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The Riemann zeta function is a continuous function which encodes the properties of the primes; string theory, a proposed theory of particles, considers continuous objects; through QM discreetness of energy levels etc. emerge from a continuous wave equation. Spacetime I think in GR is continous, but then again, any theory of everything which humans can find, or any theorem of mathematics, must be formulated using a finite set of discrete symbols, which again means any observable can be computed by a discrete Turing machine.

What arguments are there for whether discreteness or continuity (if either) is the emergent property?

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    Suggest that this may be easier to answer if you're willing to elaborate what is it about the reasoning that: "any theory of everything which humans can find, or any theorem of mathematics, must be formulated using a finite set of discrete symbols, which again means any observable can be computed by a discrete Turing machine." supports the idea that discreetness or continuity 'is the emergent property'? – Dr Sister Mar 24 '13 at 3:33
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    Im not looking for easy answers, because there probably is none. But someone has surely considered this question to greater depth then me. Most real numbers are uncomputable, hence if hte universe is continuous some parameters must take on these values. But if we can simulate the universe on a computer, I have trouble imagining it to be continuous. – 4real Mar 24 '13 at 4:10
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    +1. Nice question. Bear in mind that if we only care about finite approximation (as how can we not be), the uncomputability of almost all real numbers is less of a concern. Bearing in mind that any systematic observation may be construed as a computation, it is not clear how uncomputable numbers could actually get in our way, unless it were to actually render physics unintelligible by giving rise to fundamental unreliability of experimental outcomes. (Hmm.) – Niel de Beaudrap Mar 24 '13 at 10:29
  • Ok, it probably boils down to whether the universe is continuous or discrete, so string theory vs quantum gravity, in 100 years we have the answer. But some mathematical arguments can be made for either side. – 4real Mar 27 '13 at 22:45
  • And how do natural numbers come into this? Do they emerge from introducing algebraic structures such as fields or are they more deeply woven into the fundamentals? They apparently introduce (discrete) structure on a continuum. Is there a meaningful subset of mathematics which gets by without referring to them (as cardinality of sets, say)? – Drux Oct 2 '13 at 6:47
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You seem to be appealing to some sort of Platonic ideal of what is really there. There are simply two equivalent ways to define things: starting from continuous functions and making step-like functions arbitrarily steep to get discreteness; or starting from discrete sets and allowing arbitrarily many states to approximate an continuum.

As you note, the physical world seems to contain things that in some sense are seem more intrinsically one or the other (quantized energy levels, continuum of momentums). Thus, using one model is often more natural than the other, but that doesn't mean that it's necessarily more fundamental. It's not clear that asking which is more fundamental--once we recognize that there is an equivalence between the two models and that certain fundamental phenomena can fall into either category more naturally--is even a sensible question. "Both", "models are a property of what's in your head, not necessarily reality", "discrete, because we have better theorems there" and various other things seem equally reasonable.

  • By the way, downvotes are more useful if you explain what's wrong. Knowing why something is wrong is much more informative than simply knowing that it is wrong. – Rex Kerr Mar 24 '13 at 1:14
  • Sure, there may be an equivalence class on descriptions, but we live in only 1 universe, it must work 1 way only. Its not always easy to know which is the more fundamental one, hence my question. Whether space-time is discrete or not is a question of physics. – 4real Mar 24 '13 at 4:00
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    @4real - What do you mean by "work one way only"? Between QM and Turing computability we've pretty much proved that we couldn't know the difference; and when something is unknowable in principle it begs the question of what "is" means. – Rex Kerr Mar 24 '13 at 14:30
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    lol, you cant test the matrix. And tell me, what is the difference between "to test", and "to test in principle"? – 4real Mar 24 '13 at 23:13
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    @4real - I will grant that if there is a Matrix then the analog/digital question may make sense (e.g. we are implemented on a digital machine). In the absence of information about things we are pretty sure we cannot have information about, however, the two views are equivalent to us, and it's not clear that the two views are exhaustive, so we can't rule out "something else" either (e.g. maybe operations on transfinite sets will do the trick also). The issue is not whether one can invent a scenario where it's a sensible question, but whether it is with the simplest interpretation. – Rex Kerr Mar 24 '13 at 23:20
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Hidden discreteness may yield obvious discreteness

I would describe most of your examples as demonstrating discreteness at a higher level, arising from discreteness at a lower level.

  • The Riemann zeta function is a continuous function which encodes the properties of the primes — but it is defined as an analytic continuation of an infinite series summing over the integers; the relationship between the Riemann Zeta and the primes may be seen as arising due to the fact that for real input values for which the series is convergent, the Riemann Zeta describes something of a probability distribution over the integers in terms of the probability that it is divisible by any particular prime power, via the Fundamental Theorem of Arithmetic.

  • String theory, a proposed theory of particles, considers continuous objects — which have a finite length, such as closed loops, which define a length scale and determines the possible vibrational modes.

  • Through QM discreetness of energy levels etc. emerge from a continuous wave equation — in particular because the electrons are attracted by protons; where in particular all electrons have the same charge, all protons have the same charge. We would still have discrete energy levels for any one particular atom if this were not the case, but it would be much harder for us to discover that it were so, because we discovered it through bulk properties of matter (e.g. experiments with heated gasses). Still more remarkably, protons and electrons both have the same magnitude of charge.

In each case, the discreteness which we find to be "emerging", may only be emerging because of discreteness which is present at another lower level. Simple continuous mathematical objects, such as the real number line or an exponential curve, don't betray any particular sign of discreteness at all; although our only tools to describe the concept of the real line is through discrete symbols — and by demonstrating its failure to conform to any naive model of discreteness.

Discreteness and continuousness as heuristic concepts

Perhaps your remark

any theory of everything which humans can find, or any theorem of mathematics, must be formulated using a finite set of discrete symbols

is the more interesting: is it the case that any theory that we are capable of formulating will involve discreteness, and if so is it because of our notational systems? I think that mathematics — both as an example of language, and as an extension of our interest in quantity and structure — involves discreteness at its deepest levels, possibly because discretization is fruitful for solving problems quickly and well enough in the several million years in which our ancestors lived.

But it is not clear that this means that we are bound to perceive discreteness where there is none. Indeed, it took us a long time to perceive discreteness where evidently it did exist, in the form of the atomic theory of matter; and in some branches of mathematics it is much easier to approximate a large discrete object (such as an infinite series) by a continuous one (in this case, an integral), than to evaluate the discrete object directly. Here we see that sometimes it is more fruitful a strategy to compute things as though they were a continuum.

Speech recognition is an interesting example: speech elements are discrete phonemes, which are a coarse-grained description of a time-evolving frequency spectrum representing pressure waves in air, which is actually composed of individual atoms, but which themselves are peaks in the continuous wave-function, and so forth. Our best descriptions of phenomena equivocate between whether it is better to model that phenomenon as a continuum or as discrete. Furthermore the choices are essentially made by us as a matter of computational (and conceptual) convenience.

In view of the lesson of quantum mechanics, we might ask whether there is an excluded middle of something which is in some way neither discrete nor a continuum, but has features of both. Perhaps such a notion would be more informative of the way that the world works and explains why we have a tendency to alternate between continua and quanta for describing the world. But more to the point, the very notions of discreteness and continuum are tools which we use to understand the world, and our notions of what the signs of 'continuousness' and 'discreteness' are suggest to us ways in which to try to understand the world, which will not necessarily provide us insight into its fundamental nature unless we have been lucky enough that those practical concepts relate to fundamental features of the world.

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I don't think without more specific context one can say which of the two is emergent. What is emergent may also be historically contingent too.

Discreteness and continuity are implicated in each other. Take the real line, it is made of discrete points. Now of course we say that the real line has a topology, the topology just given is the discrete topology - but in fact this doesn't reproduce the geometric properties of the real line as we understand them - we require instead the interval topology. Of course the points of the line remain there as the set-theortic substrate; but in so-called pointless topology we can get rid of this substrate and consider only the topology (technically this theory is called the theory of locales). It may appear that we have gotten ridden of discreteness; but this loss is only apparent. We have gotten ridden of discreteness when considered as points; but not as discreteness when considered as the elements of the topology. What we have done, in this perspective, is to expand the notion of discreteness so that it is not synonymous with the notion of point.

But when we consider the line in euclidean geometry (synthetic geometry) we see that it is not made of points, the line is a given. However where two lines cross we have a point marked on each line. This is the beginning of analytic or cartesian geometry. Here discreteness emerges out of the continuous. Or the analytic out of the synthetic.

Rather than say one is prior to the other, and the other emerges out of it; its better to say that either can emerge from the other; and neither are prior.

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