I was re-reading unit GTB1 tonight, of the Open University course M208 Pure Mathematics.
I just thought I would note down something that I have noticed, but which I have only just realized, on reviewing the material, this year.
Just a bit of background; M208 was the first pure mathematics course that I completed, and part of it contained sections on Group Theory. As any diligent group theorist would do, I began by learning the Axoims that lead to the definition of a 'Group', from a set of any given elements.
Now, one of those axioms is the property of associativity. That is:
For all g1, g2, g3 elements of G,
g1 o (g2 o g3) = (g1 o g2) o g3.
(the numbers are supposed to be subscript, but my Latex editor is currently 'up the Shoot'.)
Whilst I was studying the course, I had assumed that a group probably needed three elements in a set, in order to meet this axiom's criteria.
I say I assumed this, however, to be honest, I had never actually given it any real thought.
So, on revision of this material in 'slow time'; this is the sort of question, that seems to be popping into my head. A promising sign, in my eyes.
So, I glanced at the issue tonight and, of course, I quickly concluded that the axioms do not state that you need three elements in a set, in order to meet their requirements.
For example, if I have a set {1}; this could be a group under multiplication. Why? Well, it meets the axioms of Closure, Identity, Inverses and Associativity; where g1, g2 and g3 are all the set element {1}. There are no axioms that say that the elements in a set must be different, or that the elements must be distinct.
Why? Well, the axioms do not go into numbers of elements, or other topics, for that matter. The way that they are written, means that they do not assume that there is more than one element or even that the elements actually exist.
Having said that the Axioms certainly do not state, that there should be a none existent element, or that there should be a group that exists, which contains less than three elements.
So, the axioms certainly don't prove that there is a set of less than three elements that can form a group.
Logical, but confusing, I think.
Saturday, 8 June 2013
Wednesday, 5 June 2013
Pure Mathematics
Good lord, has it really been 8 weeks since my last post? I've had a bit of a break from all things studying for the last two months, as I had to defer my O.U module, Astrophysics, whilst I recovered from some health issues.
Before I left, I managed a couple of TMA's and also some of the research elements; and whilst I am sad to leave this interesting subject, I can't help but feel that it is probably actually for the best; as I can now, truly concentrate, on my new found love affair with pure mathematics. My degree profile will also probably look a a little more focussed, since it will contain all level 3 maths modules.
Also, in a strange twist of fate; whilst I have just effectively added another year to my level 3 study schedule; I hadn't realized that by doing Astrophysics, I would have been unable to complete Complex analysis, before moving onto the MSc. This would have been quite foolhardy, in my opinion. So, this could be a blessing in disguise.
It now means that I will take two pure maths modules in October, as planned (number theory and logic, with groups and geometry), but because I will now need to study more level 3 modules in 2014, I can now add complex analysis, to my quiver. I had planned to self study the complex analysis books, that were kindly sent to my by Chris F; but I think that Chris was quite right when he suggested, some time ago, that any future sponsor for post-graduate maths research / work, would question why one did not have such an important module.
Anyway, the study break has given me a good opportunity to start slowly re-reading some of my O.U module Units from M208, Pure Mathematics, which I completed last year. I have forgotten a fair bit, and there was some of it that I never entirely understood, the first time around (epsilon-delta, I'm looking at you!)
So, I have a rather lazy summer ahead, with some hopefully enjoyable revision of pure maths. I also plan to dust off my copies of Hardy's - Course in Pure Mathematics; Brannan's - Geometry and Spivak's - Calculus (Which should really be called Real Analysis, judging by the content of the book).
Before I left, I managed a couple of TMA's and also some of the research elements; and whilst I am sad to leave this interesting subject, I can't help but feel that it is probably actually for the best; as I can now, truly concentrate, on my new found love affair with pure mathematics. My degree profile will also probably look a a little more focussed, since it will contain all level 3 maths modules.
Also, in a strange twist of fate; whilst I have just effectively added another year to my level 3 study schedule; I hadn't realized that by doing Astrophysics, I would have been unable to complete Complex analysis, before moving onto the MSc. This would have been quite foolhardy, in my opinion. So, this could be a blessing in disguise.
It now means that I will take two pure maths modules in October, as planned (number theory and logic, with groups and geometry), but because I will now need to study more level 3 modules in 2014, I can now add complex analysis, to my quiver. I had planned to self study the complex analysis books, that were kindly sent to my by Chris F; but I think that Chris was quite right when he suggested, some time ago, that any future sponsor for post-graduate maths research / work, would question why one did not have such an important module.
Anyway, the study break has given me a good opportunity to start slowly re-reading some of my O.U module Units from M208, Pure Mathematics, which I completed last year. I have forgotten a fair bit, and there was some of it that I never entirely understood, the first time around (epsilon-delta, I'm looking at you!)
So, I have a rather lazy summer ahead, with some hopefully enjoyable revision of pure maths. I also plan to dust off my copies of Hardy's - Course in Pure Mathematics; Brannan's - Geometry and Spivak's - Calculus (Which should really be called Real Analysis, judging by the content of the book).
Wednesday, 3 April 2013
A Question of Pure Mathematics
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!
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!
First Taste of Real Scientific Research
Okay, TMA02 for my current course Astrophysics, was successfully completed and pinged to my tutor via the enigma, that is, the electronic TMA service with the Open University.
Actually, I like the fact that I can just upload my coursework as an electronic file and send it, without having the last-minute scramble to the post office, several days before the cut off date. Life is difficult enough, without having to give up 48-72hrs of prep time to the damn, inefficient and utterly unreliable Royal Mail service; so sending my work as 'naughts and ones', via the phone-line is much easier and a lot less stressful.
I think that one of my better decisions in the recent-past, was to anticipate having to send in TMA's as an electronic document, by starting to learn and use Latex in my coursework. I started doing so as early as my first level 1 maths course MST121, Using Mathematics. It wasn't necessarily needed; and I believe that I was in a minority, doing so. However, taking so much time to write out my answers using LaTex, at such an early stages, allowed me to practice and become adept at using the system; thus, writing level 3 physics TMA's and research or post graduate mathematics scripts, has not become prohibitively slow and cumbersome.
It's worked fairly well, as I've now committed many of the keyboard short-cuts to long term memory, and I can 'knock out' an in-line formula in seconds, rather than tens of minutes or hours. Any advantage at this level, needs to be grasped firmly, as the study material is difficult enough!
I say LaTex; but actually, I cheat by using Mathtype which is a 'what you see is what you get' equation editor; but it is fantastic and does a really professional job. I can't recommend it highly enough.
Anyway, having completed TMA02, I now have my sights rising to meet the next important step in my scientific journey this year. I am due to complete my first piece of real research within astrophysics.
I don't know all of the details yet; but I do know that I will be crunching large amounts of astrophysical data that have been collected from the Sloan Digital Sky Survey (SDSS) archive. I will then use this data and proceed to devise a project concerning the optical spectroscopy of previously unstudied quasars; a prospect that will make my recent hard slog and study of stellar evolution and nucleosynthesis, well worth the effort.
Actually, I like the fact that I can just upload my coursework as an electronic file and send it, without having the last-minute scramble to the post office, several days before the cut off date. Life is difficult enough, without having to give up 48-72hrs of prep time to the damn, inefficient and utterly unreliable Royal Mail service; so sending my work as 'naughts and ones', via the phone-line is much easier and a lot less stressful.
I think that one of my better decisions in the recent-past, was to anticipate having to send in TMA's as an electronic document, by starting to learn and use Latex in my coursework. I started doing so as early as my first level 1 maths course MST121, Using Mathematics. It wasn't necessarily needed; and I believe that I was in a minority, doing so. However, taking so much time to write out my answers using LaTex, at such an early stages, allowed me to practice and become adept at using the system; thus, writing level 3 physics TMA's and research or post graduate mathematics scripts, has not become prohibitively slow and cumbersome.
It's worked fairly well, as I've now committed many of the keyboard short-cuts to long term memory, and I can 'knock out' an in-line formula in seconds, rather than tens of minutes or hours. Any advantage at this level, needs to be grasped firmly, as the study material is difficult enough!
I say LaTex; but actually, I cheat by using Mathtype which is a 'what you see is what you get' equation editor; but it is fantastic and does a really professional job. I can't recommend it highly enough.
Anyway, having completed TMA02, I now have my sights rising to meet the next important step in my scientific journey this year. I am due to complete my first piece of real research within astrophysics.
I don't know all of the details yet; but I do know that I will be crunching large amounts of astrophysical data that have been collected from the Sloan Digital Sky Survey (SDSS) archive. I will then use this data and proceed to devise a project concerning the optical spectroscopy of previously unstudied quasars; a prospect that will make my recent hard slog and study of stellar evolution and nucleosynthesis, well worth the effort.
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