Our question this week was
We use different types of numbers in mathematics. For example, we have the integers, the rational numbers, the real numbers, and the complex numbers. Furthermore, there is a distinct hierarchy among number types - the hierarchy for the four types above is:
\(integers \subset rational ~numbers \subset real ~numbers \subset complex~ numbers\).
Physics, however, should only need to use a certain type of number to describe the universe, and which type should be dictated by nature. Which number type is required for our current physical theories, and why?
Let me preface this answer by noting that as far as I know, there is no "right" answer for this question, in the sense that there is an answer everyone will automatically agree on. I think this viewpoint is reflected in the varying answers from everyone on this topic. Constructing "an" answer relies on further assumptions about what one means by need, and how willing one is to define number types as independent from each other. I will spell out my concrete and boring answer and the corresponding assumptions that go into it. Then, I'll give a more speculative idea that I hope will be less boring.
Assumption 1: I am not going to worry about constructions of one number type from another. Yes, one can construct the rationals from the integers by using equivalence classes of integer pairs, and the reals from the rationals using completion or Dedekind cuts. Hence the idea that one only needs natural numbers is a valid viewpoint, and there are some great posts initial thread on this topic, but it's not what I am going to consider (initially). I could have rephrased the whole question as what algebraic structures are required for physics if I had wanted to take a constructivist viewpoint. There are people who think about these questions, and try to use very fundamental mathematical concepts like category theory to explain physics(and I love the example of categories as what you can do to a potato!), but these constructions are harder to wrap one's head around and require a lot of effort to get anywhere meaningful. Therefore I will treat each number type as a separate mathematical structure and not worry about constructing one type from another.
Assumption 2: I will restrict to current physical theories, basically classical mechanics or quantum mechanics.
The boring answer about classical mechanics is that we model space and time by \(R^N\), it works, and so we need the real numbers. Quantum mechanics is only slightly less boring, and I direct you to either the post by Mursalin in the initial thread about quantum mechanics or this lecture by Scott Aaronson on why complex numbers must appear in quantum mechanics. I highly recommend reading this lecture by the way, it talks about all sorts of other issues like linearity, probabilities, etc. in a nice and entertaining way. Therefore using just the "these are our physical theories that experimentally work" criteria, then our answer is: we need the complex numbers.
Alright, on to less boring ideas. First off, what does physics do? Well, physics predicts - give me the measured initial information at some point in time and I will tell you what the result of a measurement will be at a later time. That's really all physics is supposed to accomplish. So one can think of physics as a big map \(P: X\rightarrow Y\) and the question becomes: what is the domain of X and Y? Key to figuring out the domain is the notion of measurement, after all physics takes measured initial data, not just numbers we get to make up. Let's think about the measurement of length. A typical operational definition is that the measurement of the length of an object corresponds to taking a meter stick, laying it down beside the object, and determining what fraction of the meter stick the object covers. So, do we need the reals for this measurement? Yes, if we model the stick as a segment of the real line. How accurately did we measure the length, though? Did we count the number of millimeter lines on the stick to get the fraction? If so, then we're really not using the full reals, but in fact only a subset: we're counting. We could subdivide the millimeters further, into nanometers, picometers, etc. but no matter what the division is eventually we reach the limit of precision of our measuring device. Hence at the end of the day, from a measurement perspective all we can do is count. It's the same with measurements of time, a second is defined as a count of physically measurable oscillation of a cesium isotope. You literally make an atomic clock by constructing a device that counts these oscillations. From this standpoint the domain of X and Y are really just the natural numbers, and all physics does is say "Give me this set of natural numbers and I'll tell you what the value of that set will be." From this viewpoint, the real numbers of classical mechanics are just a continuum approximation to a very finely distributed set of natural numbers, which is the true required number system.
At this point you should say "Wait! This logic is fallacious!" as after all, just because we can't measure to infinite precision doesn't mean that physics doesn't require an infinitely precise set of intial data. We just have error bars in our in our in data and our out prediction. This is true, if physics doesn't have a stopping point as to how much data one can actually put in a region of space. Whether there is a limit to how much data one can put in a region is unknown. If it is true, then there is some small region of space where the most information that can be put in is one bit and physics as applied to finite experiments really would just require a finite set of initial data. Then the "number system" required is simply 0 and 1 for each small region, plus the natural numbers so we can label the different small regions that constitute an arbitrarily big experiment. There are theoretical arguments that this might be the case based on the behavior of black holes and how much information one can cram into a very small black hole. For a synopsis of this method of thinking see here. On the other hand, one can also construct theories where physics doesn't have such a cutoff in information and in those theories the domain of X,Y would still the real (complex for QM) numbers. Nobody really knows for sure and this is an open question in theoretical physics.
Finally, I will confess that after writing this post I find it mildly suspicious that our preferred number system for macroscopic experiments, the reals, are derivable from the natural numbers (relaxing assumption 1). If one had a number system from which the reals could not be derived by any method, would that imply that that number system could never be used to describe fundamental physics? In other words are these speculative ideas about the role of discrete number systems in fundamental physics only possible because one can get to the reals from the natural numbers mathematically? I'll have to think about it more.