A Mind for Madness

It is half-way through the year, so I’ll sum up the best and worst albums that have released so far. Of course, this is my opinion and I’ll get on to NCG tomorrow. Just thought I’d list without too much commentary. If you want to contest one of them just comment.

Great:
Bon Iver: For Emma, Forever Ago
Elbow: The Seldom Seen Kid
Lightspeed Champion: Falling off the Lavender Bridge
Portishead: Third
Son Lux: At War with Walls and Mazes

Good:
Coldplay: Viva La Vida
Extra Life: Secular Works
Mates of State: Re-arrange Us
Wolf Parade: At Mount Zoomer

Disappointments:
Black Mountain: In the Future
Death Cab for Cutie: Narrow Stairs
Nada Surf: Lucky
The Black Keys: Attack and Release
Vampire Weekend: Vampire Weekend

So this was off the top of my head. I may be missing stuff. Things to note. This is of the stuff I listen to, so none of it is truly bad. That’s why I labeled it “disappointments.” It is more they didn’t live up to their past accomplishments, or the hype was more than the delivery.

Other notes: The Black Keys, Black Mountain, and Wolf Parade are usually spoken of together. I find the first two to be rehashing of prog rock in an unoriginal and boring way. The Wolf Parade style is very original and complicated at times allowing it into the “good” section. I’d say that of the greats, Bon Iver and Portishead deserve an “extra great” section for defying all convention and being at the same time able to move you beyond belief. Yes, the new Death Cab album is horrible.

Things I haven’t heard, but probably should due to hype: The Dodos, The Magnetic Fields, Devotchka, Panic! at the Disco, Sigur Ros, She & Him, and uh, Scarlett Johansson recorded an album?!?

There is still time to vote! I think this will only take one post, so where I go next can be decided by you.

Background: I’ve done a few phil math posts before, so many of you know that I tend toward the humanist/social construct/embodied mind/quasi-empiricist side. This is meant as a disclaimer that 99.9% of mathematicians will disagree with me (and to the point of storming out in order to not become violent). Math tends to be a touchy subject. This probably has to do a lot with the romance of mathematics.

In that post I talk about why the romance of mathematics is mistaken, but not why it is so important to mathematicians that they can’t let go. I think there are deep psychological problems that most mathematicians have to overcome. The first of which has to do with how hard it is to become a mathematician. You spend K-12 learning math, then do an undergrad degree for 4 years, then do something so much more incredibly hard and spend probably 5 or more years for the Ph.D., then maybe a 3 year post-doc, then a professor position. After some 25 years of studying like you wouldn’t believe, people still don’t make it. Others still have many years ahead to become well grounded in their field. So…there is a lot of time invested that most other disciplines can’t claim. I think this makes mathematicians feel like they have some sort of superiority to other disciplines, and they want that “permanent” label placed on their work. When stuff comes up that contradicts that, it is just ignored. The other problem is that mathematicians tell their students that they earn this label throughout all those years. It is hard to reverse that much brainwashing.

Here (mainly under Chapter 11) I talk about the rejection of Platonism and the inexactness of math. One more thing to mention is the social construct aspect. Standards of math change and conceptualizations change as you go through time and to different cultures. This means that proofs that were once correct can become incorrect. This should not be threatening to mathematicians. It opens up new doorways. I don’t want to go into empiricism, but I am a firm believer that computer proofs should be accepted as proof. (Yes, empiricism is involved there. People will argue with that, but don’t listen to them).

It has been suggested that I talk about a recent post biological proof. So that preface was to say that unlike 99% of mathematicians, I am open to the idea that this could be a legitimate proof method. But I’m going to put on my philosopher’s hat and look at some of the assumptions that could be problematic.

First off, I’m not a huge fan of the term of Isoperimetric Inequality, but I’ll ignore that and see if the statement: The solid that minimizes the ratio of surface area to volume (SA/V) in Euclidean 3-space is the sphere is proven.

1. Animals overwhelmingly assume a spherical shape when they are cold. A lot could be said about this. Just look at the picture given. Is that a sphere? It looks more ellipsoidal to me. In not so sarcastic terms, how do we rule out things that are rather close to the sphere? No animal actually makes it to a sphere.

2. If animals overwhelmingly assume some shape when they’re cold, then it is because that shape is the warmest. Granting that we somehow accomplish 1, we now have this being the absolute crux of the proof, yet I feel this has some major hidden baggage. I’m guessing this is some sort evolutionary claim. If some animals assume a shape that warmest and some assume one that is not warmest, then the not warmest ones do not have as high evolutionary fitness and will not continue on. Well, granting that evolution is true (sort of a jump), I still think there are problems. What about other warming mechanisms like fur and fat. What about animals that don’t live in regions that get extremely cold (in that it would be nice to assume warmest positions when cold, but not threatening to survival if you don’t quite get there).

What have we learned? I’d say that it is most likely that any biological proof is going to have assumptions that can be picked apart and thus not be a legitimate proof. But, I do not rule out the possibility that one could exist and be at the same “level of convincing” as a standard proof.

Let’s go back to my age old claim, though. Mathematics is NOT about answers, calculation, and results. In general, mathematicians like to read about and hear results, but in general they could care less. They care about ideas. Mathematics IS ideas. We do learn a lot from that proof in terms of ideas (and what purpose does a proof serve other than to present new ideas?). We get a conjecture from the proof. We get a couple of notions on how to go about proving it (although as it turns out they probably won’t work). So in terms of importance, I think proofs such as these carry equal weight to make and examine as standard proofs. In fact, they may carry more weight, since they are more universal than a typical proof. You don’t have to be an expert to read it, and once you understand the principle you can translate it to your specialty.

OK. I’ll stop babbling. Note: I didn’t want to be influenced, so I did not read the comments (in fact, I caught this blog post for the first time before any comments were made, so it was weird to see them there this time), so I apologize if I rehash something that has been said. Also, I feel bad for not actually addressing this in the original post, but this would be a rather annoying comment to receive I imagine.

There are many ways I could go for the next couple of weeks. All are fascinating to me, so I’ll let you decide. Vote now!

1. I could lay out in simple terms my favorite Millennium Problem: The Hodge Conjecture. I wrote this up last winter, and it is for all levels. It is conceptual and basically no details rear their ugly head. So if you’re interested in what these million dollar prize problems are like vote 1.

2. Several art related things that are well worth analyzing/discussing have come up.
2a. Literature: My Gravity’s Rainbow Challenge is well under way or I never really discussed my thoughts on my first Haruki Murakami experience.
2b. Film: I saw my first Pedro Almodovar film. Other directors worth discussing that have come up recently are Harmony Korine, Werner Herzog, Shinya Tsukamoto, and Shyamalan’s newest The Happening.
2c. Music: Who has popped up this year as exceptional (Bon Iver, Son Lux, Extra Life, etc) and who has let me down (Death Cab). I have a harsh opinion people don’t want to hear.

3. Philosophy: The standard philosophy of mind and language that I’ve been reading, or some ethical debates (more on Sam Harris maybe?).

4. Choose your own adventure: Anything you’ve seen or heard of lately pertaining to math, physics, philosophy, or art that you think I may be able to shed some light on. I have an article entitled “Noncommutative Geometry for Pedestrians” that I’ve been looking for an excuse to read. Also, I have a library system and netflix, so basically any book or film you bring up I should be able to get my hands on.

Just because a good deal of my readers probably don’t look at any of the blogs on the blogroll under the category of Math, I will share some news that is spreading like wildfire. It seems to be the cool thing to do right now to point out that someone submitted a proof of the Riemann Hypothesis to the arxiv.

Hmm…some terms should be defined I guess. The arxiv is a preprint server for math and physics (and computer science, etc?). It allows you to “publish” your work before it gets picked up by a journal which could take years.

The Riemann Hypothesis is probably the most well-known problem in math. It has to do with the zeros of the Riemann zeta function. Blah, blah, blah…I know you don’t care, but!!! You get a million dollars if you solve it. (It is one of the seven Millenium Problems).

So the supposed solution has already had two people find holes. This type of thing happens all the time. The main reason for the blowing up of this is that the submitter is Xian-Jin Li, a professor at BYU. Usually the “solvers” are amateur mathematicians trying for a million dollars. A move like this is very embarrassing to an established research mathematician.

If you are unfamiliar with Sam Harris, I highly recommend him. He has ethical concerns for people who consider themselves to be rational human beings, but do not stand up to unreasonable action and thought. I won’t go into the details or else this could turn into a thousands of words post.

Dear Sam,

I am a big fan of yours and align myself pretty completely with your views. I am sure that you get tons of letters everyday from people criticizing your ideas, so it is possible you won’t even read this. Following your lead, I feel that I need to point out a flaw in your argument for upholding spirituality (in the form of isolated meditation). Now I just saw you roll your eyes, since this has probably been your most received letter since your talk at the 2007 AAI Conference.

First off, I said I aligned with you, and this is true on the point of spirituality in that I am a practitioner of Zen forms of meditation. My point is that you cannot use the argument you do to uphold this position. My argument comes from the theory of cognitive dissonance, so if you do not subscribe to this, then my point may be moot.

Your seem to have two main claims. One being that through experiences such as isolation and meditation, we can achieve greater awareness of ourselves and our nature. Or, before you jump on that, at the very least, it is possible for every human to test for themselves whether or not this is a true statement. If you find that it isn’t true, then we can’t totally dismiss it, because as with a great athlete, the training required might be much more than what the individual put in. The second claim being that people that claim to have this higher awareness, are in general happier, selfless, etc making the pursuit a worthy one.

Please keep reading even if the details of the above are not correct, as I feel my argument is not in the details but the general. Suppose someone goes off to live in isolation for a year. He meditates for 15 hours every day. Nothing is happening. Isn’t it possible that some cognitive dissonance is building up? The person is told that they will become enlightened if they put the time and effort in. After months of nothing happening, there is a change. The things that people said will happen are happening: the loss of the sense of self, and more. Isn’t it possible that to resolve the cognitive dissonance that this person has sat alone in isolation for a year with no results, the brain has decided to make the fiction a reality?

If the above is even of slightest, remotest, possibility, then we have a problem. This is quite possibly what goes on in other religions as well. You pray every day of your life, then one day you have a life altering religious experience and “know the truth.” This sounds very similar to the person that meditates and has a life altering experience. According to your own viewpoint, if we condone the “non-religious” spirituality, then we necessarily cannot make an arbitrary distinction and condemn the other (especially if the same cognitive mechanism drives both of them).

What about the outcomes? I believe that it is also your opinion that outcome is irrelevant to justification. Even if it is, there are plenty of Christians (insert religion of choice) that due to their very belief in God do positive things in the world. “It gives meaning to their life.” Can you explain how it is different from when the ascetic gains meaning to their life through meditation?

To end on a more positive note, I still consider myself in alignment with you. But as someone who practices rationality, I feel that I cannot use the reasons you have provided to justify my spirituality. They seem to be too close a variant on the irrational arguments (and possibly the same mechanism) that organized religious people use.

Sincerely,
HilbertThm90

I am going to send this to him, but I would first like to hear if I am missing something major. If you are unfamiliar with Sam Harris, then you probably will have trouble following this post.

So I’m sure I’m losing readers/upsetting some readers with my recent mumbo-jumbo, and I shall continue in that vein. I will answer a long standing question: What is the Poisson Integral? Before I begin, I will not be doing this in all its glory.

So let’s clear up what I consider to be a misnomer. This should not be called the Poisson integral, but rather the Poisson integral representation. Let U be the unit disc in the complex plane. Let T be the unit circle. Then if A is a vector space of continuous complex functions on the closed unit disc , A contains all polynomials and for every , then the Poisson integral representation

where is valid for every and every .

OK. Let’s break this down. It isn’t so bad. (Why is this in my real analysis textbook?!?) So first note that A contains all polynomials. It may not contain anything extra than that, and that is OK. If we throw extra stuff in, then we need to make sure it is still a vector space. That next condition can be stated in a different way. It is sort of a “maximum modulus” type condition. We want , where we use the typical norm for an operator on a Banach space ().

Now to get the integral representation, fix a z in U. Now be the Riesz Representation Theorem we have that there is a Borel measure, , such that . Since z is complex we can write it as for some |r|<1. Let , then we have the that the collection is dense in A by Stone-Weierstrass, so let’s see what we can get out of the integral representation for this collection.

for n=0,1,2,… and since on T, we have now for .

Now define where t is real. But by the Dominated Convergence Theorem we have that , but here we have constructed what we want .

But you say this doesn’t look the same as you said. Well, let’s deconstruct which is called the Poisson kernel. Since the series is symmetrical about zero, we can break it into . But we only care about the real part (why?), so , i.e. the Poisson integral in the form first stated!

Also, if anyone actually read all this and understood it, please leave a comment that you did.

I know this isn’t going to make much sense to anyone, but if you want to say:
あなたの男の子がかわいいです。

Then you probably shouldn’t say:
あなたの男の子が嫌いです。

They may not look close, but it is the difference between kawaii and kirai (or your boy is cute vs I hate your boy).

I had many things that I wanted to talk about, but when I read this theorem, it was so shocking that I just had to post it. Now from a general intuition standpoint, you might think this theorem to be quite natural. But remember, most of us have been trained to think Fourier series are extremely nicely behaved. In fact, if I had read the wikipedia article when I was learning Fourier series years ago, I wouldn’t be surprised as much since under Divergence, it tells us this stuff.

So secretly we always are working with , or square-integrable -periodic functions. When we think in this way, we have the Fourier series of f at x given by the partial sums where . It turns out quite simply that in -norm the partial sums converge to f quite quickly. This nice convergence tricks us into thinking that we will have nice convergence all the time.

So if we switch to just continuous -periodic functions (which is a dense subset of ), do we get something as simple as point-wise convergence (it would surely be too much to ask for uniform convergence)? Well, from a common measure theory theorem, since the sequence converges in -norm we have a subsequence converging point-wise almost everywhere. But this leaves much room for error. How much error you ask?

It turns out that there is is a set which is a dense in C(T) which has the following property: For each , the set is a dense in . Note that .

This is pretty rough considering it means in non-technical terms that continuous functions are completely filled with functions for which the points where the Fourier series behaves badly is almost everywhere. Also, note that a “dense ” is uncountable (nice little topological proof if you want to try it), so this isn’t like some minimally dense set we’re talking about.

I was going to prove the theorem, but now I don’t think I will because, there may be at best one person that has made it this far. If interested drop a comment and I’ll gladly add the proof here, though. I just need to make sure there is an interest before going forth with it.

We should note that it doesn’t take much to correct the problems stated here. If we just make sure that our function is Lipschitz of some order, then we have a convergent Fourier series.

So a nice discussion on the extra spatial dimensions post has led me to elaborate on what the fuss over gravity is. I have a wide range of audience, and most are probably in the philosophy school due to Philosophers’ Carnival. I think it is sort of important to know the controversy in physics right now, especially as a philosopher pondering the nature of the universe, and with CERN nearing its completion.

It is true that this fuss is over gravity, but it isn’t in an extra-dimensional sense. The four fundamental forces are the strong force, electromagnetic force, weak force, and gravity. The first three fit perfectly together through quantum mechanics. Then gravity has its own theory: general relativity. I actually listed them in order of strength. Gravity is by far the weakest and the strong force is the strongest (hence the name).

The problem is to get gravity into a quantum mechanics theory. The problem has to do with the uncertainty principle. Essentially, gravity is nonrenormalizable. This is terribly challenging to explain, but when you attempt to correct for small errors, those errors have to go to zero, but with gravity the errors build up and actually go to infinity. In a sense gravity “blows up.” Now if we try the other way, to put the really small into general relativity, we get what are known as singularities. This may be possible to overcome, but the postulates of quantum mechanics cannot be carried over to curved space-time (which is the whole point of GR).

So instead of trying to put one into the other people are trying completely new things and trying to get QM and GR as special cases of a more general theory. There are many attempts to do this. There is loop quantum gravity, topological quantum field theory, string theory, etc. It turns out that so far string theory is the only one that can get gravity into it successfully. It also turns out that string theory demands extra dimensions.

Disclaimer: Before some crazy physicist yells at me that other theories have fit gravity into their picture, I am saying this from my own knowledge. I know that there are many many different attempts at this, it is to my knowledge that none have successfully incorporated gravity without severe compromises somewhere else.

I lied to you again. It turns out that Gravity’s Rainbow has been delayed by a week. I started looking up influences and references in it, and it turns out I have a lot of reading to do before starting the novel. Second, the book club I joined meets this Sat, and I haven’t even started the book. The book is Edith Wharton’s The House of Mirth.

OK. Since I’m interested in post-modernism, I would never dream of reading Wharton on my own. She writes social realism, which seems to be a wasted genre. Realism definitely has its pluses, since it points out faults with society, but to be considered literary enough for a book club is beyond me. When your main concern is to get things right, how can you include the millions of other aspects of a work of literature that actually requires interpretation and hence would make good discussion.

I started it, and it is quite enjoyable. At the same time it is highly frustrating. Wharton’s intent is to point out many dreadful aspects of society (of her time), but unintentionally she is pointing out something much bigger in my mind. The clearest theme (and I’m quite sure it is unintentional) that comes across is how unreflective western society is. These people are going through their entire lives and not once asking themselves any of what I consider the “big questions.” They don’t even realize that there could be something to think and talk about outside of their shallow thoughts.

I could be wrong about whether Wharton wanted this to come across or not, but I don’t think so. Her themes (as realism dictates) are quite obvious. Since she is portraying realistically, though, the theme I just discussed inevitably comes across as well to someone concerned with that aspect of society.

This just reminds me how far art has come in such a short period of time. This was only about 20 years before Faulkner, but Faulkner is at such a higher level. Other than theme, I can’t really think of good things to talk about in Wharton. If we were to discuss Faulkner, we would have to start at such a more fundamental level. What is the plot of the novel? There would be disagreement. What are all these crazy devices being used? Why? Look at all these different levels and patterns emerging, etc.

When we move even further along the literary timeline we get even crazier things like: Character A is thinking about Character B. Suddenly the narrative shifts to B, and travels via analepsis back in time to her life with Character C. Character C takes up the narrative and analeptically shifts the focus back in time farther to an event that shaped his life; then the focus returns to C’s “present.” The prose is then recycled back to its starting point with Character A, who is currently inhabiting the reader’s “real-time” present.

I just don’t understand why a book club would pick a novel that is so one-dimensional with so many other novels out there that have extreme interpretations worth discussing.