Wednesday, 28 March 2012

Some grammatical features of Inuktitut (pt. ii)

Indicative and Interrogative verb endings.

These series should have started with the Indicative mood, pronominal verb endings. These endings indicate or state progressive or perfected present (depending on…), and their forms conjugate thus:
first person, Indicative
singular:

[-junga]*
/[-tunga] – ‘I…                        isumajunga      ‘I think (that)…’
                                                  isiqtunga          ‘I enter’

dual:

[-juguk]
/[-tuguk] – ‘You and I…’          anijuguk          ‘You and I exit’
                                                   pisuktuguk       ‘You and I walk’

plural:

[-jugut]
/[-tugut] – ‘We (many)…’         tukisijugut        ‘We (many) understand’
                                                   siniktugut         ‘We (many) sleep’

 *some schools of thought contend that this morpheme [-junga] varies with [-vunga], but I say that the bilabial form, [-vunga]/[-punga], is a symbol of “formality” when not to ask but to state something as in [isuma-vunga] which is clearly different from the interrogative.

The interrogative forms look like this:

first person, Interrogative

singular:

[-vungaa]
/[-pungaa] – ‘Do I..?’           takuvungaa     ‘(You ask) if I see?’                                                    
                                            isiqpungaa       ‘(You ask) if I (have) enter(ed)?’

dual:

[-vinuuk]
/[-pinuuk] – ‘Do You and I...?’              qaujivinuuk     ‘Do you and I (now) know?’
                                                               nattiqpinuuk    ‘Did you and I catch a seal?’

plural:

[-vuguut]
/[-puguut] – ‘Do we (many)..?’              pivuguut          ‘Do we (many) get some?’
                                                               tusaqpuguut     ‘Did we (many) hear?’

-all of which I consider differently from:

first person, Formal/Indicative

singular:

[-vunga]
/[-punga] – ‘I am…’                              apirivunga       ‘I am asking (you)’
                                                              tusaqpunga      ‘I hear (that)…’

dual:

[-vuguk]
/[-puguk] – ‘You and I…’                      tukisivuguk      ‘You and I (now) understand (that)…’
                                                               isiqpuguk         ‘You and I enter’

plural:

[-vugut]
/[-pugut] – ‘We (many)…’                    takuvugut        ‘We see (that)…’
                                                              nirijariiqpugut ‘We (just) finish(ed) eating’


-we may now surmise from the examples above that its grammatical pattern seems to go like this:

-the formal form is used for reporting on or stating a present, progressive/perfective fact – as in: nirijariiqpugut ‘We (just) finish(ed) eating’ ; otherwise,

-using such a morpheme creates a transitive phrase (ie, the main phrase needs another phrase to complete the grammaticality) – as in: apirivunga tavva ilinnit… ‘And I ask you (now)…’

-this class may be differentiated from the “pure” Indicative morphemes, such as: [-junga] which are always intransitive phrases (ie, grammatically complete, in and of themselves) – pisuktunga ‘I walk(ed)’ – the use of (while) appositional may be converted as pisukłunga ‘while I walked…’ in North Baffin, but that deviates from the argument of intransitivity of [-junga].

These are highly restrictive constraints governing linguistic analysis. The formal form [-vunga] when an intransitive construct may denote progressive/perfective tense, but; when it functions in the transitive present progressive tense, it is always occurs with the main verb. The form, pisukłunga ‘while I walked…’ in North Baffin, is an “appositional” tense (while I…) and, therefore, treated as a different class by virtue of its tense designation.

The forms slowly begin to stand out if one persists in seeking order from these conjugations/declensions. The use of practical logic is easier to do than to discern its origins, its technical statements – what I’m saying is that order is easier to see here than to connect it to an abstract name. We see and discern things long before we name them.

Logic gives beauty and form to the possible formulations and interpretations of these structures. The mathematical principles are sound but it will always be possible to formulate a more elegant statement of these facts using logic and reason alone.

Jay

Some grammatical features of Inuktitut (pt. i)


There are certain grammatical features of Inuktitut that I’d like to describe here – namely, what are referred to as “ergative” markers (a case where the action on the object is “owned” by the actor). There are a few of these as far as I can estimate. Their conjugations may be organized thus:



first person, singular

[-jagit] – ‘I… you’                                 takujagit                      ‘I see you’

[-jara] – ‘I… it’                                     takujara                       ‘I see it’



first person, dual and plural

[-javuk] – ‘You and I… it’                     takujavuk                    ‘You and I see it’

[-javut] – ‘We (many)… it’                    takujavut                     ‘We see it’



second person, singular

[-jannga] – ‘You… me’                         takujannga                  ‘You see me’

[-jait] – ‘You… it’                                 takujait                         ‘You see it’



-just remember that you can insert any verb of perception root and other certain verb classes in front of these pronominal endings: tusaqtagit – ‘I hear you’; tukisijannga – ‘You understand (comprehend) me’, and so on.



second person, dual

[-jassinnga] – ‘You (two)… me’             tissigijassinnga           ‘You (two) laugh at me’

[-jaaksi] – ‘You (two)… it’                    nanijaaksi                    ‘You (two) find it’



second person, plural

[-jasi] – ‘You (many)… it’                     tarrijaqtasi                  ‘You (many) see a film (of it)’



This class of grammatical functions (ie, being pronouns + ergativity) seem to be denoted by [-ja-] – which, as you see from some examples above, interchange with [-ta-] after consonants. Elegant.

I shall write up some more articles like this on Inuktitut grammar in the future.

Jay

Tuesday, 20 March 2012

Utilitarian vs Conceptual philosophy of learning

In Timothy Findley's novel, Not Wanted on the Voyage, Lucifer is portrayed in the "trickster" role as a seven foot tall geisha drag queen named Lucy whose fall from divine grace is not so much the traditional telling of Satan's lot but for simply asking "Why?". At the risk of being misunderstood completely, and being accused of unduly stretching the analogies, I think Findley's novel is a parable on the limits of arbitrary authority and the sad consequences of it. In this regard, I think it appropriate case-study for the Nunavut (read: aboriginal) experience, especially in terms of alienation and dehumanization of Inuit.

I don't think there's ever really been a sustained Inuit (read: aboriginal) discourse on our own experience as a colonialized society. The intergenerational trauma has overpowered any semblance of space for reasoned, detached discussion of where we are and how we got here. The hurt and strong emotions are either ignored completely or expressed at the cost of everything else.

Take the issue of education. Any thoughtful person could not deny how pathetic our lot is. One either needs connections, a willingness to play token Inuitism, or extreme native intelligence to transcend the shit hole in which we start our lives - the squalor, poverty and paucity of intellectual and physical prospects.

I know that there is an obsession for "education" in the aboriginal communities. But I think the expectations have been so far unrealistic. I'll try and state why that is: there are choices of differing philosophies of pedagogy out there. These choices may be classed into two broad types: utilitarian and conceptual approaches to education.

The utilitarian types tend to be prescriptive and rote-memory based - the types described by Dickens' in his role as a social critic of Victorian England. These are exclusionary and bureaucratic, and their systems of control and rewards are intended to assert the value system of the "ruling class". Ideally, one remains where one is born into - if not in theory, then in practice.

Granted, there are different outcomes in the utilitarian class, and this type should not be interpreted as being monolithic in nature. For example, Napoleon created polytechincal institutions which are considered world-class even unto today. His approach was motivated by the desire to overthrow the ancien régime and, expressly, as a means of realizing his grand vision of new France. Not only were these colleges meritocratic drawing talent from every class of society but the emphasis was on education and science especially serving the greater good of France's glory. Everything became "Republican" this or that - as in "Republican mathematics" or "Republican science". Though theoretical scientific investigation was de-emphasized, it was understood as being necessary if its practical applications were immediate and/or useful for advancement of Napoleon's military machine. This was "enlightened" despotism on steroids.

In this respect, utilitarian approach to education is not a bad thing for nation-building purposes. But for a society considered at worse as an inconvenience the notion of theoretical discourse is entirely missing while institutionalization predominates.

At the beginning of the 19th century Wilhelm von Humboldt became the minister of education for Prussia. As a humanist thinker and civil service reformer Humboldt created new schools called Gymnasiums which favoured a shift from science education as a means to an end towards a more classical tradition of pursuit of knowledge for its own sake. This meant a more conceptual outlook of literary analysis/criticism, philosophy, mathematics and science as studies worthy simply as they are and because they are.

The blossoming of modern deutschland as a litrary, scientific, and technical powerhouse was almost immediate given that german-speaking peoples already had a long tradition of celebrating their gifted. Einstein himself has said that technical knowledge itself is not enough for the advancement of scientific knowledge, that (informed) imagination plays a significant role in discovery. This is conceptual-based pedagogy at work.

Carl Jacobi, one of the countless beneficiaries of Humboldt's program, wrote criticising the great French scientist, Joseph Fourier, who looked down upon the German approach to education:

It is true that Fourier was of the opinion that the principal object of mathematics is public use and the explanation of natural phenomena; but a philosopher like him ought to have known that the sole object of the science is the honour of the human spirit, and that on this view a problem in the theory of numbers is worth as much as a problem of the system of the world. (Marcus du Sautoy, The Music of the Primes, p. 60, New York, 2003)

It was Einstein of Germany who put together his revolutionary theories of relativity, and not a French contemporary of his. This is not to say that France is a slacker as many, many exceptional and original thinkers and creators (including Fourier) also come from that great country.

Nunavut (read: aboriginal communities) have proven time and again that the sheer exercise to acquire tools is not enough; conceptual know-how along with the cultivation of capability to play with first principles is the "true honouring of the human spirit".

Jay

Saturday, 17 March 2012

The tyranny of Science

As a free thinker, I love science and believe in the power and grace of reason. I think demonstrable truths through the use of logic systems alone simply beautiful. This is one of the reasons why I love thinking about axiomatic mathematics, linguistics, philosophy, etc. Not just arithmetic, not just eloquence and demagoguery...

Though I find myself at odds trying to find fault in intelligent discourse, I try and follow its logical and practical consequences. It's been my observation that we rarely have our cake and eat it too.

In his book, A History of Reading, Alberto Manguel wrote a passage that got me thinking hard about the "cake adage". In a chapter titled, Ordainers of the Universe, he writes about the origins of bibliographical organization:

Rooms, corridors, bookcases, shelves, filing cards and computerized catalogues assume that the subjects on which our thoughts dwell are actual entities, and through this assumption a certain book may be lent a particular tone and value. Filed under Fiction, Jonathan Swift's Gulliver's Travels is a humorous novel of adventure; under Sociology, a satirical study of England in the eighteenth century; under Children's Literature, an entertaining fable about dwarfs and giants and talking horses; under Fantasy, a precursor of science fiction; under Travel, an imaginary voyage; under Classics, a part of Western literary canon. Categories are exclusive; reading is not - or should not be.  Whatever classifications have been chosen, every library tyrannizes the act of reading, and forces the reader - the curious reader, the alert reader - to rescue the book from the category to which it has been condemned. (p. 199)

How very Weberesque.

"Categories are exclusive..." and, therefore, convenient for scientific/mathematical investigation; being human (and human cultures) is not because life itself is not amenable to exclusivity treatment. Is the physical universe continuous or discreet? It depends: particulate matter is discreet (quantum physics) but the description of motion (and curvature) seems impossible without the notion of a continuum (theory of relativity). Some people have spent their entire scientific and philosophical careers trying to reconcile the dichotomy of continuity and discreetness. But it seems to me like tongueing a cut in the roof of your mouth; it heals only after you've stopped.

The axioms of continuity and discreetness cannot be regarded as mutually exclusive principles without reaching Zeno-type paradoxes: is the tortoise winning the race? - the apparent absurdities of Zeno's conclusions are reconciled by the concepts of infinitesimals and the limit in calculus. In sociological, biological and statistical analyses, a similar (granted, at a higher level) lasting, intellectually tenable peace has never really been sought. Though people like the great Gregory Bateson have tried.

That recent controversial piece by a respected climate scientist that compared the burning of all coal and fossil fuels in the world did not look into how the biosphere (including climate stability) and life in it would be affected, as if only the relative content of gases in the atmosphere mattered. I call this the DSM syndrome.

The Diagnostic and Statistical Manual of Mental Disorders (DSM) is a categorizing exercise gone horribly wrong whose roots lay not in the advancement of quality of life and human knowledge but in the thinly-veiled economic interests of corporatism. There are historic precursors to this DSM syndrome: namely, the malleus maleficarum (the hammer of witches). Like the DSM after it, this ideological/theological tool was used to hunt down and eliminate "witches" in the 15th century Europe.

Sadly, there are many examples of applying arbitrary categories as if they were actual entities in human history. The categorizing imperative makes fools of them who have the presumption to weld them like weapons. When Hurricane Katrina hit New Orleans one televangelist made a definitive statement to the effect that Jesus hated that grand old city. Little did he know (nor would he have the intelligence to seek out) that no one on the planet is immune to climate change. Does Jesus also "hate" the mid-western states who've recently suffered the devastation of tornadoes? How should the victims there be judged and categorized? Methinks them God-fearing peoples down that way. No?

The irresistible power of science (even psuedo-science) knows no bounds. Because it comes from the human mind like the seven cardinal vices all thoughtful people must remain always vigilant lest it starts to appeal to the least common denominator and wreaks devastation upon us all.

We must never forget that we are, all of us, more than the sum of our parts, that what we think we've mastered and categorized may be just an illusion, an unfounded presumption on our part. In the I Ching, the image of the wind carries both good and ill with it. This is science and technology par excellence: the wind itself is impervious but may carry with it stronger pathogenic agents invisible to the unweary.

Jay

Tuesday, 13 March 2012

Aphorisms = wisdom?

I'm often struck by the philosophical similarities between Inuit Qaujimaningit (IQ, or Inuit Knowledge) and Taoism (as written by Lao Tzu). In fact, I think ancient IQ comes from the same source as Taoism. Taoism is not a religion - its spirituality arises out of the recognition that human beings are subject to The Way (that which blunts the sharpness; untangles the knot; merges with the dust; which is hidden but ever present...).

I think there is a certain degree of mistranslation of "the way". I think it is closer to Inuit conception of piqqusiq - which has many hues of meaning but all stemming from the concept of essence or esprit. Lao Tzu's writings imply that "the way" is "the way things are in Nature/Universe" - ie, as being outside of human foibles and immature expectations.

The Tao is an empty vessel; it is used but never filled
Unfathomable source of ten thousand things

in semiological terms, I'd propose to analyse it thus:

wisdom is particular and consequential: it is thinking/acting that is guided by proven principles (the empty vessel)
Nature uses wisdom (as above) and everything it creates has its place and significance accorded to its own within the larger context.

Like the principle of divine love in Christianity, the Tao asks not our empty worship and empty words; it demands action and consistent (ethical and reasonable) behaviour. Empty words and worship are like security blanket for an infant - The Tao that can be told is not the eternal Tao; the name that can be named is not the eternal name. The nameless is the beginning of heaven and earth; the named is the mother of the ten thousand things.

In Inuktitut, the word for wisdom has the same root as Nature/Universe: sila-: silatuniq (wisdom); silarjuaq (Nature and its ways). In other words, to be wise is to emulate the inherent justice and maturity of all that is around us. This inherent justice and maturity appeared to Darwin cruel and dark because he believed in unfounded notions of privileged position (of humanity), permanence and empirical positivism; Taoism and IQ see and expect impartiality and must needs humility of us so as no undue conflict and maladjustment befalls us.

Another thorny translation I find in the English versions of the Tao Te Ching is "non-action". I think the original notion has more to do with "non-interference" or "non-manipulation" rather than "inaction". This notion - at least in IQ - has to do with the concept of respect for the equality of beings.

The bible says that we are created in the image of God. I always take this as we are endowed with reason to deliberate the possible consequences of our actions and thinking (both good and bad). As an Inuk, my notions of silatuniq play into this assumption; so also my admiration of great Western thinkers who've concluded that we can come to know and understand the universe.

Where I claim issues with "science" is that I, like Aristotle, believe complete human knowledge to comprise of sophia and phronesis (or wisdom and knowledge). To paraphrase Aristotle, phronesis is:

"wisdom to take counsel, to judge the goods and evils and all the things in life that are desirable and to be avoided, to use all the available goods finely, to behave rightly in society, to observe due occasions, to employ both speech and action with sagacity, to have expert knowledge of all things that are useful". see: http://grammar.about.com/od/pq/g/Phronesis-term.htm

I think all conscientious free thinkers come to arrive at similar conclusions.

Now, having said that wisdom is particular and consequential but guided by principles, I want to differentiate this notion from aphorisms. Often I have seen people trying to equate wisdom with aphorisms. But wisdom, as I'm trying to describe it here, is made manifest particular to a given context of having to make a deliberated decision or choice, whereas an "aphorism" is defined as: a pithy observation that contains a general truth.

The two (ie, the general and particular forms of wisdom) are aspects of philosophical discourse but an aphorism appeals to impervious "authority" (of experience and knowledge, mind); whereas, wisdom (phronesis) proper appeals to reason and deliberation on part of the thinker. Or, one is a guide (aphorism) - passive, in and of itself - and the other (wisdom) is an act or behaviour (piqqusiq).

The problem with the majority of scientific or technical knowledge is that it tends to have extremely narrow interests (which is both its strength and weakness). Science by itself, like capitalism, has a tendency to subsume and consume everything around it like a mindless monster whose only purpose is to self-justify by virtue of acquiring more and more power.

The beautiful irony of "conscientious" empirical science is that it is "based on, concerned with, or verifiable by observation or experience rather than [just] theory or pure logic". In this regard science may claim its uniqueness from mathematics.

Jay

Saturday, 10 March 2012

Pronominals in Inuktitut

There are always some irregularities in any conjugation of verb forms in any language. Inuktitut is no different. There are beautiful patterns to see, to be sure, but the underlying structures are not always obvious.

In Inuktitut, there are two basic types for each pronominal ending dependent upon whether the preceding morpheme ends in a vowel or in an allowable consonant in the final position (there are three such consonants in Inuktitut: t, k, and q - ie, all phrases in Inuktitut either end with a vowel or with any one of the three consonants).

1st person, singular, indicative (I am):
[-junga] after morphemes that end with a vowel;
[-tunga] after consonants.

2nd person, singular, indicative (you are):
[-jutit] after vowels;
[-tutit] after consonants.

3rd person, singular indicative (he/she/it is):
[-juq] after vowels;
[-tuq] after consonants.

This type of phonological patterning is simple and elegant (mathematically-speaking) because the variation of the first segment is regular and predictable: /j/ after vowels; /t/ elsewhere. This all happens at the subconscious level and intuitively for native speakers. Before I became aware of it, it never occurred to me that there was such a phonological variation so I think a better way of describing the base pronominal morphemes is - for 'I am' - C+unga, with a capital C to indicate the initial consonant slot. But I digress...

Now, for the first two forms ('I' and 'you') there seem to be no examples of them occurring medially but the third one is definitely allowed by the grammar to occur in the middle of a phrase, but, it seems, only it and not the others.

ikuallak[-tuq] - 'it is (starting to) burn'

ikuallaktuviniq - 'it (started) burning' or 'it burned down', which may be analysed thus:

ikuallak+tuq+viniq
catch fire + it + past tense

but not

*isiqtungaviniq, because it should come out as isilauqtunga 'I came in' and neither

*isiqtutiviniq because it comes out as isilauqtutit 'you came in'.

The first two pronominal forms cannot occur at middle of a phrase, but the third person form may occur at the middle of a phrase. It's as if the first two forms were "different" from the third insofar as the Inuktitut grammar is concerned.

Now, I think what is happening here (and, I've always maintained) is that the notions of formality and politeness are extended only to the first two ('I' and 'you') and not to 'she/he/it', whose presence is not considered to be obligatory after all. This fact may be enough to put the morphemic class of the third person, indicative, into another grammatical category in much the same fashion as 'it' in English: 'it is sunny out today'.

In other words, the third person singular may act as a purely grammatical function indicating the subject of the sentence - ie, without specifying who or what that is - whereas the first and second person forms always refer to a specific person whose presence or (tacit) acknowledgement is obligatory.

Of course, there are many more pronominal conjugations in Inuktitut for the various case and mood inflections for all the person and number forms. But the /C+uq/ morpheme seems to be the only one allowed to act in the function of a (purely) grammatical subject, and perhaps the only one allowed to occur medially.

Jay

Wednesday, 7 March 2012

Acquiring and cultivating analytical and critical thinking skills

One of the aspects of education that is most difficult to consciously think about is how do we teach (and learn) critical thinking skills. Without a doubt, critical thinking skills are key to good education. But what are they?

Critical thinking cannot be taught the same way mechanical rote learning can be "taught". It requires a certain amount of comprehension and active engagement and, ultimately, imagination is demanded of the student. For mathmatical (or analytical) thinking, memorization of multiplication tables will not do. Analytical thinking requires freedom to explore and experiment to test the boundaries of accepted forms.

Analytical thinking is an ability to use the constraints and restrictions of a discourse/study and its first principles to arrive at a tenable position and/or original insights. This is true for constitutional politics, policy development, science and maths. Analytical thought does not resort to "dirty tricks" or "sophistry"; the ends do not justify the means, rather, the means are demanded to justify their efficacy and intellectual "purity" for arriving at a certain end.

Having said that, mistakes and errors in judgement are unavoidable in analytical thinking and a certain amount of self-confidence is required to make and defend arguments, and one must be willing to admit error so one may advance (to learn from one's mistakes). It takes character. This character is a cultivation of critical and analytical thinking.

When I started thinking about linguistics and, later, number analysis I was drawn in by the fact that one can make general statements about things and processes using the principle of parsimony, and the veracity of these statements may be judged according to the particular logical consequences of them. This is critical thinking par excellence.

Doing arithmetic is boring - nay, mind-numbing. There is a better way to teach maths after the student has acquired and mastered the basic arithmetic operations of adding, subtracting, multiplying and dividing. This better way is to examine and explore the structure of numbers.

For instance, I modified some equations, such as: 4n - 1; 4n + 1 into

4n - 2n +/- 1 (for odd numbers)
4n - 2n + 0 or 2 (for even numbers)

and started generating numbers starting from 1 to infinity, thus:

4 - 4 + 1 = 1
4 - 4 + 2 = 2

8 - 4 - 1 = 3
8 - 4 + 0 = 4
8 - 4 + 1 = 5
8 - 4 + 2 = 6

12 - 4 - 1 = 7
12 - 4 + 0 = 8
12 - 4 + 1 = 9
12 - 4 + 2 = 10...

This is a bit more interesting than just teaching simple arithmetic. Using variations of the equations and colour-coding by whether the component asks to subtract or add a number also brings out some very interesting patterns which have regularity but some magical disproportionality about them as well, especially the prime numbers whose waves upon waves one may see and appreciate but never quite predict (or, at least, I couldn't). These types of exercises can make maths fun.

In my career as an analyst, I've sought to perceive underlying structures of things and processes, and how rearranging them affects the truth of them (whether it perverts or brings out its beauty). I have, in fact, spent many hours mesmerised by these thoughts. When I perceive a pattern, I try and figure out what the motivating principles are for what I see.

For example, I tried to capture the patterns in the periodic table of elements on my own. I didn't succeed obviously but the mere exercise of contemplation made me appreciate its inner order all the more. It's the same with music... with physics.

Jay

Saturday, 3 March 2012

On equations intended to generate only prime numbers

Like most people who are fascinated by mathematics and analysis, I enjoy contemplating the nature of prime numbers. Prime numbers are those that cannot be factored into rational numbers other than themselves or by one.

Throughout maths long history there have been many, many attempts at creating equations that will produce only prime numbers and nothing else. Some famous (or infamous) examples include such equations as:

P(x) = 2x + 1 who more generalized linear form is P(x) = px + q

but there others, such as

the quadratic P(x)  = x2 + 1,

which stipulates that a necessary condition that in order for P(x) to be prime is that x must end in the digits 4 or 6; (16) + 1 = 17; (36) + 1 = 37, and so on...

then there is the Mersenne function: P(x) = 2x - 1

or the Fermat function: P(x) = 2x + 1.

There are no shortages of very productive equations, including ones created by Euler.

The thing I want to say here is that basic mathematical concepts like the restriction principle and closure of number domains with respect to the fundamental operations of arithmetic do not seem to allow the generation of only prime numbers. To quote, again, Tobias Dantzig:

For instance, we could terminate the natural sequence at the physiological and psycholoogical limits of the counting process, say 1,000,000. In such an arithmetic addition and multiplication, when possible, would be associative and  commutative; but the operations would not always be possible. Such expressions as (500,000 + 500,001) or (1000 x 1001) would be meaningless, and it is obvious that the number of meaningless cases would far exceed those which have a meaning. This restriction on integers would cause a corresponding restriction on fractions; no decimal fraction could have more than 6 places, and the conversion of such a fraction as 1/3 into a decimal fraction would have no meaning. Indefinite divisibility would have no more meaning than indefinite growth, and we would reach the indivisible by dividing any object into a million equal parts. (p. 76)

The two basic conditions alone (the restriction principle and closedness of number domains) seem to preclude equations that could generate only prime numbers.

But that is not to say it is futile to contemplate the prime numbers. Far from it. These are mathematical entities which belie such "restrictions" by the simple virtue of their significance to number theory. Prime numbers are fundamental to number analysis and have far-reaching and interesting ramifications both within pure and applied mathematics.

Besides, any intelligent person can perceive the unique and fascinating qualities of such numbers. Mathematical entities invariably hold infinite possibilities for those with the perseverance and imagination to pursue them, those with the chutzpah to negotiate with "the legendary dragon guarding the entrance to the enchanted garden". (p. 64)

Jay

Mistaking a symbol for the name

One of the issues in the Inuit Language discourse in Nunavut that kind of concerns me is that, every once in a while, there is a call to "reform" the syllabic and "roman" orthographies. For eg, during the last Inuit Language week in Nunavut, I heard someone talk about the "need" for adding a specific symbol for the circumflex 'j' - ie, a consonantal blending of r with j.

There are various technical issues with such demands that have to be seriously considered: if such reforms were initiated this would create expensive issues for all concerned - the need to rewrite the bible being the least of them but the greatest expense would be to add a new symbol in the Inuktitut keyboard which now often creates unicode compliance issues for Inuktitut fonts; and, then, it would also create a need to reflect the change in the education system. The implications are actually quite far-reaching.

Both of the Inuktitut writing systems were created to account for and balance phonetic and phonemic variations inherent in the Inuit Language (or, that which results in dialectal differences at the phonological level). What I mean is this:

The very first Inuktitut linguistics paper that I ever wrote was to describe the [l] to /r/ rule in North Baffin. This phonological phenomenon is similar to the Asian tendencies to convert 'l' to 'r'. But there is a beautiful pattern that is not immediately obvious why this happens in North Baffin dialects. And it has to do with the quality and placement of the preceding vowels:

for 'iqaluk', the 'l' is changed to 'iqaruk';
for 'ulu', the change is 'uru';
but, when it comes to 'illu', there is no change; or 'ullu', no change
'piluk' also remains 'piluk'.

As I said, the quality or place of articulation of the vowel in the mouth cavity is that, for the first two examples, the vowels 'a' and 'u' are produced towards the back of the mouth cavity; whereas, the vowel 'i' is produced towards the front. At the deep psychological level, North Baffin speakers do not consider, nor normally perceive, the phonetic change of [l] to /r/ as something "real", and the consonantal shift is systematic - where the single [l] occurs after vowels 'u' and 'a', shift the lateral consonant to /r/; elsewhere, do not change single [l] - all of which happens at the subconscious level.

The same sort of phenomenon is happening with not only the [j] to /r/ but also [s] to /h/, but these changes are, more or less, phonetic (surface level) and not phonemic (psychological level), and can be accounted for and predicted as to where they normally occur, all of it subconsciously by the speakers of the particular dialects.

This phenomena occurs to in all languages, including English. For eg, there is no /j/ consonant symbol in the word "fire" but the word is pronounced [faijr]... and, because the /j/ is non-phonemic it is never perceived to be produced in the actual pronunciation of 'fire'.

For non-native speakers of a language, these phonetic changes seem very obvious and are, at first, treated as real and significant. For eg, the insertion of /r/ to words that end in a vowel, in some English dialects, is very obvious to most who speak a different variation of English. To Boston speakers, the word [idea], for eg, comes out as /idea(r)/ but Boston-version of English speakers do not perceive that superfluous /r/ at the end of the word though it still comes out where it should occur in actual speech.

Just as English can live with wide dialectal variations in pronunciation but still have a standardized spelling system, the Inuktitut syllabics and roman script writing systems are intended to have the same flexibility for differences in pronunciation while retaining a basic system of writing that is common to all writers and readers. Since Inuit of North Baffin dialects do not have to consciously think about the pronunciation of /r/ for the phoneme [l], the writing system is not asked to change to accomodate for the phonetic change.

The same goes for the "nasalization" rule for final consonants in the end of words in most Inuit dialects: [inuk] tends to come out as /inung/ in normal speech but it is spelt "inuk" nonetheless... Just as it should be: unique pronunciations are allowed by the system while the basic phonemic rules of the language are obeyed. The Inuktitut writing systems, in other words, account for but do not confuse the symbol (writing conventions) for reality (dialectal variations).

Jay