Part 03.
Dialectics and Nonlinearity Series.
and
SINGULARITY.
Dear Reader,
It
is my pleasure,
and my honor, as an elected member
of the Foundation Encyclopedia Dialectica [F.E.D.]
General Council, and
as a voting member of F.E.D., to share, with you, from time to time, as they are approved
for public
release by
the F.E.D. General Council, Seldon’s commentaries on key Encyclopedia
Dialectica concepts of Seldonian Theory.
This next text in
this new such
series is posted herewith
[Some E.D.
standard edits have been applied, in the version presented below, by the editors
of the F.E.D. Special Council for the Encyclopedia,
to the direct transcript of our co-founder’s
discourse].
Seldon
–
Sometimes
it is ultra-simplification that yields the new insights needed to handle the
full[er] complexity of a Domain of Nature, or of Nature as a totality.
The
core nonlinear dynamical equations of modern physics – Newton’s equations for
gravitational “forces”, Coulomb’s equations for electrodynamic forces, and the Einstein
equations of General Relativity, offer astounding quantitative precision of
prediction for pairs of interacting bodies, but, especially in the cases of the
former two, contain immanent failures that manifest especially in cases of three
or more interacting bodies, but that even manifest egregiously in cases of just
two bodies, if they collide.
The
Newton and Coulomb equations model their interacting gravitating bodies, and
interacting charged bodies, respectively, not as extended, e.g.,
three-dimensional-volumed bodies of substance, but as mathematical points. These bodies are thus replaced, in these
equations, by zero-dimensional, zero-volume, extension-less, infinitesimal
points. But these ‘physical nothings’
are somehow associated with finite masses or with finite electric charges,
respectively: they are construed as “mass-points” or as ‘charge-points’, respectively’
A
heavy price is paid by projecting these mathematical – mentally-internal, unphysical
– idea-objects onto physical reality, where they do not exist, and do not
belong.
When Newton’s
equations predict a collision of two planets, modeled as mass-points, such
collision means that the “two” mass-points coincide; combine into just one
point. This means that the distance
between them becomes zero.
The
denominator of the Newton gravitational “force” equation quantifies the
distance between the two mass-points as a dynamical function, a “function of
time”, squared, when they model the gravitational interaction between,
e.g., two planets that those two mass-points “represent”.
Thus,
when the two “mass-points” collide, and coincide, the denominator of their
Newtonian “force” equation model “experiences” a division by zero: 02 = 0.
Divisions
of, e.g., a Rational-Number finite numerator by zero produce either a value
which is “undefined”, or “infinity”.
Both of these values are outside the closure of the Rational Numbers.
Both
are predictively meaningless in terms of any accurate quantitative description
of what actually happens when, e.g., two real, physical planets collide.
What
actually happens in such a real collision is typically, initially, some
combination of fragmentation into solid-phase fragments, liquification into
molten liquid-phase fragments, vaporization into gaseous-phase vapors, and ‘plasma-ization’
into plasma-phase charged fire-balls.
What
the Newton equations predict in cases of such collisions is an “infinite”
gravitational force “between” the “two” collided mass-points, per the usual
interpretation of the value that results from a zero denominator “divided” into
a finite, rational or “real” numerator.
Thus,
the Newtonian collision predictions is “infinitely” wrong.
No “infinite”
force ever manifests empirically. Such
physical collisions typically result in the “two” formed bodies ceasing to
exist as such, and, at first at least, in their part-coalescence and part
disintegration: a finite result.
The
difference between the “infinite” force-value predicted, and the finite actual
result is, per the prevailing “arithmetic of infinite magnitudes”, again “infinity”. That is, the Newtonian gravity equation’s collision-prediction
exhibits an “infinity residual”, i.e., is an “infinitely” erroneous prediction!
Something
similar happens, with the Coulomb equations, when they are used to model two charged
particles, say two oppositely charged particles, that mutually attract,
which the Coulomb equations model as ‘charge-points’.
As in
the case of the Newton gravitational “force” equations, the Coulomb
electrodynamic “force” equations have, for their denominator, a “function of
time”, that quantifies the distance between the two ‘charge-points’ as a “function
of time”, squared, when they model the electrodynamic interaction
between, e.g., two charged particles that those two ‘charge-points’ “represent”.
Let’s
even consider the case of two particles of like charge, that the Coulomb
equation generally predicts will mutually repel, but that, under the conditions
in the cores of “main sequence” stars, actually “collide”, forming transient “di-protons”,
on the way to becoming elemental Helium nuclei – two proton “particle” units,
and two neutron “particle” units, coalesced into a single new kind of, “atom”,
unit, beyond the older, “particles” kind of units.
When
its – unrealistic, unphysical – ‘point-charges’
collide, hence coincide, their inter-“point” distance vanishes, leading, again,
to a division-by-02 of the Coulomb “force” finite numerator, which is the
product of the two electrodynamical charges further multiplied by a “constant of
proportionality”.
The
Coulomb electrodynamic “force” equation for such a two-“charged particles”
collision again predicts an “infinite” electrodynamical “force” at the instant
of their collision, which represents, again, and “infinite” error in relation
to the actual finite results of a real such collision, albeit which involves a
considerable release of photonic energy, but still a finite release.
The,
far more complex, and far more predictively powerful Einstein General Relativity
equations – Einstein’s ten, coupled, “simultaneous” nonlinear partial
differential equations, more compactly represented as a single tensor equation –
has similar problems when the mass of a body exceeds a certain – still finite –
mass magnitude.
In such
cases, the Einstein equations predict that this body will collapse, under the
force of its own self-gravitation, into a volume-less, extension-less,
zero-dimensional, “infinitesimal” “mass-point”, forming a “black hole”. At that point, Einstein’s “laws of physics”
break down, “predicting” meaningless – that is, “infinite” – values.
Sometimes,
ultra-simplification can yield new insights that point a way out of immanent mathematico-scientific
contradictions, paradoxes, anomalies, errors and impasses.
Such, we believe, is the case with the proneness of contemporary nonlinear, vis-à-vis linear, “law of nature” equations to division-by-zero “singularities” – for reasons that we have detailed elsewhere* – if one contemplates the simplified, “purely”-qualitative, “purely”-ontological model of “singularity” provided by the NQ algebra.”
For more
information regarding these
Seldonian insights, and to read and/or download, free
of charge, PDFs and/or JPGs of Foundation books, other texts, and images, please see:
and
https://independent.academia.edu/KarlSeldon
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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