Saturday, September 24, 2011

Part I.: Intimations of a faster than light Interstellar Drive -- An F.E.D. Hypothesis

Intimations of a faster-than-light Inter-Stellar Drive:: "Side-Stepping" the V = c Singularity -- An F.E.D. Hypothesis, Part I.

[¿A "Non-Celeritarian" Pathway from the "Tardyonic Meta-Phase", through the "Luxonic Meta-Phase", to the "Tachyonic Meta-Phase", and back again?]


[Note: as usual in this blog, I use visible-light-spectrum color-ordinality to highlight categorial, or system, dialectical ordinalities -- "ROY G. BIV"].



Dear Readers,

I have, in previous posts to this thread, noted the "incompleteness" of all of the present "Standard" systems of arithmetic, with respect to division by zero, and the connexion of that "incompleteness" to the problem of "singularities" in differential equations, to which the nonlinear partial and total differential equations -- deeply linked to dialectics, per F.E.D., and the latter of which, in particular, are native to "complexity theory", or "nonlinear dynamical systems theory" -- are especially prone.

I have also here alluded to, and linked to, works, by F.E.D., which show a -- to my knowledge, at least -- never before broached pathway to the "semantification" -- the "making meaningful, and empirically fitting" -- of the otherwise "meaningless", or "supposedly empirically infinite in magnitude" [and therefore "infinitely erroneous"], problematics of finite-time zero-division, in the unsolved [and usually nonlinear] differential equations themselves, and/or in their solution-functions, in those few cases of nonlinear differential equations where their solution-functions are known.


This mathematical capability for this new "pathway to semantification" first emerges, in the F.E.D. meta-systematic dialectical progression of the F.E.D. dialectical arithmetics, in the seventh system of dialectical arithmetic, corresponding to their generic "dialector" q/7 in the NQ_-algebraic dialectical "Meta-Model" of that systems-progression.


That seventh system of dialectical arithmetic is the second "full synthesis" system, or second "grand uni-system", of this potentially infinite dialectical systems-progression of dialectical-arithmetical axioms-systems.

That seventh system combines the first two "antithesis systems", "anti-systems", or "contra-systems", of that progression, with its 
"<< arche'>>-thesis" <<arche'>>-system, or beginning system, that of the "Natural Numbers" –
N = {1, 2, 3, 4, . . .}

-- into a "complex unity", denoted by Nq/MQN, or by Nq/MU.

The Nq/MU "grand uni-system" is alternatively viewable as a "meta-synthesis", arithmetical system, one which "combines", "hybridizes", or "unifies" the "first synthesis" system, Nq/U, or Nq/QN, with the "second antithesis system", Nq/M, the "antithesis" system to the Nq/Q, or NQ_, system.

The NM_, or Nq/M, system, is the "second antithesis"/"anti-system" arithmetical axioms-system, that of Metrical qualifiers, whereas NQ_, or Nq/Q, denotes the first "antithesis" "anti-system" arithmetical axioms-system, that of ontological Qualifiers, and NU_, or Nq/U, denotes the first "full synthesis" system, or first "grand uni-system" axioms-system of dialectical arithmetic, which unites the NQ_ and N_ axioms-systems into its predecessor "complex unity" system of "quantifiable ontological qualifiers", or, equally, of "ontologically-qualifiable quantifiers".

The fuller applicability of this 7th system to the differential equations / systems dynamics domain, and to the systems "meta-dynamics" domain, requires a further dialectical systems-progression, "orthogonal" to the main "direction" of progression, from Nq/MQN, to Wq/MQN, to Zq/MQN, to Qq/MQN, and, thence, to, at least, Rq/MQN, wherein W denotes the "Standard" higher-[than-first-]order-logic axioms-system of the "Whole" numbers, Z that of the "Integers" [which includes, for the first-time in this sub-progression, the "new ideo-ontology" of the "negative numbers"], Q that of the "Ratio-nal" [i.e., of the "Quotient"] numbers, and R that of "Real" numbers.


I have several examples which I would like to share in this thread, illustrating how this new "pathway to semantification" works to "de-fang" differential equation, zero-division singularities.

However, before I do so, I shall first share, in this post, an example of such "singularity de-fanging" which arises, not via the use of the new, F.E.D.
"meta-numbers", but via the use of one of the Musèan "hypernumbers", one which is even still a "linear operator", isomorphically expressible, for example, by a 2-by-2 matrix "containing" only entries from Z, the number-space, or number-set, of the integers.


This example is remarkable in that it points the way -- if the alternative, singularity-obviating, "hypernumber"-qualifier-valued solutions are have not just mathematical meaning, but also physical meaning [meaning that would manifest empirically under the correct physical conditions] -- to a design principle for "an inter-stellar drive"; for the possible "translation" of spacecraft to effective velocities far in excess of the speed of light, c.


The "Musèan hypernumbers" are a spectrum of quantifiable arithmetical qualifiers, mostly with "convolute" rules of "multiplication", that were discovered by one Dr. Charles Musès, during the 1970s and 1980s, and about which he published papers in, principally, the Journal of Applied Mathematics and Computation.

I know that his work is known to F.E.D. -- and that he "gave fits" to at least some of their membership -- because, in the "dedications" to their book
A Dialectical Theory of Everything, they include the following as the final dedication in a series of a dozen such dedications: "To Dr. Charles Musès [1919 to 2000 C.E.] -- friend, teacher, partial co-thinker, and partial adversary to key member's of the Foundation's first, founding generation." [emphasis added by M.D.].

I also know that the late Dr. Musès's work has influenced that of F.E.D. because, in the old on-line forum of their original website -- www.dialectics.org -- a forum that was closed down many years ago, after it was "hacked" -- they outlined the example that I am about to share.


The singularity in question is that of the Einsteinian, special-relativistic momentum equation -- really, that of Einstein's corrected version of the classical Newtonian momentum equation [although, in truth, Newton did not typically express his physics findings in the form of equations, but, rather, in the form of "proportions" -- of "ratios of ratios". This fact has important "psychohistorical-dialectical" implications, implications which accrue to the very heart of the F.E.D. opus, but which we must pass over, for now, with no more mention than this].


So, first, what, pray tell, is "momentum"?


"Momentum", or "quantity of motion" [Newton] is not simply "velocity", nor is it "force".

"Momentum" is a concept that combines both "Velocity" magnitude [V] and "Mass" magnitude [M], multiplicatively, in the formation of its magnitude [P]: "the momentaneous magnitude of the momentum [P] of a body of mass[-energy] equals the magnitude of the mass of that body as of that moment, "times" the magnitude of the velocity of that body as of that moment."

Or, ideogramically [using phonograms converted into mnemonic "quantifier" ideograms]: P = MV.

That is, the nature of the Momentum concept is such that, e.g., a small, standard baseball would have the same magnitude of Momentum as a gigantic, massive, open-pit mine ore-hauler, if the Velocity-magnitude of that baseball were high enough to compensate for its relatively tiny Mass-magnitude, and if the Velocity-magnitude of the ore-hauler were tiny enough to compensate for that hauler's humongous Mass-magnitude:
Mb..x..Vb.....=.....Mh..x..Vh.


"Momentum" is, however, closely related to "Force".

You may be used to hearing the phrase "Force [magnitude: F] equals Mass [magnitude: M] times Acceleration [magnitude: A]": F = MA, i.e.,

F = M x (dV/dt).

However, that formulation presumes that the Mass magnitude, M, is constant though time.

If the Mass magnitude of a body varies in parallel with time's [apparent] "self-variation", especially if its Mass magnitude varies in a way which is causally connected to how its Velocity magnitude, and/or how its Velocity direction, varies in parallel with time's "self-variation" -- as in the example of a "self-propulsion-by-mass-expulsion" rocket -- then it is the Momentum-magnitude time-function that must be differentiated with respect to time to derive the Force function, not just the Velocity-magnitude time-function, and, 
if both the Mass magnitude function and the Velocity magnitude function are unknown functions, then the resulting differential equation is a "quadratic" -- 
or degree 2 -- differential equation, i.e., a nonlinear differential equation.

That is, if M = m(t), as well as if V = v(t), then [per the "product rule" of time-function differentiation] --

F = f(t) = dP/dt =

dp(t)/dt =

d(MV)/dt =

(dM/dt)V + M(dV/dt) =

(...(..dm(t)/dt)..x..v(t)..)...+...(..m(t)..x..(dv(t)/dt)..)...)....=

M'V + MV' =

M'V + MA

-- or "Force-magnitude as a function of time-magnitude equals the sum of the time-rate-of-change-of-Mass-magnitude times the Velocity-magnitude, plus the Mass-magnitude times Acceleration-magnitude".


Now, Einstein, in his "special theory of relativity" -- the theory that abstracts from the existence of accelerated motion, an abstraction that was redressed in his later "general theory of relativity" -- multiplicatively applied a crucial, "Lorentzian", correction-factor, namely, a factor of --

( 1 - (V^2)/(c^2) )^( -1/2)

-- to the Newton-based momentum equation just discussed above.

The need for this correction becomes extremely evident in human experiences/experiments with bodies moving with very high velocity-magnitudes, much higher velocity-magnitudes than those to which Newton, or others of his time, had any ready experiential/experimental access.

The result of this correction is the new, "Einsteinian, special-relativistic momentum equation" --

P = MV / ( 1 - (V^2)/(c^2) )^(1/2)

-- i.e., [special-]relativistic Momentum-magnitude equals Mass-magnitude times Velocity-magnitude, divided by the square-root of the difference in magnitude between 1 and the quotient of the Velocity-magnitude squared, divided by the [constant] speed of light squared.

It may be interesting to contemplate the "degree" of this equation -- to contemplate the question as to whether or not this equation is "nonlinear", i.e., is of "degree" other than 1 -- if we treat this equation as, e.g., an algebraic equation, with V as the unknown magnitude to-be-solved-for.

In that case, V, the unknown, occurs in the numerator of the Right-Hand-Side [RHS] of that equation in the linear "degree", i.e., to the "first" "power" or "degree"; in the "degree" 1.

That unknown, V, occurs in the denominator of the RHS of that equation to a "nonlinear degree" -- the "degree" of 2, due to its "squaring" -- but that "squaring" is, in a sense, "undone" by the "power" of 1/2; by the "square-root-taking", or "unsquaring", of the whole expression in which the V-"squared" occurs.

In any case, the Newtonian Momentum-function, given a constant Mass-magnitude, M, and a monotonically rising Velocity-magnitude function-value, produces a straight-line, or "rectilinear", graph, of slope M, in Momentum-magnitude-versus-Velocity-magnitude graph-space.

The Einsteinian special-relativistic Momentum-function, given a monotonically rising Velocity-magnitude function-value, produces a non-rectilinear, "curvilinear", rising-slope graph, but one which has a "zero-division singularity" -- a point of apparent "infinite magnitude" [herein denoted by "oo"] on the Momentum-magnitude axis -- associated with the point on the Velocity-magnitude axis where V = c -- with c denoting the [constant] Velocity of Light -- and also given the assumption that the square-root of 0 is also, and only, 0 again, itself.

Why this "zero-division singularity"?

You can see why, arithmetically, by substituting the value c in place of every occurrence of V in the Einsteinian special-relativistic version of the momentum equation, and by noting that, for any non-zero r in R, its "self-division" yields 1, r/r = 1, which means that, in particular --

(c^2)/(c^2) = 1

-- thus leading to a "self-subtraction" of 1, which yields 0, in the denominator of that equation's RHS momentum-calculator --

for V = c: P = p(V) = P(c) = Mc/( 1 - (c^2)/(c^2) )^(1/2) =

Mc/( 1 - 1 )^(1/2) =

Mc/( 0 )^(1/2) =

Mc/0...."="...."oo"?


This result is typically interpreted as meaning, physically, that, as the Velocity of a body increases toward a Velocity-magnitude of V = c, that body appears to become rapidly heavier and heavier, so that ever greater energy is needed to further increase its speed toward the speed of light, c.

Thus, for bodies which have "non-zero rest-mass" magnitude, e.g., positive "rest-mass" magnitude, the velocity of light appears to be an absolute velocity barrier: the Velocity-magnitude of such bodies can never exceed c. Indeed, their velocity can never even attain c.

Given that kind of barrier, one can only forget about any possibility of effective human[oid] travel between separate planetary-systems, or about any possibility of a galactic-scale [meta-]human[oid] civilization.


Now, some of you may have encountered the mathematical proposition "Every set is also an improper sub-set of itself."

To be a "proper" sub-set of a set, the sub-set must be different from that set [indeed, qualitatively different from that set, for, e.g., sub-set {a, b} is not greater than, nor is it equal to, nor is it less than, set {a, b, c}].


Well, in that same semantic spirit, and even though 0 x 0 = 0 [and even though 1 x 1 = 1], isn't 0 an "improper" square-root for 0 itself [and isn't 1 an "improper" square-root for 1 itself?]?

Is there not also one, or even more than one, "proper" square root(s) of 0 [and one, or even more than one, "proper" square root(s) of 1?]?

I.e., is there not a number that is different from 0, but that, "times" itself, equals 0 [and a number that is different from 1, but that, "times" itself, equals 1?]?

Dr. Charles Musès was one of those who, indeed, discovered new kinds of numbers -- "hypernumbers" -- that are different from 1 -- even "qualitatively" different from 1 -- but which, when "squared", produce 1.

One of those "square-root-of-unity" "hypernumbers" then led directly to his discovery of a -- "compound" -- "hypernumber" that is different from 0 -- even "qualitatively" different from 0 -- but which, when "squared", i.e., when "self-multiplied", produces 0. Such a [hyper-]number is called a "nilpotent".

Musès denoted his "proper-square-root-of-unity" hypernumber with the Greek letter "Epsilon", which I will denote here by E, given the absence of a "Symbol font".

This "hypernumber" has many intriguing "properties", out of which, for the purposes of this post, we shall invoke only a few, including --

EE = E^2 = +1;

1/E = EE/E = EE/E = E

[i.e., E is its own "multiplicative inverse", a new solution to the equation –
x = 1/x].



Now, when we additively "mix" E with another "hypernumber" -- the earliest one discovered, and one which was not discovered by Dr. Musès -- namely, with i, the "proper square root of -1", such that ii = -1, and given, already, that EE = +1, we discover a "proper square root of 0" --

(i + E)^2 =

(i + E)(i + E) = [using the "distributive" "law", or rule, for "mixing" multiplication and addition]

i^2 + iE + Ei + E^2 =

ii + iE + Ei + EE =

-1 + iE - iE + 1 =

-1 + 0 + 1 =

-1+1

= 0.

Thus, the square-root of 0 is "multi-valued", and also equals (i + E), as well as equaling 0 itself, similar to, e.g., the square-root of 2, which has both a -1.41421... and a +1.41421... value.

That is, (i + E)^2 = 0, so, therefore, ( (i + E)^2 )^(1/2) = (i + E) = 0^(1/2), a "square-root" of 0.

A "computational behavior", or "ideo-phenomenon", of i "mixed" with E that is key to the calculation above, is that i and E are multiplicatively non-commutative with one another, in the special sense of being "anti-commutative" with one another [as we shall later show, by a matrix representation of i and E, below] --

iE = -Ei

-- and, multiplying both sides of the above equation by -1, and, the, applying the "reflexive" rule of equality --

Ei = -iE.

Now, first-off, please note that both i and Musès's E are, in F.E.D. 's sense -- and as are also F.E.D. 's q/n -- arithmetical "qualifiers", not arithmetical "quantifiers", whereas all of the "numbers" of the "Standard" axioms-systems for arithmetics -- W, for the "Whole" numbers, Z, for the "Integers", Q for the "Ratio-nal" [i.e., for the "Quotient"] numbers, and R for the "Real" numbers -- are "pure arithmetical quantifiers".

The "units", or "unities", i and E are also "quantifiable qualifiers", unlike F.E.D. 's q/n "unquantifiable qualifiers", i.e. --

1i < 2i < 3i < 4i < . . .

-- and --

1E < 2E < 3E < 4E < . . .

-- but, by themselves, "unquantified", or, rather, in "unit" quantification by the "unit" 1 [1i = i = i1, & 1E = E = E1], they are each "qualitatively" different from -- not quantitatively different from -- "ordinary", or "Real", "unity", and from one another:

i is not greater than 1, & i is not equal to 1, & i is not less than 1; therefore i is "qualitatively" unequal to 1;

E is not greater than 1, & E is not equal to 1, & E is not less than 1; therefore E is "qualitatively" unequal to 1;

E is not greater than i, & E is not equal to i, & E is not less than i; therefore E is "qualitatively" unequal to i.

Please note also that i arises immanently within [psycho-]historical development of arithmetic, from within the "Real" axioms-system of arithmetic, R [or, even, potentially, already from within the "Integer" axioms-system of arithmetic, Z].

It arises out of their "Goedelian incompletenesses", as manifested in the algebraic, "diophantine equation" --

x^2..+..1....=....0, or xx..+..1....=....0

-- which uses only "ideo-ontological symbols" completely familiar from "Real" arithmetic, but which, nonetheless, "drops" a veritable "bombshell" upon previous presumptions about "arithmetic" [whether or not the "Goedel [prime] number" assignments to the symbols of symbolic logic can be contrived so as to make that "diophantine equation" the one to which a "Goedel formula" for R [or for Z] "deformalizes"], an equation which has no solution in R [or in Z], but which does have solutions in the successor axioms-system, of yet-further-expanded number "ideo-ontology" -- the system of the "Complex numbers", C.

A little algebraic transformation of the algebraically nonlinear equation --

xx..+..1....=....0

-- will "explicitize" that this equation implicitly asserts a rather startling arithmetical "ideo-phenomenon", one totally unprecedented in the "ideo-phenomena" of the Z, Q, and R [axioms-]systems of "Standard" arithmetic --

xx..+..1....=....0,........===>

xx..+..1..-..1....=....0..-..1,........===>

xx..+..0....=....-1,........===>

xx....=....-1,........===>

xx/x....=....-1/x,........===>

x....=....-1/x,........===>

-x....=....-..-1/x,........===>

-x....=....+1/x

-- which asserts that the number(s) which solve(s) the equation x^2..+..1....=....0, will be (a) number(s) whose "additive inverse", 
-x, is equal to its "multiplicative inverse", +1/x [ or, equivalently, x^(-1) ].

Ordinary numbers just will not "fill the bill" for this x, e.g. --

-0 is not equal to +1/0;
-1 is not equal to +1/1;
-2 is not equal to +1/2;
-3 is not equal to +1/3;
-4 is not equal to +1/4;
-5 is not equal to +1/5;
-6 is not equal to +1/6; . . .

But i....= ....(-1)^(1/2) will [given that --

+1....=....iiii....=....(ii)(ii)....=....(..-1)(..-1)....=....+1] --

-i....=....+1/i....= .... +iiii/i....=....+iii....=....(ii)i....=....(..-1)i....=....-i.

The "hypernumber [kind of] unit[y]" i is essential to standard algebra.

The very "Fundamental Theorem of Algebra" -- the proposition that all "algebraic" equations are solvable, and have a count of solutions equal to their degree -- is not true unless the i-based numbers are admitted together with, and combined with, the r = 1 based numbers, to form the set C of "Complex" numbers as the general source for all "algebraic" equation solution-sets.


Because i and E are both "contra-Boolean" arithmetical "qualifier" unit[ie]s, unlike the "ordinary", "Real" unit[y], r = 1 [for which 1^2 = 1] in that i and E "behave" in accord with the "contra-Boolean" fundamental "law" of dialectical thought --

x^2 is qualitatively unequal to x

-- viz. --

i^2 is qualitatively unequal to i, in that ii = -1, and;

E^2 is qualitatively unequal to E, in that EE = +1

-- such that --

i is not greater than -1, & i is not equal to -1, & i is not less than -1, and;
E is not greater than +1, & E is not equal to +1, & E is not less than +1

-- I will, hereinafter, in this thread, follow the -- "non-Standard" -- F.E.D. convention of representing these "arithmetical qualifiers" with an underscore, although neither i nor E exhibit the "evolute" [or "double-<<aufheben>>] operatorial character that the F.E.D. q/n, & sequel, "evolute dialectors" exhibit, such that --

x^2 = x + delta[x] --

-- in that --

ii does not equal i + (-1), or -1 + i, and;

EE does not equal E + 1

-- so that F.E.D. terms both i and E "convolute" arithmetical "qualifier" operators, not "evolute" ones.

For more about "evolute" versus "convolute" operations, see --



The key question for this present sub-series of posts is: ¿Are these "convolute" arithmetical "qualifiers", i and E, also "behavioral qualifiers", i.e., "phenomenological qualifiers", in a physically meaningful sense?



So, suppose that we now, in the Einsteinian, special-relativistic momentum equation for V = c, substitute, for --

( 0 )^(1/2) = 0

-- this alternative, "compound hypernumber", value for that square-root of 0, i.e., a "proper" value for the square-root of 0, namely --

( 0 )^(1/2) = ( i + E ),

viz. --

for V = c: P = p(V) = P(c) = Mc/( 1 - (c^2)/(c^2) )^(1/2) =

Mc/( 1 - 1 )^(1/2) =

Mc/( 0 )^(1/2) =

Mc/( i + E ) = ?


So, with this alternative valuation -- of ( i + E ) for ( 0 )^(1/2) -- we obtain

P(c) = Mc/(i + E)

-- a Momentum magnitude, or Momentum "quantifier", Mc, "qualified" by the multiplicatively-inverted, "compound hypernumber", quantifiable qualifier 1/(i + E), whenever V = c, i.e., for any body of mass-energy traveling at exactly the speed of light.

Such "fractions", e.g.,1/(i + E), or Mc/(i + E), do not further "amalgamate", or "reduce", given the [qualitative] heterogeneity between their "numerators"
1 and Mc -- and their "denominator" -- (i + E).


But what, if anything, could this "qualified" Momentum magnitude -- the Mc "quantifier", "qualified" by the hypernumber factor 1/(i + E) -- possibly signify physically?

To what kind of physical phenomena, or characteristics, if any, can we associate the "hyper-numerical qualifier" value 1/(i + E)?


CONTINUED NEXT POST



Regards,

Miguel

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