Continuation Causes Superior but Unrealistic Ambiguity by Eckard Blumschein

Dear Eckard,

While there is no negative distance in reality, there definitely is a left and right, an up and down and a backward and forward, relative to a selected position, and since space is usually defined as a set of positions satisfying the postulates of geometry, I think it important to not throw the baby out with the bathwater.

Sir W. R. Hamilton was an English astronomer and mathematician who coined the term "vector," first explained complex numbers as coordinates on a graph (or rotation made possible by an imaginary number), and is credited with the invention of quaternions (even though a Frenchman beat him to it and understood the true nature of quaternions better).

It was based on his ideas that Pait, arguing for Hamilton's quaternions, took on Heaviside and Gibbs, and lost the battle with their vector algebra, which has dominated ever since.

However, quaternions found a new life in computer calculations, since it greatly simplifies the manipulations of 3D rotations and avoids gymbal lock. They were mostly revived by David Hestenes, who first recognized the value of the ideas of Grassmann and Clifford and popularized them through his modification of Grassmann's geometric algebra (GA) (see here).

GA is slowly gaining acceptance, since it greatly simplifies vector algebra through the use of new concepts and definitions that combine scalars and vectors in something called the geometric product. It has to do with rotation and an inner and outer product.

As I mentioned, it was Hestenes' work on GA that drew attention to the work of Hamilton, Grassmann and Clifford, and it was mostly centered on the set of Clifford algebras upon which his work sheds so much light.

This lead me to a little known essay by Hamilton that is called "Algebra as the Science of Pure Time." Most people think that it was based on Kant's ideas of time. I don't think it has anything to do with the Greek concept of the continuum as a moving point.

It focuses on the comparison of moments of time, where we can say that two moments may be coincident, or else one is later or earlier than the other. Of course, his development was confined to the observer's frame of reference in the abstract.

Dear Eckard,

When we consider the approach of the pendulum to the plumb line, the plumb line is definitely in its future. If the plumb line and the arc of the pendulum's swing are divided into n and m parts, respectively, then we can describe the projected position of the pendulum on the plumb line in terms of x/n, as x --> n, and the actual position of the pendulum in terms of y/m, as y --> m.

I think that we can all agree that a "now" moment arrives when x = n and y = m. However, the plumb line is still in the future of the pendulum at this point, since both n and m cannot be points of no spatial extent, but only magnitudes of some extent, ad infinitum. Clearly, we cannot say that the two are coincident, until both 1/n and 1/m --> 0.

Yet, if we are to measure the diminishment of these two remaining units, we have to further sub-divide them into n and m parts, and the process starts over, ad infinitum. Hence, we see that Zeno's paradox is in full force, in this case, even though we know that x --> 0 and y --> 0, eventually.

How do we resolve this paradox? I submit that one way is to transform the remaining y unit into its inverse. Since x is wholly dependent on y, it can serve as a measure of y's transformation into its inverse. When y's transformation is complete, x = 0.

But what is the inverse of y, if not -y? and doesn't x --> 1, in the other "direction," as y --> -y? How, then, does x --> 0 and x --> 1, simultaneously? The only way to resolve this new paradox, as far as I can tell, is to admit that x too has an inverse, namely -x. Consequently, as y --> 0, -y --> 1, simultaneously, and as x --> 0, then -x --> 1, simultaneously.

Such a transformation, in both cases, involves an instant change in "direction," at all "points" along the length of the units. It is only at this boundary between the two, opposite, "directions" that we can consistently define a point of no spatial extent and an instant of no temporal duration.

The physicists describe just such a constant transformation, as the potential energy of position is transformed into its inverse, the kinetic energy of motion and vice-versa. Therefore, this approach not only seems reasonable to me, but exhibits a hint of that unreasonable effectiveness of math in the natural sciences, that Wigner points out.

Warm regards,

Doug

Dear Doug,

You intended to demonstrate that negative quantities are necessary. Well, this is true for mathematical physics with emphasis on mathematical in a derogative sense. I see so called harmonic oscillator as well as your pendulum as examples for strictly speaking unrealistic models. In reality, no oscillator oscillates for good.

You might argue that for instance sound pressure is the alternating components of atmospheric pressure. Maybe you heard of mammals blooding out of their ears because they were exposed to 210 dB re 20 microPa. Yes, I refer to whales in water. Air cannot convey symmetrical waves of such SPL because there the negative half wave is limited to the atmospheric pressure.

In your example, neither kinetic not potential energy is negative. If I recall correctly, energy typically corresponds to distance from origin in phase plane, and realistic trajectories of stable systems are strictly speaking inward directed spirals. What about your definitions of x and y, aren't they arbitrary?

You wrote:"Such a transformation, in both cases, involves an instant change in "direction," at all "points" along the length of the units. It is only at this boundary between the two, opposite, "directions" that we can consistently define a point of no spatial extent and an instant of no temporal duration."

Really? Isn't this at odds with Euclid's definitions of a point as having no extension in general and of a number as always denoting a measure, not a point?

While quantities like distance and elapsed time do not have an inverse, differences between distances or between elapsed times can of course be negative. Consequently, the logarithm of e.g. a relation between two distances can also be negative.

As long as physicists always prefer the "more general" R instead of the tailor-made R, people may continue mocking: mathematical physics is if three out of two men left a room and therefore one man has to come in as to make the room empty.

Warm regards,

Eckard

Dear Eckard,

You wrote:

"In your example, neither kinetic not potential energy is negative. If I recall correctly, energy typically corresponds to distance from origin in phase plane, and realistic trajectories of stable systems are strictly speaking inward directed spirals. What about your definitions of x and y, aren't they arbitrary?"

Well, of course, but we need labels. No one is arguing the ontology of negative cabbages. However, if the point of the plumb line, which the location of the pendulum crosses, is designated '0' then, according to normal convention, we can label the top of its swing to the right as positive 1, and the top of its swing to the left as negative 1.

Even though both of these poles are positive, in the sense that the rise of their swings is in opposition to gravity, the location of the pendulum at either pole is the result of motion in two, opposed, "directions." If we plot the motion, and correlate it with potential and kinetic energy, we can see how the two types of energy are inverses:

Position of Pendulum: -1, 0, +1

Energy of Position: -1, 0, +1

Energy of Momentum: 0, 1, 0

Thus, the fact that kinetic energy is the inverse of potential energy does not require negative reality, only negative labels to describe the reality.

Wouldn't you agree?

Regards,

Doug

Dear Eckard,

You wrote:

"This does of course not answer my original question, which was already answered by your hint with the "point in the middle left over". Nonetheless it shows to me that mathematicians like John Baez do not devote the due attention to what I consider important: the question of redundancy and ambiguity."

The reason this is so, I believe, is due to the fact that they have had to construct the 1D, 2D and 3D inverses by inventing real numbers and imaginary numbers. They invented real numbers via logic and symbols, and imaginary numbers via rotations.

In a sense, by combining these ad hoc inventions, they have been able to increase the algebra of 20 (the reals, R), to 21 (the complexes, C), to 22 (the quaternions, H), and to 23 (the octonions, O), but the O algebra is non-associative. Hence, they run into problems with trying to use it in physical theory.

Since the algebra of C only lacks the distributive property, they mostly use it, but evidently they can use the R and the H algebras, as well (the latter probably as H+H).

But what they would really like is an R3 algebra that has all three division algebra properties intact. The fact that R- is redundant to R+ and that the arrow of time has to be added to remove ambiguity in physical interactions modeled with DEQs, is not immediately relevant to their challenge of pathological algebras that they are having to cope with, though it might still be after a solution is found.

Regards,

Doug

Sorry, "1D, 2D and 3D inverses" should read "1D, 2D and 3D algebras."

Hi Eckard.

I will clarify;

"The LT is not required as the light signal from the centre of the stream only does c"

The LT was essentially used by Einstein to prove the limit 'c' for all (co-moving) observers. That is what I refer to it as being "REQUIRED" for. . - BUT, if an observer only receives light at 'c', and the light 'pulse' he observes is only MOVING at 'c', the LT is simply superfluous! It was a mistake not to understand the point, which Georgina has taken up and proved well, that 'apparent' velocity from another reference frame is not the same as actual velocity. The light signals from the pulse to the observer are new, 'scattered' (and polarised) signals, in sequence one after the other, giving only the 'impression' of something moving at c

Eckard

My clarification /answer in the above thread missed this;

"we must now resort to the big gun with a curved trajectory to falsify our model; GR."

It was a bit of a play on words, a 'big gun' as the ruling paradigm of General Relativity, as both big guns and GR have curved trajectories! but also to give it the sternest test; How would the DFM stand up under the test of GR and gravity!

It's opened up an astonishingly simple solution, and I've since gone further; Have you ever wondered how a passing asteroid can exert more gravity on us if it moves faster? That's what inertial/gravitational mass equivalence says. Stupid? Perhaps not, as the DFM shows how it works. Add up all the plasma ions (photo-electrons) with inertial mass but zero rest mass (they evaporate at rest), that grow around matter accelerated in a void (i.e. the LHC pipe) proportionally to speed, and we find it perfectly equivalent!

It was also found that the (quantum particle) plasma coma/tail of a comet did have that gravity as it 'bent' and dilated space time (diffracted through the plasma halo) by 2 arc secs, exactly as predicted.

Of course this may look, waddle, swim and quack like a duck but not be one. So do we all ignore it? I anyway decided that while everyone else was away hungrily searching for quantumish relatives of my similitudinal 'ducky' thing I'd try roasting it with some orange. It tasted perfect! I offer everyone as much as they wish, (including as sandwiches while off looking for those mysterious ducks if they wish).

I hope you might try a taste?

Peter

PS.

Arthur Eddington was leading the duck hunting party so they're in safe hands. But you'll be pleased to know the LT was plucked out with all the other feathers, down and entrails.

I'm trying to find the comet Ref. but here are 2 of dozens to be going on with in case you wish to check out the scientific basis. Just ask for any more.

http://sci.esa.int/jump.cfm?oid=48440

Anderes E Knox L, van Engelen A. Lense mapping

against CMBR ion scattering. Phys.Rev.D (Accepted

Mon Jan 31 2011). (no hyperlink yet)

Xing-Hao Y, Quiang L. Gravitational lensing analysed by the

graded refractive index of a vacuum J. Opt. A: Pure Appl. Opt. 10

075001. http://dx.doi.org/10.1088/1464-4258/10/7/075001

Best wishes

Peter

Dear Eckard,

You wrote (for some reason, this doggone editor eliminates all plus signs for me!!!):

"Just add what you wrote:

Energy of Position: -1, 0, +1 plus Energy of Momentum: 0, 1, 0

The sum is strange: -1, 1, +1

I consider the sum of kinetic and potential energy in a closed system constant: sin^2(x) plus cos^2(x) = 1."

As you point out, the sum of the kinetic and potential energy remains a constant value, as their relative values change inversely, but when the same relationship is expressed in terms of sine squared and cosine squared, using the Pythagorean theorem, the algebra eliminates the negative sign, because it has two square roots for -1: -12 = +12 = 1.

Thus, the sum is strange, but only because of the limitation of the algebra. In this case, it appears that what might normally be considered a vice in the algebra of R, is now regarded as a virtue!

Regards,

Doug

Dear Doug,

At first I would like to again express my gratitude for giving me the correct hint to Baez's "point in the middle left over".

I wrote: Nonetheless it shows to me that mathematicians like John Baez do not devote the due attention to what I consider important: the question of redundancy and ambiguity."

You explained:"The reason this is so, I believe, is due to the fact that they have had to construct the 1D, 2D and 3D inverses by inventing real numbers and imaginary numbers. They invented real numbers via logic and symbols, and imaginary numbers via rotations."

I guess, they simply took the real line R between minus infinity and plus infinity for granted. Descartes reportedly hesitated to adopt R, and you are familiar with Hamilton: On one hand, he "felt his notation [of a couple (a,b)] somehow better since it avoids the absurd" [a+ib]. On the other hand he meant "... these definitions are really not arbitrarily chosen" (cf. Nahin p. 81 who commented "indeed not").

Really not? I do not share Nahin's opinion. Let's begin with a real function f(x)=cos(x) that describes a measured quantity and with Euler's identity F(x)=exp(ix)=cos(x)+i sin(x). In order to relate f(x) to the anti-clockwise rotating phasor in complex plane we may either arbitrarily add +i sin(x) or split cos(x) into exp(x) + exp(-x) and arbitrarily omit exp(-x). In both cases f(x) and F(x) are related to each other not via an identity but an arbitrary transformation. We may benefit from the chosen complex plane, no matter whether we gave preference to the positive or negative sign. If we intend to interpret a result we got in complex plane, then we are well advised to perform the due inverse transformation.

You wrote: "... the O [octonions] algebra is non-associative. Hence, they run into problems with trying to use it in physical theory. Since the algebra of C only lacks the distributive property, they mostly use it, but evidently they can use the R and the H algebras, as well (the latter probably as H+H).

But what they would really like is an R3 algebra that has all three division algebra properties intact. The fact that R- is redundant to R+ and that the arrow of time has to be added to remove ambiguity in physical interactions modeled with DEQs, is not immediately relevant to their challenge of pathological algebras that they are having to cope with, though it might still be after a solution is found."

I do not deny additional problems with algebras of further extended numbers. My focus is on the peculiarity of all elapsed or as elapsed anticipated time as well as of all distance: They are always real, and they always have a natural point zero. In other words: They are real and one-sided. So far I see there aspects persistently ignored in mathematics. As a consequence, many experts did not believe that the results of my real-valued calculations of a spectral analysis can be correct. They even ignored compelling arguments: Their own ears are not synchronized to our agreed time scale, and they cannot perform complex calculus. The moving pictures expert group gave preference to MP3, which turned out to excellently work with cosine transform instead of Fourier transform.

Restriction to one-sided real functions is partially quite accepted. Nobody is using a negative radius, and negative temporal distance is also rather abstruse. Some theorists seem to enjoy worrying people with negative probability, negative energy, or the like. Electrical engineers remain calm. They are familiar with negative (differential) resistance. I blame most physicists for still sticking on not just Pauli's belief in a mystery.

Best regards,

Eckard

Eckard,

You also wrote:

"Isn't there still growing confusion not just among students who have to swallow an abundance of allegedly counter-intuitive theories from Cantor's transfinite alephs up to white holes? I observed even here in the contest a widespread readiness among laymen to spontaneously welcome new suggestions ranging from a variable Planck's constant by Constantinos Ragazas to Peter Jackson's exciting ideas and the tendency among experts to favor overly formalized "serious" speculations. Not even Georgina Parry did consequently challenge idolized theories. I consider my arguments in Appendix C strong ones. I just avoided to open one more frontal attack."

I concur with your line of thought on this. You may may be interested in this quote from David Hestenes:

"...there is a great proliferation of different mathematical systems designed to express geometrical ideas - tensor algebra, matrix algebra, spinor algebra - to name just a few of the most common. It might be thought that this profusion of systems reveals the richness of mathematics. On the contrary, it reveals a wide-spread confusion - confusion about the aims and principles of geometric algebra."

Doug

Dear Eckard,

I almost missed your reference to me in your post below! It would be helpful if there were automatic 'alerts' send to those affected by such comments!

Clearly, from your quote on my thinking, or more accurately your understanding of this,

"... a widespread readiness among laymen to spontaneously welcome new suggestions ranging from a variable Planck's constant by Constantinos Ragazas to ..."

it is clear to me that you don't understand my thinking! Not that this should be a surprise - that being the nature of new ideas. But you could have addressed your misgivings about my ideas directly with me! I would have clarified!

Simply, dear Eckard, it is NOT my position that Planck's constant (and all that is associated with this in Physics) can be considered as a variable! Rather, the quantity that Planck's constant is a value of ( the time-integral of energy) CAN be considered as a variable! Consider integrating energy at a point over a time interval from 0 to t. Wont that quantity (that integral) be a 'variable'?

There is nothing inconsistent (mathematically or physically) in everything that I demonstrate in my essay. In fact, most all of these will appear in a chapter in a book on Thermodynamics this July. The coauthor that has invited me to this project is Hayrani Oz, a Professor of Aerospace Engineering at Ohio State University. He finds my work (including the 'quantity eta', same type as Planck's constant) commendable and in total agreement with methods and ideas he has been using in his own work and teaching for many years. Whereas I use 'eta' for the time-integral of energy, he uses what he calls 'enerxaction'.

In my essay I do, however, present an argument as to why Planck's constant exists, and what it really means! Read my essay more carefully, and if you have questions PLEASE address these to me instead!

I have greatly valued your past support and contributions in these FQXi blogs, Eckard, and I continue to do so. But that comment took me by completely surprise!

Best regards,

Constantinos

Dear Doug,

My dictionary tells me: A vice is a moral fault in someone's character or behavior. I would blame contradictory arrows of positive directions rather than a limitation of the algebra.

I only very vaguely recall, in connection with the notion of action, that the Lagrangian is not the sum of but the difference between kinetic and potential energy. Maybe I am wrong here. Nonetheless I expect confusion if, e.g., the positive incrementation dx of x remains unchanged while the direction of force changes its sign at zero. I also recall oddities with modal analysis in cylindrical coordinates.

Just for fun I will quote from Nahin, p. 14, how already John Wallis confused himself:

Since a/0, with a larger than 0, is positive infinity, and since a/b, with b smaller than 0 is a negative number, then this negative number must be greater than positive infinity because the denominator in the second case is less than the the denominator in the first case. This left Wallis to the astounding conclusion that a negative number is simultaneously both less than zero and greater than positive infinity ...".

Best regards,

Eckard