I just noticed this and I have to wonder if it's a common thing or not: When looking at the number of posts with a given tag on my tag list, it has a vaguely exponential distribution. Discrete, of course, but it has that same downward slope and everything when the tags are ordered by number of posts. I wonder if the histogram of number of tags with a given post count is Poisson?
I may just take the current counts of this page, run them through R, and post the results. It would be really cool if lots of other people did this and posted the results below or emailed theirs to me for inclusion.
I wonder if it says anything about human behavior? Do we tend to clump most things into the same few bins and have lots of smaller ones?
Showing posts with label beauty. Show all posts
Showing posts with label beauty. Show all posts
Sunday, February 3, 2008
Friday, January 18, 2008
Mathematics as Just a Game
There is a quote attributed to Hilbert that " Mathematics is a game played according to certain rules with meaningless marks on paper." It is perhaps the most cynical characterization of the whole field I have ever encountered.
Pure mathematics, in its modern form, is a race to find prove theorems on the basis of some set of axioms, some set of definitions and a given system of logic. Unlike a science or medicine in which a result must be repeatedly checked against experience and observation (and there is a good career to be had in doing so), in mathematics once a theorem is proven there are only three ways to work with it:
On the other hand, those theorems can describe all manor of objects (and in fact describe anything which meet the hypotheses). One simple example, set theory, works fine for piles of sand or stacks of money. Geometry we see around us all the time. The Navier-Stokes Equations govern, on the applicable scales, the flow of every fluid known to man.
The point here is simple: given a system of logic, a set of axioms, and a set of definitions power sets of theoretical tools for describing not only other theoretical objects but myriad real world systems can be effectively analyzed. While, yes, mathematics for its own sake does resemble a game on paper with meaningless symbols, it is those powerful tools which set it apart.
Pure mathematics, in its modern form, is a race to find prove theorems on the basis of some set of axioms, some set of definitions and a given system of logic. Unlike a science or medicine in which a result must be repeatedly checked against experience and observation (and there is a good career to be had in doing so), in mathematics once a theorem is proven there are only three ways to work with it:
- To prove another theorem.
- To get some applied work done.
- Find another, better way to prove it.
On the other hand, those theorems can describe all manor of objects (and in fact describe anything which meet the hypotheses). One simple example, set theory, works fine for piles of sand or stacks of money. Geometry we see around us all the time. The Navier-Stokes Equations govern, on the applicable scales, the flow of every fluid known to man.
The point here is simple: given a system of logic, a set of axioms, and a set of definitions power sets of theoretical tools for describing not only other theoretical objects but myriad real world systems can be effectively analyzed. While, yes, mathematics for its own sake does resemble a game on paper with meaningless symbols, it is those powerful tools which set it apart.
Saturday, January 12, 2008
Why I Try to CrossFit
The reason is simple: I want to get much more fit, and to do that requires a pretty hardcore routine. The thing I've found about CrossFit is that once I get into it for a few weeks it gets much easier, and I usually at that point hit midterms or something that keeps me out of the gym.
Why do I say I only try? Because I'm a thin runner-type not a lifter.
Why do I say I only try? Because I'm a thin runner-type not a lifter.
Sunday, December 16, 2007
Haiku #1 and #2
I must admit that aside from my otherwise aesthetic feelings towards the format particularly, poetry holds a certain cathartic value for me. Some feelings, some thoughts are so deeply held it seems they are only fully recognized in poetic forms.
At some point, I suppose, I’ll have to reckon with my own little contradictions—a man who strives after the rigorous logic of mathematics as the key to life simultaneously giving voice and comfort to his deepest passions in poetry, not exactly the most logical or deterministic form of expression. Perhaps if I wrote in Lojban . . .
Regardless, here are number 1 and 2, in that order:
A vibrant blossom.
softly its gentle beauty fades,
withering to dust.
Hope of love and joy,
once cast aside by fearful hearts,
crumbles into dust.
Another interesting feature of haiku and poetry generally is the need to encode information though subtext, metaphor, and other lingual constructs in order to convey the complete message. For instance, while neither of the above are per say particularly cryptic, unless one knows me and my life at time of writing (within a month of this post), what's really being spoken to will remain unclear. Presuming the analogy to cryptography can be sufficiently fleshed out, one interesting information theoretic question might be this: Given a particular amount of poetry, how much total information could be unambiguously conveyed? The idea certainly lends a new twist to "literary analysis".
At some point, I suppose, I’ll have to reckon with my own little contradictions—a man who strives after the rigorous logic of mathematics as the key to life simultaneously giving voice and comfort to his deepest passions in poetry, not exactly the most logical or deterministic form of expression. Perhaps if I wrote in Lojban . . .
Regardless, here are number 1 and 2, in that order:
A vibrant blossom.
softly its gentle beauty fades,
withering to dust.
Hope of love and joy,
once cast aside by fearful hearts,
crumbles into dust.
Another interesting feature of haiku and poetry generally is the need to encode information though subtext, metaphor, and other lingual constructs in order to convey the complete message. For instance, while neither of the above are per say particularly cryptic, unless one knows me and my life at time of writing (within a month of this post), what's really being spoken to will remain unclear. Presuming the analogy to cryptography can be sufficiently fleshed out, one interesting information theoretic question might be this: Given a particular amount of poetry, how much total information could be unambiguously conveyed? The idea certainly lends a new twist to "literary analysis".
Monday, December 10, 2007
Haiku
I've decided one thing to do before I die is learn and practice the art of Haiku.
Why Haiku? Three reasons. Those are the challenge, the experience, and the convenience.
As self expression goes, a single thought into 17 syllables is fairly trivial, craming it into exactly 17 is non-trivial, and doing it well is very challenging. Mix with my own semi-perfectionist nature, and this gets interesting fast. Example: "W.T.F" is a complete thought, but how would you express the same thing in 17 syllables?
I can't explain it, but there's something about the experience of Haiku which is simply amazing from start to finish. The setup, the suspense, the contrast . . . it's just awesome to finally look at even a decent haiku (much less a good one) and simply enjoy the way it works on me.
Third, poetry takes lots of itterations before it's anywhere near good. Haiku's being as short as they are, the cyclic rate for a complete set of revisions is high (even if the number of revisions is longer). This is more compatible with my lifestyle as I much more frequently have 15 minutes (which can be enough to ponder a single word choice) here and there than the hours it would take to consider the implications of a single work or comma on a larger poem.
Why Haiku? Three reasons. Those are the challenge, the experience, and the convenience.
As self expression goes, a single thought into 17 syllables is fairly trivial, craming it into exactly 17 is non-trivial, and doing it well is very challenging. Mix with my own semi-perfectionist nature, and this gets interesting fast. Example: "W.T.F" is a complete thought, but how would you express the same thing in 17 syllables?
I can't explain it, but there's something about the experience of Haiku which is simply amazing from start to finish. The setup, the suspense, the contrast . . . it's just awesome to finally look at even a decent haiku (much less a good one) and simply enjoy the way it works on me.
Third, poetry takes lots of itterations before it's anywhere near good. Haiku's being as short as they are, the cyclic rate for a complete set of revisions is high (even if the number of revisions is longer). This is more compatible with my lifestyle as I much more frequently have 15 minutes (which can be enough to ponder a single word choice) here and there than the hours it would take to consider the implications of a single work or comma on a larger poem.
Thursday, November 15, 2007
Beauty of Mathematics Part One--No Right Way to Think About Mathematics
I call this Part One because it will probably be a recurrant theme in my posts for a VERY long time. The rest of the title should be self-explanitory.
Saying there is a right way to think about mathematics is a little like saying there is a right way to think about sculpture or any other form of art. It just doesn't work. Film (to pick a relatively ubiquitous medium for the modern human) speaks uniquely to all who view it. The Godfather Trilogy, for instance, is to me a very sublime commentary on life and a moving tragedy in three parts. To another it may just be another crime film. Math is the same way. For many of my aspiring engineer friends, math is a set of tools for getting a job done. To me it is, amoungst other things, art.
Needless to say, it takes a particular breed to be passionate about what to many is a collection of formulas or a set of rules for balancing your checkbook.
But mathematics transcends this simple formulation as sculpture transcends the aftermath of attacking a rock with hammer and chisel. Think of something classical, like the Pythagorian Theorem. In its simplest form, it speaks of the length of the hypoteneus of a right triangle. In a broader sense, it is the shortest distance between two points in Euclidean space with the usual Euclidean distance. One geometric proof derives it from properties not of triangles, but circles. It has intimate connections to sine and cosine--the length of the opposite side of over the square of the length of the hypoteneuse is by definition sine. In calculus, it is critical in solving certain classes of integrals. It's the simplest non-trivial form of the dot product from vector analysis of a vector with itself (which is just the square of its usual Euclidean length). Each context worries about a version of what could be called a Pythagorean Theorem, which is simply a special case of more general ideas about metric or distance and each is equally valid.
Saying there is a right way to think about mathematics is a little like saying there is a right way to think about sculpture or any other form of art. It just doesn't work. Film (to pick a relatively ubiquitous medium for the modern human) speaks uniquely to all who view it. The Godfather Trilogy, for instance, is to me a very sublime commentary on life and a moving tragedy in three parts. To another it may just be another crime film. Math is the same way. For many of my aspiring engineer friends, math is a set of tools for getting a job done. To me it is, amoungst other things, art.
Needless to say, it takes a particular breed to be passionate about what to many is a collection of formulas or a set of rules for balancing your checkbook.
But mathematics transcends this simple formulation as sculpture transcends the aftermath of attacking a rock with hammer and chisel. Think of something classical, like the Pythagorian Theorem. In its simplest form, it speaks of the length of the hypoteneus of a right triangle. In a broader sense, it is the shortest distance between two points in Euclidean space with the usual Euclidean distance. One geometric proof derives it from properties not of triangles, but circles. It has intimate connections to sine and cosine--the length of the opposite side of over the square of the length of the hypoteneuse is by definition sine. In calculus, it is critical in solving certain classes of integrals. It's the simplest non-trivial form of the dot product from vector analysis of a vector with itself (which is just the square of its usual Euclidean length). Each context worries about a version of what could be called a Pythagorean Theorem, which is simply a special case of more general ideas about metric or distance and each is equally valid.
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