samedi 3 mars 2012

Could division be fast subtraction?

The expression multiplication is fast addition suits the ones for whom an interesting insight is expressed. I admit the expression had me wonder during the late 1980s why we do not say division is fast subtraction. Rather than wondering how the calculations work, I was wondering why the inverses were not lining up.

Alrighty then, let us review how the calculations work. Matrices and Cartesian coordinates are nice introductions to what multiplication could be. Essentially, handed a finite set of compartmentalizable stuff, we partition stuff, and when we partition stuff as though with a cookie-cutter, then we make available to ourselves relatively fast computation of the volume of stuff as the quantity of cookies cut times the volume of stuff in each cookie. Inside such contexts, some dual concepts are introduced together, including variables + constants, addition + subtraction, multiplication + division, and = seeming to be the verb to be, and seeming to be a full declaration of sameness in all attributes. The 1965 edition of Mathematical Logic by R. L. Goodstein distinguishes in the introduction The Function of Mathematical Logic the conceptual differences between knowing how to compute the partitioning of r by q for some real number r and some fraction q versus describing what has the methodology work, versus winning a dispute about that. Familiar to anyone is the locution, to divide by a fraction, multiply by the reciprocal; however, who has the capacity to say what has this work without algebraic reliance on the calculi and without the presumption of duality being complete includes mathematical logicians. The answer to my question, how division differs from fast subtraction, is division is a partition whereas subtraction is displacement, a translation from observed-event-space to private-event-space. Pardon me for inventing names for some of these concepts while working through this.

The distinction in a nutshell is, when we see a mature elephant who very convincingly seems to weigh 10 pounds on Earth, rather than concluding we have found our first light elephant, instead we conclude the scale was miscalibrated. Ditto with time; if/when anyone wakes inside 1990s time management systems near the 49th parallel on Earth with all geographically local time declaration devices claiming 10:00 am yet the night sky is observed, then the time declaration devices are miscalibrated, regardless of how much they agree with each other. In grade school when teaching subtraction, we traditionally teach take away 3 from 5 rather than teaching translate 5 to have 3 displaced to the other side of 0 from which we agree to look away since the other side of 0 is private. 

I have been considering how I prefer to have us get the most out of my review of logic, calculi and related concepts without umbrage. Sure, we have the question of binary and ternary calculations however, most striking to me is the defense (against what conversation?) displayed in expositions of mathematical logic, leading me to wonder how straightforwardly these same concepts might be eloquently expressed were we to have mathematical logicians conversing in safety, without experiencing our existence nullified as though at war in every step we take. Who put such semblance of war in our conversation space I do not know, yet, Goodstein visibly expresses the entire self-defense structure, as does Bertrand Russell in The Principles of Mathematics. Let us examine the self-defense of Goodstein's presentation.

I confess some familiarity with Sociology and Dramaturgy introduced to me by my adoptive father Michael Overington whose books I omitted reading, and with whom I debated reality partitions and reality management for 16 years. These fond memories include my fascination with observing a Phi Beta Kappa mind who claims and produces convincing evidence of absence of mathematical and basic logic skill yet does so with algebraic calculations of basic logic and fallacy formulations perfectly intact. Aristotle gave us a bit of a conundrum when he rejected India's concept, the number zero, thereby handing our species our traditionally most coveted and most lucrative construction: nested binary debates. Our persuasions seem convincingly of two types: logic else fallacies. Do logical and fallacious people belong together else apart? Are fallacious arguments right else wrong? Let us bridge the conceptual gap between logic and fallacies as persuasive devices, for world peace. See the economic driver in the algorithm: scan the world, select two examples of whatever, hire lawyers to debate via a series of binary locutions which example of whatever is correct; repeat. Dick Cheney in In My Time reveals the related economic driver of top-down control applied to the masses as the cooking process of punching down risen dough in yeast breads, necessitating massive job creation for supervision of arbitrary control devices, pp 58 to 62. Ostensibly Aristotle wanted to prove God's existence thus by coincidence gave Galileo a challenge. Thank goodness Aristotle's accidental refutation of the physical existence of zero coincidentally drives money creation in western civilization. Examples of valid interpretations of physical instantiations of zero include our synapses which are nor axons nor dendrites; the spaces on roads into which we drive our vehicles else collide; empty food dishes; empty bank accounts; space available in tax-free savings accounts; absence of inclusion in society; the gap between bones in our joints; and for anyone still counting fingers, the spaces in between our fingers really does have measurable, observable existence. Matryoshka dolls solve Zeno's Paradox; the point being, the limit infimum of [0, 1] does exist and is zero regardless of any so-called mathematician's opinion otherwise, and by reverse direction of theoretical proof and physical observation, the limit supremum of [0, 1] does exist and is one, again regardless of any so-called mathematician's opinion otherwise. Incorrect calibration during physical observation combined with Appeal to Authority yields the marriage of Appeal to Ignorance with Appeal to Force, which exemplifies that sort of persuasive technique Aristotle establishes differs from logic together with correctly calibrated physical observation. A positive outcome of this exercise includes the convincing observable semblance of the force of gravity per se in Economics, admittedly a conceptual re-categorization which may have surprised Newton and which has me curious about the theory of gravity.

History informs us humanity apparently felt like spontaneously generating logical definitions of numbers, for communicating with unambiguous clarity what a number is thus also nailing what a number ain't, in the 1890s, exemplified by Gottlob Frege's introduction in 1894. (Goodstein, p2) Specifically, prior to Frege's publication, there were social conversations providing sensible context for Frege's publication, without which Frege's publication could have occurred as irrelevant, nonsensical, disobedient, nonconformist, nit-picky. Prior to our human minds exerting control are indicators of some absence of and some necessity for control, whether the inspiration to exert control comes from fear, desire to drive money, desire to have power, and alternative quests. Frege sacrificed some opportunities expressing his preference and his priorities among his options; he opted to exert his control of the definition of numbers. Something relevant happened prior, for example, something which could have questioned the reality construct of numbers, such as the seemingly benign and semi-competent random locution of numbers and real life being allegedly unrelated, and perhaps the social context in which Frege's publication occurs as acceptable has threats upon the reality construct of numbers posed as jokes. Without knowing Frege's societal context, from being human and knowing humans, clearly we normally have absence of spontaneously feeling like logically defining numbers unambiguously without any motivation, merely on a whim. IE minds in societal contexts are analogous to particles in waves, cf deBroglie. For example, my 2010 solution to Wikipedia's public claim of the openness of Waring's problem has been silently ignored since 2010, showing my absence of inclusion in peer review publication has been driven by societal context.  If Frege had been interpreted as suitable exclusively for slave labour, a forced idea generator for the benefit of the work on paper of lesser minds, empty-headed, off-kilter and irrelevant, then his logical definition of numbers could have remained unpublished and thus unread by Bertrand Russell in 1903. Via defining numbers, Frege and Russell disambiguate equality from similarity and establish the importance of one-to-one conceptual correspondence as a valid and real constructive method. How tight, how strong this conceptual, logical insistence on clear and honest correlation between physical observation and words for human communication expresses the existence of the unnamed unwelcomed uninviteds infiltrators in human conversation, existence proven by shifted demand curves of mathematical logicians responding, demonstrating the will and importance of this human defense.

Mathematical Logic, R. L. Goodstein, Leicester University Press.
Conversations en famille from 1986 to 2003.

Several classes in the honours mathematics stream, Dalhousie University, Halifax, Canada.
Zero, Charles Seife.

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