. Last I heard they were working pretty well, he'd published a new algorithm that worked a bit better than the ones using more classical techniques.

Something that might interest you if you have a strong desire to do math and get direct "help others" consequences: mathematics applied to medicine and to biology is all the rage currently. There's some books up on Google Books on those subjects, you could always take a glance and see if that strikes a chord with you.
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Re: Should I study math?

Postby gorcee » Thu Mar 17, 2011 5:35 pm UTC

Quaternia wrote:Others have pretty much said it all already, so I just wanted to give an example.
I have a friend who's in mathematical logic, namely category theory. Now, that didn't look remotely applicable to anything other than pure math problems to me, but then he explained that his research is in computational linguistics, and he needs category theory to design new algorithms. Last I heard they were working pretty well, he'd published a new algorithm that worked a bit better than the ones using more classical techniques.

Something that might interest you if you have a strong desire to do math and get direct "help others" consequences: mathematics applied to medicine and to biology is all the rage currently. There's some books up on Google Books on those subjects, you could always take a glance and see if that strikes a chord with you.


There's also quite a bit of interest in applications in large scale non-uniform data structures, such as social network analysis, etc. in which researchers are applying previously unrelated branches of mathematics to obtain useful and remarkable results. There's also research money in these fields, which is important because at the end of the day, someone's gotta pay for your lunch.

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Re: Should I study math?

Postby MartianInvader » Thu Mar 17, 2011 8:28 pm UTC

Okay, so I'm going to go against the grain here and say a lot of higher math is pretty useless in "the real world", at least in my experience. I majored in math in college, minored in computer science. I went on to finish my Ph.D. in math about a year ago.

Unfortunately, with the economy the way it was, most US universities didn't have much in the way of post-doctorial positions available. I probably could have found something, but I got married during grad school, and the odds me and my wife both getting positions close to each other were not good.

So I went into industry. I now work in a tech company where I do mathematical modeling, mostly of marketing data, and also make use of my computer science skills.

The thing is, though, that I don't actually use any of the math that I studied in school. I studied a lot of topology and abstract algebra, and wrote my thesis in geometric group theory. While this stuff does have some application for physics, cryptography, etc., I wasn't about to compete with people who had actual degrees in physics or cryptography. I still keep up with what's going on in the world of geometric group theory, but I don't think I've used a single thing I've learned in grad school for my new job.

That said, I'm still very glad I got the degree. I really enjoyed studying math, and even if the specific stuff I learned doesn't apply to where I am now, I acquired less tangible skills, like the ability to think abstractly, come at problems from different angles, and quickly synthesize new ideas and quantitative structures. It kept my mind "in shape", and I learned the more applied math extremely quickly once I started my new job. And I really enjoy my job! The work still tickles the part of my brain that made me enjoy math, and having the PhD on my resume helped me land a position I enjoy.

So my answers are: Does pure math have applications in the real world? Only a tiny, tiny fraction of the time. But should you study math? Absolutely!
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Re: Should I study math?

Postby radams » Fri Mar 18, 2011 3:02 pm UTC

Paul Graham wrote an interesting essay that's relevant here:

http://www.paulgraham.com/hs.html

Especially his advice to stay upwind. Pure maths is upwind of applied maths, mathematical modelling, programming, and all the other "useful" disciplines; that is, it's relatively easy to move from pure maths to a useful area, relatively hard to go the other way.

If you're good at maths and want to do something useful, you should certainly look into biomathematics. Biology, biomaths and bioinformatics are the exciting areas in science right now - biology is undergoing a boom every bit as big as the physics boom in the 50s, and it's needing new tools from maths and computer science to cope. It's certainly useful; if the cure for cancer is going to come from anywhere, it's here.

If you do end up in pure maths research, you can expect to see much of what you produce turn out to be useful, albeit indirectly. But you can never predict in advance how useful solving a problem will be. Be prepared for the problem you sweated over for years to turn out to be useless, while a trivial little intermediate result you took five minutes to produce ends up being cited hundreds of times.

But to echo what others have said: go to Cambridge. Go, go, go. You'll love it, and it will open many, many doors; pure maths research is by no means your only next step. You'd be crazy to turn the opportunity down.

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Re: Should I study math?

Postby OverBored » Fri Mar 18, 2011 3:50 pm UTC

I don't have much to add in the way of commenting on Mathematics as a discipline, but as a current Cambridge Maths undergrad, I'd be happy to answer any questions about the course and Cambridge in general (just PM me if you are interested). I've never been too worried about direct applicability of what I learn, but I can understand why it might be an issue.
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Re: Should I study math?

Postby confused314 » Thu Mar 24, 2011 2:18 pm UTC

@Greengiant: I hope that too! :)
@eSOANEM: I think I understand the point - but my brain keeps looking for some tangible evidence, but thanks for pointing that out, I guess I will come to realise it in the future a bit more
@Quaternia and gorcee: Thanks for the posts - I found this field called theoretical biology that appears to be quite interesting
@Martian Invader: Your post helped me a lot. I see your point that you obviously cannot compete with physicist in applying your math knowledge. But, you can apply thinking techniques of mathematics. I keep listening to mathematicians saying that and it has to be true. :) Also, you got to study really beautiful theoretical math and you found an interesting job later. This is cool.
@radams: I read the essay. Indeed very interesting. (It’s a pity he did not delivered the actual speech. Biomathematics appears as a very interesting domain – I will start reading biomath
@overbored: I appreciate that you see my point. (I will PM you about the course – thanks for that)

Thank you all for the posts. The discussion is definitely helping me.

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Re: Should I study math?

Postby andrewxc » Thu Mar 24, 2011 3:26 pm UTC

Some of my friends at the Univ of New Hampshire, Durham, ended up figuring out a new coding method for .mp3s that allowed them to compress .mp3s to a few hundred KB, rather than 8MB. Granted, the quality was lower, but they ended up patenting the coding system for a new generation of phones and their ring tones.
They weren't originally intending to do this; their aim was to minimize corruption to data files, when sent over wifi / cell signals, and ended up creating something ingenious. Theoretical mathematicians were on their team during the entire process.

I went to school for Math Education, and studied mostly theory up until Linear Algebra (Senior Level) and went into a Master's program in Applied Math. I must say that this path seems the more popular one.
In addition, I did a LOT of programming, which included several classes where I used Neural Networks to solve complex problems. I think that you would want a strong theoretical base, before you moved to applied math, since it may be easier to see the connections between subjects once you have had time to learn how they work.
And theory classes usually will show some applications at some point.
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Re: Should I study math?

Postby Robo Connery » Sun Mar 27, 2011 3:21 am UTC

Judging by what you've said, I would recommend looking at either an single honours in applied maths or a combined in maths and physics. Read up whatever material cambridge has on the various maths degrees they offer and see if any of those sound better than just mathematics.


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