Is there some sort of disconnect between the math we use and our "observed phenomena" in reality?

Question

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

Answer 1

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.

Answer 2

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.

Answer 3

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

Answer 4

It might be suggested that we need a "better"/clearer grasp of the continuous/discrete distinction itself, before we can make the kind of headway on this question that we'd like to. For a long time, the mass/count noun distinction was a qualitative way of interpreting the difference; as Conifold remarks:

[Those two distinctions] are more or less the same, and go back to the age-old distinction between counting and measurement that predates Cantorian dissociation of the continuum into an uncountable multiplicity. The latter, while technically convenient in mathematics, is rather artificial conceptually, and linguistic use is likely informed by Aristotelian intuitions of the continuum as not assembled from points and not being a multiplicity at all.

C.f. the SEP article on continuity and infinitesimals (we will be quoting from that text much in this answer):

Atomism was challenged by Aristotle (384–322 BCE), who was the first to undertake the systematic analysis of continuity and discreteness. A thoroughgoing synechist, he maintained that physical reality is a continuous plenum, and that the structure of a continuum, common to space, time and motion, is not reducible to anything else. His answer to the Eleatic problem was that continuous magnitudes are potentially divisible to infinity, in the sense that they may be divided anywhere, though they cannot be divided everywhere at the same time.

It continues:

Aristotle identifies continuity and discreteness as attributes applying to the category of Quantity.

So just as the alchemy-chemistry transition was mediated by a certain prioritization of quantitative over qualitative understanding (not an exclusion of qualitative reasoning, to be sure), we can see a shift in attempts to understand continuity attendant upon the emergence of the infinitesimal calculus as an understanding of continuity in terms of how many points are connected in a space, rather than a "hazier" sense of how the points are connected "intuitively."

But so theories of infinitesimals still involved qualitative variations. The default "image" of an infinitesimal was "the reciprocal of infinity," but compelling geometrical reasoning suggested nilsquare terms as well:

Additionally, the old "doctrine of neglect" (that infinitesimal terms can be treated as zeroes at the "end" of local calculations) seemed to be somewhat clarified by this idea of nilpotence: for example, in the typical expression ^(x+e)2-x2/_e, treating e² as 0 and ^e/_e as 1 lets us collapse the expression to 2x without an otherwise seemingly unmotivated decision to "ignore" e's presence in the sequence of expressions.

Nowadays, we have a multitude of theories of continuity still. Smooth infinitesimal analysis works primarily off nilpotent infinitesimals operating in a paracomplete logical background. Hyperreal analysis focuses on reciprocals of infinities and makes use of a "standard part function" in place of the doctrine of outright neglect (a sensible maneuver from the model-theoretic point of view, since we would of course imagine functions in model theory that go to and from various standard and nonstandard inputs). And now while Bishop may have lamented, "Now we have a [theory of calculus] that can be used to confirm [novice students'] experience of mathematics as an esoteric and meaningless exercise in technique," I daresay that standard calculus might be taken for an esoteric matter of "technique" as well, or indeed any theory of calculus must be technically precise! So Bishop's complaints are not sufficiently insightful to decide the issue.

For better or worse, though, even if we stick to the mainline representation of continuity in terms of sets of real numbers, we still lack a determinate enough representation of "the" continuum. Set-theoretically, the powerset axiom can be combined with the axiom of infinity to deliver the term ℘(ℕ) apart from the axiom of choice and hence apart from the well-ordering lemma. So it is not essential to the natural powerset, that it must be a well-ordered set with an aleph for its cardinality. But even when a natural powerset is formed in a strictly well-ordered world of sets (such as any ZFC world, then), absent peculiar assumptions like the axiom of constructibility or those covered in Moore[00?], next to nothing can be said about which aleph "the" continuum's cardinality partakes of. For all we know (modulo ZFC), 2^ℵ₀ = 2^ℵ₁ = 2^ℵ₂, among other things, for example.

Furthermore, it is not very difficult to venture into other worlds of sets where the natural powerset is inflated to the size of a proper class. This happens in a sort of trivial way in intuitionism and predicativism, but in Matthews[21] we have:

Or consider the surreal numbers. There is a surreal reciprocal of any transfinite ordinal, so there are surreal terms ¹/_ωa for any ordinal a. So then as far as the surreal number line goes, there are absolutely infinitely many infinitesimals between every single point on this line. (C.f. Ehrlich[10].) Moreover, we can then take a surreal reciprocal of infinity and use it as the power of a finite number, e.g. 2^1/ω, which goes to some x such that x^ω = 2. As the limit of the square, cube, etc. roots of 2, this x is some 1 + y such that (1 + y)^ω = 2. But then that addend y represents an infinitesimal extension past the number 1, on the number line. Then there are also 2^1/ω₁, etc. So besides the local contingua of surreal reciprocals (summing to unity), there is this multiplicative continuum of infinitesimals among the surreal numbers.

That, in turn, faintly echoes Cantor's unusual claim that infinitesimals would be x such that x times A, for some infinite ordinal A, is still less than any finite number. But whereas Cantor used such a definition in an attempt to disprove infinitesimals, we can now recognize that he accidentally stumbled upon merely another kind of these things.

So is continuity best understood first in qualitative or quantitative terms? If a curve is an infinilateral polygon, composed of infinitely small straight lines—or if the downward Löwenheim-Skolem theorem allows us to collapse a theory with uncountably many propositions to one with countably many, even if one of the propositions retained is a claim that uncountable sets exist—whence the strength of the discreteness/continuity distinction? If the cardinality of "the" continuum is greater than ℵ₁, then between the cardinal form of a contiguum and the form of a continuum, aren't there intermediary forms? When is an infinitesimal term an image of something plausibly apparent in physical (or at least a priori) intuition, and when is it an abstract fiction, a sample of mere formal "technique"? Without more resolute answers to such questions, and to the subquestions that can be extracted therefrom, it is perhaps not quite possible to know whether physically observed phenomena do or do not well-satisfy the descriptions of these matters embedded in these many and varied theories.

Answer 5

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.