Real numbers assume we can have infinite precision and some of the theory behind them uses infinite processes to establish certain proofs. A small band of mathematicians–eg ultrafinitists–disagree with this and deny the possibility of infinite processes or the logical structure of proofs that require them.
I always assumed that this was an interesting but faddish idea.
But I was recently reading a book by the very influential mathematician and computer scientist Donald Knuth (incidentally one of the popularisers and inventors of surreal numbers, a totally different way of establishing a theory of numbers) and though he didn't doubt the theory of real numbers he made the argument that it is wrong to assume they apply to the real world.
He argues (in chapter 6 of Things a Computer Scientists rarely Talks About, bold highlight mine, italics Knuth):
When I say that the question "finite or infinite?" is a red herring, I don't mean simply that philosophers and theologians have often been arguing about an unimportant issues. I also mean that physicists and scientists fail to realise this. For example, take the literature of chaos theory: Hundreds of papers have been written about the behaviour of solutions to unstable recurrences, by people who assume that real numbers are real.
...But it is a tremendous leap of faith to assume that real numbers apply perfectly to the real world...
...It seems to me that a new branch of physics is needed, called maybe "discrete physics" or something like that, to study the effects of the assumption that parameters can be infinitely precise and to consider instead that the universe probably has only a finite–but extremely large–number of states.
I'm not interested in the philosophical implication or speculations, or whether ultrafinitism has a point. But the question of whether we misunderstand some physics because we assume real numbers apply to the mathematics underpinning physical theory sounds about as important as realising that the universe is not based on flat Euclidian geometry.
So what parts of mathematics relevant to physics would be different if the concept of real numbers doesn't work for physical theory? And what are the implications for physics?