Sunday, 16 August 2015

Imaginary numbers

When I first came across the notion of the, so-called, 'imaginary' number it was within the context of physics so it seemed totally natural to me—ie, the imaginary unit, in particular, not as a value needed to solve certain equations, but as part of a dynamic geometric point needed besides the x and the y specifications to complete the description. It was until much later that I realized what unreasonable gauntlet it had had to take (as a 'number', per se) in order for it to be accepted as having the proper bona fides of a number.

Number, in fact, as a primitive concept, is always more than what we bargain for. The hapless Pythagoreans and the unruly diagonal of the unit square (an irrational number written as, √2)*; the negative numbers that just lingered on like persistent haze after every, innocent commercial transaction (post-dark ages); the number zero (an assault on commonsense, surely) that was needed to complete a new 'orthography' of numbers (the Hindu-Arabic numerals: 0-9). The internecine wars, the highly-personal gouge-out-eye brawls, intrigue and betrayal—not to mention the seemingly intractable cultural, and, even, sectarian strife—the history of mathematics is brutal (to say the least!).

*double irony: the number is derived by applying their famous theorem.

The imaginary unit (the one that satisfies the equation: x2 = −1) is no different than the other numbers that have preceded it down through human history. It, like the other numbers, is more than is obvious and it, like the other numbers, serves many functions. It is utterly invaluable to the two great pillars of our current understanding of the universe: classical and quantum physics. It is in this respect that I admire the imaginary number the most—ie, as a geometric construct.

The two famous numbers of the Cartesian coordinate system (x,y) may be interpreted as extending not only east and west (x) but also up and down (y) which together make up a two-dimensional plane that is sagittally-oriented (ie, facing us directly and flatly); the less-famous number (often written as: √-1) is required to describe the electromagnetic field more completely, say—which, as it happens, occurs in more than just a two dimensional plane (x,y) but needs the north/south extension to fully 'account' for its allowable and observed orbits.

The graph itself is rather unremarkable at first sight:


where z denotes the north-south coordinate that results from a strange-looking (what is called) 'complex number':

x + iy.

As a number x + iy, the concept is a bit more complex (excuse the pun) than casual conversation can ever hope to disclose justly, but as a geometric concept there is no 'addition' involved. The entirety of the number acts exactly like a point described by (x,y)—only the imaginary plane extends the x axis north-south and not up-down, thus opening up a 3D landscape (x,y,z) that is more intuitive than what we'd expected.

It does more than just sit there: the imaginary unit (i) is also known as the rotational operator—meaning that it has the capacity to describe periodicity of a trigonometric function!


and how it does this 'rotating' is by applying 'multiplication' on i like this:
The powers of i
return cyclic values:
... (repeats the pattern
from blue area)
i−3 = i
i−2 = −1
i−1 = −i
i0 = 1
i1 = i
i2 = −1
i3 = −i
i4 = 1
i5 = i
i6 = −1
... (repeats the pattern
from the blue area)

The number 1 (the right side of x) is the starting point in the cycle above where y = 0—as in: 1 + iy; and the next result, x = 0—as in: x − i1, which is 90 degrees from the starting point; and, continuing down the return cyclic values, the same process (with opposite signs) is repeated for the negative values.

Once I learned the general forms of the complex number, I linked the arithmetic operations to their geometries and learned to visualize these wonderful numbers like one would learn to read music and hear the sounds.

I'm no calculator but a highly visual person. I know it's primitive way of regarding such beauty but it has its certain charms: the insights (or, perhaps they are delusions) that come up come like a picture show. Mesmerizing.


Friday, 14 August 2015

Recasting Inuit Knowledge: Honouring Our Elders

I was recently asked at work to look into some climate change stuff. I didn't really get a chance to look into the work itself but it got me thinking about the whole academic research into Inuit knowledge on climate change discourse. There seems to be two commonly-employed stratagems for dealing with this type of research: sociology and as a component of 'native studies'.

Of the two, I'd say that sociology is the kinder discipline - for the simple fact that it is something that feels more familiar to me as having done academic research myself. Also, I'm a huge fan of Max Weber and the social criticism of obscurantist philosophers like Wittgenstein, Camus, etc. (ie, those thinkers who see real world linkages between language and social justice).

The native studies variety (no moral judgement, I assure you), on the other hand, is a bit further behind (by virtue of still needing a common ground, a common framework, and quite possibly nothing else).

This thing, this common framework, is a very important detail that decides the fate of all formal forms of discourse - from the religious to the cultural (and I mean here, say, a hunting culture (like Inuit) and its intellectual/epistemological justifications for its modes of being). These 'justifications', I think I can show, need not be pedantic and/or academic, but practical and highly intuitive (ie, comprehensible to 'outsiders' without much need for explanation).

I was recently shown a video interview of the father of one of my oldest and dearest friends. I was paying particular attention to Jaypiti Palluq's responses to questions what he was doing just now right before the interview (and what he'd be doing were it the past this time of the year). He said he had been checking daily the passage for whales that opens this time of the year (describing in great and wonderfully-useful detail how the conditions and ways of the broken sea ice behaved as he knew it in the past and how it seems to act now).

The description is all-encompassing. He links the changes he sees in his daily wanderings with changes in animal behaviour, with shifts and forced-adaptations in our own behaviours, in turn. But it requires a refocus in how we listen and watch out for certain data. My name-sake is describing the life activities of the socio-economic structures of his culture - the trick is to understand that its embedded in the ecology and seasonal conditions that he is already intimately familiar with, and it is up to us (as researchers) to figure it all out.

The past is the baseline; Jaypiti's invaluable and totally trust-worthy insights into and descriptions of his contemporaneous observations are right there (he is talking to us). There are many different ways of laying out his irreplaceable data and thus obtain an overview and perspective that may honour his gift to us: Inuit science.


Tuesday, 11 August 2015

wanna see something freaky I did?

I think I've just made a great discovery.

I was able to generate two circles (step 1 - as you may discern the 1 and the label 'circle(s)' in the picture); step 2, I made the small square (labelled 'square 2'); step 3, i made the triangle by describing two angles equal using the base of the square to the center of the first circle; step 4, i did the parallel lines; step 5, the hexagon; and, finally, step 6, the ratio of the golden mean by surpassing the hexagon's inferior to include a small hat to cap the structure above.

What do you think that final step also generated?

I kid you not. I can demonstrate it again in person (or, I'll make a powerpoint presentation and post it here where you can watch the steps). I really did make this beautiful geometric structure. And all in one sitting in the crapper. It was G*d's glory I saw and now I'm frightened.



I was quite possibility wrong about generating the pentagon in the above construct. but the other geometric shapes are really there. i've been attempting to derive it but how it would arise naturally is kind of putting up a fight. my apologies.

I just realized the impossibility of generating the right intersections for a regular pentagon using only one compass. One needs at least two compasses (compii?) because the initial point of the pentagon starts outside the center of the circle and finding it requires adjusting to at least one other radius to generate the starting point.

Monday, 10 August 2015

He causes the sun to rise on the wicked and the good

Matthew 5: 45 is not just an admonition from our Lord Savior, Jesus Christ; He is prophesying the coming of His Kingdom, where the oppressive sky of G*d's glory (blessed be the Ancient One) that radiates down on evil as it does on the undeserving turns to life-giving rain that falls on the righteous and the unrighteous.

I ask: who is 'righteous'? certainly not I: I'm a convicted sinner. It be G*d, the Father; G*d, the Son; and, G*d, the Holy Ghost, Whose בינה and חכמה (Sword and Shield) swoop me up from the depths whence I cry out to Him.

Saturday, 25 July 2015

"by virtue of the absurd"

When I first read Kierkegaard I just didn't get him at all - snippets of brilliant insights and disturbing truths, to be sure, but the 'found' quality of the book kind of jarred me. I needed its roots and foundations if I ever hoped to gain purchase.

Soren Kierkegaard (5 May 1813 - 11 November 1855) was a Danish philosopher, poet and theologian, and is considered to be the first 'existentialist' writer/commentator of some renown. His Christianity, his philosophy, the way his thoughts and ideas express themselves, is all surprisingly simple and even rustic. Rustic, in the sense that he took the Christ's Gospel at face value (who and what he must be, personally, to attain discipleship - his 'knight-hood of faith') and even in the way his central philosophical program is impelled:

What I really need is to get clear about what I must do, not what I must know, except insofar as knowledge must precede every act. What matters is to find a purpose, to see what it really is that God wills that I shall do; the crucial thing is to find a truth which is truth for me, to find the idea for which I am willing to live and die. (Søren Kierkegaard's Journals & Papers IA Gilleleie, 1 August 1835)

In his commentary on who our neighbour is (to us personally) he clearly seeks to find "a truth which is truth for me". There is nothing to suggest he didn't take his words and ideas seriously. He was critical of the very idea of a state religion, especially one that recognized his precious Christian faith as such, not because it was in vogue to hold such 'enlightened' views but because what exists 'by virtue of the absurd' (faith) cannot be objectified without consequences (more precisely: faith cannot be objectified, period):

Not until a man has inwardly understood himself and then sees the course he is to take does his life gain peace and meaning; only then is he free of that irksome, sinister traveling companion — that irony of life, which manifests itself in the sphere of knowledge and invites true knowing to begin with a not-knowing (Socrates) just as God created the world from nothing. But in the waters of morality it is especially at home to those who still have not entered the tradewinds of virtue. (ibid)

Kierkegaard (and his faith) is only seen in true light when we consider that he truly believed that he had 'earned' G*d's wrath:

He is said to have believed that his personal sins, perhaps indiscretions such as cursing the name of God in his youth or impregnating Ane out of wedlock, necessitated [...] punishment. (

To him, faith is not some whimsy, it is an idea for which he is willing to live and die for (having suffered profound angst by the apparent meaninglessness of an unexamined, unclaimed life):

Frequently a person feels his very best when the illness is the worst, as in tuberculosis. In vain he tries to resist it but he has not sufficient strength, and it is no help to him that he has gone through the same thing many times; the kind of practice acquired in this way does not apply here. (Kierkegaard, 1 August 1835)

In the book, Batman and Philosophy, one of the contributors, Christopher M Drohan, writes of this 'owning up' to life:

While the knights of infinite resignation [ie, Batman] are always waiting for some future ideal state, the knights of faith [ie, Alfred the butler] have found it, and are living it presently. Their eternity is not to come, but is found in the moment, as they realize that in loving and serving others they exercise a kind of fellowship that will infinitely sustain humanity. For them, peace on earth must be made with every gesture and every action. And is starts by committing ourselves to another person and by helping that person in every way that we can.

Alfred knows that if we treated others in this way there would be no need for Batman, or for any type of coercive justice for that matter. (Batman and Philosophy, Christopher M Drohan, 2008, p. 194)

We've all suffered (or, been blessed by) moments of crises in our lives that cannot be characterized as anything but 'spiritual'. I've had my share. I think what has saved my rationality from the experience of seeing pure ignorance and want assert themselves in this life and realizing that they are me unchecked, unhindered selfishness that births and sustain's humanity's ignorance and want:

"'Forgive me if I am not justified in what I ask,' said
Scrooge, looking intently at the Spirit's robe,' but I see
something strange, and not belonging to yourself, protruding
from your skirts. Is it a foot or a claw.' 

'It might be a claw, for the flesh there is upon it,' was
the Spirit's sorrowful reply. 'Look here.'

From the foldings of its robe, it brought two children;
wretched, abject, frightful, hideous, miserable. They knelt
down at its feet, and clung upon the outside of its garment.

'Oh, Man. look here. Look, look, down here.' exclaimed the Ghost.

They were a boy and a girl. Yellow, meagre, ragged, scowling,
wolfish; but prostrate, too, in their humility. Where
graceful youth should have filled their features out, and
touched them with its freshest tints, a stale and shrivelled
hand, like that of age, had pinched, and twisted them, and
pulled them into shreds. Where angels might have sat
enthroned, devils lurked, and glared out menacing. No
change, no degradation, no perversion of humanity, in any
grade, through all the mysteries of wonderful creation, has
monsters half so horrible and dread.

Scrooge started back, appalled. Having them shown to him
in this way, he tried to say they were fine children, but
the words choked themselves, rather than be parties to a lie
of such enormous magnitude.

'Spirit. are they yours.' Scrooge could say no more.

'They are Man's,' said the Spirit, looking down upon
them. 'And they cling to me, appealing from their fathers.
This boy is Ignorance. This girl is Want. Beware them both,
and all of their degree, but most of all beware this boy,
for on his brow I see that written which is Doom, unless the
writing be erased. Deny it.' cried the Spirit, stretching out
its hand towards the city. 'Slander those who tell it ye.
Admit it for your factious purposes, and make it worse.
And abide the end.' 

'Have they no refuge or resource.' cried Scrooge.

'Are there no prisons.' said the Spirit, turning on him
for the last time with his own words. 'Are there no workhouses.'" (A Christmas Carol, Stave 3: The Second of the Three Spirits)

Becoming Scrooge is a terrifying proposition (the pre- and post-visitation Scrooge, warts and all). I think what Kierkegaard is saying is that becoming Scrooge is G*d's will, the wicket gate.


Saturday, 11 July 2015

I nail ducks to the wall

I think most people who know me would agree that I am a loner. Surrounded by love and affection as much as lack of amity (from both or either side at some points in time) I never really realized loneliness, per se. I've never really been alone and isolated. I think this type of isolation is deadly; what I thought was 'lonerhood' was actually 'heavy into solitude'. And, after every bout of solitude, I've always had the luxury to share things I think about with others.

Everyday, I wake up around 4:30am-ish. I watch the world outside and inside unfold. I have a lot of space and time to draw in, what I call, 'strength'. Regardless of the weather and season outside, each morning is a time for me to appreciate the universe as evidence of divine power and majesty. In the darkness of winter this divine presence is most real. The frozen expanse abides: there, creative potentiality still pulses, awaiting the inevitable coordination of celestial mechanics to bring forth new life.

I read somewhere that the British, after the horrors of the second world war, took up a tradition of nailing painted duck figures onto their walls to symbolize their resolve to reclaim the idyllic life. I took the title of this blog entry from a line in Roger Waters', Pros and Cons of Hitchhiking.

I've always thought the Waters' album was one long, rambling rant of a lunatic. But, as with all remarkable works of art, it is a bit more complex than that: the darkness is superficial. It is merely a shadow rolling across the landscape (lived, individual life against the backdrop of history).

There is something of Max Weber in Roger Waters' writing. He gives a diagnosis but we don't know what to make of it. We feel it in our bones. We assume it is external—the source of this feeling—, but perhaps these dark prophets are adumbrating that all 'feelings' are internal. If, then, it is an issue of interiority (as I think it is) the challenge is personal.

In the Islamic tradition, it is said that the archangel Gabriel's first word to Muhammad was: "Read". Then, after the third admonition to read, Gabriel spoke:

Proclaim! (or read!) in the name of thy Lord and Cherisher, Who created-
Created man, out of a (mere) clot of congealed blood:
Proclaim! And thy Lord is Most Bountiful,-
Who taught (the use of) the pen,-
Taught man that which he knew not. (Quran 96: 1-5)

Clearly, G*d knew that Muhammad was illiterate so the '(use of) the pen' and 'learning' cannot possibly mean our conventional notions of 'learnedness' (where ego figures large) but a state of being that is receptive and devoted to the 'greater' something that is everywhere around us: necessity (that which he knew not, or, the real) and coordination (the use of the pen, or, divine guidance).

Submission to this necessity and coordination is "the peeling away of feeling" (ie, ego) and the realization of the truth: we are submerged in an ocean of love and life. Life seems dramatic, unpredictable and chaotic but that is an illusion: it is an outcome of our ego-striving (ego—itself dramatic, unpredictable and chaotic) rather than these negative features being inherent to the realities of our existence.

Recently, I've brought about factors in my life that made me realize how much I love and am loved. This insight was rather accidentally-obtained because it was an onset of a deep sense of loneliness and its obligatory depression that realized it in my mind. I somehow looked outside of myself and realized the evidence everywhere of how much I'm loved, and have loved.


Tuesday, 30 June 2015

Moral exemplarhood

I'm reading a very interesting article written by Ryan Indy Rhodes and David Kyle Johnson in the Batman and Philosophy book called, What Would Batman Do? Bruce Wayne as an Exemplar (2008, pp. 114-125), where the authors argue that the fictional Batman character has as much right to 'belong' to a list of moral exemplars that have actually existed in history (Buddha, Jesus, Mother Teresa, Gandhi, etc.):

Although most readers probably don't need reminding, let's consider some examples of how Batman exemplifies moral virtues. Justice is a constant aim of his activities, not only in the general sense of fighting crime and protecting the innocent, but in more particular endeavors...(p. 115)

-the authors list an impressive demonstration of Batman's commitment to a 'moral code' in dealing with 'real', individual people as well as the villains (symbols) he is sworn to struggle against. But...

Since Batman is a fictional character, it would seem that he cannot be referenced by language. That is, because Batman is not real, sentences about him do not operate in the same way as they do about things that really exist. (p. 117)

The finer points of the authors' argument are compelling and interesting; I would tend to root for their argument whole-heartedly. The two lines of reasoning above are similar to the ones I proposed in Socrates the Man somewhere in this blog.

But there is a sweet irony in this particular paper: if we should deny Batman membership to this 'elite' list of real historical exemplars for the simple fact that Batman is fictional, perhaps we should re-examine historical figures in light of the inevitable embellishments of tradition and demand the same standards of them.

An 'exemplar', you see, is best interpreted semiotically rather than lexically: it is more an embodiment of a set of idealized characteristics than it is in reference to a person. After all, "What is a pond"? -depending on who and what you are (a dragonfly, a person, a frog, a bird, etc.), what makes a pond a 'pond' changes because your needs and emphases change (even from season to season).

Yeshua ben Yosef is my Lord and Savior, and no historical treatment of His figure can change that: He is an embodiment; He is an inevitability; He is expressed in the Nazarene (John 14:6). Period. Full stop.

But I also admire the Batman character. And, for different reasons. The Batman is not only psychologically-real, but psychologically-real that instantly makes him more than a fancy, a whimsy but a well-developed and highly adaptable (ie, stable) characterization of moral exceptionalism.

He has a certain integrity that is preserved across a broad spectrum of backgrounds and re-tellings. Miller's Batman (though the most original telling, to be sure) is inherent and adumbrated in the earlier iterations by other writers. There is an ineffable pedigree linking everything. Even Adam West's Batman has his place in this substantive mythos.

In the Inuit tradition, the Batman figure is very familiar, more so than the Christ figure (though Christ is readily accepted by Inuit): Kiviuq, for eg, is a 'normal' human being cast into exceptional circumstances. He is flawed but ultimately perseveres. And the story continues...

However, Inuit do not somehow mistake the Kiviuq character as someone real: he is an exemplar of human curiosity and ingenuity. Where he goes and what he does is restrained by the seasons and geography. These features are not obvious but may be justifiably extrapolated in each 'episode': it is autumn when the story begins because Kiviuq becomes lost in a sea storm and lands to a place where the bumble-bee woman ultimately 'invents' ice that traps him before he can qajaq safely away; in one episode, Kiviuq has to traipse across a gigantic cooking pot (ie, hunting area) where in one end  he jumps onto seal blubber (ie, Marble Island) and hops over to a series of ribs (ie, Belcher Islands on the other side of the Hudson Bay).

Each telling is different but, throughout, Kiviuq exhibits exquisite sensitivity and sensibility towards animals that he encounters (another hint to times of year) that justify Inuit hunting culture as much as are expressed by a realistic 'person' in Kiviuq that we may try and emulate and strive to be.