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.
A Mathematician’s objection to Real Numbers
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.
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…