The Discreet Charm of the Discrete by Paul Halpern

Dear Professor Halpern,

Let me start by saying that I enjoyed the mix of the historical with modern conjectural views toward physics displayed by your essay. One question. Have you considered the possible role of The Extended Theories of Gravity in which the cosmos possesses an intrinsic curvature as an alternative to the Standard Model with respect to the many open cosmological questions we now face? I would point you toward Christian Corda's excellent essay as well as own. In my own simple approach, I show that the cosmos must have an intrinsic curvature for a very elementary reason. As for the observations that lead to the DE hypothesis, they have to rate as the most exciting discovery since Hubble.

Have a great day,

Dan

Sorry, that last anon. post was mine.

Dan

Dear Paul,

I enjoyed reading your essay and I like how you use the idea of lowest wavelength fields to discuss discrete aspects of the physical law. Congratulations for the very well-written essay and the profound exploration of the implications of the idea of holons.

Best regards,

Cristi

Dear Paul Halpern,

Twenty years ago, when I heard for the first time of CA, they were used almost like FEM by those who model plasma, heat conduction and the like. Since then I did not get aware of reported convincing benefits. Your essay makes understandable to me chains of more or less speculative hope for foundational insight that might be confirmed in the very far future.

Let me frankly admit that I am ready to accept the big bang as an exiting hypothesis but not yet as proven for good. What about the attribute foundational, I prefer to judge it as did you from experience over centuries. I appreciate that you avoided technicalities that would perhaps anyway not be convincing.

Most appealing to me was your question for upper and lower frequencies of in particular electromagnetic waves. You will certainly not need reading my essay in order to understand that discrete frequencies correspond via cosine transform with continuous functions of temporal or spatial distance and nice versa. Cosine transform differs from Fourier transform in that performing it twice yields the original function. In other words, it is its own inverse.

Not just in CA the notion neighbor plays a role. Hausdorff topology does also consider a neighbor each to the left and to the right. I found out that this is based on a non-Euclidean notion of number and at odds with a lot. For instance, it is to be blame for trouble at zero, for the unwillingness to accept R as sufficient in case of non negative spatial or temporal distance, and even with the usual unilateral attribution of the increment dx to the direction of x.

Regards,

Eckard

Hi Paul,

interesting essay!

You write:

'One measuring of entropy in network theory is the growth of complexity in link structure.'

Are you referring to a specific definition of entropy for networks? Any pointers? Would this be something like the log of the number of graph automorphisms?

You also write:

'Given the promise of simple, discrete algorithms, it is interesting to consider a dynamic digital model of fundamental interactions based on holon states. Because holons would have the minimal wavelength of all fields, they might be thought of as 'cells' in a dynamic grid. It is possible that simple digital rules, akin to SDCA algorithms, could transform both the values (quantum states, such as polarization) and linkage of sites. This would render this stratum geometrically dynamic, offering a digital basis for distortions in spacetime'.

I am interested in SDCA type of models, where interconnection patterns evolve, although I claim that having a dynamic interconnection pattern should suffice for building everything, without need of an additional layer of labels, or cell states. The interconnection pattern could indeed code for local states. In the planar case, for example, one has polygonal faces, each with its size, and this generalizes to simplices in higher dimensions. Depending on whether or not these 'atoms' have and preserve their own identity, one can conceive two different types of field/particle (I see analogies with fermion vs. boson behaviour here).

Did you run computer experiments in which both the background cell structure and the additional layer of state information you place on top of it give rise, separately, to the emergence of solitons or other patterns?

Thanks and good luck!

Tommaso

Paul,

From my perspective you have successfully moved a "holon" from the list of zeroes (could exist but don't exist) to the list of ones (exists). Also from the list of fuzziness to the list of clarity. Now, I am interested in more movement from zero to one. If I pointed to some point in space with my finger...would a holon exist at the end of my point? If not, what would increase the possibility of a holon being there? Expansion of the space pointed to...and what about expansion of the time allowed for observation at the intitial point at the end of my finger? Would expansion of space searched and time observed increase my chances of locating a holon? Why is the possibility increasing with a change in either condition? Or, is there actually only holons everywhere which make my questions mute?

Irvon

Paul,

So, are we willing to attach the fuzziness of nothingness to create meaningfulness in the sensory world (sensed objects, forces and relationships that we actually can point to and even get close enough to touch)?

Here I would suggest what I think is an appropriate poem:

V

I don't know which came first

After the chicken

And sex

And the egg

And the empty can of soup

There's no need for

An opinion

Based on observed facts

Not in this day of

Revealing ignorance

What's left for modern man

Isn't even a bone

There's just the opportunity

To scavenge

To catch a glimpse of

A suggestive piece of evidence

An imprint exists

Where it once was

Substance

And now

Even

Ideas are Art

--------------------------

Paul, you have created a great piece of art!

Respectfully,

Irvon Clear

Paul,

I was looking forward to your entry in this contest, and I was not disappointed. You are so very expert at drawing a picture of abstract models in sensuous terms. I especially appreciate your comparison of an office tower to the possibility of a finite field theory (in itself a concept to launch a thousand dissertations). I was reminded of Jorge Luis Borges's short story, "The Library of Babel."

Interesting, informative and original. Just excellent.

Good luck in the contest, and I hope you get a chance to visit my essay, which shares in common with yours an emphasis on information theory.

All best,

Tom

Dear Paul,

Thank you for your very kind remarks on my essay. I had also read your essay, some days ago, and very much appreciated your holon idea.

With my best wishes,

Tejinder

Dear Paul,

I liked very much the historical part of your essay, which is much more developed than mine. Also, I liked the analogy of the information contained in an envelope and the size of the characters.

On the other hand, a minimal wavelength seems to me counternatural as this would mean an upper limit of the wavenumber. Intuitively both ends (linear and inverse linear) are unbound.

Best wishes for this contest.

Arjen