Smoothness is an Abstraction that Could be Real
Hope you appreciate the little play on words in the title. This post is dedicated to Professor Normal Wilberger who taught at the University of Newcastle in Australia and who is now in retirement but still active teaching via YouTube his ideas on rational calculus and history of mathematics.
Just for the pure intellectual enjoyment alone his series on rational trigonometry is worth following. The idea of RT being that one tries to work at the level of quadrances (areas) rather than lengths, since that removes most of the irrational surds, and allows one to formulate geometry almost entirely in terms of rational numbers, which can be computed dead accurate.
Here is where I part ways with Norman, but I think on good terms. Wildberger takes the power of the rational number field (that power being that any real number can be arbitrarily accurately approximated by a rational number) and takes it as a philosophical rationale for believing that Real numbers are "fake".
"How so?" you might ask. Well, the Reals are just the Rationals appended with all the irrational numbers (numbers with non-repeating decimal expansions, or, if you prefer, Dedekind Cuts of the Rationals). But, so Wildberger argues, the Irrationals cannot be computed. We can name algorithms for generating their decimal digits, but we can never actually write down finite expressions for most irrationals. The algorithms can by finite, but the digits churned out by any algorithm are never-ending, and so take an infinite amount of computation to display.
To the "finitist" mathematical philosopher, who take a somewhat logical positivist point of view, such things just do not exist in our "real" world. "Real world" of course confusingly being totally different to anything associated with "Real numbers". One should think of "real world" in this context as the "physical world" and then you will not go astray. Indeed, to a Finitists the term "Real number" is grotesque, since to the finitist like Wildberger, the Real numbers are pure fictions, pure abstractions, and in the physical world we can only refer to them indirectly by some algorithm which might compute them, or failing that, some metaphysics that say only 'they exist".
The latter point is actually important to note. The vast majority (uncountably many) so-called Real numbers cannot be computed. We can only offer abstract "proofs of existence". Such uncomputable Real numbers are in a sense truly random.
From the transcendentalist perspective (people like myself who believe in the mathematical utility of the infinite, and more generally, Hyperreal numbers --- the Hyperreals include infinitesimals, numbers that are closer to zero than any Real number) the fact we have well-defined algorithms for generating at least the computable Reals is good enough reason to believe they are "real".
The question then is, what do we mean by "real" in a mathematical context? One cannot say. The metaphysical notion of reality is not a mathematical concept. Thus, shifting gears to metaphysics and metamathematics, I would offer the provisional meaning that something, even an idea, is "real" is we are able to conceive of it as an abstraction that can be defined consistently with other notions (other abstractions). This is the basic program for axiomatic mathematics, but it was also the rough idea of classical mathematics all along. We never pretend we are dealing with the physical world. mathematics is an abstraction beyond the plane of the physical.
Finitists usually disagree with this abstract conception of mathematics, and so, to the Finitist, mathematics is by definition tied and bound to the physical world. Since they see no actual smooth lines and curves or planes or volumes in the physical world (drawing upon quantum mechanics these days as a rough justification), they justify their belief that infinite numbers are not "real". To the finitist, or to most of them (most that I've heard talk or seen write about such stuff), the physical world is all that exists.
A transcendentalist, or Platonist mathematician, says, no, we can think of concepts beyond the realm of the physical, and the algorithms that we are sure are well-defined but cannot ever be completed by any finite time machine, are pointing to such unphysical realities.
Of course, none of this establishes real numbers are not implicit in the physical world. Einstein's General Relativity (GR), Maxwell's Electromagnetism (EM), and most of Quantum Mechanics (QM) deals with real or complex number fields. No contradiction arises from taking the scalar fields to be real or complex. The finitist case that the physical world is finite and discrete is based on an erroneous interpretation of the Heisenberg Uncertainty Principle (HUP).
According to HUP together with GR we are unable to make arbitrarily accurate measurements of lengths and time intervals. To do so would require enormous amounts of energy packed into small regions of space, eventually collapsing spacetime into black holes. This is true. But it does not any any way imply spacetime itself is discrete and point-like finite. It is an energy and measurement limit, not a discreteness assumption.
This brings me to the thought for today that I tried to squeeze int the title of this post. A lot of quantum philosophers take the leap in thought that spacetime is fundamentally discrete. Smoothness of spacetime in GR and EM is then regarded only as an approximation we get by ignoring the fine structure of spacetime which, they claim, is actually discrete.
To my mind however, there is nothing in modern quantum physics that tells us spacetime is discrete. The smoothness assumption might be an abstraction, and might not be accurately describing the physical world, but it might also be exactly the correct description of the spacetime manifold. We just do not have enough empirical evidence to know either way, and the "weirdness" of QM does absolutely nothing to prove spacetime is non-smooth, nor the converse. Most QM employs the smoothness assumption. The only exception being Quantum Chromodynamics (QCD) which makes a lot of use of "lattice models". But Lattice QCD is a mere computational convenience, not a metaphysical assumption of a discrete spacetime. Indeed, it would break Lorentz invariance if the "lattice" in QCD computations was taken to be physical. (You do not want to break Lorentz invariance.)
The famously "fractal-like" trajectories of virtual particles in the Feynman Sum-Over-Histories (SOH) interpretation of QM is no exception either. Those trajectories can be regarded as arising from non-trivial spacetime topology, but in a smooth 4-manifold.
The lesson is, when we make mental abstractions, most often we know we are making simplifications and idealizations of the "real (physical) world". But there is a greater multiverse, consisting of the abstract Platonic realm of all possible thoughts and ideas, and that "world" is not physical. Whether it has a reality independent of a sentient brain that can think about such things is a deeper philosophical question. We are free to believe such non-physical realms "exist" independent of anyone around to think about them. This puts us in metaphysical opposition to Finitists like Normal Wildberger.
As far as practical finite computations go though, we can be in complete agreement. No transcendentalists believe we can actually physically compute Real numbers. Physically real numbers and spacetime smoothness are indeed abstractions that we cannot precisely measure. But on the other hand nothing in Nature tells us the Finitists are correct and that Transcendentalists are wrong. Taken as metaphysics, one school of thought is indeed "more right than wrong" but based only on empirical evidence we cannot tell.
I can go further: one can potentially prove Finitism is wrong (perhaps, by coming up with some definitive empirical evidence that smoothness or Real numbers like the irrationals are absolutely necessary in physics). But conversely no amount of empirical data can ever disprove Transcendentalism (among which is the smoothness abstraction). That is because one can never survey all of physical reality. You cannot write a list of all phenomena and tick off each one as finitely describable, (a) because there is no such list, and (b) you cannot guarantee the list itself is not infinite.
This is not a blanket tick in favour of Non-finitist metaphysics though. Transcendentalism (here meaning blandly just the utility and consistency of infinite and Hyperreal numbers) can be rejected in other ways if it could be shown no theory of infinite numbers can be consistent (free of logical contradictions).
At the present date there is no reason to think the theory of infinite numbers is inconsistent. It is a realm of mathematics with bizarre theorems, like the Banach-Tarksi Theorem, but it is not an inconsistent branch of mathematics. It may even apply to our physical universe: quantum field theory does not make any sense without infinite quantities. The infinities arising in QFT in Dirac and Feynman's day, used to be thought of as proof of absurdity, that QFT could not be an accurate description of our physical universe, but more modern thought tells us the infinities arising in QFT play a vital role, and if ignored cause great problems.
I only came across this modern view that the QFT infinities are essential in the last year when studying QFT (for the third sustained period in my life, one must often come back to things you know). Physicists have gotten used to the idea QFT is not the theory of physics, so it is alwyas possible a better theory could displace the infinities and bring us back to finitism. (Actually, we've never really had finitism in physics, so I guess it'd be the case that dispelling infinities from fundamental theoretical physics would not "bring us back" to finitism it would just start us on a possible path to finitism).
If that ever happened I would thoroughly enjoy revisiting Wildberger's lectures again. Whether you agree with Wildberger's metaphysics or not (I do not), you have to admit that putting on the straight-jacket of finitism is quite an intellectually fruitful thing to do, for a time. By so constraining your playground you get to discover more insights that immediately escaping into the sloppy forgiving realm of the Reals would perhaps allow you to gloss over. And for that I salute you Norman.