As a relief from all the angst about Pure maths and Topology which seems to be getting quite a few of us down. Radio 3 has decided to spend the whole of this week playing continuous Schubert.
http://www.bbc.co.uk/radio3/
Schubert is amongst my favourite composers. Some years ago I wrote this little biography and I share it with my friends
http://dl.dropbox.com/u/16049029/Schubert_bio_cdf.pdf
For those who don't know Schubert then you don't know what you've been missing. I would recommend starting with the last 3 piano sonatas preferably played by Alfred Brendel.
http://www.amazon.co.uk/Schubert-Piano-Works-Franz/dp/B00000417C/ref=sr_1_1?ie=UTF8&qid=1332792862&sr=8-1
The last quartets and String quintet by the Lindsay quartet
http://www.amazon.co.uk/Schubert-Late-String-Quartets-Quintet/dp/B0001Z2RSW/ref=sr_1_1?s=music&ie=UTF8&qid=1332792942&sr=1-1
Finally the three great song cycles with Fischer Dieskau and Gerald Moore.
http://www.amazon.co.uk/s/ref=nb_sb_noss?url=search-alias%3Dpopular&field-keywords=Fischer+Dieskau+schubert+song+cycles+
Anyway I hope you enjoy exploring Schubert as much as I have. Take the opportunity to listen to Radio 3 this week especially the main evening concerts. I've just finished hearing a performance by Paul Lewis, one of Alfred Brendel's pupil's of the two A major sonata's.
It's quite surprising just how quickly time flies I can't believe it's almost 10 years since I wrote the short biography. Hopefully it wont be another 10 years before I write another one.
Monday, 26 March 2012
Thursday, 22 March 2012
That sinking feeling
Help, I think topology must be one of the densest subjects to get into. I have to say I'm really struggling with this one as it's quite alien to anything I've come across. Still I'm not alone that does seem to be the consensus of a number of people on the forum.
I think part of the problem, is that one is dealing with general relations between functions, which aren't specified at all. In the analysis sections of M208 there was usually a function which was specified and it was possible, no matter how strange the function to say something about it. For example, the notorious Blancmange function of M208, yes proving it's continuous is a real pain in the neck (See the discussion I've been having with Duncan on this )
http://matrices-reloaded.blogspot.co.uk/2012/03/all-downhill-from-now-on.html
but at least you had something to get your teeth into. The function is defined, I more or less know the epsilon delta defintion of continuity and how to test if a function is differentiable or not. OK I might not grasp all the details of the proof but I can get the gist and I hope finally to consolidate my understanding with a few pints with Duncan next week.
With topology it's quite different, the definitions are about relations, between relations, between relations (alright I exaggerate a bit). It's very often definitiion 1, definition 2, combine definition 1 and 2 to get theorem 1. Definition 3 definition 3.1 Definition 3 and Definition 3.1 gives us theorem 2 and on and on, unfortunately there is no real road map as to where all this is going so it seems like wading through treacle.
Having said that, I think I've finally developed some sort of strategy to cope with this course. (a bit too late for the second TMA due in a couple of weeks time), but I think the only way to get a handle on this course is to try and learn the defintions by heart, at least in the first instance, testing yourself at odd moments during the day once those are grasped then you can begin to follow the theorems. This seems to have worked so far for unit A2. At first I was totally baffled, by the defintion of continuity for metric spaces, in terms of balls, (Indeed one can legitimately say topology or at least metric is a load of balls !!) A ball is essentially a generalisation of an open interval around a point in two or higher dimensional space. However having written it out a few times and more or less learnt it off by heart, it now begins to make sense and I can move on. Hopefully I've not left it too late to catch up. Fortunately we have a tutorial on Saturday and that should give me confidence to tackle the TMA over the next week. In the mean time I'll keep plodding away.
On a brighter note, I got my first TMA back from the Waves course, grade 1, but I dropped a few marks because of silly sign errors and also I had totally missed out a small section of one of the questions. TMA02 of this course is tempting, but as the school teacher said in the Pink Floyd's the Wall, "You can;t have your pudding until you've had your meat, how can you have your pudding if you don't eat your meat? "
So I have to eat the meat of TMA02 for the topology course before I can have my pudding of the waves course.
Finally in a slightly amusing way, I've left my small mark in Edinburgh. I was out drinking with some colleagues from work in the Blue Blazar a local pub just off Lothian Road, when a new guy who's just joined went to the toilet. He came up and said "Thats what I really like about Edinburgh, where else would you see Maxwell's equations scrawled next to some graffiti slagging off Hibs supporters". What is really amusing is that I had scribbled those on the toilet door about six years ago after a rather boozy Christmas do. I was amazed that they are still there.
Hopefully the mist will begin to clear vis a vis topology and I can't wait till my pudding but I must be careful not to gobble it all down quickly otherwise I'll drop some marks for silly mistakes.
I think part of the problem, is that one is dealing with general relations between functions, which aren't specified at all. In the analysis sections of M208 there was usually a function which was specified and it was possible, no matter how strange the function to say something about it. For example, the notorious Blancmange function of M208, yes proving it's continuous is a real pain in the neck (See the discussion I've been having with Duncan on this )
http://matrices-reloaded.blogspot.co.uk/2012/03/all-downhill-from-now-on.html
but at least you had something to get your teeth into. The function is defined, I more or less know the epsilon delta defintion of continuity and how to test if a function is differentiable or not. OK I might not grasp all the details of the proof but I can get the gist and I hope finally to consolidate my understanding with a few pints with Duncan next week.
With topology it's quite different, the definitions are about relations, between relations, between relations (alright I exaggerate a bit). It's very often definitiion 1, definition 2, combine definition 1 and 2 to get theorem 1. Definition 3 definition 3.1 Definition 3 and Definition 3.1 gives us theorem 2 and on and on, unfortunately there is no real road map as to where all this is going so it seems like wading through treacle.
Having said that, I think I've finally developed some sort of strategy to cope with this course. (a bit too late for the second TMA due in a couple of weeks time), but I think the only way to get a handle on this course is to try and learn the defintions by heart, at least in the first instance, testing yourself at odd moments during the day once those are grasped then you can begin to follow the theorems. This seems to have worked so far for unit A2. At first I was totally baffled, by the defintion of continuity for metric spaces, in terms of balls, (Indeed one can legitimately say topology or at least metric is a load of balls !!) A ball is essentially a generalisation of an open interval around a point in two or higher dimensional space. However having written it out a few times and more or less learnt it off by heart, it now begins to make sense and I can move on. Hopefully I've not left it too late to catch up. Fortunately we have a tutorial on Saturday and that should give me confidence to tackle the TMA over the next week. In the mean time I'll keep plodding away.
On a brighter note, I got my first TMA back from the Waves course, grade 1, but I dropped a few marks because of silly sign errors and also I had totally missed out a small section of one of the questions. TMA02 of this course is tempting, but as the school teacher said in the Pink Floyd's the Wall, "You can;t have your pudding until you've had your meat, how can you have your pudding if you don't eat your meat? "
So I have to eat the meat of TMA02 for the topology course before I can have my pudding of the waves course.
Finally in a slightly amusing way, I've left my small mark in Edinburgh. I was out drinking with some colleagues from work in the Blue Blazar a local pub just off Lothian Road, when a new guy who's just joined went to the toilet. He came up and said "Thats what I really like about Edinburgh, where else would you see Maxwell's equations scrawled next to some graffiti slagging off Hibs supporters". What is really amusing is that I had scribbled those on the toilet door about six years ago after a rather boozy Christmas do. I was amazed that they are still there.
Hopefully the mist will begin to clear vis a vis topology and I can't wait till my pudding but I must be careful not to gobble it all down quickly otherwise I'll drop some marks for silly mistakes.
Sunday, 4 March 2012
Progress
Well sorry for not posting for a while anyway progress of sorts has been made on M338 I got my first TMA back and did reasonably well but lost a few marks for taking short cuts. Also there was some difficulty in notation hopefully the wrinkles will be ironed out in time for the exam. Have to admit some of this course seems like wading through treacle and it's difficult to see the point of a lot of it. Still just keep carrying on I know a knowledge of topology is an essential pre-requisite if one wants to understand things like the singularity theorems of General Relativity proved by Penrose and Hawking so mustn't get too disheartened.
Also finished the first assignment for MS324 which was on the whole a joy to do I'm definitely an Applied mathematician. The questions were essentially a review of mathematical techniques one should be aware of with one or two wrinkles thrown in. The first question was a question on Partial Fractions and an integration based on taking the limit of the integral to infinity. This involved some tricky manipulations of Logartihms but I think I got there. The second question was a straightforward and pleasurable solution of an inhomogeneous second order differential equation. The third question was a straightforward but tedious question on multivariable calculus and finding the stationary points. What made this question a bit tedious was the fact that the nature of the points varied for a wide range of conditions and you had to be careful not to miss out any combinations. The final question was on the deformation of springs more of a modelling question than a mathematics question I think I got most of it correct but I admit the very last part seemed badly phrased so I'm not sure if I got it correct or not. Still it's only 3 marks so should be on target for a good stab at this.
Both courses haven't really got started yet I really hope I can penetrate the fog of definition and theorem that is plaguing the topology units but at the minute I really can't see the wood for the trees. Conversely I know that the first real block of MS324 is on one of my favourite topics namely the wave equation and it's solutions I and feel reasonably confident I can polish off the next TMA in the next two weeks. The problem is it would be like a kid who eats his dessert before his main course instead of leaving it to last but what the heck.
As a final point at the risk of overloading people I've found yet another book on mathematics as applied to physics.
http://www.amazon.co.uk/Mathematics-Classical-Quantum-Physics-Dover/dp/048667164X/ref=sr_1_1?ie=UTF8&qid=1330880090&sr=8-1
Although quite old it covers all the branches of mathematics (apart from Topology) and their direct relevance to physics. It has quite an accessible introduction to the Hilbert space formulation of quantum mechanics stresses the key analogy between geometric and vector spaces which is so important to physics as well as having lots of concrete and challenging examples. I intend to try at least some of the questions which involve the separation of variables of pde's in spherical coordinate systems something not really covered in MS324 but which plays an important part in both classical and quantum physics. Anyway nice to see a book covering both classical and quantum physics in the same volume. A bit less abstract than say Szekres
http://www.amazon.co.uk/Course-Modern-Mathematical-Physics-Differential/dp/0521829607/ref=sr_1_sc_2?s=books&ie=UTF8&qid=1330880446&sr=1-2-spell
but not just another recipe book, Szekres and Byron complement each other nicely in my honest opinion.
Also finished the first assignment for MS324 which was on the whole a joy to do I'm definitely an Applied mathematician. The questions were essentially a review of mathematical techniques one should be aware of with one or two wrinkles thrown in. The first question was a question on Partial Fractions and an integration based on taking the limit of the integral to infinity. This involved some tricky manipulations of Logartihms but I think I got there. The second question was a straightforward and pleasurable solution of an inhomogeneous second order differential equation. The third question was a straightforward but tedious question on multivariable calculus and finding the stationary points. What made this question a bit tedious was the fact that the nature of the points varied for a wide range of conditions and you had to be careful not to miss out any combinations. The final question was on the deformation of springs more of a modelling question than a mathematics question I think I got most of it correct but I admit the very last part seemed badly phrased so I'm not sure if I got it correct or not. Still it's only 3 marks so should be on target for a good stab at this.
Both courses haven't really got started yet I really hope I can penetrate the fog of definition and theorem that is plaguing the topology units but at the minute I really can't see the wood for the trees. Conversely I know that the first real block of MS324 is on one of my favourite topics namely the wave equation and it's solutions I and feel reasonably confident I can polish off the next TMA in the next two weeks. The problem is it would be like a kid who eats his dessert before his main course instead of leaving it to last but what the heck.
As a final point at the risk of overloading people I've found yet another book on mathematics as applied to physics.
http://www.amazon.co.uk/Mathematics-Classical-Quantum-Physics-Dover/dp/048667164X/ref=sr_1_1?ie=UTF8&qid=1330880090&sr=8-1
Although quite old it covers all the branches of mathematics (apart from Topology) and their direct relevance to physics. It has quite an accessible introduction to the Hilbert space formulation of quantum mechanics stresses the key analogy between geometric and vector spaces which is so important to physics as well as having lots of concrete and challenging examples. I intend to try at least some of the questions which involve the separation of variables of pde's in spherical coordinate systems something not really covered in MS324 but which plays an important part in both classical and quantum physics. Anyway nice to see a book covering both classical and quantum physics in the same volume. A bit less abstract than say Szekres
http://www.amazon.co.uk/Course-Modern-Mathematical-Physics-Differential/dp/0521829607/ref=sr_1_sc_2?s=books&ie=UTF8&qid=1330880446&sr=1-2-spell
but not just another recipe book, Szekres and Byron complement each other nicely in my honest opinion.
Sunday, 12 February 2012
General Relativity and Neutron Stars
Hi I've been trying to consolidate some of my physics Notes that I wrote (over 10 years ago How frightening) on general relativity. I've put a couple on my dropbox public folder
A general review of General Relativity
And an application to Neutron Stars
This gives the background and then goes through the original calculation that Oppenheimer and Volkoff did just before the Start of the second world war. I wrote it as at the time there did not seem to be a clear account of the actual calculation in the public literature. I was also gratified to find a misprint in the original paper.
I wrote this stuff as at work I used to convene a 'Failed physicists' society our work has lots of people who did PhD's and even post doctorates in physics or in my case a humble MSc but were brutally ejected from Academia. It arose out of a sense of frustration that the popular literature gave no hint of the actual maths and physics behind key caclulations or that people were doing calculations for themselves but had no means of sharing them with other people. Anyway in the hope that other people find this useful I have posted the results here.
As for the maths courses I had my first tutorial on topology last week with Alan and it was good to meet up with Neil. Alan's tutorials are always useful in that they clear up ambiguities, for example when defining a set which covers all of the Real numbers should we include infinity. The answer is no so an interval R is the open set (-Infinity, Infinity) and not [-Infinity, Infinity]. Despite this procrastination has meant that I haven't yet tidied up the TMA for final send off. Still I'm not the only one in this position
http://learn.open.ac.uk/mod/oublog/view.php?user=108287
Must be something to do with the gloomy and rather oppressive winter in Edinburgh
http://learn.open.ac.uk/mod/oublog/view.php?user=108287
Must be something to do with the gloomy and rather oppressive winter in Edinburgh
Also have been racing ahead with the MS324 course jumping straight into one of my favourite topics of Applied maths namely solutions to partial differential equations by separation of variables. For this topic unlike Topology or pure maths I feel like I'm coming home, or to use a crude expression 'like a pig in shit'.
Topics like this remind me why I got interested in maths in the first place. I love going to the pub, notepad in hand doing such problems and I find they just flow. Guess that means I'll always be an Applied mathematician/theoretical physicist, rather than a Pure mathematician/ Mathematical physicsist. I'll put another post explaining the difference.
Finally last week I got my essay back that I did for the Edinburgh university course on Hume. It was essentially a synthesis of ideas that I explored on my posts about Hume. I was pleased to get 85% and my tutor seemed impressed. He said he didn't have much to add except some criticism of some stylistic features. I tend when laying out the points of a philosophical argument to list the points separately (see some of my earlier posts). Anyway thats another 10 points towards my philosophy education I have 100 so far, and I look forward to the Philosophy of Art course in the Summer given by the same tutor Dr John Gordan.
As far as further studies go, It looks like I could have a unique opportunity to gain more than the usual 120 points for philosophy via the OU. The course I did A211 has been replaced with a more traditional course which is not an excluded combination with A211,
http://www3.open.ac.uk/study/undergraduate/course/a222.htm
Also the current third level course AA308
http://www3.open.ac.uk/study/undergraduate/course/aa308.htm
is due to be replaced with one entitled 'Key Questions in Philosophy' in October 2014 but given the way the OU works with replacement courses I suspect it wont be until 2015 till such a course appears on the horizon. So it looks like I could do the other second level philosophy course, AA308 and the new course by 2015 which would mean getting a total of 240 points in philosophy before embarking on postgraduate work.
That would suit my other ambitions by then I would hope to be well into the Maths MSc and have finished my compositon courses.
So must stop procrastinating and write up the Topology TMA this week and do the first TMA for MS324.
As far as further studies go, It looks like I could have a unique opportunity to gain more than the usual 120 points for philosophy via the OU. The course I did A211 has been replaced with a more traditional course which is not an excluded combination with A211,
http://www3.open.ac.uk/study/undergraduate/course/a222.htm
Also the current third level course AA308
http://www3.open.ac.uk/study/undergraduate/course/aa308.htm
is due to be replaced with one entitled 'Key Questions in Philosophy' in October 2014 but given the way the OU works with replacement courses I suspect it wont be until 2015 till such a course appears on the horizon. So it looks like I could do the other second level philosophy course, AA308 and the new course by 2015 which would mean getting a total of 240 points in philosophy before embarking on postgraduate work.
That would suit my other ambitions by then I would hope to be well into the Maths MSc and have finished my compositon courses.
So must stop procrastinating and write up the Topology TMA this week and do the first TMA for MS324.
Sunday, 29 January 2012
M338 and MS324 progress so far
Just a short post here, I've more or less finished the first TMA for M338. The first question seems apparently straightforward. A skectch of a simple function, specifying the subsets over a specified interval, and then finding the image set of a function which isn't quite an inverse as the function is many valued. Whilst straightforward it's probably a warm up getting us used to the specification of continuous functions in terms of the behavior of the inverse of open sets of a function. Sort of backwards but seems fundamental in topology.
The second question is a standard proof that a composite function involving trigonometric functions is continuous.
Then another question involving proving that a function is continuous on the real line using the epsilon delta definition of continuity. This is then combined with the first new stuff namely the extension of continuity to Euclidean space namely a vector space of N dimensions. We live in three dimensional space but it is of course possible to conceive of N dimensional space. The definitions of continuity in N dimensional space involve the Pythagorean distance between two points instead of the modulus function. Anyway much of the same ideas to continuity apply to N dimensional space with this distance function. The questions seem relatively straightforward but as with all pure maths it's important to learn the phraseology and make sure all aspects are specified. It is tricky to get this correct and I find it takes two or three goes to get it right. Sometimes it feels more like writing an essay rather than doing maths anyway it's all part of getting in to how Pure mathematicians think.
So it's a relief to get back to 'real' maths with MS324. So far the website hasn't opened but I've been working my way through unit 0. This is a revision unit to get one up to speed the first part covers Integration and differential equations. The second part concerns multi valued calculus and a bit of vector calculus leading up to the divergence theorem. So far so good, I've not found the problems too difficult and I think I stand a good chance of 'winging' this one namely just concentrating on the TMA's and past exam papers as a light relief to all the heavy stuff on M338. There are end of section questions and I will see if I can do those first before tackling the main units. I can't wait till I get the TMA's which should be available once the website opens tomorrow.
Friday, 27 January 2012
More books for a budding Physicist the (relatively) easy way
Hi Prompted by Keiths question on my recent blog as to whether or not there was one text covering supersymmetry, quantum field theory for some one with MST209 and M208 under their belt. The quick answer is not really apart from the route I mapped out in my two reading lists for General relativity and particle physicists.
However a more straightforward route to get a glimpse would be to take advantage of the following texts, as before the best way to get a backgound in classical physics prior to study of quantum physics is via Longair's book
http://www.amazon.co.uk/Theoretical-Concepts-Physics-Alternative-Reasoning/dp/052152878X
If one wanted some background on Maxwell's equations one could go also get hold of
http://www.amazon.co.uk/Students-Guide-Maxwells-Equations/dp/0521701473/ref=sr_1_1?s=books&ie=UTF8&qid=1327684577&sr=1-1
Then one would be ready to start on quantum physics and general relativity and a good overview but which skimps on calculational detail is given by the 'Demystified series' so in order of reading
Quantum Mechanics
http://www.amazon.co.uk/Quantum-Mechanics-Demystified-David-Mcmahon/dp/0071455469/ref=sr_1_1?s=books&ie=UTF8&qid=1327684714&sr=1-1
Relativity
http://www.amazon.co.uk/Relativity-Demystified-David-Mcmahon/dp/0071455450/ref=sr_1_2?s=books&ie=UTF8&qid=1327684746&sr=1-2
(This book does actually have quite an accessible introduction to modern coordinate free differential geometry)
Then Quantum Field theory
http://www.amazon.co.uk/Quantum-Field-Theory-Demystified-Mcmahon/dp/0071543821/ref=sr_1_4?s=books&ie=UTF8&qid=1327684746&sr=1-4
This is good as an overview but it really skimps on calculational detail.
Then you can try Superstings if thats your bag (It certainly isn't mine !!)
http://www.amazon.co.uk/String-Theory-Demystified-David-Mcmahon/dp/0071498702/ref=sr_1_5?s=books&ie=UTF8&qid=1327684746&sr=1-5
I have it on my shelf and whilst I can understand the maths I'm still mystified as to the relevance of superstrings to anything but worth having to get a glimpse of what all the fuss is about.
Finally and I wasn't aware of this there is one on supersymmetry (which I've just ordered)
http://www.amazon.co.uk/Supersymmetry-DeMYSTiFied-Patrick-Labelle/dp/0071636412/ref=sr_1_1?s=books&ie=UTF8&qid=1327685098&sr=1-1
The one thing that the series doesn't cover is applications of group theory to physics so my recommendation is still Jones
http://www.amazon.co.uk/Groups-Representations-Physics-H-F-Jones/dp/0750305045/ref=sr_1_1?s=books&ie=UTF8&qid=1327685218&sr=1-1
This could profitably be read after the quantum mechanics book.
Added 28th JAn 2012
I forgot to mention the other field of current research namely quantum information theory and quantum computing, the best starting point is Mermin's lecture notes and they could be started straight away (Although the quantum mechanics book could be read beforehand)
http://people.ccmr.cornell.edu/~mermin/qcomp/CS483.html
I do find it remarkable that such a productive field of current research makes no mention of wavefunctions the distinction between classical computing and quantum computing seems to be the extension to q-bits rather than c-bits being the fact that q-bits are complex probability amplitudes rather than just binary numbers.
Those who want the real McCoy however are referred to my other reading lists
http://chrisfmathsphysicsmusic.blogspot.com/2011/10/general-relativity-reading-list.html
http://chrisfmathsphysicsmusic.blogspot.com/2011/06/reading-list-for-budding-particle.html
However a more straightforward route to get a glimpse would be to take advantage of the following texts, as before the best way to get a backgound in classical physics prior to study of quantum physics is via Longair's book
http://www.amazon.co.uk/Theoretical-Concepts-Physics-Alternative-Reasoning/dp/052152878X
If one wanted some background on Maxwell's equations one could go also get hold of
http://www.amazon.co.uk/Students-Guide-Maxwells-Equations/dp/0521701473/ref=sr_1_1?s=books&ie=UTF8&qid=1327684577&sr=1-1
Then one would be ready to start on quantum physics and general relativity and a good overview but which skimps on calculational detail is given by the 'Demystified series' so in order of reading
Quantum Mechanics
http://www.amazon.co.uk/Quantum-Mechanics-Demystified-David-Mcmahon/dp/0071455469/ref=sr_1_1?s=books&ie=UTF8&qid=1327684714&sr=1-1
Relativity
http://www.amazon.co.uk/Relativity-Demystified-David-Mcmahon/dp/0071455450/ref=sr_1_2?s=books&ie=UTF8&qid=1327684746&sr=1-2
(This book does actually have quite an accessible introduction to modern coordinate free differential geometry)
Then Quantum Field theory
http://www.amazon.co.uk/Quantum-Field-Theory-Demystified-Mcmahon/dp/0071543821/ref=sr_1_4?s=books&ie=UTF8&qid=1327684746&sr=1-4
This is good as an overview but it really skimps on calculational detail.
Then you can try Superstings if thats your bag (It certainly isn't mine !!)
http://www.amazon.co.uk/String-Theory-Demystified-David-Mcmahon/dp/0071498702/ref=sr_1_5?s=books&ie=UTF8&qid=1327684746&sr=1-5
I have it on my shelf and whilst I can understand the maths I'm still mystified as to the relevance of superstrings to anything but worth having to get a glimpse of what all the fuss is about.
Finally and I wasn't aware of this there is one on supersymmetry (which I've just ordered)
http://www.amazon.co.uk/Supersymmetry-DeMYSTiFied-Patrick-Labelle/dp/0071636412/ref=sr_1_1?s=books&ie=UTF8&qid=1327685098&sr=1-1
The one thing that the series doesn't cover is applications of group theory to physics so my recommendation is still Jones
http://www.amazon.co.uk/Groups-Representations-Physics-H-F-Jones/dp/0750305045/ref=sr_1_1?s=books&ie=UTF8&qid=1327685218&sr=1-1
This could profitably be read after the quantum mechanics book.
Added 28th JAn 2012
I forgot to mention the other field of current research namely quantum information theory and quantum computing, the best starting point is Mermin's lecture notes and they could be started straight away (Although the quantum mechanics book could be read beforehand)
http://people.ccmr.cornell.edu/~mermin/qcomp/CS483.html
I do find it remarkable that such a productive field of current research makes no mention of wavefunctions the distinction between classical computing and quantum computing seems to be the extension to q-bits rather than c-bits being the fact that q-bits are complex probability amplitudes rather than just binary numbers.
Those who want the real McCoy however are referred to my other reading lists
http://chrisfmathsphysicsmusic.blogspot.com/2011/10/general-relativity-reading-list.html
http://chrisfmathsphysicsmusic.blogspot.com/2011/06/reading-list-for-budding-particle.html
What type of mathematician/physicist are you? Part II
In a post I wrote many months ago I asked the question what type of mathematician are you ? I have been having various debates on the Fora about this, Including one with Nilo
The big divide is between those who think (like Nilo) that we discover maths and those like myself that we use maths in order to make sense of the world around us. On my view I have no problem with seeing maths essentially as a construction like a work of Art or music. It's relationship with the real world being an accident rather than some god given laws. Indeed when one applies mathematics to physics there will always be a gap. The Simple Harmonic Oscillator only works well when applied to small amplitudes. If one applies group theory to particle physics it only really works if we assume the particles have the same mass but of course they don't. Nevertheless it is still productive to see how far this can be taken.
Anyway an interesting article I came across in the Physics Arxiv concerns a debate between 3 physicists/mathematicans which delineates three views
1) The Fundamentalist view articulated by Max Tegmark who seems to be prominent on the TV more and more. This essentially on the extreme view that Max Tegmark asserts that to every mathematical concept there is a corresponding element of reality if not in this universe then in another universe. It has it's roots in Platonism and many mathematicians have been attracted to Platonism such as G H Hardy and in our own time Penrose and Marcus Du Satoy are prominent examples although they probably would not be as extreme as Tegmark. For them as I have commented before mathematics is about uncovering the deep nature of reality so that whenever a mathematical theorem is deduced no matter how trivial something fundamental is discovered about the nature of reality. On this view mathematics seems to be akin to a new form of mysticism and it is interesting that most TV programmes which talk about maths always seem to have people who express the Platonist view. Very rarely if at all are other views
2) The mystical view represented by Peter Hut which claims that in 500 years time the laws of physics and the principles of mathematics will have changed beyond recognition so that we are only getting glimpses of what is really the case. I'll leave you to read the article to find out more
3) Finally the secular view represented by Marcus Alford and which I would identify with. For us secularists mathematics is essentially a construction of the human mind which we impose on the world around us in order to make sense of it. There are only a limited subset of mathematical concepts which are applicable to the world around us and there will always be a gap between the real world and the mathematics that is used to understand it. For example (and this is the point I was making to Nilo) the symmetries of particle physics assume that all particles have the same mass which they don't. Nevertheless they are useful in classifying particles and making some order out of the apparent chaos. Again not all mathematics has to apply to the real world but one shouldn't dismiss out of hand their success when it occasionally works. This is not to denigrate maths at all but to see it as one amongst many of humanities greatest achievements.
Going back to Hume's distinction between relationships between ideas and matters of fact mathematics concerns itself with relationship between ideas and not necessarily matters of fact. On the one hand there is this pristine pefect world in the realm of ideas as exemplified by mathematics and on the other hand there is the messy world which aided by mathematics and our knowledge of the laws of physics we can begin to make a little sense of. But we have to acknowledge that there is no route to absolute truth even if humanity occasionally wins some battles.
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