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So my question comes from two questions on this site one of them being my own, How small can we measure space? and Is the (surprising) applicability of mathematics to the physical world a brute fact or something that cries out for a (theistic) explanation? (the first comment specifically)

I see a central and connecting idea in both the question and the comment raised which brings me to the formulating of a slightly more appealing yet slightly difficult question that has always stumped me and taken my peace of mind. I didn't know how to properly state the question before but I think I kind of have a grasp on it but I think I still might improve it.

The question goes as follows, "Is any fundamental phenomena that we observe in the 'natural' world i.e., natural science in particular physics discrete or continuous? Phenomena like space, time, length, size, ... and how does that fit with the continuity of numbers? Is there some sort of disconnect between the math we use and our observations?"

I understand that much of physics has to usually quantize and approximate in order to have an understanding and progress and make calculations and observations more humane.

What feels weird for me is that I have a strong intuition that reality is completely continuous. For example when discussing "now" one cannot say this instance as in this second, or microsecond or picosecond or 0.00000001 picosecond which shows the ultimate continuity of time from intuition and the same argument might be applied to space, but is this intuition wrong; I understand this question also relates to the continuity of consciousness but does that directly affect our measurements and observation of such phenomena? Does it stem from the fact that I know the continuity of different categories of numbers such as rational, irrational and complex numbers?

But again, naturally looking at the development of numbers and mathematics, we can see that we started out by discreteness i.e., the development from natural numbers, integers, to continuous(rational and irrational) numbers and again I think that this was more of an intuitive revelation of numbers to humans

Infinity, continuity, discreteness, natural numbers and reality

Julius Hamilton
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How why e
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    Well, we just don't know. The wavefunction is continuous. Even particles like photons are really just excitations in a continuous field. But that doesn't mean reality works like that all the way down. I would lean towards yes, maybe 70:30 odds that it's continuous, on the basis that physics equations seem to be simpler that way, e.g. the Hamiltonian requires continuity. However, Stephen Wolfram has a very interesting theory of how it could be fundamentally discrete. – causative Mar 29 '24 at 09:08
  • @causative thank you for your honest insight. My follow up question goes, "Can we know (probably sometime in the future, is there some sort of inclination), if so how?" – How why e Mar 29 '24 at 09:18

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You have a strong intuition that reality is continuous because it is continuous at a scale that is relevant to your powers of perception, and your mind has evolved to make sense of your surroundings at that scale. The conscious human brain cannot differentiate intervals of time smaller than a few hundredths of a second, so there should be no surprise whatsoever that you take time to be continuous given that experiments have shown that it is continuous over intervals much smaller than that by many orders of magnitude. Some work in theoretical physics suggests that spacetime might be granular, but the scale of the granularity is probably thirty or more orders of magnitude smaller than anything your mind can detect, so for all practical purposes you can consider space and time to be continuous. Indeed, not only is the suspected granularity far too small for humans to detect, it is far too small to be directly measured by any current technology and there are good reasons to suppose we will never be capable of developing technology to observe it directly.

Marco Ocram
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  • I actually think my intuition for continuity stems from a different standpoint rather than one rooted in real-time experience, it is rather rooted in an exercise I do in my head as mentioned in my question where I tend to apply the continuous division of the number line to reality; an almost "illusory" kind of continuity that makes sense about time in relation to the number line. You might be right it might actually stem from the reason given in your answer but I dont really know – How why e Mar 29 '24 at 14:45
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According to modern physics, reality cannot be continuous

Other answers do a better job of addressing the philosophical and mental aspects of this, but I'd like to chime in with some physics.

I understand that much of physics has to usually quantize and approximate in order to have an understanding and progress and make calculations and observations more humane.

This isn't the whole story.

Modern physics doesn't just quantize and approximate in order to make computations. Physicists discovered that the equations we've come up with for how much energy is carried by light do not work for short wavelengths if we allow infinite subdivisions. In order for our equations (which accurately predict the energy for everything from the visible spectrum and below) to accurately predict the energy carried by ultraviolet light, we had to assume that a time range is a sum of finitely many non-infinitesmal moments (a classical sum), rather than the limit of that sum as the size of each moment goes to zero (an integral).

This discovery is known as the Ultraviolet Catastrophe and was the genesis of quantum physics - which is (loosely speaking) an examination of what it means if space, time, and energy are discrete (or "quantized," hence, quantum) rather than continuous. Quantum physics has had enough successful predictive power that we can say its central assumption of a discrete reality makes a better model than continuous reality does.

Tim C
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  • By far one of the best answers! – How why e Mar 29 '24 at 19:22
  • @Howwhye - I'm flattered, but from a philosophical standpoint, I don't even consider this a proper answer - it's a comment that was too long for the comment box. Even if one of the priors that led you to ask the question was incomplete/flawed, challenging that prior is not an answer to the question. (If this were asked on physics stackexchange, that would be another story) – Tim C Mar 29 '24 at 19:31
  • I know that this doesn't fully answer my question but it is an insight into developing a bigger picture that may lead me to a realization of an answer. I, personally believe in a holistic approach of inquiry and therefore a similar answer, so your feedback is highly important, also any answer helps me build on existing question so. thank you for your insight. As for physics SE, they would have dismissed my question entirely that is why I didn't bother. @TimC – How why e Mar 29 '24 at 19:45
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But again, naturally looking at the development of numbers and mathematics, we can see that we started out by discreteness i.e., the development from natural numbers, integers, to continuous(rational and irrational) numbers and again I think that this was more of an intuitive revelation of numbers to humans

The historical development is not so neat: the greeks had non-zero natural numbers and line segments, and their theory of proportions provided for positive rationals and some irrationals, but 0 and negatives took a while longer to appear and 'be accepted' (can you imagine?) in european mathematics. I sadly don't know about parallel developments in other places. From another front, some people - maybe including Poincaré, I'm not so sure - understood that we do hold a primitive/intuitive conception of 'geometric' continuums, as exemplified, for example, in euclidean geometry, that is prior to and independent from its 'codification' as a number system

What feels weird for me is that I have a strong intuition that reality is completely continuous.

All aspects of reality? I can understand your comment about time, but would you say the same thing applies to mass/matter?

It also seems a bit weird to talk about 'continuity' of the rationals, but anyway

I understand that much of physics has to usually quantize and approximate in order to have an understanding and progress and make calculations and observations more humane.

If anything, sometimes pretending some atomizable things are/behave as if they were continuous is what makes life easier

ac15
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  • Great answer, thank you for the insight. Yes even for mass after the decimal point it is definitely infinitely continuous; this actually pose a slightly even more interesting question that matter could be divided indivisibly just like how democritus found the atom and we then later found the sub-atomic partice but then infinitely go down until we meet gluons and fancy particles – How why e Mar 29 '24 at 15:07
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The answer to this ultimately depends on whether the way we categorize objects in the universe is mind dependent or mind independent. Certain categories of items seem “natural” to put into classes and categories, such as atoms.

If these categories at the lowest level of reality are mind independent, it could be argued that the world is made of discrete elements. If they are not, then all such divisions do not inherently exist in reality. There is just…reality as a whole, and divisions cease to exist except in the mind.

Baby_philosopher
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