[Lewk]

Right, just to make things clearer, I am currently in second year mathematics. After thinking about what to do for my final year project which i must decide on quite soon, I have been wondering what area to specialise in. Not just for my project, but in general. And when I say 'pure maths' I mean numerical methods, logic, calculus, algebra etc since I don't really know the exact definition of pure mathematics :P

There is statistics & analysis, which I find quite easy, and there is plenty you can do in the real world with that but I just don't find it very interesting. Statistics would also be the 'easy route' for my final year project as well but what I'd like to do is something to do with pure maths, but I am worried it will be too challenging and that I will regret it incase I get stressed with it being my final year & all. Not only that, I can't think of many real world applications for when i finish my degree.

I'd also like to do programming & computing but that would take a lot of practice in my own time since I am not doing a computing or programming degree, there's also plenty of real world applications for that too.


At the end of the day, I know it's all my choice and that what other people say with have little impact on my decision, but it would be nice for people to comment on my situation, and I'd also like to know if there are many jobs available for pure mathematicians.

To summarise:

Statistics- easy, boring and quite highly sort after in the real world (I think)

Pure mathematics - Fun, hard, no idea about if there are much prospects in the real world in this field though. It's also where my true expertise lie, i.e I am fast at solving challenging problems in this area and catch up quickly compared with my coursemates. But if there are few job prospects then I'd rather not.

Computing/programming - Fun, challenging, highly sort after in the real world, but I have no knowledge in this area since my degree doesn't cover it, so would have to learn it by myself and even then I would have no proper certificate to prove my knowledge in the area like I would with maths.

finance - Just no.

[CHY872]

I had it put best to me at one of my university open days. 'At some point during your time, you might be in a lecture, or reading a book, or just thinking, and you'll realise that what you're thinking about is what you want to spend the rest of your life doing'.

I don't think anyone here would be able to give you good advice on your final year project, since you'll want to make it as interesting as possible for yourself - and different people have different definitions of interesting.

I mean, if I were in your position, I'd probably go about finding things that are contradicted by Godel's Incompleteness Theorems. We're different people - so that probably wouldn't be interesting for you.

[Lewk]

(Original post by CHY872)
I had it put best to me at one of my university open days. 'At some point during your time, you might be in a lecture, or reading a book, or just thinking, and you'll realise that what you're thinking about is what you want to spend the rest of your life doing'.

I don't think anyone here would be able to give you good advice on your final year project, since you'll want to make it as interesting as possible for yourself - and different people have different definitions of interesting.

I mean, if I were in your position, I'd probably go about finding things that are contradicted by Godel's Incompleteness Theorems. We're different people - so that probably wouldn't be interesting for you.

That's the kind of thing that interests me as well but what career prospects are there in the real world for mathematicians who specialise in that area?

[CHY872]

(Original post by Lewk)
That's the kind of thing that interests me as well but what career prospects are there in the real world for mathematicians who specialise in that area?

No clue. My guess is that if you do good enough work in any area, you'll be able to get someone to pay you for it. There's a really good chance you'd end up with absolutely no dissertation though, having failed to identify anything. Personally, I'd just go with what you enjoy the most - this is university, some of the final formative years of your life. Someone I knew went to do an accountancy degree simply because an accountancy company was paying his fees, and I never could understand why...

[Raiden10]

(Original post by CHY872)

I mean, if I were in your position, I'd probably go about finding things that are contradicted by Godel's Incompleteness Theorems. We're different people - so that probably wouldn't be interesting for you.

How do you mean? Godel's Incompleteness Theorem is a theorem of mathematics that is accepted as canonical (i.e. it has been proven). The only things that are contradicted by Godel's Incompleteness Theorem must then be false.

[Raiden10]

(Original post by Lewk)
Right, just to make things clearer, I am currently in second year mathematics. After thinking about what to do for my final year project which i must decide on quite soon, I have been wondering what area to specialise in. Not just for my project, but in general. And when I say 'pure maths' I mean numerical methods, logic, calculus, algebra etc since I don't really know the exact definition of pure mathematics :P

...

Statistics- easy, boring and quite highly sort after in the real world (I think)

Pure mathematics - Fun, hard, no idea about if there are much prospects in the real world in this field though. It's also where my true expertise lie, i.e I am fast at solving challenging problems in this area and catch up quickly compared with my coursemates. But if there are few job prospects then I'd rather not.

Computing/programming - Fun, challenging, highly sort after in the real world, but I have no knowledge in this area since my degree doesn't cover it, so would have to learn it by myself and even then I would have no proper certificate to prove my knowledge in the area like I would with maths.

finance - Just no.

Perhaps listing modules that you have done, or have not done yet but look interesting to you, would be a good idea. Then we can say things about those modules, and what "areas" of maths that they represent, while also giving you some idea of what "pure maths", is supposed to mean. Statistics is extremely useful in the real world and in science, job prospects in pure mathematics tend to be in academia and perhaps in GCHQ or something like that. I would treat programming as an "extra" thing you can do, plenty of maths graduates go into it, but it doesn't make much sense to compare it to stats or pure maths.

In the course of a maths degree you could specialise in pure maths areas, or stats areas. If your degree had a computer science component you could specialise in the mathematical aspects of that, but you said they didn't. So surely anything in computer science would just be a job after graduating, not something you specialise in during your degree. So it's a different type of choice. It refers to job hunting, not to your degree in any particular way besides the fact that it's a maths degree.

[CHY872]

(Original post by Raiden10)
How do you mean? Godel's Incompleteness Theorem is a theorem of mathematics that is accepted as canonical (i.e. it has been proven). The only things that are contradicted by Godel's Incompleteness Theorem must then be false.

What I meant by that was things that contradict Godel, thus making themselves unprovable.

For example, the Strengthened finite Ramsey Theorem

For any positive integers n, k, m we can find N with the following property: if we color each of the n-element subsets of S = {1, 2, 3,..., N} with one of k colors, then we can find a subset Y of S with at least m elements, such that all n element subsets of Y have the same color, and the number of elements of Y is at least the smallest element of Y.

implies the consistency of Peano arithmetic if proved in Peano arithmetic - thus contradicting Godel's second incompleteness theorem and proving itself unprovable in first order logic. This is an interesting result (it's the Paris-Harrington theorem).

[Raiden10]

(Original post by CHY872)
What I meant by that was things that contradict Godel, thus making themselves unprovable.

For example, the Strengthened finite Ramsey Theorem


implies the consistency of Peano arithmetic if proved in Peano arithmetic - thus contradicting Godel's second incompleteness theorem and proving itself unprovable in first order logic. This is an interesting result (it's the Paris-Harrington theorem).

Well the theorem itself doesn't contradict Godel's Theorem. The provability of it in Peano Arithmetic does that.

You refer to Godelian undecidability results (the axiom of choice doesn't count). If you're interested in that then perhaps you should read up on Hercules and the Hydra.

[CHY872]

(Original post by Raiden10)
Well the theorem itself doesn't contradict Godel's Theorem. The provability of it in Peano Arithmetic does that.

I fail to see how what I wrote is incorrect. All I said (in less detail) was that you can find a contradiction to Godel's second incompleteness theorem through other branches of maths (i.e. prove Peano arithmetic in Peano arithmetic) - thus making what you've used unprovable (in Peano arithmetic).

[Raiden10]

(Original post by CHY872)
I fail to see how what I wrote is incorrect. All I said (in less detail) was that you can find a contradiction to Godel's second incompleteness theorem through other branches of maths (i.e. prove Peano arithmetic in Peano arithmetic) - thus making what you've used unprovable (in Peano arithmetic).

I don't understand.

[Txi]

Possible explanation of multiple dimensions and how to get there by the law of powers.