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In mathematics, continuity is a core concept, especially in calculus, where a function is considered continuous if it lacks discontinuities such as jumps or gaps. This mirrors the completeness attribute of real numbers, asserting that there is no smallest real number immediately succeeding any given number, like 2. For example, although 2.0000001 might appear minuscule, 2.0000000000001 is tinier, and this pursuit of ever-smaller numbers greater than 2 is unending. Consequently, there is no minimal real number exceeding 2, encapsulating continuity's essence.

Grasping the mathematical definition of continuity raises the question of its tangible existence within the universe's measurable dimensions, such as time and space. Does the cosmos have a minimum measurable time interval?

Consider the case of 2 seconds. If time were continuous, moments would exist between 2 and 2.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 seconds. Thus, before any moment after 2 seconds, there would be a smaller one. Yet, this idea faces a logical conundrum: if continuity were factual, then before reaching 2.1 seconds, 2.01 seconds must occur, followed by 2.001 seconds, and so forth. This suggests an infinite sequence of moments, implying that not even a single second could pass, which challenges the concept of time. Is this reasoning valid? Does continuity not exist in the universe?

If this argument holds, it implies that a continuous cosmos is logically implausible.

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    Your argument is similar to Zeno’s Paradoxes: https://iep.utm.edu/zenos-paradoxes/#:~:text=Zeno's%20paradoxes%20of%20motion%20are,on%20all%20kinds%20of%20plurality. – PW_246 Mar 11 '24 at 13:52
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    You are talking about infinite divisibility, not continuity, and your argument is a variant of Zeno's Dichotomy paradox. Before a whole, its half must be passed and so the passing never starts. But there is no logical problem with infinitely many "moments" passing in finite time, infinite series can add up to finite sums after all, so this argument is invalid. Infinite division of physical time/space is surely impossible in practice, but not because of any "logic". – Conifold Mar 11 '24 at 13:56
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    Still, your explanation of completeness is not quite right, since there is no smallest rational number greater than 2 either. What the completeness of the Reals is saying is that for any two subsets A,B of the Reals where a<b for every a∈A and b∈B, then there is some c for which a<c and c<b for every a∈A and b∈B. What this is basically saying is that you can’t “exclude” irrational numbers and form a complete line, since an irrational real number, e.g. e lies exactly between a subset of rationals all of which are strictly lesser than e and a subset of greater rationals. – PW_246 Mar 11 '24 at 14:00
  • @PW_246 Yes, because I explained it intuitively and simply, which is not an accurate explanation. –  Mar 11 '24 at 14:47
  • Please, be aware that - as QM has showed (but already Newtonina gravitation law, with tha "action at distance") - "logically implausibility" (whatever it means) does not mean false. – Mauro ALLEGRANZA Mar 12 '24 at 08:02
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    @Conifold But if you take the specific infinite series of "Before a whole, its half must be passed", which is 0.5 + 0.25 + 0.125 + ... then 1 will be its limit, not its sum. Which means that it will never pass. So, the fact that infinite series can add up to finite sums, is irrelevant to this specific argument. – Sam Mar 12 '24 at 09:11
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    "Will be its limit, not its sum. Which means that it will never pass" is a non-sequitur. Whether you call it "limit" or "sum" has no relation to "passing". Logic does not even prevent infinitely many actions from concluding in finite time if they take time amounts from your series, for example, see supertasks. And generally, logic does not contain any information about time and space that you need to derive anything at all about them. – Conifold Mar 12 '24 at 10:20
  • @PW_246 It's definitely Zeno's Paradox, but I like this formulation: it makes it easier to show why it isn't really a paradox. Before 1 second can pass, 0.5 seconds must pass. It takes 0.5 seconds for 0.5 seconds to pass. Now we have 0.5 seconds left, which takes another 0.5 seconds. For a grand total of 1 second. And dividing it further just gets us 4 * 1/4 seconds = 1 second, or more generally, dividing it into any k equal-sized chunks takes 1/k seconds each, for k * 1/k = 1 second. We don't need to explicitly take a limit; we just observe that dividing it further never changes the total. – Ray Mar 12 '24 at 15:57
  • ...that argument only holds for rational numbers, of course, but by the time someone understands that rational numbers and real numbers don't have the same cardinality, they should be able to handle a proof involving limits. – Ray Mar 12 '24 at 16:04
  • An epistemological constructivist might add that the "real world" is governed by how we construct a description of it to some extent, and that continuity is perfectly at home in visualization, for instance, where we see things move continuously. It is the continuity of the visual system that makes the "real world" continuous when the description is visual therefore. A verbal description of vision, however, is discrete because language functions in discrete units. Thus, the world is both continuous and discontinuous depending on the mode of description. – J D Mar 18 '24 at 20:05

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This is actually an open question in physics - it's currently unclear if physics requires any kind of spatial infinite continuum, or if all of spacetime may be discrete. Physicists take both possibilities seriously, and some prefer the idea of discrete for various reasons, and others prefer to think that it is a genuine continuum, and many are just uncertain and undecided and comfortable saying "I don't know".

If you have a strong preference one way or the other, keep that preference, and work on articulating it, but at this point it's wise to hedge that preference with "... but I'm not certain this is how reality must work".

TKoL
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    Thank you, but if this is an open question in physics, then what exactly is Planck time? Doesn’t that answer this question? –  Mar 11 '24 at 14:26
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    @Amin The Planck time is the smallest duration of time that can be meaningfully measured, and doesn't imply discreteness. It does not imply that no smaller quantity of time can exist or that all durations are integer multiples of Planck units. It just means we can't tell the difference between a duration of 0.1 Planck-times and one of 0.5 Planck-times, not that the duration must be either 0 or 1 Planck-time. – Nuclear Hoagie Mar 11 '24 at 14:36
  • @Amin it's potentially the case that time in quantised while other things, like spatial distance, are not. – TKoL Mar 11 '24 at 15:01
  • @TKoL In the same way, for spatial distances, we have Planck length. –  Mar 11 '24 at 15:06
  • @Amin: But the Planck unit is definitely not the smallest unit of everything (e.g. the Planck mass is about 10^22 times larger than the mass of an electron) – Micah Mar 12 '24 at 16:33
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    @Amin: If every distance must be an integer multiple of Planck length, and every time must be an integer multiple of Planck time, then either every speed must be an integer multiple of the Planck speed (clearly false, since the Planck speed is the speed of light), or objects moving slower than that must be alternating between moving at the Planck speed and being at rest (probably false, since any motion at the speed of light is problematic for massive objects, no matter how brief). – Kevin Mar 12 '24 at 19:05
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"Infinitesimally small" is an abstraction. As is "infinity". As is all of mathematics. Abstractions help us understand things as we pick out interesting features of reality without having to talk about the messiness of the real world.

Quantum mechanics taught us that many things come in discrete packages.

Planck time is 5.391 10^-44 s. So one would expect that things get messy for shorter time scales. Planck length is 1.61×10^−35 m. That is 10^20 smaller then the size of a proton. No one really knows about reality below those scales. There is e.g. Wolfram Physics which explores the idea that reality is a graph of discrete nodes with sizes that could be in the scale of 10^-100m. This could be the "real" world. We do not know, at least yet.

Nonetheless all these abstractions still make sense as a means of understanding the world. When you e.g. talk about a "cat" you use an abstraction. You have a picture in your had and there is a more or less agreed upon definition that is shared by others, so using the abstraction allows you to communicate.

Sometimes it seems that the abstractions of mathematics become to abstract to be useful. E.g.: for centuries "number theory" was not that much useful up until the point where it is now the basis of our understanding for cryptography and where it holds the internet together..

mond
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Where is the contradiction? Like you could take any arbitrary length n and divide it into x fractions of 1/x length. Now in the limit of the infinite you'd multiply infinity with 0 and get n. But if you stop just short of infinity as a concept and just take any existing x between 1 and infinity it still works.

Also in the real world we'd have a completely different problem and that is that we can't even measure things that accurately. Like a first approach to measuring stuff is by comparing things to other things. Now if they are large enough the inaccuracy of the ruler isn't relevant to the thing that is measured (as it's orders of magnitude smaller). Similarly the relative motion of the 2 things is not relevant yet. But if you go ever smaller, you run out of things to compare to each other and all these inaccuracies become crucially relevant. Often you proceed by measuring indirectly via making use of physical properties, but that kinda introduces hard limits as to how small our measurements can get. Stuff like Heisenberg's uncertainty relation that argues that the product of the uncertainty in space and momentum is a constant. So if you want to measure the location of a particle you do no longer know how fast it is going and how massive it was and if you want its momentum you won't know where it is.

Or you know Planck units which mark the smallest units of space and time which are physically useful, because there's currently no physical effect that is relevant on time scales smaller than that. So there's simply no "ruler" small enough to measure things below that.

So even if time would be continuous and would not be this weird thing that is actually relative to the observer, but an absolute property. Even then we would run out of clocks fast enough to measure it, meaning for all intents and purposes you'd measure it in distinct Planck time ticks.

On top of that we have already found other things that are quantitized like charge which naturally occurs not continuous but in multiples of the charge of the electron, so while a current (a stream of electrons) appears to be continuous, you could still find single "droplets" of charge once you reduce the stream to a dripping faucet.

Though again nothing of that is relevant to whether you can divide a number ever further. There's nothing stopping you from doing that, until you run into the point where you can no longer measure it. Now whether that change below that measuring accuracy is continuous or discrete... we might never know.

haxor789
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  1. A principle ascribed to several physicists and philosophers, e.g., to Newton and Leibniz says

    Natura non facit saltus (nature does not jump)

    Leibniz calls the statement his principle of continuity.

    Nevertheless physicists of the 20th century had to base the theory of quantum theory on the existence of smallest discrete quantities for energy, time, and space. While the Planck time and Planck length are still too far away from the domain of our measurements, the Planck constant ‘h’ of energy is well-established. It is considered a constant of nature, quantifying the smallest energy. Its value is independent from our physical theories.

  2. It is an ongoing discussion in physics whether mathematical physics should rely on calculus and its concept of differentiability, or on a discrete formulation with lattice points as a mathematical basis. But there is no a-priori reason why the concept of continuity and why calculus does not apply to formulate the laws of science. Of course one has always to keep apart nature itself and our description of nature.

Aside: The mathematical concept of continuity applies to functions of real numbers. The precise definition is the so called "epsilon-delta" definition, e.g., see definition of continuity. Calculus requires that the functions under consideration are even differentiable, which is stronger than being continuous. Continutiy is not a property of numbers, but of functions.

Jo Wehler
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We don't know whether time is continuous or proceeds in tiny jumps. The shortest time measured, according to Wikipedia, is about a quarter of an attosecond, or 0.000000000000000000247 seconds, give or take some zeros I might have missed. Some theories being investigated by physicists consider spacetime to be quantised; apparently, there are some good reasons to expect that time should become granular once you get down to intervals of around 0.0000000000000000000000000000000000000000000005 seconds (again give or take a few zeros), which is known as the Planck time, although I don't understand the reasons sufficiently to explain them (the result of being as thick as two short Plancks).

Your concern about a second never passing, owing to an infinite number of intervening intervals, is misplaced. The sum of an infinite number of infinitely small fractions of a second is a second, so no problem there.

Marco Ocram
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I think it's a bit like trying to cut a cake into the smallest possible piece. You can keep cutting it in half, and then in half again, and so on. But is there a point where you can't cut it any further? That's the big question.

In physics, there's this thing called the Planck time, which is super tiny, like 10^-43 seconds tiny. It's named after Max Planck, a German physicist. This is considered the smallest meaningful unit of time. But here's the kicker, it doesn't mean that time is necessarily broken up into these tiny, discrete chunks. It's more like our current understanding of physics can't really deal with anything smaller.

Your argument about an infinite number of moments in time reminds me of Zeno's paradoxes. These are ancient philosophical problems that deal with the idea of infinity. One of them, called Zeno's Dichotomy paradox, is similar to what you're saying. It suggests that you can't ever get anywhere because you always have to get halfway there first, and then halfway again, and so on. But we know that's not how things work in the real world, right? We can walk from one side of the room to the other without getting stuck in an infinite loop of half-steps.

So, in my view, while it's true that the mathematical concept of continuity suggests an infinite number of points between any two points, it doesn't necessarily mean that the universe isn't continuous. It might just be that our understanding of the universe and the tools we use to describe it, like mathematics and physics, have their limits. And that's okay. It's part of what makes exploring these questions so fascinating.