As a fairly straightforward application of linear algebra over the next few posts I want to show how linear algebra can be applied to any simple two state system I will then extend the analysis given here to quantum systems. I want to stress the formalism given here when applied to classical systems may seem cumbersome and artificial, on the other hand it is readily extendable to quantum systems. What is remarkable in what follows (and it's extension to quantum systems) is that the physics of a given situation seems completely irrelevant. I want to stress that all we will be dealing with is a systematic calculus which gives the probabilities of a certain event occuring, a coin tossed, a beam of particles passing through two slits or a beam splitter. When we consider the beam splitter for example, it's only function in the quantum analysis is to allow particles or even a single particle through 1 path or another with a certain probability, there is no attempt to actually consider how the beam splitter functions in terms of the interaction of the beam particles with the material of the beam splitter. However remarkably that is all one needs to predict all the allegedly strange effects associated with quantum interference. I say allegedly as I believe so much balderdash has been written about 1 photon or large molecules such as a buckyball splitting in two as it passes through a beam splitter and then magically reforming at the detector so that it appears as a local entity.
I must admit when I read such accounts I feel like General Grant in the battle of the Wilderness who when exasperated with his brigadier's accounts of what they thought General Lee was upto said "I am heartily tired of hearing what Lee is going to do, some of you always seem to think he is suddenly going to turn a double somersault and land on our rear and on both our flanks at the same time!!".
The aim of this post and the ones that follow are to stress the analogies between quantum physics and classical physics. By doing so I hope that I can clarify why I find such phraseology such as 'The collapse of the wave packet' , A particle (like General Lee) must travel down two paths simultaneously and all the stuff of popular books (and even reputable textbooks) so misleading. There are important differences between quantum physics and classical physics, but they are not such as to make quantum physics, the strange almost voodoo like thing that seems to be beloved of our popular culture today.
A two state system |1> , |2> can be represented by a set of two column vectors (called kets by Dirac)
$$|1> = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$ and
$$|2> = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$
Here 1 and 2 are just labels for a property held by an object it might be Heads or Tails for a coin, spin orientations for an electron, or polarisation vectors for a photon. If analysing a two path system they can be seen as indicating which path a particle has passed through or slit if it's a two slit experiment. I really want to stress that these are just labels nothing else.
A dual vector space formed by the possible states of a system is formed by taking the transpose of the column vectors these are called bra's by Dirac and are denoted as
$$<1| = \begin{pmatrix} 1 & 0 \end{pmatrix}$$ and $$<2| = \begin{pmatrix} 0 & 1 \end{pmatrix}$$
If the state is represented by a complex number (which as we will see is necessary to describe many quantum
systems) then the complex conjugate of the ket vector must be taken, A scalar product can be defined by multiplying a bra with a ket, to form a bra-ket or bracket. This is denoted as follows
$$<1|1>=\begin{pmatrix} 1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = 1 $$
It is relatively straightforward to show also that
$$< i | j > = \delta_{ij}$$
Where $$\delta_{ij}$$ is the so called Kronecker delta function which = 1 if i equals j and 0 if i is not equal j. So that
<1|1> = <2|2> = 1 and <1|2> = <2|1> = 0.
In quantum mechanics we are often interested in the transition between states specifically if given an initial state what is the probability that a certain final state is will be achieved. This probability is given by the Born rule
$$p_{fi} = |< f | i >|^{2} $$ where <f| is the final state of the system and |i> is the initial state of the system
If for example we toss a coin then as we do not know the initial state of the coin, we write the probability state vector of the coin as
$$|C> = \frac{1}{\sqrt{2}}(|1> + |2>) $$ where |1> represents the state of the coin being a head and |2> being the state of the coin being a tail of course these labels are arbitrary. Also please note that the state vector is not a physical superposition of the coin being in some limbo state. It is just a weighted average of the possible final states of the coin. Applying the Born rule to say getting the probability of the coin ending up heads gives
$$|<1|C>|^2 = |\frac{1}{\sqrt{2}}(<1|1> + <1|2>)|^2 = 1/2 $$ and (exercise for the reader ) a similar result would apply if I wanted the probability of getting a tail.
This formalism can be extended to a system with N outcomes, Thus for a dice the probability state vector would be
$$|D> = \frac{1}{\sqrt{6}}(|1>+|2>+ ... |6>)$$ this time the individual states would be described by a 6x1 column vector whose entries would be zero apart from the entry corresponding to the state thus eg
$$|3>=\begin{pmatrix}0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} $$
Again the Born rule could be used to calculate the probabilities of getting a 5 and so forth. This would (exercise for the student) simply be 1/6. Also the states are orthogonal so that
< 3 | 3 > = 1 whilst < 3 | 5> = 0 etc.
Another example, suppose our coin was biased so that the probability of getting a head was 3/4 and the probability of getting a tail was 1/4. then the probability state vector of the bent coin is
$$|BC> = \frac{\sqrt{3}}{2}|1> + \frac{1}{2}|2>$$
And the reader can verify that the Born rule gives the correct probabilities.
Well so what why can't we just stick to normal probability, why bother with all this stuff about probability state vectors and so forth ?. The point is that by recasting the language of ordinary probability in the language of quantum physics, we can see how quantum theory is essentially a statistical theory. The probability state vectors, encapsulating via the Born rule what happens when a dice or bent coin is thrown, are just that they aren't anything physical. The same would apply to a cat which we don't know is alive or dead to quote one famous example beloved of the popular literature. The same state vector is used only now |1> represents a cat being dead and |2> represents a cat being alive or vice versa, the labels do not matter.
I spoke in an earlier post how people were mislead as to what quantum physics is all about by concentrating on Schrodinger's wave equation and it's solutions. The probability state vectors, I have described above are often referred as wave functions and the probability state vector is said to collapse when a transition between 1 state and another is realised. As this is thought to be a physical process much ink has been spilled as to what a collapse can actually mean. At least for the simple cases I've shown above there is nothing physical. All that happens is that one of the various possibilities encapsulated in the probability state vector is realised in practice. If I perform a large number of runs then I will get the statistics represented by the probability state vector nothing more nothing less. Cats, coins or dice are not in limbo and there is nothing physical about the so called collapse of the probability state vector. All this seems so obvious when we translate the language of quantum mechanics to classical probability theory. I really don't understand why the popular literature plays up the alleged mysterious nature of quantum theory there is no basis for this whatsoever.
In the next post I will show how a simple extension of the state vector to complex numbers enables the quantum interference of two state systems to be modelled.
Tuesday, 24 May 2011
Sunday, 8 May 2011
Linear Algebra
So I'm near the end of this block starting to pick up speed a little just need to put the finishing touches to the TMA. The unit is a game of two halves as a foothall manager would say. Blocks one and two are fairly straightforward summaries of vectors, solving equations by row reduction and a bit of coordinate geometry. Then the next three units lay the foundations of vector spaces and linear algebra. It covers quite a lot of ground in a short space of time. Introducing the concept of ortho normal basis vectors and showing how matrices can be diagonalised. It also hints briefly (but not nearly enough for my tastes) that vector spaces are nore than just relations between geometric vectors but can be applied to functions as well. Indeed it was the realisation by Heisenberg and Pauli that the theory of vector spaces could be applied to quantum mechanics that led to a breakthrough in the field. I have commented on this in earlier posts.
In this post I will briefly expand on some of the hints given in the M208 about the relationship between geometric vector spaces and function spaces. The simplest example to illustrate the analogy are the trigonometric functions
if I differentiate $$cos(\lambda x)$$ where $$\lambda$$ is a scalar then I get
$$\frac{d^2}{dx^2} cos(\lambda x) = -\lambda^{2}cos(\lambda x) $$
Thus we can treat $$\lambda^2$$ as an eigenvalue of the eigenfunction cos(\lambda x) but instead of matrices we now have a differential operator. So that $$cos(\lambda x)$$ is an eigenfunction of the differential operator $$\frac{d^2}{dx^2}$$ with eigenvalue $$-\lambda^2$$
As the trigonometric functions obey the following integrals where m does not equal n and m and n are integers
$$\int_{-\pi}^{\pi} cos(mx)cos(nx) dx = 0 $$
$$\int_{-\pi}^{\pi} sin(mx)sin(nx) dx = 0 $$
$$\int_{-\pi}^{\pi} cos(mx)sin(nx) dx = 0 $$
We can define a set of orthogonal vectors in function space involving the trigonometric functions with the scalar product being defined as
$$\int_{-\pi}^{\pi} e_{m}e_{n} dx $$
where $$e_{m},e_{n}$$ are vectors defined taken from the set
$${1,cos(x),sin(x),cos(2x),sin(2x) ........cos(mx), sin(nx)...}$$
To make this set orthonormal note that
$$\int_{-\pi}^{\pi}1 dx = 2\pi$$
and
$$\int_{-\pi}^{\pi}sin^{2}mx dx = \int_{-\pi}^{\pi}cos^{2}mx dx = \pi $$ if m > 0
Hence the functions
$$\frac{1}{\sqrt{2\pi}},\frac{cos x}{\sqrt{\pi}}....\frac{cos nx}{\sqrt{\pi}},
\frac{sin x}{\sqrt{\pi}}, ....\frac{sin nx}{\sqrt{\pi}}$$
Form an orthonormal basis set
Well so what you might say all this shows is an analogy, however this analogy is quite profound, recall that given a vector space and an orthonormal set of basis vectors any function in that space can be expanded as a sum over the basis vectors. In physics this means that any complicated wave form can be reduced to a sum over all the basis vectors. So for example a square wave can be seen as a complicated sum over its component waveforms taken from the basis set given hear. Musical synthesis is essentially based on this profound yet simple idea. Such a technique is called Fourier decomposition for more details see eg
http://en.wikipedia.org/wiki/Fourier_series
Next post on this topic I'll show how Gram Schmidt othogonalisation can be extended to Polynomials.
The relationship between vectors, differential operators and orthonormal basis vectors is called linear analysis. Many years ago the Open University offered a course based on this topic called M201 it is a sad sign of the times that this course is no longer available. However if you search on Amazon you might be able to pick up the text book on which the course was based. It is called "An Introduction to Linear Analysis" by Krieder, Kuller, Ostberg and Perkins. This book is a synthesis between the Applied maths of MST209 and the vector space and analysis parts of M208 showing how both pure and Applied maths combine together. It shows how the concept of linear vector spaces and it's analogies can be used to illuminate the structure behind the solutions of both ordinary and Partial differential equations. It is probably my favourite book on maths at the minute and is strongly recommended.
In this post I will briefly expand on some of the hints given in the M208 about the relationship between geometric vector spaces and function spaces. The simplest example to illustrate the analogy are the trigonometric functions
if I differentiate $$cos(\lambda x)$$ where $$\lambda$$ is a scalar then I get
$$\frac{d^2}{dx^2} cos(\lambda x) = -\lambda^{2}cos(\lambda x) $$
Thus we can treat $$\lambda^2$$ as an eigenvalue of the eigenfunction cos(\lambda x) but instead of matrices we now have a differential operator. So that $$cos(\lambda x)$$ is an eigenfunction of the differential operator $$\frac{d^2}{dx^2}$$ with eigenvalue $$-\lambda^2$$
As the trigonometric functions obey the following integrals where m does not equal n and m and n are integers
$$\int_{-\pi}^{\pi} cos(mx)cos(nx) dx = 0 $$
$$\int_{-\pi}^{\pi} sin(mx)sin(nx) dx = 0 $$
$$\int_{-\pi}^{\pi} cos(mx)sin(nx) dx = 0 $$
We can define a set of orthogonal vectors in function space involving the trigonometric functions with the scalar product being defined as
$$\int_{-\pi}^{\pi} e_{m}e_{n} dx $$
where $$e_{m},e_{n}$$ are vectors defined taken from the set
$${1,cos(x),sin(x),cos(2x),sin(2x) ........cos(mx), sin(nx)...}$$
To make this set orthonormal note that
$$\int_{-\pi}^{\pi}1 dx = 2\pi$$
and
$$\int_{-\pi}^{\pi}sin^{2}mx dx = \int_{-\pi}^{\pi}cos^{2}mx dx = \pi $$ if m > 0
Hence the functions
$$\frac{1}{\sqrt{2\pi}},\frac{cos x}{\sqrt{\pi}}....\frac{cos nx}{\sqrt{\pi}},
\frac{sin x}{\sqrt{\pi}}, ....\frac{sin nx}{\sqrt{\pi}}$$
Form an orthonormal basis set
Well so what you might say all this shows is an analogy, however this analogy is quite profound, recall that given a vector space and an orthonormal set of basis vectors any function in that space can be expanded as a sum over the basis vectors. In physics this means that any complicated wave form can be reduced to a sum over all the basis vectors. So for example a square wave can be seen as a complicated sum over its component waveforms taken from the basis set given hear. Musical synthesis is essentially based on this profound yet simple idea. Such a technique is called Fourier decomposition for more details see eg
http://en.wikipedia.org/wiki/Fourier_series
Next post on this topic I'll show how Gram Schmidt othogonalisation can be extended to Polynomials.
The relationship between vectors, differential operators and orthonormal basis vectors is called linear analysis. Many years ago the Open University offered a course based on this topic called M201 it is a sad sign of the times that this course is no longer available. However if you search on Amazon you might be able to pick up the text book on which the course was based. It is called "An Introduction to Linear Analysis" by Krieder, Kuller, Ostberg and Perkins. This book is a synthesis between the Applied maths of MST209 and the vector space and analysis parts of M208 showing how both pure and Applied maths combine together. It shows how the concept of linear vector spaces and it's analogies can be used to illuminate the structure behind the solutions of both ordinary and Partial differential equations. It is probably my favourite book on maths at the minute and is strongly recommended.
Monday, 25 April 2011
It's official I've got Asperger's
Courtesy of Nilo it would appear according to this test I've got Asperger's syndrome
http://www.piepalace.ca/blog/asperger-test-aq-test/
I got 34 definitely in the Asperger's zone according to the test. A lot of my colleagues, joke at work that they have Asperger's, indeed one couldn't achieve anything in our work if we didn't have the ability to work on our own, and put up with a lot of boring tedious and repetive tasks and pay attention to detail. On a more general point if one wants to achieve anything intellectual, then one needs to spend a lot of time on one's own trying out ideas and reading. If one has a full time job then it stands to reason that juggling the demands of OU study means that one's social life is bound to take a back burner.
As I commented on Nilo's blog it seems to me that the rise of this syndrome is partly due to the lack of appreciation of a person's intellectual abilities. Also it reflects societies fear of people who are quite happy on their own and have an intellectual skill which they do not. It's interesting that it's scientist's mathematicians and engineers that are usually accused of having Asperger's syndrome. Whilst artist's, composer's and writer's who have the same lack of social skill's as mathematicians scientists and engineers are not considered Aspergic.
Our society seems to place a high value on superficial beauty, or wealth and a low value on things that really matter. Labelling people as Aspergic simply because they do not fit in with the norms of society does not help. As a final point muddying the waters between Asperger's and Autism will not help find a cure for Autism.
http://www.piepalace.ca/blog/asperger-test-aq-test/
I got 34 definitely in the Asperger's zone according to the test. A lot of my colleagues, joke at work that they have Asperger's, indeed one couldn't achieve anything in our work if we didn't have the ability to work on our own, and put up with a lot of boring tedious and repetive tasks and pay attention to detail. On a more general point if one wants to achieve anything intellectual, then one needs to spend a lot of time on one's own trying out ideas and reading. If one has a full time job then it stands to reason that juggling the demands of OU study means that one's social life is bound to take a back burner.
As I commented on Nilo's blog it seems to me that the rise of this syndrome is partly due to the lack of appreciation of a person's intellectual abilities. Also it reflects societies fear of people who are quite happy on their own and have an intellectual skill which they do not. It's interesting that it's scientist's mathematicians and engineers that are usually accused of having Asperger's syndrome. Whilst artist's, composer's and writer's who have the same lack of social skill's as mathematicians scientists and engineers are not considered Aspergic.
Our society seems to place a high value on superficial beauty, or wealth and a low value on things that really matter. Labelling people as Aspergic simply because they do not fit in with the norms of society does not help. As a final point muddying the waters between Asperger's and Autism will not help find a cure for Autism.
Tuesday, 19 April 2011
M208 M346 M337 I saw one go out and another Two come back in
Today 1 day late, I posted my off my first TMA for M346 linear statistical modelling. I felt it was fairly straightforward essentially being exercises in interpretation of output from GENSTAT and a review of some statistical tests. Essentially revision there were one or two points I hadn't come across before. So quietly confident of a reasonable grade although one can never be sure whether I've missed out something fundamental.
So thats one out. Also on my door step I found my returned TMA for complex variables slightly disappointed as I didn't make distinction due to some careless mistakes also some times I was quite rigorous in my arguments other times I missed out key justifications thus omitting to mention that a sequence tended to zero as it was a null sequence just stating that the sequence tended to zero. I think it's a question of learning how to phrase arguments in analysis properly. Some times I'm guilty of making to many points other times I miss out key steps. Still one can only go on.
In contrast I seemed to have done really well with the TMA for M208 which I received earlier this week . I wonder if it will continue when I get to the analysis sections. Hopefully some of the lesson's I'm learning from M337 will help here.
As a final point talking with one of my mates Neil at the tutorial
http://neilanderson.freehostia.com/thoughts/degree/m208/
We both worried that in the chase to get good marks for the TMA's we were skimping on details and understanding. Still need's must Alan my tutor himself said that as far as the first two units of linear algebra were concerned we should just concentrate on whether we have learnt the basic techniques and then concentrate on the more abstract stuff in the latter part of the block.
I think it's a question of concentrating on what you feel you need to know in detail for me this is group theory and analysis as far as M208 is concerned specifically learning how to deal with the dreaded
$$\epsilon - \delta$$ definitions of continuity and it's applications to pathological functions such as the so called blancmage function which is an iterated sequence of triangles. This function is every where continuous and no where differentiable. I'll speak more about this on another post. Also most of M337 even though it is currently the most challenging part of pure mathematics I have tackled so far. Still as I said to Neil we are not here for an easy ride.
This means that M346 will have to take a back burner and I will be fairly pragmatic about the work I need to do for this course which means essentially working backwards from the TMA questions.
So thats one out. Also on my door step I found my returned TMA for complex variables slightly disappointed as I didn't make distinction due to some careless mistakes also some times I was quite rigorous in my arguments other times I missed out key justifications thus omitting to mention that a sequence tended to zero as it was a null sequence just stating that the sequence tended to zero. I think it's a question of learning how to phrase arguments in analysis properly. Some times I'm guilty of making to many points other times I miss out key steps. Still one can only go on.
In contrast I seemed to have done really well with the TMA for M208 which I received earlier this week . I wonder if it will continue when I get to the analysis sections. Hopefully some of the lesson's I'm learning from M337 will help here.
As a final point talking with one of my mates Neil at the tutorial
http://neilanderson.freehostia.com/thoughts/degree/m208/
We both worried that in the chase to get good marks for the TMA's we were skimping on details and understanding. Still need's must Alan my tutor himself said that as far as the first two units of linear algebra were concerned we should just concentrate on whether we have learnt the basic techniques and then concentrate on the more abstract stuff in the latter part of the block.
I think it's a question of concentrating on what you feel you need to know in detail for me this is group theory and analysis as far as M208 is concerned specifically learning how to deal with the dreaded
$$\epsilon - \delta$$ definitions of continuity and it's applications to pathological functions such as the so called blancmage function which is an iterated sequence of triangles. This function is every where continuous and no where differentiable. I'll speak more about this on another post. Also most of M337 even though it is currently the most challenging part of pure mathematics I have tackled so far. Still as I said to Neil we are not here for an easy ride.
This means that M346 will have to take a back burner and I will be fairly pragmatic about the work I need to do for this course which means essentially working backwards from the TMA questions.
Tuesday, 12 April 2011
What do I want ??
A little time to reflect from the pressures of TMA deadlines made me think what is it I really want to achieve having taken the libertry of telling someone on the M337 forum what makes a good PhD it's a question of 'physician heal thyself'. If one wants to achieve anything in life it's a question of gettting enough background knowledge to articulate a certiain insiight which no one else has thought of, but on the other hand not getting so bogged down in a lack of confidence of one's own abilities. Obviously this meams that one has to have a certain degree of self confidence, which I lack and also a degree of realism, as to what can be achieved, Thus it is highly unlikely that I am going to solve the Riemann hypothesis, or any of the Seven Millennia problems. It is highly unlikely that I am goiing to come up with a Grand unified theory of everyhiing and it is highlly unlikely that I am going to solve the problem of a coherent quantisiation of Einstein's theory of relativity. Given that better people have been there before me and have tried to solve the problem and have not achieved anything lasting.
Ok so what is it I think I can at least aritculate, it is, it is whether or not there is a connection between the philosophy of language and the rejection of bivalence as enshrined with Michael Dummet ideas
and the fact that the mathematical stucture of quantum mechanics seems to imply a similar rejection of bivalence as enshrined in the Kochen Specker paradox. Both would seem to imply a rejection of realism but how far does this go ?
See for example Dummett
http://www.iep.utm.edu/dummett/
and Kochen Specker Paradox
http://en.wikipedia.org/wiki/Kochen%E2%80%93Specker_theorem
For a current interpreation of quantum mechanics which tries to circumvent this dilemma see Topos theory the simplest introduction to which I know of can be accessed here.
http://topos-physics.org/
It seems to me at least articulating what the possible connections between the three issues namely whether
1) Does Micheal Dummet's assetion that realist theories of epsitemology or metaphysics are commited to bivalence hold water
2) Does the fact that quantum mechancs seems to imply an adherence to non realism via the Kochen Specker paradox spell the death knell for realist interpreations of quantum physics such as Bohm's intepretation or even worse the Many worlds interpretation ( I hope so).
3) How far do theories such as Topos theory circumvent the problem of realism and the implications of the Kochen Specker paradox . It seems to me that they go some way but not far enough.
would be worthy of an MSc by research in philosophy as I've not seen any full lengrh account of the issue, but it seems to be mentioned in passing in a number of papers.
So why don't I take a breather and write a research proposal for the Logic and language department of philosophy at Ediunbrgh University ?
http://www.ppls.ed.ac.uk/philosophy/groups/logic-language-edinburgh
and see whether they would take me on part time, and then get accepted by say Octobet this year or certainly October 2012 ? Instead of getting bogged down both time wise and financially in pursuing formal qualifications in mathematics and philosophy via the Open university or other sources. After all I do not need a knowledge of complex analyisis, Kant's critiique of pure reason or especially Heidegger's Being and Time or Nietszche's Genealogy of morals interesting as it would be to pursue these issues in some depth, to be able to pursue the question as to whether there is a link between Dummet's rejection of Bivalence and realism and wheher or not quantum mechanics in it's present form implies the rejection of bivalence and as a consequence lends some support to Dummet's ideas. I do not need say for example the course in functional analysis offered by the OU MSc especially as it explicitly states that it does not deal with any concrete applications of the subject eg to say quantum physics. I feel I can get such knowledge from say Kreyszig or even tackle Von Neumann directly now that being embroiled in M337 and M208 has given me a degree of confidence in handling pure mathematical concepts which I didn't have before.
Clearly I need to invesitgate these issues further the question is given time and finances whether or nor I can do this in a reasonable amount of time and financial cost ? Given that fee's are going to skyrocket over the next few years then it would seem reasonable to cut one's losses and just go for it.
On the other hand I am handicapped by a lack of self confidence in putting this forward given the as yet lack of formal qualifications in both pure Maths and philosophy. On the other hand waiting 5 years or so before I can gain the qualifications to get me there isn't going to help me either. So as a compromse I wait a couple of years whilst I finish the degree in maths and my open degree in social sciences the Arts and philosophy supplemented by Geoffrey Klempner's pathway courses and his associateship. He is an expert in Dummet so at least from the philosophical side I should get some idea as to whether or not my ideas are on the right track or whether or niot I'm just talking rubbish.
http://www.philosophypathways.com/programs/pak2.html
After that just go for it rather than handicap myself with yet another financial and time burden just chasing after qualifications which whilst definitely worth it distract one from achieving one's real goals. So yes I will complete my current courses and do the other pure maths course to complete the degree and also My open degree with the philosophy of mind course and Geoffrey Klempner;s pathways modules but then instead of spending circa £12000 on getting the MSc in maths and a correspoinding MA in continental philosophy spending what will be £6,000 on a parttime MSc by research (assuming I get accepted) will save me a lot of money and time to achieve the goal I really want. So I'll finish the two degrees I'm aiming for but not bother with the MSc's and take my chances, sooner or later you have to break away, at least I am aware of what I should know and also what I want to explore.
Ok so what is it I think I can at least aritculate, it is, it is whether or not there is a connection between the philosophy of language and the rejection of bivalence as enshrined with Michael Dummet ideas
and the fact that the mathematical stucture of quantum mechanics seems to imply a similar rejection of bivalence as enshrined in the Kochen Specker paradox. Both would seem to imply a rejection of realism but how far does this go ?
See for example Dummett
http://www.iep.utm.edu/dummett/
and Kochen Specker Paradox
http://en.wikipedia.org/wiki/Kochen%E2%80%93Specker_theorem
For a current interpreation of quantum mechanics which tries to circumvent this dilemma see Topos theory the simplest introduction to which I know of can be accessed here.
http://topos-physics.org/
It seems to me at least articulating what the possible connections between the three issues namely whether
1) Does Micheal Dummet's assetion that realist theories of epsitemology or metaphysics are commited to bivalence hold water
2) Does the fact that quantum mechancs seems to imply an adherence to non realism via the Kochen Specker paradox spell the death knell for realist interpreations of quantum physics such as Bohm's intepretation or even worse the Many worlds interpretation ( I hope so).
3) How far do theories such as Topos theory circumvent the problem of realism and the implications of the Kochen Specker paradox . It seems to me that they go some way but not far enough.
would be worthy of an MSc by research in philosophy as I've not seen any full lengrh account of the issue, but it seems to be mentioned in passing in a number of papers.
So why don't I take a breather and write a research proposal for the Logic and language department of philosophy at Ediunbrgh University ?
http://www.ppls.ed.ac.uk/philosophy/groups/logic-language-edinburgh
and see whether they would take me on part time, and then get accepted by say Octobet this year or certainly October 2012 ? Instead of getting bogged down both time wise and financially in pursuing formal qualifications in mathematics and philosophy via the Open university or other sources. After all I do not need a knowledge of complex analyisis, Kant's critiique of pure reason or especially Heidegger's Being and Time or Nietszche's Genealogy of morals interesting as it would be to pursue these issues in some depth, to be able to pursue the question as to whether there is a link between Dummet's rejection of Bivalence and realism and wheher or not quantum mechanics in it's present form implies the rejection of bivalence and as a consequence lends some support to Dummet's ideas. I do not need say for example the course in functional analysis offered by the OU MSc especially as it explicitly states that it does not deal with any concrete applications of the subject eg to say quantum physics. I feel I can get such knowledge from say Kreyszig or even tackle Von Neumann directly now that being embroiled in M337 and M208 has given me a degree of confidence in handling pure mathematical concepts which I didn't have before.
Clearly I need to invesitgate these issues further the question is given time and finances whether or nor I can do this in a reasonable amount of time and financial cost ? Given that fee's are going to skyrocket over the next few years then it would seem reasonable to cut one's losses and just go for it.
On the other hand I am handicapped by a lack of self confidence in putting this forward given the as yet lack of formal qualifications in both pure Maths and philosophy. On the other hand waiting 5 years or so before I can gain the qualifications to get me there isn't going to help me either. So as a compromse I wait a couple of years whilst I finish the degree in maths and my open degree in social sciences the Arts and philosophy supplemented by Geoffrey Klempner's pathway courses and his associateship. He is an expert in Dummet so at least from the philosophical side I should get some idea as to whether or not my ideas are on the right track or whether or niot I'm just talking rubbish.
http://www.philosophypathways.com/programs/pak2.html
After that just go for it rather than handicap myself with yet another financial and time burden just chasing after qualifications which whilst definitely worth it distract one from achieving one's real goals. So yes I will complete my current courses and do the other pure maths course to complete the degree and also My open degree with the philosophy of mind course and Geoffrey Klempner;s pathways modules but then instead of spending circa £12000 on getting the MSc in maths and a correspoinding MA in continental philosophy spending what will be £6,000 on a parttime MSc by research (assuming I get accepted) will save me a lot of money and time to achieve the goal I really want. So I'll finish the two degrees I'm aiming for but not bother with the MSc's and take my chances, sooner or later you have to break away, at least I am aware of what I should know and also what I want to explore.
Sunday, 10 April 2011
Amazing
Wow and I don't lknow how I did it but MathJAX now seems to work just as in the Forums of Moodle you have to mark off your LATEX input with two dollar signs so
$x^2$ just produces the crap you see here whereas enclosing it with two dolllar signs
where as $$x^2$$ now seems to work
Have amended my group theory post of March 6th to take into account the New LATEX facility.
Apologies if the blog takes longer to load that's the price you pay for sophistication.
$x^2$ just produces the crap you see here whereas enclosing it with two dolllar signs
where as $$x^2$$ now seems to work
Have amended my group theory post of March 6th to take into account the New LATEX facility.
Apologies if the blog takes longer to load that's the price you pay for sophistication.
Math JAX
As some of you will have seen I've tried to incorporate math JAX on this blog however I can't get it to work
thus what should appear as a nice x squared term comes out simply as $x^2$ or even $$x^2$$
Post 11/04/11 In light of the partial success I removed the rather defunct script but I stiil hate Computers !!
For reasons only known to the guru's of cypberspace Whilst Math JAX appears to work for the visible pages it doesn't seem to be able to update old posts even though when I preview the amendments it seems to incorporate the LATEX. As a result my old post on group theory is even more incomprehensible than ever.
On a brighter note finished both TMA's for M337 and M208, but still need to make inroads into the statistics. I shall have to take at least one day off next week.
Best wishes Chris
thus what should appear as a nice x squared term comes out simply as $x^2$ or even $$x^2$$
Post 11/04/11 In light of the partial success I removed the rather defunct script but I stiil hate Computers !!
For reasons only known to the guru's of cypberspace Whilst Math JAX appears to work for the visible pages it doesn't seem to be able to update old posts even though when I preview the amendments it seems to incorporate the LATEX. As a result my old post on group theory is even more incomprehensible than ever.
On a brighter note finished both TMA's for M337 and M208, but still need to make inroads into the statistics. I shall have to take at least one day off next week.
Best wishes Chris
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