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The idea that there is some space between any two spaces is somewhat related to continuity, but the mathematical term for this is "dense". The rationals are dense, as there is some rational between any two, but they are not continuous.

The topological definition of a continuous function is that for any open set in the co-domain, its inverse image is open in the domain. This is equivalent to the epsilon-delta definition, but can we say that any non-mathematical construct is truly continuous? What exactly would that mean?

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  • We have the geometric intuition of the continuum : a straight line. We have the intuition of the continuity of the "time-line" and we have the 19th century arithmetization of the continuum that showed - as you said - that density of rationals were not enough to characterize arithmetically the continuum.
    – Mauro ALLEGRANZA
    Apr 16 '18 at 12:38
  • See the post what does Weyl mean by this remark for Weyl's quote : "let us stick to time as the most fundamental continuum."
    – Mauro ALLEGRANZA
    Apr 16 '18 at 12:44
  • You could just as well ask whether anything is truly discrete. Are the boundaries between objects truly discontinuous thresholds or just very steep continuous changes?
    – Mitch
    Apr 16 '18 at 13:27
  • We can not apply mathematical predicates to non-mathematical constructs, and mathematical models of reality that involve "true continuity" are equivalent to arbitrary precision to models that are discrete. In other words, the question about "real existence" of mathematical continuity is vacuous.
    – Conifold
    Apr 16 '18 at 20:10
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A beam of light may be considered continuous from source to destination. The reason is simple: any discontinuity would cause it not to touch the destination. The beam only exists if it is continuous at the measured point. This even complies with the math function continuity rules.

I'd also say a basic particle is a continuous construct. Otherwise, the particle would lose integrity.

Note that math exists to quantify things, not the other way around (as some scientist today try to imply).

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  • I'm not a physics major, but a beam of light is really made up of photons, isn't it? So maybe it just appears continuous.
    – the Diog
    Apr 16 '18 at 12:39
  • As an independent manifestation it is a continuous construct. Everything is made of something else, that does not mean that a steel bar has no integrity just because atoms in it actually do not physically touch each other.
    – Overmind
    Apr 16 '18 at 12:49
  • Then can I say that the bristles on a toothbrush are continuous, even though there's space between them, because they are part of the same construct--the toothbrush?
    – the Diog
    Apr 16 '18 at 12:52
  • You can for a purpose point of view (it will have a continuous interaction with the tooth). If you are referring to the fact that none are missing between the front point of the brush and the back of it's head, nothing wrong here, although the example is exaggerated as they are quite far apart from each other. If we were to extrapolate this downwards to particle level, then we can say that the most continuous thing that exists is the field forming the basic particles. But one day we may discover that it also have sub-components forming it.
    – Overmind
    Apr 16 '18 at 12:59
  • +1 Your statement about the particle losing integrity got me thinking. My answer is based on yours, but slightly different.
    – Frank Hubeny
    Apr 16 '18 at 15:53
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This is a partial answer with the goal of clarifying the question.

The idea of continuity is not the same as the idea of a continuum. See Peter Smith’s answer to iblue’s question, "Why are the rational numbers not continuous".

As Smith mentions one can have a continuous function such as the identity function going from a domain containing only rationals to a co-domain also only containing rationals. However, the rationals are not a continuum because the set of rationals do not contain all their limit points. For example, we can construct a sequence of rational numbers approaching an irrational number that would not be in the set of rationals as closely as we please. Since the irrational number is not in the set of rationals, the rationals are not a continuum. They do not contain all their limit points or targets of sequences.

To see how this might work in the real world consider Zeno paradoxes. Although Zeno’s arrow gets closer to the target at each point of measurement, will the arrow actually reach the target? If the travelling arrow contained all its limit points then the target event should occur. Otherwise the moving arrow would be like the rational numbers lacking a limit point in the irrationals. For a mathematical presentation of this see Nathan Pflueger’s lecture “Convergence of series”.

Overmind’s answer to the question contains this idea of limit point. When the beam of light is measured it reaches the target, that is, the quantum system of the beam collapses. This collapse is certainly part of the real world since it is the only part of the beam we actually see.

Overmind makes the following philosophical assumption that is at the heart of the question:

I'd also say a basic particle is a continuous construct. Otherwise, the particle would lose integrity.

The beam would be continuous, or rather a continuum, if that statement were true, but is it?

Unlike Zeno’s arrow which we can watch going to the target, we can’t see the beam go to the measuring device. Is the reality surrounding the measurement of the beam also a “basic particle”? If it isn’t then the target photon and whatever that beam was prior to acting like a photon at the target are not the same. If the beam does not contain its limit point as a set of reality like itself, then it would be like the rational numbers lacking an irrational limit point and the beam would not be continuous.

So to answer the question, Is anything truly continuous?, would require answering the question about what happens during a quantum collapse.

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  • The photon itself is made out 2 basic particles (today's physics would call them quantas or something like that), one with the positive pole on the outside, one with the negative pole on the outside (like the electron) that interact with each other at the frequency that actually gives us the wavelength. What actually happens when you use solar cells is that the photon splits, the negative pole on the outside becomes the actual electron useful in our circuit and the other one loses integrity and dissipates as heat.
    – Overmind
    Apr 17 '18 at 5:02
  • @Overmind A photon, whatever its characteristics, is what is measured. It might not be the "wave" or whatever it is prior to measurement. The reason there is doubt is because of the indeterminacy at measurement. For continuity, the two realities (wave and particle) have to be the same. If one looks at the measurement as the limit point whether there is continuity in the real world depends on whether this reality is the same both prior to measurement and at the collapse. My suspicion is that it is not the same and hence it is not continuous. Your answer does point the OP in the right direction.
    – Frank Hubeny
    Apr 17 '18 at 13:59
  • The photon is a particle. Its wavelength is actually it's pulse frequency (how fast it's 2 parts inter-change).
    – Overmind
    Apr 18 '18 at 12:59

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