25 February, 2006

Update (2013): **Roger Penrose** discusses similar ideas
(and refers non-specifically to suggestions from other authors) in
section 16.2 **A finite or infinite geometry for physics?** in his book
**The Road to Reality**.

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
**2 ^{1000}**. The laws of arithmetic tell us that

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 2^{1000} *atoms* in the
universe, in which case it is not possible to count 2^{1000} of
*anything*.

And if we cannot count to 2^{1000}, then any statement of
arithmetic involving numbers equal to or larger than 2^{1000} does
not have an operational meaning, in other words, it is not
*falsifiable*. Which means that it could be true, for example, that
2^{1000} = 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:

- The atomic theory of matter taught us that continuous "fluids" are actually collections of countable molecules.
- Quantum mechanics taught us that changes in angular momentum are countable.

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.

(Updated in response to comments on Digg at 2.57pm 26 Feb NZ DST.)