Is our universe a mix of both continuous and discrete elements?

[Civilized]

According to Chaitin it does.

His work is considered to be controversial, it is of a philosophical nature, and it is not widely accepted by mathematicians. Fortunately we do not need to argue about the existence of uncountable sets, since the rationals are countable, and the set of analytic functions from Q to Q is countable, and everything we do with the continuum in physics could be translated to smooth rational functions. (Discrete Or Uncountable) is a false dichotomy, the rationals are a counterexample.

and as far as we know the physical world is computable.

I disagree, you keep coming back to the same circular assumption. To the contrary, as far as we know the universe is best described by quantum field theory. Please tell me what theory captures the world more accurately than QFT and suggests that the "physical world is computable."

 

[turbo]

We may have gotten to the point at which the determination of whether space is discontinuous on small scales is testable. A prediction of LQG (or a variant of LQG) is that very high-energy gamma rays will interact with the fine structure of the space through which it propagates, and will be slowed more than lower-energy gamma rays. It is entirely possible that such an effect (hinted at by a few observations so far) might be produced by variables at the source of the GRB, but if it can be shown that such delays are proportional to the redshift of the source, LQG and discretization of space will have gained a lot of traction. If delays are proportional to redshift, then the idea that all the gamma rays are emitted at the same time, and that dispersion effects cause the delays gains credence. Fermi hasn't been observing all that long - lots more operational time and observations should shed some light on this question.

 

[apeiron]

Again, either/or is the conventional bind people get themselves into. That would be the false dichotomy. If we listen to what dichotomies really tell us, we would instead be interested in understanding how opposing extreme can both be true, both be fundamental, both offer a vantage point on the production of realities.

Take for example category theory which breaks the mathematical world into the dichotomy of objects and morphisms (the discrete object/the continuous morphism). After many years floundering around with set theory and null sets, maths gave up either/or to embrace a fundamental duality (in interaction).

I could also mention the dualities that have emerged as central to string theory - Civilised may appreciate the dichotomy inherent there. And the switch in view from the local/discrete to the global/continuous.

You cannot turn in any direction in maths, science or philosophy without bumping into lurking dualism or dichotomies. Which is why it is really important to understand there are other choices of reaction apart from getting stuck in the eternal rut of either/or.

And I may as well say, from the view of modern epistemology, another way this discussion is getting bogged down is the confusion between models and simulations. Count Ibis is mostly thinking about simulations.

Simulation is about recreating what is "out there". So you have to represent both the general AND the particular. All the information contained in a system must be recreated.

But models are instead about the extraction of generals. Particulars are discarded in the creation of the models.

So for example any physical equation standing for a natural law. The general relationship is represented by something like E=mc^2. Informationally this is so compact it can go on a t-shirt. Then to use the model, you plug in measurements. You plug in local particulars like the energy or mass that is relevant to the prediction in hand.

Models involve a reduction of represented information. And the more that can be discarded, the better the model.

Simulation goes the other way. In principle, you would want to represent every last bit of information and so fully recreate some actual (particular) system. There then arises a practical cut-off question because our information representing resources are usually discrete (digital) and so we face an infinite trajectory to arrive at a representation of something (infintesimally close even) to the continuous (or analog).

Following this, perhaps we ought to ask whether the number line is a model or a simulation?

 

[Count Iblis]

You go from finite to countable in the infinite volume limit if you have a cut-off at some momentum: The number of physical states of a system contained in a finite volume is finite.

QFT is computable, in the sense that you can simulate it on a computer to any desired degree of accuracy.

 

[Naty1]

I was checking on another subject and came across the following: Light is quantized; Agree or disagree? Anyone care to suggest what it means?
Wikipedia, http://en.wikipedia.org/wiki/Photons#Early_objections, last paragraph

A few physicists persisted[39] in developing semiclassical models in which electromagnetic radiation is not quantized, but matter appears to obey the laws of quantum mechanics. Although the evidence for photons from chemical and physical experiments was overwhelming by the 1970s, this evidence could not be considered as absolutely definitive; since it relied on the interaction of light with matter, a sufficiently complicated theory of matter could in principle account for the evidence. Nevertheless, all semiclassical theories were refuted definitively in the 1970s and 1980s by photon-correlation experiments.[Notes 2] Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven.

 

[Hurkyl]

The continuum hypothesis is the claim that the cardinality of the real numbers is equal to the cardanality of the power set (set of all subsets of) the natural numbers.

No, that is known for sure. The continuum hypothesis is that there are no cardinalities between that of the naturals and that of the reals.

 

[Hurkyl]

Take for example category theory which breaks the mathematical world into the dichotomy of objects and morphisms (the discrete object/the continuous morphism). After many years floundering around with set theory and null sets, maths gave up either/or to embrace a fundamental duality (in interaction).

I can't make heads nor tails of what you're trying to say here.

 

[apeiron]

I can't make heads nor tails of what you're trying to say here.

Just making the point that everywhere you turn when people are trying to make deep distinctions, you find dichotomies emerging. As in category theory. Definitions by mutal exclusion, followed by the interaction of what has been created.

You could argue that object/morphism is not exactly a discrete/continuous distinction. But it is close in spirit. And then discrete/continuous is not itself the most fundamental dichotomy. Local/global would be a "deeper" level of generalisation.

If every field depends on dichotomies - science, metaphysics, maths - why are people not more familiar with the logical principles involved here? Why this mania for either/or when the disciplines themselves rely on "both"?

 

[Gil Grissom]

What happened to cellular automata models of physics?

 

[IsotropicSpinManifol]

I think these questions are not necessarily opposed. We could for instance live in a "pythagorean universe", a universe in which the simplest objects are neither discrete nor continuous, but "pythagorean".

I wrote a paper in philosophy that addressed this question recently.

 

[mbacon22]

I think that David Deutsch generally has the right approach to this question. In short, he thinks that “within each universe all observable quantities are discrete, but the multiverse as a whole is a continuum. When the equations of quantum theory describe a continuous but not-directly-observable transition between two values of a discrete quantity, what they are telling us is that the transition does not take place entirely within one universe. So perhaps the price of continuous motion is not an infinity of consecutive actions, but an infinity of concurrent actions taking place across the multiverse.”