In the past months, I have pointed out a few times that real numbers have no reason to exist in physics, if only because our measurement instruments don’t have an infinite precision and range. In general, this point, which I believe should make any physicist nervous, does not get much traction. Maybe most people simply believe that it does not matter, but I believe on the contrary that one can show how it has a significant impact on physics.

In mathematics, Skolem’s paradox for instance can be used to invalidate some far-fetched claims. But I was not considering that real numbers in mathematics were otherwise a problem.

#### A Mathematician’s objection to Real Numbers

Well, it looks like at least one mathematician is also making similar points, but for mathematics instead of physics. Prof. Wildberger writes:

Why Real Numbers are a Joke

According to the status quo, the continuum is properly modelled by the `real numbers’.[…]

But here is a very important point: we are not obliged, in modern mathematics, to actually have a rule or algorithm that specifies the sequence In other words, `arbitrary’ sequences are allowed, as long as they have the Cauchy convergence property. This removes the obligation to specify concretely the objects which you are talking about. Sequences generated by algorithms can be specified by those algorithms, but what possibly could it mean to discuss a `sequence’ which is not generated by such a finite rule? Such an object would contain an `infinite amount’ of information, and there are no concrete examples of such things in the known universe. This is metaphysics masquerading as mathematics.

#### Contrarian view

I reached this post through one of MarkCC’s posts, where he criticizes Prof. Wildberger’s arguments. Both points of view are worth a read (although both are a bit long).

In general, MarkCC is a no-nonsense guy, and I tend to side with him. But for this once, I disagree with his criticism. As I understand it, Prof. Wildberger main point is that real numbers are defined in a way that is rather fuzzy. If it does not allow us to prove uniqueness, maybe something is flawed in the logic leading to how we build them? Again, it all boils down to the problem of defining specific, non-trivial real numbers. There are too many of them, so the set of mathematical symbols is not sufficient: it only allows us to make a countable number of mathematical propositions, and the set of real numbers is provably not countable.

And real numbers are only the beginning…

Opinions?

And yet most physicists seem to believe in real numbers, and that that truly continuous variables exist.A localized physical realization of a real number would seem to me to allow the encoding of infinite information in a finite region of space, violating the Bekenstein bound, no?As a layman I ask that you excuse any stupid question I ask here: I’m wondering whether, since that limit is depends on surface area, a region of space with fractal surface would not allow infinite information since the surface area of a true fractal can be infinitely large,o r would the Plank length be a limit to the lowest scale in which surface features can be realized?Your post also made me think of Mohrhoff’s Pondicherry interpretation, specifically his claim that spacetime is not infinitely differentiable. I’m wondering whether this objective “fuzziness” can in some way be mapped to the so called “digital physics” ideas of a fundamentally discrete/computable universe, given that it also appears to say infinite information density is an impossibility.I noticed Mohrhoff cited in your paper on the Theory of Incomplete Measurements, so I’m curious if you can provide some comments on his interpretation. I haven’t found much since I came across his papers on arxiv some years ago. I found it made at least the most intuitive sense to me as a layman, from among the various interpretations I have been exposed to, though I haven’t read yours in detail yet (I just came across your site a few hours ago 🙂 ).

Hi Quince,A localized physical realization of a real number would seem to me to allow the encoding of infinite information in a finite region of space, violating the Bekenstein bound, no?Another very interesting objection to the idea of continuum…I’m wondering whether, since that limit is depends on surface area, a region of space with fractal surface would not allow infinite information since the surface area of a true fractal can be infinitely large, or would the Plank length be a limit to the lowest scale in which surface features can be realized?This is where I differ a little from Nottale. Nottale sees fractality as a “first principle”, I see it as a mathematical tool that we can use because it is a consequence of some property of the physical measurements. As such, it is valid only within acceptable physical limits.For example, if you increase the frequency (energy) of photons, you reduce the wave length, so you increase the resolution. Consequently, you actually expose more features, more “information”. But at some point, there is enough energy in the photon to see new particles emerge, so that the photon participates in the features you observe.I noticed Mohrhoff cited in your paper on the Theory of Incomplete Measurements, so I’m curious if you can provide some comments on his interpretation.I think that what he calls “facts” is pretty close to what I call “measurement results”, and I think that we are relatively close regarding the meaning of state vectors.