Wednesday, 30 November 2011

Reflections, a slight slowing down of pace and an expansion of aims

Ok so having had time to digest the exam results I've become slightly reflective. It seems to me than this year was dominated by Pure maths with a lot of scratching of my head and a lot of what the f**k does this mean. Why ? because unlike say Applied maths I found that Pure Maths does just not flow easily. So my first step is to do just 1 pure maths course per time block and also to have some relief from the incessant puzzlement that I faced this year. I got away with skimming of some of the more difficult proofs in M208 but think I became unstuck on M337.

To this end I feel that as a course such as M338 promises to be quite conceptually demanding (even more so than M337) It deserves my full attention. As light relief :) I intend to do MS324 (Waves etc) as preparation for MST326 and postpone M336 till October 2013. This means that I will have to postpone the MSc in maths till October 2014 (1 year later than I had hoped for). But I did find it quite frustrating having to skim over proofs. However there is another reason why I want to give myself some extra space and that is because I discovered this website over the weekend.

One of the frustrating things for me after having got my diploma in  music was that there appeared to be no follow up in terms of getting to grips with the theory or any way to learn composition. So this discovery is a God send for some one like me. The chance to get coaching in composition culminating in a year woking on large scale work with an established composer seems to good to miss.

I will need about 6 months to revise A214 the OU music theory course and get myself upto speed. I will also continue to do the odd philosophy course at undergraduate level either the OU the Oxford courses or the dept of continuing education here at Edinburgh but I'll put the aim of doing an MA in philosophy  on the back burner for now.

So the timetable for the next few years looks like

Amendments (26th Dec 2011) Have decided to drop M336 as it would take my second Open degree to over 360 points. I can easily get the background from other sources and my interest in group theory is applications of symmetry to problems in Physics and Chemistry which isn't really covered by M336, also that means I can start the MSc in 2013 as originally intended, so the timetable looks like

                     Pure Maths     Applied Maths   Music           Philosophy             
Feb  2012             M338        MS324                                Various  
 June 2012                                                     Comp 1             
Oct  2012             M381       MST326         
June 2013                                                      Comp  2
Oct  2013    Start MSc                                                       AA308
June 2014                                                      Comp  3           
Oct 2014                                                                           Philosophy MA

Prior to Oct 2014 doing 2 or 3 10 pointers per year of  philosophy either via Edinburgh University or Oxford dept continuing education, although the Edinburgh ones are considerably cheaper than Oxford and also involve live interaction. Next one on the agenda is Philosophy of the Arts starting in April 2012.

That seems to cover my main interests in a reasonable amount of time. I'm really thrilled that I've found a way to continue my studies in music. Although I'll always be a Salieri rather than a Mozart. I spent three hours trying to harmonise a 16 bar melody and trying to avoid parallel fifths and octaves. Very difficult if you use triads in root position 135 followed by another 135 (eg CEG followed by DFA) seems almost guaranteed to give you a parallel fifth. Still practice practice.

Tuesday, 29 November 2011

Wow Results Out Amazing

Well I'm flabbergasted the results have been published today two weeks before I was expecting them.
Anyway as expected I got grade 2 for M208 (4 short of a distinction) and just scrapped a grade 3 for Complex Analysis.

So reasonably satisfied with my grade 2 for M208 but a bit disappointed with my grade 3 for M337 still it's a fair reflection of how I performed on the exam in both cases

Sunday, 20 November 2011

Hume on Induction

Hi I have been catching up on Hume over the past two weeks. I have an exam in two weeks, just a single question to answer in 45 minutes. and then an essay to write before the 20th.  We have been told the types of questions in advance and I have chosen to do one the topic of Induction. Here is a short essay on the subject. I had known vaguely that Hume was the first person to think that there could be no rational justification for the principle of induction. On the other hand I had never realised just how powerful his arguments were. Here is a short OU TMA type essay on the subject which I wrote to clarify my ideas on the subject (although a bit short by TMA standards). I hope you find it interesting.

I will write an essay on Hume's Scepticism for the seen part of the assessment

Hume on Induction

This essay will summarise Hume’s arguments as to why there cannot be any rational argument or argument from experience (which Hume call’s probable or moral reasoning) to justify our use of Induction. Thus we can only justify our inductive reasonings by custom or habit. Given Hume’s empiricist views, that Hume thinks there can be no rational justification for induction is not so surprising. However it is quite surprising that according to Hume there can be no empirical justifications for induction either. The argument that follows is based on section 4 of Hume’s Enquiry (Hume, 1748 pp 20-29). Peter Millican (2002 ) has shown that the argument presented in the enquiry is quite different from that in the Treatise (Hume, 1739) and that Hume introduces a new premise in the argument of the Enquiry that is not present in the argument of the treatise, namely the appeal to the uniformity of nature. However as Hume points out this cannot be used to justify the principle of induction on appeal to experience as it involves circular reasoning. Neither can the uniformity of nature be justified on rational grounds (In what follows the numbers in square brackets refer to the relevant paragraph in section 4 of the Enquiry)
Induction, as Hume defines it [16], is based on a move from the fact that an object has in the past been associated with a particular effect to the fact that is reasonable to assume that similar objects will in the future be associated with similar effects. Note that Hume is not saying that from an observation of many white swans, all swans are white as traditional accounts of the problem of induction would have it. He only wants to make a move from previous observations of an object to the behaviour or properties of a similar object presented to us. The dilemma is that Hume cannot see a way to justify this principle apart from an appeal to custom or habit.
A simplified structure of Hume’s argument is as follows (For a more detailed analysis see Millican 2002 Ch 4)
Premise 1: (Hume’s Fork) All reasoning is either about relations between ideas or of matters of fact. [1]
Premise 2: Reasoning about relations between ideas are based on deductive or a priori reasoning.[1]
Premise 3: Reasoning about matters of fact are based on probable reasoning or experience. [2]
Premise 4: In order to extend our reasoning about matters of fact beyond our immediate sense experience or memory we must appeal to reasonings based on cause and effect. [4]
Premise 5: Induction attempts to move beyond our immediate sense experience or memory to predict what will happen in the future, thus if there is a justification for induction it can only be by an appeal to cause and effect. (This is implicit in Hume’s argument)
Premise 6: All reasonings concerning cause and effect cannot be based on deductive reasoning. Therefore (Intermediate conclusion): from Premises 1 and 3 reasonings based on cause and effect must involve an appeal to experience [7].
Premise 7: All reasoning based on cause and effect involves an appeal to the uniformity of nature. However the uniformity of nature cannot be justified by an appeal to experience as this presuspposes the very thing we wish to assume. Neither are there any rational grounds for this proposition [19].

Conclusion: The principle of Induction cannot be justified either by an appeal to deductive reasoning or an appeal to experience. Hence we can only appeal to custom or habit to justify our inductive reasonings.

It should be pointed out that Hume is not denying the usefulness of inductive reasoning. Indeed he appeals to it in the latter half ot the Enquiry he just wants to alert people that it is impossible to justify this principle apart from an appeal to custom or habit.In what follows I will expand on the basic structure given above.

Hume begins section IV by introducing what has been labelled as Hume’s fork namely that reasoning can be divided into two types That concerning relationships between ideas and that which can be divided into relationships between matters of fact. By the first Hume has in mind reasoning involving mathematical or geometrical truths. Thus a proof that the internal angles of a triangle add up to 1800 would be achieved by deductive reasoning from Euclid’s axioms. There are a number of important features of this type of reasoning

1) Once demonstrated a theorem achieved by deductive reasoning will always be true.

2) It is impossible to conceive of a contradiction, all triangles in Euclidean space have Internal angles adding up to 180 degrees.

3) It only takes one example of a proof by deductive reasoning to establish it as true for all cases.

On the other hand reasoning concerning matters of fact are not ascertained by deductive reasoning, but by what Hume calls probable reasoning. It is reasonable to assume that the sun will rise tomorrow, but it is possible to conceive that it will not rise without contradiction. Thus reasoning concerning matters of fact differ from those concerning relationships between ideas.

If we want to extend our knowledge regarding matters beyond immediate sense experience or memory we must make an appeal to cause and effect. If I’m asked why I know my friend is in France I will say because he told me or that I have a postcard which arrived from him this very morninng. The question thus arises, as to what is the type of reasoning concerning cause and effect. Hume argues that it cannot be by an appeal to deductive reasoning for the following reasons:

1) If a person is presented with a new object despite their having extemely good powers of reason he or she cannot, without appeal to experience predict any of its effects. Thus from the colour and consistency of claret, alone it is impossible to predict in advance its restorative or pleasurable effects.

2) It is perfectly possible to imagine a number of possibilities in any given situation without prior experience. Thus to use one of Hume’s favourite examples, when considering the collision of two billiard balls without prior experience it is quite possible to conceieve that one will avoid the other or that the first one will stop whilst the second one moves. However deductive reasoning based on say a knowledge of what Hume calls an objects secret powers will only be able to deduce one possibility.

3) Relationships of cause and effect require many observations to establish the relationship whereas deductive reasoning only requires one demonstration to establish the truth for all time.
Thus given the logic of Hume’s fork, relationships based on cause and effect can only be based on Experience [14].

This raises the key dilemma for Hume, because if we now ask what is the foundation of all our arguments from experience we are faced with a more difficullt question. Hume [15] claims that he can find no reason either from deductive or probable arguments. Hume argues that when we see for example, a glass of claret, without much knowledge of it’s ‘secret powers’ we immediately, based on our previous experience expect it to give us pleasure, but if drunk to excess will make us drunk and give us a hangover in the morning. Our reasonings for this and similar examples are based on an appeal to the uniformity of nature. However if we were to further ask what is the justification for this assumption, we can only appeal to experience. Thus the argument would be circular. It is expected that an object with a similar appearance to one we have previously encountered will behave in a similar manner, by appealing to the uniformity of nature, but our only foundation for this belief is based solely on our previous experience.

Again it is not possible to justify the uniformity of nature on deductive grounds, because we know that in practice whilst similar objects have similar effects they are never quite the same. ‘Nothing so like as eggs’ [20] as Hume points out, but each egg we taste will taste slightly different. On the other hand whenever a proof of deductive reasoning occurs it will be true once and for all. Thus if it were possible to deduce the properties of eggs in a purely rational manner, it would apply to all eggs without fail and there would be no variation between them. Furthernore it takes experience of a number of eggs to deduce their properties whereas a theorem produced by deductive reasoning only requires one demonstration to prove it for all time. As a final point it is quite possible to conceive that the laws of nature might change or that they would be different to what they are now, but this contradicts the feature of deductive reasoning that it cannot allow of any contradictions.

Thus there is a real dilemma, Hume has shown that inductive reasoning only works by an appeal to the uniformity of nature, on the other hand this principle cannot be justifed by an appeal to experience as this would involve us in circular reasoning. Neither can an appeal to deductive reasoning help to justify this principle. Thus there can be no form of reasoning either deductive or probable which will enable us to justify our inductive experiences. It would seem that we can only appeal to custom or habit.
Hume D 1739 ‘Treatise of Human Nature ‘
Hume D (1748) ‘An Enquiry concerning Human Understanding’ (Oxford Classics edition edited by Peter Millican Oxford 2007)
Millican P ‘Humes Sceptical doubts concerning Induction’ Ch 4 of Reading Hume on Human Understanding Oxford 2002.

Monday, 14 November 2011

M208 Review Part II

This is the second part of my review of M208 covering both linear Algebra and Analysis

Linear Algebra covers 1 block of 5 units. The first part should be fairly familiar to those who have done MST221 as it covers some coordinate geometry a bit of vector algebra mostly concerned with the vector equation of a straight line and some properties of conics (ellipses, hyperbolae and parabola's) Then there is a bit on linear equations and their solution using matrix techniques and Gaussian elimination this extends work in MS221 to three dimensions. A slightly tedious and tricky aspect is row reduction but if care is taken it is quite straightforward. There is also an introduction to the calculatrion of eigenvalues and eigenvectors of a matrix.

Then there is a brief introduction to what I feel is one of the most fascinating branches of mathematics the extension of Vector techniques to general spaces including functions. This fascinating area leads eventually to the mathematical structure of quantum mechanics (But none of this is mentioned in the unit alas). I have remarked in an early post how linear algebra can be extended to functions with the notion of a scalar product and othogonality being generalised to functions that can be defined as orthogonal by an integral over a suitable range see my earlier post for further details

As I've said before in the early days of the OU there was a whole course devoted to linear analysis M203 which developed the analogy covering both the pure aspects and it's applications to the classical differential equations of classical physics namely the heat, diffusion and wave equation. (And there was no pretending that a second level course is really a third level course as is currently done with a lot of the current third level courses). Anyway for those who want more on this subject as I've recommended before this is the set book on which the course was based is called an Introduction to linear analysis

and has the right combination of pure and Applied maths really worth investing in as a prelude to Functional Analysis. Finally as I've also mentioned those who want to understand the mathematical structure of quantum mechanics Linear Algebra and Analysis is a necessary prerequisite. Essentially the mathematical structure of quantum mechanics is that of a linear vector space over the field of complex numbers. I believe the second part of the OU quantum mechanics covers this. Although obviously it does not cove Functional analysis. So a good block just slightly disappointed that it didn't take things further. I'll have to wait until the MSc to pursue my interest or may be I'll stick with Krieder for now.

Finally we come to the best part of M208 namely Analysis. A lot of people found this difficult and it probably is the hardest part of M208. Analysis is the attempt to make calculus rigorous and it can be truly said to divide Pure mathematicians from Applied mathematicians, Physicists and Engineers. I have spent most of my career ignoring this whilst developing my caclulus skills. Part of the problem is that 30 years ago the books available to explain analysis were a very concise, just presented the theorems with out much motivation and it seemed really difficult to see what was going on. Fortunately times have changed the best introductory book on analysis for those wanting to do M208 is by Brannan and I would strongly recommend those contemplating doing M208 this year to invest in it as quickly as possible

It contains all the analysis covered in M208 but has the convenience of not being split up into units and there are no irritating breaks for video programmes or CD's.

Ok so what does analysis cover it starts off with the real number system and care is taken to define upper and lower bounds of real numbers. It is possible to construct the real numbers from the rational numbers by a procedure due to Dedekind but this is not covered in M208 partly because introduced right at the begining of a course it would be even more off putting than it is already.

Then a study is made of convergence of sequences and series culminating in the first definition of continuity based on convergence of series. It is quite straightforward to master the techniques for testing whether or not a  series convergence or not but remembering the details of each proof is quite tricky but these are never examined. One does have to be careful to introduce all the definitions carefully and examine the conditions of each part but as there are structured answers one (eventually) gets used to what is needed.

In the second part another definition of continuity is introduced the notorious epsilon delta definition of continuity this puts analysis on a really rigorous basis. Fortunately in M208 and Brannan this is introduced gently and with plenty of examples. These culminate in the testing for continuity of pathological functions such as the Blancmange Function which is a function which is everywhere continuous and nowhere differentiable. This proof is quite tricky but well worth reading through once or twice. (You don't have to understand it to be able to answer either the TMA or exam questions but if you understand it you will really have achieved something. I can't claim to understand it fully yet but I do hope to look at it again until I do)

After that then the climax comes when it is explained how to differentiate and integrate rigourously, There is not much extension of techniques that have already been learnt. The focus is on understanding how integration and differentiation can be defined rigourously not on developing techniques of integration.
the final unit develops more on Taylor series which some people might find a bit dry but remember this technique is used in numerical analysis so worth getting to know if you are interested in mathematical modelling.

So Analysis is probably the most challenging part of M208 which is why I recommed anyone thinking of doing M208 to get Brannan as quickly as possible, you will not unless you are really dedicated have time to do justice to the analysis units in 8 weeks and exam revision. Investing in Brannan will buy you time so you can get used to the abstract nature of analysis, the first step to maturity as a Pure mathematician and I only wish I had studied analysis earlier.

In conclusion then M208 is a curates egg of a course, the pressure of TMA deadlines means that a lot of the formal stuff will be bypassed. This could of course be remedied by the OU making the TMA's at least focus more on the conceptual aspects of the subject. Like everything you get as much as you put into it. Technically it is possible to do well by repeating almost by rote the answers given at the back of the books and by annotating the handbook with set answers. If's that what you want fine, but whether you deserve to be called a mathematician by following that course is another matter.

Tuesday, 8 November 2011

Review of M208 Part 1 Introduction and Group Theory

Ok as promised here is my review of M208. My friend and M208 colleague Neil has already posted his review

So mine will be complimentary to his. In this post I will review the group theory and Introductory sessions

M208 Covers 3 main areas of pure maths with an introductory section. The areas are Group theory, Linear Algebra and Analysis. It  covers the pure maths element of a typical first year undergraduate degree in maths. The Applied stuff is covered by MST209. This may surprise some people as these are designated second level courses but It's one of the things one has to get used to with the OU that their open door policy means that a lot of catching up has to be done but realisitically MST121 and MS221 are really only up to A level. So in that sense M208 will be the first course beyond A level Maths that one encounters for many people and for me it was certainly the first time I had encountered the Pure maths topics especially analysis.

My motivation for doing this course was to finally understand the language of pure maths especially analysis despite having an MSc in theoretical physics which involved quite a lot of linear algebra and group theory analysis had always been a blind spot and like most practising physicists and Engineers when confronted with a proof involving say the epsilon delta definition of continuity I would either ignore it or scratch my head in puzzlement as to what they were getting at. So my main motivation was to understand real analysis as a prelude to other topics in analysis such as topology. Linear Analysis, Lebesgue Integration and Functional Analysis and in that I believe I succeeded.

Anyway here is a review of the topics


This covers the basics of functions, a systematic account of curve sketching. An overview of mathematical language and proof including proof by induction and a small amount of logic, finally an introduction to the number system including an introduction to complex numbers. Most of this has been covered in the last part of MS221 so those who have done this course should find this a bit straightforward.
The curve sketching is treated in a systematic way by the use of sign tables essentially breaking up a function into it's different parts and working out where a part is negative or positive. Whilst tedious this is a really good method for sketching a curve.

Group Theory

This was divided into two parts and for convenience I'll cover both parts in this paragraph. The first part is quite straightforward introducing the concept of a group, a subgroup left and right cosets and Lagranges theorem which is that the order of a subgroup must be a factor of the group as a whole. There is a lot of tedious stuff about the symmetries of planar figures and to be quite frank I would have appreciated it, had they taken a slightly more abstract approach introducing the concept of a dihedral group first of all and then showing how the symmetry operations of rotation and reflection can be related to each other in general. But no we were expected to be able to work out  what happens when a planar figure is rotated through a given angle then rotated about a given axis. Unless you are adept at something like rubiks cube this is a real pain. Yet as I explained in a previous post the rules are really quite simple.

The other topic covered in the first part was permutations again these can be a bit tricky but practice makes perfect. Also a small amount on matrix groups.

The second part builds on the first part developing the idea of conjugacy classes and extending symmetries to three dimensions. Here again this was a real pain, you now have to think in three dimensions and it took me ages to do a particular TMA question. There is in fact a systematic method for working out the symmetry group of a solid regularly used by Chemists and it would have been much better if the course introduced that rather than leaving everything to chance. Having had a chance to look at the handbook of M336 the next group theory course it seems to cover that there.

The second unit in the last part was probably the most abstract of the group theory units. Introducing the concept of homomorphisms and the relationship between the Kernel of a group and its image set. I need to go back and look at this again, as there was some pretty deep stuff there.

It then ended with the counting theorem again I need to look at this again I gave my rather strong views on the examples they chose namely colouring flags or rotating chessboards which I thought were rather infantile

I also got my fingers burnt on the M208 forum for pointing this out as some people found it easy.

So overall a mixed unit and probably the least enjoyable from my point of view it seems to me that the underlying structure of group theory is obscured by all the examples they choose and for people like me who find it difficult to visualise 3 d rotations or reflections I felt I wasted a substantial amount of time trying to work out what happens in such cases. Time which would have been better put to use by concentrating on the abstract structure of the subject.

The TMA questions were on the whole applications of concepts rather than conceptual ones testing your ablility to understand the concepts and your ability to understand proofs. Indeed that seems to be the general approach of M208 so although I can apply group theory to problems to say working out the cosets of a group and classifying it's subgroups, I'm still not that confident of my ability to generate proofs of theorems in group theory, for that I'll need a more abstract approach hopefully M336 will help me develop  a better feel for the abstract aspects of group theory.