Thursday, 29 December 2011

Hume on Scepticism Part II

Greetings to my fellow readers and bloggers. I hope you all had a good Christmas, I managed to watch Lohengrin an intriguing mix of styles between the Ring cycle and Parsifal. Indeed Lohengrin is in fact Parsifal's son. Which may raise a few eyebrows as Parsifal in the opera devoted to him has commited himself to celibacy. Still consistency was never a strong point of mythology. Those who don't know Lohengrin should get to know it.  Act II is the best, the music looking forward to the Ring Cycle. Anyway this is not the main point of the post, which is to finish off my examination of Hume's scepticism.

Recall that Hume is arguing for a mitigated scepticism which recognises the limits of Human Knowledge, we saw in the first post on this that one of the benefits of Hume's scepticism is that it would avoid dogmatism. The other benefit Hume sees is that it will act as a brake on speculative philosophy by acknowledging that there are limits to what philosophy or any branch of human knowledge can achieve and we should take care to recognise speculation for what it is and concentrate on problems which we can actually solve.

It seems to me that Hume's lesson needs to be taken on board in a number of areas not least in some developments of physics. There is an interesting tension, on the one hand given Heisenberg's uncertainty principle  and Godel's theorem. we seem to be acknowledging that even our most rigorous methods of obtaining knowledge about the world have severe limitations, something which Hume would be delighted to

On the other hand some aspects of modern physics do seem to border on the realms of speculation to which
there can be no testable consequences. I refer of course to the speculation that seems rife in quantum physics such as  Many Worlds theories and also that attempts to quantise gravity seem to be inherently untestable.

Readers of earlier posts will know that I tend to view claims that the wavefunction in quantum physics as a real field with a healthy degree of scepticism. The simplest explanation of it's role is as a mathematical device for generating the probabilites of quantum effects to occur. Thus when a measurement occurs the wave function does not physically collapse, all that happens is that one of the possibilities is realised. This pragmatic and slightly unromantic view of the situation cuts out all the speculation associated with what for want of a better word is the mystical or science fiction interpretation of quantum mechanics. The Many worlders would agree with me that the wavefunction does not physically collapse, but they would argue (if I understand it correctly) that because the wavefunction is a real field every time a measurement is made the wavefunction splits creating another universe, but of course we have no means of communication with that other universe.
(How convenient),   If that is one of the prices one has to pay for seeing the wavefunction as a real field then I'm afraid it's not worth it. By definition such a hypothesis is untestable and therefore cannot be good science as it is 'Not even wrong'.

A similar criticism could be made of the current attempts to quantise gravity. Indeed such a criticism has been made of superstring theory by Peter Woit.

I first read this book about 6 years ago. In it he argues that superstring theory is inherently untestable as the methodology used there does not lead to anything which can be tested in the lab. For example it's not clear that it does contain the standard model of particle physics as a lower limit. In this it is quite different from say quantum physics or relativty which do reduce to Newtonian physics in the limit. Woit rightly in my opinion criticises the amount of money that has been spent on such research.
Woit has devoted a blog which challenges the claims of those who would over exaggerate the importance
of areas of research such as superstrings.

And I recommend it to my readers as an antidote to all the hype about modern physics one consistently gets in the media.

Part of the problem, it would seem is that superstrings and the mystical/science fiction interpretation of quantum mechancis seem to tap into some part of the human pschye for a modern form of mysticism. Well fine, but don't claim that this is real science and certainly don't expect the taxpayer to fund your lifestyle.

Why not admit that superstrings or many worlds theories are speculations, that quantum physics far from being a mystery is in fact a well developed tool for predicting the behaviour of atomic, molecular and nuclear systems with applications to astrophysics, particle physics, chemistry and other fields. There is more than enough awe and wonder to be found in the day to day applications of science to solve currently intractable problems and given that one can go out and test such applications I would have thought more rewarding.     

I'll let Hume have the last word, here he is referring to the typical 'proofs' current in his time, that the Supreme Being is actively causing everything (even my finger movements over the keyboard as I type this post).

"Though the chain of arguments, which conduct to it, were ever so logical, there must arise a strong suspicion, if not absolute assurance, that it has carried us quite beyond the reach of our faculties, when it leads to conclusions so extraordinary, and so remote from common life and experience. We are got into fairy land."

It seems to me that talk of many worlds or us living in a world of 10,11 or 26 dimensions is just as likely to have led us into fairy land as anything else.


Saturday, 24 December 2011

A Defence of Liberal Secularism for Xmas

Ok well as promised here is my defence of liberal secularism, call it Chris's Christmas Sermon if you wish. Today is Christmas Eve, and no doubt this evening and tomorrow, there will be many churches and other religious instittuions, denouncing liberal secularism, as it has led to a decline in moral standards as exemplified by the toleration of behaviour of the rioters last Summer. The true meaning of Christmas being obscured by the rampant consumerism of the past months. Moral relativism, leading to a disrespect for authorities and tolerating any form of behaviour, even that which casues harm to other people. There will no doubt be an attack on multiculturalism and the usual guff that councils up and down the land are banning the word Christmas for fear of causing offence to other religions. Finally, the claim, that without a belief in God there can be no basis for morality. We should remember, so the argument goes,  as the Prime Minister has recently stated that the foundations for our morality lie in Christianity and we as a nation should return to it.

In what follows I want to challenge 4  basic premises that form the basis of such beliefs these are

1) A belief in God is necessary for morality.
2) The basis for western morality lies with Christianity.
3) Liberal secularism is the same as moral relativism hence liberal secularists can have no values.
4) Non believers in God must be living impoverished lives as they are 'spiritually dead'.

In doing so I will draw on an Early dialogue by Plato called the Euthyphro dialogue and Mill's Essay on Liberty.

To take the first premise, that a belief in God is necessary for morality. This was shown to be fallacious by Plato in an Early dialogue which he attributes to Socrates on his way to his trial for impeiety. He gets talking to a young man Eutyhphro (hence the name of the dialogue) who is also on his way to prosecute his father for murder. This leads to a general discussion on the nature of piety which for the Greeks was considered synomonous with morality or Ethics. Socrates poses the dillema, Is morality or ethics, good initself  or is it  only good because God (or the Gods) have ordained them. If the former, then obviously one does not need God, or a belief in God, as a  foundation for  our ethical principles. If the latter, then what guarantee do we have that the commandments of God are in fact good. To say that God, defines goodness is to beg the question, especially as in some of the texts God is seen ordering the Israelites to smite other nations, or more pertinately in todays climate, it is the duty of Islamists to wage violent Jihad on the decadent West. Given the tendency of religious institutions to want to impose a theocracy on non believers, then they must presumably think that Ethical principles are good in themselves, but then you don't need a belief in God to justify them. It is quite remarkable that a dialogue written 2,400 years ago asks such a pertinent question. There is no justification for the view that ethical or moral principles require a belief in God.

I come to the second argument that our ethical views are based in Christianity. I would argue that 400 years before Christianity, the foundations of our ethical perspective was given by the Greek philosophers especially Plato and Aristotle. The difference between the Greeks and Faith based views of ethics is that whereas for the most part faith based ethics says simply these principles were ordained by God and are enshrined in the texts. (As if one could simply write down a system of ethics, valid for eternity 2000 years ago and simply apply it to another age, without regard to context or the changes in society) Philosophers would say that you must provide a good argument for your case.  Instead of basing our ethics on religious texts I would suggest that a study of the main philosophical ethical texts is a far better way to learn ethical principles. The benefits will be  learnng how to argue for your case, rather just 'It's in the Bible, Koran or whatever religious text therefore it must be correct'. Also a certain modesty (cf Yesterdays post) and an ability to revise ones position, when confronted with something that doesn't quite fit the usual ethical percepts. To give a trivial example, whilst we can probably agree, that in general lying is a bad thing, it would be a brave man  who when asked 'Does my bum look big in this?' by his girlfriend said yes. Obviously there are more serious cases does one want to reveal where ones friends are to an enemy who wants to kill them for example.  Anyway the point is that it was the Greeks who invented philosophical ethics 400 years before Christianity was even heard of and it is those principles which forms the basis of our society not Christianity.

This brings me to my third point the claim that  liberal secularists have no values. Anyone who claims this has obviously not read Mill's essay on liberty. There Mill introduces a very simple principle namely that any lifestyle belief or cultural practice is fine provided it does not harm any other person. This is a two edged sword, on the one hand diversity in life is to be encouraged and welcomed. On the other hand if cultural practices do cause harm to other people then those practices are to be condemned. Thus for example liberal secularists who follow Mill in the harm principle would have no problem in, attacking for example genital mutilation and it is not cultural imperialism to do so.  A further pertinent point that Mill makes is that offence is no harm. Religious people who are often vociferous in banning plays which criticise religion or make fun of it often claim that they are offended by this. Well fair enough you have a right to express your view but it is not your right to close down plays by threat of violence, neither is it your right to seek to kill those who write books criticising your religion. You want to attack liberals for having no morals or values, well then learn to accept criticism yourself. 

Far from liberals having no values I would argue that liberals possess a set of core values, firstly the harm principle as enunciated above, but furthermore we believe that a diversity of lifestyles is good for society we also believe that discrimination against people on the basis of ethnicity, gender and sexual orientation is also a bad thing. If your religious institution or way of life maintains discrimination against women or gay people then it's about time you changed your position. Furthermore if you are a public servant and you discriminate against ethnic minorities, women or gay people then you deserve to lose your job. The nub of the battle between liberals and the Pope for example isn't that liberals don't have ethical values whereas instituitionalised religion does. It is that there is a real clash of values and in the case of institutions which refuse to let women or gay people rise to the top based on their merits such behaviour is essentially racist, misoygnistic and bigoted. The more you denounce certain ways of life which you don't approve of, then the more you will be justly criticised and the more you just show your institutionalised bigotry. I leave it to you to ponder, whether or not such attitudes are consistent with the teachings of a Galilean peasant, who says you should love your neighbour as yourself.  

To my final point the alleged lack of fullfilment of non religious people as they are spiritually dead. I would argue that this is clearly not the case. The secular world has much to offer in terms of fulfillment, I prefer to use the word aesthetic rather than spiritual. Here it is abundantly obvious that the Arts have much to offer here. I would say from personal experience that for exampel the symphonies of Mahler, the operas of Wagner, the novels of Tolstoy, Dostoevsky and the paintings of Picasso have far more to offer than a few hymns or carols and the  continual regurgitation of the same old text week in week out. For other people it may simply be enjoying a good walk in the country or just cultivating friendships and family. Also one must not forget the joy that the study of the sciences and mathematics can give.

So, I long for the day when this world tolerates other lifestyles, when people can rise to the peak of their abilities, without being discriminated against, because of their gender, ethnicity or sexual orientation. I long for the day when people don't harm anyone else. I also long for the day when religious institutions live up to their founders principles instead of indulging in institutional bigotry as most of them seem to, 

Have a good Christmas Winter Solstice, excuse to get drunk whatever way you want to call it

Best wishes Chris 

Friday, 23 December 2011

Hume on Scepticism

First of all apologies for not posting for a while, my main concern has been with finishing off my Hume studies (although informally I will continue to read him as he is definitely one of the greatest philosophers of all time).
After a delay caused by gale force winds. I sat the exam which consisted of three questions of which we had to answer one in 3/4 hour. There was a question on induction and it's importance in the rest of the Enquiry. I think I answered the first part quite well, but did not spend much time on the second one. In answering the first part I essentially rehashed the post below. I wonder if you can be charged with plagiarism if you quote yourself ? I then spent last weekend writing an essay on Hume's scepticism, the question was is there a good Humean case for Scepticism ? The relevant section of the Enquiry is section 12.

Hume distinguishes three types of scepticism, the first due primarily to Descartes is called antecedent scepticism. In the first part of the meditations Descartes attempts to purge his mind of anything that is possible to doubt so that he can found a sure foundation from which he then move forward. As is well known
Descartes thought he could find an indubitable principle in the proposition 'I think therefore I am' (the cogito) but this does nothing really to help Descartes. His notorious dilemma that we may be deceived by an evil demon isn't really addressed, but left hanging. Invoking a benevolent God as Descartes does just begs the question. On the other hand even if we have such an indubitable principle such as the cogito, and development of this principle to some of Descarte's notorious conclusions such as the existence of God or the separation of mind and body can only be achieved by use of the faculties which Descartes has already condemned as faulty.

On the other hand Hume says that some form of scepticism is necessary to any good philosophy that we should always subject our initial premsises to scrutiny and careful check of our reasoning is a vital process. Indeed we may be so convinced of the rightness of our conclusion that we may have overlooked some faulty step in our reasoning. Hence the need to subject our ideas to external criticism.

The second form of scepticism is consequential scepticism, this starts from the premise that our experience of the external world is essentially through images present to the mind. But whereas the image we see varies from which angle or distance we see it, The object itself remains the same. Thus there is a difference between the world as we perceive it and the world as it is initself. The problem is that if all we have our images in the mind how can we be said to have real knowledge of the objects or even worse  how do we have any guarantee that there is a world independent of our experience of it. Hume admits that this problem so contrary to common sense seems insoluble, but Hume makes an important distinction between the philosopher in his study who is interested in the limits of Human reason to find out what the justification of our every day assumptions are and every day life. No one could live their lives as the Pyhrronean sceptics would have us do taking this seriously. The problem has to be acknowledged as a real one which may or may not be resolved one day but should not affect our every day lives.

Hume argues for a mitigated scepticism which acknowledges the limits of Human reasoning both deductive and inductive to provide a justification for many of our commonly held ones. The main argument is essentially the one he gives against our ability to justify  induction and in the essay I essentially reiterated this argument (plagiarising myself again ;).   There is a second less formal argument namely the benefits that would accrue to society if an acknowledgement of the limits of Human reasoning were to be taken on board.

Hume claims there are at least two, the first would be as a check against all forms of Dogmatism, which Hume sadly acknowledges affects a lot of human beings. The claim to have access to absolute truth quite often leads to seeing those who disagree with a particular viewpoint as enemies who should be destroyed.

As is well known this seems to have infected institutional religion which currently seems to brand anyone not religious as immoral and destroying the fabric of our society. Indeed our prime minister despite having a degree in philosophy, politics and economics seems to have jumped on this band wagon. I'll try and write another post in detail why I think this is totally wrong. But one of the main reasons why it is wrong to accuse liberal secularists of tolerating anything is given by Mill in his wonderful book An Essay on Liberty where he introduces a simple principle the so called Harm principle, namely any life style or belief can be tolerated provided it does not cause harm to other people. Thus liberals would not tolerate the behavior of the rioters in London as they definitely caused harm to other people. Thats not to say that liberals would not try to understand some of the underlying causes or condemn some of the punishments currently being meted out for example two years for stealing a bottle of water as unduly harsh. Or even condemn a society which tolerates Bullingdon bullies such as Boris Johnson or David Cameron himself in their undergraduate years smashing windows in a restaurant and then being let off with a caution and a small fine. As a final ad hominem point, the behavior of some of the Catholic Churches priests and even more worrying the attempted cover up in the child abuse scandal would show at the least that there is no strong link between belief in God and personal morality.
(I can see a secular Christmas sermon coming on watch this space !!)

I digress, Hume argues that instead of seeing those who disagree with us as enemies if we acknowledge the limits of Human reason we will have a certain modesty about our claims and see those who differ from us as fellow seekers after truth. Thus by dialogue rather than confrontation we may learn from each other and that way advance the search for truth in an open manner. Of course the realisation that there are limits to Human Knowledge may be troubling to some but we have to learn to live with it. Indeed as Locke says in his own Essay concerning Human understanding

"It will be an unpardonable, as well as childish peevishness (what wonderful language !!) if we undervalue the advantages of our knowledge, and neglect to improve it to the ends for which it was given us, because there are somethings set out of the reach of it. .... The candle, that is set up in us, shines bright enough for all our purposes. The discoveries we can make with this, ought to satisfy us... If we will disbelieve everything, because we cannot certainly know all things; we shall do muchwhat as wisely as he, who would not use his legs, but sit still and perish, because he had no wings to fly"

I'll talk a bit more about the second benefit that Hume thinks that will accrue by acknowledging the limits of Human reason and it's surprising relevance to current debates in modern physics in another post.
Hope you all have a good Christmas, Winter  Solstice, excuse to pig out, whatever and look out for my Christmas secular sermon

Thanks to all my followers over the past year.

Best wishes Chris

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.

Sunday, 23 October 2011

Normal service will be resumed in two weeks

Hi I apologise for not been able to post anything of substance over the next two weeks but I'm afraid I'm having to devote a lot of my spare time to work at the minute. The crisis should be resolved in two weeks time so I'll resume normal service. I think this week is this blogs 1st birthday. I would like to thank all who find this blog interesting and for all the new friends I've made. Who says mathematicians/physicists are anti social. It's only that it's difficult to find like minded people I wish more people were interested in maths as  opposed to Soap Operas or Football the world would be a much more interesting place.

Friday, 14 October 2011

General Relativity a reading list

Prompted by a discussion on the Maths Choice Forum I'm afraid I couldn't stop myself so here is a reading list for budding General Relativists. The question was what is a good mathematical approach to General Relativity. One of the contributors seemed a bit sniffy about the first book of the physics course The Relativistic Universe. Ok it's not the most mathematical approach but the basics are there namely introduction to relativity and the main solutions namely the Schwarzschild metric and the Robertson Walker metric. So a good introduction for those wanting more here is a brief survey.  

"Of the books I'm familiar with the most mathematical is Hawking and Elliss The large scale structure of space time but you do need a background in Topology and differential geometry before it becomes even remotely comprehensible. It deals with the singularity theorems that Hawking and Penrose discovered. There is little direct application to astrophysics.
And although there is a chapter on cosmological models you would have to spend a lot of time filling in the gaps

The monster which does justice to both the mathematics and astrophysical applications is Misner Thorne and Wheeler. Although some might find it a bit long winded more for reference than reading through
A reasonable compromise between concrete applications and abstract mathematics is Wald
All of these books are probably a bit dated an uptodate  version but more geared to applications than mathematics is
A book on mathematical physics which includes the background to differential geometry and also functional analysis for those interested in quantum physics is
Finally one of the first books on Geometric Algebra and its application to physics  is
Coauthored by the same Lasenby who wrote General Relativity an Introduction for Physicists.

Insertion added 15th October
Forgot to add these two classics
Chandresekhar on Black Holes

This has the encouraging paragraph at the end of chapter 9

"Every effort has been taken to present the mathematical developments in this chapter in a comprehensible
logical sequence. But the nature of the development simply does not allow a presentation that can be followed in detail, with modest effort:. The reductions that are necessary to go from one step to another are often very elaborate and so on occasion may require as many as ten, twenty or even fifty pages !! (my exclamation marks) "

Doesn't exactly make you want to read it does it.

And Weinberg's First book on General Relativity

All of these books have been sitting on my shelves and I occasionally take a peek sometimes even work through a chapter or two.

I'm in two minds about the usefulness of a purely abstract approach to General Relativity. Undoubtedly the Hawking and Penrose theorems regarding singularities are of great significance but the Topological Approach seems a bit like existence theorems in Pure maths. Yes they have been shown to exist, but what causes them or what are the consequences for that you need the detailed mathematical modelling that Astrophysicists use to construct what happens when stars collapse or work out the implications of a given cosmological model for say elemental abundances. For that the abstract mathematics is an elegant means to an end, be it differential forms or now geometric algebra, namely the components of the Riemann Tensor. Once you have those and solve the Einstein Field equations you end up with differential equations and the standard methods of solving them.
I'm currently trying to understand  the applications of variational methods to the standard solutions to general relativity especially the big bang. This seems a reasonable compromise between physical insight and the abstract methods certainly beats the usual slog of computing 128 Christoffel symbols only to find out most are zero.
I think to get the full picture you need both approaches but whether it is possible for a person to get the full picture in the typical lifetime of an undergraduate or a 1 year MSc course is another question.

Anyway hope to get back to my studies on the Variational method as applied to the Robertson Walker metric sorted by Christmas.

Wednesday, 12 October 2011

M208 and M337 Exams Debriefing

Well what a day I started with M337 I have to say it was a fair exam even though I probably wont get higher than grade 3.
Part 1 has 8 questions the topics were

1) Allegedly simple questions on algebraic manipulation of complex numbers. However the last part was a fiendish raising a number to a complex power. In general for a complex number z
$$z^{\alpha} = exp(\alpha Log z) $$
where Log z = log|z| + iArg z

So it should have been simple to work out but I got bogged down

2) Sketching some sets and their difference and deciding whether or not the sets are regions, compact etc
Straightforward but I froze and for the life of me couldn't after about 2 attempts sketch the difference between the two sets.
It took me about 1/2 an hour to do these questions which should have been a simple warm up was not feeling very happy.

Then the core 3 questions on Complex Integration my favourite topic think I managed to do justice to all three parts and finished these in the next half hour so calmed down and felt a bit better,

Question 6 involved Rouche's theorem which is a method of finding zero's of a complex function within a given interval managed to get the first part out which is quite straightforward but unable to answer the last part as you had to work out how many of the zero's lay in the upper half and I couldn't see how to do it.

Question 7 was a standard one on fluid flows and their complex potentials and sketching the stream lines
I got the complex potential but couldn't sketch the stream line as it involved a complicated circle and I couldn't remember where the centre of a circle of the form

    $$x^2 + y^2 + y = c$$ was

Question 8 was a question on Complex Iteration and determination of whether or not a point lay in the Mandelbrot set got about 3/4 of this question out

So reckon probably about 3/4 of part 1 answered correctly

Part II there were 4 questions
9 Part 1 was on the Cauchy Riemann conditions which I think I managed to answer more or less correctly
  Part II was on curves and the effects of a transformation. Normally the curves are quite straightforward but this time they weren't and so only managed to answer about 1/2 of the second part

Then my last question 10 should have been a straightforward derivation of a Laurent Series for the function

$$\frac{z}{1-cosz} $$

however I couldn't get it out all the usual tricks didn't work and I couldn't see how to proceed

I abandoned this and was able to answer a question on the Complex Integration of the series as they gave the answer. The second part was about the singularities of the function but by this time my brain was frazzled so I gave up.

So about 1/2 of part 2 answered. So not a satisfactory paper. I've done enough to pass probably grade 3
but certainly no more.

M208 This went much better
Part 1 had twelve questions
1 Sketch of a graph.
2 Exercise on proof involving the converse of a statement
3 Question on whether or not two sets were groups
4 Another question on groups and their cosets
5 A question on row reduction
6 A question on basis for a linear transformation I did not answer this as I hadn't revised this so left it
7 A question on the solution to an inequality
8 A question on whether or not two sequences were convergent or not
9 A question on the symmetry groups of a hexagon
10 A question on homomorphisms
11 A question on L'Hopital's Rule
12 A question on showing that an Integral was less than a certain value
    Think I got most of this correct 
So apart from question 6 managed to give reasonably full answers to most of the questions but may have lost one or two marks here and there

Part II had 5 questions of which I answered 2
13 A question on the diagonalisation of a 3x3 matrix. Seemed to come out so confident that I got most of the marks for this question
17 A question on the Taylor polynomial of a function which I got most of

     A question on the epsilon delta definition of continuity. By this time it was the last 15 minutes and I was feeling quite tired. Also as I did not write out a sample problem of this type in my handbook unlike some people I didn't phrase the answer properly so will probably only get a few marks for this question.

Still having answered most of part 1 and 3/4 of part 2 I think I've done enough to get grade 2 and if the examiners are feeling generous might get distinction.

Some people might think I'm being perverse in refusing to annotate the handbook. My argument is that the exam is a test of how much you know without any crutches. Not an ability to spot model answers to a question and then simply copying them out. There was much heated debate on this in the forum. In my opinion annotation should not be allowed however no doubt other people will think differently.

So a bit disappointed by my performance on M337 but reasonably satisfied with my performance on M208
Will try and review both courses in the next week or two and also how I plan to use down time productively.

Monday, 10 October 2011

Tick Tock E Day approaches

Well tomorrow is E day for me I have Complex Analysis in the morning followed by Pure Maths in the afternoon. Having been through two or three past papers for both courses. I seem able to do most of the
questions in Part 1 and about 1 and a half questions in Part 2 in the time available. So distinction is probably not likely and certainly not for Complex Analysis. Anyway I'll be happy with grade 2. I'm not going to do anymore revision. I'm just going to try to relax till tomorrow. It's a bit like a Kid waiting for Christmas Eve to be over so that he can open his presents.

Good luck to all my fellow course mates for tomorrow

I'll report back on Wednesday when I'll either be over the moon or down in the dumps

Till then Bye for now.

Friday, 30 September 2011

M337 TMA04 back and Hume

So got my last TMA for M337 back overall average in the high seventies so looking for grade 2 pass if the exam goes well. Have to say don't really feel I own the subject in the way I do for M208 where my average was in the mid nineties. Still we will see. Not long now D day or should it be E day is the 11th October Complex Analysis in the morning and Pure Maths in the afternoon. This weekend will be exclusively on Complex Analysis revision I hope to work through two papers thoroughly.

Anyway started my Hume course last night. The focus is on Hune's enquiry which is a smaller version of his magnum opus A Treatise on Human Nature. Hume attempts to apply the experimental method so successfully applied by Newton to the study of human nature, in a way it's a precursor to both psychology and sociology and it is arguable that from these perspectives that Hume has been superseded by developments in say cognitive psychology, or sociology. However Hume has an importance in establishing the limits of human knowledge, which Kant was able to develop further.

Hume as is well known is an empiricist, for him the rationalist speculation of Descartes or Leibniz over exagerates what can be achieved by reason alone. They sought to try to base human knowledge on what could be proved purely by reason in order to arrive at absolutely certain knowledge. Hume and other empiricists such as Locke  argue that what cannot be based on human experience, is speculation and they would argue the dangers of exaggerating what can be achieved purely by the use of reason. A bit like the debate that still goes on today between Platonists in mathematicians for whom mathematics uncovers the hidden secrets of a world of which the empirical reality is a mere copy. In contrast to  those who see us using mathematics as an aid to understanding, but would not make want to make the mistake of assuming that to every mathematical concept there must correspond an element of reality.

Hume begins his enquiry by contrasting what he calls the easy philosophy with more abstruse philosophy. The 'easy' philosophy purports to be  an extension of common sense and attempts to describe the importance of say virtue in ways which can easily be understood. By the use of poetry or literature. In Hume's day novelists such as Addison or the poet Pope would be adherents to such philosophy, in our day think of Alain de Boton.

Hume does not dispute the popularity of such philosophy but 'edifying philosophy' has it's limits and indeed there is a tedency for those who prefer 'edifying' philosophy to denigrate the more abstract philosophy because it is obscure and can only be understood by a few people. Hume offers three main arguments against this

1) A knowledge of abstract knowledge can help poets and artists, for example a painter gains much from a knowledge of anatomy.

2) Abstract knowledge satisfies human curioisity and should be encouraged even if it does not benefit mankind directly. Somewhat optimisitically, Hume claims that politicians and lawyers would benefit from a study of abstract philosophy as they would have a better understanding of how government works. Here Hume approaches Plato's ideas that the guardians of the state should be trained in abstract philosophy.

3) Perhaps his strongest argument is that quite often rational principles are misused as a cloak to justify all sorts of superstition. Indeed it is quite surprising that so called analytic philosophers of religion such as eg Plantiga or Peter Geach, see their main task to use modern techniques of analytical philosophy to maintain standard Christian doctrine no matter how obnoxious. For example the specious justifications often given for why God if he is all powerful and knowing allows evils such as Auszchwitz or Tsunami's to happen. (I might sound off about this in another post). For Hume the only way such specious reasoning can be countered is to master the abstruse metaphysics so that it's weaknesses can be exposed for all to see. Hume offers the hope that by enquiring seriously into the nature of Human Understanding it can be shown by an exact analysis of the powers and capacities of the Human mind that it is not fitted for such remote and abstruse subjects. He hopes to undermine the foundations of obscure metaphysics which has in the past served only to shelter superstition and absurdity.

In the next section Hume discusses the Origin of Ideas of which I will give a basic summary of next week.
If I get time this weekend I shall as promised review Richard II. I'll be watching Henry IV part 1 this weekend.   

Tuesday, 27 September 2011

M208 TMA07 back and Gotterdammerung

Got My last TMA for M208 back yesterday just made 90 due to rushing some group theory questions. Think I definitely prefer analysis to group theory or at least to OU group theory. As I've said before the parts that are relevant to physics is representation theory basically concentrating on representing group operations by matrices but this does not seem to play any part in the OU treatment.

Saturday was spent on a last tutorial for M208 Alan my tutor has been wonderfully supportive and I'm lucky to have him. He will be tutoring me for Topology and Group theory next year. Revision for M208 is under control but have to admit M337 needs some attention. I shall focus on this till Monday hopefully I can do two or even three papers by then.

On Sunday as a complete contrast I went to a concert performance of my favourite Wagner Opera
Gotterdammerung played by the Edinburgh Players Opera group.

Of which one of my work colleagues is the Leader. This has been an annual concert in the late summer the players meet for a whole weekend and friends are invited to come along on the Sunday.

It provides an opportunity for both amateur orchestral players and up and coming singers to perform in roles which they would not otherwise have the opportunity to do so. Ok so there were a few squeeks and squawks but there were no hitches. It is an amazing experience and I still have the music ringing in my head.


Friday, 16 September 2011

TMA06 M208 Back

Well got the 6th TMA for M208 back not bad in the 90's but the elusive 100% escapes me it highlighted one point which I had totally missed. Also it just goes to show how counterintuitive analysis can be.

The series

$$\sum_{0}^{\infty}\frac{1}{n}$$  is divergent but the sum

$$\sum_{0}^{\infty}\frac{-1^n}{n}$$ is convergent

to some people on the forums this seems obvious but to me given that the second series has alternating terms I would think it diverges to two different series. Anyway at least I know now.

Started revising my plan is to do 1 paper per week per course under exam conditions at the start of each weekend and then to consolidate my answers revise topics I'm not sure about until, the last few days before the exams. Which I will take off and try and do two papers for each course on the last four days. That will make a total of five past papers per course which should be enought. At least the exams are on the same day so I can divide my time equally on each course before each exam.

Anyway I'll review Richard 2nd later on this weekend and also summarise the first section of Hume's Enquiry.

Sunday, 11 September 2011

M337 TMA04 finished

After a fairly intensive couple of days I've managed to complete the final TMA04 for M337 hooray.
The questions were

1) Questions on conformal mapping alright once one had grasped the basic idea.

2) Questions on Fluid Mechanics
This is actually a continuation of part 1 as a large part of the fluid dynamics of an aereofoil can be seen as a conformal transformation discovered by Joukowski. There were two parts

  a) Some relatively straightforward questions on Complex Potentials and Stream Functions

  b) A quite tricky question on the flow past an obstacle involving the Joukowski transformation
     and the Flow ,mapping theorem.

3) Questions on Complex iteration including the properties of the Mandelbrot set

   Potentially this should be one of the most interesting units. However short of time I had to skim through the  material and did not do justice to the question. Espescially as the question on fixed points seem to give rise to a particularly awful set of fixed points. Also unlike M208 there is no fixed strategy given for say finding the Keep set of a complex function

I think potentially the second unit of this block is the most interesting and I hope to revisit it during revision

On the whole M337 has been a very stimulating course and I'm looking forward to doing the MSc course on Complex Variables. I'll post a full review after the exam.  I have the book for the MSc course already and will try and fit some of it in during down time. Tomorrow I'm taking the day off to finish off the last TMA for M208. Then it's a question of trying to do as many past papers under exam like conditions as possible.

 For now I'm just going to relax and have a few beers


Sunday, 4 September 2011

This and that and a bit on Hegel.

Not much happening really, ended up with a severe cold  on Thursday so brain hasn't been entirely in gear.

Finished TMA06 for M208 and safely posted it away had a wake up call when trying to the revision TMA blind ie with just the handbook in exam conditions. Only managed about 1/2 still this is usual when revision starts should do enough to finish it this week then revision starts.

Bit bogged down on unit D of M337, after the joys of residue calculus, this is rather tedious stuff on conformal transformations and Mobius transformations. Have finally finished the TMA question but had my head scratching quite a while. Doesn't help that key points are buried in the Audio Visual units. Looks like I'll be winging it for the last two units one on fluid dynamics and the other on the Mandelbrot set. A pity really as I was looking forward to the fluid mechanics part.

Other maths projects on hold until end of next week whislt I crack on with the TMA's which have to be submitted.

As a preparation for my projected MA in Contintental philosophy, and to complement my reading of Hume have started to read Hegel's Phenomenology of Mind or Spirit as some translators have it. Must admit to being deeply biased against it as one of my first books I read on Philosophy was Karl Popper's Open society and it's enemies. This remarkable book makes a link between Plato, Hegel and Marx and the rise of totalitarianism Hegel being one of the main targets of Popper. Popper's line if I recall it was that any attempt to see an underlying pattern in history is doomed to failure as it isn't scientific as it can't be falsified. However given Hegel's importance in 19th Century German philosophy he influenced Marx and it is claimed existentialism, it is important to have an understanding of his ideas. Also Popper's idea of science as essentially that of physics narrows the scope of a wide range of19th century  philosophical texts for them science is more like systematic knowledge rather than something which can be quantified. I can't believe it's all to be dismissed simply because it doesn't conform to the standards of physics. Even Hume's main works despite his dictum to consign any thing that didn't contain 'abstract reasoning concering quantitiy and number or experimental reasoning concerning matter of fact and existence' to the flames doesn't conform to the standards he set himself.  

I'll post more later indeed I want to provide a section by section summary of the main philosophical works on this blog however this is just first impressions.  Having skimmed through the preface, it's quite interesting. Hegel thinks he has found a method of breaking the dualisms associated with philosophy up to Kant namely the dualism between subject and object. he seems to want to go back to an Aristotlean notion of substance which he calls Nous and is translated as Notion but this is now a dynamic thing instead of the static thing of the meadival times. A naive interpretation would say that this is similar to the existentialist idea of being indeed Hegel uses terms like being in itself and being for itself which play a part in Sartre's philosophy

For Hegel science and mathematical truths only describe the appearance of things and provides a static view of the world. Nous however is always changing only presenting part of itself to us. The dynamic of history is driven by Nous's attempt to reconcile it's multifarious appearances with it's underlying being. At different times in something aking to a collective conscious Nous manifests itself. However it is never content to reconcile itself with one particular era or epoch. Or at least that was Hegel's view at the time he wrote the Phenomenology. Of course as is well known Hegel 15 years later was to claim (just as Fukyama claimed in the 1990's) that History had reached it's final destiny in the establisment of the Prussian State. He was wrong then and Fukyama has been shown to be wrong now. So I must confess to some ambivalence about Hegel. We'll see.

As a break from maths and philosophy going to watch Richard II tonight with a bottle of wine I'll let you know what I think Derek Jacobi is in the title role so should be worth watching. I hope also to do Henry IV part I and part II
during the rest of September.   

Tuesday, 23 August 2011

More on the Bisection Method

Error analysis and convergence for bisection method

Ok here is a basic summary of the error analysis associated with the bisection method. Let $$\delta$$ be the required tolerance and let $$e_{n}$$ be the error after the nth iteration.
Then as the bisection method  reduces the error by 1/2 at each iteration  interval, we have the basic recurrence relation $$e_{n+1}=\frac{1}{2}e_{n}$$

As this is linearly dependent  on the previous error the convergence is linear (unlike Newton Rhapshon) and can be comparatively slow to higher order methods as hopefully I will be able to show when I consider the Newton Rhapshon method.

So in terms of the original error e = |a-b|  (where a and b) are the points of the original interval, we have after n iterations
$$e_{n+1}=\frac{|a-b|}{2^n}$$    (2)
Now if $$\delta$$  is the desired tolerance then we can use equation 2 to get an estimate of how many iterations it will take to reach the tolerance call this N. So we have
$$\delta > \frac{|a-b|}{2^N}$$
Rearranging gives
$$2^N >\frac{|a-b|}{\delta}$$
Taking logarithms (it doesn't matter which)
gives$$N log2 > log|a-b| - log{\delta}$$ or
So if |a-b| = 1 say and $$\delta=0.5 . 10^{-5}$$
then $$N >\frac{log1-log{\delta}}{log2}$$
or N > 17.6

Hence the number of iterations it takes to converge to less than $$\delta$$  is 18.
I checked this for various intervals and amazingly it worked each time.
I love the way in this method it's the interval that determines the convergence it has nothing to do with the function at all. Simple and Brilliant.

Sunday, 21 August 2011

Cambridge Computing Projects 1 Root finding Bisection Method

This is just a short post on the current status of the Cambridge projects, there are about 18 members so far and counting. Four are active and others are just lurking for now which is fine.

We have started to look at the first project, automated root search, ie to find where f(x) = 0 , and their associated errors. There are three methods:

1) Bisection
2) Direct Iteration (which we have covered in MS221)
3) Newton Rhapshon (again which we have covered in MS221)

In this post I will discuss the method of Bisection. This  is really quite clever and simple. If an interval [a,b]
brackets a root of a function f(x0),  then f(a) and f(b) must have different signs, so that the product f(a)f(b) is less than zero. Let c lie half way  between a and b ie c = 1/2(a+b), Then evaluate the function at c

Now if f(a)f(c) < 0 then the root lies in the interval [a,c], else it lies in the interval [c,b]. Replace a or b accordingly and then bisect the new interval. Keep doing this until the error |a-b| is less than a predefined tolerance, then f(c) is the root of the function within that defined tolerance.

The convergence of the error is linear, as can be seen as follows. Given that the method guarantees that the root lies within a given interval, then the error for a given interval must be no greater than

                          e < |a-b|

As the error is halved for each interval, we must have the error at the n+1 th step is given by

                      e(n+1) = 1/2e(n).                       

 As this is linearly dependent on the previous error, the convergence is said to be linear. Whilst the method is guaranteed to work, provided the intial interval brackets the root,  it is in fact much preferable to have faster convergence. Of the techniques described above the Newton Rhapshon method has quadratic convergence provided the root is simple. I shall discuss this later, in the mean time I was able to write a simple programme to implement the method. The initial guess, being enabled, by getting the code to plot out the function before selecting the interval. Anyway a good start, I've nearly finished the Direct iteration method and I'll post the results of that next week, Looks like I'm in a good position to finish the initial project by the end of September.

This weekend I also completed the 6th TMA for M208 this was relatively straightforward the topics covered were as follows:

1) A question on limits and continuity including a basic question on the epsilon delta definition of continuity which I'm begining to like mainly via practice.

2) Questions on differentiability from the definition. Found the first part a bit tricky as it was a quotient and applying the definition directly led to a complete mess, so I was forced to follow the derivation of the differentiability of the quotient rule. Got there in the end but can't help think that there was an easier way.

3) Questions on partitions and lower and upper sums for a discontinuous function, alright provided you take care as to where the function is defined. Then (at last) some real calculus involving integration by substitution and derivation of a recurrence relation.

4) Questions on Taylor series, prior to embarking on the Cambridge Computing projects, this would have seemed a bit dry, but given that Taylor expansions form the basis of most numerical techniques, it has taken on a whole new interest. Those of my fellow students who are bored with Taylor series and haven't already enrolled on the Cambridge projects might like to enlist even just as a lurker.

Anyway really must get to grips with TMA04 for Complex Variables over the next week, I'm giving myself Monday and Tuesday of the week after next off, so hopefully will have finished it by then.

Also started to read Hume's enquiry in preparation for my Edinburgh Philosophy course, he has an interesting phrase for applied maths which he calls mixed maths. It's a joy to read and I'm looking forward to studying it in depth.


Friday, 19 August 2011

M337 TMA03 Back and Plans for next year

Got my third TMA for M337 back up in the higher end of grade 2 so consistent with my other TMA's. Just couldn't see how to do a question on analytic continuation, my tutor has given me some clues so I'll try and do that question again. Nearly finished TMA06 for M208 hopefully I'll polish it off over the weekend and I want to get stuck in to the final TMA for M337 which I hope to finish by the end of next week.

Anyway plans for next year have crystallised As already stated I'll be doing
M338 Topology
M336 Group theory and Geometry

and also
MS324 Waves, Diffusion and  Variational Principles

Also I'll be doing a little philosophy I'm registered for a course on David Hume at Edinburgh University Dept Continuing Education this courses counts for 10 CATS points and so is equivalent to 10 OU points.

Next year I'll be doing two more philosophy courses from Oxford University dept continuing education

Namely Epistemology and Philosophy of science again these are 10 points.

Come October I shall be doing

M381 Number theory and Logic
MST326 Fluid Dynamics and Mathematical methods
and the OU Philosophy of mind course.

So by June 2013 I should have finished my undergraduate studies in both Philosophy and Mathematics and then we'll see. I want to do the MSc in maths but I haven't quite worked out how to continue my philosophy studies. The Open University is not the only university to change it's course structure. Lampeter University has now been amalgamated into St Davids university and their philosophy courses have changed their structure they now offer three courses:

An MA in European philosophy (Kant, Nietszche, Foucault, the Frankfurt school etc). An MA in general philosopy and  an MA in Applied Philosophy. I'm leaning towards the European philosophy course,

as St David's have a large track record of providing distance learning MA's and PhD's it would seem the best place to do philosophy. That would take 3-4 years alongside the 4 years I envisage to do the MSc in maths

I feel that whilst Contintental philosophy is usually frowned upon by most Anglo - American philosophy as it's supposed to be nonsense eg A J Ayer once dismissed Existentialism as a misuse of the verb to be. It deals with more concrete issues than current analytic philosophy. Also I get the feeling that Analytic philosophy suffers from 'Science envy' it's trying to do science without actually being scientific.

Also, it's not clear to me that analytic philosophy is immune from nonsense,. There is a large school of thought stemming from Lewis that takes the idea of all possible worlds having an existence. This has had (in my view) a pernicious influence in the philosophy of science especially enshrined in the Many Worlds interpretation of quantum mechanics. I find incredible that this idea is taken seriously however that is the current state of play.

I'm not convinced that I can understand mathematics or the foundations of physics better by studying the philosophy of it any more than I'm doing now. Also I want philosphy to represent another side of my personality.

 Given the current failure of capitalism it seems to me that the insights of the Critical school be it either French (Foucault etc) or German (Habermas, Adorno, Horkheimer etc) have something to offer and it would be worth trying to understand their ideas. 

In the next post I'll talk about the Cambridge Computing project which has about 12 recruits. Hopefully on Sunday evening