Personal Blogs

I'm going all applied and physical.

In parallel I'm going to do M820 (on my own, not officially with the OU) and study physics, particularly quantum mechanics, a bit of electromagnetism, and relativity.  I've done all the maths to enable me to do this now.  I've previously covered vector calculus which is required for electromagnetism, and this year I've covered differential geometry in a fairly abstract way, such that the way it's presented for general relativity seems pretty easy.  M820 will cover Lagrangian formalism, and once I've done that I'll go back to my M827 text and re-read the bit about Lagrangians and Hamiltonians in the setting of a differentiable manifold.

All this stems from buying Roger Penrose's 'Road to Reality' tome, which I foolishly thought I could just read and then know everything.  It's taken me 5 years of reasonably intense mathematical studying to get to the point where most of it makes sense, although I've still got a long way to go.

Year after that?  Probably start working on my proof of the Riemann hypothesis

So chapter 10 of Pirani and Crampin is where it all comes together.  I haven't studied it properly yet but did read through it quickly - it basically goes over most of the work done in the first 9 chapters of the book and puts it into the context of an abstract differentiable manifold, rather than either an affine space or an embedded surface in 3D space.  And the results are all the same!  There's a bit of abstraction and technical detail using bump functions to justify this, but in reality and from a calculation perspective I don't think there's anything new.  It's a good revision chapter.

Before taking up the baton of differential geometry again this week I all but finished the analytic number theory course, completing the proof of the prime number theorem.  Feels like an achievement.  I've understood evey bit and followed all the proofs, but if I was asked to write it out from memory - I'd have no chance!

So now I know how it's all proved.  Not yet halfway through M829 and we've reached the climax.  Well perhaps not, but I undersand it all in principle - I think.

In M823 we covered a few equivalent statements to the actual theorem.  The prime number theorem states that for large .  This can be shown to be equivalent to or where and is its integral.  Oh and is Mangoldt's function where whenever and zero otherwise.

Next, an integral expression is constructed that is equal to .  This expression contains the zeta function and its derivative in the form .  Using the residue theorem, and knowledge of the function's poles, this can be integrated and shown, as x approaches infinity, to equal .

And the prime number theorem is proved.  That only took about 4 and a half years of studying maths to understand.

Just found an old notebook from before I started my OU studies, and in it were my plans for my open degree.

2008/2009 - MST121 and MS221

2010 - M208

2011 - MST209

2012 - DD202

2013 - M337 and SM358

2014 - M338 and S357

So what happened to all that?  If I'd stuck to that plan I'd have done a 60 point economics course last year and would now be starting complex analysis and quantum mechanics.  Next year would have to have changed as M338 is no more, like S357.

It started off right, but I found the workload in the first year so light that I did 208 and 209 together, advancing things by a year.  I then decided that I'd just do maths, as they dropped the old relativity course and I wasn't that interested in economics anyway.  I have the DD202 course books (£10 from eBay) and may read them one day.  I still plan to learn quantum mechanics and relativity one day, probably once I've got the MSc.

First take a sphere S₂ in 3D space.  Next consider the free group G generated by 2 elements, i.e. <a,b> which consists of all possible combinations of a, b, a^-1, b^-1.  Each element of G can be written in a unique way.  G can be split into 5 disjoint sets: G = {e} U S(a) U S(a^-1) U S(b) U S(b^-1) where S(x) represents the set of elements in G whose 'string' starts with x.

It is clear that aS(a^-1) is equal to {e} U S(a^-1) U S(b) U S(b^-1) (the a 'cancels' the a^-1 at the start of each element in S(a^-1) leaving all elements starting with either a^-1, b or b^-1.  Similarly bS(b^-1) is equal to {e} U S(b^-1) U S(a) U S(a^-1).

So aS(a^-1) U S(a) = G.  Same for bS(b^-1) U S(b).

Now consider a and b to be rotations of the sphere about orthogonal axes by some irrational number of degrees - say 1 radian.  Every element of G is a combination of rotations about these 2 axes, and is thus a rotation.  Each element of G (except e) will therefore have 2 fixed points on the sphere.  Let the set of all these fixed points be Q, which is countable (as the elements of G are countable, being...equinumerous with N?...Wait!  This is wrong!

I thought I understood this, but it doesn't work.  The elements of G aren't countable, are they?  Perhaps they are.  Back to the drawing board.  Rest of the proof involves the action of G on the set of points in the sphere, writing the set of points as a disjoint set of the orbits of G acting on S-Q, invoking the axiom of choice to select one point from each orbit which are put into a set M.  This set is then composed with the disjoint bits of G to decompose the point-set of the sphere into 4 bits, two of which are translated across (say MS(a) and MS(a^-1)) and then MS(a^-1) is rotated by rotation a, and hey presto you've got your original sphere back.  Do the same with the non-translated bits and you've got 2 spheres.

Few technical details with regard to the set Q - you have to rotate it by something not a member of G or something like that, and to extend the transformation to a solid ball rather than a hollow sphere you have to include the centre as a separate point and use the projections of each point to the centre, but it's largely the same.

It's all nonsense anyway - axiom of choice is clearly false, certainly when it comes to maths that could be applied to the real world.  At least that's my opinion.

I've been doing well with my private study - am making a proper go of learning Galois theory and boning up on set theory, which leads me to the title of this post.

The Axiom of Choice (AC).  Introduced initially by Zermelo when formulating the axioms of set theory, so that they would cover all of Cantor's work on infinite sets and avoid paradoxes such as Russell's.  It soon became apparent that mathematicians had been using AC for years without really knowing it.

I don't accept it.  If accepting it means you can prove that a sphere can be cut into no more than 9 pieces which are then reassembled into 2 spheres the same size as the first one (Banach-Tarski paradox/theorem whichever way you look at it), then it either has to be wrong or its level of abstraction is so far removed from reality that it's irrelevant.

You need the axiom of choice to fully appreciate Cantor's infinite cardinals.  Again, there's got to be something wrong here, and it must be with the priniciple that a bijection between infinite sets means they are equinumerous.  How can the set of natural numbers be equinumerous with the set of even natural numbers?  I know there's a bijection n -> 2n, but by any reasonable reckoning there are clearly half as many even natural numbers as there are natural numbers.

I've come to the conclusion that infinity is just infinity, which just means without bound.  This works for epsilon-delta treatments in real analysis (but for sequential continuity you need AC).  Complex analysts happily define the set and then divide by zero.  I'm happy with that.  I'm not happy with saying there are more irrational numbers than rationals on the one hand, and then saying that between any two rational numbers there is an irrational, and between any two irrationals there is a rational.  Makes no sense.  I follow the proof, just don't accept the philosophy.

I'll soon finish the set theory book.  I'm then going to study the Banach-Tarski proof (which is on about page 2 of my Axiom of Choice book) just so I understand what I don't believe.  Mohammed put it best when he said (probably in the Koran somewhere) that one should study witchcraft, but not practice it.

More on Galois theory anon - algebra gets more interesting, and so far seems devoid of some of the nonsense that seems to plague foundational mathematics such as set theory and logic.