SOLVING the “Universally-UNsolvable” '''Diophantine Equation'''* via Common-Sense, ‘Realistic Mathematics’ – i.e., via ‘‘‘Finitization’’’.

  

 

SOLVING

 

the “Universally-UNsolvable”

 

‘‘‘Diophantine Equation’’’*

 

via

Common-Sense, 

‘Realistic Mathematics’ –

 

i.e., via ‘‘‘Finitization’’’.

 

GLOBAL STRATEGIC HYPOTHESES.

 

 

 

 

 

 

 

Dear Reader,

 

The ‘Nonlinearity Barrier’ – the incapacity of current mainstream mathematics to solve, “in closed, analytical form” nonlinear, i.e. self-reflexive [i.e. dialectical] integrodifferential equations that constitute that mathematics’ best formulations to-date of the “laws” of Nature – blocks the progress of humanity’s social forces of production, and of our sciences, around our entire perimeter with today’s unknowns.

 

The problem of finite-time division-by-zero singularities – of “meaningless infinities” – is a key sub-problem within this ‘Problem of Nonlinearity’, whose solution would help, most signally, to serve the demystification of the mathematical-ideological mystifications that help so cripplingly to sustain ‘The Nonlinearity Barrier’.

For technical reasons which I will not address in this blog-entry, nonlinear polynomial dynamical integrodifferential equations are especially prone to finite-time, division-by-zero singularities, e.g., relative to linear differential equations, the latter being the only kind of integrodifferential equations that can be generally solved in analytical, closed form within the capabilities of present-day mainstream mathematics.

 

Machine learning, e.g., neural network training, can capture de facto solutions to otherwise “unsolvable” nonlinear equations for specific “points”/loci within their parameter-spaces, but not in closed form – only in “black box” form – and at substantial costs in term of the high energy requirements of “training”, as well as in training time.

 

A step forward toward a general breach of ‘The Nonlinearity Barrier’ would be achieved if the division-by-zero singularity problem – if the problem of spurious and “meaningless” finite-time infinite predicted values of the state-metrics – could be put to rest.

Such singularities represent “infinity residuals” – infinite errors – in such dynamical differential equation models; a “residual” being the difference between a model-predicted value and the actually-measured value of that model-predicted state-metric.  “Infinity”, minus any actual, measured – i.e., finite – state-value (f) of any real, physical system, is still “infinity” per the ‘infinities-arithmetic’ of contemporary mainstream mathematics:  

¥ – f  =  ¥.

 

 

Thus, when, using the differential equation for the Newtonian gravitic interaction of multiple planets, if that equation predicts that two of those planets will collide, and then they do collide, Newton’s equation gives an infinitely-wrong answer for the magnitude of the gravitational force between those two planets “at” the moment of their collision.

 

The Newtonian equation models each planet as a “mass-point” – as an “infinitesimal”, zero-dimensional mathematical point, with a finite mass somehow associated with that ‘extension-less’ point.  The denominator of that equation is the difference between the 3-D positions of the two destined-to-collide planets, squared.  When those planets do collide, that denominator of that Newton equation becomes zero: (position coordinates minus same position coordinates)²   =   0²  =  0.

 

The result is that the Newton equation predicts a division-by-zero singularity for that collision, which means that the Newton equation predicts an infinite force between the two planets “at” collision. 

 

What really happens is that the gravitational force between these two planets rises as they converge, reaches a maximum, and then vanishes as they collide, explode, partially fragment, partially coalesce, partially melt, partially vaporize, etc., i.e., as they ‘dis-exist-ent-tiate’ – all real, physical processes which the Newtonian “point-mass” prediction-language can neither encompass nor describe.

 

Another example of the division-by-zero “infinite singularity” problem is that of the breakdown of the General Relativity Theory “law of Nature” when that theory’s mathematical model of gravity is used to describe the state of matter-energy density at the core of a “black hole”.  Since “density” means mass divided by volume, if a finite mass where compressed into the “zero volume” of a mathematical point, its mass-energy density would “become infinity” –

d = m/V; m/0 “=” “¥”.

To transcend the “infinite singularity” component of “The Nonlinearity Problem”, we need a mathematical language that can “tell it like it is” in such situations; a mathematical language that has the linguistic, symbolic wherewithal to assert a mathematical language failure when such a failure arises.

 

 

We have already seen, elsewhere in this blog, that, beginning with the axioms-systems category – in the categorial progression of the axioms-systems ‘ideo-ontological categories’ of the Seldonian arithmetics and algebras for modeling dialectics, corresponding to the generic Seldonian ‘meta-number’ q₇ – a mathematical 

language emerges that is able to explicitly articulate such mathematical language failures, or mathematical language ‘‘‘incompletenesses’’’ [not unrelated to Gödelian incompleteness].

 

This capacity first arises in that 7^th arithmetic/algebra, named – 

q _MQN    ¶|-º    q _MU    |-º    m_   

– by means of the ‘metrologicalization’ of the ‘higher meta-numbers’, that therein arise, and that therein arises.

 

The ‘‘‘nullification’’’ of a metrological unit arithmetical qualifier, via its multiplication by zero, whether that metrological unit qualifier is also multiplied by an ontological arithmetical qualifier, and/or by an arithmetical quantifier [other than unity], intuitively suggests the nullification/‘dis-exist-ent-tiation’ of the physical [ev]entity that this metrological qualifier models mathematically.

 

That intuition is codified in the following axioms of the m  axioms-systems category –

– wherein the ‘full zero’ value symbol is that ideographical, or ideogramic, symbolic expression asserting mathematical language failure. 

 

Such language failure can be due, of course, to the inadequacy of the mathematical language – the language that was adequate up to [near to] the timing of the singularity – to describe the physical state of affairs at and/or after that moment of singularity.

 

But it can also be due to cases where the nullification reflects a change in the ‘‘‘ontology’’’ that the mathematical model had presumed and was predicated upon – i.e., where that nullification reflects the ‘dis-exist-ent-tiation’ of one [or more] of the model-assumed kinds of [ev]entities, and, perhaps, therefore also, the transformation of [part of] the ‘dis-exist-ent-tiated’ [ev]entit(y)(ies) into (a) new kind(s) of [ev]entit(y)(ies); e.g., the irruption into existence of a new kind of ontology, e.g., of a new kind of [ev]entity that had not been included in the [implicit] model specification, and which that model’s mathematical language is therefore incapable of describing.

 

 

However, even prior to the level of complexity of the m__ arithmetic’s solution to the division-by-zero singularity sub-problem of ‘The Nonlinearity Problem’, the adoption of [what we call] the [demystified] ‘Realistic Mathematics Programme’ provides a ‘‘‘Realistic’’’, finitary solution to the ‘universally “unsolvable” division-by-zero ‘‘‘diophantine equation’’’ problem, about an “unsolvable” ‘‘‘diophantine equation’’’ that points to a very deep ‘‘‘Gödelian incompleteness’’’, not just in one or another of the “standard” arithmetics – whose number-spaces are standardly denoted by N, W, Z, Q, R, C, H, O**, S**, … – but to an egregious incompleteness shared by ALL of their arithmetical axioms-systems, and an incompleteness which points beyond ALL of them, because, for all of these number-spaces, or number-sets, mathematical “infinity” is not “in” them, in not one of their elements, is not among their constituents –

 

¥ Ï N: If x = n/0, x = ?

¥ Ï W: If x = w/0, x = ?

¥ Ï Z: If x = z/0, x = ?

¥ Ï Q: If x = q/0, x = ?

¥ Ï R: If x = r/0, x = ?

¥ Ï C: If x = (a + bi)/0, x = ?

…

*[the triple quote marks herein surrounding the phrase ‘‘‘diophantine equation’’’ signify that we are using this term with modification to its standard meaning.  Such equations are standardly limited to integer coefficients [and are considered solved only if a solution exists that is expressible in integers only].  Thus, algebraic equations with, e.g., fractional -- “rational” -- or “irrational”, i.e., “Real”, coefficients are excluded.  In the equations of the form x = y/0, we are admitting, as ‘‘‘diophantine’’’ in our usage of the term, algebraic equations involving (1/0) as a coefficient.].

**[The higher standard arithmetics, and also, e.g., the Grassmann arithmetics, inherently contain “nilpotents”, hypernumbers x such that x raised to the nth power equals 0, where n is a “Natural” number [i.e., n is not equal to 0], such that the value (1/x^n) might be considered a solution to y = z/0, albeit (1/x^n) would not be an integer.].

This simpler approach to defanging the division-by-zero problem involves, prior to any, more-complex, ‘metrologicalization’ of arithmetic, its finitization.

 

This solution involves the defining of a new arithmetical variable, ‘/|\’, to stand for the highest [positive] value in a given “standard” number-set that can be realistically, practically represented – e.g., by the computer in use to facilitate calculation and collaboration – and to re-designate, via ‘/|\’ as a ‘pre-subscript’, the, e.g., supposedly “aleph-null infinitary” standard arithmetics’ number-sets’ symbols, as the ‘actually-finitary’ sets _/|\N, _/|\W, and _/|\Z, such that –

 

N = {1, 2, 3,…, /|\_N}

W = {0, 1, 2, 3,…, /|\_W}

Z = { -/|\_Z,…, -3, -2, -1,  ±0, +1, +2, +3,…, +/|\_Z }

etc.

 

Practically – “actually”, concretely, realistically – no known physical system can represent an infinite number of numbers; “aleph-null” of them, or otherwise.

 

The, realistic, number-spaces _/|\N, _/|\W, _/|\Z, etc., provide a realistic, if imperfect, solution to the ‘division-by-zero ‘‘‘diophantine equation’’’ ’, and, thereby also, to the ‘division-by-zero singularity problem’.

 

Viz., in _/|\N, _/|\W, _/|\Z, we have, replacing the “infinity” or “indeterminate”, or “undefined” value of the “[non-]solution” to algebraic, ‘‘‘diophantine equation’’’ like –

 

x = n/0 

x = w/0 

x = ±z/0  

x = ±q/0

x = ±r/0 

and, e.g., 

x = (±Ö2)/0 

– the realistic solutions –

/|\_N  =   n/(1//|\_N)

  

/|\_W  =  w/(1//|\_W)

  

/|\_Z  =   ±z/(±1/±/|\_Z)

  

etc.

 

– with the 0 divisor replaced by the smallest numbers expressible in _/|\N and _/|\W, respectively, and by the closest-to-±0 number expressible in _/|\Z 

[since –/|\_Z is the smallest number expressible in _/|\Z].

 

Note also that, then, the /|\ values “behave infinity-like”, in terms of the prevailing “standard” arithmetic of infinity, in that we would probably want to hold that, due to “overflow” –

 

/|\ + 1  =   /|\,    a la    ¥ + 1  =  ¥

 

/|\ + /|\   =   /|\,    a la    ¥ + ¥  =  ¥

 

/|\²  =  /|\,    a la            ¥²  =   ¥

 

etc.

 

 

Returning to our Newtonian gravitic nonlinear equation example, then, we would have, “at” the moment of the two planets’ collision –

 

F  =  GM₁M₂/(x - x)²

 

    »  GM₁M₂/(1//|\)²

 

    =  GM₁M₂/(1²//|\²)

 

    =  GM₁M₂/(1//|\)

 

    =  /|\GM₁M₂

 

    =  /|\, 

and NOT:

 

F  =  ¥ .  

 

 

This solution is still imperfect, however, e.g., because the value of F “at” collision, /|\, is not tied to the objective physical process of that collision – e.g., does not reflect that actual maximum value of F as the collision is neared – but is arbitrary with respect to that actual, physical process, reflecting in fact, e.g., the limit of the numerical representation capability of the computer in use, and not the physical ‘existential limit’ of the collision-destined planets.

 

 

 

 

 

 

 

For more information regarding the Seldonian insights, including for free-of-charge downloading of F.E.D. books, other texts, and images, please see --

 

www.dialectics.info

 

 

 

 

 

 

 

For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of ‘dialectical art’ – as well as dialectically-illustrated books, published by the F.E.D. Press, see –

 

https://www.etsy.com/shop/DialecticsMATH

 

 

 

 

 

 

 

 

 

 

 

¡ENJOY!

 

 

 

 

 

 

 

 

 

 

 

Regards,

 

 

 

 

Miguel Detonacciones,

 

Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];

Elected Member, F.E.D. General Council;

Participant, F.E.D. Special Council for Public Liaison;

Officer, F.E.D. Office of Public Liaison.

 

 

 

 

 

 

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