What is the Point of Reality? by Doug Bundy

Ekard,

The trouble is not with Euclid's definition, but with the modern definition as something that does not have parts, yet has the properties of spin, charge and magnetic moment. See Feynman's Lectures on Physics.

Hestenes wrote:

"Now, the Dirac electron theory and its extension to quantum electrodynamics is universally recognized as the most well substantiated domain of physics. Strangely, however, it is rarely involved in the discussions of the foundations of quantum mechanics. This is a grievous error, for the Dirac theory entails an irreducible relation between spin and complex numbers with undeniable implications for the interpretation of quantum mechanics. Analysis of this relation strongly suggests that the complex phase factor in the complex function describes a kinematic feature of electron wave motion and therefore has a physical, rather than statistical, origin."

How can a point particle (a term that is an oxymoron in and of itself), as something with no parts, have properties of any kind, let alone a kinematical feature?

The practical benefit is gained in understanding the point and the elementary particles in some consistent manner. As Einstein said, "You know, it would be sufficient to really understand the electron." Likewise, it would be nice to understand quantum spin and isospin, but we haven't a clue as to their physical meaning, understood in terms of complex numbers. Yet, a school child can be taught the physical meaning of a 3D oscillation cycle and why it's equivalent to a 4 pi rotation.

I'm not a mathematician, but I think the search for an intuitive understanding of algebra is related to an intuitive understanding of the point. The idea of zero, in a quantitative sense is related to the idea of unity in an operational sense, which we can see when we realize that there is zero difference between the numerator and denominator of the rational number, n/n. This leads to the two interpretations of number enabling us to use them to form a field from natural numbers, which includes 0D, 1D, 2D and 3D scalars.

In my paper's end notes, I explain a little how this can lead to higher dimensional groups nested within the zero-dimensional group, when those numbers are derived from the 3D oscillations of space and time. It involves replacing the unit 1 = 12 = 13 with R not equal to R2 not equal to R3, where R = 21/2.

Regards,

Doug

Thanks Alan. Best wishes to you also.

Thanks George,

There's certainly a long way to go, but the idea of 3D numbers is about to get some powerful exposure, I hope!

Dear Eckard,

You wrote: "Thank you for your hint to John Baez. Unfortunately, he is an overly prolific interpreter of sometimes rather unrealistic mathematics in terms of physics, and I did not yet find his "clarification" you are alluding to. Hopefully someone else can give me a clue. Is a Baez always correct?"

Most of the things he writes about, I can't understand anyway, but he has a fascination with Octonions, and that got my interest. I believe that the ad hoc invention of imaginary numbers in algebra, while useful in some ways, is counter-productive in the end, as particle theorists found out in SU3 studies, but, what I like to call the tetraktys, the binomial expansion, up to dimension 3, is key to understanding a true R3. Among other things, it has an inverse!

You wrote: "When you compared my essay with a whirlwind tour in a museum, I sadly did not reach you. I tried to investigate where mathematics started to become arbitrarily rather than logically founded."

Please don't misunderstand me. I did not mean your essay was museum-like. I meant that my reading of it was tour-like. My daughter is having her baby today and that's just one of many pressing things I have to attend to, which doesn't leave me enough time to study your essay, but I will! I read the two Joyce documents and can hardly wait to comment further.

You wrote: "I am not surprised that even the very cautiously thinking Ian Durham in his new essay ignored the possibility of mistakes when he wrote: "... while results from ... WMAP have demonstrated that the geometry of the universe must be flat ...and thus 'Eucildean,' we of course have long known that it is locally curved.""

In the new physical system of theory that I advocate, the major assumption is a space/time progression, but it included the assumption that the universe was flat and I had to defend that assumption vigorously. The WMAP news was very welcome on that score.

You wrote: "What about your multidimensional small-signal numbers I recall a Dutch outsider who seems to be close to your approach. I got aware of him when he was quoted from a participant of a previous FQXi contest. I also vaguely remember of a peculiarity in theory of acoustics waves: Solutions for even spatial dimensions (0, 2) behave differently from those for odd (1, 3) spatial dimensions. Did you know that?"

No, I do not know about that. I also don't know of the Dutch outsider you refer to.

I have to run now, but I will be back tomorrow. I just wanted to tell you how much I appreciate your knowledge of things and your belief and attitude toward things fundamental. I support your insistence on re-instating Euclid's view of number as measure and Peirce's view of the continuum as infinitely divisible, full heartedly.

Regards,

Doug

The above anonymous post is from me, in case it's not obvious. I forgot to log in, before posting it.

This info was posted in a thread earlier, but people had a hard time finding it, so I am re-posting it here:

A lot of the essays in this contest have to do with the issue of mathematics versus reality. While numbers can be used to count things, they are not those things, but I'm not sure why it's important to make that obvious distinction. I don't recall that it was ever a big issue in geometry.

In my case, what is being counted are units of space and time. The 3D expansion of space upon which it is based is an observed physical phenomenon. The operational interpretation of the mathematical expression s3/t = 23/1 describes how many measures of space per measure of time are generated from a given point, as the progression continues. Thus, after n linear measures of time pass, (n*2)3 volume measures are generated, giving us the mathematical progression, 8, 64, 216, 512, ... quantitatively, corresponding to the linear march of time, 1, 2, 3, 4, ....

Geometrically, the time units are constructed as the radius of the inner circle of figure 1 in my essay (considering only one of the two constructions there). The volume units are the volume of the 2x2x2 stack of 8 one-unit cubes, which grows to a 4x4x4 stack of 64 one-unit cubes, a 6x6x6 stack of 216 one-unit cubes, an 8x8x8 stack of 512 one-unit cubes, and so on, ad infinitum, as time marches on.

Admittedly, these discrete volume units in the progression correspond to nothing real, since space doesn't expand outward cubically, in only eight "directions," but outward radially, in an infinite number of directions. However, the inner and outer radial volumes are determined by the 3D stack of one-unit cubes and the ratio of their volumes is equal to R3, where R = 21/2. Hence, the progression of the ratio of the two volumes of the expanding spheres (technically balls) is, (n*21/2)3, or 81/2, 5121/2, 58321/2, 327681/2..., as time marches on.

It may not appear at first that these numbers correspond to anything real, but the fact that their squares are integers, and very familiar integers at that, indicates otherwise. Indeed, when we look at the ratio of the surface areas of the spheres, the formula for the progression of which is (n*R)2, the numbers are very curious: (1*21/2)2 = 2, (2*21/2)2 = 8, (3*22)2 = 18, (4*21/2)2 = 32.

These are the numbers of elements in the half-periods of the periodic table, something we might be tempted to dismiss as conincidental, smacking of numerology, if it weren't for the fact that Le Cornec has made what C.K. Whitney calls "a stunning demonstration:" Le Cornec discovered a previously un-noted pattern underlying all ionization potential data: the square roots of all ionization orders, when plotted as a function of atomic number Z, lie on a straight line. (See Whitney's "Relativistic Dynamics in Basic Chemistry")

That the IPs are something real is undeniable, but that their order appears closely related to a natural numerical progression is astonishing, in my view.

More on this later.

The feedback I'm getting privately indicates that people do not understand the concept illustrated in figure 1 of my essay, in several respects. Given the constraints of the contest rules, I found it very difficult to include all the explanatory detail I would have liked to have included.

One of the difficulties has to do with background. There is no background. There are two, reciprocal aspects of one component, motion, which are 3D space and 3D time, but since time has no direction in space, and space has no direction in time, one of the reciprocal aspects of motion is always zero-dimensional.

However, just as, strictly speaking, natural numbers and energy are zero-dimensional measures, and, to be useful, they have to be regarded as one-dimensional, so also the 0D aspect of motion has to be one-dimensional.

In the case of numbers, any number raised to the zero power, n0, is equal to 1, because any number divided by itself is equal to 1, by virtue of the law of exponents. Hence, n/n is actually n1/n1 = n(1-1) = n0.

In the case of energy, any magnitude that does work has to be one-dimensional, because magnitudes that have no direction are scalar (i.e. 0D), while magnitudes with direction are vectorial (i.e. 1D). Energy per se is scalar. It has no specific direction in space, but in its form as work, it must have direction, and therefore it is regarded as a one-dimensional quantity.

In the geometric constructions of figure 1 in my essay, we see the discrete (digital) and the continuous (analog) expansion of 3D space (or 3D time), after one unit of 0D time (or 0D space) has elapsed. Though the 0D time (space) aspect of the expansion cannot be represented geometrically in any direct fashion, the 1D radius of the inner circle is equal to the time duration, in the same sense that a time line on a space and time graph, or the 1D sweep of an oscilloscope, represents a duration of time.

With this much understood, the radii of the inner circles are 1D representations of the 0D time (space) of the expansion. Now, the digital and analog representations of the two, inverse, expansions in the figure, each contain the 1D, 2D and 3D components. The digital representations are contained in the 2x2x2 stack of one-unit cubes, while the analog representations are contained in the ratio of the two balls that are determined by the stack of cubes.

In the digital case, the 1D magnitude is the width (or the height, or the depth) of the stack, while the 2D magnitude is one of the faces of the stack, and the 3D magnitude is the volume of the stack.

In the corresponding analog case, the 1D magnitude is the ratio of ball diameters, the 2D magnitude is the ratio of the spherical surfaces (or else the cross-sections of the balls), and the 3D magnitude is the ratio of the volumes of the balls.

The reason that the analog magnitudes are taken as the ratio of the two balls is because the magnitudes of the inner ball are necessarily less than the magnitudes of the stack, while the magnitudes of the outer ball are necessarily greater than the magnitudes of the stack. It turns out, however, that their ratios are integers and square roots of integers, which means that the digital magnitudes can be directly related to the analog magnitudes, not just approximated!

As the expansion continues beyond the one unit elapsed time stage, the digital and analog magnitudes follow their respective geometric progressions, part of which is explained in the previous post above. The key point is that these units can be arranged in the customary mathematical forms called groups. The usual 1D digital groups of integers and rational numbers apply, plus a new group with the square root of 2 as the 1D unit, instead of the number 1 as the 1D unit, applies.

However, this new group contains 2D and 3D elements as well as 1D elements, which provides for 1D, 2D and 3D scalar algebras called division algebras, which have all the properties an algebra needs to have to be used in physics.

Ultimately, this means that the 3D oscillations of these units, and their mathematical combinations and relations between their combinations, can be used as building blocks, called preons, to build the particles of the standard model of physics, at least in part. The hope is that the entire model will eventually emerge, including gravity. If this turns out as hoped, it will be very strong evidence that the physical universe consists of one component, motion, existing in three dimensions, in discrete units, with two reciprocal aspects, space and time.

I hope this helps.

I've been asked to provide a graphic depicting what I mean by the "common origin" of the 3D oscillation. Here is the best I can do for now:

3D Oscillation

I'll try to improve it when I get time, but it should be good enough to convey what is meant by the common origin of the two, reciprocal, volumes that define a true point of no spatial extent and no duration.

BTW one cycle of this motion is equivalent to 4pi rotation (something inexplicable until now.