Saturday, 11 January 2014

The arithmetic of infinity

One of the most bizarre and wondrous concepts of maths is the arithmetic of infinity (or Cantorian set theory). It was developed by Georg Cantor and Richard Dedekind in the late 19th century with Cantor at the helm.


It seems absurd on the surface that one can do arithmetical operations on infinities but it is a rigorous enterprise (ie, set theory) that discovered that there are different sizes/types of infinities, that, in fact, without these notions/perspectives of set theory it would be pretty difficult (if not impossible) to define what an irrational number is (what "number" is for that matter - at least, axiomatically), what a transcendental number is, whether infinitesimals and limits are reliable concepts in calculus...


There is a wonderful book called, A Tour of the Calculus by David Berlinski (A Tour of the Calculus (Vintage, 1996) ISBN 0-679-42645-0), that describes some of the evolution of calculus from the set theoretical perspective. The incomparable Berlinski presents a vignette of Dedekind as he defines what an irrational number is that is most memorable. I highly recommend Berlinski to anyone interested in the foundational concepts of mathematics as he is an undeniable master of historical pedagogy on the subject.


Now, I'm going to make a conceptual leap that reeks of Cantor's descent into madness in appealing to the more "theological" aspects of the arithmetic of the infinity. I've always been struck by the lengths in which fundamentalists are willing to take to do a literal interpretation of the creation stories of the bible, and have never been able to rationally accept the magical thinking that goes on in the minds of the so-called creationist "theorists", and am put off much by the blazon anti-intellectualism thereof.


I'm a believer in Christ; let me be clear from the onset of this strand of reasoning. I see no irreconcilable differences between the findings of science and the tenets of Judeo-Christian belief systems. In fact, I'm informed by Aristotle's notions of "sophia" and "phrenosis" and consider human knowledge incomplete with one without the other.


My line of reasoning stems from the fact that I can distinguish between the "orthographical" conventions of language and the (almost) Platonic reality of "number". In Peano's axioms of what a "number" is, he starts out by stating that zero is a number and from that all other numbers may be generated. Here is the rub: there is a difference between the place-holder function of zero in the Hindu-Arabic numeric system (the orthographical conventions) and zero as a true number in its own right.


I have found that infinity divided by half is exactly zero (the number) if we cancel out every positive number with its negative counterpart on the other side of zero. This odd result is a concrete example of why Peano's axioms would work (though I've never been able to find anywhere in Peano's writings how he's generated the number zero). Excuse the apparent tautological aspects of the definition for the moment...


Cantor also found that there are infinities embedded within infinities all along the real number line; that the space between zero and one is as "everywhere dense" as say the space between zero and two, three...and so on to infinity...and there will always be a one-to-one correspondence between these different "lengths". In fact, the center point of a circle generates the circle itself and can be made to correspond one-to-one thusly. Freaky but true: a line of whatever length is equal to all the infinite points in a plane as even unto n-dimensional spatially extended structures. There is a thing called the Riemann sphere that captures this notion beautifully, and, in fact, extends the real line to the imaginary plane.


Going back to the theological aspects that set me off on the tangent: The mind of G*d is utterly inscrutable and infinitely creative. From the one-to-one correspondences we may now say, with all logical/spiritual rigor, that every point is unique and particular no matter its distance from the point zero - the only thing that matters is the spatio-temporally undefined null point (ie, the zero that is often used as the symbol of G*d - ein sof is endless not as a myriad but as a simple root of all things).


This means that a Mind that has the quality and ability to generate these fundamental precepts (that define reality) is not beholden to our notions of time and space, and whether this current year, by human reckoning, is numbered in the billions since the ignition of our sun (in fact, the time-span since the Big Bang), or 5774 according to the Hebrew calendar, or 2014 by the Christian counting of years, makes no iota of difference when human beings actually arrogated themselves or were granted/graced a place of divine privilege in creation.


The point is that everything may and could exist before humanity came along (as science says) but that does not in any way detract from the miracle of conscious awareness and intelligence capable of perceiving the "speech of the heavens" (cf. psalm 19) as well as having the capacity to wreak utter evil and destruction (which, in my mind, necessitates the Lord Saviour most cogently and elegantly).


The infinite mind of G*d that has created such numbers as π which in fact does not settle down to a neat ratio and is in fact the very definition of uniqueness, is said to be holy (holy, holy, holy is the Lord almighty; the heavens and the earth are full of His glory). The number zero holds a truly unique position than every other number in that it is without quantity but like the center point of a circle the origin of every number along the arc.


Jay

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