Discrete/Continuous: Two sides of the same coin?

[Kherubin]

To quote David Bohm,

However, on closer investigation it would appear that any theory of the continuous nature of matter can in fact be based upon an opposing theory involving discrete matter that is so fine as to have never manifested its nature up to the present time. Conversely, any theory of the discontinuous structure of matter can be explained as arising through the localization and concentration of a continuous background. -- Science, Order & Creativity, pp. 72-3

Does anyone have any evidence, thoughts or opinions that lend credence to this statement (or contradict it)?

Are discrete/continuous phenomena, in actuality, two sides of the same coin?

Thanks,
Kherubin

 

[SpectraCat]

To quote David Bohm,

However, on closer investigation it would appear that any theory of the continuous nature of matter can in fact be based upon an opposing theory involving discrete matter that is so fine as to have never manifested its nature up to the present time. Conversely, any theory of the discontinuous structure of matter can be explained as arising through the localization and concentration of a continuous background. -- Science, Order & Creativity, pp. 72-3

Does anyone have any evidence, thoughts or opinions that lend credence to this statement (or contradict it)?

Are discrete/continuous phenomena, in actuality, two sides of the same coin?

Thanks,
Kherubin

That quote doesn't actually say that they are two sides of the same coin. What it says is that, if you take a discrete theory and make the step-sizes small enough, you will arrive at something that is fundamentally indistinguishable from a continuous theory. To me, this seems particularly true if you take the HUP into account, since any real measurement of a physical property will reflect the finite uncertainty of the underlying quantum description (i.e. the wavefunction).

This idea crops up fairly frequently in physics ... one version would be the Bohr correspondence principle. Another would be the evolution of the continuous band structure of bulk metals from the discrete energy levels of small metal clusters.

 

[Kherubin]

Thank you for your reply.

Isn't the point that Bohm is making that if you obsevre a 'continuous' description of reality with sufficient fidelity, you arrive at a 'disecrete' model, and if you go on to investigate this undestanding in sufficient detail you once again are faced with a 'continuous' account, potentially ad infinitum?

Thanks,
Kherubin

 

[SpectraCat]

Thank you for your reply.

Isn't the point that Bohm is making that if you obsevre a 'continuous' description of reality with sufficient fidelity, you arrive at a 'disecrete' model, and if you go on to investigate this undestanding in sufficient detail you once again are faced with a 'continuous' account, potentially ad infinitum?

Thanks,
Kherubin

Hmmm ... to be honest, I didn't really focus on the second part of the quotation, and I do see your point. I guess I would say that the first case resonates better with my own understanding of Q.M. ... i.e. that the universe is fundamentally fine-grained at the quantum level. I guess the second point seems more theoretical to me .. i.e. it hypothesizes that there are some "ideal" continua representing position and momentum, which become quantized at very small scales according to QED.

But I now see why you said "two sides of the same coin", and I agree that it what the quotation seems to be saying. However, I would also say that the first part of the statement is more reflective of the experimental reality of the universe, while the second is more theoretical ... but that my just be my own personal bias showing.

 

[alxm]

I haven't read the quote in context, but given the author, I think he's making another point here: Namely that physics ultimately can't determine questions that are fundamentally metaphysical. An example of that is that whether things are discrete vs continuous.

You can take your pick from the philosophies of science, but a physical theory must in some way describe something which is observable. A theory explaining that theory, is a metaphysical theory if it doesn't affect anything that's observable. So he's pointing out that any physical theory claiming matter is continuous may in turn be explained by a metaphysical theory where matter is discrete in some unobservable way and so only apparently continuous. And vice-versa.

Bohm was hardly a stranger to that kind of thing, since he did do just that when he showed how the "apparent" indeterminism of quantum mechanics could arise from a deterministic model - Bohmian mechanics. (Which is generally considered an 'interpretation' of QM rather than a physical theory, for the reasons stated)

There are objections on actual physical grounds against Bohmian mechanics, although this board hardly needs another thread on that topic, but I think Bohm's point still stands: Physics can't answer questions that are ultimately metaphysical, and that if you embrace the prevailing physical theory as 'reality', you should be aware you're taking a philosophical/ontological position and not a scientific one. (although scientifically-minded folks as a rule do tend to eschew metaphysics, myself included)

 

[maverick_starstrider]

To quote David Bohm,

However, on closer investigation it would appear that any theory of the continuous nature of matter can in fact be based upon an opposing theory involving discrete matter that is so fine as to have never manifested its nature up to the present time. Conversely, any theory of the discontinuous structure of matter can be explained as arising through the localization and concentration of a continuous background. -- Science, Order & Creativity, pp. 72-3

Does anyone have any evidence, thoughts or opinions that lend credence to this statement (or contradict it)?

Are discrete/continuous phenomena, in actuality, two sides of the same coin?

Thanks,
Kherubin

Well David Bohm wasn't so active during the rise of the computer. As a computationalist I can say that, metaphysics aside, not every equation can be quantized, even in theory, and produce identical results to the continuous case. Simply google "numerical stability" and I'm sure you'll find thousands of discussions about how truncating or discretizing various partial differential equations will lead to them exploding with instability. Ultimately this is an artifact of non-linearity and chaos theory. Take any of a whole host of mathematical functions and impose hard limits on the precision of initial conditions (and time steps forward) and you will find that its behavior will wildly diverge from the continuous case. Thus, even if only through that lens alone, I believe we can discount the possibility that all relations in nature can be discretized (no matter how small the discrete packet) without producing drastically different results.