The discovery of non-Euclidean geometry highlighted the difference between mathematics and physics. As mathematics, the axioms of Euclidean geometry are a set of propositions deemed to be true, but as physics, the axioms become scientific hypotheses, verifiable by experimentation and observation. For example, a theorem of Euclidean geometry tells us that the inner angles of a triangle add up to 180 degrees. Scientifically, this theorem defines an experiment that we can actually do, i.e. we can measure the inner angles of a real physical triangle, and see if they add up to 180 degrees. If they add up to some other total, then we have discovered that Euclidean geometry is not the geometry of the real world.
If the validity of a geometrical system cannot be confirmed physically, then how can we confirm it? It's all very well saying that a geometry follows a certain set of axioms, but how do we know that the axioms make sense?
The modern approach is to justify a geometry by creating a numerical model for it. For example, in the case of Euclidean geometry, this consists of defining points in terms of numerical coordinates, and then defining the distance between two points as a function of their coordinates, according to a formula which is consistent with Pythagoras Theorem. We conclude that the geometry is a valid mathematical system, because it can be defined in terms of numbers.
But is our concept of number purely mathematical, or is it also physical? The most basic concept of number is that of the natural numbers and the operations of arithmetic defined on them. The standard physical interpretation of natural numbers is that of counting. For example, I might count the number of beans in a bag. If I calculate mathematically that, for instance, 23 + 45 = 68, this has a physical interpretation, which is that if I count 23 beans into a bag, and count another 45 beans into the bag, then I will find that I have 68 beans in the bag.
Suppose, however, that we consider a very large number, like 21000. The laws of arithmetic tell us that 21000 + 21000 = 21001. But we can't verify this using beans in a bag, for a very simple reason: there aren't that many beans in the universe.
Actually, we can't be sure that there aren't that many beans in the universe, but it is certainly possible that there aren't. If the universe is finite, and not too much bigger than the universe that we can observe directly, then there will be fewer than 21000 atoms in the universe, in which case it is not possible to count 21000 of anything.
And if we cannot count to 21000, then any statement of arithmetic involving numbers equal to or larger than 21000 does not have an operational meaning, in other words, it is not falsifiable. Which means that it could be true, for example, that 21000 = 0.
If the arithmetic of the universe is not standard arithmetic, what else could it be? It would have to be an arithmetic that produced results very similar to the results that standard arithmetic gives for all the arithmetical sums that we normally do.
One promising candidate is arithmetic modulo N, where N is some very large number. "modulo N" means that N is deemed equal to zero. Arithmetic modulo N is indistinguishable from normal arithmetic, if we only do addition, subtraction and multiplication with numbers much smaller than N. (Division is more problematic, but we can define division of "small" numbers that is required to have a "small" answer.)
So if N was, for instance, 1000000, then 1000000 would be equal to zero. If this was the arithmetic of the universe, then 1000000 would be "physically" equal to zero. In such a universe, if we put 500000 beans into a bag, and then we put another 500000 beans in, we would look inside and find that the bag was empty, because 500000 + 500000 = 1000000 = 0 (modulo 1000000). If we only put small numbers of beans into our bags, we would never discover that 1000000 was equal to zero, because we would never have that many beans in a bag at once.
Of course we know that 1000000 is not physically equal to zero in our universe, because we don't routinely observe that millions of objects suddenly disappear. If there is a large natural number N physically equal to zero, it must be a number much larger than 1000000.
It might seem a bit naive to consider counting as the only scientific test of arithmetic. There are many experiments and results that involve measuring rather than counting, and measurements are specified in terms of real numbers. Real numbers are a different kind of number to natural numbers. However, the standard mathematical approach to defining real numbers is to define them as the limits of certain sequences of rational numbers (i.e. fractions), where each rational number is defined as the quotient of two natural numbers.
To give a specific example, consider the number e. e can be defined as the infinite sum 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + .... We can write out some explicit fractions in the convergent sequence defined by this series, e.g. 1, 4/2, 15/6, 64/24, 325/120, 1956/720, 13699/5040, 109600/40320 ...
None of these partial sums actually represents the number e. Notionally e is represented by the full infinite sequence of partial sums. But an infinite sequence is not something we can practically create, because it would take an infinitely long time to create it. And more importantly, whenever we calculate a real number for the purposes of making a scientific prediction, we have to use an actual calculated rational approximation.
This actual calculated rational approximation consists of a pair of integers (like one of the partial sums that I just gave for e). The integers themselves are just natural numbers with an optional minus sign (or, if we want to be sophisticated, we can define each integer as the result of subtracting one natural number from another). These natural numbers can be shown to be the result of a series of arithmetic operations which only ever involve natural numbers and natural number operations. (For a detailed understanding of why this is so for all real numbers encountered in practical calculations – and not just for e – you need to study constructive real analysis.)
It follows that every practical calculation involving real numbers can in principle be restated as a calculation involving natural numbers. And for every such restatement, there will be some largest natural number contained in the calculation. If we consider the largest natural number contained in all the restatements of all the calculations that we will ever do (in relation to the verification of scientific theories and involving real numbers), and choose a natural number N which is even larger than this number, then we have discovered a natural number N such that no scientific observation can prove that N is not equal to 0. (In other words, we don't know what N is, but we have a lower bound for N, if N exists at all.)
Another more interesting possibility is that so far we have not made any scientific observation which contradicts the assumption that N is not equal to zero, but that we might make such an observation in the future (as we do more calculations and make more observations). This discovery might correspond to the development of a theory that the universe is more discrete than we thought it was. (And it might also give us a specific value for N.)
Of course discovering discreteness is nothing new:
However, we have not yet discovered that all measurable quantities are discrete. In particular, current standard theories of physics assume that time and space are continuous.
But if all arithmetic in the universe is modulo N for some natural number N, then there cannot be any continuum, and anything that appears continuous must be discrete. Including even time and space, which would have to form some kind of lattice.
