Sunday, 27 April 2014

Elliptic curves and modular forms

Simon Singh wrote a book called, Fermat's Enigma (1997). In it he describes in beautifully rendered prose a mathematician's program (Robert Langlands) to prove that different realms of maths had a deeper unity by using the Taniyama-Shimura Conjecture (the link between elliptic curves and modular forms) as the starting point.

Whatever these are - the "Taniyama-Shimura Conjecture" or "elliptic curves and modular forms" - they must be something magical and wondrous to inspire someone so (mathematically- and aesthetically-speaking, of course). There is something divine about mathematical structures (mathematical -arithmetic, -geometric, -physics and linguistics, to be sure): impartial, apparently eternal, inscrutable, unified, elegant, non-dimensional (ie, a structural relationship whose intrinsic beauty and perfection is "recoverable" only outside the mundane, arbitrary units of measure humans may come up with).

In the Gospel of John, the writer of the Gospel says that "In the beginning was the Word, and the Word was with G*d; the Word was G*d..." The four canonical Gospels are all unique yet have a simple unity that history, the petty ambitions of men, and the eroding influence of time have not done away with. To quote Finn and Jake: Algebraic.

When I became obsessed with prime numbers I tried to play around with many different equations: some whose truth of them I unfolded without interrupting them; some whose equations I played with like a fugue (variations upon variations 'til they made no more sense).

One of these I played around with is in the exponential form:

2n"+1 plus or minus 2n" plus or minus 2n' plus or minus 1 (excuse the clumsy attempt as I'm not a mathematician). Using this equation (I looked for the person who came up with the form but couldn't find it but I leave the credit to him/her) I started playing and color-coding and came up with this:


The prime numbers go from 3 or (8 - 4 - 1) to 673 or (1024 - 512 + 256 - 128 + 64 - 32  + 1) - the empty white spaces exclude the unnecessary exponential sequences (ie, for 673, the numbers 16, 8 and 4 are unnecessary and, therefore, comprise the white space in the row).

Immediately, one can see some (crude) fractal scaling of a webbed T whose left side is blue and whose right is red. There is a pattern of growth and repetition but there is not discernible regularity to the patterns. The growth in height (or width) looks like it follows a logarithm. Some proportionality is at play here, but there is a rhythmic limp that goes from the blue to the red side of the T and red to blue that is chaotic and unpredictable. But I only played with the prime numbers and do not know if there is more regularity to the pattern of color-coding of all numbers than the rendering of primes only (I doubt it).

Now, why did I bring up the holy scriptures? There is a saying called, the "Anna Karenina principle" (from the Leo Tolstoy novel) that goes like: "Happy families are all alike; every unhappy family is unhappy in its own way". Variations of this "principle" are replete in the Bible. The most famous one is in the first part of Isaiah 53:6

We all, like sheep, have gone astray,
each of us has turned to our own way

The Anna Karenina principle, in statistical terms says: "there are any number of ways in which a dataset may violate the null hypothesis and only one in which all the assumptions are satisfied" (http://en.wikipedia.org/wiki/Anna_Karenina_principle)

The Lord Saviour said that he is the Way, the Truth and the Life. I believe this statement to be the literal and figurative truth:

“Holy, holy, holy is the Lord Almighty;
    the whole earth is full of his glory.”

Isaiah 6: 3

Jay

Saturday, 26 April 2014

How smart is "smart"?

Many people tell me that I'm smart. Given the length of my life, up to the present, just as many people have told me that I'm not smart (to put it politely). Using a double-entry accounting method, I'd guess I'm pretty close to zero either way.


I'm not being self-denigrating; looking at life processes or quantum physical principles, I'd say all is well; at the very least, I have a pulse.


There is currently a promotional on CBC that says: "It isn't how smart you are, but how you are smart" that got me thinking about the subject of this entry, about the nature of intelligence.


To be sure, there are many different types of intelligence, many of which I fall short of being called "smart". I'm good at abstract thinking, not so good with emotional thinking. I'm good at formulating questions and phrases, but I have found this is often not enough.


Much like the Sheldon character in the TV show, The Big Bang Theory, I can appear "freakishly" smart sometimes. Unlike Sheldon, I don't have eidetic memory. Over the years of studying whatever interests me, I've learned to use an organic, idiosyncratic logic to draw out aspects of "knowledge" which often appears as eidetic memory (to a limited range of topics). I can even pull of a trick of generating original insights, given enough time.


I'm a great admirer of Shakespeare and of the enlightenment writers/thinkers. The world of knowledge that captures my imagination tends to the "classical" including the phraseology and humanity of the Bible. I believe in the ancient methodology of teaching called, the liberal arts. The aspect of the liberal arts that I believe in completely is the study of "first principles". It is what I seek out in the study of music, literature, mathematics, etc.


The study and contemplation of phronesis (or, ethical thinking) is to me my current obsession. This is where I've come to realize the single dimensionality of my kind of smarts, that I sorely lack perspective and depth. In a couple of words: I have a highly developed sense of aesthetics (formulation of ideas), but a pathetically poor developed sense of perspective (phronesis).


Since wanting to become a disciple of Christ's principle teachings (Mark 12: 29-31), I have come to find that inserting compassion, humility and agape into my relationship with social/ecological reality a real struggle, to live the talk. "The fear of the Lord is the beginning of wisdom" is not just a beautifully-phrased string of words but a spiritual challenge to admit to self (ie, vanity and pride) there is much that is bigger and deeper outside of us; that we are subjects to lessons that may break and destroy us if we do not have the proper perspective for the precise reason that humility is not a natural thing for us.


Andrew Murray, one of the Christian thinkers I take seriously, wrote a whole series of devotionals on the humility of Christ. In another work, he says:


Anyone grasping the promise only when he wants something very special for himself will be disappointed, because he is making Jesus the servant of his own comfort. (Murray, With Christ in the School of Prayer)


It is often observed that the culture of Jazz is an all-encompassing attitude. Another American strand that I deeply admire is what is all-purposely called, the American Gothic. It isn't just the literature I'm talking about here, but the culture in which Johnny Cash, Stevie Ray Vaughn, etc. belong: a personal journey of discovery of the limits and consequences of self-indulgence to the attitude of love.


The choice of subject matter of these artists changes and evolves in the course of their careers. An exquisite sense of irony and humility (cannot think of a better term) becomes apparent in their craft: their impressive talent takes on a greater depth. A personal reflection leads to a choice: seen-it-done-that, or realizing that a persistent insistence on self-will leads only to undignified parody of self.


In the Murrayian interpretation of Christ's lessons to humanity: "Humility is simply the disposition which prepares the soul for living on trust." (Murray, Humility)


Jay

Sunday, 20 April 2014

WFFs, or P versus NP

I was surfing the BBC website the other day when I came across this interesting link: http://aeon.co/magazine/world-views/what-is-left-for-mathematics-to-be-about/


In and of itself the author says nothing new in the article - in fact, it's about one of the oldest arguments about mathematics since the Greeks took it out of everyday, particular world into something abstract and general: what is mathematics? The main premise of James Franklin, the author, says there is a link between mathematics and the real world, and that this link is often overlooked in the discourse of metaphysics of mathematics.


Don't get me wrong. I'm not dismissive of Franklin's article. I appreciate his arguments, and he does say more than is apparent.


But it got me thinking about the similarities and differences between the language of everyday and the language of mathematics. And there are similarities between the two, to be sure. But the are also profound differences: the notion of "closed rings" for instance.


A closed ring, in mathematics, is an important concept that allows orderedness and constraint (ie, goes here but no further) that are so important to why we can trust results and derivations in mathematical operations, or, more precisely, arithmetical operations: for addition and multiplication, the positive, whole numbers are sufficient (a closed ring); for subtraction and division, we need to expand the notion of number to negative and ratio numbers (the positive and the other two "new" notions of numbers, in turn, comprise a larger closed ring for the four fundamental operations of arithmetic); geometry requires more than just that though, because some geometric constructs/relations require the notion of irrational numbers for some of its results to make sense (ie, the square root of two, the ratio between the circumference of a circle to its diameter - to cite just two famous irrational numbers). There are more closed rings beyond these numbers, even unto the "imaginary" numbers which are so useful for engineers and physicists...


But I digress.


To those initiators of our modern notions of mathematics (ie, the ancient Greeks), mathematics, ethics and metaphysics comprised only parts of a larger, more encompassing discourse called, philosophy. Mathematical notions were used to "prove" philosophical arguments, and vice versa. All serious, all the time. The abhorrence of beans as much as the abhorrence of irrational numbers - all fit into the worldviews of some cults that gave birth to august academies.


One of the unfortunate artifacts of this is the contamination of mathematics by unfortunate choice of terminologies - namely, the notion of "truth". In the canon of humanistic corpus, the notion of "truth" is, ironically, the first word of its satanic verse - the father of sophists (ie, postmodern criticism), as far as I'm concerned. The insidious nature of "truth", in this sense, is that it is an undefined assumption and people just never bothered to ask critically. Yeah, eh!


There is another mathematical paper that I tried to read and didn't give up on - just put aside for now if only to contemplate on it and compare it to other works. The paper was written by Max Tegmark and called, Is "the theory of everything" merely the ultimate ensemble theory?


In it Tegmark says a lot of interesting things. But the most interesting one (which makes part of his premise) is a mathematical notion called, a "well-formed formula" (the WFF in the title of this entry) and pronounced "woof" by logicians - I kid you not.


A well-formed formula is an important concept in mathematics in much the same way as a syntactic tree is to linguistics. A syntactic tree - in terms of structural principles - allows you to analyse elements of a phrase to determine whether it is grammatical or not. It makes no claims to the truth of a phrase, and that is its beauty, its strength. It has allowed linguistics to figure out that a well-formed phrase does not necessarily mean that it'll make any sense at all.


For instance, "little, green ideas dream furiously" is a well-formed phrase but makes absolutely no sense at all. It is what is called, a linguist accident. "Linguistic accidents" are very useful things: At the phonological level, it is exploited especially well by marketing agencies to come up with brand names.


Another phrasal accident that has stymied logicians since the ancient Greeks is "this sentence is a lie" (or, "all Cretins are liars"). Linguistics has developed a rather more sophisticated regard of such WFFs than mathematics, and, in fact, has developed a component of linguistic analysis called, Pragmatics.


This analytical component in the linguistics toolbox allows the formal study of how social, cultural and even temporal/geographical context and semiotics contribute to enciphering and deciphering meaning:


In this respect, pragmatics explains how language users are able to overcome apparent ambiguity, since meaning relies on the manner, place, time etc. of an utterance. (http://en.wikipedia.org/wiki/Pragmatics)


Unlike the apparent "timeless profundity" of results/insights in mathematics, everyday language is dynamic, mutable, chaotic, and utterly supple and living-breathingly alive. And, thus, linguistics so.


Surely, conceptual cognates of this nature must exist in mathematics. One would think that this conceptual notion has some bearing upon the P vs NP question, whether a given problem is a waste of polynomial time...an analytical pons asinorum perhaps.


Jay

Sunday, 13 April 2014

Micah 6:8

Ever since I learned how to read I've been fascinated by scientific knowledge. There is something of elegance and grace and simplicity in the expression of its principles (whether symbolic or linguistic). Other-worldly. In my adult life, the one thing that distracts me completely still is the contemplation of abstract structures and their first principles.


There is such a thing as "mathematical" linguistics, with its own symbolic logic language and rules of constraint that allow objective conclusions/results that rival the best of physics and maths. It was my path to appreciating physics and math.


I am no calculator so every insight I've gained has been laborious blessedness. Sometimes it takes me years for things to fall into place but, invariably, there is no satisfaction like reverse engineering an insight and realizing its impeccable pedigree is and has always been first principles.


It is near impossible to appreciate Schrodinger's probability wave equation (other than the beautiful string of mathematical symbols) until one sees the possible (allowable) shapes that it produces - see http://en.wikipedia.org/wiki/Atomic_orbital to get an idea. Often it's been my experience that when I have no clue what an equation means there is sure to be a geometric (ie, graphic) rendering that illustrates its outcomes and vice-versa.


For instance, I have little appreciation of graphs of Einstein's theory of general relativity but the Lorentz equations are more intuitive for me being that they are expressed as roots of relative velocities/energies in relation to the speed of light, c. The notion of singularities (and the attendant diminishing returns) really stand out here because in extremis one is forced to try and divide by zero (ie, subtracting 1 from the ratio between the speed of light and mass/energy in the bottom part of the equation(s), which also amount to 1 when speed of light is reached). Beautiful.


Seen globally, the laws of nature are such that nothing can be added to or taken out of the system (ie, the universe), only transformed from one state of energy/matter to another. Whether the information survives intact the transformation is really a matter of metaphysics and philosophy. Or is it?


Regarding the theological/doctrinal essence of the Judeo-Christian faith, it appears that even the moral laws of G*d may be patterned like the physical laws of conservation and entropy. Just indulge me for a moment: let us imagine that the fall from grace was a real event in time. At the very worst, it is a simple change in perspective that irrevocably perverts the intent, the meaning of human life, from ecological/social/philosophical/moral justice to purely egotistical, short-sighted, short-term gratification.


We cannot change the past, only choose between right and wrong here and now (with the realization that every act has inescapable consequences). Belief in the teachings of Christ is not just wishful/magical/talismanic thinking, not the attainment of social/historical standing, but an acceptance of responsibility and purchase onto and embracing of something deeper and bigger than ourselves. There is no intent outside of the human heart, good or bad (at least in this world). The Hopi say: all is beautiful, all is beautiful. Only the self and vanity stand in the way of appreciating the glory of G*d (literally, Isaiah 6:3).


At the bottom of it, "the meaning of life" may just be a subjective phenomena. But its emotional/psychological/spiritual impacts are very real nonetheless even unto the subconscious level. We forgive, turn the other cheek (rather than conspire to strike back), because there is Providential healing in the promise and perspective spoken of in Micah 6:8:


He has shown you, O mortal, what is good.
And what does the Lord require of you?
To act justly and to love mercy
and to walk humbly with your God.


The Gospel says nothing of changing the world and others, only the self. In my mindful meditations and self-monitoring, I've come to realize how hard and arduous the apparently simple often is. Faith and trust in Divine admonishments/promises often seem to run contrary to common sense and self-interest, but nursing our notions of entitlement with selfish intents and ever-present threats of resentment only detract from the "meaning of life", which, in the end, is the only thing that matters.


Upon the death of someone we often feel an impulse to heap praise and honor upon the dead. I highly doubt the cessation of my life and consciousness will allow me to appreciate my new-found status and esteem from my fellows. My faith in Christ is here and now, the desire to pattern my life in the humble Spirit of Christ is here and now. I've yet to experience holiness and enlightenment, but I doubt it'll be an instantaneous, magical transformation but rather a slow evolution of spiritual learning from subjective experience.


Jay