Alshia said:
Hey chiro and thank you for the excellent response (from what I can gather).
If an undergraduate program is simply called 'Mathematics' does it necessarily imply that it is a Pure Mathematics program? I used the term 'General Mathematics' because in some Mathematics program (like MIT, from what little I've read) it says that students will be speciallizing in General, Pure or Applied Math in the final years, and if a Mathematics program is already a Pure Mathematics program from the start it makes no sense to specialize in Applied Mathematics.
Secondly, the issue is that I need to be able to prove that a formula works (which is what much of Pure Mathematics is about) to convince myself that it's true. But at the same time I am compelled to apply the knowledge to solve tons of practical real world problems.
Based on my past experience with how Mathematics is taught (all below college level by the way), if one takes some kind of applied course, the proof for the formulas and methods will not be studied. This will stunt my mathematical development because I need to understand the formulas and method to their very core to appreciate them.
Thus, I am hesitant about going into an Applied Mathematics program right off the bat. I would rather apply for a General Math course before specializing.
My default choice is to go into finance and other business-like industry, as the compensation will likely be more than satisfactory (just an intuition, correct me if I'm wrong.) However ideally I would like to involve myself in social science settings, doing research and applying the results to create innovative products for people.
I do not mind learning how to communicate my methods in laymen's terms. I enjoy conveying what I've learned or discovered (hence why I also wish to teach part time). Your description of the criteria of an applied mathematician is exciting for me because those are the kind of things I wish to become good at.
As far as my experience and understand goes, you will pick a specialization for mathematics that roughly corresponds to pure, applied, statistics or some combination thereof (for your information I am a math major).
All mathematics majors will have to do the fundamental subjects of the calculus series and will have to gain at least an exposure to the other fields of non-specialization at some level.
The major will concentrate usually on getting exposed enough to the fundamentals of the specialization.
In statistics this means doing a year long intro university sequence in probability and statistics and then later doing sequences in inference, regression modelling, experimental design, and applied probability.
For pure mathematics, this concerns getting a solid grounding in analysis, topology, and algebra at differing levels (pure mathematics like every field has its own unique specializations). These subjects have their own issues, but these form the core of the pure mathematics in many respects.
Applied carries in a lot of domain dependencies, but there are non-domain subjects like optimization, numerical analysis, dynamics and chaos theory, computational issues in respect to applied mathematics and general mathematical modelling. You will also look at the 'applied results' that have been worked out which include differential equations, integral transforms, and other similar kinds of things in an applied context. Integral transforms can be thought of as in terms of decomposing things into independent characteristics and then using those for whatever purpose. Signal processing is just really a huge application of this kind of thing.
For the second issue, this will depend on your focus. You can, even in an applied context understand enough about why something works without knowing the specifics, and a good teacher should be able to tell you this kind of thing, even an applied one. Many applied programs and sciences often have hand-wavy arguments for things that are usually good enough for the purposes of that particular focus.
It's really up to you to establish what the focus is: this will help you decide specifically where you wish to end up eventually. Also remember that the lines between pure, applied, and statistics get blurry so don't think that you have to belong to only one or even two of these groups. Also real world problems are, by their nature, applied however lots of things that have a pure designation (like number theory) are still applied in some form. You should also note that typically what happens is that pure mathematics ends up being applied, and Terry Tao's comment was that the delay is about 50 years. With the way things are going now, I wouldn't be surprised this reducing to 10 or even 5 years in the future (possibly being nearly instantaneous at some point).
With regards to proofs and such, I think that yes for many purposes this is fair characterization in one way or another. Again though, you have to be aware that the focus for each stream is different and there is a reason for this: each career has different goals in mind and usually what works is that if you need something complex done, you typically get people all with different focii to work together to solve something that typically one person could not do alone, even if they were competent in all areas. Think of it as a jack of all trades, master of none times n people vs n people who are really good in something distinct working together.
There are advantages and disadvantages of the two approaches and ultimately it depends on the task at hand, but eventually this is the way it is because there is a tendency for specializations working together to do complex things being the way to go.
Also you need to think about the kinds of problems we have nowadays: we are designing computers with god knows how many transistors, design issues, physics issues and so on as well as jets, bridges and everything else and this is why specialization is required by default because the spectrum of what is needed to know knowledge and experience wise is huge and also very specific and subtle.
For your comment about social science, you should note that human beings are scientists by default. Most of us don't quantify our experience, place strict controls on the apparatus, or experimental conditions and write papers for journals, but we are still kind of 'naive' scientists at heart (I use the analogue of 'naive psychologist': if you haven't heard of the term, it's based on a psychological theory in the early days of psychology).
People that have a lot of experience in a particular area are taken seriously and sometimes in some areas and industries, experience is a necessary collateral to be taken seriously and academia is actually in many cases not taken seriously.
So you don't necessarily need to be a scientist by formal designation to get experience and use that either in an advisory or speculative capacity: people do this all the time and they make all kinds of decisions about the direction of projects or even the direction of entire companies: their collateral for doing this is mostly based on experience and this is a natural thing across the board, and it should be intuitive to understand why this is the case.
If you want to communicate things in laymans terms, you are going to be a good applied scientist. If you end up working for a business or doing consultancy, this is definitely a requirement for the job.
The people that want applied mathematicians want them to solve real world problems, or at least give a structured set of advice for solving them (typically decision makers end up choosing how to solve it, but because of their scarce time and importance for decision making, they get analysts and other people to do the grunt work). Definitely I think you are in the right area given this statement.
