How Real Are Real Numbers? Challenging Continuity with Discreteness
How real are real numbers? (2004)
I explore mathematical and physical arguments questioning the existence of continuity in our universe, advocating instead for a discrete reality. Drawing heavily on the pioneering ideas of Emile Borel, I examine why the traditional concept of real numbers may be a mathematical fiction rather than a physical truth. This perspective suggests that the infinite precision implied by real numbers does not align with the observable, finite nature of our world.
The real numbers are not real; they are a mathematical abstraction that fails to capture the discrete nature of physical reality.
- ttctciyf
I've long admired Chaitin for his original thinking and especially his ability to clearly convey his ideas about foundations, complexity and information in concise and digestible short proofs.
I'm a little surprised, however, to see him here proselytizing for a particular side in the constructivism debate. I associate him more with what he has described as a "quasi empirical" approach to mathematics[0] where the adoption of new axioms (such as for example the axiom of choice) is justified by their resulting in new, interesting mathematics.
But here, it seems his goal is to arrive at a somewhat Platonic conclusion, that either the reals are valid numbers or (seemingly he prefers) not.
My lay and naive take would be: if you adopt these rules (this Formal Axiomatic System) then you can have Big Fun in the playground of ever more esoteric and complex infinite cardinals, or if you adopt this other FAS you get to discover which results can and can't be obtained under a strict constructivist regime, and whichever FAS you choose it's just the same process of choosing axioms and applying valid deductive steps to arrive at a result you find interesting, with no more "existence" implied than the thoroughly non-Platonic existence of a solution to a problem, which can be demonstrated by solving it: if you take such-and-such steps then such-and-such result will follow.
I suppose the point being made is that avoiding axioms which imply the "existence" of the reals is more useful for doing […]
- dhosek
I’ve had this paper downloaded for about a decade and haven’t gotten around to reading it, but thinking about it, especially if space and time are quantized (an undetermined question last I checked and almost certainly still so), there would exist numbers in ℝ that cannot be expressed as physical quantities, even with an infinite universe. It’s possible that even the algebraic numbers include numbers that are non-physical (although it might be a larger subset of numbers than the constructible numbers depending on the structure of space-time’s quantization).
- jonahx
Norman Wildberger is a required mention on this topic.
Here's a great discussion on Curt Jaimungal's podcast:
https://www.youtube.com/watch?v=l7LvgvunVCM
And a good debate on the topic with Daniel Rubin, who takes the more orthodox position:
https://www.youtube.com/watch?v=edh5bbgSKqo
Wildberger has tons more on this topic on his own channel. His arguments are thought-provoking, even if you don't agree with them.
- Nevermark
It is too bad we don't have a pithy familiar term for the uncomputable / constructible reals.
I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.
- andrewla
I think people overindex on the continuity problem with the reals. I'm personally a bit of real-number denier myself as a constructivist / intuitionalist.
But when we say things like "the rationals are discrete" or "the computable numbers are discrete" these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense. Similarly when people say "the rationals are countable" and "the computable numbers are countable" this is taking for granted the Cantor notion of measuring cardinality by bijective correspondence, once again, a theory that yields nothing of value except endless paradoxes and naval-gazing nonsense.
In the practical sense the rational numbers are quite continuous -- between any two rational numbers there are an infinite (unbounded) number of rational numbers -- there's no notion of a "leap" the way there is with the integers. And any useful number can be approximated arbitrarily closely by rationals.
And for computable numbers there's even less of a gap. With rationals you can only approximate. But you can have a computable number that is exactly equal to the square root of 2, because a computable number is the algorithm by which you form arbitrarily close approximations. The square of that computable number is itself computable and is exactly equal to 2.
What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers t […]