Archive for the ‘mathematics’ Category


It’s nearly, if not totally, impossible to remove our deeply held biases, values and contextual history from our raw, sensory perceptions of the world. The difficulty to sense more objectively is what perpetuates so many non-truths about the world. Nearly everything we sense and think is distorted by the biological patterns shaped within us by the world and our interaction within the world. It is within this frame of reference I seek to put down, in an obviously flawed ways, what I think to be at least less non-truth than other theories and thoughts floating about out there.

Our most basic means of communication, the words, sounds, gestures and pictures, we use are so filled with bias it’s impossible to commit to their use in an objective way. The best hope I have is to present as many variations across mediums so that what emerges from these communications is perhaps, if not objective, at least more fully representative of various perspectives that at least the trap of obvious one-sided subjectivity is avoided.

And with that warning, let us proceed.

A first exercise is of definitions and clarifying of terms.

Everything is information. From the most basic particles of existence to governments to rocket ships to the abstractions of mathematics – everything is information. Information looped and entangeled within other information. Information trapped within patterns of information by other patterns of information. Particles trapped into behaviors dictated by the laws of physics. Proteins and chemicals replicating into biological entities by the encodings of genetic instruction. Objects of pure quantity expressed in combinations dictated by rules of provable logical inference. Symbols imbued with meaning combined to form words and sentences and stories that stick in the brains of people and come out of their mouths to be reinterpreted over the eons. Faintly remembered events strung together by stories to form history and imagined events of some time that has not come to pass forming a future hope.

More fundamentally… space and time and causality and logic and being itself. All are matters of information. The casual ordering of events in relation to what is different based on the difference of another entity forming the conception of time and space. The coherence within a frame of reference of words strung together with symbols for equal, not equal, for all, and the such coalescing into logic. What is and what isn’t in reference to what’s logically or causally sensible to us becomes the notion of being.

But this is not quite enough.

Recently various categories of research, science and/or philosophic discussion have added ’emergence’ and ‘complexity’ to the pantheon of fundamental concepts from which we can chart our maps of existence and meaning. The unseen in the parts that only shows itself in the collective – the multitude – the interactive, this notion of emergence.

All in – meaning. Meaning is a vague notion of symbolics and representation within the ontological dimensions of space, time, cause, logic, emergence and being. Meaning is proximal, local phenomenon of pattern. In totality, all things considered, that is all of infinity, there is no meaning – there is no pattern. That is, all patterns at play is pure entropy and no meaning is possible on a universal, infinite scale. (As if we can even imagine such a concept). On a local, limited frame of reference meaning emerges from patterns (people, computers, plants, etc) pattern matching (sensing, perceiving, transforming, encoding, processing).

I propose a phrase: existential equivalence. Every investigative thought, every scientific gesture, every act of art, every attempt to send a message, every ritual, every interaction at all with the world at any level is all of similar thing: the encoding and decoding of information within information. This is not a reduction or a reductionist exercise. Quite the contrary. The varieties of symbolic expression in all of existence is REAL, it is a thing. That existence is expressible in an infinite variety is necessary. and it can only be known, even in a limited way, by actual variety of expression. If anything is to exist, it must exist in infinite variety and multiplicity. Everything that exists has existential equivalence. The entirity of existence is relational.

For instance if there is such a concept and sensation of color it must have expression in physical and artistic and literary terms. It exists at all levels implicated there. If a wavelength of light is able to generate a visual and neuronal concept we called red, then red isn’t just the wavelength, nor the wave of light, nor the eye, nor the brain, nor the word… it is all of those things and all of the things we do not yet think or talk or gesture about.

Or consider a computer program. Its existence is a string of words and phrases transcoded into 1s and 0s and into physical logical gates transmitting electrons and back around and on itself into monitor LEDs into human eyes and brains into motor movements of mouse and keyboard and so on. A computer program is the interaction of all the information.

But surely there are such simple things that do not have a universal relationship – an existential equivalence? what is the simplest thing we can think or speak of? a boson? the number 1? a dot? just an abstract 1? It is impossible to wipe the complexity of existence from even these pure abstractions. We only conceive of their simplicity in relation to other concepts we find complex. Their simplicity must be weighed against everything that isn’t simple.

And so here we have a collosal contradiction. Patterns are a local phenomenon. They aren’t the entirety. And yet I’ve suggested that patterns are existence – all that exists. Unraveling this I am directly saying that patterns interpreting/transcoding/sensing patterns is what exists – creates th world – at all levels. Pure relation, which is only possible at a local level, is existence. Particles only exist in relation to other particles – a gradient. Humans to other humans, to animals, to the planet, to particles. Planets to other massive bodies… and so on, and on, up and down, left to right, back and forward, in and out….

herein lies a beautiful thing – mathematics and computation are a wonderfully efficient symbolic translation methods. This is why computers and mathematics always creep their way into our efforts to make things and make sense of the world. It is why our brains are so damn useful. complex abstract pattern recognizing patterns – these networks of neurons. It is why DNA is so proficient at replication. a “simple”, resilient substrate carrying everything necessary to generate and regenerate these networks of neurons that can then make synthetic networks of pure relation. Whether particles or quantum or digital or biological or chemical there is pure relation, pure patterns among patterns – there is math. It matters not and is completely the point that math and computation can be done in any substrate – between proteins, with pen and paper, on a calculator, in a quantum computer.


why is that? WHY?

In a feat of complete and utter stupid philosophy and unlogic… because it cannot be any other way. Positing a god doesn’t escape this. Positing a multi-verse doesn’t escape this. If any of those things are to exist, they must exist still in relation – they are relation! It’s borderline mystical. Of course it is!

And why does any of this matter? is this just another sound of one hand clapping? a tree falls in a forest does it make a sound? Yes. yes indeed. Those, while used to dismiss the question from the outset actually do call attention to the entirety of the situation. What we conceive of as existence and existing is usually reductively done in by our discrete categorization and our failure to continuously review and revise our categories. The practical implications of this adherence to categories (zoology, isms, religion, gender, nations, science disciplines, etc) is what stunts our path towards knowledge and keeps us in fear.

If we don’t lean into the idea that everything has an existential equivalence we are simply deciding to be ignorant. And in that ignorance we trend towards non-existence. In every day terms if we see the human population only by the color of skin we diminish human existence. If we say and take for truth all of the -isms, reductions, and arbitrary definitions we snuff out relation. If we make any assumptions at all and refuse to question those assumptions, even what we think are so obvious and so simple, we move closer to entropy. If we want to exist at all, we must be mystical and fanatical about sensing relation, resensing it, re-interpreting it. This is not a moral argument. Existence is no more moral than non-existence – except as a local conception.

It really does come down to this (and this is very Camus-like):

If you care at all to exist as you, you must question/express/relate to everything as much as you can before your pattern is fully transcoded into something not you. (we are just food for worms…)

So yes, ask yourself and answer it in infinite variety over and over “if a tree falls in a forest does it make a sound?” This is life – it is your existential equivalence to everything else. You relate, therefore, you are. I relate, therefore I am. X is, therefore X relates.

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In Defense of The Question Is The Thing

I’ve oft been accused of being all vision with little to no practical finishing capability. That is, people see me as a philosopher not a doer. Perhaps a defense of myself and philosophy/approach isn’t necessary and the world is fine to have tacticians and philosophers and no one is very much put off by this.

I am not satisfied. The usual notion of doing and what is done and what constitutes application is misguided and misunderstood.

The universe is determined yet unpredictable (see complexity theory, cellular automota). Everything that happens and is has anticedents (see behaviorism, computation, physics). Initiatial conditions have dramatic effect on system behavior over time (see chaos theory). These three statements are roughly equivalent or at least very tightly related. And they form the basis of my defense of what it means to do.

“Now I’m not antiperformance, but I find it very precarious for a culture only to be able to measure performance and never be able to credit the questions themselves.” – Robert Irwin, page 90, seeing is forgetting the name of thing one sees

The Question Is The Thing! And by The Question that means the context or the situation or the environment or the purpose. and I don’t mean The Question or purpose as assigned by some absolute authority agent. It is the sense of a particular or relevative instance we consider a question. What is the question at hand?

Identifying and really asking the question at hand drives the activity to and fro. To do is to ask. The very act of seriously asking a question delivers the do, the completion. So what people mistake in me as “vision” is really an insatiable curiousity and need to ask the right question. To do without the question is nothing, it’s directionless motion and random walk. To seriously ask a question every detail of the context is important. To begin answering the question requires the environment to be staged and the materials provided for answers to emerge.

There is no real completion without a constant re-asking of the question. Does this answer the question? Did that answer the question?

So bring it to something a lot of people associate me with: web and software development. In the traditional sense I haven’t written a tremendous amount of code myself. Sure I’ve shipped lots of pet projects, chunks of enterprise systems, scripts here and there, and the occassional well crafted app and large scale system. There’s a view though that unless you wrote every line of code or contributed some brilliant algorithm line for line, you haven’t done anything. The fact is there’s a ton of code written every day on this planet and very little of it would i consider “doing something”. Most of it lacks a question, it’s not asking a question, a real, big, juicy, ambitious question.

Asking the question in software development requires setting the entire environment up to answer it. Literally the configuration of programmer desks, designer tools, lighting, communication cadence, resources, mixing styles and on and on. I do by asking the question and configuring the environment. The act of shipping software takes care of itself if the right question is seriously asked within an environment that let’s answers emerge.

Great questions tend to take the shape of How Does This Really Change the World for the User? What new capability does this give the world? How does this extend the ability of a user to X? What is the user trying to do in the world?

Great environments to birth answers are varied and don’t stay static. The tools, the materials all need to change per the unique nature of the question.

Often the question begs us to create less. Write less code. Tear code out. Leave things alone. Let time pass. Write documentation. Do anything but add more stuff that stuffs the answers further back.

The question and emergent answers aren’t timeless or stuck in time. The context changes the question or shape of the question may change.

Is this to say I’m anti shipping (or anti performance as Irwin put it)? No. Lets put it this way we move too much and ask too little and actual don’t change the world that much. Do the least amount to affect the most is more of what I think is the approach.

The question is The Thing much more than thing that results from work. The question has all the power. It starts and ends there.

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Mathematical Inspiration from Mathematicians: An Outer View of the Inner World, Mariana Cook, Princeton University Press, 2009

“In the past, mathematicians always tried to solve problems exactly. Now we realize that most problems will never have an exact solution. Nonetheless, we can hope to understand the general shape of a solution, and topology gives a language for talking about these shapes. Topology gives a new point of view on all kinds of physical phenomena: the collapse of a bridge that vibrates too much, the tangling of strands of DNA, and so on. But I have to admit that my own interest is based on the joy of understanding shapes rather than on any particular applications.”

– Burt Totaro, page 64 (a professor I had in college, FWIW)


“As a grad student I had become interested in the annulus conjecture. Saunders Mac Lane advised me that it was a bit hard for a thesis problem (it was), but I thought about it whenever I had an idea. In 1968, while looking after my four-month-old son, an idea occurred to me, now called the “torus trick.” It only took a few days to realize that I had reduced the annulus conjecture to a problem about PL homotopy tori, and in a different direction had proved the local contractibility of the space of homeomorphisms of n-space.”

– Robion Kirby, page 62


“The imprint of the world in our minds is not photographic; all the brain knows of the outside world is a chaotic sequence of electric impulses and out of these it creates a structural entity: our perception of what we see and hear. Most of the time, an adult’s brain talks to itself and creates more and more refined structures within itself. The word “structure” means a mathematical structure, something which becomes more and more abstract and better and better logically organized in the course of this self-conversation….

We are all fascinated with structural patterns: periodicity of a music tune, a symmetry of an ornament, self-similarity of computer images of fractals. And the structures already prepared within ourselves are the most fascinating of all. Alas, most of them are hidden from ourselves…. Brains are our masters, with only 2 percent of our body weight, they take 20 percent of the oxygen resources of our bodies; you cannot cannot resist their commands. You become a mathematician, a slave of this insatiable hunger of your brain, of everybody’s brain, for making structures of everything that goes into it.”

– Mikhael Leonidovich Gromov, page 34


“I often think of cats. I think of trees. I think of dogs occasionally but I don’t think of them all that much because dogs are agreeable. They do what you want them to do to some extent. Some people believe that mathematics is what we think it is and it’s created by our thoughts. I don’t. I’m a Platonist at heart, although I know there are a very great difficulties with that view.”

– John Horton Conway, page 18


“At the moment, one of the things I’m working on understanding is the total wavelength of a surface like a sphere or something of greater complexity, such as the surface of a bagel or a pretzel. What is the total wavelength? … I first became interested in the total wavelength as a model related to a question which can be roughly stated as, can one hear the shape of the universe?”

– Kate Abedola Okikiolu, page 98


“For example, the “Ode to Joy” would be 334554321123322 for the right hand, and 332112345543344 for the left, with corresponding digits always adding up to 6. Soon music became a passion itself, on a par with my passion for numbers, though on its own terms.”

– Noam K Elkies, page 158


“In mathematics, there are not only theorems. There are, what we call, “philosophies” or “yogas,” which remain vague. Sometimes we can guess the flavor of what should be true but cannot make a precise statement. When I want to understand a problem, I first need to have a panorama of what is around it. A philosophy creates a panorama where you can put the things in place and understand that if you can do something here, you can make progress somewhere else. This is how things begin to fit together.”

– Viscount Pierre Deligne, page 156


” I prefer to close my eyes when I think about mathematics. The best work is done by night, in half sleep. Sometimes I go to bed thinking, “Ah, I have a nice lemma to prove–or disprove.” (Should I explain what a lemma is? A mountain climber needs holds to get from one level to the next one. Lemmas are the the holds of a mathematician.)”

– Jean-Pierre Serre, page 144


“How to define the roughness of rusted iron, of broken stone, metal, or glass? What shape is a mountain, a coastline, a river, or a dividing line between two watersheds? That is, can geometry deliver what the word seems to promise, namely, truthful measurements of untamed Earth? How fast does the wind blow during a storm? what shape is a cloud, a flame, or a welding? What is the density of galaxies in the universe? What is the volatility of the prices quoted on financial markets? How to compare and hopefully even measure different writers’ vocabularies?”

– Benoit Mandlebrot, page 94

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I really like this post on Good Math, Bad Math.

Beyond being mildly humorous in that cranky math person non-funny kinda way, it touches on lots of my favorite subjects: enumeration, Cantor, classic proofs, cranky math people.

The catch – and it’s a huge catch – is that the tree defines a representation, not an enumeration or mapping. As a representation, taken to infinity, it includes every possible real number. But that doesn’t mean that there’s a one-to-one correspondence between the natural numbers and the real numbers. There’s no one-to-one correspondence between the natural numbers and the nodes of this infinite tree. It doesn’t escape Cantor’s diagonalization. It just replaces “real number” with “node of this infinite tree”. The infinite tree contains uncountably many values – there’s a one-to-one correspondence between nodes of the infi To see the distinction, let’s look at it as an enumeration. In an enumeration of a set, there will be some finite point in time at which any member of the set will be emitted by the enumeration. So when will you get to 1/3rd, which has no finite representation as a base-10 decimal? When will you get to π?


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An interesting approach to knowledge mentioned in Stephen Wolfram’s blog:

But what about all the actual knowledge that we as humans have accumulated?

A lot of it is now on the web—in billions of pages of text. And with search engines, we can very efficiently search for specific terms and phrases in that text.

But we can’t compute from that. And in effect, we can only answer questions that have been literally asked before. We can look things up, but we can’t figure anything new out.

Let’s see where this goes!

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Reproduced from  a private email by Mahesh Johari:

For most of the last 25 years we (as a nation) have been sold a story about investing in stocks for the long run.  Invest steadily, mindlessly, and over the long run stocks will earn almost 10% annual returns.  By the time this bear market has ended, this notion will be questioned by a great many.

I’ll give all of you a head start.

Let’s quickly review one of the greatest achievements of the past 15 years – the rise of the personal computer and the growth of the Internet.  During this time we saw two giants dominate this market – Intel (NASDAQ: INTC) and Microsoft (NASDAQ: MSFT).  They were the veritable Pippen & Jordan of the tech Bulls dynasty: nearly pure monopolists with gigantic profit margins, huge revenues, and fantastic cash flows.

Today, Intel’s share price closed at $12.08, the same level it was at when I turned 25 years old.  That was almost 12 1/2 years ago.  Let’s look at Intel closely and see what they have to show for this incredible run.

At the end of September in 1996, Intel’s share price was $12.08.  Adjusted for splits, there were 7.14 billion diluted shares outstanding.  The book value per diluted share was $2.09.

We could go into a brief academic debate about why I’m using book value instead of some other measure.  Book value is the accounting net worth of the company.  With some caveats, it is a reflection of value that takes into account all of what the company owns and all of its obligations.  The book value reflects the amount of capital the company has available to deploy productively in the course of business.

I use book value per share because one share of Intel essentially grants you ownership to that amount of book value.  If you simply hold that share, you could imagine that the value backing that share of Intel is growing by the same rate as the book value.  Book value is not influenced by the share price, which can fluctuate wildly with the market.

Compare it to your own situation.  If I asked how you have done financially over the last 12 years, I could look at your net worth 12 years ago, compare to what it is today, and have a pretty good idea of how you fared financially.  It’s the same idea.

Today, Intel’s book value is $6.80 per diluted share.  Over the last 12 1/2 years, it means that Intel has grown book value per diluted share at an annual rate of 9.94%.  Some of you will argue that I am not including dividends that were paid out.  Those dividends have totalled $2.25 over that time frame.  Including dividends, Intel generated annual rates of return of 12.50% over the last 12 1/2 years (assuming you didn’t reinvest the dividends).

That number sounds pretty good.  Until you realize that Intel was a near monopolist operating through one of the highest growth phases of their industry.  Think about that for a minute – a monopolist in a boom was only able to generate 12.50% per year in returns.  What does this imply for the 495 companies in the S&P 500 that are NOT monopolies, and are NOT going to be going through an incredible boom in demand?  A 10% annualized rate of return for the entire market suddenly sounds like a fantasy, doesn’t it?

So what the heck happened?  What about all those studies that touted how stocks would make 9-10% over long periods?  Is 12 1/2 years not long enough?

What happened is what always happens when people blindly follow historical statistics en masse.  The underlying behavioral model changes.  As naive shareholders piled in and stopped paying close attention to how the company was run, profits got transferred from shareholders to employees through stock options and bonuses.  Additional billions were blown on share buybacks at much higher prices.  Looking at the result after more than a decade of shareholder un-friendly behavior, it’s no wonder Intel’s results are mediocre.

The time to buy stocks for the long run will be when those that bought for the long run realize they have been fleeced.  When those people are so disgusted that they sell at low prices and the shareholder outrage forces companies to change their behavior – that is the time to buy stocks for the long run.  Price matters.  That’s how it has always been.

For those who like to check the numbers, I suggest the following:
Intel 2008 Annual 10-K:
Intel Q3 2006 10-Q:
Intel splits (shown on chart):

As a side note, if you actually bought that share of Intel in September of 1996, your rate of return was not 12.50% but 1.38% annually, assuming you didn’t reinvest dividends.  Ouch!

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Getting webMathematica working on MacOS X was not entirely trivial, even with decent install instructions.

Here’s what you’ll need to avoid wasted time in getting set up:

  1. Install Mathematica first, ideally the latest version
  2. Your java should be fine provided you’re on OS X 10.4.11+, but double check your java -version looking for 1.5+
  3. Use tomcat6.  I tried glassfish, it was kinda a pain (i.e. it wasn’t drag and drop like tomcat)
  4. get the webMathematica.zip or .war file and deploy within the webapps folder in tomcat
  5. create a mathpass file and put it in the /conf folder in the webmathematica web app.  Follow this formating.   Be sure to register you webMathematica with register.wolfram.com to get your mathID and all that.
  6. Grab the J/Link jar from your current Mathematica.app/SystemFiles/Links/JLink/JLink.jar  and dump it into your tomcat/webapps/webmathematica/lib/JLink.jar — maybe this isn’t necessary, but i figured it would be best to match the JLink that came with the kernal to the one used in the local webMathematica (I couldn’t get it to work with the .jar on the webMathematica disk)
  7. start tomcat.  try the examples.

Sadly there are very few other places to get webMathematica troubleshooting tips.  The FAQs aren’t too deep and the forums have nothing.  Generally there aren’t a whole lot of people using webMathematica (should be more!) so community support suffers.  Also, those who are using it generally aren’t on Mac OS X 10.5.6+.

Post a comment if you changes or suggestions or your own experience.

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