Experimental Mathematics and the Digital Nature of Reality by Fredric Alan Litt

Thank you Alan,

The computer's "random number generator" is intended as a metaphor: We KNOW that there is an algorithm that the computer uses to generate a "random" series but, because we don't know the algorithm, the numbers can be accepted as random. We could, in principle, predict the numbers that follow if we were given the algorithm and a sufficient portion of the pseudorandom series. If so, the series would no longer be useful as "random" numbers. Fortunately, the algorithm is sufficiently complicated that we are not able to reverse engineer it.

In looking at the interval between natural events such as fluorescence or radioactive decay of single emitters, we certainly don't have such an algorithm. We can therefore assume the assume randomness and derive such (longer time period) properties such as half-life or fluorescence efficiency. My point is that using the series of emission events as a random series does NOT prove that the series is truly random; it shows only that we don't know the underlying algorithm (which might or might not exist).

When I read the subject of this essay contest literally only moments after reading an obituary of Benoit Mandelbrot, it occurred to me that the emission events need not be truly random; they might instead be chaotic and we might not be able to discern the difference using space and time as continuous functions.

I was therefore driven to find an example of a difference function that was IDENTICAL (not merely asymptotic) to the continuous function over part of its range but was also capable of giving chaotic behavior over another part.

Dear Dan,

Please also see my reply to Alan, above.

You comment, "However, we are never in a position to establish absolutely that a pattern is "truly" random." I fully agree. The point is that we can deduce larger-scale properties if we assume true randomness. Once we can predict the series however, these deductions are only approximately valid since our treatment of the data assumes randomness that really isn't there.

You comment further, "Secondly, we can only absolutely know that a pattern is pseudo-random if the generating algorithm is already known, which seems to beg the question." I don't agree that it begs any question. Again using the computer's random numbers as an example, we KNOW that they're pseudorandom because we know that an algorithm exists, even if we don't know what the algorithm is.

As I replied to Alan, we can reach certain larger-scale conclusions by treating the series as random, and we have done so through nearly the entire twentieth century. I'm suggesting that such series that we treated as random might instead be causal (one number deriving from the previous) if the relationship produced a mathematically chaotic output that we could not distinguish from random. This causality is present even in the chaotic portion of the range of the difference function, even where the equivalent continuous function still produces stable results.

I'm merely speculating whether solution of the differential equations of quantum physics as continuous functions hides the chaotic or pseudorandom behavior that might arrive from treating them instead as a discreet difference equation.

(Sorry, that might have been a telling slip-up. Of course I meant a discrete function, not a discreet one. Nothing indiscreet about the alternative.)

Hi, Paul

"Again using the computer's random numbers as an example, we KNOW that they're pseudorandom because we know that an algorithm exists, even if we don't know what the algorithm is."

Yes, but my question is whether can we assume the analogy of this example applies in the case of the natural world?

Dan

Hello again Dan,

It was an analogy to illustrate the difference between random and pseudorandom numbers or intervals.

I didn't intend it to be a metaphor for real events. Whether it is or not is not the subject of the essay; the point is that the chaotic part of the series is completely causal, yet produces a series that is pseudorandom. The implication is that, confronted with a series of events that APPEARS random, you cannot assume that there is no causality underlying the series. The apparent random behavior does not mean that there is no such detailed explanation at a lower level. In other words, one CAN imagine the existence of a difference equation at a level below that of elementary particles, one that would predict the exact time intervals between such events. This potential ability compares with the lack of evident success using differential equations of continuous variables.

The difference between random and pseudorandom is important only if you know the generating algorithm. With the chaotic part of the difference equation we do. But to an observer who is confronted only with the series, he can treat the events as random because he doesn't know the generating algorithm; if he did, his conclusions from his assumption of randomness would be only an approximation, although it might be a very good one.

Fredric Litt

Yes Fredric, I think I see where you are coming from with this statement "..after reading an obituary of Benoit Mandelbrot, it occurred to me that the emission events need not be truly random; they might instead be chaotic and we might not be able to discern the difference using space and time as continuous functions". It's a lack of understanding of the physical dynamical system which is paramount imo. Changing the subject slightly, I have a simple new idea in foundational physics which I would like your opinion on. It is the use of an Archimedes screw as a model for the graviton at the time of Newton. If he or one of his contemporaries had hit upon this simple idea then Einstein would never had thought of a spacetime continuum! We would have had an answer for dark energy as well! It's a lot to explain in one go and requires careful tactile consideration. It makes a lot of sense to have a mechanical particle as a force carrying entity imo. What do you think to this heretical idea?

Hi again, Fredric

I guess the question remains, for me, whether there is any such thing as a "truly random" event or process in nature. As you say, it may not be the issue you are addressing. I seem to believe that neither determinism nor indeterminism is a description applicable to the real world--only within mathematics, i.e. as decidability or computability. But I am open to arguments to the contrary...

Thanks,

Dan

Hi Dan, I guess we are using 'determinism' and 'causality' synonomously. In general, I believe that we can keep looking for causes at lower and lower levels until we are eventually stymied. At this point we need to explore a lower-level system -- molecules to atoms to atomic particles to quarks, etc. Philosophically this might go on indefinitely; maybe in fact, it MUST due to Godel's theorem implying that you can't prove a system by limiting yourself to properties within the system (one way Godel's theorem is explained to us mortals).

Similarly, in heat radiation theory, Planck succeeded where Boltzmann failed, by considering radiation incremental (ie: quanta) rather than continuous (waves). Photons became irreducible units of light just as atoms became irreducible units of chemical elements. These examples analogize discreetness of properties such as time and length. That's the background underlying my search for a function that would be identical to common (continuous) time and space, while still underlying a discrete or digital reality, that might predict a different behavior at really small units of time or space.

Fred

yeah, maybe Goedel has the last word...

thanks again, Fred, and best wishes,

Dan

Dear Fredric,

I wanted to say hello and let you know I enjoyed your essay discussing the randomness of quantum events. I have been analyzing a model for continuous internal motion and have been thinking about randomness that would naturally arise. Even though you are suggesting discrete behavior, I like how you logically analyze the problem, and wanted to say thanks for an interesting and thought provoking essay.

Kind regards, Russell Jurgensen