My mentor, Chris F, made a comment back on January 3rd, which I didn't get around to answering, but it is an important question and one that helps to explain what has been driving me in my study experiment that 'kicked-off' this blog, a couple of years ago.
Chris's comment is below:
"Changes of direction are always likely as you find out more about the subject and what it can involve. You haven't really explained what it was that has given such an antipathy to Applied maths or physics especially as you seemed so enthusiastic at the start."
I don't think I am feeling antipathy, as such, towards science or applied maths.
It is much more a case of self-examination /understanding and a little bit of discovering what makes me tick. And while it may sound a little trite, I really do, now, understand myself a bit better than when I first started.
From a pragmatic perspective, I look back at my premise for this experiment when I decided to pursue a PhD in physics. It appeared to me, to be the most difficult of human pursuits and I wanted to see if I could stay the course; and if I couldn't? I wanted to know at what point I dropped off.
The experiment progressed and I began to find that I wasn't enjoying the applied areas of mathematics, as much as areas such as number theory or analysis. I don't know if it was the way that the Open University presented applied maths that started my dislike of the subject, or whether it is just the way that I am wired up? I find that if I can understand something axiomatically; from first principles, then I seem to better understand and enjoy the subject on a very fundamentally level.
I do think that there might be a little bit of being a 'control freak', that is causing my problems. Meaning, that I generally struggle to accept something, and by definition, I tend to wrestle with it, if I am told just to 'accept' that the foundations that something is built on, are correct.
As an example, I know that a lot of differential equations just 'work' rather beautifully and can be used to describe some of the most elegant of scientific ideas. But there is a grumbling part of me, that doesn't like accepting some of these equations, because I haven't understood them from first principles. For example, as I read through some of the MST209 maths units, I felt like I was being taught the odd tool to tackle certain types of differential equations; but only if they were of a certain type.
However, when I studied group theory, despite it being utterly frustrating when being asked to apply this maths to wallpaper patterns or polyhedra etc... I knew that I could just follow back through the unit's axioms and always come out the back of it, with a very clear understanding of why, and exactly how, it worked.
Please don't misunderstand me; this post is not about me having a go at applied maths, or in some way stating that it is inferior. In many ways applied maths and science is clearly breathtaking.
My problem is, that I don't have the time (or the brains!) to go back to first principles in learning applied maths, that will provide me with a sufficient understanding of the mathematical background to allow my brain to accept and understand some of the tools and techniques that are taught.
Just to be clear; to try and go back to first principles in much of the applied maths and science that is needed to tackle real world problems, would be wholly unproductive, unless you were trying to understand the subject, for its own sake; rather than use it in a real and practical way.
I have to say, that I just don't think that I have the special type of abilities or intelligence that allows one to tackle, use and expand on applied maths and scientific principles.
Pure maths, I can do, I can understand (mostly) and I crave it, when I am not studying it. It may not be much of an explanation, but it just 'feels' right, somehow.
It is this craving, that is probably going to be the biggest and most important factor in keeping me studying into and beyond postgraduate work. Without it, I am surely doomed to failure?
So, as a reality check:
Do I now believe that I will be able to successfully work towards (and enjoy) postgraduate Physics studies?
Regrettably, no.
Do I believe that I will be able to successfully work towards (and enjoy) postgraduate studies in Pure Mathematics?
Absolutely, yes.
So then, comes the question about my blog. I am not into revisionist practices that wipe away previous paths and dreams; and part of the experiment embodied within this blog, is in keeping a diary of my path regardless of which direction it takes. So the blog will remain in its current format, and I will continue to contribute posts without much change in style and content.
I am a little worried that my blog précis and personal statement may confuse readers, as I begin to lean more towards pure mathematics, from October this year.
So I may remove a few words in the 'about me' section, to make my current goals a little clearer. But I shan't be adding any new ones.
And lets not forget, that I still have seven months of astrophysics ahead of me for which I will still need those Jedi powers, to keep me on track!
Interesting and thank you for your detailed response. I guess it's horses for courses really I love say solving a complicated differential equation or a knotty Integral and then being able to relate the solutions to something in physics.
ReplyDeleteIt is possible to make Applied mathematics axiomatic the book on linear analysis by Krieder shows how this is done. There is also a general theory of solutions to differential equations called Sturm Liouville theory. Unfortunately the OU in it's infinite wisdom abandoned a course based on Krieder and replaced it with the current MST209 which can give the appearance of just being a rag bag of tricks although I do think solving linear differential equations is more than that.
Anyway I'll be doing Number theory and logic later on in the year as well as Will Duncan who probably has finished the course in it's entirety by now. Looks like you'll have to make him your mentor now.
It's a bit early to decide what area of pure maths you want to specialise in but the area of computability seems one in which you might be able to get a career if the academic route doesn't work out. Also applications of say Galois theory and group theory to coding etc. All very interesting Anyway good luck in your chosen path and look forward to joining you for number theory and logic in October
Best wishes Chris
Well done Dan for making that kind of decision. Sometimes you need to be brave to do this and I applaud the fact that you are able to do it and are listening to your own gut feelings. I am sure you have made the right decision. If you are craving pure maths then you should definitely go with it. It will still be difficult, but if you always have that overriding feeling that it makes you tick, then you will get on better in the end. Some of the toughest frontiers in theoretical physics probably require a more pure maths background to understand them anyway, so the chance to cross over again will always be open to you, I am sure.
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